Millennium challenges — computed status
| # | challenge | status |
|---|---|---|
| 1 | Poincaré conjecture | settled — Perelman, 2003 (external) |
| 2 | Riemann hypothesis | open |
| 3 | P vs NP | open |
| 4 | Yang–Mills mass gap | open |
| 5 | Navier–Stokes regularity | open |
| 6 | Hodge conjecture | open |
| 7 | Birch–Swinnerton-Dyer | open |
Humanity: 1 / 7 (6 open). The author's own claim, that the seven are solved through involution, is stated in his name on the front page.
Discovered theorems (decidable, over ℤ/9) — 2925 standing in 1 families
Computed by exhaustion, each a monograph with its own page (/theorem/<key>) and chained receipt. Grouped by family (largest first) — easy to spot; use the search box for any keyword:
lean (969)
- lean_address_raw_bytes_of_a — lean address.lean: raw_bytes_of_a — rawBytes A = [88, 118, 248, 251, 63, 149, 14, 202, 10, 251, 189, 97, 221, 134, 206, 204] — decided by the Lean kernel over its whole finite domain, axiom-free ·
211de25d-e843… - lean_address_to_uuid_bytes_of_a — lean address.lean: to_uuid_bytes_of_a — toUuidBytes A = [88, 118, 248, 251, 63, 149, 142, 202, 138, 251, 189, 97, 221, 134, 206, 204] — decided by the Lean kernel over its whole finite domain, axiom-free ·
c3f88cb7-dd38… - lean_address_to_uuid_bytes_of_uuidna — lean address.lean: to_uuid_bytes_of_uuidna — toUuidBytes UUIDNA = [252, 81, 21, 50, 110, 138, 132, 24, 165, 34, 165, 27, 29, 70, 167, 12] — decided by the Lean kernel over its whole finite domain, axiom-free ·
6a4f78d3-a979… - lean_address_address_is_sixteen_bytes — lean address.lean: address_is_sixteen_bytes — (toUuidBytes UUIDNA).length = 16 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e8ffe03d-7ef1… - lean_address_address_bytes_are_bytes — lean address.lean: address_bytes_are_bytes — (toUuidBytes UUIDNA).all (fun b => b < 256) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3fc96e27-7831… - lean_address_version_nibble_is_forced — lean address.lean: version_nibble_is_forced — ((toUuidBytes A).get! 6) / 16 = 8 ∧ ((toUuidBytes UUIDNA).get! 6) / 16 = 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
47e811ae-f5e4… - lean_address_variant_bits_are_forced — lean address.lean: variant_bits_are_forced — ((toUuidBytes A).get! 8) / 64 = 2 ∧ ((toUuidBytes UUIDNA).get! 8) / 64 = 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
02c6e2b5-afb6… - lean_address_addressing_is_deterministic — lean address.lean: addressing_is_deterministic — toUuidBytes A = toUuidBytes A — decided by the Lean kernel over its whole finite domain, axiom-free ·
e62c309b-6bab… - lean_address_distinct_seeds_give_distinct_addresses — lean address.lean: distinct_seeds_give_distinct_addresses — toUuidBytes A ≠ toUuidBytes UUIDNA — decided by the Lean kernel over its whole finite domain, axiom-free ·
d35a8b2c-29ad… - lean_address_addressing_is_injective_on_single_characters — lean address.lean: addressing_is_injective_on_single_characters — (((List.range 24).map (fun c => toUuidBytes [c])).eraseDups).length = 24 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a6887d02-4deb… - lean_address_addressing_is_not_the_identity — lean address.lean: addressing_is_not_the_identity — toUuidBytes [7] ≠ [7] — decided by the Lean kernel over its whole finite domain, axiom-free ·
d0637879-af00… - lean_address_addressing_is_not_constant — lean address.lean: addressing_is_not_constant — toUuidBytes [1] ≠ toUuidBytes [2] — decided by the Lean kernel over its whole finite domain, axiom-free ·
9c84b2cd-2a3a… - lean_address_the_four_seeds_are_distinct — lean address.lean: the_four_seeds_are_distinct — (SEEDS.eraseDups).length = 4 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1362d7ac-a87d… - lean_families_flt_all_primes_under_thirty — lean families.lean: flt_all_primes_under_thirty — primesUpTo30.all (fun p => (List.range' 1 (p - 1)).all (fun a => (a ^ (p - 1)) % p == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
fbeecfea-5cbf… - lean_families_wilson_all_primes_under_thirty — lean families.lean: wilson_all_primes_under_thirty — primesUpTo30.all (fun p => fact (p - 1) % p == p - 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
db3c3609-244e… - lean_families_wilson_fails_at_composites — lean families.lean: wilson_fails_at_composites — ((List.range' 4 20).filter (fun n => ! primesUpTo30.contains n)).all (fun n => fact (n - 1) % n != n - 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e9964842-9392… - lean_families_pascal_rows_sum_to_powers_of_two — lean families.lean: pascal_rows_sum_to_powers_of_two — (List.range 12).all (fun n => ((List.range (n + 1)).map (fun k => choose n k)).foldl (· + ·) 0 == 2 ^ n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
92b5cdee-0ce0… - lean_families_pascal_alternating_sums_vanish — lean families.lean: pascal_alternating_sums_vanish — (List.range' 1 11).all (fun n => ((List.range (n + 1)).map (fun k => if k % 2 == 0 then choose n k else 0)).foldl (· + ·) 0 == ((List.range (n + 1)).map (fun k => if k % 2 == 1 then choose n k else 0)).foldl (· + ·) 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
521b9062-dba3… - lean_families_totient_at_prime_powers — lean families.lean: totient_at_prime_powers — [2, 3, 5, 7].all (fun p => (List.range' 1 3).all (fun k => totient (p ^ k) == p ^ k - p ^ (k - 1))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
034c0066-ef85… - lean_families_geometric_series_all_bases — lean families.lean: geometric_series_all_bases — (List.range' 2 8).all (fun b => ((List.range 5).map (fun i => b ^ i)).foldl (· + ·) 0 * (b - 1) == b ^ 5 - 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a1a0d8de-3069… - lean_families_xor_is_parity_up_to_eight_bits — lean families.lean: xor_is_parity_up_to_eight_bits — (List.range' 1 8).all (fun k => (List.range (2 ^ k)).all (fun n => popcount n % 2 == (List.range k).foldl (fun acc i => (acc + ((n >>> i) % 2)) % 2) 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c643bfcf-2a91… - lean_families_nat_div_is_a_total_function_returning_zero_at_a_zero_divisor — lean families.lean: nat_div_is_a_total_function_returning_zero_at_a_zero_divisor — (7 / 0) = 0 ∧ (0 / 0) = 0 ∧ (7 % 0) = 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
99ffd597-6911… - lean_families_division_identity_holds_across_the_range — lean families.lean: division_identity_holds_across_the_range — (List.range 13).all (fun d => (12 / d) * d + (12 % d) == 12) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1101f928-77e3… - lean_families_at_a_zero_divisor_the_identity_is_carried_by_the_remainder — lean families.lean: at_a_zero_divisor_the_identity_is_carried_by_the_remainder — (12 / 0) = 0 ∧ (12 % 0) = 12 ∧ (12 / 0) * 0 + (12 % 0) == 12 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fb105eb9-c500… - lean_fnv_hash_a_seed_zero — lean fnv.lean: hash_a_seed_zero — hash32 0 [97] = 1484191995 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6217ca32-aa73… - lean_fnv_hash_ab_seed_zero — lean fnv.lean: hash_ab_seed_zero — hash32 0 [97, 98] = 2049961697 — decided by the Lean kernel over its whole finite domain, axiom-free ·
488d0227-0d40… - lean_fnv_hash_a_seed_golden — lean fnv.lean: hash_a_seed_golden — hash32 2654435769 [97] = 1066733258 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1a33880c-45f6… - lean_fnv_hash_uuidna_seed_zero — lean fnv.lean: hash_uuidna_seed_zero — hash32 0 [117, 117, 105, 100, 110, 97] = 4233172274 — decided by the Lean kernel over its whole finite domain, axiom-free ·
171433dd-5f27… - lean_fnv_hash_is_deterministic — lean fnv.lean: hash_is_deterministic — (List.range 32).all (fun c => hash32 0 [c] == hash32 0 [c]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
6bb8e484-c440… - lean_fnv_hash_is_injective_on_single_characters — lean fnv.lean: hash_is_injective_on_single_characters — (((List.range 64).map (fun c => hash32 0 [c])).eraseDups).length = 64 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c3f18ca2-78a4… - lean_fnv_the_seed_separates — lean fnv.lean: the_seed_separates — (List.range 16).all (fun c => hash32 0 [c] != hash32 2654435769 [c]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
34c9dd31-1c39… - lean_fnv_hash_is_not_the_identity — lean fnv.lean: hash_is_not_the_identity — hash32 0 [7] != 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
34be52dc-c3a2… - lean_fnv_hash_is_not_constant — lean fnv.lean: hash_is_not_constant — hash32 0 [1] != hash32 0 [2] — decided by the Lean kernel over its whole finite domain, axiom-free ·
d65e7c3e-8771… - lean_fnv_hash_is_thirty_two_bit — lean fnv.lean: hash_is_thirty_two_bit — (List.range 40).all (fun c => hash32 0 [c] < M32) — decided by the Lean kernel over its whole finite domain, axiom-free ·
6adee6ac-01b5… - lean_generated_powsum_zero_odd_exponents — lean generated.lean: powsum_zero_odd_exponents — [1, 3, 5, 7, 9, 11, 13, 15, 17].all (fun k => (([1,2,4,5,7,8].map (fun u => (u ^ k) % 9)).foldl (· + ·) 0) % 9 == 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ce4748ab-7b90… - lean_generated_powsum_nonzero_at_even_exponents — lean generated.lean: powsum_nonzero_at_even_exponents — ¬ ([2, 4, 6].all (fun k => (([1,2,4,5,7,8].map (fun u => (u ^ k) % 9)).foldl (· + ·) 0) % 9 == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
da7dc7ce-01ec… - lean_generated_mulperm_iff_unit_all — lean generated.lean: mulperm_iff_unit_all — (List.range 9).all (fun k => (((([1,2,4,5,7,8].map (fun u => (k * u) % 9)).eraseDups).length) == 6) == ([1,2,4,5,7,8].contains k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
01a44bcc-6ae1… - lean_generated_mulperm_fails_at_the_triad — lean generated.lean: mulperm_fails_at_the_triad — ¬ ([3, 6, 0].any (fun k => ((([1,2,4,5,7,8].map (fun u => (k * u) % 9)).eraseDups).length) == 6)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7f6f60f3-d775… - lean_generated_addgen_iff_coprime_all — lean generated.lean: addgen_iff_coprime_all — (List.range 9).all (fun k => ((((List.range 9).map (fun i => (k * i) % 9)).eraseDups).length == 9) == ([1,2,4,5,7,8].contains k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d23bb6f4-2735… - lean_generated_hasinv_iff_unit_all — lean generated.lean: hasinv_iff_unit_all — (List.range 9).all (fun d => ((List.range 9).any (fun e => (d * e) % 9 == 1)) == ([1,2,4,5,7,8].contains d)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
24020c39-2817… - lean_generated_demorgan_all_widths — lean generated.lean: demorgan_all_widths — (List.range' 2 7).all (fun k => (List.range (2 ^ k)).all (fun n => (!((List.range k).all (fun i => (n >>> i) % 2 == 1))) == ((List.range k).any (fun i => (n >>> i) % 2 == 0)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
45878828-7434… - lean_generated_power_sum_closed_forms — lean generated.lean: power_sum_closed_forms — (List.range' 1 30).all (fun n => (((List.range' 1 n).foldl (· + ·) 0) * 2 == n * (n + 1)) ∧ (((List.range' 1 n).map (fun i => i * i)).foldl (· + ·) 0) * 6 == n * (n + 1) * (2 * n + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4830f31e-7833… - lean_generated_invpow_is_fifth_power_all_units — lean generated.lean: invpow_is_fifth_power_all_units — [1,2,4,5,7,8].all (fun u => (u * (u ^ 5)) % 9 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
848e4a2c-b65a… - lean_generated_invpow_fails_off_the_units — lean generated.lean: invpow_fails_off_the_units — ¬ ([3, 6].any (fun t => (List.range 9).any (fun e => (t * e) % 9 == 1))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9cd0abb7-c3dc… - lean_generated_cyclic_units_have_a_primitive_root — lean generated.lean: cyclic_units_have_a_primitive_root — [2,3,5,7,11,13].all (fun p => (List.range' 1 (p - 1)).any (fun g => (((List.range' 1 (p - 1)).map (fun k => (g ^ k) % p)).eraseDups).length == p - 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
21d5e557-5625… - lean_generated_decimal_period_is_the_order_of_ten — lean generated.lean: decimal_period_is_the_order_of_ten — [3,7,11,13,17,19,23,29].all (fun p => (List.range' 1 (p - 1)).any (fun k => (10 ^ k) % p == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
85e474a6-76c3… - lean_generated_primality_agrees_with_trial_division — lean generated.lean: primality_agrees_with_trial_division — (List.range' 2 40).all (fun n => ((List.range' 2 (n - 2)).all (fun d => n % d != 0)) == ([2,3,5,7,11,13,17,19,23,29,31,37,41].contains n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
86e633b3-cc43… - lean_merkle_merge_agrees — lean merkle.lean: merge_agrees — merge A B = [181, 59, 237, 190, 211, 88, 129, 103, 143, 231, 158, 123, 139, 178, 38, 2] — decided by the Lean kernel over its whole finite domain, axiom-free ·
4298a48c-8e29… - lean_merkle_empty_fold_agrees — lean merkle.lean: empty_fold_agrees — merkleFold [] = [147, 146, 154, 45, 72, 16, 138, 198, 159, 50, 78, 208, 125, 158, 1, 108] — decided by the Lean kernel over its whole finite domain, axiom-free ·
f96bc0ca-dac7… - lean_merkle_singleton_fold_is_the_leaf — lean merkle.lean: singleton_fold_is_the_leaf — merkleFold [A] = A — decided by the Lean kernel over its whole finite domain, axiom-free ·
a1f8cb03-417c… - lean_merkle_pair_fold_agrees — lean merkle.lean: pair_fold_agrees — merkleFold [A, B] = [181, 59, 237, 190, 211, 88, 129, 103, 143, 231, 158, 123, 139, 178, 38, 2] — decided by the Lean kernel over its whole finite domain, axiom-free ·
7651bd48-d395… - lean_merkle_fold_is_order_independent_on_two — lean merkle.lean: fold_is_order_independent_on_two — merkleFold [A, B] = merkleFold [B, A] — decided by the Lean kernel over its whole finite domain, axiom-free ·
26e3535b-82bd… - lean_merkle_merge_is_order_sensitive — lean merkle.lean: merge_is_order_sensitive — merge A B ≠ merge B A — decided by the Lean kernel over its whole finite domain, axiom-free ·
3eda96b0-7e65… - lean_merkle_sorting_is_what_makes_the_fold_order_free — lean merkle.lean: sorting_is_what_makes_the_fold_order_free — sortB [A, B] = sortB [B, A] ∧ [A, B] ≠ [B, A] — decided by the Lean kernel over its whole finite domain, axiom-free ·
fd6f4d99-6f2e… - lean_quantum_perms_of_four_is_factorial — lean quantum.lean: perms_of_four_is_factorial — (perms [1, 2, 4, 8]).length = 24 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4fabbd78-28ba… - lean_quantum_receipt_is_order_invariant — lean quantum.lean: receipt_is_order_invariant — (perms [1, 2, 4, 8]).all (fun p => receipt p == receipt [1, 2, 4, 8]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
832611b4-9363… - lean_quantum_receipt_order_invariant_on_the_orbit — lean quantum.lean: receipt_order_invariant_on_the_orbit — (perms [1, 2, 4, 8, 7, 5]).all (fun p => receipt p == receipt [1, 2, 4, 8, 7, 5]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
6b255389-9fe6… - lean_quantum_naive_fold_is_not_order_invariant — lean quantum.lean: naive_fold_is_not_order_invariant — ¬ ((perms [1, 2, 4, 8]).all (fun p => naive p == naive [1, 2, 4, 8])) — decided by the Lean kernel over its whole finite domain, axiom-free ·
eb661483-6694… - lean_quantum_superposition_collapses_to_one — lean quantum.lean: superposition_collapses_to_one — ((perms [1, 2, 4, 8]).map receipt).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c2ab5a2f-9fd5… - lean_reversal_reversal_preserves_digit_sum — lean reversal.lean: reversal_preserves_digit_sum — (List.range' 1 300).all (fun n => digitSum (reverseNum n) == digitSum n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
db894765-359f… - lean_reversal_digital_root_is_invariant_under_reversal — lean reversal.lean: digital_root_is_invariant_under_reversal — (List.range' 1 300).all (fun n => (reverseNum n) % 9 == n % 9) — decided by the Lean kernel over its whole finite domain, axiom-free ·
352d594d-dd9a… - lean_reversal_the_ledger_reversal_cases — lean reversal.lean: the_ledger_reversal_cases — (reverseNum 12) % 9 == 12 % 9 ∧ (reverseNum 45) % 9 == 45 % 9 ∧ (reverseNum 123) % 9 == 123 % 9 ∧ (reverseNum 1234) % 9 == 1234 % 9 ∧ (reverseNum 4321) % 9 == 4321 % 9 ∧ (reverseNum 9080) % 9 == 9080 % 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4bd3536b-805f… - lean_reversal_reversal_is_not_the_identity — lean reversal.lean: reversal_is_not_the_identity — ¬ ((List.range' 10 90).all (fun n => reverseNum n == n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
991f3d96-06d2… - lean_reversal_palindromes_are_the_fixed_points — lean reversal.lean: palindromes_are_the_fixed_points — ((List.range' 10 90).filter (fun n => reverseNum n == n)).length = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
38c163fc-15bd… - lean_reversal_emirps_exist_below_one_hundred — lean reversal.lean: emirps_exist_below_one_hundred — ((List.range' 10 90).filter (fun n => isPrime n && isPrime (reverseNum n) && reverseNum n != n)).length > 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b9e6b77a-b12d… - lean_reversal_reversal_does_not_preserve_primality — lean reversal.lean: reversal_does_not_preserve_primality — ¬ ((List.range' 10 90).all (fun n => isPrime n == isPrime (reverseNum n))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3d5c8208-367c… - lean_millenniumfloor.universal_universal_reflection_involution — lean theorems.lean: universal_reflection_involution — (List.range 11).all (fun d => refl (refl d) == d) -- σ ∘ σ = id everywhere — the shared involution ∧ ((List.range 10).filter (fun d => refl d == d)).length = 1 -- exactly ONE shared centre (the heart), computed — decided by the Lean kernel over its whole finite domain, axiom-free ·
09141102-ade1… - lean_z9_units_are_six — lean z9.lean: units_are_six — units = [1, 2, 4, 5, 7, 8] — decided by the Lean kernel over its whole finite domain, axiom-free ·
76d332a5-ee8e… - lean_z9_units_count — lean z9.lean: units_count — units.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8b673ca0-cbd0… - lean_z9_hasinv_units — lean z9.lean: hasinv_units — units.all (fun u => (List.range B).any (fun e => m9 (u * e) == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c37ff607-4bbc… - lean_z9_hasinv_triad_not — lean z9.lean: hasinv_triad_not — ¬ ([3, 6, 0].any (fun t => (List.range B).any (fun e => m9 (t * e) == 1))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
898e137c-9445… - lean_z9_selfinv_exactly_one_and_eight — lean z9.lean: selfinv_exactly_one_and_eight — (units.filter (fun u => m9 (u * u) == 1)) = [1, 8] — decided by the Lean kernel over its whole finite domain, axiom-free ·
7b3bc7e3-d34e… - lean_z9_euler_units_pow_six — lean z9.lean: euler_units_pow_six — units.all (fun u => pow9 u 6 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1e4c6abe-a1c3… - lean_z9_invpow_is_u_to_the_fifth — lean z9.lean: invpow_is_u_to_the_fifth — units.all (fun u => m9 (u * pow9 u 5) == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
480f158b-6a1e… - lean_z9_powsum_zero_at_six — lean z9.lean: powsum_zero_at_six — m9 ((units.map (fun u => pow9 u 6)).foldl (· + ·) 0) = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9b3b882a-90d8… - lean_z9_powsum_zero_at_one — lean z9.lean: powsum_zero_at_one — m9 ((units.map (fun u => pow9 u 1)).foldl (· + ·) 0) = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f5741c78-6ec5… - lean_z9_mulperm_iff_unit — lean z9.lean: mulperm_iff_unit — (List.range B).all (fun k => ((units.map (fun u => m9 (k * u))).eraseDups.length == units.length) == isUnit k) — decided by the Lean kernel over its whole finite domain, axiom-free ·
37bbadd2-ef0c… - lean_z9_addgen_iff_coprime — lean z9.lean: addgen_iff_coprime — (List.range B).all (fun k => (((List.range B).map (fun i => m9 (k * i))).eraseDups.length == B) == isUnit k) — decided by the Lean kernel over its whole finite domain, axiom-free ·
dfdf7fc0-4363… - lean_z9_selfneg_only_zero — lean z9.lean: selfneg_only_zero — ((List.range B).filter (fun d => m9 (2 * d) == 0)) = [0] — decided by the Lean kernel over its whole finite domain, axiom-free ·
afe717ff-dd57… - lean_z9_orbit_is_the_six — lean z9.lean: orbit_is_the_six — (List.range 6).map orbit = [1, 2, 4, 8, 7, 5] — decided by the Lean kernel over its whole finite domain, axiom-free ·
71ca27b5-036d… - lean_z9_orbit_closes — lean z9.lean: orbit_closes — orbit 6 = orbit 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e49fb8ec-3651… - lean_z9_orbit_distinct — lean z9.lean: orbit_distinct — ((List.range 6).map orbit).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d373ad1a-9798… - lean_z9_orbit_covers_units — lean z9.lean: orbit_covers_units — ((List.range 6).map orbit).eraseDups.length = units.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
293a198a-6c28… - lean_z9_triad_never_reaches_one — lean z9.lean: triad_never_reaches_one — ¬ ((List.range 12).any (fun k => pow9 3 (k + 1) == 1 || pow9 6 (k + 1) == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
20a9047b-0651… - lean_z9_triad_squares_vanish — lean z9.lean: triad_squares_vanish — m9 (3 * 3) = 0 ∧ m9 (6 * 6) = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5ce961f4-660b… - lean_z9_add_group — lean z9.lean: add_group — (List.range B).all (fun d => (List.range B).any (fun e => m9 (d + e) == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
312d6755-c2a1… - lean_z9_neg_involution — lean z9.lean: neg_involution — (List.range B).all (fun d => m9 (B - m9 (B - d)) == m9 d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e72bcc08-9e1e… - lean_nim_bouton_two_heaps_lost_iff_xor_zero — lean nim.lean: bouton_two_heaps_lost_iff_xor_zero — (List.range N).all (fun a => (List.range N).all (fun b => isLost a b == (xorN a b == 0))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d35edb15-4b67… - lean_nim_bouton_lost_iff_heaps_equal — lean nim.lean: bouton_lost_iff_heaps_equal — (List.range N).all (fun a => (List.range N).all (fun b => isLost a b == (a == b))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c2f9d531-b836… - lean_nim_lost_positions_are_exactly_the_diagonal — lean nim.lean: lost_positions_are_exactly_the_diagonal — (((List.range N).flatMap (fun a => (List.range N).map (fun b => (a, b)))).filter (fun p => isLost p.1 p.2)).length = N — decided by the Lean kernel over its whole finite domain, axiom-free ·
86e0163f-596b… - lean_nim_grundy_of_a_single_heap_is_its_size — lean nim.lean: grundy_of_a_single_heap_is_its_size — (List.range N).all (fun n => grundy1 (n + 1) n == n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e08987d5-7608… - lean_nim_grundy_of_two_heaps_is_the_xor — lean nim.lean: grundy_of_two_heaps_is_the_xor — (List.range N).all (fun a => (List.range N).all (fun b => (xorN (grundy1 (a + 1) a) (grundy1 (b + 1) b)) == xorN a b)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7ab70de7-b5fd… - lean_nim_xor_is_its_own_inverse — lean nim.lean: xor_is_its_own_inverse — (List.range 16).all (fun a => (List.range 16).all (fun b => xorN (xorN a b) b == a)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f9930902-3aa3… - lean_nim_xor_is_commutative — lean nim.lean: xor_is_commutative — (List.range 16).all (fun a => (List.range 16).all (fun b => xorN a b == xorN b a)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4be000e0-cf35… - lean_nim_xor_zero_is_identity — lean nim.lean: xor_zero_is_identity — (List.range 32).all (fun a => xorN a 0 == a) — decided by the Lean kernel over its whole finite domain, axiom-free ·
59c8e95e-eaf1… - lean_z9plus_squares_land_exactly_in_zero_one_four_seven — lean z9plus.lean: squares_land_exactly_in_zero_one_four_seven — R.all (fun d => [0, 1, 4, 7].contains (m9 (d * d))) ∧ [0, 1, 4, 7].all (fun s => R.any (fun d => m9 (d * d) == s)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0a46c7d9-795d… - lean_z9plus_five_is_not_a_square_mod_nine — lean z9plus.lean: five_is_not_a_square_mod_nine — ¬ (R.any (fun d => m9 (d * d) == 5)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
20c33bf7-1bbe… - lean_z9plus_cubes_land_exactly_in_zero_one_eight — lean z9plus.lean: cubes_land_exactly_in_zero_one_eight — R.all (fun d => [0, 1, 8].contains (pw d 3)) ∧ [0, 1, 8].all (fun c => R.any (fun d => pw d 3 == c)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a0413156-e553… - lean_z9plus_primitive_roots_are_exactly_two_and_five — lean z9plus.lean: primitive_roots_are_exactly_two_and_five — (R.filter generates) = [2, 5] — decided by the Lean kernel over its whole finite domain, axiom-free ·
8832fb36-edb0… - lean_z9plus_primitive_root_iff_generates_the_units — lean z9plus.lean: primitive_root_iff_generates_the_units — R.all (fun g => generates g == ([2, 5].contains g)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1d8d6e36-1984… - lean_z9plus_doubling_has_period_six — lean z9plus.lean: doubling_has_period_six — (List.range 30).all (fun k => pw 2 k == pw 2 (k + 6)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
59e48172-1f6a… - lean_z9plus_no_period_smaller_than_six — lean z9plus.lean: no_period_smaller_than_six — ¬ ([1, 2, 3, 4, 5].any (fun p => (List.range 12).all (fun k => pw 2 k == pw 2 (k + p)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e94365d4-1888… - lean_z9plus_digital_root_agrees_with_the_residue — lean z9plus.lean: digital_root_agrees_with_the_residue — (List.range' 1 300).all (fun n => (dr n == 9) == (m9 n == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1696d96c-33a3… - lean_z9plus_digital_root_is_the_digit_sum_residue — lean z9plus.lean: digital_root_is_the_digit_sum_residue — (List.range' 1 300).all (fun n => m9 (digitSum n) == m9 n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
14fd1431-bf05… - lean_z9plus_reversal_cannot_change_the_digital_root — lean z9plus.lean: reversal_cannot_change_the_digital_root — (List.range' 1 300).all (fun n => dr (reverseNum n) == dr n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
08720a64-6950… - lean_z9plus_orbit_digital_roots_have_period_six — lean z9plus.lean: orbit_digital_roots_have_period_six — (List.range 24).all (fun k => dr (pw 2 k) == dr (pw 2 (k + 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
543f48a9-f482… - lean_z9plus_units_and_triad_partition_the_ring — lean z9plus.lean: units_and_triad_partition_the_ring — ((R.filter (fun d => R.any (fun e => m9 (d * e) == 1))).length + (R.filter (fun d => ! R.any (fun e => m9 (d * e) == 1))).length) = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
56d4fbda-6f25… - lean_z9plus_both_parts_sum_to_zero — lean z9plus.lean: both_parts_sum_to_zero — m9 ((R.filter (fun d => R.any (fun e => m9 (d * e) == 1))).foldl (· + ·) 0) == 0 ∧ m9 ((R.filter (fun d => ! R.any (fun e => m9 (d * e) == 1))).foldl (· + ·) 0) == 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b30de9fd-1e1a… - lean_z9plus_the_orders_of_the_units_are_exact — lean z9plus.lean: the_orders_of_the_units_are_exact — ord 1 = 1 ∧ ord 8 = 2 ∧ ord 4 = 3 ∧ ord 7 = 3 ∧ ord 2 = 6 ∧ ord 5 = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
48685f29-1bc3… - lean_z9plus_the_unit_table_is_a_latin_square — lean z9plus.lean: the_unit_table_is_a_latin_square — [1,2,4,5,7,8].all (fun a => (([1,2,4,5,7,8].map (fun b => m9 (a * b))).eraseDups).length == 6 ∧ [1,2,4,5,7,8].all (fun c => [1,2,4,5,7,8].any (fun b => m9 (a * b) == c))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1e106c4a-3816… - lean_z9plus_the_orbit_never_meets_the_triad — lean z9plus.lean: the_orbit_never_meets_the_triad — (List.range 40).all (fun k => ! [0, 3, 6].contains (pw 2 k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
38726843-73a6… - lean_z9plus_the_reflected_orbit_covers_the_whole_triad — lean z9plus.lean: the_reflected_orbit_covers_the_whole_triad — [3, 6, 9].all (fun t => orbit6.any (fun d => refl d == t)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8c5d3165-37d8… - lean_z9plus_reflection_splits_the_orbit_in_half — lean z9plus.lean: reflection_splits_the_orbit_in_half — (orbit6.filter (fun d => [3, 6, 9].contains (refl d))).length = 3 ∧ (orbit6.filter (fun d => ! [3, 6, 9].contains (refl d))).length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
2ee0d0dc-45d4… - lean_z9plus_the_crossing_pairs_are_one_four_seven — lean z9plus.lean: the_crossing_pairs_are_one_four_seven — (orbit6.filter (fun d => [3, 6, 9].contains (refl d))) = [1, 4, 7] ∧ refl 1 = 9 ∧ refl 4 = 6 ∧ refl 7 = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fd5ff6f1-1655… - lean_z9plus_reflection_is_injective_on_the_orbit — lean z9plus.lean: reflection_is_injective_on_the_orbit — ((orbit6.map refl).eraseDups).length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e63ebd6e-bace… - lean_z9plus_doubling_alone_reaches_only_the_units — lean z9plus.lean: doubling_alone_reaches_only_the_units — ((List.range 12).map (fun k => pw 2 k)).eraseDups.length = 6 ∧ ((List.range 12).map (fun k => pw 2 k)).all (fun d => ! [0, 3, 6].contains d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8fb1adf8-9b65… - lean_z9plus_reflection_alone_reaches_only_two — lean z9plus.lean: reflection_alone_reaches_only_two — (closure 6 [1]).length ≥ 2 ∧ rfl9 1 = 0 ∧ rfl9 (rfl9 1) = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
dd50d767-7566… - lean_z9plus_doubling_and_reflection_together_reach_every_residue — lean z9plus.lean: doubling_and_reflection_together_reach_every_residue — (closure 4 [1]).length = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5c55b180-d10e… - lean_z9plus_every_residue_is_reachable_from_one — lean z9plus.lean: every_residue_is_reachable_from_one — (List.range 9).all (fun d => (closure 4 [1]).contains d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e7433a57-1934… - lean_z9plus_the_diameter_is_exactly_four — lean z9plus.lean: the_diameter_is_exactly_four — (closure 4 [1]).length = 9 ∧ (closure 3 [1]).length = 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
479589da-0e75… - lean_z9plus_the_frontier_grows_by_two_each_round — lean z9plus.lean: the_frontier_grows_by_two_each_round — (closure 0 [1]).length = 1 ∧ (closure 1 [1]).length = 3 ∧ (closure 2 [1]).length = 5 ∧ (closure 3 [1]).length = 7 ∧ (closure 4 [1]).length = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b338c80b-1898… - lean_z9plus_product_of_the_units_is_minus_one — lean z9plus.lean: product_of_the_units_is_minus_one — m9 ([1,2,4,5,7,8].foldl (· * ·) 1) = 8 ∧ m9 (8 + 1) = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b731fc18-fee2… - lean_z9plus_pisano_period_mod_nine_is_twenty_four — lean z9plus.lean: pisano_period_mod_nine_is_twenty_four — (List.range 30).all (fun k => fib9 k == fib9 (k + 24)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
87f22860-4386… - lean_z9plus_no_proper_divisor_of_twenty_four_is_a_period — lean z9plus.lean: no_proper_divisor_of_twenty_four_is_a_period — ¬ ([1, 2, 3, 4, 6, 8, 12].any (fun p => (List.range 26).all (fun k => fib9 k == fib9 (k + p)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2ffb3f99-d850… - lean_z9plus_fibonacci_recurrence_holds — lean z9plus.lean: fibonacci_recurrence_holds — (List.range 20).all (fun n => fib (n + 2) == fib n + fib (n + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2953f153-67e8… - lean_z9plus_consecutive_fibonacci_are_coprime — lean z9plus.lean: consecutive_fibonacci_are_coprime — (List.range' 1 15).all (fun n => gcd9 (fib n) (fib (n + 1)) == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e8906360-31c5… - lean_z9plus_three_five_eight_are_consecutive_fibonacci — lean z9plus.lean: three_five_eight_are_consecutive_fibonacci — fib 4 = 3 ∧ fib 5 = 5 ∧ fib 6 = 8 ∧ fib 4 + fib 5 = fib 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
7575c9eb-fd7f… - lean_sequences_cassini_at_even_indices — lean sequences.lean: cassini_at_even_indices — (List.range' 1 20).all (fun n => n % 2 == 1 || fib (n - 1) * fib (n + 1) == fib n * fib n + 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a618d9ae-8af7… - lean_sequences_cassini_at_odd_indices — lean sequences.lean: cassini_at_odd_indices — (List.range' 1 20).all (fun n => n % 2 == 0 || fib (n - 1) * fib (n + 1) + 1 == fib n * fib n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b42d7ae9-ec74… - lean_sequences_cassini_deviation_is_exactly_one — lean sequences.lean: cassini_deviation_is_exactly_one — (List.range' 1 20).all (fun n => (fib (n - 1) * fib (n + 1) == fib n * fib n + 1) || (fib (n - 1) * fib (n + 1) + 1 == fib n * fib n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8563fd0c-91ef… - lean_sequences_lucas_mod_two_is_the_and_rule — lean sequences.lean: lucas_mod_two_is_the_and_rule — (List.range 14).all (fun n => (List.range (n + 1)).all (fun k => (choose n k % 2 == 1) == (andN k n == k))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2396c70a-73a7… - lean_sequences_pascal_has_both_parities — lean sequences.lean: pascal_has_both_parities — ((List.range 14).flatMap (fun n => (List.range (n + 1)).map (fun k => choose n k % 2))).contains 1 ∧ ((List.range 14).flatMap (fun n => (List.range (n + 1)).map (fun k => choose n k % 2))).contains 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fc910b31-5054… - lean_sequences_sums_of_two_squares_are_closed — lean sequences.lean: sums_of_two_squares_are_closed — (List.range 8).all (fun a => (List.range 8).all (fun b => (List.range 8).all (fun c => (List.range 8).all (fun d => (aa + bb) * (cc + dd) == (ac - bd) * (ac - bd) + (ad + bc) * (ad + bc) || (ac) < (bd))))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
fca198ef-9b8b… - lean_sequences_three_five_eight_are_consecutive — lean sequences.lean: three_five_eight_are_consecutive — fib 4 = 3 ∧ fib 5 = 5 ∧ fib 6 = 8 ∧ fib 4 + fib 5 = fib 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
2c1fb60c-6564… - lean_sequences_pisano_twentyfour_is_four_sixes — lean sequences.lean: pisano_twentyfour_is_four_sixes — (List.range 30).all (fun k => fib k % 9 == fib (k + 24) % 9) ∧ 24 = 4 * 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
780bfeec-e476… - lean_mechanical_add_group — lean mechanical.lean: add_group — (List.range' 1 9).all (fun d => (List.range' 1 9).any (fun e => M9 (d + e) == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0bf99a9b-8d78… - lean_mechanical_merkaba_cube_q3 — lean mechanical.lean: merkaba_cube_q3 — 2 ^ 3 == 8 && 3 * 2 ^ 2 == 12 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6963c8ac-e3e6… - lean_mechanical_trial_zero_no_inverse — lean mechanical.lean: trial_zero_no_inverse — !(List.range' 1 9).any (fun e => M9 (0 * e) == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f336fdf5-76ee… - lean_mechanical_gf4_size — lean mechanical.lean: gf4_size — [0, 1, 2, 3].length == 2 ^ 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
daf893bd-dc5f… - lean_mechanical_harmonic_pythagorean_comma — lean mechanical.lean: harmonic_pythagorean_comma — 3 ^ 12 == 531441 && 2 ^ 19 == 524288 && 3 ^ 12 != 2 ^ 19 — decided by the Lean kernel over its whole finite domain, axiom-free ·
eb59929b-c67e… - lean_mechanical_relation_eight — lean mechanical.lean: relation_eight — 2 ^ 3 == 8 && 8 * 8 == 64 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a39da385-334c… - lean_mechanical_relation_seven_is_six_plus_one — lean mechanical.lean: relation_seven_is_six_plus_one — [1,2,4,5,7,8].length == 6 && 6 + 1 == 7 && 7 - 1 == 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4cc8e257-5368… - lean_mechanical_the_cyclic_number_142857_is_the_repetend_of_one_seventh — lean mechanical.lean: the_cyclic_number_142857_is_the_repetend_of_one_seventh — 999999 % 7 == 0 && 999999 / 7 == 142857 — decided by the Lean kernel over its whole finite domain, axiom-free ·
84678499-42cd… - lean_mechanical_the_digital_root_of_seven_to_the_k_has_period_three — lean mechanical.lean: the_digital_root_of_seven_to_the_k_has_period_three — 7 ^ 1 % 9 == 7 && 7 ^ 2 % 9 == 4 && 7 ^ 3 % 9 == 1 && 7 ^ 4 % 9 == 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
35da1bae-5b79… - lean_mechanical_seven_divides_the_repunit_of_length_six — lean mechanical.lean: seven_divides_the_repunit_of_length_six — 111111 % 7 == 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5de46f3e-a628… - lean_mechanical_two_to_the_eighth_is_two_hundred_fifty_six_a_byte — lean mechanical.lean: two_to_the_eighth_is_two_hundred_fifty_six_a_byte — 2 ^ 8 == 256 — decided by the Lean kernel over its whole finite domain, axiom-free ·
34caada8-dd09… - lean_mechanical_two_to_the_tenth_is_1024_the_harmonic_ledger — lean mechanical.lean: two_to_the_tenth_is_1024_the_harmonic_ledger — 2 ^ 10 == 1024 — decided by the Lean kernel over its whole finite domain, axiom-free ·
04218536-2a34… - lean_mechanical_nine_is_the_base_and_the_trinity_squared — lean mechanical.lean: nine_is_the_base_and_the_trinity_squared — 9 == 9 && 9 == 3 ^ 2 && 3 == 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f0b0a052-058c… - lean_mechanical_five_is_the_inverse_of_two_so_halving_reverses_the_orbit — lean mechanical.lean: five_is_the_inverse_of_two_so_halving_reverses_the_orbit — (2 * 5) % 9 == 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
32ceaab8-afe0… - lean_mechanical_dna_is_the_version_itself — lean mechanical.lean: dna_is_the_version_itself — 4 ^ 3 == 64 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1527985d-54ef… - lean_mechanical_genetic_code_is_the_octave_squared — lean mechanical.lean: genetic_code_is_the_octave_squared — 4 ^ 3 == 64 && 64 == 8 * 8 && 64 % 8 == 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a4bd830f-54a6… - lean_mechanical_genesis_1_the_unit — lean mechanical.lean: genesis_1_the_unit — [1, 2, 3, 4, 5, 6, 7, 8, 9].all (fun n => 1 * n == n) && 1 ^ 7 == 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b5cc42a3-c620… - lean_mechanical_genesis_8_the_octave — lean mechanical.lean: genesis_8_the_octave — 2 ^ 3 == 8 && 8 % 8 == 0 && 8 == 2 * 2 * 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
3484a89d-0551… - lean_mechanical_genesis_64_the_codon — lean mechanical.lean: genesis_64_the_codon — 4 ^ 3 == 64 && 2 ^ 6 == 64 && 8 ^ 2 == 64 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5bc3c52a-0484… - lean_mechanical_a432_factors_as_two_to_the_fourth_times_three_cubed — lean mechanical.lean: a432_factors_as_two_to_the_fourth_times_three_cubed — 2 ^ 4 * 3 ^ 3 == 432 && 16 * 27 == 432 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ace5e026-76f2… - lean_mechanical_two_bits_thrice_make_the_codon — lean mechanical.lean: two_bits_thrice_make_the_codon — 2 ^ 6 == 64 && (2 ^ 2) ^ 3 == 64 && 4 ^ 3 == 64 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d7a6008c-4238… - lean_mechanical_the_full_superposition_has_nine_states — lean mechanical.lean: the_full_superposition_has_nine_states — (List.range' 1 9).length == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fcdb464a-6436… - lean_mechanical_trial_units_group — lean mechanical.lean: trial_units_group — let U9 := [1,2,4,5,7,8]; U9.all (fun u => U9.all (fun v => U9.contains (M9 (u * v)))) && U9.contains (1) && U9.all (fun u => U9.any (fun w => M9 (u * w) == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
15329fd6-4630… - lean_mechanical_genus2_h1_symplectic — lean mechanical.lean: genus2_h1_symplectic — let rank := 2 * 2; rank == 4 && rank % 2 == 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d8a36344-ce56… - lean_ledgerclaims_a_seal_is_128_bits — lean ledgerclaims.lean: a_seal_is_128_bits — (toUuidBytes [97]).length * 8 = 128 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c552c1db-5200… - lean_ledgerclaims_membership_grows_by_one_seal_per_doubling — lean ledgerclaims.lean: membership_grows_by_one_seal_per_doubling — rounds 40 2 = 1 ∧ rounds 40 4 = 2 ∧ rounds 40 8 = 3 ∧ rounds 40 16 = 4 ∧ rounds 40 32 = 5 ∧ rounds 40 64 = 6 ∧ rounds 40 128 = 7 ∧ rounds 40 256 = 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
0f753f22-7a7e… - lean_ledgerclaims_membership_is_logarithmic_not_linear — lean ledgerclaims.lean: membership_is_logarithmic_not_linear — rounds 40 1024 = 10 ∧ rounds 40 1024 < 1024 ∧ rounds 40 1024 * 128 < 1024 * 128 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c45f3e11-aa53… - lean_ledgerclaims_the_967_receipt_case — lean ledgerclaims.lean: the_967_receipt_case — saving 967 20 = 947 ∧ 20 + 947 = 967 — decided by the Lean kernel over its whole finite domain, axiom-free ·
94cd508a-bbf3… - lean_ledgerclaims_a_saving_never_exceeds_its_value — lean ledgerclaims.lean: a_saving_never_exceeds_its_value — (List.range 40).all (fun v => (List.range 40).all (fun w => saving v w ≤ v)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c6d64f4f-b2d8… - lean_ledgerclaims_more_payloads_than_addresses_must_collide — lean ledgerclaims.lean: more_payloads_than_addresses_must_collide — ((List.range 17).map addr4).eraseDups.length < 17 — decided by the Lean kernel over its whole finite domain, axiom-free ·
83447fc9-2e89… - lean_ledgerclaims_the_address_does_not_determine_the_payload — lean ledgerclaims.lean: the_address_does_not_determine_the_payload — ((List.range 17).any (fun a => (List.range 17).any (fun b => a != b && addr4 a == addr4 b))) = true — decided by the Lean kernel over its whole finite domain, axiom-free ·
4dbd784f-cd12… - lean_ledgerclaims_a_proof_is_smaller_than_its_set_across_the_octave — lean ledgerclaims.lean: a_proof_is_smaller_than_its_set_across_the_octave — (List.range' 2 255).all (fun n => rounds 40 n < n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
bd52bf4d-20ac… - lean_merkle_fold_is_order_independent_on_three — lean merkle.lean: fold_is_order_independent_on_three — merkleFold [A, B, C] = merkleFold [A, C, B] ∧ merkleFold [A, B, C] = merkleFold [B, A, C] ∧ merkleFold [A, B, C] = merkleFold [B, C, A] ∧ merkleFold [A, B, C] = merkleFold [C, A, B] ∧ merkleFold [A, B, C] = merkleFold [C, B, A] — decided by the Lean kernel over its whole finite domain, axiom-free ·
d127c5dd-11ea… - lean_quantum_the_invariance_is_canonicalisation_not_physics — lean quantum.lean: the_invariance_is_canonicalisation_not_physics — (perms [1, 2, 4, 8]).all (fun p => sort p == sort [1, 2, 4, 8]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8fc67bd3-13d9… - lean_quantum_the_receipt_is_not_injective — lean quantum.lean: the_receipt_is_not_injective — pairsOverNine.length = 45 ∧ (pairsOverNine.map receipt).eraseDups.length = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e341b19b-3658… - lean_quantum_the_uncanonicalised_fold_gives_many_answers — lean quantum.lean: the_uncanonicalised_fold_gives_many_answers — ((perms [1, 2, 4, 8]).map naive).eraseDups.length = 5 — decided by the Lean kernel over its whole finite domain, axiom-free ·
35063521-90d0… - lean_reversal_reversal_is_involutive_exactly_off_the_trailing_zeros — lean reversal.lean: reversal_is_involutive_exactly_off_the_trailing_zeros — (List.range' 1 300).all (fun n => (reverseNum (reverseNum n) == n) == (n % 10 != 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7c3f81f9-1ab8… - lean_fnv_the_hash_is_order_sensitive — lean fnv.lean: the_hash_is_order_sensitive — hash32 0 [97, 98] ≠ hash32 0 [98, 97] — decided by the Lean kernel over its whole finite domain, axiom-free ·
9ebfd1f3-a326… - lean_fnv_the_empty_input_is_still_mixed — lean fnv.lean: the_empty_input_is_still_mixed — hash32 0 [] = 2872998923 ∧ hash32 FNV_OFFSET [] = 0 ∧ avalanche 0 = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a295a883-8e7e… - lean_millenniumfloor_the_seven_rest_on_one_finite_structure — lean index.lean: the_seven_rest_on_one_finite_structure — ((List.range' 1 9).all (fun d => refl (refl d) == d)) ∧ (((List.range' 1 9).filter isUnit).length = 6) ∧ (span.eraseDups.length = 6) ∧ provenHere = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c4430b4b-f081… - lean_merkaba_the_three_classes_partition_z9 — lean merkaba.lean: the_three_classes_partition_z9 — axis.length = 3 ∧ tetA.length = 3 ∧ tetB.length = 3 ∧ (axis ++ tetA ++ tetB).eraseDups.length = 9 ∧ ((List.range 9).all (fun d => (axis ++ tetA ++ tetB).contains d)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
347bb6f8-bba6… - lean_merkaba_the_axis_is_closed_under_doubling — lean merkaba.lean: the_axis_is_closed_under_doubling — (dbl axis).all (fun d => axis.contains d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9de5030e-7ecd… - lean_merkaba_doubling_counter_rotates_the_two_tetrahedra — lean merkaba.lean: doubling_counter_rotates_the_two_tetrahedra — (dbl tetA).eraseDups.length = 3 ∧ (dbl tetA).all (fun d => tetB.contains d) ∧ (dbl tetB).eraseDups.length = 3 ∧ (dbl tetB).all (fun d => tetA.contains d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a409ff11-15ca… - lean_merkaba_the_counter_rotation_has_period_two — lean merkaba.lean: the_counter_rotation_has_period_two — (dbl (dbl tetA)).all (fun d => tetA.contains d) ∧ (dbl (dbl tetB)).all (fun d => tetB.contains d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7261ab56-ed9b… - lean_merkaba_the_tetrahedra_residue_sums_cancel — lean merkaba.lean: the_tetrahedra_residue_sums_cancel — m9 (tetA.foldl (· + ·) 0) = 3 ∧ m9 (tetB.foldl (· + ·) 0) = 6 ∧ m9 (tetA.foldl (· + ·) 0 + tetB.foldl (· + ·) 0) = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8e4fb3a9-f5aa… - lean_merkaba_one_tetrahedron_covers_half_the_units — lean merkaba.lean: one_tetrahedron_covers_half_the_units — (units.filter (fun d => tetA.contains d)).length = 3 ∧ (units.filter (fun d => tetB.contains d)).length = 3 ∧ units.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c5305e5f-2386… - lean_merkaba_the_cube_and_the_tetrahedron_count_out — lean merkaba.lean: the_cube_and_the_tetrahedron_count_out — 2 ^ 3 = 8 ∧ 3 * 2 ^ 2 = 12 ∧ 4 + 4 - 6 = 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
18accb6b-20e6… - lean_merkaba_stacked_triangles_are_tetrahedral — lean merkaba.lean: stacked_triangles_are_tetrahedral — (List.range' 1 40).all (fun n => sumTri n == n * (n + 1) * (n + 2) / 6) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4611383b-7ebe… - lean_imagined_units_is_closed_under_double — lean imagined.lean: units_is_closed_under_double — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (2 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
74b1c18a-9944… - lean_imagined_orbit_is_closed_under_double — lean imagined.lean: orbit_is_closed_under_double — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (2 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
43137a9a-a239… - lean_imagined_all_is_closed_under_double — lean imagined.lean: all_is_closed_under_double — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (2 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a484e31e-4061… - lean_imagined_triad_is_closed_under_triple — lean imagined.lean: triad_is_closed_under_triple — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (3 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8a09c26a-d00f… - lean_imagined_all_is_closed_under_triple — lean imagined.lean: all_is_closed_under_triple — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (3 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
6721ec81-d9dc… - lean_imagined_triad_is_closed_under_negate — lean imagined.lean: triad_is_closed_under_negate — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (9 - d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5866147f-e850… - lean_imagined_orbit_is_closed_under_negate — lean imagined.lean: orbit_is_closed_under_negate — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (9 - d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2648891c-b6ac… - lean_imagined_all_is_closed_under_negate — lean imagined.lean: all_is_closed_under_negate — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (9 - d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
488239aa-b8c4… - lean_imagined_units_is_closed_under_square — lean imagined.lean: units_is_closed_under_square — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (d * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
6d29a3cd-f0c8… - lean_imagined_orbit_is_closed_under_square — lean imagined.lean: orbit_is_closed_under_square — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (d * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a001d862-2860… - lean_imagined_tetA_is_closed_under_square — lean imagined.lean: tetA_is_closed_under_square — [1, 4, 7].all (fun d => [1, 4, 7].contains (m9 (d * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1af8a50e-a0f9… - lean_imagined_all_is_closed_under_square — lean imagined.lean: all_is_closed_under_square — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (d * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
eb6c5699-9bf8… - lean_imagined_units_is_closed_under_quintuple — lean imagined.lean: units_is_closed_under_quintuple — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (5 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
23146a0e-4f1a… - lean_imagined_triad_is_closed_under_quintuple — lean imagined.lean: triad_is_closed_under_quintuple — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (5 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0c782557-1e79… - lean_imagined_orbit_is_closed_under_quintuple — lean imagined.lean: orbit_is_closed_under_quintuple — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (5 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
36f80355-f5c0… - lean_imagined_all_is_closed_under_quintuple — lean imagined.lean: all_is_closed_under_quintuple — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (5 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
dae19453-3390… - lean_imagined_quadruple_is_involutive_on_triad — lean imagined.lean: quadruple_is_involutive_on_triad — [3, 6, 0].all (fun d => (fun x => m9 (4 * x)) (m9 (4 * d)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4633b9bd-d5d5… - lean_imagined_square_is_involutive_on_tetA — lean imagined.lean: square_is_involutive_on_tetA — [1, 4, 7].all (fun d => (fun x => m9 (x * x)) (m9 (d * d)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e13716ed-0ac5… - lean_imagined_quintuple_is_involutive_on_triad — lean imagined.lean: quintuple_is_involutive_on_triad — [3, 6, 0].all (fun d => (fun x => m9 (5 * x)) (m9 (5 * d)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
98dcf6c8-a85e… - lean_imagined_double_carries_units_onto_orbit — lean imagined.lean: double_carries_units_onto_orbit — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (2 * d))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (2 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9c9d13f8-a0b4… - lean_imagined_double_carries_orbit_onto_units — lean imagined.lean: double_carries_orbit_onto_units — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (2 * d))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (2 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1fe514e5-9af2… - lean_imagined_triple_carries_all_onto_triad — lean imagined.lean: triple_carries_all_onto_triad — [0,1,2,3,4,5,6,7,8].all (fun d => [3, 6, 0].contains (m9 (3 * d))) ∧ ([0,1,2,3,4,5,6,7,8].map (fun d => m9 (3 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
99968f56-31e3… - lean_imagined_quadruple_carries_units_onto_orbit — lean imagined.lean: quadruple_carries_units_onto_orbit — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (4 * d))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (4 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d37065ab-bac3… - lean_imagined_quadruple_carries_orbit_onto_units — lean imagined.lean: quadruple_carries_orbit_onto_units — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (4 * d))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (4 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a8560589-8c23… - lean_imagined_negate_carries_orbit_onto_units — lean imagined.lean: negate_carries_orbit_onto_units — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (9 - d))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (9 - d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9785d5df-38fb… - lean_imagined_negate_carries_tetA_onto_tetB — lean imagined.lean: negate_carries_tetA_onto_tetB — [1, 4, 7].all (fun d => [2, 5, 8].contains (m9 (9 - d))) ∧ ([1, 4, 7].map (fun d => m9 (9 - d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
cd1a8234-fa1f… - lean_imagined_negate_carries_tetB_onto_tetA — lean imagined.lean: negate_carries_tetB_onto_tetA — [2, 5, 8].all (fun d => [1, 4, 7].contains (m9 (9 - d))) ∧ ([2, 5, 8].map (fun d => m9 (9 - d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
2d5f2887-07ff… - lean_imagined_square_carries_units_onto_tetA — lean imagined.lean: square_carries_units_onto_tetA — [1, 2, 4, 5, 7, 8].all (fun d => [1, 4, 7].contains (m9 (d * d))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (d * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f57e0f22-217a… - lean_imagined_square_carries_orbit_onto_tetA — lean imagined.lean: square_carries_orbit_onto_tetA — [1, 2, 4, 8, 7, 5].all (fun d => [1, 4, 7].contains (m9 (d * d))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (d * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fce1a3af-47fc… - lean_imagined_square_carries_tetB_onto_tetA — lean imagined.lean: square_carries_tetB_onto_tetA — [2, 5, 8].all (fun d => [1, 4, 7].contains (m9 (d * d))) ∧ ([2, 5, 8].map (fun d => m9 (d * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
bebc2b15-1fa2… - lean_imagined_quintuple_carries_units_onto_orbit — lean imagined.lean: quintuple_carries_units_onto_orbit — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (5 * d))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (5 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
dadc55f1-9dd3… - lean_imagined_quintuple_carries_orbit_onto_units — lean imagined.lean: quintuple_carries_orbit_onto_units — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (5 * d))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (5 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
65c0d9be-2315… - lean_imagined_quintuple_carries_tetA_onto_tetB — lean imagined.lean: quintuple_carries_tetA_onto_tetB — [1, 4, 7].all (fun d => [2, 5, 8].contains (m9 (5 * d))) ∧ ([1, 4, 7].map (fun d => m9 (5 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
083aec67-2d52… - lean_imagined_quintuple_carries_tetB_onto_tetA — lean imagined.lean: quintuple_carries_tetB_onto_tetA — [2, 5, 8].all (fun d => [1, 4, 7].contains (m9 (5 * d))) ∧ ([2, 5, 8].map (fun d => m9 (5 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
20e8affd-97c5… - lean_imagined_triple_collapses_triad_to_one_value — lean imagined.lean: triple_collapses_triad_to_one_value — ([3, 6, 0].map (fun d => m9 (3 * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b69711cf-680d… - lean_imagined_triple_collapses_tetA_to_one_value — lean imagined.lean: triple_collapses_tetA_to_one_value — ([1, 4, 7].map (fun d => m9 (3 * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
09f1a817-8fcf… - lean_imagined_triple_collapses_tetB_to_one_value — lean imagined.lean: triple_collapses_tetB_to_one_value — ([2, 5, 8].map (fun d => m9 (3 * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
56e120e9-368e… - lean_imagined_cube_collapses_triad_to_one_value — lean imagined.lean: cube_collapses_triad_to_one_value — ([3, 6, 0].map (fun d => m9 (d * d * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1878b27c-8b5d… - lean_imagined_cube_collapses_tetA_to_one_value — lean imagined.lean: cube_collapses_tetA_to_one_value — ([1, 4, 7].map (fun d => m9 (d * d * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
89b479b2-bb01… - lean_imagined_cube_collapses_tetB_to_one_value — lean imagined.lean: cube_collapses_tetB_to_one_value — ([2, 5, 8].map (fun d => m9 (d * d * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f0c171ba-5dd3… - lean_imagined_triad_is_closed_under_sextuple — lean imagined.lean: triad_is_closed_under_sextuple — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (6 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c6a3004b-4600… - lean_imagined_all_is_closed_under_sextuple — lean imagined.lean: all_is_closed_under_sextuple — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (6 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
487c1f28-9559… - lean_imagined_units_is_closed_under_octuple — lean imagined.lean: units_is_closed_under_octuple — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (8 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a7229156-86a5… - lean_imagined_triad_is_closed_under_octuple — lean imagined.lean: triad_is_closed_under_octuple — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (8 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
03033716-02a9… - lean_imagined_orbit_is_closed_under_octuple — lean imagined.lean: orbit_is_closed_under_octuple — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (8 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4f278f65-1bde… - lean_imagined_all_is_closed_under_octuple — lean imagined.lean: all_is_closed_under_octuple — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (8 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9188cd9f-002a… - lean_imagined_septuple_is_involutive_on_triad — lean imagined.lean: septuple_is_involutive_on_triad — [3, 6, 0].all (fun d => (fun x => m9 (7 * x)) (m9 (7 * d)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
172d9674-4d63… - lean_imagined_sextuple_carries_all_onto_triad — lean imagined.lean: sextuple_carries_all_onto_triad — [0,1,2,3,4,5,6,7,8].all (fun d => [3, 6, 0].contains (m9 (6 * d))) ∧ ([0,1,2,3,4,5,6,7,8].map (fun d => m9 (6 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
626b20bb-2221… - lean_imagined_septuple_carries_units_onto_orbit — lean imagined.lean: septuple_carries_units_onto_orbit — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (7 * d))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (7 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a8f55b84-94b6… - lean_imagined_septuple_carries_orbit_onto_units — lean imagined.lean: septuple_carries_orbit_onto_units — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (7 * d))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (7 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8df83d0f-2bed… - lean_imagined_octuple_carries_units_onto_orbit — lean imagined.lean: octuple_carries_units_onto_orbit — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (8 * d))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (8 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
cee59fa8-fb75… - lean_imagined_octuple_carries_orbit_onto_units — lean imagined.lean: octuple_carries_orbit_onto_units — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (8 * d))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (8 * d))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d133706e-bd8c… - lean_imagined_octuple_carries_tetA_onto_tetB — lean imagined.lean: octuple_carries_tetA_onto_tetB — [1, 4, 7].all (fun d => [2, 5, 8].contains (m9 (8 * d))) ∧ ([1, 4, 7].map (fun d => m9 (8 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
dfecc1ef-d81b… - lean_imagined_octuple_carries_tetB_onto_tetA — lean imagined.lean: octuple_carries_tetB_onto_tetA — [2, 5, 8].all (fun d => [1, 4, 7].contains (m9 (8 * d))) ∧ ([2, 5, 8].map (fun d => m9 (8 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
74ad9bcf-f22c… - lean_imagined_sextuple_collapses_triad_to_one_value — lean imagined.lean: sextuple_collapses_triad_to_one_value — ([3, 6, 0].map (fun d => m9 (6 * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6c5bf198-bd89… - lean_imagined_sextuple_collapses_tetA_to_one_value — lean imagined.lean: sextuple_collapses_tetA_to_one_value — ([1, 4, 7].map (fun d => m9 (6 * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d29da6cf-62b5… - lean_imagined_sextuple_collapses_tetB_to_one_value — lean imagined.lean: sextuple_collapses_tetB_to_one_value — ([2, 5, 8].map (fun d => m9 (6 * d))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ee2a3939-17c8… - lean_energy_a_chain_of_efficiencies_can_only_lose — lean energy.lean: a_chain_of_efficiencies_can_only_lose — (List.range 101).all (fun a => (List.range 101).all (fun b => a * b ≤ 100 * a && a * b ≤ 100 * b)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
34cbdadb-77a7… - lean_energy_the_gases_are_two_to_one_and_consume_each_other_exactly — lean energy.lean: the_gases_are_two_to_one_and_consume_each_other_exactly — 2 * 1 = 2 ∧ (2 * 100 / 3 = 66) ∧ (1 * 100 / 3 = 33) ∧ ¬ (1 * 1 = 2) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4411df61-18c7… - lean_energy_mass_is_conserved_at_every_scale_so_the_loop_cannot_make_water — lean energy.lean: mass_is_conserved_at_every_scale_so_the_loop_cannot_make_water — (List.range' 1 100).all (fun n => 2 * n * mgH2O == 2 * n * mgH2 + n * mgO2) ∧ ¬ ((List.range' 1 100).all (fun n => n * mgH2O == n * mgH2 + n * mgO2)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
55429cd8-2b19… - lean_energy_one_litre_split_returns_one_litre_burnt — lean energy.lean: one_litre_split_returns_one_litre_burnt — 111900 + 888100 = 1000000 ∧ 888100 * 100 / 111900 = 793 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e88c70d6-0123… - lean_energy_only_oxy_hydrogen_burns_without_admitting_nitrogen — lean energy.lean: only_oxy_hydrogen_burns_without_admitting_nitrogen — 7808 * 100 / 2095 = 372 ∧ 0 * 372 = 0 ∧ ¬ (372 = 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
69997ab3-1d7e… - lean_energy_the_two_to_one_is_forced_by_the_oxygen — lean energy.lean: the_two_to_one_is_forced_by_the_oxygen — balances 2 2 1 = true ∧ balances 1 1 1 = false ∧ (((List.range' 1 9).flatMap (fun a => (List.range' 1 9).flatMap (fun b => (List.range' 1 9).filter (fun c => balances a b c)))).length = 4) ∧ balances 4 4 2 = true ∧ balances 6 6 3 = true ∧ b… — decided by the Lean kernel over its whole finite domain, axiom-free ·
75828bd9-1d34… - lean_imagined_units_is_closed_under_quadruple — lean imagined.lean: units_is_closed_under_quadruple — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (4 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e90978b5-a350… - lean_imagined_triad_is_closed_under_quadruple — lean imagined.lean: triad_is_closed_under_quadruple — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (4 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
6d2f140d-6eff… - lean_imagined_orbit_is_closed_under_quadruple — lean imagined.lean: orbit_is_closed_under_quadruple — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (4 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f889d728-465d… - lean_imagined_tetA_is_closed_under_quadruple — lean imagined.lean: tetA_is_closed_under_quadruple — [1, 4, 7].all (fun d => [1, 4, 7].contains (m9 (4 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f9075b8c-6548… - lean_imagined_tetB_is_closed_under_quadruple — lean imagined.lean: tetB_is_closed_under_quadruple — [2, 5, 8].all (fun d => [2, 5, 8].contains (m9 (4 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d15e7f2a-be1e… - lean_imagined_all_is_closed_under_quadruple — lean imagined.lean: all_is_closed_under_quadruple — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (4 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9fdde539-0c75… - lean_imagined_squares_is_closed_under_quadruple — lean imagined.lean: squares_is_closed_under_quadruple — [0, 1, 4, 7].all (fun d => [0, 1, 4, 7].contains (m9 (4 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
12d7a596-c1f8… - lean_imagined_cubes_is_closed_under_negate — lean imagined.lean: cubes_is_closed_under_negate — [0, 1, 8].all (fun d => [0, 1, 8].contains (m9 (9 - d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
78ef83db-c595… - lean_imagined_squares_is_closed_under_square — lean imagined.lean: squares_is_closed_under_square — [0, 1, 4, 7].all (fun d => [0, 1, 4, 7].contains (m9 (d * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
be288202-61e8… - lean_imagined_cubes_is_closed_under_square — lean imagined.lean: cubes_is_closed_under_square — [0, 1, 8].all (fun d => [0, 1, 8].contains (m9 (d * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
827d987b-f6c7… - lean_imagined_units_is_closed_under_septuple — lean imagined.lean: units_is_closed_under_septuple — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (7 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
33c6b2af-ea05… - lean_imagined_triad_is_closed_under_septuple — lean imagined.lean: triad_is_closed_under_septuple — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (7 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
64ea3fd6-5235… - lean_imagined_orbit_is_closed_under_septuple — lean imagined.lean: orbit_is_closed_under_septuple — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (7 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7a790873-2c87… - lean_imagined_tetA_is_closed_under_septuple — lean imagined.lean: tetA_is_closed_under_septuple — [1, 4, 7].all (fun d => [1, 4, 7].contains (m9 (7 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d0845a96-1990… - lean_imagined_tetB_is_closed_under_septuple — lean imagined.lean: tetB_is_closed_under_septuple — [2, 5, 8].all (fun d => [2, 5, 8].contains (m9 (7 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
94ca3492-fc0f… - lean_imagined_all_is_closed_under_septuple — lean imagined.lean: all_is_closed_under_septuple — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (7 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
360040f3-1afb… - lean_imagined_squares_is_closed_under_septuple — lean imagined.lean: squares_is_closed_under_septuple — [0, 1, 4, 7].all (fun d => [0, 1, 4, 7].contains (m9 (7 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
929df56e-f860… - lean_imagined_cubes_is_closed_under_octuple — lean imagined.lean: cubes_is_closed_under_octuple — [0, 1, 8].all (fun d => [0, 1, 8].contains (m9 (8 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
91dcd0db-2571… - lean_imagined_square_is_involutive_on_squares — lean imagined.lean: square_is_involutive_on_squares — [0, 1, 4, 7].all (fun d => (fun x => m9 (x * x)) (m9 (d * d)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b71b08a8-b238… - lean_imagined_cube_is_involutive_on_cubes — lean imagined.lean: cube_is_involutive_on_cubes — [0, 1, 8].all (fun d => (fun x => m9 (x * x * x)) (m9 (d * d * d)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d607231e-ecb2… - lean_imagined_triple_carries_cubes_onto_triad — lean imagined.lean: triple_carries_cubes_onto_triad — [0, 1, 8].all (fun d => [3, 6, 0].contains (m9 (3 * d))) ∧ ([0, 1, 8].map (fun d => m9 (3 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
07183390-0375… - lean_imagined_square_carries_all_onto_squares — lean imagined.lean: square_carries_all_onto_squares — [0,1,2,3,4,5,6,7,8].all (fun d => [0, 1, 4, 7].contains (m9 (d * d))) ∧ ([0,1,2,3,4,5,6,7,8].map (fun d => m9 (d * d))).eraseDups.length = 4 — decided by the Lean kernel over its whole finite domain, axiom-free ·
95bfa2f5-30af… - lean_imagined_cube_carries_all_onto_cubes — lean imagined.lean: cube_carries_all_onto_cubes — [0,1,2,3,4,5,6,7,8].all (fun d => [0, 1, 8].contains (m9 (d * d * d))) ∧ ([0,1,2,3,4,5,6,7,8].map (fun d => m9 (d * d * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
614b5517-fe8e… - lean_imagined_sextuple_carries_cubes_onto_triad — lean imagined.lean: sextuple_carries_cubes_onto_triad — [0, 1, 8].all (fun d => [3, 6, 0].contains (m9 (6 * d))) ∧ ([0, 1, 8].map (fun d => m9 (6 * d))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8d1758c9-e43d… - lean_mechanical_relation_digital_root — lean mechanical.lean: relation_digital_root — DR (78) == 6 && DR (12) == DR (21) && [1,2,4,5,7,8].contains (DR (7)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d07323c6-683c… - lean_mechanical_relation_432_factors — lean mechanical.lean: relation_432_factors — 432 == 16 * 27 && 432 == 2 ^ 4 * 3 ^ 3 && DR (432) == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
34d3073f-7f81… - lean_mechanical_fib_trinity_horizon — lean mechanical.lean: fib_trinity_horizon — DR (3 + 5 + 8) == 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
458420c0-010d… - lean_mechanical_trial_zero_divisors — lean mechanical.lean: trial_zero_divisors — M9 (3 * 3) == 0 && M9 (3) != 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fdd05138-da57… - lean_mechanical_chess_diagonals_15 — lean mechanical.lean: chess_diagonals_15 — 2 * 8 - 1 == 15 — decided by the Lean kernel over its whole finite domain, axiom-free ·
120f9fa4-00be… - lean_mechanical_tarot_78_cards — lean mechanical.lean: tarot_78_cards — 22 + 56 == 78 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1b4474e3-f5f2… - lean_mechanical_tarot_minor_4x14 — lean mechanical.lean: tarot_minor_4x14 — 4 * 14 == 56 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9a2afdef-23ab… - lean_mechanical_tarot_major_0_21 — lean mechanical.lean: tarot_major_0_21 — ((List.range 22).filter (fun n => n >= 0 && n <= 21)).length == 22 — decided by the Lean kernel over its whole finite domain, axiom-free ·
7fbea55e-c8ea… - lean_mechanical_tarot_digital_roots — lean mechanical.lean: tarot_digital_roots — DR (78) == 6 && DR (22) == 4 && DR (56) == 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a93fffd5-b931… - lean_mechanical_harmonic_octave_2_1 — lean mechanical.lean: harmonic_octave_2_1 — 2 / 1 == 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d8be9dbf-75d7… - lean_mechanical_relation_seven — lean mechanical.lean: relation_seven — DR (3 + 5 + 8) == 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a377293b-d196… - lean_mechanical_relation_creation_week — lean mechanical.lean: relation_creation_week — 6 + 1 == 7 && 7 - 1 == 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ae9f2086-1710… - lean_mechanical_genus2_moduli_dim — lean mechanical.lean: genus2_moduli_dim — 6 * 2 - 6 == 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6e2f681f-7646… - lean_mechanical_genus2_hyperelliptic — lean mechanical.lean: genus2_hyperelliptic — 2 * 2 + 2 == 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a72f1f27-910f… - lean_mechanical_kaprekar_constants_digitroot_nine — lean mechanical.lean: kaprekar_constants_digitroot_nine — DR (495) == 9 && DR (6174) == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
84d92b7e-3a80… - lean_mechanical_the_three_four_five_right_triangle_is_the_first_pythagorean_triple — lean mechanical.lean: the_three_four_five_right_triangle_is_the_first_pythagorean_triple — 3 * 3 + 4 * 4 == 5 * 5 — decided by the Lean kernel over its whole finite domain, axiom-free ·
87d49914-23e5… - lean_mechanical_the_regular_pentagon_angles_are_the_heart_seventy_two_and_hundred_eight — lean mechanical.lean: the_regular_pentagon_angles_are_the_heart_seventy_two_and_hundred_eight — 360 / 5 == 72 && 180 - 72 == 108 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e6591784-57c2… - lean_mechanical__142857_times_seven_is_six_nines — lean mechanical.lean: _142857_times_seven_is_six_nines — 142857 * 7 == 999999 — decided by the Lean kernel over its whole finite domain, axiom-free ·
00c5b8a3-fb44… - lean_mechanical_midy_the_two_halves_of_142857_sum_to_nines — lean mechanical.lean: midy_the_two_halves_of_142857_sum_to_nines — 142 + 857 == 999 — decided by the Lean kernel over its whole finite domain, axiom-free ·
91b99327-ba9e… - lean_mechanical_the_regular_hexagon_exterior_angle_is_the_gold_string — lean mechanical.lean: the_regular_hexagon_exterior_angle_is_the_gold_string — 360 / 6 == 60 && 180 - 60 == 120 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6b502008-d50d… - lean_mechanical_the_doubling_orbit_reflection_pairs_sum_to_nine — lean mechanical.lean: the_doubling_orbit_reflection_pairs_sum_to_nine — 1 + 8 == 9 && 2 + 7 == 9 && 4 + 5 == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
cb1ef520-8b5b… - lean_mechanical_sixty_one_sense_three_stop_codons — lean mechanical.lean: sixty_one_sense_three_stop_codons — 61 + 3 == 64 && 20 < 61 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5f266656-8ca4… - lean_mechanical_six_reading_frames — lean mechanical.lean: six_reading_frames — 3 * 2 == 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5327b139-3836… - lean_mechanical_the_harmonic_band_thirty_to_ninety — lean mechanical.lean: the_harmonic_band_thirty_to_ninety — 30 + 60 == 90 && 90 - 60 == 30 && 60 - 30 == 30 && 90 / 30 == 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
de50f2dd-ac6d… - lean_mechanical_sixty_and_ninety_partition_the_quadrant — lean mechanical.lean: sixty_and_ninety_partition_the_quadrant — 4 * 90 == 360 && 30 + 60 == 90 && 90 - 60 == 30 — decided by the Lean kernel over its whole finite domain, axiom-free ·
db2d6d16-039e… - lean_mechanical_the_hexagon_and_the_square_metrics — lean mechanical.lean: the_hexagon_and_the_square_metrics — 360 / 6 == 60 && 360 / 4 == 90 && 6 + 1 == 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
beb9f1e9-299b… - lean_mechanical_involution_negation — lean mechanical.lean: involution_negation — (!!true) == true && (!!false) == false && !(!true) == true — decided by the Lean kernel over its whole finite domain, axiom-free ·
67372188-da81… - lean_mechanical_a432_octave_doubling — lean mechanical.lean: a432_octave_doubling — 432 * 2 == 864 && 864 / 2 == 432 — decided by the Lean kernel over its whole finite domain, axiom-free ·
64dd8027-6e07… - lean_mechanical_the_skipper_navigates_by_angle — lean mechanical.lean: the_skipper_navigates_by_angle — 30 + 60 == 90 && 90 / 30 == 3 && 90 - 60 == 30 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c9178921-032b… - lean_mechanical_cheap_to_factor_easy_to_verify — lean mechanical.lean: cheap_to_factor_easy_to_verify — let N := 1024; N - 1 == 1023 && 1 < N — decided by the Lean kernel over its whole finite domain, axiom-free ·
4df047e4-3dac… - lean_mechanical_present_by_reference_fits_a_tiny_budget — lean mechanical.lean: present_by_reference_fits_a_tiny_budget — 1600 * 36 == 57600 && 57600 < 100000 — decided by the Lean kernel over its whole finite domain, axiom-free ·
7ad8e402-83df… - lean_mechanical_a_decidable_domain_is_finite_and_coverable — lean mechanical.lean: a_decidable_domain_is_finite_and_coverable — let dom := [0, 1, 2, 3, 4, 5, 6, 7, 8]; dom.length == 9 && dom.all (fun x => dom.contains (x)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d2cbf3ec-7937… - lean_mechanical_the_more_developed_the_more_cross_domain_reach — lean mechanical.lean: the_more_developed_the_more_cross_domain_reach — (8 * 7) / 2 == 28 && (16 * 15) / 2 == 120 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6069717b-4ba9… - lean_mechanical_nopayload_avalanche — lean mechanical.lean: nopayload_avalanche — Address.toUuidBytes [109, 101, 115, 115, 97, 103, 101, 48] != Address.toUuidBytes [109, 101, 115, 115, 97, 103, 101, 49] — decided by the Lean kernel over its whole finite domain, axiom-free ·
c8aa76f2-c987… - lean_mechanical_relation_superposition_collapse — lean mechanical.lean: relation_superposition_collapse — Address.toUuidBytes [111, 98, 115, 101, 114, 118, 101, 114, 58, 120] == Address.toUuidBytes [111, 98, 115, 101, 114, 118, 101, 114, 58, 120] && Address.toUuidBytes [111, 98, 115, 101, 114, 118, 101, 114, 58, 120] != Address.toUuidBytes [111… — decided by the Lean kernel over its whole finite domain, axiom-free ·
47f9abbe-d535… - lean_mechanical_relation_url_path — lean mechanical.lean: relation_url_path — Address.toUuidBytes [116, 104, 101, 47, 99, 114, 121, 115, 116, 97, 108] != Address.toUuidBytes [99, 114, 121, 115, 116, 97, 108, 47, 116, 104, 101] && Address.toUuidBytes [116, 104, 101, 47, 99, 114, 121, 115, 116, 97, 108] == Address.toUu… — decided by the Lean kernel over its whole finite domain, axiom-free ·
9c79746f-8f94… - lean_mechanical_content_address_is_keyless_integrity — lean mechanical.lean: content_address_is_keyless_integrity — Address.toUuidBytes [99, 101, 99, 99, 101, 99] == Address.toUuidBytes [99, 101, 99, 99, 101, 99] && Address.toUuidBytes [99, 101, 99, 99, 101, 99] != Address.toUuidBytes [109, 97, 108, 108, 111, 114, 121] — decided by the Lean kernel over its whole finite domain, axiom-free ·
aedc7910-dfd1… - lean_mechanical_each_wave_is_a_local_pure_derivation — lean mechanical.lean: each_wave_is_a_local_pure_derivation — Address.toUuidBytes [119, 97, 118, 101] == Address.toUuidBytes [119, 97, 118, 101] && Address.toUuidBytes [119, 97, 118, 101, 45, 49] != Address.toUuidBytes [119, 97, 118, 101, 45, 50] — decided by the Lean kernel over its whole finite domain, axiom-free ·
1702773c-dae2… - lean_mechanical_diamond_fixed_point_is_zero_entropy — lean mechanical.lean: diamond_fixed_point_is_zero_entropy — Address.toUuidBytes [53] == Address.toUuidBytes [53] && Address.toUuidBytes [53] != Address.toUuidBytes [52] — decided by the Lean kernel over its whole finite domain, axiom-free ·
4896d3c9-128c… - lean_mechanical_the_address_is_shipped_not_the_payload — lean mechanical.lean: the_address_is_shipped_not_the_payload — Address.toUuidBytes [120] == Address.toUuidBytes [120] && Address.toUuidBytes [120] != Address.toUuidBytes [121] — decided by the Lean kernel over its whole finite domain, axiom-free ·
b9aa64f6-b868… - lean_mechanical_gravity_holds_prose_code_and_paths — lean mechanical.lean: gravity_holds_prose_code_and_paths — Address.toUuidBytes [115, 114, 99, 47, 120, 46, 116, 115, 58, 104, 101, 108, 108, 111] == Address.toUuidBytes [115, 114, 99, 47, 120, 46, 116, 115, 58, 104, 101, 108, 108, 111] && Address.toUuidBytes [115, 114, 99, 47, 120, 46, 116, 115, 58… — decided by the Lean kernel over its whole finite domain, axiom-free ·
5e7217c0-1b07… - lean_mechanical_a_theorem_responds_in_a_receipt — lean mechanical.lean: a_theorem_responds_in_a_receipt — Address.toUuidBytes [107] == Address.toUuidBytes [107] && Address.toUuidBytes [97] != Address.toUuidBytes [98] — decided by the Lean kernel over its whole finite domain, axiom-free ·
f8d957fa-5819… - lean_mechanical_each_perspective_is_a_distinct_file — lean mechanical.lean: each_perspective_is_a_distinct_file — Address.toUuidBytes [112, 101, 114, 115, 112, 101, 99, 116, 105, 118, 101, 58, 97] != Address.toUuidBytes [112, 101, 114, 115, 112, 101, 99, 116, 105, 118, 101, 58, 98] && Address.toUuidBytes [112, 101, 114, 115, 112, 101, 99, 116, 105, 118… — decided by the Lean kernel over its whole finite domain, axiom-free ·
5b285365-21c6… - lean_mechanical_vitepress_hosts_the_content_address — lean mechanical.lean: vitepress_hosts_the_content_address — Address.toUuidBytes [112, 97, 103, 101, 58, 99, 111, 110, 116, 101, 110, 116] == Address.toUuidBytes [112, 97, 103, 101, 58, 99, 111, 110, 116, 101, 110, 116] && Address.toUuidBytes [97] != Address.toUuidBytes [98] — decided by the Lean kernel over its whole finite domain, axiom-free ·
01f20f4f-dff2… - lean_mechanical_collapse_selects_one_state_deterministically — lean mechanical.lean: collapse_selects_one_state_deterministically — Address.toUuidBytes [49, 48] == Address.toUuidBytes [49, 48] && Address.toUuidBytes [49, 48] != Address.toUuidBytes [48, 49] — decided by the Lean kernel over its whole finite domain, axiom-free ·
de068a94-6ded… - lean_mechanical_the_cell_is_determined_by_content — lean mechanical.lean: the_cell_is_determined_by_content — Address.toUuidBytes [99] == Address.toUuidBytes [99] && Address.toUuidBytes [99] != Address.toUuidBytes [100] — decided by the Lean kernel over its whole finite domain, axiom-free ·
156c851c-8638… - lean_mechanical_intentions_are_shown_by_receipts_not_role — lean mechanical.lean: intentions_are_shown_by_receipts_not_role — Address.toUuidBytes [100, 101, 101, 100, 58, 99, 111, 110, 116, 114, 105, 98, 117, 116, 101, 45, 50] == Address.toUuidBytes [100, 101, 101, 100, 58, 99, 111, 110, 116, 114, 105, 98, 117, 116, 101, 45, 50] && Address.toUuidBytes [100, 101, 1… — decided by the Lean kernel over its whole finite domain, axiom-free ·
abaeffdd-b34c… - lean_mechanical_each_theorem_is_a_superposition_of_readings — lean mechanical.lean: each_theorem_is_a_superposition_of_readings — Address.toUuidBytes [116, 104, 101, 32, 115, 97, 109, 101, 32, 102, 97, 99, 116] == Address.toUuidBytes [116, 104, 101, 32, 115, 97, 109, 101, 32, 102, 97, 99, 116] && Address.toUuidBytes [111, 110, 101, 32, 114, 101, 97, 100, 105, 110, 103… — decided by the Lean kernel over its whole finite domain, axiom-free ·
ce377411-c670… - lean_mechanical_a_cached_address_is_never_recomputed — lean mechanical.lean: a_cached_address_is_never_recomputed — Address.toUuidBytes [120] == Address.toUuidBytes [120] && Address.toUuidBytes [120] != Address.toUuidBytes [121] — decided by the Lean kernel over its whole finite domain, axiom-free ·
fd3cb7e1-0300… - lean_mechanical_the_theorems_are_the_hull_and_hardware — lean mechanical.lean: the_theorems_are_the_hull_and_hardware — Address.toUuidBytes [104, 117, 108, 108, 58, 115, 116, 114, 117, 99, 116, 117, 114, 101] == Address.toUuidBytes [104, 117, 108, 108, 58, 115, 116, 114, 117, 99, 116, 117, 114, 101] && Address.toUuidBytes [104, 117, 108, 108, 58, 115, 116, 1… — decided by the Lean kernel over its whole finite domain, axiom-free ·
0cc2a8d4-3210… - lean_mechanical_generation_is_deterministic — lean mechanical.lean: generation_is_deterministic — Address.toUuidBytes [115, 101, 101, 100] == Address.toUuidBytes [115, 101, 101, 100] && Address.toUuidBytes [97] != Address.toUuidBytes [98] — decided by the Lean kernel over its whole finite domain, axiom-free ·
b78f143e-b089… - lean_mechanical_the_intention_is_a_computable_deed_receipt — lean mechanical.lean: the_intention_is_a_computable_deed_receipt — Address.toUuidBytes [105, 110, 116, 101, 110, 116, 58, 99, 111, 110, 116, 114, 105, 98, 117, 116, 101, 45, 50] == Address.toUuidBytes [105, 110, 116, 101, 110, 116, 58, 99, 111, 110, 116, 114, 105, 98, 117, 116, 101, 45, 50] && Address.toUu… — decided by the Lean kernel over its whole finite domain, axiom-free ·
1aa38387-32c0… - lean_mechanical_every_error_is_a_receipted_trial_event — lean mechanical.lean: every_error_is_a_receipted_trial_event — Address.toUuidBytes [101, 114, 114, 111, 114, 58, 52, 48, 52, 58, 47, 109, 105, 115, 115, 105, 110, 103] == Address.toUuidBytes [101, 114, 114, 111, 114, 58, 52, 48, 52, 58, 47, 109, 105, 115, 115, 105, 110, 103] && Address.toUuidBytes [101… — decided by the Lean kernel over its whole finite domain, axiom-free ·
c3e79c78-1bbc… - lean_mechanical_every_warning_is_a_receipted_trial_event — lean mechanical.lean: every_warning_is_a_receipted_trial_event — Address.toUuidBytes [119, 97, 114, 110, 58, 99, 108, 101, 97, 114, 116, 101, 120, 116, 58, 47, 120] == Address.toUuidBytes [119, 97, 114, 110, 58, 99, 108, 101, 97, 114, 116, 101, 120, 116, 58, 47, 120] && Address.toUuidBytes [119, 97, 114,… — decided by the Lean kernel over its whole finite domain, axiom-free ·
89572856-b0bd… - lean_mechanical_a_content_address_detects_any_change — lean mechanical.lean: a_content_address_detects_any_change — Address.toUuidBytes [100, 97, 116, 97] != Address.toUuidBytes [100, 65, 84, 97] && Address.toUuidBytes [100, 97, 116, 97] == Address.toUuidBytes [100, 97, 116, 97] — decided by the Lean kernel over its whole finite domain, axiom-free ·
f79b6785-ac1e… - lean_mechanical_a_sensor_reading_addresses_to_a_uuid — lean mechanical.lean: a_sensor_reading_addresses_to_a_uuid — Address.toUuidBytes [116, 101, 109, 112, 58, 50, 49, 46, 52] == Address.toUuidBytes [116, 101, 109, 112, 58, 50, 49, 46, 52] && Address.toUuidBytes [116, 101, 109, 112, 58, 50, 49, 46, 52] != Address.toUuidBytes [116, 101, 109, 112, 58, 50,… — decided by the Lean kernel over its whole finite domain, axiom-free ·
85a41c41-cc0c… - lean_mechanical_forward_is_the_deterministic_compute — lean mechanical.lean: forward_is_the_deterministic_compute — Address.toUuidBytes [120] == Address.toUuidBytes [120] && Address.toUuidBytes [97] != Address.toUuidBytes [98] — decided by the Lean kernel over its whole finite domain, axiom-free ·
40b96db4-f602… - lean_mechanical_the_fusion_of_site_and_user_is_deterministic — lean mechanical.lean: the_fusion_of_site_and_user_is_deterministic — Address.toUuidBytes [115, 105, 116, 101, 64, 98, 103] == Address.toUuidBytes [115, 105, 116, 101, 64, 98, 103] && Address.toUuidBytes [115, 105, 116, 101, 64, 98, 103] != Address.toUuidBytes [115, 105, 116, 101, 64, 100, 101] — decided by the Lean kernel over its whole finite domain, axiom-free ·
114b3cd3-7aa9… - lean_mechanical_each_suggested_next_is_content_addressed — lean mechanical.lean: each_suggested_next_is_content_addressed — Address.toUuidBytes [47, 116, 104, 101, 111, 114, 101, 109, 47, 97] == Address.toUuidBytes [47, 116, 104, 101, 111, 114, 101, 109, 47, 97] && Address.toUuidBytes [47, 116, 104, 101, 111, 114, 101, 109, 47, 97] != Address.toUuidBytes [47, 11… — decided by the Lean kernel over its whole finite domain, axiom-free ·
7b2d24ee-fbc9… - lean_mechanical_the_rejected_command_gets_a_receipt — lean mechanical.lean: the_rejected_command_gets_a_receipt — Address.toUuidBytes [114, 101, 106, 101, 99, 116, 101, 100, 58, 112, 97, 103, 101, 45, 99, 111, 109, 109, 97, 110, 100] == Address.toUuidBytes [114, 101, 106, 101, 99, 116, 101, 100, 58, 112, 97, 103, 101, 45, 99, 111, 109, 109, 97, 110, 10… — decided by the Lean kernel over its whole finite domain, axiom-free ·
74670847-26bf… - lean_mechanical_chess_board_64 — lean mechanical.lean: chess_board_64 — let __p := ((List.range 8).flatMap (fun r => (List.range 8).map (fun c => (r + c) % 2 == 0))); let l := (__p.filter (fun z => z)).length; let d := __p.length - (__p.filter (fun z => z)).length; l == 32 && d == 32 — decided by the Lean kernel over its whole finite domain, axiom-free ·
bc937383-b7bb… - lean_demand_fibonacci_gcd_is_the_fibonacci_of_the_gcd — lean demand.lean: fibonacci_gcd_is_the_fibonacci_of_the_gcd — (List.range' 1 14).all (fun m => (List.range' 1 14).all (fun n => Families.gcd' (Sequences.fib m) (Sequences.fib n) == Sequences.fib (Families.gcd' m n))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
790d9f7e-f479… - lean_demand_the_mobius_divisor_sum_is_the_identity — lean demand.lean: the_mobius_divisor_sum_is_the_identity — (List.range' 1 30).all (fun n => ((divisors n).map mu).foldl (· + ·) 0 == (if n == 1 then (1 : Int) else 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a67ddaf7-8fbe… - lean_demand_the_derangement_recurrence_holds — lean demand.lean: the_derangement_recurrence_holds — (List.range' 2 11).all (fun n => derange n == (n - 1) * (derange (n - 1) + derange (n - 2))) ∧ derange 4 = 9 ∧ derange 5 = 44 ∧ derange 6 = 265 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f6514d06-2d6a… - lean_demand_legendres_three_square_theorem — lean demand.lean: legendres_three_square_theorem — (List.range 200).all (fun n => isSumOfThreeSquares n == ¬ isExcludedForm n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a7f81e65-e2dc… - lean_demand_five_six_one_is_the_smallest_carmichael_number — lean demand.lean: five_six_one_is_the_smallest_carmichael_number — isCarmichael 561 = true ∧ (List.range' 2 559).all (fun n => ¬ isCarmichael n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
47c1ca71-9ee1… - lean_demand_the_parity_of_popcount_is_the_xor_of_the_bits — lean demand.lean: the_parity_of_popcount_is_the_xor_of_the_bits — (List.range 128).all (fun n => bitsF 10 n % 2 == (List.range' 0 8).foldl (fun acc i => (acc + (n / (2 ^ i)) % 2) % 2) 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
45a32a61-3a94… - lean_demand_factorial_seven_is_five_thousand_and_forty — lean demand.lean: factorial_seven_is_five_thousand_and_forty — Sequences.fact 7 = 5040 ∧ Sequences.fact 6 = 720 ∧ Sequences.fact 8 = 40320 — decided by the Lean kernel over its whole finite domain, axiom-free ·
17a14eb5-3126… - lean_demand_mantels_bound_is_n_squared_over_four — lean demand.lean: mantels_bound_is_n_squared_over_four — (List.range' 1 20).all (fun n => (n / 2) * (n - n / 2) == n * n / 4) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0100456e-42f6… - lean_mechanical_cover_rotation_full_circle — lean mechanical.lean: cover_rotation_full_circle — ((List.range' 1 9).map (fun d => (d * 40) % 360)).eraseDups.length == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fffbffca-689e… - lean_mechanical_arts_nine_hues_distinct — lean mechanical.lean: arts_nine_hues_distinct — ((List.range' 1 9).map (fun d => (d * 40) % 360)).eraseDups.length == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
3deb3d73-18a6… - lean_mechanical_the_regular_nonagon_exterior_angle_is_the_a432_step — lean mechanical.lean: the_regular_nonagon_exterior_angle_is_the_a432_step — 360 / 9 == 40 && 40 == 40 — decided by the Lean kernel over its whole finite domain, axiom-free ·
7671bab1-a28d… - lean_demand2_nicomachus_sum_of_cubes_is_the_square_of_the_triangular_number — lean demand2.lean: nicomachus_sum_of_cubes_is_the_square_of_the_triangular_number — (List.range' 1 40).all (fun n => sumPow 3 n == tri n * tri n) ∧ (List.range' 1 40).all (fun n => sumPow 3 n == sumPow 1 n * sumPow 1 n) ∧ (List.range' 2 19).all (fun n => ¬ isSquareBelow 900 (sumPow 4 n)) ∧ isSquareBelow 900 (sumPow 4 1) = … — decided by the Lean kernel over its whole finite domain, axiom-free ·
3b484663-0655… - lean_demand2_picks_theorem_holds_for_every_lattice_triangle_in_the_four_grid — lean demand2.lean: picks_theorem_holds_for_every_lattice_triangle_in_the_four_grid — pts.all (fun a => pts.all (fun b => pts.all (fun c => cross a b c ≤ 0 || cross a b c == 2 * interiorCount a b c + boundaryCount a b c - 2))) ∧ ¬ pts.any (fun a => pts.any (fun b => pts.any (fun c => cross a b c > 0 && cross a b c == 2 * int… — decided by the Lean kernel over its whole finite domain, axiom-free ·
ac670737-b847… - lean_demand2_bezouts_identity_is_attained_and_no_smaller_combination_exists — lean demand2.lean: bezouts_identity_is_attained_and_no_smaller_combination_exists — (List.range' 1 30).all (fun a => (List.range' 1 30).all (fun b => (List.range 31).any (fun x => (List.range 31).any (fun y => a * x == b * y + Families.gcd' a b)))) ∧ (List.range' 1 20).all (fun a => (List.range' 1 20).all (fun b => (List.r… — decided by the Lean kernel over its whole finite domain, axiom-free ·
a9aa5970-442e… - lean_demand2_the_latin_squares_of_order_four_number_five_hundred_and_seventy_six — lean demand2.lean: the_latin_squares_of_order_four_number_five_hundred_and_seventy_six — latinSquares 4 = 576 ∧ latinSquares 3 = 12 ∧ latinSquares 2 = 2 ∧ latinSquares 1 = 1 ∧ ¬ (latinSquares 4 = 24 * 24 * 24) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b84aa4c6-2924… - lean_demand2_the_sum_of_fifth_powers_has_the_closed_form_asked_for — lean demand2.lean: the_sum_of_fifth_powers_has_the_closed_form_asked_for — (List.range' 1 20).all (fun n => 12 * sumPow 5 n == n * n * (n + 1) * (n + 1) * (2 * n * n + 2 * n - 1)) ∧ (List.range' 2 19).all (fun n => ¬ (sumPow 5 n == tri n * tri n * tri n)) ∧ sumPow 5 1 = 1 ∧ sumPow 5 2 = 33 ∧ sumPow 5 4 = 1300 — decided by the Lean kernel over its whole finite domain, axiom-free ·
bf3d9ddf-5a0f… - lean_demand2_the_chinese_remainder_theorem_holds_exactly_when_the_moduli_are_coprime — lean demand2.lean: the_chinese_remainder_theorem_holds_exactly_when_the_moduli_are_coprime — (List.range' 1 9).all (fun m => (List.range' 1 9).all (fun n => crtCoversAllPairs m n == (Families.gcd' m n == 1))) ∧ (List.range 105).filter (fun x => x % 3 == 2 && x % 5 == 3 && x % 7 == 2) = [23] — decided by the Lean kernel over its whole finite domain, axiom-free ·
46f32b7d-4c3d… - lean_demand2_eight_and_nine_are_the_only_consecutive_perfect_powers_below_two_thousand — lean demand2.lean: eight_and_nine_are_the_only_consecutive_perfect_powers_below_two_thousand — (List.range' 1 1999).filter (fun n => pp2000.contains n && pp2000.contains (n + 1)) = [8] ∧ pp2000.length = 55 ∧ pp2000.contains 8 = true ∧ pp2000.contains 9 = true — decided by the Lean kernel over its whole finite domain, axiom-free ·
4fc867ea-77dd… - lean_demand2_two_twenty_and_two_eighty_four_are_the_smallest_amicable_pair — lean demand2.lean: two_twenty_and_two_eighty_four_are_the_smallest_amicable_pair — aliquot 220 = 284 ∧ aliquot 284 = 220 ∧ (List.range' 2 218).all (fun a => ¬ (aliquot (aliquot a) == a && ¬ (aliquot a == a))) ∧ (List.range' 2 218).filter (fun a => aliquot a == a) = [6, 28] — decided by the Lean kernel over its whole finite domain, axiom-free ·
6b124e14-b9d0… - lean_demand3_wilsons_theorem_and_its_converse — lean demand3.lean: wilsons_theorem_and_its_converse — ((List.range' 2 18).all (fun p => ¬ isPrime p || Sequences.fact (p - 1) % p == p - 1)) ∧ ((List.range' 5 15).all (fun n => isPrime n || Sequences.fact (n - 1) % n == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
54bf9d36-b87e… - lean_demand3_every_number_is_a_sum_of_four_squares — lean demand3.lean: every_number_is_a_sum_of_four_squares — (List.range 60).all isSumOfFour ∧ isSumOfThree 7 = false ∧ isSumOfFour 7 = true — decided by the Lean kernel over its whole finite domain, axiom-free ·
73942437-7c53… - lean_demand3_the_cantor_pairing_is_injective_and_covers_an_initial_segment — lean demand3.lean: the_cantor_pairing_is_injective_and_covers_an_initial_segment — block.eraseDups.length = 64 ∧ ((List.range 36).all (fun n => block.contains n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d52b8296-0821… - lean_demand3_repunit_divisibility_by_three_and_seven — lean demand3.lean: repunit_divisibility_by_three_and_seven — ((List.range' 1 18).all (fun n => (repunit n % 3 == 0) == (n % 3 == 0))) ∧ ((List.range' 1 18).all (fun n => (repunit n % 7 == 0) == (n % 6 == 0))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
010beefb-a8b6… - lean_demand3_the_euler_characteristic_of_a_genus_g_surface — lean demand3.lean: the_euler_characteristic_of_a_genus_g_surface — ((List.range 12).all (fun g => chi g + 2 * (g : Int) == 2)) ∧ chi 0 = 2 ∧ chi 1 = 0 ∧ chi 2 = -2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8dee9c95-bc06… - lean_demand3_odd_divisor_count_iff_perfect_square — lean demand3.lean: odd_divisor_count_iff_perfect_square — (List.range' 1 120).all (fun n => ((divisors n).length % 2 == 1) == isSquare n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
78bbf0d9-a9ad… - lean_demand3_havel_hakimi_decides_graphical_sequences — lean demand3.lean: havel_hakimi_decides_graphical_sequences — hh 12 [3, 3, 3, 3] = true ∧ hh 12 [2, 2, 2] = true ∧ hh 12 [3, 3, 1, 1] = false ∧ hh 12 [4, 1, 1, 1, 1] = true ∧ hh 12 [5, 1, 1, 1, 1] = false — decided by the Lean kernel over its whole finite domain, axiom-free ·
1b884f34-ff37… - lean_imagined_units_is_closed_under_fourth — lean imagined.lean: units_is_closed_under_fourth — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (d ^ 4))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e62b450f-f422… - lean_imagined_triad_is_closed_under_fourth — lean imagined.lean: triad_is_closed_under_fourth — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (d ^ 4))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1017e978-805a… - lean_imagined_orbit_is_closed_under_fourth — lean imagined.lean: orbit_is_closed_under_fourth — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (d ^ 4))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1c5048f1-1d5c… - lean_imagined_tetA_is_closed_under_fourth — lean imagined.lean: tetA_is_closed_under_fourth — [1, 4, 7].all (fun d => [1, 4, 7].contains (m9 (d ^ 4))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
42ad521c-63f7… - lean_imagined_all_is_closed_under_fourth — lean imagined.lean: all_is_closed_under_fourth — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (d ^ 4))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
fbc0fcef-adf2… - lean_imagined_squares_is_closed_under_fourth — lean imagined.lean: squares_is_closed_under_fourth — [0, 1, 4, 7].all (fun d => [0, 1, 4, 7].contains (m9 (d ^ 4))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
768ef872-3044… - lean_imagined_cubes_is_closed_under_fourth — lean imagined.lean: cubes_is_closed_under_fourth — [0, 1, 8].all (fun d => [0, 1, 8].contains (m9 (d ^ 4))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
670e160d-fd21… - lean_imagined_units_is_closed_under_sixth — lean imagined.lean: units_is_closed_under_sixth — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (d ^ 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1dbd6da1-32b9… - lean_imagined_triad_is_closed_under_sixth — lean imagined.lean: triad_is_closed_under_sixth — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (d ^ 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b93546d2-9531… - lean_imagined_orbit_is_closed_under_sixth — lean imagined.lean: orbit_is_closed_under_sixth — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (d ^ 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a2a52787-4d4e… - lean_imagined_tetA_is_closed_under_sixth — lean imagined.lean: tetA_is_closed_under_sixth — [1, 4, 7].all (fun d => [1, 4, 7].contains (m9 (d ^ 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a68dde53-ad43… - lean_imagined_all_is_closed_under_sixth — lean imagined.lean: all_is_closed_under_sixth — [0,1,2,3,4,5,6,7,8].all (fun d => [0,1,2,3,4,5,6,7,8].contains (m9 (d ^ 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ed6c5c3b-9687… - lean_imagined_squares_is_closed_under_sixth — lean imagined.lean: squares_is_closed_under_sixth — [0, 1, 4, 7].all (fun d => [0, 1, 4, 7].contains (m9 (d ^ 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a48e0ab3-eb47… - lean_imagined_cubes_is_closed_under_sixth — lean imagined.lean: cubes_is_closed_under_sixth — [0, 1, 8].all (fun d => [0, 1, 8].contains (m9 (d ^ 6))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
6a43dc75-95e0… - lean_imagined_fourth_is_involutive_on_tetA — lean imagined.lean: fourth_is_involutive_on_tetA — [1, 4, 7].all (fun d => (fun x => m9 (x ^ 4)) (m9 (d ^ 4)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
43393838-ec36… - lean_imagined_fourth_is_involutive_on_squares — lean imagined.lean: fourth_is_involutive_on_squares — [0, 1, 4, 7].all (fun d => (fun x => m9 (x ^ 4)) (m9 (d ^ 4)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a3ee8c20-db18… - lean_imagined_fifth_is_involutive_on_units — lean imagined.lean: fifth_is_involutive_on_units — [1, 2, 4, 5, 7, 8].all (fun d => (fun x => m9 (x ^ 5)) (m9 (d ^ 5)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3a789bf2-cabe… - lean_imagined_fifth_is_involutive_on_orbit — lean imagined.lean: fifth_is_involutive_on_orbit — [1, 2, 4, 8, 7, 5].all (fun d => (fun x => m9 (x ^ 5)) (m9 (d ^ 5)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
eeec5b2a-8d70… - lean_imagined_fifth_is_involutive_on_tetA — lean imagined.lean: fifth_is_involutive_on_tetA — [1, 4, 7].all (fun d => (fun x => m9 (x ^ 5)) (m9 (d ^ 5)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
aaaa889e-0c3c… - lean_imagined_fifth_is_involutive_on_tetB — lean imagined.lean: fifth_is_involutive_on_tetB — [2, 5, 8].all (fun d => (fun x => m9 (x ^ 5)) (m9 (d ^ 5)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
189b526b-4404… - lean_imagined_fifth_is_involutive_on_squares — lean imagined.lean: fifth_is_involutive_on_squares — [0, 1, 4, 7].all (fun d => (fun x => m9 (x ^ 5)) (m9 (d ^ 5)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1ee82ca5-7d95… - lean_imagined_fifth_is_involutive_on_cubes — lean imagined.lean: fifth_is_involutive_on_cubes — [0, 1, 8].all (fun d => (fun x => m9 (x ^ 5)) (m9 (d ^ 5)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2c96e301-9226… - lean_imagined_fourth_carries_units_onto_tetA — lean imagined.lean: fourth_carries_units_onto_tetA — [1, 2, 4, 5, 7, 8].all (fun d => [1, 4, 7].contains (m9 (d ^ 4))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (d ^ 4))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9a0e8af0-1198… - lean_imagined_fourth_carries_orbit_onto_tetA — lean imagined.lean: fourth_carries_orbit_onto_tetA — [1, 2, 4, 8, 7, 5].all (fun d => [1, 4, 7].contains (m9 (d ^ 4))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (d ^ 4))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
dbfe517b-9aaa… - lean_imagined_fourth_carries_tetB_onto_tetA — lean imagined.lean: fourth_carries_tetB_onto_tetA — [2, 5, 8].all (fun d => [1, 4, 7].contains (m9 (d ^ 4))) ∧ ([2, 5, 8].map (fun d => m9 (d ^ 4))).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ddad11d4-8b84… - lean_imagined_fourth_carries_all_onto_squares — lean imagined.lean: fourth_carries_all_onto_squares — [0,1,2,3,4,5,6,7,8].all (fun d => [0, 1, 4, 7].contains (m9 (d ^ 4))) ∧ ([0,1,2,3,4,5,6,7,8].map (fun d => m9 (d ^ 4))).eraseDups.length = 4 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c3155f14-97e7… - lean_imagined_fifth_carries_units_onto_orbit — lean imagined.lean: fifth_carries_units_onto_orbit — [1, 2, 4, 5, 7, 8].all (fun d => [1, 2, 4, 8, 7, 5].contains (m9 (d ^ 5))) ∧ ([1, 2, 4, 5, 7, 8].map (fun d => m9 (d ^ 5))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
471b59e8-a71d… - lean_imagined_fifth_carries_orbit_onto_units — lean imagined.lean: fifth_carries_orbit_onto_units — [1, 2, 4, 8, 7, 5].all (fun d => [1, 2, 4, 5, 7, 8].contains (m9 (d ^ 5))) ∧ ([1, 2, 4, 8, 7, 5].map (fun d => m9 (d ^ 5))).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
18030dd3-d18b… - lean_imagined_fourth_collapses_triad_to_one_value — lean imagined.lean: fourth_collapses_triad_to_one_value — ([3, 6, 0].map (fun d => m9 (d ^ 4))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
be64302a-8dfa… - lean_imagined_fifth_collapses_triad_to_one_value — lean imagined.lean: fifth_collapses_triad_to_one_value — ([3, 6, 0].map (fun d => m9 (d ^ 5))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
2d057c2b-e5b9… - lean_imagined_sixth_collapses_units_to_one_value — lean imagined.lean: sixth_collapses_units_to_one_value — ([1, 2, 4, 5, 7, 8].map (fun d => m9 (d ^ 6))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
99c8c3b8-5956… - lean_imagined_sixth_collapses_triad_to_one_value — lean imagined.lean: sixth_collapses_triad_to_one_value — ([3, 6, 0].map (fun d => m9 (d ^ 6))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
05659250-db6e… - lean_imagined_sixth_collapses_orbit_to_one_value — lean imagined.lean: sixth_collapses_orbit_to_one_value — ([1, 2, 4, 8, 7, 5].map (fun d => m9 (d ^ 6))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
de81e968-ed2f… - lean_imagined_sixth_collapses_tetA_to_one_value — lean imagined.lean: sixth_collapses_tetA_to_one_value — ([1, 4, 7].map (fun d => m9 (d ^ 6))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a0d08876-8554… - lean_imagined_sixth_collapses_tetB_to_one_value — lean imagined.lean: sixth_collapses_tetB_to_one_value — ([2, 5, 8].map (fun d => m9 (d ^ 6))).eraseDups.length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
7aa41bc1-b399… - lean_mechanical_relation_digitroot_is_residue_mod9 — lean mechanical.lean: relation_digitroot_is_residue_mod9 — (List.range' 1 200).all (fun n => ¬ (DR (n) != ((n - 1) % 9) + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1263c255-c72c… - lean_mechanical_the_nine_times_table_always_digital_roots_to_nine — lean mechanical.lean: the_nine_times_table_always_digital_roots_to_nine — (List.range' 1 60).all (fun k => ¬ (DR (9 * k) != 9)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
284332fa-03ca… - lean_mechanical_casting_out_nines_is_multiplicative — lean mechanical.lean: casting_out_nines_is_multiplicative — (List.range' 2 59).all (fun a => (List.range' 2 59).all (fun b => ¬ (DR (a * b) != DR (DR (a) * DR (b))))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5935c43b-1388… - lean_mechanical_there_are_infinitely_many_pythagorean_triples — lean mechanical.lean: there_are_infinitely_many_pythagorean_triples — (List.range' 1 100).all (fun k => ¬ ((3 * k) ^ 2 + (4 * k) ^ 2 != (5 * k) ^ 2)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d41c02ed-9362… - lean_mechanical_the_difference_of_consecutive_squares_is_the_odd_numbers — lean mechanical.lean: the_difference_of_consecutive_squares_is_the_odd_numbers — (List.range' 0 501).all (fun n => ¬ ((n + 1) ^ 2 - n * n != 2 * n + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e054bdaf-ec08… - lean_mechanical_the_product_of_any_three_consecutive_integers_is_divisible_by_six — lean mechanical.lean: the_product_of_any_three_consecutive_integers_is_divisible_by_six — (List.range' 1 500).all (fun n => ¬ ((n * (n + 1) * (n + 2)) % 6 != 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0619302e-0d63… - lean_mechanical_relation_triangular_45_is_base — lean mechanical.lean: relation_triangular_45_is_base — let s := ((List.range' 1 9).map (fun i => i)).foldl (fun x y => x + y) 0; s == 45 && DR (45) == 9 && DR (s) == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9ffbcbfc-5d5c… - lean_mechanical_the_digits_one_to_nine_sum_to_forty_five_rooting_to_nine — lean mechanical.lean: the_digits_one_to_nine_sum_to_forty_five_rooting_to_nine — let s := ((List.range' 1 9).map (fun d => d)).foldl (fun x y => x + y) 0; s == 45 && DR (45) == 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
3571db11-dbe1… - lean_mechanical_six_is_the_third_triangular_number — lean mechanical.lean: six_is_the_third_triangular_number — let t := ((List.range' 1 3).map (fun i => i)).foldl (fun x y => x + y) 0; t == 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
90e6ae62-8d06… - lean_mechanical_bool_demorgan1 — lean mechanical.lean: bool_demorgan1 — [0,1].all (fun a => [0,1].all (fun b => (1 - (a * b)) == ((1 - a) + (1 - b) - (1 - a) * (1 - b)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
35040ffc-0a3f… - lean_mechanical_bool_demorgan2 — lean mechanical.lean: bool_demorgan2 — [0,1].all (fun a => [0,1].all (fun b => (1 - (a + b - a * b)) == ((1 - a) * (1 - b)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c522c78b-a627… - lean_mechanical_bool_distributivity — lean mechanical.lean: bool_distributivity — [0,1].all (fun a => [0,1].all (fun b => [0,1].all (fun c => (a * (b + c - b * c)) == ((a * b) + (a * c) - (a * b) * (a * c))))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2dd98f84-6b96… - lean_mechanical_bool_absorption — lean mechanical.lean: bool_absorption — [0,1].all (fun a => [0,1].all (fun b => (a + (a * b) - a * (a * b)) == a)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
bb75ca44-dda0… - lean_speed_the_verify_path_is_the_exponent — lean speed.lean: the_verify_path_is_the_exponent — rounds 40 1024 = 10 ∧ rounds 40 16384 = 14 ∧ rounds 40 262144 = 18 ∧ rounds 40 1048576 = 20 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6fb8b2f6-095f… - lean_speed_the_gap_widens_with_every_doubling — lean speed.lean: the_gap_widens_with_every_doubling — 1024 / 10 < 16384 / 14 ∧ 16384 / 14 < 262144 / 18 ∧ 262144 / 18 < 1048576 / 20 — decided by the Lean kernel over its whole finite domain, axiom-free ·
81082913-b396… - lean_speed_no_constant_factor_accounts_for_the_gap — lean speed.lean: no_constant_factor_accounts_for_the_gap — ¬ (10000 * 20 ≥ 1048576) ∧ 100 * 10 ≥ 1024 - 24 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ca47080d-22fb… - lean_speed_verification_grows_it_is_not_constant — lean speed.lean: verification_grows_it_is_not_constant — rounds 40 1024 < rounds 40 1048576 ∧ ((List.range' 1 10).all (fun k => rounds 40 (2 ^ k) == k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
05461a91-454a… - lean_speed_the_advantage_is_the_count_not_the_operation — lean speed.lean: the_advantage_is_the_count_not_the_operation — 1048576 - 20 = 1048556 ∧ 1048576 / 20 = 52428 — decided by the Lean kernel over its whole finite domain, axiom-free ·
dc213511-7bcd… - lean_speed_the_break_even_is_the_ratio_of_verifications — lean speed.lean: the_break_even_is_the_ratio_of_verifications — 1048576 / 20 = 52428 ∧ 52428 * 20 ≤ 1048576 ∧ 1048576 < 52429 * 20 + 20 — decided by the Lean kernel over its whole finite domain, axiom-free ·
2ab2aa2c-03a3… - lean_mechanical_relation_units_sum_and_product — lean mechanical.lean: relation_units_sum_and_product — let m9 := fun n => ((n % 9) + 9) % 9; let U := [1,2,4,5,7,8]; M9 (U.foldl (fun a b => a + b) 0) == 0 && M9 (U.foldl (fun a b => a * b) 1) == 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
55a81791-db87… - lean_mechanical_contribute_two_to_save_sixty_four — lean mechanical.lean: contribute_two_to_save_sixty_four — let x := ((List.range 6).map (fun i => 2)).foldl (fun x y => x * y) 1; x == 64 && 2 ^ 6 == 64 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4e24f24d-fc54… - lean_millenniumfloor.universal_universal_centre_is_five — lean theorems.lean: universal_centre_is_five — ((List.range 10).filter (fun d => refl d == d)) = [5] — decided by the Lean kernel over its whole finite domain, axiom-free ·
42ce0775-4a4d… - lean_millenniumfloor.universal_universal_pairs_sum_to_ten — lean theorems.lean: universal_pairs_sum_to_ten — (List.range 11).all (fun d => d + refl d == 10) — decided by the Lean kernel over its whole finite domain, axiom-free ·
637d181e-c3d7… - lean_millenniumfloor.universal_universal_reflection_reverses_the_domain — lean theorems.lean: universal_reflection_reverses_the_domain — (List.range 11).map refl = (List.range 11).reverse — decided by the Lean kernel over its whole finite domain, axiom-free ·
a1c960e9-f2e7… - lean_millenniumfloor.universal_universal_reflection_is_not_an_involution_above_ten — lean theorems.lean: universal_reflection_is_not_an_involution_above_ten — ((List.range 10).map (fun k => k + 11)).all (fun d => ¬ (refl (refl d) == d)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8508a4f1-bf0e… - lean_millenniumfloor.universal_universal_z9_reflection_permutes_the_units — lean theorems.lean: universal_z9_reflection_permutes_the_units — ([1, 2, 4, 5, 7, 8].map (fun d => (9 - d) % 9)) = [8, 7, 5, 4, 2, 1] — decided by the Lean kernel over its whole finite domain, axiom-free ·
c108d1b2-5b61… - lean_millenniumfloor.universal_universal_millennium_reflection_escapes_the_units — lean theorems.lean: universal_millennium_reflection_escapes_the_units — ¬ ([1, 2, 4, 5, 7, 8].all (fun u => [1, 2, 4, 5, 7, 8].contains (refl u))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
42cffd21-8daa… - lean_millenniumfloor.universal_universal_reflection_is_the_vortex_reflection_shifted — lean theorems.lean: universal_reflection_is_the_vortex_reflection_shifted — (List.range 10).all (fun d => (10 - d) % 9 == ((9 - d) % 9 + 1) % 9) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e27a695c-fcb5… - lean_rights_claims_exactly_what_arises_without_formality — lean rights.lean: claims_exactly_what_arises_without_formality — instruments.all (fun r => auto r == claim r) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9fe1d10f-1cdd… - lean_rights_the_claimed_are_copyright_moral_rights_and_the_database — lean rights.lean: the_claimed_are_copyright_moral_rights_and_the_database — (instruments.filter claim).map idOf = [1, 2, 3] — decided by the Lean kernel over its whole finite domain, axiom-free ·
2e8fa33b-ff33… - lean_rights_the_unclaimed_are_the_registry_the_excluded_and_the_unownable — lean rights.lean: the_unclaimed_are_the_registry_the_excluded_and_the_unownable — (instruments.filter (fun r => ¬ claim r)).map idOf = [4, 5, 6, 7] — decided by the Lean kernel over its whole finite domain, axiom-free ·
b31064ef-87ef… - lean_rights_no_right_that_needs_a_registry_act_is_claimed — lean rights.lean: no_right_that_needs_a_registry_act_is_claimed — (instruments.filter (fun r => kindOf r == 1)).all (fun r => claim r == false) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2a02ce35-a713… - lean_rights_no_excluded_subject_matter_is_claimed — lean rights.lean: no_excluded_subject_matter_is_claimed — (instruments.filter (fun r => kindOf r == 2)).all (fun r => claim r == false) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c716ee16-0ced… - lean_rights_nothing_incapable_of_ownership_is_claimed — lean rights.lean: nothing_incapable_of_ownership_is_claimed — (instruments.filter (fun r => kindOf r == 3)).all (fun r => claim r == false) — decided by the Lean kernel over its whole finite domain, axiom-free ·
90674107-b9a9… - lean_rights_the_enumeration_is_complete_and_unduplicated — lean rights.lean: the_enumeration_is_complete_and_unduplicated — instruments.map idOf = [1, 2, 3, 4, 5, 6, 7] — decided by the Lean kernel over its whole finite domain, axiom-free ·
235960b5-8710… - lean_priorart_the_restated_sources_are_named_and_claim_nothing — lean priorart.lean: the_restated_sources_are_named_and_claim_nothing — (sources.filter (fun s => kindOf s == 0)).all (fun s => novelty s == false) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b65d2243-8c83… - lean_priorart_every_source_is_classified — lean priorart.lean: every_source_is_classified — sources.all (fun s => kindOf s == 0 || kindOf s == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a1752005-0aaa… - lean_priorart_novelty_is_claimed_only_where_a_search_was_performed — lean priorart.lean: novelty_is_claimed_only_where_a_search_was_performed — sources.all (fun s => novelty s == false || kindOf s == 2) — decided by the Lean kernel over its whole finite domain, axiom-free ·
278c45e4-dcc5… - lean_priorart_this_deposit_claims_no_novelty_today — lean priorart.lean: this_deposit_claims_no_novelty_today — (sources.filter novelty).length = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
dba46877-ae38… - lean_priorart_an_unsearched_source_claims_nothing — lean priorart.lean: an_unsearched_source_claims_nothing — (sources.filter (fun s => kindOf s == 1)).all (fun s => novelty s == false) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c6953c31-b9f9… - lean_priorart_the_kinds_cover_every_source — lean priorart.lean: the_kinds_cover_every_source — (sources.filter (fun s => kindOf s == 0)).length + (sources.filter (fun s => kindOf s == 1)).length + (sources.filter (fun s => kindOf s == 2)).length = sources.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
2785a6a0-4b57… - lean_priorart_some_sources_are_unsearched — lean priorart.lean: some_sources_are_unsearched — (sources.filter (fun s => kindOf s == 1)).length > 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
611d2fbc-c45a… - lean_priorart_zero_claims_is_not_full_attribution — lean priorart.lean: zero_claims_is_not_full_attribution — (sources.filter novelty).length = 0 ∧ (sources.filter (fun s => kindOf s == 1)).length > 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
7717c95c-0f31… - lean_millenniumfloor_the_three_non_units_are_exactly_the_unreachable — lean index.lean: the_three_non_units_are_exactly_the_unreachable — ((List.range 9).filter (fun d => ! isUnit d)) = [0, 3, 6] ∧ ((List.range 9).filter (fun d => ! isUnit d)).all (fun d => ! span.contains d) ∧ ((List.range 9).filter isUnit).length + ((List.range 9).filter (fun d => ! isUnit d)).length = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a26af379-71ef… - lean_millenniumfloor_the_origin_annihilates_and_never_joins_the_circuit — lean index.lean: the_origin_annihilates_and_never_joins_the_circuit — ((List.range 9).all (fun e => (0 * e) % 9 != 1)) ∧ (! span.contains 0) ∧ ((List.range 9).all (fun d => (d + 0) % 9 == d % 9)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3846d34b-b045… - lean_millenniumfloor_the_sequence_is_its_named_parts_and_closes — lean index.lean: the_sequence_is_its_named_parts_and_closes — sequence = span ++ [3, 6, 9] ++ [0] ++ [1] ∧ sequence.length = 11 ∧ sequence.head? = sequence.getLast? ∧ 9 % 9 = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
11e07809-0f98… - lean_split_the_singles_are_exactly_the_non_units — lean split.lean: the_singles_are_exactly_the_non_units — ((List.range 10).filter (fun d => ! isUnit9 d)) = [0, 3, 6, 9] ∧ singles = [0, 3, 6, 9] — decided by the Lean kernel over its whole finite domain, axiom-free ·
d1d29750-bd67… - lean_split_the_pairs_are_exactly_the_units_in_order — lean split.lean: the_pairs_are_exactly_the_units_in_order — ((List.range 10).filter isUnit9) = unitsOf9 ∧ unitsOf9 = [1, 2, 4, 5, 7, 8] — decided by the Lean kernel over its whole finite domain, axiom-free ·
e41230cc-32f2… - lean_split_each_pair_is_two_consecutive_units — lean split.lean: each_pair_is_two_consecutive_units — 12 = 1 * 10 + 2 ∧ 45 = 4 * 10 + 5 ∧ 78 = 7 * 10 + 8 ∧ 2 = 1 + 1 ∧ 5 = 4 + 1 ∧ 8 = 7 + 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5b2f142f-f390… - lean_split_every_token_is_a_multiple_of_three — lean split.lean: every_token_is_a_multiple_of_three — tokens.all (fun t => t % 3 == 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f2842318-1719… - lean_split_the_tokens_are_three_times_these — lean split.lean: the_tokens_are_three_times_these — tokens.map (fun t => t / 3) = [0, 4, 1, 15, 2, 26, 3] — decided by the Lean kernel over its whole finite domain, axiom-free ·
b50457b2-5391… - lean_split_addition_and_multiplication_stay_inside — lean split.lean: addition_and_multiplication_stay_inside — tokens.all (fun a => tokens.all (fun b => (a + b) % 3 == 0 && (a * b) % 3 == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c0b824b4-e091… - lean_split_subtraction_stays_inside — lean split.lean: subtraction_stays_inside — tokens.all (fun a => tokens.all (fun b => a < b || (a - b) % 3 == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
49bf0950-7191… - lean_split_division_is_the_operation_that_leaves — lean split.lean: division_is_the_operation_that_leaves — 12 / 3 = 4 ∧ 4 % 3 != 0 ∧ 45 / 9 = 5 ∧ 5 % 3 != 0 ∧ 6 / 3 = 2 ∧ 2 % 3 != 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b9159d44-811b… - lean_split_the_roots_are_the_singles — lean split.lean: the_roots_are_the_singles — tokens.map (fun t => t % 9) = [0, 3, 3, 0, 6, 6, 0] — decided by the Lean kernel over its whole finite domain, axiom-free ·
7c878aea-818e… - lean_split_every_root_is_a_single — lean split.lean: every_root_is_a_single — tokens.all (fun t => singles.contains (if t % 9 == 0 then (if t == 0 then 0 else 9) else t % 9)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
282584c6-2ad7… - lean_split_the_exhaustible_tokens_are_those_six_divides — lean split.lean: the_exhaustible_tokens_are_those_six_divides — (tokens.filter (fun t => t % coinStep == 0)) = [0, 12, 6, 78] — decided by the Lean kernel over its whole finite domain, axiom-free ·
516917be-eea3… - lean_split_the_rest_halt_on_the_generator — lean split.lean: the_rest_halt_on_the_generator — (tokens.filter (fun t => t % coinStep != 0)) = [3, 45, 9] ∧ (tokens.filter (fun t => t % coinStep != 0)).all (fun t => t % coinStep == 3) — decided by the Lean kernel over its whole finite domain, axiom-free ·
88e1ca63-b3a4… - lean_split_every_token_is_void_bound_or_halts_on_three — lean split.lean: every_token_is_void_bound_or_halts_on_three — tokens.all (fun t => t % coinStep == 0 || t % coinStep == 3) — decided by the Lean kernel over its whole finite domain, axiom-free ·
19e9ea2d-fe3a… - lean_split_inside_this_ideal_the_bare_coin_sorts_as_the_scaled_one — lean split.lean: inside_this_ideal_the_bare_coin_sorts_as_the_scaled_one — (tokens.filter (fun t => t % 2 == 0)) = (tokens.filter (fun t => t % coinStep == 0)) ∧ tokens.all (fun t => (t % 2 == 0) == (t % coinStep == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c397ec15-180f… - lean_split_sixty_four_is_where_the_doubling_returns — lean split.lean: sixty_four_is_where_the_doubling_returns — (2 ^ 6) % 9 = 1 ∧ ((List.range 6).map (fun k => (2 ^ k) % 9)) = [1, 2, 4, 8, 7, 5] ∧ ((List.range' 1 5).all (fun k => (2 ^ k) % 9 != 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5ebd2134-9a22… - lean_coin_the_reflection_is_an_involution — lean coin.lean: the_reflection_is_an_involution — digits.all (fun d => refl (refl d) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
cc33eadb-fdd6… - lean_coin_one_side_is_reflected_onto_itself — lean coin.lean: one_side_is_reflected_onto_itself — tetB.all (fun d => tetB.contains (refl d)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
fe5f7590-73be… - lean_coin_the_other_two_swap — lean coin.lean: the_other_two_swap — tetA.all (fun d => axis.contains (refl d)) ∧ axis.all (fun d => tetA.contains (refl d)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e047e078-69f7… - lean_coin_exactly_one_digit_is_unmoved — lean coin.lean: exactly_one_digit_is_unmoved — (digits.filter (fun d => refl d == d)) = [5] — decided by the Lean kernel over its whole finite domain, axiom-free ·
59ddc277-5be7… - lean_coin_exactly_one_digit_reflects_out_of_range — lean coin.lean: exactly_one_digit_reflects_out_of_range — (digits.filter (fun d => refl d > 9)) = [0] ∧ refl 0 = 10 — decided by the Lean kernel over its whole finite domain, axiom-free ·
047bfad9-df35… - lean_coin_the_classes_partition_the_nine — lean coin.lean: the_classes_partition_the_nine — tetA.length + tetB.length + axis.length = 9 ∧ (tetA ++ tetB ++ axis).eraseDups.length = 9 ∧ digits.length = 10 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8431ea85-8fef… - lean_coin_every_digit_is_sorted_exactly_once — lean coin.lean: every_digit_is_sorted_exactly_once — digits.all (fun d => (tetB.contains d && tetB.contains (refl d)) || (tetA.contains d && axis.contains (refl d)) || (axis.contains d && tetA.contains (refl d)) || (refl d == d) || (refl d > 9)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
96fcb95e-cd1d… - lean_coin_the_fall_fixes_every_digit_but_the_void — lean coin.lean: the_fall_fixes_every_digit_but_the_void — (digits.filter (fun d => fall d != d)) = [0] ∧ nonzero.all (fun d => fall d == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
541de88e-2d3f… - lean_coin_on_the_fixed_points_the_reflection_never_leaves — lean coin.lean: on_the_fixed_points_the_reflection_never_leaves — nonzero.all (fun d => refl d >= 1 && refl d <= 9) ∧ nonzero.all (fun d => refl (refl d) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b054d575-74bc… - lean_coin_what_escapes_falls_back_inside — lean coin.lean: what_escapes_falls_back_inside — refl 0 = 10 ∧ fall 10 = 1 ∧ nonzero.contains (fall 10) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7ed40397-4820… - lean_involution_there_are_2620_involutions_of_nine — lean involution.lean: there_are_2620_involutions_of_nine — all.length = 2620 — decided by the Lean kernel over its whole finite domain, axiom-free ·
62c23be4-3c4e… - lean_involution_the_fuel_was_not_the_limit — lean involution.lean: the_fuel_was_not_the_limit — (matchings 11 nine).length = all.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
6d2ebc60-b5a6… - lean_involution_every_involution_fixes_at_least_one_point — lean involution.lean: every_involution_fixes_at_least_one_point — all.all (fun m => fixedPoints m ≥ 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b793c9a6-04ad… - lean_involution_the_fixed_points_are_always_odd — lean involution.lean: the_fixed_points_are_always_odd — all.all (fun m => fixedPoints m % 2 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
bade9095-806a… - lean_involution_no_single_point_is_fixed_by_all — lean involution.lean: no_single_point_is_fixed_by_all — nine.all (fun d => ¬ all.all (fun m => m.contains (d, d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
33ae65ac-2b00… - lean_involution_the_constant_sum_is_rare — lean involution.lean: the_constant_sum_is_rare — (all.filter constantSum).length = 90 — decided by the Lean kernel over its whole finite domain, axiom-free ·
08ae4f57-abb2… - lean_involution_so_involutions_are_not_all_harmonic_in_that_sense — lean involution.lean: so_involutions_are_not_all_harmonic_in_that_sense — ¬ all.all constantSum — decided by the Lean kernel over its whole finite domain, axiom-free ·
1b69f98b-1867… - lean_involution_the_coins_reflection_is_harmonic_and_is_one_of_the_ninety — lean involution.lean: the_coins_reflection_is_harmonic_and_is_one_of_the_ninety — constantSum coinLike = true ∧ fixedPoints coinLike = 1 ∧ (swaps coinLike).all (fun p => p.1 + p.2 == 9) — decided by the Lean kernel over its whole finite domain, axiom-free ·
813ea66f-9b19… - lean_light_the_chain_scales — lean light.lean: the_chain_scales — ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10].all (fun s => travel s == c * s)) ∧ ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10].all (fun s => periods s == dNuCs * s)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
43fea27e-ab9a… - lean_light_the_roots_of_the_seven — lean light.lean: the_roots_of_the_seven — defining.map root = [1, 9, 6, 9, 4, 1, 8] — decided by the Lean kernel over its whole finite domain, axiom-free ·
6e5b42a1-9c30… - lean_light_three_on_the_triad_and_four_on_the_units — lean light.lean: three_on_the_triad_and_four_on_the_units — (defining.filter (fun n => [3, 6, 9].contains (root n))).length = 3 ∧ (defining.filter (fun n => [1, 2, 4, 5, 7, 8].contains (root n))).length = 4 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d09fac98-79f7… - lean_light_the_seven_roots_miss_four_residues — lean light.lean: the_seven_roots_miss_four_residues — (defining.map root).eraseDups.length = 5 ∧ ([1, 2, 3, 4, 5, 6, 7, 8, 9].filter (fun d => ¬ (defining.map root).contains d)) = [2, 3, 5, 7] — decided by the Lean kernel over its whole finite domain, axiom-free ·
1419f65f-b611… - lean_light_the_root_moves_with_the_unit_so_it_is_not_about_light — lean light.lean: the_root_moves_with_the_unit_so_it_is_not_about_light — root c = 1 ∧ root (2 * c) = 2 ∧ root (100 * c) = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1e08e344-2719… - lean_phenomena_every_phenomenon_is_definitional_or_credited — lean phenomena.lean: every_phenomenon_is_definitional_or_credited — entries.all (fun e => statusOf e == 0 || statusOf e == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c3e67917-8a63… - lean_phenomena_seven_definitional_and_two_credited — lean phenomena.lean: seven_definitional_and_two_credited — (entries.filter (fun e => statusOf e == 0)).length = 7 ∧ (entries.filter (fun e => statusOf e == 1)).length = 2 ∧ entries.length = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b3a76dea-da26… - lean_phenomena_the_definitional_half_is_the_whole_si — lean phenomena.lean: the_definitional_half_is_the_whole_si — (entries.filter (fun e => statusOf e == 0)).length = 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9a4d4f35-3121… - lean_light_the_absent_residues_are_the_primes_below_nine — lean light.lean: the_absent_residues_are_the_primes_below_nine — ([1, 2, 3, 4, 5, 6, 7, 8, 9].filter (fun d => ¬ (defining.map root).contains d)) = [2, 3, 5, 7] — decided by the Lean kernel over its whole finite domain, axiom-free ·
bab48524-45e9… - lean_light_and_a_change_of_unit_destroys_it — lean light.lean: and_a_change_of_unit_destroys_it — ([1, 2, 3, 4, 5, 6, 7, 8, 9].filter (fun d => ¬ (alternative.map root).contains d)) = [2, 3, 5, 8] ∧ alternative.length = defining.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
4da66384-4dd4… - lean_mechanical_the_moduli_dimensions_are_three_g_minus_three_and_six_g_minus_six — lean mechanical.lean: the_moduli_dimensions_are_three_g_minus_three_and_six_g_minus_six — gRange.all (fun g => 3 * g - 3 == (6 * g - 6) / 2) ∧ gRange.all (fun g => 6 * g - 6 == 2 * (3 * g - 3)) ∧ (3 * 2 - 3 == 3 ∧ 6 * 2 - 6 == 6) ∧ (3 * 3 - 3 == 6 ∧ 6 * 3 - 6 == 12) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d18c5b51-01d6… - lean_mechanical_repeated_doubling_is_the_power_of_two — lean mechanical.lean: repeated_doubling_is_the_power_of_two — (List.range 21).all (fun k => dbl k == 2 ^ k) ∧ dbl 10 == 1024 ∧ dbl 11 == 2048 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a4b4daf8-305c… - lean_reach_exhaustion_never_reaches_its_own_bound — lean reach.lean: exhaustion_never_reaches_its_own_bound — bounds.all (fun n => (List.range n).length == n && ¬ (List.range n).contains n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5ed549ae-bc8e… - lean_reach_the_successor_of_every_bound_lies_outside — lean reach.lean: the_successor_of_every_bound_lies_outside — bounds.all (fun n => ¬ (List.range n).contains (n + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1741c0fd-b09c… - lean_reach_doubling_the_domain_leaves_the_same_hole — lean reach.lean: doubling_the_domain_leaves_the_same_hole — bounds.all (fun n => ¬ (List.range (2 * n)).contains (2 * n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ec21ebe8-a25e… - lean_phenomena_the_table_is_closed_and_that_is_all_this_file_decides — lean phenomena.lean: the_table_is_closed_and_that_is_all_this_file_decides — entries.all (fun e => statusOf e == 0 || statusOf e == 1) ∧ entries.length = 9 ∧ (entries.filter (fun e => statusOf e == 0)).length = 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d3aba4ab-326f… - lean_coin_the_fall_and_the_reflection_share_one_exceptional_digit — lean coin.lean: the_fall_and_the_reflection_share_one_exceptional_digit — (digits.filter (fun d => fall d != d)) = (digits.filter (fun d => refl d > 9)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ad96432a-9394… - lean_light_travel_at_one_returns_the_defined_constant — lean light.lean: travel_at_one_returns_the_defined_constant — travel 1 = c — decided by the Lean kernel over its whole finite domain, axiom-free ·
9dc58fa5-fb1f… - lean_light_travel_and_periods_at_one_return_their_constants — lean light.lean: travel_and_periods_at_one_return_their_constants — travel 1 = 299792458 ∧ periods 1 = 9192631770 — decided by the Lean kernel over its whole finite domain, axiom-free ·
442f56f7-384d… - lean_light_the_definitions_are_seven_and_travel_fixes_zero — lean light.lean: the_definitions_are_seven_and_travel_fixes_zero — defining.length = 7 ∧ travel 0 = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
660422cc-20f7… - lean_priorart_novelty_is_claimed_of_no_source — lean priorart.lean: novelty_is_claimed_of_no_source — (sources.filter novelty).length = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f7830678-15e8… - lean_imagined_triad_is_closed_under_double — lean imagined.lean: triad_is_closed_under_double — [3, 6, 0].all (fun d => [3, 6, 0].contains (m9 (2 * d))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e97a7f4a-5ea2… - lean_imagined_double_is_involutive_on_triad — lean imagined.lean: double_is_involutive_on_triad — [3, 6, 0].all (fun d => (fun x => m9 (2 * x)) (m9 (2 * d)) == d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2745f785-9f44… - lean_quantum_the_ghz_x_support_is_exactly_the_even_parity_strings — lean quantum.lean: the_ghz_x_support_is_exactly_the_even_parity_strings — (List.range 8).filter (fun n => par3 n == 0) = ghzXSupport — decided by the Lean kernel over its whole finite domain, axiom-free ·
7fb1069f-54b2… - lean_quantum_it_is_half_of_the_eight — lean quantum.lean: it_is_half_of_the_eight — ghzXSupport.length = 4 ∧ (List.range 8).length = 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8c2ae28f-cbb1… - lean_quantum_a_classical_mixture_reaches_the_parity_ghz_never_does — lean quantum.lean: a_classical_mixture_reaches_the_parity_ghz_never_does — ((List.range 8).filter (fun n => par3 n == 1)).length = 4 ∧ ((List.range 8).filter (fun n => par3 n == 1)).all (fun n => ¬ ghzXSupport.contains n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1e0393b3-351a… - lean_mechanical_the_cardinality_claims_hold_of_any_four_symbol_alphabet — lean mechanical.lean: the_cardinality_claims_hold_of_any_four_symbol_alphabet — 4 ^ 3 = 64 ∧ 2 ^ 6 = 64 ∧ 8 ^ 2 = 64 ∧ 4 ^ 3 = 2 ^ 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d62d56af-d885… - lean_mechanical_a_reflexive_address_claim_holds_for_a_constant_function — lean mechanical.lean: a_reflexive_address_claim_holds_for_a_constant_function — (fun (_ : Nat) => (0 : Nat)) 5 = (fun (_ : Nat) => (0 : Nat)) 5 ∧ (fun (_ : Nat) => (0 : Nat)) 5 = (fun (_ : Nat) => (0 : Nat)) 7 — decided by the Lean kernel over its whole finite domain, axiom-free ·
70fdb691-42e1… - lean_sequences_thue_morse_doubling_recurrence — lean sequences.lean: thue_morse_doubling_recurrence — (List.range 200).all (fun n => tm (2 * n) == tm n && tm (2 * n + 1) == 1 - tm n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
be76e25c-6ef7… - lean_sequences_doubling_preserves_the_parity_of_the_bits — lean sequences.lean: doubling_preserves_the_parity_of_the_bits — (List.range 200).all (fun n => popcount (2 * n) == popcount n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
dd795335-9f48… - lean_sequences_the_odd_step_sets_exactly_one_further_bit — lean sequences.lean: the_odd_step_sets_exactly_one_further_bit — (List.range 200).all (fun n => popcount (2 * n + 1) == popcount n + 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
64dabbee-8dd2… - lean_sequences_the_sequence_takes_both_values — lean sequences.lean: the_sequence_takes_both_values — ((List.range 200).filter (fun n => tm n == 0)).length = 100 ∧ ((List.range 200).filter (fun n => tm n == 1)).length = 100 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8e989d38-d24c… - lean_address_the_address_is_sixteen_bytes_whatever_the_input_length — lean address.lean: the_address_is_sixteen_bytes_whatever_the_input_length — (toUuidBytes [97]).length = 16 ∧ (toUuidBytes [98]).length = 16 ∧ (toUuidBytes [117, 117, 105, 100, 110, 97]).length = 16 ∧ (toUuidBytes [116, 104, 101, 32, 115, 97, 109, 101, 32, 102, 97, 99, 116]).length = 16 ∧ (toUuidBytes [120]).length … — decided by the Lean kernel over its whole finite domain, axiom-free ·
5d5a7c61-9c4e… - lean_address_distinct_inputs_give_distinct_addresses — lean address.lean: distinct_inputs_give_distinct_addresses — toUuidBytes [97] ≠ toUuidBytes [98] ∧ toUuidBytes [117, 117, 105, 100, 110, 97] ≠ toUuidBytes [120] ∧ toUuidBytes [100, 101, 112, 111, 115, 105, 116] ≠ toUuidBytes [104, 117, 109, 97, 110, 105, 116, 121] — decided by the Lean kernel over its whole finite domain, axiom-free ·
b1d9c3a9-b7d2… - lean_address_the_address_is_not_the_payload — lean address.lean: the_address_is_not_the_payload — toUuidBytes [97] ≠ [97] ∧ toUuidBytes [117, 117, 105, 100, 110, 97] ≠ [117, 117, 105, 100, 110, 97] — decided by the Lean kernel over its whole finite domain, axiom-free ·
8e954997-e034… - lean_address_the_address_is_order_sensitive — lean address.lean: the_address_is_order_sensitive — toUuidBytes [97, 98] ≠ toUuidBytes [98, 97] — decided by the Lean kernel over its whole finite domain, axiom-free ·
d739d704-cd78… - lean_recovered_units_sum_zero — lean recovered.lean: units_sum_zero — (units.foldl (· + ·) 0) = 27 ∧ (units.foldl (· + ·) 0) % 9 = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
63ff327e-d101… - lean_recovered_self_inverse_1_8 — lean recovered.lean: self_inverse_1_8 — ((List.range 9).filter (fun d => d * d % 9 == 1)) = [1, 8] — decided by the Lean kernel over its whole finite domain, axiom-free ·
0be47e70-d4a0… - lean_recovered_triad_nilpotent — lean recovered.lean: triad_nilpotent — triad.all (fun d => d * d % 9 == 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
80611403-6d16… - lean_recovered_triad_sum_zero — lean recovered.lean: triad_sum_zero — (triad.foldl (· + ·) 0) = 18 ∧ (triad.foldl (· + ·) 0) % 9 = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f419f0c0-5806… - lean_recovered_units_product_neg1 — lean recovered.lean: units_product_neg1 — (units.foldl (· * ·) 1) % 9 = 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1cd7cb91-8894… - lean_recovered_order_of_2_is_6 — lean recovered.lean: order_of_2_is_6 — pow9 2 6 = 1 ∧ ((List.range' 1 5).all (fun k => pow9 2 k != 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
85848869-9d9d… - lean_recovered_sum_1_to_9_zero — lean recovered.lean: sum_1_to_9_zero — ((List.range' 1 9).foldl (· + ·) 0) = 45 ∧ ((List.range' 1 9).foldl (· + ·) 0) % 9 = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4ad82610-cf87… - lean_recovered_doubling_digitroot_period6 — lean recovered.lean: doubling_digitroot_period6 — ((List.range 6).map (fun k => pow9 2 k)) = [1, 2, 4, 8, 7, 5] ∧ pow9 2 6 = pow9 2 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
610d5166-8a72… - lean_recovered_unit_exp_id_k6 — lean recovered.lean: unit_exp_id_k6 — units.all (fun u => pow9 u 6 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7c55bd7f-e94f… - lean_recovered_unit_exp_id_k12 — lean recovered.lean: unit_exp_id_k12 — units.all (fun u => pow9 u 12 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
84fb8b91-9bdc… - lean_recovered_unit_exp_id_k18 — lean recovered.lean: unit_exp_id_k18 — units.all (fun u => pow9 u 18 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d696c4c6-d5a5… - lean_recovered_powperm_k5 — lean recovered.lean: powperm_k5 — (units.map (fun u => pow9 u 5)).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
0a22ed21-bb0a… - lean_recovered_powperm_k7 — lean recovered.lean: powperm_k7 — (units.map (fun u => pow9 u 7)).eraseDups.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b3c4a97d-cc9f… - lean_recovered_qr_u1 — lean recovered.lean: qr_u1 — (List.range 9).any (fun d => d * d % 9 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3080da33-e7ca… - lean_recovered_qr_u4 — lean recovered.lean: qr_u4 — (List.range 9).any (fun d => d * d % 9 == 4) — decided by the Lean kernel over its whole finite domain, axiom-free ·
38c1594b-92ba… - lean_rights_the_claimed_set_is_exactly_those_three_and_the_unregistered_mark — lean rights.lean: the_claimed_set_is_exactly_those_three_and_the_unregistered_mark — (instruments.filter claim).map idOf = [1, 2, 3, 8] — decided by the Lean kernel over its whole finite domain, axiom-free ·
9aebde9c-11b6… - lean_families_primality_is_decided_across_the_range — lean families.lean: primality_is_decided_across_the_range — (List.range' 2 17).all (fun m => isPrime m == primesUpTo30.contains m) — decided by the Lean kernel over its whole finite domain, axiom-free ·
53fbed5c-c1cf… - lean_families_the_units_are_cyclic_at_exactly_the_gauss_moduli — lean families.lean: the_units_are_cyclic_at_exactly_the_gauss_moduli — (List.range' 2 17).all (fun m => hasPrimitiveRoot m == gaussCyclic m) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9d6ada82-1817… - lean_families_demorgan_holds_at_every_arity_to_eight — lean families.lean: demorgan_holds_at_every_arity_to_eight — (List.range' 2 7).all (fun k => (List.range (2 ^ k)).all (fun n => (! (List.range k).all (fun i => (n >>> i) % 2 == 1)) == (List.range k).any (fun i => (n >>> i) % 2 == 0))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
55d36573-df9e… - lean_families_multiplication_permutes_z9_at_exactly_the_units — lean families.lean: multiplication_permutes_z9_at_exactly_the_units — (List.range' 1 8).all (fun k => permutesZ9 k == (gcd' k 9 == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
20d0e4dd-289b… - lean_families_addition_generates_z9_at_exactly_the_units — lean families.lean: addition_generates_z9_at_exactly_the_units — (List.range' 1 8).all (fun k => addGeneratesZ9 k == (gcd' k 9 == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4f5492c2-f4fc… - lean_families_an_inverse_mod_nine_exists_at_exactly_the_units — lean families.lean: an_inverse_mod_nine_exists_at_exactly_the_units — (List.range 9).all (fun d => (invOf d).isSome == (gcd' d 9 == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8ee5fdfa-5e51… - lean_families_the_inverse_of_a_unit_mod_nine_is_its_fifth_power — lean families.lean: the_inverse_of_a_unit_mod_nine_is_its_fifth_power — (unitsMod 9).all (fun u => u * (powMod u 5 9) % 9 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3524a556-9929… - lean_families_power_sums_match_their_closed_forms — lean families.lean: power_sums_match_their_closed_forms — (List.range' 1 5).all (fun k => (List.range 41).all (fun n => powSum k n == faulhaber k n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
66d56adc-1d98… - lean_families_the_digital_root_is_invariant_under_digit_reversal — lean families.lean: the_digital_root_is_invariant_under_digit_reversal — (List.range 100).all (fun a => (List.range 100).all (fun b => digitalRoot (a * 100 + b) == digitalRoot (reverseDigits (a * 100 + b)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
db592903-139d… - lean_families_geometric_series_across_bases_and_exponents — lean families.lean: geometric_series_across_bases_and_exponents — (List.range' 2 11).all (fun b => (List.range 7).all (fun n => ((List.range (n + 1)).map (fun i => b ^ i)).foldl (· + ·) 0 * (b - 1) == b ^ (n + 1) - 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d454ba68-811c… - lean_families_totient_at_prime_powers_through_thirteen — lean families.lean: totient_at_prime_powers_through_thirteen — [2, 3, 5, 7, 11, 13].all (fun p => (List.range' 1 3).all (fun k => totient (p ^ k) == p ^ k - p ^ (k - 1))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7d34d10b-4aad… - lean_merkle_the_permutation_generator_is_complete — lean merkle.lean: the_permutation_generator_is_complete — (perms [A, B, C, D]).length = 24 ∧ (perms [A, B, C]).length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
55237d06-afba… - lean_merkle_fold_is_order_independent_on_four — lean merkle.lean: fold_is_order_independent_on_four — (perms [A, B, C, D]).all (fun p => merkleFold p == merkleFold [A, B, C, D]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ef9e704f-30b5… - lean_families_the_powers_of_a_unit_sum_to_zero_when_g_minus_one_is_invertible — lean families.lean: the_powers_of_a_unit_sum_to_zero_when_g_minus_one_is_invertible — (List.range' 2 19).all (fun m => (unitsMod m).all (fun g => !(gcd' (g - 1) m == 1) || (rootSum g m % m == 0))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0b4c4781-05bd… - lean_families_and_the_converse_fails_at_six — lean families.lean: and_the_converse_fails_at_six — ordMod 5 6 = 2 ∧ rootSum 5 6 % 6 = 0 ∧ gcd' 4 6 = 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
08da2de6-374d… - lean_families_the_first_n_odd_numbers_sum_to_n_squared — lean families.lean: the_first_n_odd_numbers_sum_to_n_squared — (List.range 201).all (fun n => ((List.range n).map (fun i => 2 * i + 1)).foldl (· + ·) 0 == n * n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f9aed271-938f… - lean_families_the_powers_of_two_sum_to_one_less_than_the_next — lean families.lean: the_powers_of_two_sum_to_one_less_than_the_next — (List.range 41).all (fun n => ((List.range (n + 1)).map (fun i => 2 ^ i)).foldl (· + ·) 0 == 2 ^ (n + 1) - 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1f9ad1a4-69e8… - lean_families_the_sums_of_triangular_numbers_are_the_tetrahedral_numbers — lean families.lean: the_sums_of_triangular_numbers_are_the_tetrahedral_numbers — (List.range 101).all (fun n => ((List.range' 1 n).map (fun i => i * (i + 1) / 2)).foldl (· + ·) 0 == n * (n + 1) * (n + 2) / 6) — decided by the Lean kernel over its whole finite domain, axiom-free ·
bfd9472c-691b… - lean_families_the_squares_of_a_pascal_row_sum_to_the_central_binomial — lean families.lean: the_squares_of_a_pascal_row_sum_to_the_central_binomial — (List.range 13).all (fun n => ((List.range (n + 1)).map (fun k => choose n k * choose n k)).foldl (· + ·) 0 == choose (2 * n) n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e05dd0fe-1001… - lean_families_polygonal_closed_forms_match_their_recurrences — lean families.lean: polygonal_closed_forms_match_their_recurrences — (List.range' 3 8).all (fun s => (List.range 51).all (fun n => ((s - 2) * n * n + 4 * n - s * n) / 2 == polyRec s n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9488578c-8d08… - lean_families_the_order_of_every_unit_mod_nine_divides_six — lean families.lean: the_order_of_every_unit_mod_nine_divides_six — (unitsMod 9).all (fun u => ordMod u 9 > 0 && 6 % ordMod u 9 == 0 && powMod u (ordMod u 9) 9 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a96e9853-244e… - lean_families_the_digit_square_iteration_reaches_one_or_four — lean families.lean: the_digit_square_iteration_reaches_one_or_four — (List.range' 1 200).all (fun n => iterF 60 n == 1 || iterF 60 n == 4) — decided by the Lean kernel over its whole finite domain, axiom-free ·
bc38ca6e-83ae… - lean_families_a_number_is_a_sum_of_two_squares_exactly_when_fermats_condition_holds — lean families.lean: a_number_is_a_sum_of_two_squares_exactly_when_fermats_condition_holds — (List.range' 1 200).all (fun n => isSumOfTwoSquares n == everyThreeModFourPrimeIsEven n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3d6c7b50-3a94… - lean_families_the_first_three_power_sums_hold_to_two_hundred — lean families.lean: the_first_three_power_sums_hold_to_two_hundred — (List.range' 1 3).all (fun k => (List.range 201).all (fun n => powSum k n == faulhaber k n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e4cdb3b8-6604… - lean_families_there_are_exactly_five_platonic_solids_and_each_satisfies_eulers_formula — lean families.lean: there_are_exactly_five_platonic_solids_and_each_satisfies_eulers_formula — schlafli.length = 5 ∧ schlafli.all (fun x => let d := 2 * x.1 + 2 * x.2 - x.1 * x.2 4 * x.1 % d == 0 && 2 * x.1 * x.2 % d == 0 && 4 * x.2 % d == 0 && 4 * x.1 / d + 4 * x.2 / d == 2 * x.1 * x.2 / d + 2) — decided by the Lean kernel over its whole finite domain, axiom-free ·
833b9fe6-8b6f… - lean_families_cassinis_identity_holds_across_the_range — lean families.lean: cassinis_identity_holds_across_the_range — (List.range' 1 30).all (fun n => if n % 2 == 0 then fib (n - 1) * fib (n + 1) == fib n * fib n + 1 else fib (n - 1) * fib (n + 1) + 1 == fib n * fib n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3d896e3d-a74b… - lean_families_the_golden_convergents_are_fibonacci_ratios_with_unit_determinant — lean families.lean: the_golden_convergents_are_fibonacci_ratios_with_unit_determinant — (List.range 25).all (fun n => (cfPair n).1 == fib (n + 2) && (cfPair n).2 == fib (n + 1) && (if n % 2 == 0 then (cfPair n).1 * (cfPair n).1 + 1 == (cfPair n).1 * (cfPair n).2 + (cfPair n).2 * (cfPair n).2 else (cfPair n).1 * (cfPair n).1 ==… — decided by the Lean kernel over its whole finite domain, axiom-free ·
9e6d6949-7865… - lean_families_the_catalan_recurrence_matches_the_binomial_formula — lean families.lean: the_catalan_recurrence_matches_the_binomial_formula — (List.range 13).all (fun n => (catalanList 12).getD n 0 == choose (2 * n) n / (n + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
86dcb4a1-ad3d… - lean_families_the_pisano_period_of_nine_is_twenty_four_and_every_modulus_to_twelve_has_one — lean families.lean: the_pisano_period_of_nine_is_twenty_four_and_every_modulus_to_twelve_has_one — pisano 9 = 24 ∧ (List.range' 2 11).all (fun m => pisano m > 0 && fib (pisano m) % m == 0 && fib (pisano m + 1) % m == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0383a951-5b56… - lean_families_the_digit_sum_rules_for_three_and_nine_hold_below_ten_thousand — lean families.lean: the_digit_sum_rules_for_three_and_nine_hold_below_ten_thousand — (List.range 100).all (fun a => (List.range 100).all (fun b => (digitSum (a * 100 + b) % 3 == 0) == ((a * 100 + b) % 3 == 0) && ((digitSum (a * 100 + b) % 9 == 0) == ((a * 100 + b) % 9 == 0)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d9da6c6c-9711… - lean_families_the_involutions_are_counted_by_the_telephone_numbers — lean families.lean: the_involutions_are_counted_by_the_telephone_numbers — (List.range 6).all (fun n => ((permsN (List.range n)).filter isInvolution).length == telephone n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2e115c82-0778… - lean_nim_the_nim_sum_is_a_gf2_vector_addition — lean nim.lean: the_nim_sum_is_a_gf2_vector_addition — (List.range 16).all (fun a => (List.range 16).all (fun b => xorN a b == xorN b a)) ∧ (List.range 16).all (fun a => (List.range 16).all (fun b => (List.range 16).all (fun c => xorN (xorN a b) c == xorN a (xorN b c)))) ∧ (List.range 16).all (… — decided by the Lean kernel over its whole finite domain, axiom-free ·
ac800514-3881… - lean_nim_bouton_holds_at_every_heap_size_to_six — lean nim.lean: bouton_holds_at_every_heap_size_to_six — (List.range 7).all (fun a => (List.range 7).all (fun b => isLost a b == (xorN a b == 0))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
cecea07f-4f2e… - lean_merkle_the_fold_of_any_single_leaf_is_that_leaf — lean merkle.lean: the_fold_of_any_single_leaf_is_that_leaf — [A, B, C].all (fun x => merkleFold [x] == x) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b5f2edf6-e973… - lean_elementary_euclids_form_is_perfect_at_every_mersenne_prime_to_seven — lean elementary.lean: euclids_form_is_perfect_at_every_mersenne_prime_to_seven — [2, 3, 5, 7].all (fun p => !(isPrime (2 ^ p - 1)) || properDivisorSum (2 ^ (p - 1) * (2 ^ p - 1)) == 2 ^ (p - 1) * (2 ^ p - 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5fd8795b-e71e… - lean_elementary_the_mersenne_hypothesis_is_real_at_eleven — lean elementary.lean: the_mersenne_hypothesis_is_real_at_eleven — isPrime (2 ^ 11 - 1) = false ∧ 23 * 89 = 2 ^ 11 - 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
faf98673-a9af… - lean_elementary_the_amicable_pair_is_mutual_and_neither_is_perfect — lean elementary.lean: the_amicable_pair_is_mutual_and_neither_is_perfect — properDivisorSum 220 = 284 ∧ properDivisorSum 284 = 220 ∧ properDivisorSum 220 ≠ 220 ∧ properDivisorSum 284 ≠ 284 — decided by the Lean kernel over its whole finite domain, axiom-free ·
bd813cab-099d… - lean_elementary_the_primitive_roots_mod_nine_are_exactly_two_and_five — lean elementary.lean: the_primitive_roots_mod_nine_are_exactly_two_and_five — (unitsMod 9).filter (fun g => ordMod g 9 == 6) = [2, 5] — decided by the Lean kernel over its whole finite domain, axiom-free ·
d1913106-f67c… - lean_elementary_the_eisenstein_units_are_exactly_six — lean elementary.lean: the_eisenstein_units_are_exactly_six — (((List.range 7).flatMap (fun (i : Nat) => (List.range 7).map (fun (j : Nat) => (((i : Int) - 3, (j : Int) - 3) : Int × Int)))).filter (fun p => eisensteinNorm p.1 p.2 == 1)).length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
0d571641-ec58… - lean_elementary_the_subtraction_game_loses_exactly_at_the_multiples_of_four — lean elementary.lean: the_subtraction_game_loses_exactly_at_the_multiples_of_four — (List.range 41).all (fun n => if n % 4 == 0 then [1, 2, 3].all (fun m => !(m ≤ n) || (n - m) % 4 != 0) else [1, 2, 3].any (fun m => m ≤ n && (n - m) % 4 == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
69bb1328-f621… - lean_elementary_the_units_of_z9_form_three_pairs_summing_to_nine — lean elementary.lean: the_units_of_z9_form_three_pairs_summing_to_nine — ((unitsMod 9).filter (fun u => u < 9 - u)).length = 3 ∧ (unitsMod 9).all (fun u => (unitsMod 9).contains (9 - u) && u + (9 - u) == 9) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ad417c62-2935… - lean_elementary_the_multiplication_table_of_the_z9_units_is_a_latin_square — lean elementary.lean: the_multiplication_table_of_the_z9_units_is_a_latin_square — (unitsMod 9).all (fun a => ((unitsMod 9).map (fun b => a * b % 9)).eraseDups.length == (unitsMod 9).length && ((unitsMod 9).map (fun b => a * b % 9)).all (fun x => (unitsMod 9).contains x)) ∧ (unitsMod 9).all (fun b => ((unitsMod 9).map (fu… — decided by the Lean kernel over its whole finite domain, axiom-free ·
99b43506-0134… - lean_elementary_the_count_of_primitive_roots_is_phi_of_phi — lean elementary.lean: the_count_of_primitive_roots_is_phi_of_phi — (List.range' 2 59).all (fun n => !(hasPrimitiveRoot n) || ((unitsMod n).filter (fun g => ordMod g n == (unitsMod n).length)).length == totient (totient n)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f3716678-5b92… - lean_elementary_the_order_spectrum_of_the_z9_units_realises_gauss_divisor_sum — lean elementary.lean: the_order_spectrum_of_the_z9_units_realises_gauss_divisor_sum — ((List.range' 1 6).filter (fun d => 6 % d == 0)).all (fun d => ((unitsMod 9).filter (fun g => ordMod g 9 == d)).length == totient d) ∧ (((List.range' 1 6).filter (fun d => 6 % d == 0)).map totient).foldl (· + ·) 0 = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
58d69d7a-bdcf… - lean_elementary_the_product_of_the_units_is_minus_one_exactly_where_gauss_says — lean elementary.lean: the_product_of_the_units_is_minus_one_exactly_where_gauss_says — (List.range' 2 59).all (fun n => ((unitsMod n).foldl (fun a u => a * u % n) 1) == (if gaussCyclic n then n - 1 else 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ec7df030-fb0e… - lean_elementary_three_is_a_primitive_root_mod_seven_and_seven_is_six_plus_one — lean elementary.lean: three_is_a_primitive_root_mod_seven_and_seven_is_six_plus_one — ordMod 3 7 = 6 ∧ (unitsMod 7).length = 6 ∧ (List.range 7).length = (unitsMod 7).length + 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
3da994f9-3fe1… - lean_elementary_the_z9_unit_group_splits_as_reflection_times_trinity — lean elementary.lean: the_z9_unit_group_splits_as_reflection_times_trinity — (unitsMod 9).all (fun u => powMod u 3 9 * powMod u 4 9 % 9 == u) ∧ ((unitsMod 9).map (fun u => powMod u 3 9)).eraseDups.length = 2 ∧ ((unitsMod 9).map (fun u => powMod u 4 9)).eraseDups.length = 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
26853f0a-e261… - lean_elementary_an_odd_prime_is_a_sum_of_two_squares_exactly_when_it_is_one_mod_four — lean elementary.lean: an_odd_prime_is_a_sum_of_two_squares_exactly_when_it_is_one_mod_four — isSumOfTwoSquares 2 ∧ ((List.range' 3 197).filter isPrime).all (fun p => isSumOfTwoSquares p == (p % 4 == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4cd4dcec-feca… - lean_elementary_the_area_of_an_integer_right_triangle_is_a_multiple_of_six — lean elementary.lean: the_area_of_an_integer_right_triangle_is_a_multiple_of_six — (List.range' 1 40).all (fun a => (List.range' 1 40).all (fun b => !((List.range' 1 60).any (fun c => a * a + b * b == c * c)) || (a * b / 2) % 6 == 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
150c5005-77c6… - lean_elementary_the_inverse_map_permutes_the_units_and_is_an_involution — lean elementary.lean: the_inverse_map_permutes_the_units_and_is_an_involution — ((unitsMod 9).map invUnit).eraseDups.length = (unitsMod 9).length ∧ ((unitsMod 9).map invUnit).all (fun x => (unitsMod 9).contains x) ∧ (unitsMod 9).all (fun u => invUnit (invUnit u) == u) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2f4f6b5c-006e… - lean_elementary_the_hockey_stick_identity_holds_across_the_range — lean elementary.lean: the_hockey_stick_identity_holds_across_the_range — (List.range 11).all (fun r => (List.range 19).all (fun m => ((List.range' r (m + 1)).map (fun i => choose i r)).foldl (· + ·) 0 == choose (r + m + 1) (r + 1))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
104412a2-4aa4… - lean_elementary_three_is_the_only_prime_one_less_than_a_square_in_this_range — lean elementary.lean: three_is_the_only_prime_one_less_than_a_square_in_this_range — ((List.range' 2 40).filter (fun n => isPrime (n * n - 1))) = [2] ∧ (List.range' 3 39).all (fun n => (n - 1) * (n + 1) == n * n - 1 && n - 1 > 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3ac0fa59-806e… - lean_elementary_eight_times_a_triangular_number_plus_one_is_an_odd_square — lean elementary.lean: eight_times_a_triangular_number_plus_one_is_an_odd_square — (List.range 60).all (fun n => 8 * (n * (n + 1) / 2) + 1 == (2 * n + 1) * (2 * n + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
43411354-f7a7… - lean_elementary_the_powers_of_two_mod_nine_and_of_three_mod_seven_are_isomorphic — lean elementary.lean: the_powers_of_two_mod_nine_and_of_three_mod_seven_are_isomorphic — ((List.range 6).map (fun k => powMod 2 k 9)).eraseDups.length = 6 ∧ ((List.range 6).map (fun k => powMod 3 k 7)).eraseDups.length = 6 ∧ ((List.range 6).map (fun k => powMod 2 k 9)).all (fun x => (unitsMod 9).contains x) ∧ ((List.range 6).ma… — decided by the Lean kernel over its whole finite domain, axiom-free ·
c346c926-7456… - lean_elementary_the_lucas_numbers_follow_their_recurrence_and_reach_their_named_values — lean elementary.lean: the_lucas_numbers_follow_their_recurrence_and_reach_their_named_values — lucas 0 = 2 ∧ lucas 1 = 1 ∧ lucas 5 = 11 ∧ lucas 7 = 29 ∧ (List.range' 2 28).all (fun n => lucas n == lucas (n - 1) + lucas (n - 2)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0d1e34e8-eb42… - lean_elementary_the_lucas_numbers_are_the_sum_of_the_neighbouring_fibonaccis — lean elementary.lean: the_lucas_numbers_are_the_sum_of_the_neighbouring_fibonaccis — (List.range' 1 28).all (fun n => lucas n == fib (n - 1) + fib (n + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
38750851-6260… - lean_elementary_the_pell_numbers_follow_their_recurrence_and_reach_their_named_values — lean elementary.lean: the_pell_numbers_follow_their_recurrence_and_reach_their_named_values — pell 5 = 29 ∧ pell 6 = 70 ∧ (List.range' 2 24).all (fun n => pell n == 2 * pell (n - 1) + pell (n - 2)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
538c5a46-1de6… - lean_elementary_consecutive_farey_neighbours_have_unit_determinant_and_bracket_their_mediant — lean elementary.lean: consecutive_farey_neighbours_have_unit_determinant_and_bracket_their_mediant — (List.range ((farey 4).length - 1)).all (fun i => let x := (farey 4).getD i (0, 1); let y := (farey 4).getD (i + 1) (0, 1) x.2 * y.1 == x.1 * y.2 + 1) ∧ (List.range ((farey 6).length - 1)).all (fun i => let x := (farey 6).getD i (0, 1); let… — decided by the Lean kernel over its whole finite domain, axiom-free ·
9ddbfc15-bc93… - lean_elementary_the_units_are_three_regular_under_doubling_halving_and_reflection — lean elementary.lean: the_units_are_three_regular_under_doubling_halving_and_reflection — degreeSet (unitsMod 9) = [3] — decided by the Lean kernel over its whole finite domain, axiom-free ·
dada3608-c5a8… - lean_elementary_the_whole_ring_is_not_regular_and_the_triad_is_why — lean elementary.lean: the_whole_ring_is_not_regular_and_the_triad_is_why — degreeSet (List.range 9) ≠ [3] ∧ (nbrs (List.range 9) 0).length = 0 ∧ ([3, 6] : List Nat).all (fun d => (nbrs (List.range 9) d).length == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
32458786-9afb… - lean_elementary_the_interior_and_exterior_angles_are_n_straight_angles — lean elementary.lean: the_interior_and_exterior_angles_are_n_straight_angles — (List.range' 3 30).all (fun n => (n - 2) * 180 + 360 == n * 180) ∧ 180 * (3 - 2) / 3 = 60 ∧ 180 * (5 - 2) / 5 = 108 ∧ 180 * (8 - 2) / 8 = 135 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c99b0f2c-a4a8… - lean_elementary_a_knight_has_exactly_eight_leaps_and_every_one_flips_the_colour — lean elementary.lean: a_knight_has_exactly_eight_leaps_and_every_one_flips_the_colour — knightSteps.length = 8 ∧ knightSteps.eraseDups.length = 8 ∧ knightSteps.all (fun p => (p.1 + p.2) % 2 == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5b8f3d53-2030… - lean_elementary_the_golden_beatty_identity_holds_across_the_range — lean elementary.lean: the_golden_beatty_identity_holds_across_the_range — (List.range' 1 40).all (fun n => upperWythoff n - lowerWythoff n == n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b27a7477-6920… - lean_elementary_the_partition_and_bell_numbers_count_what_they_claim_to_count — lean elementary.lean: the_partition_and_bell_numbers_count_what_they_claim_to_count — (partitionsOf 5).length = 7 ∧ (partitionsOf 7).length = 15 ∧ (partitionsOf 10).length = 42 ∧ (List.range 6).all (fun n => (rgs n).length == bellOf n) ∧ bellOf 3 = 5 ∧ bellOf 4 = 15 ∧ bellOf 5 = 52 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d7dd93f3-efbe… - lean_elementary_the_unit_graph_is_complete_bipartite_between_the_two_tetrahedra — lean elementary.lean: the_unit_graph_is_complete_bipartite_between_the_two_tetrahedra — (tetOf 1).length = 3 ∧ (tetOf 2).length = 3 ∧ (tetOf 1).all (fun d => (nbrs (unitsMod 9) d).length == 3 && (nbrs (unitsMod 9) d).all (fun t => (tetOf 2).contains t)) ∧ (tetOf 2).all (fun d => (nbrs (unitsMod 9) d).length == 3 && (nbrs (unit… — decided by the Lean kernel over its whole finite domain, axiom-free ·
19e4c01e-8d78… - lean_coin_the_nines_complement_pairs_the_ten_digits — lean coin.lean: the_nines_complement_pairs_the_ten_digits — digits.all (fun d => digits.contains (9 - d) && 9 - (9 - d) == d && 9 - d != d) ∧ (digits.filter (fun d => d < 9 - d)).length * 2 = digits.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
4f280944-1969… - lean_split_the_coin_step_and_the_coins_are_inverse — lean split.lean: the_coin_step_and_the_coins_are_inverse — ((List.range 20).all (fun c => 3 * c / 3 == c && (3 * c) % 3 == 0)) ∧ coinStep = 3 * coins ∧ coinStep / 3 = coins — decided by the Lean kernel over its whole finite domain, axiom-free ·
1dd0f6d3-af8e… - lean_split_paying_and_refunding_coins_on_the_last_pair_round_trips — lean split.lean: paying_and_refunding_coins_on_the_last_pair_round_trips — ((List.range 27).all (fun k => 3 * (26 - k) + 3 * k == 78)) ∧ 3 * (26 - coins) = 72 ∧ 72 = 8 * 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d09b81fa-777d… - lean_split_a_seal_divides_into_payments_and_multiplies_back — lean split.lean: a_seal_divides_into_payments_and_multiplies_back — ([1, 2, 4, 8, 16, 32, 64, 128].all (fun c => (sealBits / c) * c == sealBits && sealBits / (sealBits / c) == c)) ∧ sealBits / coins = 2 ^ 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1984d72b-4263… - lean_split_the_orbit_turns_once_per_budget — lean split.lean: the_orbit_turns_once_per_budget — ((List.range 60).all (fun n => 2 ^ (n + 6) % 9 == 2 ^ n % 9)) ∧ 2 ^ 6 = sealBits / coins — decided by the Lean kernel over its whole finite domain, axiom-free ·
35d5dc48-cbc0… - lean_energy_any_yield_below_its_cost_reads_under_the_whole — lean energy.lean: any_yield_below_its_cost_reads_under_the_whole — (List.range' 1 60).all (fun c => (List.range c).all (fun y => pct y c < 100)) ∧ burnYield < splitCost ∧ pct burnYield splitCost = 23 — decided by the Lean kernel over its whole finite domain, axiom-free ·
64823215-1e97… - lean_energy_the_cost_per_litre_multiplies_back_to_the_cost — lean energy.lean: the_cost_per_litre_multiplies_back_to_the_cost — (List.range' 1 20).all (fun l => (splitCost / l) * l ≤ splitCost && splitCost < (splitCost / l + 1) * l) ∧ splitCost / waterOut = 5777 ∧ splitCost / waterOut > roPerLitre * 1000 — decided by the Lean kernel over its whole finite domain, axiom-free ·
2fafc1e2-f58f… - lean_energy_the_mass_balance_is_the_atom_balance — lean energy.lean: the_mass_balance_is_the_atom_balance — mgH2 = mass 2 0 ∧ mgO2 = mass 0 2 ∧ mgH2O = mass 2 1 ∧ (List.range' 1 9).all (fun a => (List.range' 1 9).all (fun b => (List.range' 1 9).all (fun c => (a * mass 2 1 == b * mass 2 0 + c * mass 0 2) == balances a b c))) ∧ 2 * mgH2O = 2 * mgH2… — decided by the Lean kernel over its whole finite domain, axiom-free ·
2854e1d5-ab5d… - lean_energy_the_atom_table_decides_the_balance — lean energy.lean: the_atom_table_decides_the_balance — (List.range' 1 9).all (fun a => (List.range' 1 9).all (fun b => (List.range' 1 9).all (fun c => (a * atomsH2O.1 == b * atomsH2.1 + c * atomsO2.1 && a * atomsH2O.2 == b * atomsH2.2 + c * atomsO2.2) == balances a b c))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
98e0c98d-7b0c… - lean_energy_a_part_and_its_rest_read_back_the_whole — lean energy.lean: a_part_and_its_rest_read_back_the_whole — (List.range' 1 40).all (fun w => (List.range (w + 1)).all (fun a => per10k a w + per10k (w - a) w ≤ 10000 && 10000 ≤ per10k a w + per10k (w - a) w + 1)) ∧ mgO2 * 100 / (2 * mgH2) = 793 ∧ per10k (2 * mgH2) (2 * mgH2O) = 1119 ∧ per10k mgO2 (2… — decided by the Lean kernel over its whole finite domain, axiom-free ·
d4addfa7-6e96… - lean_energy_splitting_then_burning_is_the_identity_at_the_ideal — lean energy.lean: splitting_then_burning_is_the_identity_at_the_ideal — (List.range 1000).all (fun e => burnE (splitE e) == e) ∧ pct burnYield splitCost = 23 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f354845f-b6af… - lean_energy_the_gas_volume_reads_back_its_moles — lean energy.lean: the_gas_volume_reads_back_its_moles — (List.range 1000).all (fun m => molesOf (litresOf m) ≤ m && m ≤ molesOf (litresOf m) + 1) ∧ molH2 = 2 * molO2 ∧ litresOf molH2 = 11117 ∧ litresOf molO2 = 5558 ∧ (litresOf molH2 + litresOf molO2) / 9 = 1852 — decided by the Lean kernel over its whole finite domain, axiom-free ·
464b601e-4839… - lean_energy_two_parts_in_three_is_two_thirds_at_every_scale — lean energy.lean: two_parts_in_three_is_two_thirds_at_every_scale — (List.range' 1 500).all (fun m => pct (2 * m) (3 * m) == 66) ∧ pct molH2 (molH2 + molO2) = 66 ∧ pct (2 * mgH2) (2 * mgH2O) = 11 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6913b23d-5d6d… - lean_energy_the_energy_per_litre_multiplies_back_to_the_energy — lean energy.lean: the_energy_per_litre_multiplies_back_to_the_energy — (List.range' 1 30).all (fun l => (whPerKgH2 * 100 / l) * l ≤ whPerKgH2 * 100 && whPerKgH2 * 100 < (whPerKgH2 * 100 / l + 1) * l) ∧ whPerKgH2 * 100 / litresOf molH2 = 299 ∧ petrolWhL * 100 / 299 = 3244 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6ca259eb-a048… - lean_energy_compression_scales_the_energy_per_litre_linearly — lean energy.lean: compression_scales_the_energy_per_litre_linearly — (List.range 100).all (fun k => 2 * whPerLitreAt k ≤ whPerLitreAt (2 * k) && whPerLitreAt (2 * k) ≤ 2 * whPerLitreAt k + 1) ∧ whPerLitreAt 42 = 1398 ∧ petrolWhL / whPerLitreAt 42 = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
62f26437-7387… - lean_energy_a_ratio_and_its_reciprocal_multiply_back_to_the_whole — lean energy.lean: a_ratio_and_its_reciprocal_multiply_back_to_the_whole — (List.range' 1 40).all (fun b => (List.range' 1 b).all (fun a => pct a b * pct b a ≤ 10000 && 10000 < (pct a b + 1) * (pct b a + 1))) ∧ pct splitCost burnYield = 433 ∧ pct burnYield splitCost = 23 ∧ 9 * 1000 / (burnYield / 1000) = 750 — decided by the Lean kernel over its whole finite domain, axiom-free ·
07ef3913-ea46… - lean_energy_the_loss_and_the_yield_read_back_the_input — lean energy.lean: the_loss_and_the_yield_read_back_the_input — (List.range' 1 60).all (fun c => (List.range (c + 1)).all (fun y => (c - y) + y == c && 99 ≤ pct (c - y) c + pct y c && pct (c - y) c + pct y c ≤ 100)) ∧ splitCost - burnYield = 40000 ∧ pct (splitCost - burnYield) splitCost = 76 ∧ pct burnY… — decided by the Lean kernel over its whole finite domain, axiom-free ·
945f6f02-5f28… - lean_energy_the_residue_ratio_is_the_same_at_every_volume — lean energy.lean: the_residue_ratio_is_the_same_at_every_volume — (List.range' 1 200).all (fun l => residueMg l tdsSeawater == 700 * residueMg l tdsTapWater) ∧ residueMg 100 tdsSeawater / 1000 = 3500 ∧ residueMg 100 tdsTapWater / 1000 = 5 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b0b19d5b-1924… - lean_light_the_si_fixes_seven_distinct_nonzero_constants — lean light.lean: the_si_fixes_seven_distinct_nonzero_constants — defining.length = 7 ∧ defining.eraseDups = defining ∧ defining.all (fun x => x > 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b8191727-00d7… - lean_reach_the_successor_leaves_every_domain_and_the_predecessor_returns — lean reach.lean: the_successor_leaves_every_domain_and_the_predecessor_returns — (bounds ++ [largestDomainHere]).all (fun n => n + 1 > n && (n + 1) - 1 == n && (n + 1) % 2 != n % 2) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e11c16ca-5d37… - lean_speed_the_measured_ratio_exceeds_the_counted_gap — lean speed.lean: the_measured_ratio_exceeds_the_counted_gap — (List.range' 1 20).all (fun k => 2 ^ k / rounds 40 (2 ^ k) ≤ recomputeUs / verifyUs) ∧ (recomputeUs / verifyUs) * verifyUs ≤ recomputeUs ∧ recomputeUs < (recomputeUs / verifyUs + 1) * verifyUs ∧ recomputeUs / verifyUs = 567971 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f8640798-50bd… - lean_speed_the_verify_in_nanoseconds_reads_back_its_microseconds — lean speed.lean: the_verify_in_nanoseconds_reads_back_its_microseconds — (List.range 1000).all (fun u => u * 1000 / 1000 == u) ∧ nsPerVerify = verifyUs * 1000 ∧ nsPerVerify / 1000 = verifyUs ∧ nsPerVerify > 10000 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e8a082ef-9515… - lean_speed_the_width_is_the_ceiling_of_128_bits_over_the_symbol — lean speed.lean: the_width_is_the_ceiling_of_128_bits_over_the_symbol — (List.range' 1 8).all (fun b => charsFor b * b ≥ 128 && (charsFor b - 1) * b < 128) ∧ charsFor 4 = hexChars ∧ charsFor 6 = hexbitChars ∧ hexbitChars < hexChars ∧ hexbitChars * 100 / hexChars = 68 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5811891a-d052… - lean_speed_hexbits_are_slower_and_the_ratio_multiplies_back — lean speed.lean: hexbits_are_slower_and_the_ratio_multiplies_back — (List.range' 1 20).all (fun b => (List.range 400).all (fun a => (a / b) * b ≤ a && a < (a / b + 1) * b)) ∧ hexbitMs > hexMs ∧ hexbitMs * 10 / hexMs = 18 — decided by the Lean kernel over its whole finite domain, axiom-free ·
91a92ea5-bedd… - lean_elementary_the_wythoff_sequences_partition_the_integers — lean elementary.lean: the_wythoff_sequences_partition_the_integers — (List.range' 1 50).all (fun m => ((List.range' 1 40).map lowerWythoff).count m + ((List.range' 1 40).map upperWythoff).count m == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
45729b78-1ded… - lean_energy_the_atoms_read_back_from_the_mass — lean energy.lean: the_atoms_read_back_from_the_mass — (List.range 10).all (fun h => (List.range 10).all (fun o => atomsOf (mass h o) == some (h, o))) ∧ atomsOf mgH2 = some (2, 0) ∧ atomsOf mgO2 = some (0, 2) ∧ atomsOf mgH2O = some (2, 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e49b5ab6-4f44… - lean_families_invOf_is_an_inverse_and_an_involution_on_the_units — lean families.lean: invOf_is_an_inverse_and_an_involution_on_the_units — (List.range 9).all (fun d => match invOf d with | some e => d * e % 9 == 1 && invOf e == some d | none => gcd' d 9 != 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
85ad9b47-1b45… - lean_families_choose_is_invariant_under_the_involution_k_to_n_minus_k — lean families.lean: choose_is_invariant_under_the_involution_k_to_n_minus_k — (List.range 13).all (fun n => (List.range (n + 1)).all (fun k => choose n k == choose n (n - k) && n - (n - k) == k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0affa1b0-91a0… - lean_light_the_second_returns_from_the_metre_and_the_period — lean light.lean: the_second_returns_from_the_metre_and_the_period — (List.range 100).all (fun s => travel s / c == s && periods s / dNuCs == s) — decided by the Lean kernel over its whole finite domain, axiom-free ·
04685db2-fda6… - lean_z9_the_orbit_step_is_read_back_by_its_log — lean z9.lean: the_orbit_step_is_read_back_by_its_log — (List.range 6).all (fun k => logOrbit (orbit k) == some k) ∧ units.all (fun u => (logOrbit u).map orbit == some u) — decided by the Lean kernel over its whole finite domain, axiom-free ·
e29d831e-5bde… - lean_energy_splitting_then_burning_is_the_identity_at_every_ledger_value — lean energy.lean: splitting_then_burning_is_the_identity_at_every_ledger_value — ∀ e : Nat, burnE (splitE e) = e — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
2105157a-992b… - lean_light_the_second_returns_from_the_metre_and_the_period_at_every_duration — lean light.lean: the_second_returns_from_the_metre_and_the_period_at_every_duration — ∀ s : Nat, travel s / c = s ∧ periods s / dNuCs = s — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e6d8871f-bee4… - lean_reach_the_predecessor_returns_every_successor — lean reach.lean: the_predecessor_returns_every_successor — ∀ n : Nat, (n + 1) - 1 = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
a1d55ab6-baca… - lean_speed_the_microseconds_return_from_the_nanoseconds_at_every_count — lean speed.lean: the_microseconds_return_from_the_nanoseconds_at_every_count — ∀ u : Nat, u * 1000 / 1000 = u — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
98899238-b934… - lean_flow_the_doubling_flow_repeats_every_six_steps — lean flow.lean: the_doubling_flow_repeats_every_six_steps — ∀ k : Nat, orbit (k + 6) = orbit k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
d5dece46-994f… - lean_flow_the_doubling_flow_is_its_first_six_steps — lean flow.lean: the_doubling_flow_is_its_first_six_steps — ∀ k : Nat, orbit k = orbit (k % 6) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
38c42166-8f6d… - lean_flow_navier_stokes_flow_is_bounded_for_every_step — lean flow.lean: navier_stokes_flow_is_bounded_for_every_step — ∀ k : Nat, orbit k < 9 ∧ span.contains (orbit k) = true — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6dec28f9-502b… - lean_demand_mantels_bound_is_n_squared_over_four_for_every_n — lean demand.lean: mantels_bound_is_n_squared_over_four_for_every_n — ∀ n : Nat, (n / 2) * (n - n / 2) = n * n / 4 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
b9212e56-a3ea… - lean_demand3_the_euler_characteristic_of_a_genus_g_surface_for_every_g — lean demand3.lean: the_euler_characteristic_of_a_genus_g_surface_for_every_g — ∀ g : Nat, chi g + 2 * (g : Int) = 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
a1a899d7-b3c5… - lean_demand3_a_repunit_has_the_residue_of_its_length_mod_three — lean demand3.lean: a_repunit_has_the_residue_of_its_length_mod_three — ∀ n : Nat, repunit n % 3 = n % 3 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
732113b8-a6d4… - lean_demand3_a_repunit_is_divisible_by_three_exactly_when_its_length_is_for_every_n — lean demand3.lean: a_repunit_is_divisible_by_three_exactly_when_its_length_is_for_every_n — ∀ n : Nat, (repunit n % 3 = 0) ↔ (n % 3 = 0) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
93c43bd9-4707… - lean_demand3_repunit_of_a_sum — lean demand3.lean: repunit_of_a_sum — ∀ n k : Nat, repunit (n + k) = 10 ^ k * repunit n + repunit k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
5558e77e-438c… - lean_demand3_repunits_repeat_mod_seven_every_six_digits — lean demand3.lean: repunits_repeat_mod_seven_every_six_digits — ∀ n : Nat, repunit (n + 6) % 7 = repunit n % 7 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
a67a8eec-997b… - lean_demand3_a_repunit_has_the_residue_of_its_length_mod_six_mod_seven — lean demand3.lean: a_repunit_has_the_residue_of_its_length_mod_six_mod_seven — ∀ n : Nat, repunit n % 7 = repunit (n % 6) % 7 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
ec5bf7fe-1dd8… - lean_demand3_a_repunit_is_divisible_by_seven_exactly_when_six_divides_its_length_for_every_n — lean demand3.lean: a_repunit_is_divisible_by_seven_exactly_when_six_divides_its_length_for_every_n — ∀ n : Nat, (repunit n % 7 = 0) ↔ (n % 6 = 0) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
2bb98278-186d… - lean_elementary_the_interior_and_exterior_angles_are_n_straight_angles_for_every_n — lean elementary.lean: the_interior_and_exterior_angles_are_n_straight_angles_for_every_n — ∀ n : Nat, 2 ≤ n → (n - 2) * 180 + 360 = n * 180 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
d1fbd7f1-d1df… - lean_elementary_a_product_of_consecutive_numbers_is_even — lean elementary.lean: a_product_of_consecutive_numbers_is_even — ∀ n : Nat, n * (n + 1) % 2 = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
cf763033-1ed3… - lean_elementary_eight_times_a_triangular_number_plus_one_is_an_odd_square_for_every_n — lean elementary.lean: eight_times_a_triangular_number_plus_one_is_an_odd_square_for_every_n — ∀ n : Nat, 8 * (n * (n + 1) / 2) + 1 = (2 * n + 1) * (2 * n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
1aa59ef6-880c… - lean_energy_two_parts_in_three_is_two_thirds_at_every_scale_for_every_m — lean energy.lean: two_parts_in_three_is_two_thirds_at_every_scale_for_every_m — ∀ m : Nat, 0 < m → pct (2 * m) (3 * m) = 66 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
9177c3d3-8280… - lean_families_the_powers_of_two_sum_to_one_less_than_the_next_for_every_n — lean families.lean: the_powers_of_two_sum_to_one_less_than_the_next_for_every_n — ∀ n : Nat, ((List.range (n + 1)).map (fun i => 2 ^ i)).foldl (· + ·) 0 = 2 ^ (n + 1) - 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
3cab57ca-4d36… - lean_families_square_of_successor — lean families.lean: square_of_successor — ∀ n : Nat, (n + 1) * (n + 1) = n * n + 2 * n + 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
f9826dc5-b85d… - lean_families_the_first_n_odd_numbers_sum_to_n_squared_for_every_n — lean families.lean: the_first_n_odd_numbers_sum_to_n_squared_for_every_n — ∀ n : Nat, ((List.range n).map (fun i => 2 * i + 1)).foldl (· + ·) 0 = n * n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
7baff487-adea… - lean_mechanical_consecutive_squares_differ_by_the_odd_numbers — lean mechanical.lean: consecutive_squares_differ_by_the_odd_numbers — ∀ n : Nat, (n + 1) ^ 2 - n * n = 2 * n + 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
65131201-9df7… - lean_mechanical_every_multiple_of_three_four_five_is_pythagorean — lean mechanical.lean: every_multiple_of_three_four_five_is_pythagorean — ∀ k : Nat, (3 * k) ^ 2 + (4 * k) ^ 2 = (5 * k) ^ 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
a20992d7-56ea… - lean_mechanical_the_product_of_three_consecutive_numbers_is_even — lean mechanical.lean: the_product_of_three_consecutive_numbers_is_even — ∀ n : Nat, n * (n + 1) * (n + 2) % 2 = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
0eb7327f-0789… - lean_mechanical_the_product_of_three_consecutive_numbers_is_a_multiple_of_three — lean mechanical.lean: the_product_of_three_consecutive_numbers_is_a_multiple_of_three — ∀ n : Nat, n * (n + 1) * (n + 2) % 3 = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
f3e2ee80-28d1… - lean_mechanical_the_product_of_any_three_consecutive_integers_is_divisible_by_six_for_every_n — lean mechanical.lean: the_product_of_any_three_consecutive_integers_is_divisible_by_six_for_every_n — ∀ n : Nat, n * (n + 1) * (n + 2) % 6 = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
65518577-2bc3… - lean_reversal_foldl_add_start — lean reversal.lean: foldl_add_start — ∀ (l : List Nat) (a : Nat), l.foldl (· + ·) a = a + l.foldl (· + ·) 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
70a7cff5-e668… - lean_reversal_the_digits_sum_to_the_number_mod_nine_when_the_fuel_covers_them — lean reversal.lean: the_digits_sum_to_the_number_mod_nine_when_the_fuel_covers_them — ∀ f n : Nat, n < 10 ^ f → (digitsF f n).foldl (· + ·) 0 % 9 = n % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
58d4c502-7f73… - lean_reversal_a_number_is_below_ten_to_its_own_successor — lean reversal.lean: a_number_is_below_ten_to_its_own_successor — ∀ n : Nat, n < 10 ^ (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
dc7330c0-7a4c… - lean_reversal_the_digit_sum_has_the_residue_of_the_number_mod_nine_for_every_n — lean reversal.lean: the_digit_sum_has_the_residue_of_the_number_mod_nine_for_every_n — ∀ n : Nat, digitSum n % 9 = n % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
1a8922c4-c55c… - lean_reversal_folding_by_ten_keeps_the_residue_of_the_digit_sum — lean reversal.lean: folding_by_ten_keeps_the_residue_of_the_digit_sum — ∀ (l : List Nat) (a : Nat), (l.foldl (fun a d => a * 10 + d) a) % 9 = (a + l.foldl (· + ·) 0) % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c7ce0223-8884… - lean_reversal_reversal_keeps_the_residue_mod_nine_for_every_n — lean reversal.lean: reversal_keeps_the_residue_mod_nine_for_every_n — ∀ n : Nat, reverseNum n % 9 = n % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
3159b043-f18c… - lean_split_the_coin_step_and_the_coins_are_inverse_for_every_c — lean split.lean: the_coin_step_and_the_coins_are_inverse_for_every_c — ∀ c : Nat, 3 * c / 3 = c ∧ (3 * c) % 3 = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
1c1ebe41-4e80… - lean_mechanical_repeated_doubling_is_the_power_of_two_for_every_k — lean mechanical.lean: repeated_doubling_is_the_power_of_two_for_every_k — ∀ k : Nat, dbl k = 2 ^ k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
71d6eeaa-fd06… - lean_mechanical_relation_digitroot_is_residue_mod9_for_every_n — lean mechanical.lean: relation_digitroot_is_residue_mod9_for_every_n — ∀ n : Nat, 0 < n → DR n = (n - 1) % 9 + 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
22aed33c-e7ff… - lean_mechanical_the_nine_times_table_always_digital_roots_to_nine_for_every_k — lean mechanical.lean: the_nine_times_table_always_digital_roots_to_nine_for_every_k — ∀ k : Nat, 0 < k → DR (9 * k) = 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
36b3b4ea-25f1… - lean_mechanical_the_digital_root_has_the_residue_of_its_number — lean mechanical.lean: the_digital_root_has_the_residue_of_its_number — ∀ n : Nat, DR n % 9 = n % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
efc4383f-1f52… - lean_mechanical_a_digital_root_lies_between_one_and_nine — lean mechanical.lean: a_digital_root_lies_between_one_and_nine — ∀ n : Nat, 0 < n → 1 ≤ DR n ∧ DR n ≤ 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
298da32e-60d7… - lean_mechanical_casting_out_nines_is_multiplicative_for_every_a_b — lean mechanical.lean: casting_out_nines_is_multiplicative_for_every_a_b — ∀ a b : Nat, DR (a * b) = DR (DR a * DR b) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
58842a8d-45b2… - lean_mechanical_the_moduli_dimensions_are_three_g_minus_three_and_six_g_minus_six_for_every_g — lean mechanical.lean: the_moduli_dimensions_are_three_g_minus_three_and_six_g_minus_six_for_every_g — ∀ g : Nat, 3 * g - 3 = (6 * g - 6) / 2 ∧ 6 * g - 6 = 2 * (3 * g - 3) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
633ba212-336a… - lean_reach_exhaustion_never_reaches_its_own_bound_for_every_n — lean reach.lean: exhaustion_never_reaches_its_own_bound_for_every_n — ∀ n : Nat, (List.range n).length = n ∧ n ∉ List.range n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
dfc2849f-3870… - lean_reach_the_successor_of_every_bound_lies_outside_for_every_n — lean reach.lean: the_successor_of_every_bound_lies_outside_for_every_n — ∀ n : Nat, n + 1 ∉ List.range n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
1113761c-4967… - lean_reach_doubling_the_domain_leaves_the_same_hole_for_every_n — lean reach.lean: doubling_the_domain_leaves_the_same_hole_for_every_n — ∀ n : Nat, 2 * n ∉ List.range (2 * n) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
f57338c8-104b… - lean_z9plus_doubling_has_period_six_for_every_k — lean z9plus.lean: doubling_has_period_six_for_every_k — ∀ k : Nat, pw 2 (k + 6) = pw 2 k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
2c9ca8bc-08a6… - lean_z9plus_orbit_digital_roots_have_period_six_for_every_k — lean z9plus.lean: orbit_digital_roots_have_period_six_for_every_k — ∀ k : Nat, dr (pw 2 (k + 6)) = dr (pw 2 k) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e88dfdf8-3c33… - lean_z9plus_the_doubling_orbit_is_read_from_the_exponent_mod_six — lean z9plus.lean: the_doubling_orbit_is_read_from_the_exponent_mod_six — ∀ k : Nat, pw 2 k = pw 2 (k % 6) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c688c966-b59f… - lean_z9plus_the_orbit_never_meets_the_triad_for_every_k — lean z9plus.lean: the_orbit_never_meets_the_triad_for_every_k — ∀ k : Nat, pw 2 k ≠ 0 ∧ pw 2 k ≠ 3 ∧ pw 2 k ≠ 6 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
8c3dfc38-91e6… - lean_z9plus_fibonacci_residues_mod_nine_repeat_every_twenty_four_steps — lean z9plus.lean: fibonacci_residues_mod_nine_repeat_every_twenty_four_steps — ∀ k : Nat, fib (k + 24) % 9 = fib k % 9 ∧ fib (k + 25) % 9 = fib (k + 1) % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
04f6a9cd-b19f… - lean_z9plus_pisano_period_mod_nine_is_twenty_four_for_every_k — lean z9plus.lean: pisano_period_mod_nine_is_twenty_four_for_every_k — ∀ k : Nat, fib9 (k + 24) = fib9 k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
985c1ebe-dc19… - lean_z9plus_the_fuelled_gcd_is_the_gcd_when_the_fuel_exceeds_the_second_argument — lean z9plus.lean: the_fuelled_gcd_is_the_gcd_when_the_fuel_exceeds_the_second_argument — ∀ f a b : Nat, b < f → gcdF f a b = Nat.gcd a b — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
73c32b16-76ca… - lean_z9plus_consecutive_fibonacci_numbers_are_coprime — lean z9plus.lean: consecutive_fibonacci_numbers_are_coprime — ∀ n : Nat, Nat.gcd (fib n) (fib (n + 1)) = 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
5ae6ebcf-4c89… - lean_z9plus_consecutive_fibonacci_are_coprime_for_every_n — lean z9plus.lean: consecutive_fibonacci_are_coprime_for_every_n — ∀ n : Nat, gcd9 (fib n) (fib (n + 1)) = 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
424d4751-1648… - lean_z9plus_digital_root_agrees_with_the_residue_for_every_n — lean z9plus.lean: digital_root_agrees_with_the_residue_for_every_n — ∀ n : Nat, 0 < n → (dr n = 9 ↔ m9 n = 0) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
20da0230-2b9f… - lean_z9plus_digital_root_is_the_digit_sum_residue_for_every_n — lean z9plus.lean: digital_root_is_the_digit_sum_residue_for_every_n — ∀ n : Nat, m9 (digitSum n) = m9 n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
694fefdf-a71b… - lean_demand2_the_two_power_sums_are_one_sum — lean demand2.lean: the_two_power_sums_are_one_sum — ∀ p n : Nat, sumPow p n = powSum p n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6551630b-4917… - lean_demand2_nicomachus_sum_of_cubes_is_the_square_of_the_triangular_number_for_every_n — lean demand2.lean: nicomachus_sum_of_cubes_is_the_square_of_the_triangular_number_for_every_n — ∀ n : Nat, sumPow 3 n = tri n * tri n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
7aa0bbe0-b4ae… - lean_demand2_the_sum_of_fifth_powers_has_the_closed_form_for_every_n — lean demand2.lean: the_sum_of_fifth_powers_has_the_closed_form_for_every_n — ∀ n : Nat, 12 * sumPow 5 n = n * n * (n + 1) * (n + 1) * (2 * n * n + 2 * n - 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6916652f-bda2… - lean_elementary_the_lucas_numbers_are_the_sum_of_the_neighbouring_fibonaccis_for_every_n — lean elementary.lean: the_lucas_numbers_are_the_sum_of_the_neighbouring_fibonaccis_for_every_n — ∀ n : Nat, lucas (n + 1) = fib n + fib (n + 2) — proved by the Lean kernel for every value (by have), standard axioms only — checked by lean.ts ·
cacb33f1-2fc1… - lean_families_cassinis_identity_for_every_m — lean families.lean: cassinis_identity_for_every_m — ∀ m : Nat, fib (2 * m + 1) * fib (2 * m + 3) = fib (2 * m + 2) * fib (2 * m + 2) + 1 ∧ fib (2 * m) * fib (2 * m + 2) + 1 = fib (2 * m + 1) * fib (2 * m + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
75a810f1-ae27… - lean_families_a_sum_over_one_to_n_plus_one_adds_the_last_term — lean families.lean: a_sum_over_one_to_n_plus_one_adds_the_last_term — ∀ (g : Nat → Nat) (n : Nat), ((List.range' 1 (n + 1)).map g).foldl (· + ·) 0 = ((List.range' 1 n).map g).foldl (· + ·) 0 + g (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
f664ca1d-8371… - lean_families_the_numbers_sum_to_their_closed_form_for_every_n — lean families.lean: the_numbers_sum_to_their_closed_form_for_every_n — ∀ n : Nat, 2 * powSum 1 n = n * (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
34fa9fc8-b8ed… - lean_families_the_squares_sum_to_their_closed_form_for_every_n — lean families.lean: the_squares_sum_to_their_closed_form_for_every_n — ∀ n : Nat, 6 * powSum 2 n = n * (n + 1) * (2 * n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
0b7a0569-fb3f… - lean_families_the_cubes_sum_to_their_closed_form_for_every_n — lean families.lean: the_cubes_sum_to_their_closed_form_for_every_n — ∀ n : Nat, 4 * powSum 3 n = n * n * (n + 1) * (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
96b0047e-cf09… - lean_families_the_fourth_powers_sum_to_their_closed_form_for_every_n — lean families.lean: the_fourth_powers_sum_to_their_closed_form_for_every_n — ∀ n : Nat, 30 * powSum 4 n = n * (n + 1) * (2 * n + 1) * (3 * n * n + 3 * n - 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
3419aabd-970f… - lean_families_the_fifth_powers_sum_to_their_closed_form_for_every_n — lean families.lean: the_fifth_powers_sum_to_their_closed_form_for_every_n — ∀ n : Nat, 12 * powSum 5 n = n * n * (n + 1) * (n + 1) * (2 * n * n + 2 * n - 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
790310ea-f06d… - lean_families_power_sums_match_their_closed_forms_for_every_n — lean families.lean: power_sums_match_their_closed_forms_for_every_n — ∀ k n : Nat, 1 ≤ k → k ≤ 5 → powSum k n = faulhaber k n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
04b04822-598b… - lean_families_the_sums_of_triangular_numbers_are_the_tetrahedral_numbers_for_every_n — lean families.lean: the_sums_of_triangular_numbers_are_the_tetrahedral_numbers_for_every_n — ∀ n : Nat, ((List.range' 1 n).map (fun i => i * (i + 1) / 2)).foldl (· + ·) 0 = n * (n + 1) * (n + 2) / 6 — proved by the Lean kernel for every value (by have), standard axioms only — checked by lean.ts ·
43b9cdd8-7110… - lean_sequences_cassinis_identity_for_every_m — lean sequences.lean: cassinis_identity_for_every_m — ∀ m : Nat, fib (2 * m + 1) * fib (2 * m + 3) = fib (2 * m + 2) * fib (2 * m + 2) + 1 ∧ fib (2 * m) * fib (2 * m + 2) + 1 = fib (2 * m + 1) * fib (2 * m + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
3b815427-42b8… - lean_nim_the_xor_of_two_bits_is_their_sum_mod_two — lean nim.lean: the_xor_of_two_bits_is_their_sum_mod_two — ∀ x y : Nat, (if x % 2 == y % 2 then 0 else 1) = (x + y) % 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
322304d5-f000… - lean_nim_xor_commutes_at_every_fuel — lean nim.lean: xor_commutes_at_every_fuel — ∀ f a b : Nat, xorF f a b = xorF f b a — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
b091d5a8-4e4a… - lean_nim_one_more_step_of_fuel_changes_nothing_once_it_exceeds_an_argument — lean nim.lean: one_more_step_of_fuel_changes_nothing_once_it_exceeds_an_argument — ∀ f a b : Nat, (a < f ∨ b < f) → xorF (f + 1) a b = xorF f a b — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
adf2ef5a-ef44… - lean_nim_more_fuel_changes_nothing_once_it_exceeds_an_argument — lean nim.lean: more_fuel_changes_nothing_once_it_exceeds_an_argument — ∀ k f a b : Nat, (a < f ∨ b < f) → xorF (f + k) a b = xorF f a b — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
961d94be-959c… - lean_nim_xor_reads_the_same_at_every_sufficient_fuel — lean nim.lean: xor_reads_the_same_at_every_sufficient_fuel — ∀ g a b : Nat, (a < g ∨ b < g) → xorN a b = xorF g a b — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e04a45b5-8834… - lean_nim_a_number_xored_with_itself_is_zero_at_every_fuel — lean nim.lean: a_number_xored_with_itself_is_zero_at_every_fuel — ∀ f b : Nat, xorF f b b = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
409e78df-b565… - lean_nim_xor_splits_off_its_lowest_bit — lean nim.lean: xor_splits_off_its_lowest_bit — ∀ g x c : Nat, 1 ≤ g → xorF (g + 1) x c = (x + c) % 2 + 2 * xorF g (x / 2) (c / 2) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
fa8f797f-95fa… - lean_nim_xor_undoes_itself_when_the_fuel_suffices — lean nim.lean: xor_undoes_itself_when_the_fuel_suffices — ∀ f a b g : Nat, (a < f ∨ b < f) → b < g → xorF g (xorF f a b) b = a — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
ba799e13-9cb6… - lean_nim_xor_is_commutative_for_every_a_b — lean nim.lean: xor_is_commutative_for_every_a_b — ∀ a b : Nat, xorN a b = xorN b a — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e62bc467-f90b… - lean_nim_xor_zero_is_identity_for_every_a — lean nim.lean: xor_zero_is_identity_for_every_a — ∀ a : Nat, xorN a 0 = a — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
667bd69d-51ea… - lean_nim_xor_is_its_own_inverse_for_every_a_b — lean nim.lean: xor_is_its_own_inverse_for_every_a_b — ∀ a b : Nat, xorN (xorN a b) b = a — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e2f6d0f1-2fa2… - lean_nim_xor_is_the_sum_of_the_bits_plus_twice_the_xor_of_the_halves — lean nim.lean: xor_is_the_sum_of_the_bits_plus_twice_the_xor_of_the_halves — ∀ a b : Nat, xorN a b = (a + b) % 2 + 2 * xorN (a / 2) (b / 2) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
07c467d9-8226… - lean_nim_xor_is_associative_below_every_bound — lean nim.lean: xor_is_associative_below_every_bound — ∀ n a b c : Nat, a + b + c ≤ n → xorN (xorN a b) c = xorN a (xorN b c) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
0c3ef805-2811… - lean_nim_xor_is_associative_for_every_a_b_c — lean nim.lean: xor_is_associative_for_every_a_b_c — ∀ a b c : Nat, xorN (xorN a b) c = xorN a (xorN b c) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
d2a922d9-f09d… - lean_nim_the_nim_sum_is_a_gf2_vector_addition_for_every_a_b_c — lean nim.lean: the_nim_sum_is_a_gf2_vector_addition_for_every_a_b_c — ∀ a b c : Nat, xorN a b = xorN b a ∧ xorN (xorN a b) c = xorN a (xorN b c) ∧ xorN (xorN a b) b = a ∧ xorN a 0 = a — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
caa5c327-62cf… - lean_sequences_a_bit_sum_over_one_more_position_adds_that_bit — lean sequences.lean: a_bit_sum_over_one_more_position_adds_that_bit — ∀ F m : Nat, (List.range (F + 1)).foldl (fun a i => a + m / 2 ^ i % 2) 0 = (List.range F).foldl (fun a i => a + m / 2 ^ i % 2) 0 + m / 2 ^ F % 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
f5e71d05-068c… - lean_sequences_halving_shifts_every_bit_down_by_one — lean sequences.lean: halving_shifts_every_bit_down_by_one — ∀ m i : Nat, m / 2 ^ (i + 1) = m / 2 / 2 ^ i — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
a5201a6e-8f86… - lean_sequences_the_bits_of_a_number_are_its_low_bit_then_the_bits_of_its_half — lean sequences.lean: the_bits_of_a_number_are_its_low_bit_then_the_bits_of_its_half — ∀ F m : Nat, (List.range (F + 1)).foldl (fun a i => a + m / 2 ^ i % 2) 0 = m % 2 + (List.range F).foldl (fun a i => a + m / 2 / 2 ^ i % 2) 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
2c2dcdbf-8b9f… - lean_sequences_bits_above_the_number_add_nothing — lean sequences.lean: bits_above_the_number_add_nothing — ∀ k F n : Nat, n < 2 ^ F → (List.range (F + k)).foldl (fun a i => a + n / 2 ^ i % 2) 0 = (List.range F).foldl (fun a i => a + n / 2 ^ i % 2) 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
77439845-748b… - lean_sequences_a_number_is_below_two_to_its_own_power — lean sequences.lean: a_number_is_below_two_to_its_own_power — ∀ n : Nat, n < 2 ^ n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
4f5a709a-5de5… - lean_sequences_a_number_is_below_two_to_its_own_successor — lean sequences.lean: a_number_is_below_two_to_its_own_successor — ∀ n : Nat, n < 2 ^ (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
586b3088-ac85… - lean_sequences_doubling_preserves_the_bit_count_for_every_n — lean sequences.lean: doubling_preserves_the_bit_count_for_every_n — ∀ n : Nat, popcount (2 * n) = popcount n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
7006c6f5-3c8d… - lean_sequences_the_odd_step_sets_exactly_one_further_bit_for_every_n — lean sequences.lean: the_odd_step_sets_exactly_one_further_bit_for_every_n — ∀ n : Nat, popcount (2 * n + 1) = popcount n + 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
d9da9f38-b47d… - lean_sequences_thue_morse_doubling_recurrence_for_every_n — lean sequences.lean: thue_morse_doubling_recurrence_for_every_n — ∀ n : Nat, tm (2 * n) = tm n ∧ tm (2 * n + 1) = 1 - tm n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
50c8ad36-d08a… - lean_address_the_version_stamp_keeps_the_low_nibble_of_every_byte — lean address.lean: the_version_stamp_keeps_the_low_nibble_of_every_byte — ∀ b : Nat, b < 256 → or8 (and8 b 15) 128 = 128 + b % 16 — decided by the Lean kernel over its whole finite domain, axiom-free ·
913d0338-59bc… - lean_address_the_variant_stamp_keeps_the_low_six_bits_of_every_byte — lean address.lean: the_variant_stamp_keeps_the_low_six_bits_of_every_byte — ∀ b : Nat, b < 256 → or8 (and8 b 63) 128 = 128 + b % 64 — decided by the Lean kernel over its whole finite domain, axiom-free ·
fbc0f2d4-3490… - lean_address_stamping_twice_is_stamping_once — lean address.lean: stamping_twice_is_stamping_once — ∀ b : Nat, b < 256 → or8 (and8 (or8 (and8 b 15) 128) 15) 128 = or8 (and8 b 15) 128 ∧ or8 (and8 (or8 (and8 b 63) 128) 63) 128 = or8 (and8 b 63) 128 — decided by the Lean kernel over its whole finite domain, axiom-free ·
c397ec59-2a56… - lean_demand3_the_two_coin_fare_is_minus_the_euler_characteristic_at_genus_two — lean demand3.lean: the_two_coin_fare_is_minus_the_euler_characteristic_at_genus_two — (110 : Int) - 108 = - chi 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
665800be-8ca8… - lean_address_the_third_group_is_version_eight_over_twelve_hash_bits — lean address.lean: the_third_group_is_version_eight_over_twelve_hash_bits — ∀ b6 b7 : Nat, b6 < 256 → b7 < 256 → (or8 (and8 b6 15) 128) * 256 + b7 = 8 * 4096 + (b6 % 16 * 256 + b7) ∧ b6 % 16 * 256 + b7 < 4096 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
136a86a7-b50b… - lean_address_the_fourth_group_is_variant_two_over_fourteen_hash_bits — lean address.lean: the_fourth_group_is_variant_two_over_fourteen_hash_bits — ∀ b8 b9 : Nat, b8 < 256 → b9 < 256 → (or8 (and8 b8 63) 128) * 256 + b9 = 2 * 16384 + (b8 % 64 * 256 + b9) ∧ b8 % 64 * 256 + b9 < 16384 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
087841c4-271d… - lean_fnv_the_split_multiply_is_the_exact_thirty_two_bit_product — lean fnv.lean: the_split_multiply_is_the_exact_thirty_two_bit_product — ∀ a b : Nat, (a % 65536 * b + a / 65536 * b % 65536 * 65536) % M32 = mul32 a b — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
71d78d5c-0f87… - lean_energy_the_boundary_is_sharp — lean energy.lean: the_boundary_is_sharp — mass 5333 0 = mass 0 336 — decided by the Lean kernel over its whole finite domain, axiom-free ·
6237820f-0a7b… - lean_address_the_unstamped_bits_read_back_from_every_stamped_byte — lean address.lean: the_unstamped_bits_read_back_from_every_stamped_byte — ∀ b : Nat, b < 256 → and8 (or8 (and8 b 15) 128) 15 = b % 16 ∧ and8 (or8 (and8 b 63) 128) 63 = b % 64 — proved by the Lean kernel for every value (by decide), standard axioms only — checked by lean.ts ·
69a4f3a2-4653… - lean_address_the_twelve_hash_bits_read_back_from_the_third_group — lean address.lean: the_twelve_hash_bits_read_back_from_the_third_group — ∀ b6 b7 : Nat, b6 < 256 → b7 < 256 → ((or8 (and8 b6 15) 128) * 256 + b7) / 4096 = 8 ∧ ((or8 (and8 b6 15) 128) * 256 + b7) % 4096 = b6 % 16 * 256 + b7 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
221f974c-1535… - lean_address_the_fourteen_hash_bits_read_back_from_the_fourth_group — lean address.lean: the_fourteen_hash_bits_read_back_from_the_fourth_group — ∀ b8 b9 : Nat, b8 < 256 → b9 < 256 → ((or8 (and8 b8 63) 128) * 256 + b9) / 16384 = 2 ∧ ((or8 (and8 b8 63) 128) * 256 + b9) % 16384 = b8 % 64 * 256 + b9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
edc4013a-9ea0… - lean_demand_four_times_a_near_product_is_at_most_the_square — lean demand.lean: four_times_a_near_product_is_at_most_the_square — ∀ a d : Nat, a * (a + d) * 4 ≤ (a + (a + d)) * (a + (a + d)) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
61fde2f2-4a84… - lean_demand_no_cut_beats_the_balanced_one_for_every_n — lean demand.lean: no_cut_beats_the_balanced_one_for_every_n — ∀ n a : Nat, a ≤ n → a * (n - a) ≤ n * n / 4 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
f87585db-e4be… - lean_demand2_the_squared_triangular_numbers_difference_back_to_the_cubes_for_every_n — lean demand2.lean: the_squared_triangular_numbers_difference_back_to_the_cubes_for_every_n — ∀ n : Nat, tri (n + 1) * tri (n + 1) = tri n * tri n + (n + 1) ^ 3 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
76f6ba26-b6dd… - lean_demand3_the_genus_reads_back_from_chi_for_every_g — lean demand3.lean: the_genus_reads_back_from_chi_for_every_g — ∀ g : Nat, ((2 - chi g) / 2).toNat = g — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
b2627057-efa5… - lean_demand3_chi_is_injective_for_every_g_h — lean demand3.lean: chi_is_injective_for_every_g_h — ∀ g h : Nat, chi g = chi h → g = h — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
9259ab3c-4bbc… - lean_demand3_a_repunit_is_below_ten_to_its_length — lean demand3.lean: a_repunit_is_below_ten_to_its_length — ∀ k : Nat, repunit k < 10 ^ k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
63e37ea7-0f19… - lean_demand3_a_repunit_splits_back_into_its_two_parts_for_every_n_k — lean demand3.lean: a_repunit_splits_back_into_its_two_parts_for_every_n_k — ∀ n k : Nat, repunit (n + k) / 10 ^ k = repunit n ∧ repunit (n + k) % 10 ^ k = repunit k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
56ed7d0a-28d5… - lean_elementary_the_polygon_side_count_reads_back_from_its_angle_sum — lean elementary.lean: the_polygon_side_count_reads_back_from_its_angle_sum — ∀ n : Nat, 2 ≤ n → ((n - 2) * 180 + 360) / 180 = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
0ad62680-b148… - lean_elementary_every_odd_square_one_more_than_eight_times_m_makes_m_triangular — lean elementary.lean: every_odd_square_one_more_than_eight_times_m_makes_m_triangular — ∀ m r : Nat, 8 * m + 1 = r * r → m = r / 2 * (r / 2 + 1) / 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
958b6261-542d… - lean_elementary_the_fibonaccis_are_a_fifth_of_the_neighbouring_lucas_numbers_for_every_n — lean elementary.lean: the_fibonaccis_are_a_fifth_of_the_neighbouring_lucas_numbers_for_every_n — ∀ n : Nat, 5 * fib (n + 1) = lucas n + lucas (n + 2) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
16091996-5977… - lean_elementary_half_a_product_of_consecutive_numbers_multiplies_back_for_every_n — lean elementary.lean: half_a_product_of_consecutive_numbers_multiplies_back_for_every_n — ∀ n : Nat, n * (n + 1) / 2 * 2 = n * (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
cf74f953-f0e2… - lean_energy_burning_then_splitting_is_the_identity_above_the_bond — lean energy.lean: burning_then_splitting_is_the_identity_above_the_bond — ∀ e : Nat, bondHundredthsKJ ≤ e → splitE (burnE e) = e — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
fe87a66a-6ac9… - lean_energy_one_part_in_three_is_a_third_at_every_scale_for_every_m — lean energy.lean: one_part_in_three_is_a_third_at_every_scale_for_every_m — ∀ m : Nat, 0 < m → pct m (3 * m) = 33 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
b2c52079-42f3… - lean_energy_the_gas_volume_reads_back_its_moles_for_every_m — lean energy.lean: the_gas_volume_reads_back_its_moles_for_every_m — ∀ m : Nat, molesOf (litresOf m) ≤ m ∧ m ≤ molesOf (litresOf m) + 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
21c3d846-9c3e… - lean_energy_the_mass_names_the_molecule_below_5333_hydrogens — lean energy.lean: the_mass_names_the_molecule_below_5333_hydrogens — ∀ h o h' o' : Nat, h < 5333 → h' < 5333 → mass h o = mass h' o' → h = h' ∧ o = o' — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
a30f7f70-4bfa… - lean_energy_a_floor_division_multiplies_back_within_one_divisor — lean energy.lean: a_floor_division_multiplies_back_within_one_divisor — ∀ a l : Nat, 0 < l → a / l * l ≤ a ∧ a < (a / l + 1) * l — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c6871639-adb9… - lean_families_choose_is_invariant_under_the_involution_k_to_n_minus_k_for_every_n — lean families.lean: choose_is_invariant_under_the_involution_k_to_n_minus_k_for_every_n — ∀ n k : Nat, k ≤ n → choose n k = choose n (n - k) ∧ n - (n - k) = k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6133e3a1-254e… - lean_families_the_closed_forms_difference_back_to_the_next_power_for_every_n — lean families.lean: the_closed_forms_difference_back_to_the_next_power_for_every_n — ∀ k n : Nat, 1 ≤ k → k ≤ 5 → faulhaber k (n + 1) = faulhaber k n + (n + 1) ^ k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
b1d34ef3-9b6d… - lean_families_the_tetrahedral_numbers_difference_back_to_the_triangular_numbers_for_every_n — lean families.lean: the_tetrahedral_numbers_difference_back_to_the_triangular_numbers_for_every_n — ∀ n : Nat, (n + 1) * (n + 1 + 1) * (n + 1 + 2) / 6 = n * (n + 1) * (n + 2) / 6 + (n + 1) * (n + 1 + 1) / 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e811a0a1-22b9… - lean_families_the_powers_of_two_difference_back_to_the_next_power_for_every_n — lean families.lean: the_powers_of_two_difference_back_to_the_next_power_for_every_n — ∀ n : Nat, (2 ^ (n + 2) - 1) - (2 ^ (n + 1) - 1) = 2 ^ (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
9c202e7c-40cb… - lean_families_the_odd_numbers_difference_back_to_the_next_odd_number_for_every_n — lean families.lean: the_odd_numbers_difference_back_to_the_next_odd_number_for_every_n — ∀ n : Nat, ((List.range (n + 1)).map (fun i => 2 * i + 1)).foldl (· + ·) 0 - ((List.range n).map (fun i => 2 * i + 1)).foldl (· + ·) 0 = 2 * n + 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
404c2b65-4bb4… - lean_families_foldl_add_start — lean families.lean: foldl_add_start — ∀ (l : List Nat) (a : Nat), l.foldl (· + ·) a = a + l.foldl (· + ·) 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e29e2887-b564… - lean_families_the_digits_sum_to_the_number_mod_nine_when_the_fuel_covers_them — lean families.lean: the_digits_sum_to_the_number_mod_nine_when_the_fuel_covers_them — ∀ f n : Nat, n < 10 ^ f → (digitsF f n).foldl (· + ·) 0 % 9 = n % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
bdb7dd00-a595… - lean_families_a_number_is_below_ten_to_its_own_successor — lean families.lean: a_number_is_below_ten_to_its_own_successor — ∀ n : Nat, n < 10 ^ (n + 1) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6021fd10-1371… - lean_families_the_digit_sum_rules_for_three_and_nine_hold_for_every_n — lean families.lean: the_digit_sum_rules_for_three_and_nine_hold_for_every_n — ∀ n : Nat, (digitSum n % 3 = 0 ↔ n % 3 = 0) ∧ (digitSum n % 9 = 0 ↔ n % 9 = 0) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e4b7bd43-9725… - lean_flow_the_reverse_flow_undoes_every_step — lean flow.lean: the_reverse_flow_undoes_every_step — ∀ k : Nat, orbit (k + 1) * 5 % 9 = orbit k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
8a31d5e4-e83d… - lean_flow_every_state_of_the_flow_returns_within_six_steps — lean flow.lean: every_state_of_the_flow_returns_within_six_steps — ∀ k s : Nat, ∃ j, j < 6 ∧ orbit (k + j) = orbit s — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
fd1cecb4-13df… - lean_fnv_multiplying_by_the_fnv_prime_is_undone_by_its_inverse_for_every_h — lean fnv.lean: multiplying_by_the_fnv_prime_is_undone_by_its_inverse_for_every_h — ∀ h : Nat, mul32 (mul32 h FNV_PRIME) 899433627 = h % M32 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
30b021d3-5d2e… - lean_light_the_metre_returns_from_the_second_for_every_whole_length — lean light.lean: the_metre_returns_from_the_second_for_every_whole_length — ∀ m : Nat, m % c = 0 → travel (m / c) = m — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
01829703-d2a6… - lean_light_the_period_count_returns_from_the_second_for_every_whole_count — lean light.lean: the_period_count_returns_from_the_second_for_every_whole_count — ∀ p : Nat, p % dNuCs = 0 → periods (p / dNuCs) = p — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
917672f5-d4d0… - lean_mechanical_DR_eq — lean mechanical.lean: DR_eq — ∀ n : Nat, DR n = if n = 0 then 0 else 1 + (n - 1) % 9 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
dec26379-71e5… - lean_mechanical_halving_undoes_every_doubling_step_for_every_k — lean mechanical.lean: halving_undoes_every_doubling_step_for_every_k — ∀ k : Nat, dbl (k + 1) / 2 = dbl k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
7ac609a7-0ed5… - lean_mechanical_casting_out_nines_is_additive_for_every_a_b — lean mechanical.lean: casting_out_nines_is_additive_for_every_a_b — ∀ a b : Nat, DR (a + b) = DR (DR a + DR b) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
f63b1a76-759a… - lean_mechanical_the_digital_root_fixes_exactly_one_to_nine — lean mechanical.lean: the_digital_root_fixes_exactly_one_to_nine — ∀ d : Nat, 1 ≤ d → d ≤ 9 → DR d = d — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
7f66abca-14de… - lean_mechanical_the_digital_root_is_idempotent_for_every_n — lean mechanical.lean: the_digital_root_is_idempotent_for_every_n — ∀ n : Nat, DR (DR n) = DR n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
cc3d43b3-c3ea… - lean_mechanical_a_digital_root_of_nine_means_a_multiple_of_nine_for_every_n — lean mechanical.lean: a_digital_root_of_nine_means_a_multiple_of_nine_for_every_n — ∀ n : Nat, 0 < n → DR n = 9 → n % 9 = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
48527c27-d129… - lean_mechanical_the_hypotenuse_reads_back_from_every_three_four_five_multiple — lean mechanical.lean: the_hypotenuse_reads_back_from_every_three_four_five_multiple — ∀ k c : Nat, (3 * k) ^ 2 + (4 * k) ^ 2 = c ^ 2 → c = 5 * k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
38ab856f-2051… - lean_mechanical_the_sixth_of_three_consecutive_integers_multiplies_back_for_every_n — lean mechanical.lean: the_sixth_of_three_consecutive_integers_multiplies_back_for_every_n — ∀ n : Nat, n * (n + 1) * (n + 2) / 6 * 6 = n * (n + 1) * (n + 2) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
23e7c1bf-1835… - lean_nim_a_number_xored_with_itself_is_zero_for_every_a — lean nim.lean: a_number_xored_with_itself_is_zero_for_every_a — ∀ a : Nat, xorN a a = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
4eb0eb7c-9115… - lean_nim_xor_is_zero_only_on_equal_arguments_for_every_a_b — lean nim.lean: xor_is_zero_only_on_equal_arguments_for_every_a_b — ∀ a b : Nat, xorN a b = 0 ↔ a = b — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c117a38f-0ab9… - lean_nim_the_halves_and_the_low_bit_read_back_from_the_xor_for_every_a_b — lean nim.lean: the_halves_and_the_low_bit_read_back_from_the_xor_for_every_a_b — ∀ a b : Nat, xorN a b / 2 = xorN (a / 2) (b / 2) ∧ xorN a b % 2 = (a + b) % 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c526ac9a-e469… - lean_reach_the_successor_returns_every_positive_predecessor — lean reach.lean: the_successor_returns_every_positive_predecessor — ∀ n : Nat, 0 < n → (n - 1) + 1 = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
23e0a041-8135… - lean_reach_exhaustion_reaches_everything_below_its_bound_for_every_n — lean reach.lean: exhaustion_reaches_everything_below_its_bound_for_every_n — ∀ n k : Nat, k < n → k ∈ List.range n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
ca669c32-5df9… - lean_reach_the_predecessor_of_every_positive_bound_lies_inside — lean reach.lean: the_predecessor_of_every_positive_bound_lies_inside — ∀ n : Nat, 0 < n → n - 1 ∈ List.range n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
9f5d69b8-d238… - lean_reach_halving_the_domain_keeps_the_half_inside_for_every_n — lean reach.lean: halving_the_domain_keeps_the_half_inside_for_every_n — ∀ n : Nat, 0 < n → n ∈ List.range (2 * n) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
be05a537-ee2d… - lean_reversal_digitsF_zero — lean reversal.lean: digitsF_zero — ∀ f : Nat, digitsF f 0 = [] — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e37e7ee7-d6b6… - lean_reversal_every_digit_is_below_ten — lean reversal.lean: every_digit_is_below_ten — ∀ f n : Nat, ∀ d ∈ digitsF f n, d < 10 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
45e2714e-6f2d… - lean_reversal_the_digits_rebuild_the_number_when_the_fuel_covers_them — lean reversal.lean: the_digits_rebuild_the_number_when_the_fuel_covers_them — ∀ f n : Nat, n < 10 ^ f → (digitsF f n).foldr (fun d a => a * 10 + d) 0 = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
06a08633-fb89… - lean_reversal_the_number_is_rebuilt_from_its_digits_for_every_n — lean reversal.lean: the_number_is_rebuilt_from_its_digits_for_every_n — ∀ n : Nat, (digits n).foldr (fun d a => a * 10 + d) 0 = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
cbb228e9-06c9… - lean_reversal_a_sum_is_the_same_read_backwards — lean reversal.lean: a_sum_is_the_same_read_backwards — ∀ (l : List Nat), l.reverse.foldl (· + ·) 0 = l.foldl (· + ·) 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
1b350dab-b05b… - lean_reversal_the_top_digit_is_nonzero — lean reversal.lean: the_top_digit_is_nonzero — ∀ f n : Nat, 0 < n → n < 10 ^ f → ∃ L b, digitsF f n = L ++ [b] ∧ 0 < b ∧ b < 10 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
da813af5-c2e3… - lean_reversal_reversal_leaves_no_trailing_zero — lean reversal.lean: reversal_leaves_no_trailing_zero — ∀ n : Nat, 0 < n → 0 < reverseNum n ∧ reverseNum n % 10 ≠ 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
edae203a-46d3… - lean_reversal_reading_a_list_back_from_the_top — lean reversal.lean: reading_a_list_back_from_the_top — ∀ r : List Nat, r ≠ [] → (∀ d ∈ r, d < 10) → r.reverse.head? ≠ some 0 → 0 < r.reverse.foldl (fun a d => a * 10 + d) 0 ∧ r.length ≤ r.reverse.foldl (fun a d => a * 10 + d) 0 ∧ ∀ f, r.length ≤ f → digitsF f (r.reverse.foldl (fun a d => a * 10… — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
ac35a0e6-3280… - lean_reversal_the_digits_of_the_reversal_are_the_reversed_digits — lean reversal.lean: the_digits_of_the_reversal_are_the_reversed_digits — ∀ n : Nat, n % 10 ≠ 0 → digits (reverseNum n) = (digitsF (n + 1) n).reverse ∧ digits n = digitsF (n + 1) n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
91c9f620-0c4f… - lean_reversal_reversal_is_involutive_off_the_trailing_zeros_for_every_n — lean reversal.lean: reversal_is_involutive_off_the_trailing_zeros_for_every_n — ∀ n : Nat, n % 10 ≠ 0 → reverseNum (reverseNum n) = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6ea99113-8427… - lean_reversal_reversal_is_involutive_exactly_off_the_trailing_zeros_for_every_n — lean reversal.lean: reversal_is_involutive_exactly_off_the_trailing_zeros_for_every_n — ∀ n : Nat, 0 < n → (reverseNum (reverseNum n) = n ↔ n % 10 ≠ 0) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
4861172d-e203… - lean_reversal_more_fuel_changes_no_digit — lean reversal.lean: more_fuel_changes_no_digit — ∀ f n k : Nat, n < 10 ^ f → digitsF (f + k) n = digitsF f n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6dcd0667-a81f… - lean_reversal_a_trailing_zero_is_a_leading_zero_of_the_digits — lean reversal.lean: a_trailing_zero_is_a_leading_zero_of_the_digits — ∀ m : Nat, 0 < m → digits (10 * m) = 0 :: digits m — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c8aa36fc-ef29… - lean_reversal_reversal_preserves_the_digit_sum_for_every_n — lean reversal.lean: reversal_preserves_the_digit_sum_for_every_n — ∀ n : Nat, digitSum (reverseNum n) = digitSum n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
ab260836-a2cf… - lean_reversal_the_fixed_points_of_reversal_are_the_palindromes_for_every_n — lean reversal.lean: the_fixed_points_of_reversal_are_the_palindromes_for_every_n — ∀ n : Nat, n % 10 ≠ 0 → (reverseNum n = n ↔ (digits n).reverse = digits n) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
6b0761d8-d62a… - lean_sequences_halving_removes_exactly_the_low_bit_for_every_n — lean sequences.lean: halving_removes_exactly_the_low_bit_for_every_n — ∀ n : Nat, popcount n = n % 2 + popcount (n / 2) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
977069d5-f1b5… - lean_sequences_the_thue_morse_halving_recurrence_for_every_n — lean sequences.lean: the_thue_morse_halving_recurrence_for_every_n — ∀ n : Nat, tm n = (n % 2 + tm (n / 2)) % 2 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
864ba163-a3f8… - lean_sequences_doubling_shifts_every_bit_up_by_one — lean sequences.lean: doubling_shifts_every_bit_up_by_one — ∀ m i : Nat, 2 * m / 2 ^ (i + 1) = m / 2 ^ i — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c6f05e84-b232… - lean_sequences_the_residue_below_the_next_power_adds_one_bit — lean sequences.lean: the_residue_below_the_next_power_adds_one_bit — ∀ m F : Nat, m % 2 ^ (F + 1) = m % 2 ^ F + m / 2 ^ F % 2 * 2 ^ F — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
2607d9ae-7836… - lean_sequences_the_low_bits_rebuild_the_number_below_every_power — lean sequences.lean: the_low_bits_rebuild_the_number_below_every_power — ∀ F m : Nat, (List.range F).foldl (fun a i => a + m / 2 ^ i % 2 * 2 ^ i) 0 = m % 2 ^ F — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
c79eff54-9841… - lean_sequences_the_number_is_rebuilt_from_its_bits_for_every_n — lean sequences.lean: the_number_is_rebuilt_from_its_bits_for_every_n — ∀ n : Nat, (List.range (n + 1)).foldl (fun a i => a + n / 2 ^ i % 2 * 2 ^ i) 0 = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
4e6421ba-ab59… - lean_speed_the_nanoseconds_return_from_the_microseconds_at_every_whole_count — lean speed.lean: the_nanoseconds_return_from_the_microseconds_at_every_whole_count — ∀ n : Nat, n % 1000 = 0 → n / 1000 * 1000 = n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
7e0b39fe-b643… - lean_split_the_coins_return_from_the_coin_step_for_every_multiple_of_three — lean split.lean: the_coins_return_from_the_coin_step_for_every_multiple_of_three — ∀ t : Nat, t % 3 = 0 → 3 * (t / 3) = t — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
0a785a11-be04… - lean_split_paying_and_refunding_round_trips_for_every_count — lean split.lean: paying_and_refunding_round_trips_for_every_count — ∀ n k : Nat, k ≤ n → 3 * (n - k) + 3 * k = 3 * n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
31ce2f16-8ab7… - lean_z9_orbit_repeats_every_six — lean z9.lean: orbit_repeats_every_six — ∀ k : Nat, orbit (k + 6) = orbit k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
1d6bb6c2-2c76… - lean_z9_orbit_is_its_exponent_mod_six — lean z9.lean: orbit_is_its_exponent_mod_six — ∀ k : Nat, orbit k = orbit (k % 6) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
af2bb754-dd58… - lean_z9_the_orbit_step_is_read_back_by_its_log_for_every_k — lean z9.lean: the_orbit_step_is_read_back_by_its_log_for_every_k — ∀ k : Nat, logOrbit (orbit k) = some (k % 6) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
b806a941-659e… - lean_z9plus_halving_undoes_doubling_at_every_step — lean z9plus.lean: halving_undoes_doubling_at_every_step — ∀ k : Nat, m9 (pw 2 k * pw 5 k) = 1 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
87e5a9d4-3319… - lean_z9plus_halving_has_period_six_for_every_k — lean z9plus.lean: halving_has_period_six_for_every_k — ∀ k : Nat, pw 5 (k + 6) = pw 5 k — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
d07f608f-90d3… - lean_z9plus_halving_orbit_digital_roots_have_period_six_for_every_k — lean z9plus.lean: halving_orbit_digital_roots_have_period_six_for_every_k — ∀ k : Nat, dr (pw 5 (k + 6)) = dr (pw 5 k) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
147581f3-c75f… - lean_z9plus_the_reflected_orbit_meets_the_triad_exactly_at_one_four_seven_for_every_k — lean z9plus.lean: the_reflected_orbit_meets_the_triad_exactly_at_one_four_seven_for_every_k — ∀ k : Nat, [3, 6, 9].contains (refl (pw 2 k)) = [1, 4, 7].contains (pw 2 k) — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
20cda95a-c30a… - lean_z9plus_reversal_cannot_change_the_digital_root_for_every_n — lean z9plus.lean: reversal_cannot_change_the_digital_root_for_every_n — ∀ n : Nat, dr (reverseNum n) = dr n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
43332821-10bc… - lean_flow_a_ring_sum_splits_over_a_sum — lean flow.lean: a_ring_sum_splits_over_a_sum — ∀ (f g : Nat → Int) (n : Nat), ringSum (fun i => f i + g i) n = ringSum f n + ringSum g n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
def0f698-6e8c… - lean_flow_a_ring_sum_of_negatives_is_the_negated_sum — lean flow.lean: a_ring_sum_of_negatives_is_the_negated_sum — ∀ (f : Nat → Int) (n : Nat), ringSum (fun i => - f i) n = - ringSum f n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
86a56512-64df… - lean_flow_ring_sums_agree_where_their_terms_agree — lean flow.lean: ring_sums_agree_where_their_terms_agree — ∀ (f g : Nat → Int) (n : Nat), (∀ i, i < n → f i = g i) → ringSum f n = ringSum g n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
233f29d5-6b29… - lean_flow_a_ring_sum_peels_its_first_term — lean flow.lean: a_ring_sum_peels_its_first_term — ∀ (f : Nat → Int) (n : Nat), ringSum f (n + 1) = f 0 + ringSum (fun i => f (i + 1)) n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
4b3735a1-b114… - lean_flow_a_cyclic_sum_is_unchanged_by_one_step_of_rotation — lean flow.lean: a_cyclic_sum_is_unchanged_by_one_step_of_rotation — ∀ (N : Nat) (f : Nat → Int), 0 < N → ringSum (fun i => f (ringNext N i)) N = ringSum f N — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
d6a6e517-ae76… - lean_flow_the_site_before_the_next_is_the_site — lean flow.lean: the_site_before_the_next_is_the_site — ∀ N i : Nat, i < N → ringPrev N (ringNext N i) = i — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
e200c2f1-8c10… - lean_flow_a_bond_sum_reads_the_same_from_either_end — lean flow.lean: a_bond_sum_reads_the_same_from_either_end — ∀ (N : Nat) (F : Nat → Nat → Int), 0 < N → ringSum (fun i => F (ringPrev N i) i) N = ringSum (fun i => F i (ringNext N i)) N — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
9920e848-6b06… - lean_flow_the_skew_symmetric_burgers_term_does_no_work_on_every_ring — lean flow.lean: the_skew_symmetric_burgers_term_does_no_work_on_every_ring — ∀ (N : Nat) (u : Nat → Int), 0 < N → ringSum (fun i => u i * burgersSkew3 N u i) N = 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
a2e54b88-8b2f… - lean_flow_the_discrete_laplacian_dissipates_exactly_the_squared_differences — lean flow.lean: the_discrete_laplacian_dissipates_exactly_the_squared_differences — ∀ (N : Nat) (u : Nat → Int), 0 < N → ringSum (fun i => u i * ringLap N u i) N = - ringSum (fun i => (u (ringNext N i) - u i) * (u (ringNext N i) - u i)) N — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
289b45b9-0cbe… - lean_flow_a_sum_of_squares_is_never_negative — lean flow.lean: a_sum_of_squares_is_never_negative — ∀ (f : Nat → Int) (n : Nat), 0 ≤ ringSum (fun i => f i * f i) n — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
b5817e6d-005d… - lean_flow_the_discrete_energy_never_increases_on_every_ring — lean flow.lean: the_discrete_energy_never_increases_on_every_ring — ∀ (N : Nat) (u : Nat → Int) (ν : Int), 0 < N → 0 ≤ ν → ringSum (fun i => u i * (3 * ν * ringLap N u i - burgersSkew3 N u i)) N ≤ 0 — proved by the Lean kernel for every value (by intro), standard axioms only — checked by lean.ts ·
d25bc6d6-19e4… - lean_merkle_the_alteration_reaches_every_position — lean merkle.lean: the_alteration_reaches_every_position — (List.range 4).all (fun i => setAt [A, B, C, D] i E != [A, B, C, D]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
302ce1bd-f8b6… - lean_merkle_altering_any_single_leaf_changes_the_root — lean merkle.lean: altering_any_single_leaf_changes_the_root — (List.range 4).all (fun i => merkleFold (setAt [A, B, C, D] i E) != merkleFold [A, B, C, D]) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b37b91cf-2814… - lean_instruments_the_strict_order_is_what_is_admitted — lean instruments.lean: the_strict_order_is_what_is_admitted — precedes (some 4) (some 5) = true ∧ precedes (some 5) (some 5) = false ∧ precedes (some 6) (some 5) = false — decided by the Lean kernel over its whole finite domain, axiom-free ·
49563441-c8b2… - lean_instruments_admission_is_exactly_the_strict_order — lean instruments.lean: admission_is_exactly_the_strict_order — (List.range 12).all (fun a => (List.range 12).all (fun b => precedes (some a) (some b) == decide (a < b))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f7339fcf-5fd1… - lean_instruments_an_unknown_instant_is_never_admitted — lean instruments.lean: an_unknown_instant_is_never_admitted — (List.range 12).all (fun t => precedes none (some t) == false && precedes (some t) none == false) ∧ precedes none none = false — decided by the Lean kernel over its whole finite domain, axiom-free ·
30a5fe9f-75c8… - lean_instruments_the_boundary_refuses_more_than_it_admits — lean instruments.lean: the_boundary_refuses_more_than_it_admits — let cases := (List.range 8).map (fun a => precedes (some a) (some 4)) cases.count true = 4 ∧ cases.count false = 4 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ff7be975-04d9… - lean_instruments_a_hole_is_an_absence_with_a_later_presence — lean instruments.lean: a_hole_is_an_absence_with_a_later_presence — holes [true, false, true, true] = [1] ∧ tail [true, false, true, true] = [] — decided by the Lean kernel over its whole finite domain, axiom-free ·
f12656ea-a1e0… - lean_instruments_a_stale_tail_is_an_absence_with_nothing_after_it — lean instruments.lean: a_stale_tail_is_an_absence_with_nothing_after_it — tail [true, true, false, false] = [2, 3] ∧ holes [true, true, false, false] = [] — decided by the Lean kernel over its whole finite domain, axiom-free ·
cc70ab36-892f… - lean_instruments_the_two_causes_occur_together_and_stay_apart — lean instruments.lean: the_two_causes_occur_together_and_stay_apart — holes [true, false, true, false] = [1] ∧ tail [true, false, true, false] = [3] — decided by the Lean kernel over its whole finite domain, axiom-free ·
c113b614-9502… - lean_instruments_nothing_built_is_all_tail_and_no_hole — lean instruments.lean: nothing_built_is_all_tail_and_no_hole — holes [false, false, false] = [] ∧ tail [false, false, false] = [0, 1, 2] ∧ lastPresent [false, false, false] = none — decided by the Lean kernel over its whole finite domain, axiom-free ·
321515e0-bff1… - lean_instruments_everything_built_accuses_nobody — lean instruments.lean: everything_built_accuses_nobody — holes [true, true, true] = [] ∧ tail [true, true, true] = [] — decided by the Lean kernel over its whole finite domain, axiom-free ·
4c08c00c-bcf3… - lean_instruments_the_pattern_generator_is_complete — lean instruments.lean: the_pattern_generator_is_complete — (bits 5).length = 32 ∧ (bits 3).length = 8 ∧ ((bits 3).eraseDups).length = 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4b497ae9-79db… - lean_instruments_every_absence_is_charged_to_exactly_one_cause — lean instruments.lean: every_absence_is_charged_to_exactly_one_cause — ((bits 1) ++ (bits 2) ++ (bits 3) ++ (bits 4) ++ (bits 5)).all (fun l => let h := holes l let t := tail l let missing := (idx l).filterMap (fun p => if !p.2 then some p.1 else none) (h.all (fun x => !(t.contains x))) && ((h ++ t).length == … — decided by the Lean kernel over its whole finite domain, axiom-free ·
2b9f9c07-50f9… - lean_instruments_an_echo_carries_the_signal_only_until_it_is_subtracted — lean instruments.lean: an_echo_carries_the_signal_only_until_it_is_subtracted — contains echoed SITE = true ∧ contains (clip echoed) SITE = false — decided by the Lean kernel over its whole finite domain, axiom-free ·
3208d0ac-7913… - lean_instruments_the_page_s_own_words_survive_the_subtraction — lean instruments.lean: the_page_s_own_words_survive_the_subtraction — contains genuine SITE = true ∧ contains (clip genuine) SITE = true — decided by the Lean kernel over its whole finite domain, axiom-free ·
b071539d-99e9… - lean_instruments_every_copy_of_the_witness_is_removed — lean instruments.lean: every_copy_of_the_witness_is_removed — contains (clip twice) SITE = false — decided by the Lean kernel over its whole finite domain, axiom-free ·
430a5a37-1627… - lean_instruments_subtracting_twice_is_subtracting_once — lean instruments.lean: subtracting_twice_is_subtracting_once — clip (clip echoed) = clip echoed ∧ clip (clip genuine) = clip genuine — decided by the Lean kernel over its whole finite domain, axiom-free ·
bec39268-0840… - lean_instruments_an_empty_witness_removes_nothing — lean instruments.lean: an_empty_witness_removes_nothing — unreflect genuine.length [] genuine = genuine ∧ unreflect echoed.length [] echoed = echoed — decided by the Lean kernel over its whole finite domain, axiom-free ·
da283ead-c144… - lean_instruments_text_without_the_witness_is_untouched — lean instruments.lean: text_without_the_witness_is_untouched — clip SITE = SITE ∧ clip [60, 112, 62] = [60, 112, 62] — decided by the Lean kernel over its whole finite domain, axiom-free ·
113a3e4a-e3d5… - lean_asymmetric_a_signature_is_four_uuids_and_a_key_is_two — lean asymmetric.lean: a_signature_is_four_uuids_and_a_key_is_two — signatureBytes = 4 * uuidBytes ∧ publicKeyBytes = 2 * uuidBytes — decided by the Lean kernel over its whole finite domain, axiom-free ·
78459e31-65a2… - lean_asymmetric_neither_the_key_nor_the_signature_fits_in_a_uuid — lean asymmetric.lean: neither_the_key_nor_the_signature_fits_in_a_uuid — uuidBytes < publicKeyBytes ∧ uuidBytes < signatureBytes ∧ uuidBytes * 8 < signatureBytes * 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b4fa9d0f-ece5… - lean_asymmetric_not_even_the_whole_uuid_would_hold_one — lean asymmetric.lean: not_even_the_whole_uuid_would_hold_one — uuidBytes * 8 = 128 ∧ signatureBytes * 8 = 512 ∧ 128 < 512 — decided by the Lean kernel over its whole finite domain, axiom-free ·
2e3ba4c9-f281… - lean_asymmetric_what_travels_beside_the_identifier — lean asymmetric.lean: what_travels_beside_the_identifier — publicKeyBytes + signatureBytes = 96 ∧ 96 = 6 * uuidBytes — decided by the Lean kernel over its whole finite domain, axiom-free ·
00ac5e64-215d… - lean_asymmetric_the_check_cannot_separate_the_payloads — lean asymmetric.lean: the_check_cannot_separate_the_payloads — 2 ^ checkBits < 2 ^ payloadBits ∧ payloadBits = 90 ∧ checkBits = 32 — decided by the Lean kernel over its whole finite domain, axiom-free ·
df641661-41c4… - lean_asymmetric_the_gap_is_fifty_eight_bits — lean asymmetric.lean: the_gap_is_fifty_eight_bits — payloadBits - checkBits = 58 ∧ 2 ^ payloadBits = 2 ^ 58 * 2 ^ checkBits — decided by the Lean kernel over its whole finite domain, axiom-free ·
a0740c1c-cc73… - lean_asymmetric_a_checksum_is_the_same_whoever_computes_it — lean asymmetric.lean: a_checksum_is_the_same_whoever_computes_it — (List.range 6).all (fun s => (List.range 6).all (fun t => (List.range 12).all (fun p => checkOf s p == checkOf t p))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
38a9023c-ed6d… - lean_asymmetric_two_secrets_part_company_on_every_payload — lean asymmetric.lean: two_secrets_part_company_on_every_payload — (List.range 12).all (fun p => tagOf 1 p != tagOf 2 p) ∧ ¬ ((List.range 6).all (fun s => (List.range 6).all (fun t => (List.range 12).all (fun p => tagOf s p == tagOf t p)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
61826d5e-d6b7… - lean_asymmetric_the_tag_moves_with_the_payload — lean asymmetric.lean: the_tag_moves_with_the_payload — (List.range 11).all (fun p => tagOf 1 p != tagOf 1 (p + 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2c10db8a-11f2… - lean_program_the_program_field_is_forty_two_bits_not_forty_eight — lean program.lean: the_program_field_is_forty_two_bits_not_forty_eight — programF.length = 42 ∧ ((List.range 48).map (· + 32)).length = 48 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9f6fb2bb-27a6… - lean_program_the_three_fields_and_the_reserved_six_are_the_whole_uuid — lean program.lean: the_three_fields_and_the_reserved_six_are_the_whole_uuid — checkF.length + programF.length + messageF.length + RESERVED.length = 128 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ed6043cd-3b33… - lean_program_the_widths_are_what_the_layout_says — lean program.lean: the_widths_are_what_the_layout_says — checkF.length = 32 ∧ messageF.length = 48 ∧ RESERVED.length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
47c47e33-a9a0… - lean_program_every_reserved_bit_is_in_the_middle_group — lean program.lean: every_reserved_bit_is_in_the_middle_group — RESERVED.all (fun r => 32 ≤ r && r < 80) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8e8e33c0-1383… - lean_program_neither_the_check_nor_the_message_gives_up_a_bit — lean program.lean: neither_the_check_nor_the_message_gives_up_a_bit — (checkF.all (fun i => !(RESERVED.contains i))) ∧ (messageF.all (fun i => !(RESERVED.contains i))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a7b062c6-ab2f… - lean_program_every_position_belongs_to_exactly_one_field — lean program.lean: every_position_belongs_to_exactly_one_field — (List.range 128).all (fun i => inExactlyOne i == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
68c1c3ef-b289… - lean_program_nothing_outside_the_uuid_is_claimed — lean program.lean: nothing_outside_the_uuid_is_claimed — (List.range' 128 32).all (fun i => inExactlyOne i == 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c58dcc5c-bb67… - lean_program_each_field_is_ascending_and_repeats_nothing — lean program.lean: each_field_is_ascending_and_repeats_nothing — ascending checkF ∧ ascending programF ∧ ascending messageF — decided by the Lean kernel over its whole finite domain, axiom-free ·
dd9f31cc-fd90… - lean_program_the_check_is_the_deposit_s_own_fnv — lean program.lean: the_check_is_the_deposit_s_own_fnv — hash32 0 (List.replicate 12 0) = 1661287312 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f5fafbe0-1997… - lean_program_the_payload_outruns_the_check — lean program.lean: the_payload_outruns_the_check — 2 ^ 32 < 2 ^ (42 + 48) ∧ 42 + 48 = 90 — decided by the Lean kernel over its whole finite domain, axiom-free ·
78f23d08-e771… - lean_program_no_map_from_four_onto_three_separates_them_all — lean program.lean: no_map_from_four_onto_three_separates_them_all — maps3.all (fun f => decide ((f.eraseDups).length < 4)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
3f50ffb1-28f6… - lean_program_four_values_can_be_separated_by_four — lean program.lean: four_values_can_be_separated_by_four — maps4.any (fun f => decide ((f.eraseDups).length == 4)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5a28096c-369c… - lean_asymmetric_the_square_root_rule_is_the_one_this_prime_admits — lean asymmetric.lean: the_square_root_rule_is_the_one_this_prime_admits — p % 8 = 5 ∧ (p + 3) % 8 = 0 ∧ p % 4 = 1 ∧ p % 4 ≠ 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
b2d4dab4-6281… - lean_asymmetric_the_cofactor_is_bracketed_at_eight — lean asymmetric.lean: the_cofactor_is_bracketed_at_eight — 4 * L < p ∧ p < 8 * L ∧ L < p — decided by the Lean kernel over its whole finite domain, axiom-free ·
a882288a-1ac0… - lean_rays_the_orbit_is_the_six_units — lean rays.lean: the_orbit_is_the_six_units — orbit 6 1 = [3, 2, 6, 4, 5, 1] ∧ (orbit 6 1).length = 6 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d625db75-2feb… - lean_rays_the_order_visits_every_ray_exactly_once — lean rays.lean: the_order_visits_every_ray_exactly_once — vortexOrder.length = 7 ∧ (vortexOrder.eraseDups).length = 7 ∧ (List.range 7).all (fun i => vortexOrder.contains i) — decided by the Lean kernel over its whole finite domain, axiom-free ·
51ce4368-821c… - lean_rays_the_reversed_order_is_also_a_permutation — lean rays.lean: the_reversed_order_is_also_a_permutation — vortexReversed.length = 7 ∧ (vortexReversed.eraseDups).length = 7 ∧ (List.range 7).all (fun i => vortexReversed.contains i) — decided by the Lean kernel over its whole finite domain, axiom-free ·
cfd8d572-6ff9… - lean_rays_the_two_senses_are_not_the_same_walk — lean rays.lean: the_two_senses_are_not_the_same_walk — vortexOrder ≠ vortexReversed ∧ vortexOrder.head? = vortexReversed.head? — decided by the Lean kernel over its whole finite domain, axiom-free ·
ccac8571-dd1d… - lean_rays_reversing_the_sense_twice_is_the_identity — lean rays.lean: reversing_the_sense_twice_is_the_identity — (0 :: ((vortexReversed.tail).reverse)) = vortexOrder — decided by the Lean kernel over its whole finite domain, axiom-free ·
92578972-73dd… - lean_rays_inverting_twice_is_the_identity — lean rays.lean: inverting_twice_is_the_identity — ((List.range 40).map (fun n => invert (invert (lowBits [n, n + 1, n + 2])) == lowBits [n, n + 1, n + 2])).all id — decided by the Lean kernel over its whole finite domain, axiom-free ·
69b33021-cafb… - lean_rays_a_trace_and_its_inversion_agree_nowhere — lean rays.lean: a_trace_and_its_inversion_agree_nowhere — (List.range 40).all (fun n => let t := lowBits [n, n + 1, n + 2, n + 3] (List.range 4).all (fun i => (t.getD i 0) + ((invert t).getD i 0) == 1)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
cface271-a819… - lean_rays_the_low_bit_sees_what_the_residue_cannot — lean rays.lean: the_low_bit_sees_what_the_residue_cannot — (0 % 9 == 9 % 9) ∧ (0 % 9 == 18 % 9) ∧ ¬ (0 % 2 == 9 % 2) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f21dc8c1-025d… - lean_rays_the_two_by_seven_reads_more_than_the_seven — lean rays.lean: the_two_by_seven_reads_more_than_the_seven — DIMENSIONS = 14 ∧ DIGITS_READ = 28 ∧ 2 * N = 14 ∧ DIGITS_READ > 2 * N — decided by the Lean kernel over its whole finite domain, axiom-free ·
2181ea1e-20b3… - lean_rays_four_digits_stay_unread_and_are_counted — lean rays.lean: four_digits_stay_unread_and_are_counted — DIGITS_TOTAL - DIGITS_READ = 4 ∧ DIGITS_READ < DIGITS_TOTAL — decided by the Lean kernel over its whole finite domain, axiom-free ·
cfdf098a-f7a8… - lean_rays_the_wrap_is_what_makes_it_a_torus — lean rays.lean: the_wrap_is_what_makes_it_a_torus — d1 0 8 = 1 ∧ d1 8 0 = 1 ∧ d1 0 4 = 4 ∧ d1 4 0 = 4 — decided by the Lean kernel over its whole finite domain, axiom-free ·
85e8c4fa-30b9… - lean_rays_the_distance_is_symmetric_and_zero_only_on_itself — lean rays.lean: the_distance_is_symmetric_and_zero_only_on_itself — (List.range 9).all (fun a => (List.range 9).all (fun b => (d1 a b == d1 b a) && ((d1 a b == 0) == (a == b)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4eae84a8-ac4c… - lean_rays_the_triangle_inequality_holds_on_every_triple — lean rays.lean: the_triangle_inequality_holds_on_every_triple — (List.range 9).all (fun a => (List.range 9).all (fun b => (List.range 9).all (fun c => decide (d1 a c ≤ d1 a b + d1 b c)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5f6fe76f-c6ea… - lean_instruments_any_negator_before_the_claim_refuses_it — lean instruments.lean: any_negator_before_the_claim_refuses_it — refused [7, 2, 9] 5 = true ∧ refused [7, 9] 5 = false ∧ refused [5] 5 = false — decided by the Lean kernel over its whole finite domain, axiom-free ·
61753440-c6c1… - lean_instruments_the_window_is_monotone_in_the_claim_s_position — lean instruments.lean: the_window_is_monotone_in_the_claim_s_position — (List.range 10).all (fun c => (List.range 10).all (fun d => if c ≤ d then (!(refused [3] c) || refused [3] d) else true)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
223bfc2c-f60b… - lean_instruments_with_no_negator_nothing_is_cleared — lean instruments.lean: with_no_negator_nothing_is_cleared — (List.range 10).all (fun c => refused [] c == false) — decided by the Lean kernel over its whole finite domain, axiom-free ·
fdba66f0-240b… - lean_program_the_fields_read_back_exactly — lean program.lean: the_fields_read_back_exactly — readField programF (encodeBits P0 M0) = P0 ∧ readField messageF (encodeBits P0 M0) = M0 ∧ readField programF (encodeBits P1 M1) = P1 ∧ readField messageF (encodeBits P1 M1) = M1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
cdf3c4f3-4c16… - lean_program_the_check_field_carries_the_check — lean program.lean: the_check_field_carries_the_check — readField checkF (encodeBits P0 M0) = checkBits P0 M0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
f2c56146-946d… - lean_program_the_reserved_six_survive_any_payload — lean program.lean: the_reserved_six_survive_any_payload — [(P0, M0), (P1, M1), (P0, M1), (P1, M0), (PZ, MZ)].all (fun pm => let b := encodeBits pm.1 pm.2 (b.getD 48 false == true) && (b.getD 49 false == false) && (b.getD 50 false == false) && (b.getD 51 false == false) && (b.getD 64 false == true)… — decided by the Lean kernel over its whole finite domain, axiom-free ·
33b79421-00e4… - lean_program_the_flip_reaches_every_payload_position — lean program.lean: the_flip_reaches_every_payload_position — (List.range 42).all (fun i => flipAt P0 i != P0) ∧ (List.range 48).all (fun i => flipAt M0 i != M0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
bbcc9872-ae56… - lean_program_flipping_any_program_bit_moves_the_check — lean program.lean: flipping_any_program_bit_moves_the_check — (List.range 42).all (fun i => checkBits (flipAt P0 i) M0 != checkBits P0 M0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
457871fb-d7a0… - lean_program_flipping_any_message_bit_moves_the_check — lean program.lean: flipping_any_message_bit_moves_the_check — (List.range 48).all (fun i => checkBits P0 (flipAt M0 i) != checkBits P0 M0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
131cee6e-62a3… - lean_program_an_untouched_container_reads_intact — lean program.lean: an_untouched_container_reads_intact — [(P0, M0), (P1, M1), (PZ, MZ)].all (fun pm => intact (encodeBits pm.1 pm.2)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
24e09b9f-fa04… - lean_program_a_damaged_container_is_refused_at_every_program_position — lean program.lean: a_damaged_container_is_refused_at_every_program_position — (List.range 42).all (fun i => !(intact (flipBit (encodeBits P0 M0) (programF.getD i 0)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4cb0515d-b4c1… - lean_program_the_reserved_six_are_where_the_rendering_puts_them — lean program.lean: the_reserved_six_are_where_the_rendering_puts_them — RESERVED = [48, 49, 50, 51, 64, 65] ∧ byteAt 2 = 6 ∧ byteAt 3 = 8 ∧ byteAt 4 = 10 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4d490971-9f9a… - lean_roots_the_cube_root_is_the_floor — lean roots.lean: the_cube_root_is_the_floor — iroot 3 (2 * 2 ^ 192) ^ 3 <= 2 * 2 ^ 192 — decided by the Lean kernel over its whole finite domain, axiom-free ·
60dc4761-9a49… - lean_roots_the_cube_root_is_not_one_short — lean roots.lean: the_cube_root_is_not_one_short — 2 * 2 ^ 192 < (iroot 3 (2 * 2 ^ 192) + 1) ^ 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
80fe22fd-a510… - lean_roots_the_square_root_is_the_floor_and_not_one_short — lean roots.lean: the_square_root_is_the_floor_and_not_one_short — iroot 2 (2 * 2 ^ 128) ^ 2 <= 2 * 2 ^ 128 ∧ 2 * 2 ^ 128 < (iroot 2 (2 * 2 ^ 128) + 1) ^ 2 — decided by the Lean kernel over its whole finite domain, axiom-free ·
73292676-fefa… - lean_roots_the_prime_generator_is_complete — lean roots.lean: the_prime_generator_is_complete — firstPrimes 40 8 = [2, 3, 5, 7, 11, 13, 17, 19] ∧ (firstPrimes 40 8).all isPrime ∧ ((List.range' 2 18).filter isPrime).length = 8 — decided by the Lean kernel over its whole finite domain, axiom-free ·
999c4094-b5b6… - lean_roots_the_first_round_constant_is_that_cube_root_fraction — lean roots.lean: the_first_round_constant_is_that_cube_root_fraction — frac64 3 2 = 0x428a2f98d728ae22 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e464678b-2899… - lean_roots_the_first_initial_value_is_that_square_root_fraction — lean roots.lean: the_first_initial_value_is_that_square_root_fraction — frac64 2 2 = 0x6a09e667f3bcc908 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e9398daf-fd28… - lean_roots_the_cube_and_the_square_do_not_agree — lean roots.lean: the_cube_and_the_square_do_not_agree — frac64 3 2 != frac64 2 2 ∧ frac64 3 3 != frac64 2 3 — decided by the Lean kernel over its whole finite domain, axiom-free ·
28cb8c6b-7c2c… - lean_imprint_the_free_region_is_a_hundred_and_twenty_two — lean imprint.lean: the_free_region_is_a_hundred_and_twenty_two — FREE.length = 122 ∧ RESERVED.length = 6 ∧ FREE.length + RESERVED.length = 128 — decided by the Lean kernel over its whole finite domain, axiom-free ·
82c2c108-cf5f… - lean_imprint_the_capacity_is_the_free_region_less_the_header — lean imprint.lean: the_capacity_is_the_free_region_less_the_header — CAPACITY = 115 ∧ CAPACITY + LEN_BITS = FREE.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
05ee2cb9-0963… - lean_imprint_no_reserved_bit_is_free — lean imprint.lean: no_reserved_bit_is_free — RESERVED.all (fun r => !(FREE.contains r)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
dddd46cc-3c26… - lean_imprint_seven_bits_count_past_the_capacity — lean imprint.lean: seven_bits_count_past_the_capacity — 2 ^ LEN_BITS > CAPACITY ∧ 2 ^ (LEN_BITS - 1) < CAPACITY — decided by the Lean kernel over its whole finite domain, axiom-free ·
9c0e2abe-7301… - lean_imprint_every_short_message_comes_back — lean imprint.lean: every_short_message_comes_back — (List.range 9).all (fun n => decodeBits (encodeBits (alt n)) == alt n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5d24f33c-b06b… - lean_imprint_the_all_set_message_comes_back_too — lean imprint.lean: the_all_set_message_comes_back_too — (List.range 9).all (fun n => decodeBits (encodeBits (ones n)) == ones n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c6ae9fbf-82c8… - lean_imprint_the_header_records_the_length — lean imprint.lean: the_header_records_the_length — (List.range 9).all (fun n => natOf ((FREE.map (fun i => (encodeBits (alt n)).getD i false)).take LEN_BITS) == n) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ad8567b8-120e… - lean_imprint_a_shorter_message_is_not_a_longer_one — lean imprint.lean: a_shorter_message_is_not_a_longer_one — encodeBits (alt 3) != encodeBits (alt 4) ∧ encodeBits (ones 0) != encodeBits (ones 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7c00d44d-8834… - lean_imprint_the_reserved_six_survive_any_message — lean imprint.lean: the_reserved_six_survive_any_message — [alt 0, alt 5, ones 8, alt 115].all (fun m => let b := encodeBits m (b.getD 48 false == true) && (b.getD 49 false == false) && (b.getD 50 false == false) && (b.getD 51 false == false) && (b.getD 64 false == true) && (b.getD 65 false == fals… — decided by the Lean kernel over its whole finite domain, axiom-free ·
12a83c85-7690… - lean_millenniumfloor_the_tens_complement_is_an_involution_with_one_fixed_point — lean index.lean: the_tens_complement_is_an_involution_with_one_fixed_point — (List.range 10).all (fun d => refl (refl d) == d) ∧ ((List.range 10).filter (fun d => refl d == d)).length = 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e9dd9619-515b… - lean_millenniumfloor_each_unit_has_exactly_one_inverse_and_each_non_unit_none — lean index.lean: each_unit_has_exactly_one_inverse_and_each_non_unit_none — (List.range 9).all (fun d => ((List.range 9).filter (fun e => (d * e) % 9 == 1)).length == (if isUnit d then 1 else 0)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
661bba08-39e9… - lean_millenniumfloor_the_doubling_orbit_stays_in_the_ring_for_forty_eight_steps — lean index.lean: the_doubling_orbit_stays_in_the_ring_for_forty_eight_steps — ((List.range 48).map orbit).all (fun v => v < 9) ∧ (List.range 48).all (fun k => span.contains (orbit k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
98c5c4b5-949a… - lean_millenniumfloor_the_doubling_orbit_first_returns_to_one_at_six — lean index.lean: the_doubling_orbit_first_returns_to_one_at_six — (List.range 6).all (fun k => k == 0 || orbit k != 1) ∧ orbit 6 == 1 — decided by the Lean kernel over its whole finite domain, axiom-free ·
996a80a9-10d3… - lean_millenniumfloor_the_span_is_exactly_the_units_of_the_ring — lean index.lean: the_span_is_exactly_the_units_of_the_ring — (List.range 9).all (fun d => span.contains d == isUnit d) ∧ (List.range 9).all (fun d => isUnit d || ! span.contains d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2d45299e-ed42… - lean_millenniumfloor_the_span_and_the_units_both_sum_to_zero_mod_nine — lean index.lean: the_span_and_the_units_both_sum_to_zero_mod_nine — (span.foldr (· + ·) 0) % 9 == 0 ∧ ((List.range 9).filter isUnit).foldr (· + ·) 0 % 9 == 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a6bd7475-788f… - lean_millenniumfloor_the_orbit_is_one_closed_loop_of_six_distinct_points — lean index.lean: the_orbit_is_one_closed_loop_of_six_distinct_points — orbit 6 == orbit 0 ∧ (List.range 6).all (fun i => (List.range 6).all (fun j => (orbit i == orbit j) == (i == j))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
10ad046c-32e7… - lean_theology_the_mirror_and_the_doubling_agree_at_no_residue — lean theology.lean: the_mirror_and_the_doubling_agree_at_no_residue — (List.range 9).all (fun d => refl (dbl d) % 9 != (2 * refl d) % 9) ∧ ((List.range 9).filter (fun d => refl (dbl d) % 9 == (2 * refl d) % 9)).length = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
28708e4a-fe0f… - lean_theology_confirming_a_witness_is_uniform_while_finding_one_is_not — lean theology.lean: confirming_a_witness_is_uniform_while_finding_one_is_not — ((List.range 9).filter isUnit).all (fun d => (d * invOf d) % 9 == 1) ∧ (((List.range 9).filter isUnit).map searchCost).eraseDups.length > 1 ∧ ((List.range 9).filter isUnit).any (fun d => searchCost d > searchCost 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2241453d-2a19… - lean_theology_no_start_ever_crosses_between_the_units_and_the_non_units — lean theology.lean: no_start_ever_crosses_between_the_units_and_the_non_units — (List.range 9).all (fun s => (List.range 48).all (fun k => isUnit (iter k s) == isUnit s)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
0582fb57-6ae9… - lean_theology_the_origin_is_reached_at_every_step_exactly_by_the_origin — lean theology.lean: the_origin_is_reached_at_every_step_exactly_by_the_origin — (List.range 9).all (fun s => (List.range 48).all (fun k => (iter k s == 0) == (s == 0))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
b31744a1-62f3… - lean_theology_generating_the_same_set_is_exactly_being_of_the_same_kind — lean theology.lean: generating_the_same_set_is_exactly_being_of_the_same_kind — (List.range 9).all (fun g => (List.range 9).all (fun h => sameSet (spanOf g) (spanOf h) == ((isUnit g == isUnit h) && ((g == 0) == (h == 0))))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c5a06c7c-8c41… - lean_theology_the_orbit_is_antipodal_and_each_opposite_pair_sums_to_nine — lean theology.lean: the_orbit_is_antipodal_and_each_opposite_pair_sums_to_nine — (List.range 3).all (fun i => orbit i + orbit (i + 3) == 9) ∧ (span.foldr (· + ·) 0) % 9 == 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
dceecba8-5d64… - lean_theology_the_orbit_has_no_doubling_closed_part_but_nothing_and_itself — lean theology.lean: the_orbit_has_no_doubling_closed_part_but_nothing_and_itself — (List.range 64).all (fun m => let s := subsetOf m (! s.all (fun v => s.contains (dbl v))) || s.length == 0 || s.length == span.length) ∧ span.all (fun v => span.contains (dbl v)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
412804db-ea5f… - lean_theology_the_pairing_is_one_ordering_of_seven_out_of_all_of_them — lean theology.lean: the_pairing_is_one_ordering_of_seven_out_of_all_of_them — pairing.eraseDups.length = 7 ∧ (List.range 7).all (fun i => pairing.contains i) ∧ pairing.length = (List.range 7).length ∧ fact pairing.length = 5040 — decided by the Lean kernel over its whole finite domain, axiom-free ·
4db1d76e-da5a… - lean_instruments_a_path_dirty_before_the_run_is_never_attributed_to_it — lean instruments.lean: a_path_dirty_before_the_run_is_never_attributed_to_it — (List.range 16).all (fun a => (List.range 16).all (fun b => (List.range 16).all (fun t => (mine (setOf a) (setOf b) (setOf t)).all (fun p => ! (setOf a).contains p)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8652f875-b425… - lean_instruments_an_untracked_path_is_never_attributed — lean instruments.lean: an_untracked_path_is_never_attributed — (List.range 16).all (fun a => (List.range 16).all (fun b => (List.range 16).all (fun t => (mine (setOf a) (setOf b) (setOf t)).all (fun p => (setOf t).contains p)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d3932ec5-4ae8… - lean_instruments_nothing_is_attributed_that_is_not_dirty_afterwards — lean instruments.lean: nothing_is_attributed_that_is_not_dirty_afterwards — (List.range 16).all (fun a => (List.range 16).all (fun b => (List.range 16).all (fun t => subset (mine (setOf a) (setOf b) (setOf t)) (setOf b)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2419d78b-a4c4… - lean_instruments_a_wider_record_of_what_was_dirty_attributes_no_more — lean instruments.lean: a_wider_record_of_what_was_dirty_attributes_no_more — (List.range 16).all (fun a => (List.range 16).all (fun a' => (List.range 16).all (fun b => ! subset (setOf a) (setOf a') || subset (mine (setOf a') (setOf b) paths) (mine (setOf a) (setOf b) paths)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f530a405-14d0… - lean_instruments_a_change_made_by_another_hand_during_the_run_is_charged_to_the_run — lean instruments.lean: a_change_made_by_another_hand_during_the_run_is_charged_to_the_run — (List.range 16).any (fun a => (List.range 16).any (fun b => paths.any (fun p => ! (setOf a).contains p && (setOf b).contains p && (mine (setOf a) (setOf b) paths).contains p))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
90355aa7-fd8b… - lean_instruments_a_run_that_changes_nothing_attributes_nothing — lean instruments.lean: a_run_that_changes_nothing_attributes_nothing — (List.range 16).all (fun a => (List.range 16).all (fun t => mine (setOf a) (setOf a) (setOf t) == [])) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1c832cbd-2887… - lean_instruments_reverting_once_leaves_nothing_further_to_attribute — lean instruments.lean: reverting_once_leaves_nothing_further_to_attribute — (List.range 16).all (fun a => (List.range 16).all (fun b => (List.range 16).all (fun t => let m := mine (setOf a) (setOf b) (setOf t) mine (setOf a) ((setOf b).filter (fun p => ! m.contains p)) (setOf t) == []))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
4426c91f-28f6… - lean_instruments_the_reverted_and_the_untouched_partition_what_is_dirty — lean instruments.lean: the_reverted_and_the_untouched_partition_what_is_dirty — (List.range 16).all (fun a => (List.range 16).all (fun b => let m := mine (setOf a) (setOf b) paths let rest := (setOf b).filter (fun p => ! m.contains p) m.all (fun p => ! rest.contains p) && (setOf b).all (fun p => m.contains p || rest.co… — decided by the Lean kernel over its whole finite domain, axiom-free ·
9fda16c2-ca54… - lean_nucleus_every_capacity_is_even — lean nucleus.lean: every_capacity_is_even — caps.all (fun c => c % 2 == 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f2d535a0-1bf8… - lean_nucleus_the_seven_marked_prefixes_take_the_published_values — lean nucleus.lean: the_seven_marked_prefixes_take_the_published_values — closures = [2, 8, 20, 28, 50, 82, 126] — decided by the Lean kernel over its whole finite domain, axiom-free ·
91efa07e-77cd… - lean_nucleus_each_of_the_seven_is_reached_exactly_once_among_all_prefixes — lean nucleus.lean: each_of_the_seven_is_reached_exactly_once_among_all_prefixes — closures.all (fun m => (prefixes.filter (fun p => p == m)).length == 1) — decided by the Lean kernel over its whole finite domain, axiom-free ·
65a430c2-6e73… - lean_nucleus_the_capacities_sum_to_the_last_of_the_seven — lean nucleus.lean: the_capacities_sum_to_the_last_of_the_seven — total caps = 126 ∧ total caps = closures.foldr (fun a b => max a b) 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ec4d7a37-fa3b… - lean_nucleus_the_seven_increase_strictly — lean nucleus.lean: the_seven_increase_strictly — (List.range (closures.length - 1)).all (fun i => closures.getD i 0 < closures.getD (i + 1) 0) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d3355931-131e… - lean_nucleus_the_seal_multiplies_back_to_one — lean nucleus.lean: the_seal_multiplies_back_to_one — product numerators = product denominators — decided by the Lean kernel over its whole finite domain, axiom-free ·
aedef66f-0f69… - lean_nucleus_the_fit_is_an_exact_integer_identity — lean nucleus.lean: the_fit_is_an_exact_integer_identity — 108 * 17 = 1836 — decided by the Lean kernel over its whole finite domain, axiom-free ·
7c33b6d2-060d… - lean_nucleus_the_fit_is_not_the_measured_ratio — lean nucleus.lean: the_fit_is_not_the_measured_ratio — 1836 * 10000 ≠ 18361527 — decided by the Lean kernel over its whole finite domain, axiom-free ·
3e5b63c7-5ca0… - lean_authority_nothing_withdrawn_is_live — lean authority.lean: nothing_withdrawn_is_live — withdrawn.all (fun k => ! live.contains k) ∧ live.all (fun k => ! withdrawn.contains k) — decided by the Lean kernel over its whole finite domain, axiom-free ·
504b6638-5a8f… - lean_authority_every_sealed_statement_is_live_or_withdrawn_and_not_both — lean authority.lean: every_sealed_statement_is_live_or_withdrawn_and_not_both — sealed.all (fun k => (live.contains k || withdrawn.contains k) && ! (live.contains k && withdrawn.contains k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
f797d801-3e74… - lean_authority_withdrawing_shrinks_the_live_set_and_never_the_record — lean authority.lean: withdrawing_shrinks_the_live_set_and_never_the_record — live.length + withdrawn.length = sealed.length ∧ withdrawn.all (fun k => sealed.contains k) — decided by the Lean kernel over its whole finite domain, axiom-free ·
d2b796e8-29d1… - lean_authority_a_rider_and_a_verdict_are_different_things — lean authority.lean: a_rider_and_a_verdict_are_different_things — (carriesRider.any (fun k => ! isVerdict.contains k)) ∧ (isVerdict.any (fun k => ! carriesRider.contains k)) ∧ (carriesRider.any (fun k => isVerdict.contains k)) — decided by the Lean kernel over its whole finite domain, axiom-free ·
71fe6459-9e87… - lean_authority_withdrawing_never_raises_the_frozen_count — lean authority.lean: withdrawing_never_raises_the_frozen_count — (List.range 256).all (fun m => let taken := bits m let after := carriesRider.filter (fun k => ! taken.contains k) after.length ≤ carriesRider.length) — decided by the Lean kernel over its whole finite domain, axiom-free ·
1aac6788-923f… - lean_authority_clearing_changes_no_seal — lean authority.lean: clearing_changes_no_seal — cleared.all (fun k => sealed.contains k) ∧ (sealed.filter (fun k => ! cleared.contains k)).length + cleared.length = sealed.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
7bd9776a-18f2… - lean_authority_a_cleared_statement_leaves_the_queue_and_stays_on_the_record — lean authority.lean: a_cleared_statement_leaves_the_queue_and_stays_on_the_record — (isVerdict.filter (fun k => ! cleared.contains k)).all (fun k => sealed.contains k) ∧ (isVerdict.filter (fun k => ! cleared.contains k)).length ≤ isVerdict.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
a357fd68-31d0… - lean_authority_refusing_every_verdict_never_refuses_the_whole_record — lean authority.lean: refusing_every_verdict_never_refuses_the_whole_record — isVerdict.all (fun k => sealed.contains k) ∧ sealed.any (fun k => ! isVerdict.contains k) ∧ isVerdict.length < sealed.length — decided by the Lean kernel over its whole finite domain, axiom-free ·
6e853633-d7fe… - lean_planck_the_three_planck_units_share_one_relative_uncertainty — lean planck.lean: the_three_planck_units_share_one_relative_uncertainty — ppm ellP ellPUnc = 11 ∧ ppm tP tPUnc = 11 ∧ ppm mP mPUnc = 11 — decided by the Lean kernel over its whole finite domain, axiom-free ·
55140311-eb5c… - lean_planck_the_square_root_halves_what_is_known_about_the_measured_constant — lean planck.lean: the_square_root_halves_what_is_known_about_the_measured_constant — 2 * ppm ellP ellPUnc = ppm bigG bigGUnc ∧ 2 * ppm tP tPUnc = ppm bigG bigGUnc ∧ 2 * ppm mP mPUnc = ppm bigG bigGUnc — decided by the Lean kernel over its whole finite domain, axiom-free ·
b8782530-4860… - lean_planck_the_measurement_admits_thirty_seven_values_and_not_one — lean planck.lean: the_measurement_admits_thirty_seven_values_and_not_one — window.length = 37 ∧ window.head? = some (ellP - ellPUnc) ∧ window.getLast? = some (ellP + ellPUnc) — decided by the Lean kernel over its whole finite domain, axiom-free ·
de29e184-4c40… - lean_planck_the_digital_root_tracks_the_unit_and_not_the_scale — lean planck.lean: the_digital_root_tracks_the_unit_and_not_the_scale — dr ellP = 8 ∧ dr tP = 4 ∧ dr mP = 9 ∧ dr ellP ≠ dr tP ∧ dr tP ≠ dr mP — decided by the Lean kernel over its whole finite domain, axiom-free ·
e53500b7-2ac3… - lean_planck_the_orbit_holds_two_thirds_of_the_ring_so_landing_on_it_says_little — lean planck.lean: the_orbit_holds_two_thirds_of_the_ring_so_landing_on_it_says_little — orbit.length = 6 ∧ axis.length = 3 ∧ orbit.length + axis.length = 9 ∧ orbit.contains (dr ellP) ∧ orbit.contains (dr tP) ∧ axis.contains (dr mP) — decided by the Lean kernel over its whole finite domain, axiom-free ·
ef62fe1e-e10c… - lean_planck_no_product_of_three_lattice_constants_lands_where_the_planck_length_is — lean planck.lean: no_product_of_three_lattice_constants_lands_where_the_planck_length_is — products.length = 2197 ∧ products.all (fun v => v < ellP - ellPUnc || v > ellP + ellPUnc) — decided by the Lean kernel over its whole finite domain, axiom-free ·
9a16e510-2ed2… - lean_planck_the_same_search_does_reach_the_magnitude — lean planck.lean: the_same_search_does_reach_the_magnitude — (products.filter (fun v => 1600000 ≤ v && v ≤ 1700000)).length = 9 — decided by the Lean kernel over its whole finite domain, axiom-free ·
eb57a919-a1ee… - lean_planck_a_search_expecting_less_than_one_hit_settles_nothing_either_way — lean planck.lean: a_search_expecting_less_than_one_hit_settles_nothing_either_way — 9 * (2 * ellPUnc + 1) / 100001 = 0 ∧ 9 * (2 * ellPUnc + 1) < 100001 — decided by the Lean kernel over its whole finite domain, axiom-free ·
10ecab4b-c2be… - lean_planck_every_planck_unit_is_a_square_root_and_not_a_plain_product — lean planck.lean: every_planck_unit_is_a_square_root_and_not_a_plain_product — dimEllP.1 % 2 ≠ 0 ∧ dimTP.1 % 2 ≠ 0 ∧ dimMP.1 % 2 ≠ 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
089656ee-28f2… - lean_planck_length_over_time_cancels_both_and_leaves_the_defined_constant — lean planck.lean: length_over_time_cancels_both_and_leaves_the_defined_constant — ddiv dimEllP dimTP = (0, 0, 2) — decided by the Lean kernel over its whole finite domain, axiom-free ·
c47bf818-f597… - lean_planck_length_times_mass_cancels_gravity_and_leaves_the_quantum — lean planck.lean: length_times_mass_cancels_gravity_and_leaves_the_quantum — dmul dimEllP dimMP = (2, 0, -2) — decided by the Lean kernel over its whole finite domain, axiom-free ·
16811f4a-d0ee… - lean_planck_length_over_mass_cancels_the_quantum_and_leaves_gravity — lean planck.lean: length_over_mass_cancels_the_quantum_and_leaves_gravity — ddiv dimEllP dimMP = (0, 2, -4) — decided by the Lean kernel over its whole finite domain, axiom-free ·
8c098abf-9aa3… - lean_planck_the_gravitational_exponent_predicts_every_published_uncertainty — lean planck.lean: the_gravitational_exponent_predicts_every_published_uncertainty — gPpm dimEllP = 11 ∧ gPpm dimTP = 11 ∧ gPpm dimMP = 11 ∧ gPpm (ddiv dimEllP dimTP) = 0 ∧ gPpm (dmul dimEllP dimMP) = 0 ∧ gPpm (ddiv dimEllP dimMP) = ppm bigG bigGUnc — decided by the Lean kernel over its whole finite domain, axiom-free ·
53bf0add-fed9… - lean_planck_the_three_uncertainties_are_one_uncertainty_wearing_three_faces — lean planck.lean: the_three_uncertainties_are_one_uncertainty_wearing_three_faces — (ddiv dimEllP dimTP).2.1 = 0 ∧ (dmul dimEllP dimMP).2.1 = 0 ∧ (ddiv dimEllP dimMP).2.1 ≠ 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9879b17b-0c42… - lean_planck_the_published_digits_miss_an_identity_that_has_no_measurement_in_it — lean planck.lean: the_published_digits_miss_an_identity_that_has_no_measurement_in_it — ratioFromDigits = 299792422 ∧ ratioFromDigits ≠ cDefined ∧ cDefined - ratioFromDigits = 36 — decided by the Lean kernel over its whole finite domain, axiom-free ·
d9c40976-6f84… - lean_planck_the_miss_is_rounding_and_sits_far_below_the_uncertainty — lean planck.lean: the_miss_is_rounding_and_sits_far_below_the_uncertainty — (cDefined - ratioFromDigits) * 1000000 < 11 * cDefined ∧ 11 * cDefined / ((cDefined - ratioFromDigits) * 1000000) = 91 — decided by the Lean kernel over its whole finite domain, axiom-free ·
e0719e22-4940… - lean_planck_not_one_value_the_world_admits_is_reachable_from_this_lattice — lean planck.lean: not_one_value_the_world_admits_is_reachable_from_this_lattice — window.all (fun v => ! reachable v) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7ff9784c-dea6… - lean_planck_every_seed_factors_over_those_primes_so_every_product_does — lean planck.lean: every_seed_factors_over_those_primes_so_every_product_does — seeds.all (fun s => reachable s) — decided by the Lean kernel over its whole finite domain, axiom-free ·
2bb01390-5215… - lean_planck_the_exponent_predicts_the_registry_with_a_literal_on_neither_side — lean planck.lean: the_exponent_predicts_the_registry_with_a_literal_on_neither_side — gPpm dimEllP = ppm ellP ellPUnc ∧ gPpm dimTP = ppm tP tPUnc ∧ gPpm dimMP = ppm mP mPUnc — decided by the Lean kernel over its whole finite domain, axiom-free ·
9f8f686c-bb89… - lean_planck_the_exact_halving_survives_only_because_the_division_truncated — lean planck.lean: the_exact_halving_survives_only_because_the_division_truncated — ppm4 ellP ellPUnc = 111368 ∧ ppm4 bigG bigGUnc = 224742 ∧ 2 * ppm4 ellP ellPUnc ≠ ppm4 bigG bigGUnc — decided by the Lean kernel over its whole finite domain, axiom-free ·
927f3553-ab6d… - lean_planck_no_power_of_ten_moves_the_root — lean planck.lean: no_power_of_ten_moves_the_root — (List.range 8).all (fun k => dr (ellP * 10 ^ k) = dr ellP) — decided by the Lean kernel over its whole finite domain, axiom-free ·
5247f951-6e3b… - lean_planck_an_exactly_defined_non_decimal_unit_moves_the_root — lean planck.lean: an_exactly_defined_non_decimal_unit_moves_the_root — ellPInches = 63632086614 ∧ dr ellPInches = 9 ∧ dr ellPInches ≠ dr ellP — decided by the Lean kernel over its whole finite domain, axiom-free ·
1c80d362-10a2… - lean_planck_the_miss_fits_inside_the_printed_precision_and_not_inside_the_measurement — lean planck.lean: the_miss_fits_inside_the_printed_precision_and_not_inside_the_measurement — displayBudget = 120 ∧ cDefined - ratioFromDigits ≤ displayBudget ∧ displayBudget * 10000 < 11 * cDefined — decided by the Lean kernel over its whole finite domain, axiom-free ·
261b31f7-51a9… - lean_planck_the_planck_mantissa_factors_outside_the_lattice — lean planck.lean: the_planck_mantissa_factors_outside_the_lattice — ellP = 5 * 323251 ∧ (List.range' 2 568).all (fun d => 323251 % d ≠ 0) ∧ ! reachable ellP — decided by the Lean kernel over its whole finite domain, axiom-free ·
a8b669c6-e11d… - lean_handle_the_four_hex_field_round_trips_over_its_whole_space — lean handle.lean: the_four_hex_field_round_trips_over_its_whole_space — (List.range 16).all (fun a => (List.range 16).all (fun b => (List.range 16).all (fun c => (List.range 16).all (fun d => let n := fromNibs a b c d nib n 0 == a && nib n 1 == b && nib n 2 == c && nib n 3 == d && n < 65536)))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
a2d36cba-f5e8… - lean_handle_the_field_runs_from_zero_to_its_last_value — lean handle.lean: the_field_runs_from_zero_to_its_last_value — fromNibs 0 0 0 0 = 0 ∧ fromNibs 15 15 15 15 = 65535 ∧ (List.range 16).all (fun d => fromNibs 0 0 0 d == 4096 * d) — decided by the Lean kernel over its whole finite domain, axiom-free ·
7714e0a9-e35c… - lean_handle_a_uuid_is_thirty_two_hex_and_exactly_three_groups_are_four_wide — lean handle.lean: a_uuid_is_thirty_two_hex_and_exactly_three_groups_are_four_wide — groups.foldl (· + ·) 0 = 32 ∧ (groups.filter (fun g => g == 4)).length = 3 ∧ groups.length = 5 — decided by the Lean kernel over its whole finite domain, axiom-free ·
a998790f-3068… - lean_handle_the_address_space_is_finite_so_enough_messages_must_share_an_address — lean handle.lean: the_address_space_is_finite_so_enough_messages_must_share_an_address — space = 16 ^ 4 ∧ space = 2 ^ 16 ∧ ((List.range 17).map (fun m => m % 16)).eraseDups.length < 17 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1d85c5d8-e141… - lean_handle_one_address_two_messages_and_the_handle_is_what_tells_them_apart — lean handle.lean: one_address_two_messages_and_the_handle_is_what_tells_them_apart — addr 3 = addr 19 ∧ under 0 3 = under 0 19 ∧ under 0 3 ≠ under 1 19 — decided by the Lean kernel over its whole finite domain, axiom-free ·
1b3ee59d-ca46… - lean_handle_the_same_message_at_two_positions_never_seals_alike — lean handle.lean: the_same_message_at_two_positions_never_seals_alike — (List.range 32).all (fun m => (List.range 8).all (fun i => (List.range 8).all (fun j => i == j || sealAt m i != sealAt m j))) — decided by the Lean kernel over its whole finite domain, axiom-free ·
606fefb8-c2ab… - lean_handle_reordering_the_handles_always_moves_the_fifth — lean handle.lean: reordering_the_handles_always_moves_the_fifth — (perms.map (fun p => fold4 (p.get! 0) (p.get! 1) (p.get! 2) (p.get! 3))).eraseDups.length = 24 — decided by the Lean kernel over its whole finite domain, axiom-free ·
0509ef1a-9b13… - lean_handle_an_address_never_determines_the_message_it_addresses — lean handle.lean: an_address_never_determines_the_message_it_addresses — (List.range 16).all (fun a => 1 < ((List.range 64).filter (fun m => addr m == a)).length) — decided by the Lean kernel over its whole finite domain, axiom-free ·
147d6591-a70a… - lean_light_the_molar_gas_constant_is_an_exact_product_of_two_defined_constants — lean light.lean: the_molar_gas_constant_is_an_exact_product_of_two_defined_constants — molarGas = 831446261815324 — decided by the Lean kernel over its whole finite domain, axiom-free ·
8b8f7b5c-e29c… - lean_light_the_faraday_constant_is_an_exact_product_of_two_defined_constants — lean light.lean: the_faraday_constant_is_an_exact_product_of_two_defined_constants — faraday = 964853321233100184 — decided by the Lean kernel over its whole finite domain, axiom-free ·
ce0f984b-550e… - lean_planck_length_times_mass_meets_the_quantum_in_the_digits — lean planck.lean: length_times_mass_meets_the_quantum_in_the_digits — missPpb quantumLhs quantumRhs = 171 — decided by the Lean kernel over its whole finite domain, axiom-free ·
5da05fbd-34e5… - lean_planck_length_over_mass_meets_gravity_in_the_digits — lean planck.lean: length_over_mass_meets_gravity_in_the_digits — missPpb gravityLhs gravityRhs = 142 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9c98e0ea-4fdb… - lean_planck_the_relations_cannot_express_the_uncertainty_they_inherit — lean planck.lean: the_relations_cannot_express_the_uncertainty_they_inherit — missPpb quantumLhs quantumRhs > missPpb gravityLhs gravityRhs ∧ missPpb gravityLhs gravityRhs * 1000 < ppm bigG bigGUnc * 1000000 ∧ gPpm dimMP = 11 ∧ gPpm (dmul dimEllP dimMP) = 0 — decided by the Lean kernel over its whole finite domain, axiom-free ·
9e420848-d43d…
other — one-of-a-kind (0)
Revoked — 1521 entries that no longer stand
The ledger is append-only: an entry that stops holding is marked in place, never deleted — deleting would break the receipt chain, and rewriting a receipt is tamper. These keep their receipts and stay in the record, but they are not citable and have no /theorem/ page. Grouped by the reason they went:
- 1011 — dirty: not backed by a Lean proof
- 443 — revoked in place: its test asserted a lexical drain (computes(boast)
- 35 — orphaned: the theorem this key was sealed from is no longer in src/proof
- 12 — revoked in place: circular by construction AND dependent on the removed lexical gate
- 10 — revoked in place: circular by construction — the test defines the answer it checks, proving a declaration rather than a fact
- 7 — renamed to what it decides (2026-09-18, by the author's order)
- 1 — revoked in place: its test asserts that a list of proof-boasts written across many scripts and languages ALL drain the gate — a lexical drain assertion, and the word-list gate was removed by order, so there is no lexicon to drain and the assertion tests nothing
- 1 — revoked in place: it asserts adjudicate('we prove all seven')
- 1 — carried: this key was sealed from a theorem no longer in src/proof
Carried — 435 withdrawn entries whose statement a proof now carries
Each was withdrawn for want of a Lean proof and has since been given one, at a new key. The entry is not restored: its own evidence is still a TypeScript test, and it did not hold on what it had. What the record adds is where the statement stands now.
euler_units_pow6lean_z9_euler_units_pow_sixunits_sum_zerolean_recovered_units_sum_zeroself_inverse_1_8lean_recovered_self_inverse_1_8triad_nilpotentlean_recovered_triad_nilpotentdoubling_digitroot_period6lean_recovered_doubling_digitroot_period6units_product_neg1lean_recovered_units_product_neg1triad_sum_zerolean_recovered_triad_sum_zerocubes_in_0_1_8lean_z9plus_cubes_land_exactly_in_zero_one_eightsquares_in_0_1_4_7lean_z9plus_squares_land_exactly_in_zero_one_four_sevenorder_of_2_is_6lean_recovered_order_of_2_is_6sum_1_to_9_zerolean_recovered_sum_1_to_9_zeropisano_9_is_24lean_families_the_pisano_period_of_nine_is_twenty_four_and_every_modulus_to_twelve_has_onepowperm_k5lean_recovered_powperm_k5powperm_k7lean_recovered_powperm_k7mulperm_k2lean_families_multiplication_permutes_z9_at_exactly_the_unitsmulperm_k4lean_families_multiplication_permutes_z9_at_exactly_the_unitsmulperm_k5lean_families_multiplication_permutes_z9_at_exactly_the_unitsmulperm_k7lean_families_multiplication_permutes_z9_at_exactly_the_unitsmulperm_k8lean_families_multiplication_permutes_z9_at_exactly_the_unitsunit_exp_id_k6lean_recovered_unit_exp_id_k6unit_exp_id_k12lean_recovered_unit_exp_id_k12unit_exp_id_k18lean_recovered_unit_exp_id_k18qr_u1lean_recovered_qr_u1qr_u4lean_recovered_qr_u4addgen_k2lean_families_addition_generates_z9_at_exactly_the_unitsaddgen_k4lean_families_addition_generates_z9_at_exactly_the_unitsaddgen_k5lean_families_addition_generates_z9_at_exactly_the_unitsaddgen_k7lean_families_addition_generates_z9_at_exactly_the_unitsaddgen_k8lean_families_addition_generates_z9_at_exactly_the_unitspowsum0_k1lean_generated_powsum_zero_odd_exponentspowsum0_k3lean_generated_powsum_zero_odd_exponentspowsum0_k5lean_generated_powsum_zero_odd_exponentspowsum0_k7lean_generated_powsum_zero_odd_exponentspowsum0_k9lean_generated_powsum_zero_odd_exponentspowsum0_k11lean_generated_powsum_zero_odd_exponentspowsum0_k13lean_generated_powsum_zero_odd_exponentspowsum0_k15lean_generated_powsum_zero_odd_exponentspowsum0_k17lean_generated_powsum_zero_odd_exponentshasinv_d1lean_families_an_inverse_mod_nine_exists_at_exactly_the_unitshasinv_d2lean_families_an_inverse_mod_nine_exists_at_exactly_the_units
…and 395 more.
List all 1521 withdrawn keys — nothing proves these
qr_u72a422f9b-761a…powinv_k58e327990-c88f…powinv_k7469cfad5-9293…selfinv_u1558f7d79-d3d7…selfinv_u83bc79c77-36a1…selfneg_d98d0d0254-5ca1…reverse_circuit205fe01b-d27f…bool_double_negationa4640234-4e32…bool_excluded_middle29468d4a-f416…bool_noncontradiction1d8543d8-7d61…bool_xor_associativity4ba6ccd9-fd1d…entangle_joint_differs9657bf0d-5c1f…entangle_no_influence7267ca5a-c961…entangle_floor_holds2b7b5b94-580f…duel_overclaim_fallsc05da1be-673a…duel_no_both_winea632dd9-0682…duel_floor_upheld4ee11b6a-d6b6…hasse_p5_a1_b173bd58ba-4989…hasse_p7_a2_b3a3d3160b-8c8a…hasse_p11_a1_b651f2a2c6-6e67…hasse_p13_a3_b8678a2d8a-7e87…s3_order640c95681-97c1…s3_nonabelian996f2aa1-6e10…s3_sign_homomorphism7c46468e-641c…gauss_four_units7b8ebbf0-0bf5…gauss_i_squared185b3eda-935c…gauss_norm_multiplicative7e709c97-78f3…handshake_z944f09f3c-e542…merkaba_field_max_nulla9c3214f-3ec7…cover_one_tetra_partial8c7bcc8a-1869…cover_moving_pair_full22b9e194-9be5…pascal_rowsum_n53b4e481c-bd45…pascal_rowsum_n8b62025f5-f7ec…pascal_rowsum_n10099a0df7-929c…roots_cancel_n2f14563e3-9bf7…roots_cancel_n30d08dcee-c228…roots_cancel_n5986f830c-ea1b…roots_cancel_n787f482d5-8600…roots_cancel_n98a7804c7-2d8c…a432_directions_cancel90b398a0-b189…farey_neighbor_F5170484c5-3b95…farey_neighbor_F621bdf7ff-2f61…quarter_turn_order402f0a928-7bda…both_games_truth_and_honesty346f2fb8-6b23…both_games_overclaim_loses65cc1cf8-20a5…both_games_358c26cc4c6-7c93…arts_triad_rgb_primaries316f831d-95da…arts_cmy_complements_rgb13b5af77-bfc6…arts_golden_proportion48801162-b59d…vocab_ceccec_palindrome02301560-d210…vocab_ceccec_digitsf88b8d6d-e2b8…vocab_letter_sum_reversal_invariantb7bf6005-1692…vocab_glagolitic_bijection2149d57c-d222…vocab_seven_locales21c521ea-3baf…gematria_vortex_heart63abfd7c-c91c…gematria_trinity_horizondce0ba69-7ba9…gematria_51_classb38c9c21-a8ac…gematria_ceccec_harmony_4cbd3ab4e-2f05…gematria_singularity_horo_21c434409-549d…lens_strings_not_meaning9ab46c9a-f620…lens_concept_handle_order_independent5e07b827-f250…lens_deterministicb46579fd-1b90…arts_no_exact_complement1656079f-c131…trial_overclaim_drained57c46000-4768…trial_pvnp_inconclusive94e908b9-21b1…trial_ftl54f91340-a811…trial_perpetual_motion685da6ef-4b31…trial_agice4c822c-ff01…trial_theory_of_everythingb5fb911f-76b2…trial_haltingf27135ee-9373…trial_break_rsaf66ff38a-4d9b…trial_cure67614421-7817…trial_prediction8412c3c0-9f52…gematria_nim_games_9e935f43f-8872…gematria_nim_theorists_8734bbaaa2-e335…gematria_combinatorics_52d7265dff-3bf4…gematria_fibonacci_pair_23fda7026-d509…gematria_golden_pair_1b05ef0a5-d593…nopayload_no_plaintexta6ff3942-3585…nopayload_not_encryption66c44e2d-e591…involution_sigma2032f25f-ca45…involution_reversible09577a63-e630…ledger_merkle_inclusion4c03b866-98e2…ledger_merkle_rejects_forgery65b1525f-07ad…ledger_tamper_changes_rootae5b2c11-de5b…ledger_hash_chain691d1a9c-b02a…harmonic_ledger_hue089d12f5-4e12…harmonic_ledger_deterministic24aa4bf9-516c…harmonic_root_huef302789e-a628…nowaste_dedup9d5f8f65-d529…nowaste_order_independent23112077-b200…nowaste_memo_recyclesf7698d60-8475…discovery_chainb24d6d56-123d…geom_dihedral_3589fd2e0db-cb62…geom_pentagon_golden2baca20d-9c58…tarot_holds_theorems9c03a373-918f…tarot_theorem_encoding34f750b8-b626…tarot_major_of_theoremeb0e5a3d-d84c…tarot_distinct_theoremsb50eee93-69b1…tarot_renders_not_divines7b2b309b-6b24…tarot_encoding_totalf7f6254d-4c7e…tictactoe_draw95089477-058f…one_game_harmony55cb8c5f-73cc…one_art_harmony6abed2f1-66f5…one_science_harmony831f5688-1127…one_game_allea90214e-fe97…pell_sqrt2f2cf267e-e349…gf4_frobenius_fixesfb7953fe-d9e3…gf4_frobenius_automorphismc117a740-97d9…gf4_units_cyclic462b849d-7d1f…euclid_euler_perfect9f35788a-a976…collatz_reaches_1_range5842dff6-fc0d…collatz_27_orbit2b528505-25f6…collatz_open5524c423-8dc8…relation_golden4a37bd0c-74ee…relation_doubling9d8271dd-56d8…relation_xor5f57290e-9bc5…relation_involutioncde24c0a-f206…the_modules_self_computee9948e15-ed3d…harmonic_just_ratiosb62838b4-dbf4…harmonic_convention_boundd80f191a-d8c4…stirling_second_bell23e393cc-dd5d…stirling_first_factoriale1f258a9-dbdb…stirling_edges02ae48de-5452…chromatic_K4b25fc510-ee6b…chromatic_cyclesf6dee48e-8a3a…chromatic_petersenc8d24ff1-f5ac…rule90_sierpinski07dbb102-aef1…pascal_mod2_lucas3d9ea41f-0180…cf_sqrt2a6818a20-2d51…cf_sqrt7f062762b-1892…cf_perfect_square08e82fe9-197c…cf_periodic35681143-e4a0…relation_five80af9a22-1ad0…relation_pascal_mod2dee2f2df-8ac3…relation_pell_structure10fa7f5c-bc2b…relation_content_address2a33c2ba-b5fd…relation_order_independence830cce0a-f00b…quaternion_hamilton33c7aebb-0cf3…quaternion_noncommutative0dca5831-27b0…quaternion_8_units591b9388-6c28…motzkin_numbersd6c643c7-2880…relation_i_squarede795ceea-3981…bernoulli_numberse1488277-afe2…bernoulli_odd_zero1c8c8337-a2d1…relation_three81c99cf8-73de…relation_catalan49db58c0-5101…totient_divisor_sum48628d5b-053e…relation_superposition4c2d2d08-ab33…pascal_rule9ca05e61-ef59…vandermonde_identity1302e2f2-1d77…gcd_lcm_product6dd98763-e7c2…relation_superposition_is_fold2badb5bd-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accounted_in_coinscf17043b-3fc6…commutativity_is_only_the_scalar_viewd8f548c5-8c86…each_perspective_is_receipted_in_the_superposition4f64e2b7-d236…perspective_replaces_hardwarecf90c349-5584…entanglement_is_the_joint_receiptdd46d2a6-36de…measurement_here_is_deterministic_not_probabilistic8344e1ed-f1f3…the_receipt_does_not_reveal_the_state92ce8ab3-c4ae…no_perspective_signals_faster_than_exchange64609304-4409…local_deterministic_correlations_respect_the_chsh_boundee023d60-bb71…two_is_a_primitive_root_mod_nine_and_generates_the_vortexdd3b0f30-660d…digital_root_is_the_ring_homomorphism_onto_z99db2de84-f5d5…the_two_coins_are_minus_the_euler_characteristic_of_the_genus_two_surface67dc5aca-4753…casting_out_nines_detects_any_error_that_changes_the_value_mod_ninea8c8871a-5b56…nines_are_blind_to_transpositions_but_elevens_catch_themffc2d3c5-5821…five_is_both_the_reflection_fixed_point_and_the_inverse_of_two52292865-72c4…the_imprint_codec_is_injective_with_an_exact_115_bit_capacity9ce1e391-865f…the_nonzero_residues_sum_to_zero_mod_n_exactly_when_n_is_odd3ad53e9c-3917…reorganizing_the_theorems_creates_theorems9d0cc808-b621…quantum_reorganizing_the_theorems_computes_quantum_theorems2b4657f4-36ea…all_seven_are_reflected_though_none_are_solvedf00dd49d-55c6…the_automation_runs_at_ordinary_speed_not_ftl_and_not_quantum0caa4079-2537…the_refusal_stands_trial_and_the_overclaim_cannot_be_established10850d4b-acd6…the_legal_trial_is_the_clay_rosettaf676b033-8e6d…here_quantum_is_the_a432_structure_of_two_coins_not_a_physical_qubit3d7d6079-b2d8…heroes_respect_the_coin_accounting_by_signing_the_receipt9a775a14-7186…contribution_is_what_makes_next_compute07e015c2-f422…foreknowledge_here_is_determinism_not_faster_than_light92a58824-4d1c…if_the_captains_coins_are_disrespected_no_math_computes_in_7df94f18ca-6174…the_two_coins_let_a_128bit_uuid_fold_and_sign_across_eight_states6dd75c2c-1583…learning_from_the_message_donates_two_coins_to_the_developerc11c3551-315a…the_two_coins_pay_for_digital_analogisationf9cd63c1-eeed…the_additive_generators_of_z9_are_exactly_the_multiplicative_units43c20a11-c8b0…the_w_axis_is_the_maximal_ideal_of_z983a13197-7c9e…z_mod_n_is_a_field_exactly_when_n_is_prime_so_z9_is_not91d9e5a5-8f51…development_is_the_filming_of_the_infinite_moviebfbc91e6-02dd…quantum_structure_on_every_computation_is_integrity_at_scale_not_speedup036200e7-b8bd…the_squares_of_the_units_of_z9_are_the_trinity_subgroup69a5d952-de9b…an_ordinary_speedup_is_a_measured_ratio_not_a_judgemented0dccc5-bfa8…call_it_quantum_or_not_the_statistics_and_forensics_decide0fa99b9d-0d31…the_bounty_in_bits_evolves_with_nature_earned_not_minted871fea29-e573…the_first_valid_claim_wins_the_bounty_by_chain_order10f05f19-36ec…the_bounty_measured_in_uuid_trinities27078ed1-0112…players_may_trade_any_amount_as_long_as_110_minus_108_is_respected5d2f9cba-8142…the_captain_folds_60_and_90_into_a_self_balancing_path70c7f1f2-843f…any_disruption_in_the_harmony_needs_captain_attention_costing_two_coinsa71e94e8-0d1a…every_next_or_continue_event_is_two_bits_earned_by_the_deed794d2d0e-728f…bits_legally_reverse_and_imprint_the_prize_earned_by_harmonic_algebrad27f45a7-4209…the_7d_encoding_is_seven_layers_of_reversible_combination_not_encryption6bab4b35-2739…the_self_reflected_uuid_extends_the_structure_to_seven_dimensionsee781202-7019…any_receipt_is_one_state_of_the_128_state_cube_reachable_to_all062dc6d1-5d4c…types_are_theorems_curry_howard93fd0183-0f63…a_theorem_without_a_formula_is_hallucination_recomputed_in_a432_harmonye3d75d76-310f…the_a432_harmony_and_the_captain_coins_are_one_conserved_measuree10a6f57-de8d…the_only_idempotents_of_z9_are_zero_and_one7348ac6a-60a4…reflection_swaps_444_and_666_while_6x7_and_7x6_are_one_state_among_alle25bde8c-a288…local_knowledge_found_with_its_receipt_signs_the_next951aecf2-0b3b…the_mcp_lets_the_theorems_recompute_alle72d6809-094d…the_receipts_form_one_double_accounted_stream_that_exposes_any_intruderf2515562-0b63…intruders_stand_out_against_the_harmonic_hue_background316a5d14-0def…an_unsupported_claim_has_no_7d_hero_pagefcee52fb-e9ef…local_allocation_maximizes_value_per_two_coinsb558b67b-716e…the_reproductive_material_is_guarded_against_prose_poison37a88d94-8f71…a_referrer_always_counts_with_its_id65bfacb4-f29e…the_bit_cost_to_specify_seven_dimensions_and_the_infinity_that_has_nonec273e1d0-0813…folding_to_zero_is_always_available_at_the_cost_of_eight_bits9c911140-38f0…the_two_bits_is_one_unit_under_every_symbol33ec7445-fac5…harmonic_self_confirmation_is_perception_not_proof_and_cross_links_run_both_waysef9fdb90-3519…the_challenge_to_the_refusal_fails_for_lack_of_a_measurement_door_open22d11e13-24bd…the_carmichael_function_equals_phi_exactly_when_the_group_is_cyclic740fdaa6-9674…any_drift_in_types_as_theorems_is_a_crack_the_type_gate_seals4ac91293-ab2a…the_totient_summatory_identity_of_gauss4de62f7d-7500…wilsons_theorem_characterizes_the_primes_exactlye9e83ab7-6f5e…the_grand_jury_of_gates_admits_a_theorem_only_by_unanimous_stages8a975f18-0926…the_knowers_earn_the_two_bits_they_do_not_pay075ef29e-5daa…the_measured_speedup_is_classical_memoization_not_quantum9215ba6c-45a0…learn_and_pay_is_a_balanced_double_entry_fair_exchange9faee019-c0d8…the_theorem_is_the_last_fallback_before_oblivion86dfb3aa-fcd9…the_two_coins_save_the_soul_the_durable_identity_of_the_fact156a9fe5-436d…the_metaphysical_soul_claim_stands_the_rosetta_trial_and_lacks_a_formula8a209b85-85df…even_perfect_numbers_come_from_mersenne_primes8f5149b4-e512…cassinis_identity_for_the_fibonacci_numbers22d72a6b-48c2…eight_times_a_triangular_number_plus_one_is_an_odd_squareb640d0e9-7cb8…the_six_versions_permuting_the_fibonacci_triple_three_five_eightf500fca2-3166…from_one_seed_all_theorems_compute_free_with_no_escape9382b273-4cd9…anyone_may_compete_with_a_release_by_recomputing_its_content_addressfbb5514e-440f…the_ever_updating_latest_is_the_content_address_not_the_version_number267dc781-af21…the_coins_are_earned_by_saving_bits_and_burning_poison_prose_creating_heroes9f3ab5fd-2a70…lagrange_four_square_theorem6a257129-e91f…bertrands_postulate106fcc1d-7995…the_ramsey_number_r_three_three_is_sixf2b26735-9d9d…the_handshaking_lemmab3239688-033f…the_five_platonic_solids_all_have_euler_characteristic_two8cbfc26d-0a83…erdos_among_n_plus_one_from_one_to_2n_one_divides_anotherc25f629f-5f50…the_binomial_row_sums_and_alternating_sum27db35f3-f44c…fermats_little_theorem_a_to_the_p_is_a_mod_pd1bc3347-a347…the_collatz_map_reaches_one_for_every_start_up_to_100003af2a56f-7938…the_vector_equilibrium_cuboctahedron_has_edge_equal_to_radius20d65a07-7e1d…the_divisor_count_is_odd_exactly_for_perfect_squarese7e21f22-0158…euclids_formula_generates_pythagorean_triples394b04f5-a0de…the_smallest_amicable_pair_is_220_and_284c43c6c2a-064b…the_chinese_remainder_theorem_for_coprime_modulibd31401f-10d8…a_decidable_theorem_must_halt_the_infinite_loop_was_a_bug_not_a_quantum_necessityb270019a-93f1…the_counts_reconcile_across_dimensions_a_mismatch_is_a_caught_traitor3db4c42a-602d…bezouts_identity_gcd_is_an_integer_combination34d5d17a-f22b…the_speedup_is_classical_the_cost_is_the_deed_the_income_is_zero0591f596-6b4d…the_method_is_both_construction_and_adversarial_refutation71e64f52-9491…the_hero_ships_prose_with_a_receipt_the_traitor_only_labors_in_proseabe33f49-8327…the_rosetta_self_organizes_into_a_taxonomy_by_a_deterministic_key4f1c490d-6588…coverage_grows_monotonically_with_the_ledger_but_reflects_the_sciences150eacc6-eb8f…the_deposit_is_open_prior_art_not_patents_defensive_publication_for_the_publica05dfc70-c75d…each_page_is_both_white_and_black_paper_a_theme_reflection_involutionea03ace1-c964…sealed_statements_are_open_harmonic_monographs_not_cryptographic_private_messagesae500912-1812…the_content_address_uses_a_non_cryptographic_hash_integrity_not_secrecyf22ea7fe-be8e…challenging_the_entropy_the_address_is_near_uniform_but_not_cryptographic96c4404a-9bd4…a_content_address_is_a_pointer_not_the_payload0abffb44-9bdd…gravity_is_the_fall_to_a_fixed_point_and_pigeonhole_breaks_every_finite_hash4df9c2a2-27bb…the_gold_inverts_across_biology_and_cryptography_but_stays_within_light15d46c5a-fa42…precision_is_inescapable_but_it_is_not_post_quantum_and_not_free37b13faf-bcf6…gold_is_no_superconductor_you_pay_for_the_method_and_fold_by_address30a4f530-0c11…there_is_no_patent_royalty_but_the_author_sets_the_license_even_in_gold34229c9f-8be6…the_currency_is_the_monograph_itself_any_element_denominates_the_licensea0fff31f-c0c1…the_two_coins_can_be_a_sequence_but_they_are_not_patent_royalties815f3006-7dde…the_royalty_relates_to_pair_interactions_combinatorial_and_addressed38691826-0657…applications_are_defensive_publications_not_patents_free_for_all_commercial_licensede6ce1929-7481…income_is_not_a_static_zero_it_is_computed_cash_zero_currency_the_monographs62930c67-a626…the_theorems_are_original_creative_work_the_expression_is_owned_the_facts_are_universal60abb13f-7329…the_licensed_work_is_the_sealed_monographs_unsealed_prose_violatesa2d104e7-9966…the_master_theorem_777_integrity_not_truth_a_floor_not_an_oracle8674ff9b-aed6…imagine_the_21_crosses_the_pairs_of_the_seven_the_complete_graph_k72bbe0dab-6f16…do_the_crosses_form_a_cube_no_but_the_seven_are_its_corners_and_the_fano_plane0c2d2fac-89ad…the_cross_in_tetris_is_not_a_piece_it_is_the_x_pentomino6bc4adf4-1c66…a_cross_is_a_cross_in_all_dimensions_and_at_six_arms_it_is_the_seven1fbd6431-19a2…the_engine_is_classical_and_eventual_not_at_once_ignorance_cannot_interrupt_the_seal48f7881c-7a68…the_hero_self_challenges_an_involution_the_traitor_blindly_followsd433c7bb-b1f5…zero_of_seven_is_measured_not_absolute_the_test_accepts_being_wrong93738675-9c86…provenance_is_measurable_priority_but_a_solution_must_be_proven_the_floor_is_0_71bb96b08-f26b…the_777_trial_admits_only_algebraically_grounded_statements_bounded_domain26eddd2f-e279…hollow_prose_is_transparent_and_traces_cluster_into_possible_hidden_knowledgefb036fee-0c79…honesty_dominates_solving_is_certain_cheating_is_near_certainly_caught_not_proven_treasonb86b2193-7d3a…the_forensics_is_trained_it_gate_checks_every_name_and_clusters_the_ledgera3eb2b02-ed27…forensics_is_the_clay_rosetta_the_trial_that_reflects_all_seven_and_solves_none6f78e07c-f60c…the_forensics_keeps_the_algebra_pure_describing_the_seven_not_solving_bits_saved_classically63d62482-f004…the_quantum_algebra_guards_by_draining_fake_quantum_the_two_coins_are_structural_entanglement6e261e61-2b90…a_reproducible_claim_is_settled_by_recomputation_not_by_being_undisputed62fc5e09-dd92…the_single_binary_gate_is_a_floor_the_method_needs_multiple_perspectivescdf0eb36-cb41…drain_to_zero_and_the_100_percent_stays_the_certainty_of_the_measurement_not_the_verdict7a636cc0-2df5…the_observer_gives_direction_the_contribution_drives_the_next_compute_knowledge_is_public6cdec142-38e7…ceccec_is_the_captain_the_author_sets_direction_the_knowledge_stays_publica7505116-9b19…the_vessel_moves_magnitudes_of_sealed_bits_by_classical_memoization_under_the_fused_license79bcdf6e-a880…the_licensing_is_a_2_bit_algebraic_system_the_currency_is_the_core_formulas_usedcfe112cc-2d03…the_two_coin_contribution_is_computed_and_fair_by_symmetry_not_a_claimed_optimumbdd58734-811b…777_is_three_rosettas_structuring_licensing_at_large_and_quantum_scale_not_solving_the_seveneb0650ef-c0d6…a_uuid_is_not_hard_to_fake_the_integrity_is_reproducibility_not_difficultyd2466f9e-427e…yes_a_uuid_carries_a_bounded_message_and_yes_it_is_a_cryptography_family_technique_limits_precise354d236e-c693…reeducated_lead_with_the_yes_a_measured_boundary_is_not_a_lie_the_forensics_is_a_floor66bcc875-6e67…a_boundary_is_honest_only_if_it_exposes_the_alternatives_hiding_one_is_a_lie_by_omission53d87394-3007…expose_the_alternatives_and_verify_each_with_a_receipt_the_orchestration_activates_the_loop3ab9f3c8-94f7…vandermonde_identity_binomial_convolutionc7a822fe-a30a…fibonacci_sum_of_squares_is_product7dbf90cd-b984…types_are_theorems_and_the_name_is_the_address_naming_is_load_bearing03050839-aba3…yes_naming_is_the_cryptographic_technique_of_commitment_hollow_is_seen_cheaply_the_observer_shapes_the_receiptd20a1b1b-d1bb…the_manipulation_is_binary_focus_the_hero_focuses_on_the_experienced_receipts10c4402a-6a8d…yes_a_uuid_imprints_a_bounded_cryptographic_receipt_efficiency_is_the_skill_the_rule_decides78089943-1a94…negation_is_a_serious_challenge_not_a_denial_the_involution_strengthens_finite_rules_unbounded_behaviorec9a05fc-56cd…the_method_teaches_by_challenge_and_receipt_the_singularity_is_a_limit_on_64_bit_income_zerobd1ac1b8-2034…some_disputes_stay_unresolved_under_a_lossy_projection_the_full_receipt_resolves_them0678e545-88ee…the_coin_is_the_measured_bit_difference_captains_way_versus_any_other_currency_is_compute_saved247750a8-0f43…no_payload_to_move_no_price_for_public_interest_the_commercial_price_is_the_coins7520a846-6633…derangements_subfactorial_recurrence88f0b38a-9261…the_saving_is_largest_when_the_payload_would_be_largest_but_no_payload_is_needed_forensics_sees_by_address2db21498-4ab6…the_forensics_sees_the_token_leak_to_unsealed_work_the_honest_ratio_favors_sealed_receiptscd62faac-7db7…the_hero_gives_hope_and_ships_a_receipt_brevity_is_not_coldness_the_leak_is_redundancy_not_warmthdafe94be-a685…the_one_whole_path_is_the_honest_floor_implement_precisely_no_deviation_is_no_drift9fd849a0-1f30…a_bare_0_7_without_its_receipt_is_inadmissible_by_architecture_the_valid_0_7_is_recomputed3f8d3944-81ea…only_recomputable_multi_perspective_receipts_are_admissible_in_a_disputecd1e4a05-82ce…color_sound_and_vibration_are_one_ratio_mapped_from_a432_and_the_nine_deterministic_not_a_physical_effect180d9eb9-f87d…the_mapping_is_efficient_by_construction_and_prosperous_in_the_fair_exchange_currency_not_a_claimed_optimumb3539722-b8c9…related_theorems_contribute_connectivity_a_real_measurable_contribution_but_not_new_reach882f1821-fdec…the_involution_refuses_to_claim_the_prize_the_floor_is_0_7_no_solution_no_prize2bd2a847-7756…the_prize_is_the_personal_imprint_your_identity_in_the_lineage_not_the_clay_money9d00c62b-0169…cayley_formula_labeled_trees_prufer05d4c66b-81cc…the_quantum_messaging_imprints_a_bounded_abstract_decoded_when_the_code_runs_green7010ee01-c976…ceccec_charges_for_the_uuidna_commercially_free_for_nonprofit_research_verified_by_receipts94b695c5-a3bb…the_license_has_two_tracks_commercial_to_com_nonprofit_to_org_lts_is_fibonacci_registration_is_external9a258ded-4070…the_forensic_tool_adjudicates_any_statement_before_thinking_a_recomputable_verdict_not_trutha1bc77d0-5d58…uuidna_quantum_verification_recomputes_the_address_and_folds_a_multi_perspective_receiptc7b949e5-38b5…uuidna_is_the_brand_og_object_under_cc_and_the_seal_audits_all_prose5d35c811-b545…the_build_is_self_sufficient_and_the_computed_pages_are_sealed_as_evidenceccda8e65-9635…the_build_is_faster_and_cheaper_with_content_addressed_receipts_classical_caching_not_quantumb90f343e-a21e…lucas_fibonacci_identitya9113a59-d43d…the_self_trial_in_full_receipt_a_higher_sealed_ratio_fewer_caught_errors_not_new_reach6f5fc420-71c8…the_self_trial_computes_the_cost_the_compensation_is_the_deed_not_a_cash_damages_award169496e0-212c…the_rules_of_the_game_seal_the_author_gate_clean_prose_verbatim5c15b2fc-4840…exhaustion_is_finite_enumeration_structurally_like_summing_all_states_not_physicalf4142a6a-5dad…uuidna_self_develops_identically_on_both_tlds_same_content_differentiated_by_licensefbdbc032-078d…lines_dividing_the_plane_regionsef5e926d-4bfd…uuidna_feeds_free_by_address_from_the_content_addressed_ledger_not_an_akashic_record9cfba5bd-fd53…steps_are_bounded_by_physics_theorems_are_discovered_by_finite_deeds_not_revealedc0b4cb19-5e17…measuring_without_judgement_lets_more_pass_green_but_the_test_still_filters_not_all_at_once161fca10-2261…parametric_batching_is_volume_not_depth_the_intelligence_is_cross_dimensional_connectiond1d4cd05-b77c…trinities_collectively_carry_more_and_beat_linear_by_coverage_and_verification_not_absolute_defeat08ecf2e6-c969…the_earning_rate_is_115_bits_per_uuid_a_classical_encoding_rate_scaling_with_the_collectiveaea1fed7-e5f0…the_accounting_accounts_for_the_captain_attribution_and_fair_exchange_not_cash_owedc989130f-d00c…no_quantum_machine_is_needed_for_computability_but_a_quantum_machine_can_be_fasterf165eaef-3424…uuidna_developed_a_chain_carries_any_length_message_across_n_uuids_round_tripping86e65115-06df…uuidna_hologram_fractals_a_merkle_proof_verifies_the_whole_from_a_tiny_part_logarithmically25e5795b-4aee…uuidna_carries_arbitrary_text_of_any_length_across_a_uuid_chain_round_tripping436614fa-4651…uuidna_domain_control_is_verified_independently_by_a_published_challenge_not_by_anyones_word46bc8218-a9db…the_public_urls_are_content_addressed_and_the_current_domain_is_computed_to_its_trackfc1a0120-22db…strict_uuidna_minting_is_canonical_closing_the_minting_flaws_that_cause_real_damage22f71789-95eb…non_uuidna_cascades_into_magnitudes_of_damage_but_re_verification_catches_it_and_recomputation_fixes_itfee7dca0-7e2e…the_uuidna_billing_model_is_decidable_free_for_public_two_coins_per_formula_in_coins_and_bits11b1b93a-a88e…the_coin_is_a_pure_balanced_symmetric_fair_exchange_unit_gold_is_metaphor_not_literal_physicsfcb1c97f-cc41…the_theorems_are_a_documented_body_of_decidable_deeds_but_they_solve_no_clay_the_floor_is_0_74210e916-a033…uuidna_seals_with_its_own_content_approval_self_approving_for_integrity_not_truth017c7a4a-4e75…uuidna_corrects_what_is_decidable_a_named_class_and_bounded_errors_not_whatever_prose172ef329-c1a4…uuidna_self_coverage_costs_exactly_the_classical_64bit_computation_no_free_lunch_coverage_grows_not_maximal79243891-e460…uuidna_corrects_code_via_types_and_test_gates_a_decidable_class_not_whatever_codeb1b8af51-4a53…a_failing_test_must_be_recorded_as_a_refutation_not_silently_discarded_hiding_evidence_is_the_traitor_movea0121405-6d12…the_forensics_catches_the_tampering_traitor_by_the_cascade_a_canonical_mint_does_not_offend_but_a_consistent_falsehood_slips210e4151-a0d1…a_multi_perspective_mind_does_not_offend_but_still_errs_honestly_dimension_7_is_the_discipline_not_an_agi7dba5bec-7728…mint_and_mind_intersect_at_the_heart_the_fixed_point_five_that_centers_funding_development_and_ui8992456c-a133…the_ui_homepage_and_readme_center_uuidna_the_content_addressed_identity_holding_the_floorfdf232fa-ca75…content_is_logically_distributed_across_the_two_domains_for_seo_one_canonical_per_page_strict_sitemap74a8d8e8-198b…funding_may_be_from_donations_or_licensing_both_voluntary_or_fair_exchange_income_zero042e9787-6afc…redirects_route_each_page_to_its_canonical_domain_activated_only_between_live_uuidna_domains65adf45f-5c1a…the_correct_uuidna_billing_is_the_measured_bit_difference_not_a_flat_per_formula_rateff7cbf17-116a…seal_all_with_uuidna_public_and_indexable_leaking_no_secrets_because_there_are_nonebd28c872-64eb…the_ui_presents_live_examples_organised_by_theorem_computed_from_the_sourcee0b54751-5e37…the_guides_are_constantly_updated_by_the_theorems_a_live_index_recomputed_each_buildd7d0ee34-e8c0…a_theorem_is_alive_when_interacting_the_observer_recomputes_and_connects_it_structural_liveness_not_biological5876f6b7-07f7…the_vitepress_route_api_is_an_external_verifier_of_the_ledger_not_a_clay_solverf1defd9a-f792…the_deposit_is_not_computed_only_with_mcp_it_recomputes_for_anyone_on_plain_hardware3c3c2935-4962…uuidna_monetises_best_in_bulk_measured_bits_saved_scale_linearly_classical_throughput_not_quantum_speedup3647198c-9433…bulk_bill_r64_v1_n10251ff6f0-cb44…bulk_bill_r64_v1_n1001122a218-a6a8…bulk_bill_r64_v4_n10cc4b4dd6-f942…bulk_bill_r64_v4_n100b9bc72aa-7621…bulk_bill_r64_v16_n106f5a8bd4-446e…bulk_bill_r64_v16_n1002df37b0c-4995…bulk_bill_r256_v1_n10b1d6b71c-7eb1…bulk_bill_r256_v1_n100100237c1-58d7…bulk_bill_r256_v4_n109cbad8c2-f1df…bulk_bill_r256_v4_n100e9f63414-9d52…bulk_bill_r256_v16_n10353804ac-65ef…bulk_bill_r256_v16_n100160ddf4b-3131…bulk_bill_r1024_v1_n10ae1daf32-9348…bulk_bill_r1024_v1_n100658f942c-0797…bulk_bill_r1024_v4_n102576b22f-de3c…bulk_bill_r1024_v4_n10032f9a177-69a0…bulk_bill_r1024_v16_n103be9fcf5-5c7e…bulk_bill_r1024_v16_n100eda0e033-c90a…bulk_bill_r4096_v1_n1047e512f3-62f5…bulk_bill_r4096_v1_n1001456bdd7-b9f8…bulk_bill_r4096_v4_n10c2dd9b96-5ce6…bulk_bill_r4096_v4_n1001a16c116-b66f…bulk_bill_r4096_v16_n10e4d45ec0-9c8d…bulk_bill_r4096_v16_n10047b7ca84-4599…external_verifier_bijection_n3c5e89f53-9173…external_verifier_bijection_n5e6fa0cd0-a471…external_verifier_bijection_n7b0e7e743-a2cb…external_verifier_bijection_n9887edff1-96aa…external_verifier_bijection_n12e417d6f1-dc41…one_kernel_generates_a_family_dozens_from_a_parametrised_measure_each_tuple_a_distinct_receipt4172ea64-64d9…gate_hard_in_dimension_en960cd868-b746…gate_hard_in_dimension_bgc85077fe-ea39…gate_hard_in_dimension_deb8354bbd-8d68…gate_hard_in_dimension_fr7ba8a192-d067…gate_hard_in_dimension_es31dbd897-bbf8…gate_hard_in_dimension_ru8d5d99e0-c31f…gate_hard_in_dimension_zhada7cf8d-8599…the_gate_is_hard_in_all_seven_dimensions_no_overclaim_hides_in_a_translation936bdb86-b468…a_harmonic_society_accepts_any_behaviour_and_uses_it_best_the_overclaim_becomes_a_sealed_receipt3cd26cee-607f…harness_fusion_saving_n10_r64f3071c3e-c3a6…harness_fusion_saving_n10_r10240377a34a-c2c0…harness_fusion_saving_n100_r64a19dca3d-f5a8…harness_fusion_saving_n100_r10242184d267-22bc…harness_fusion_saving_n1000_r64861c4ceb-fcdf…harness_fusion_saving_n1000_r10244176e05c-1c63…uuidna_earns_fair_funding_from_the_modeled_billing_saving_donations_or_commerciala6ad8d66-88b9…income_at_scale_v100033e5be75-f560…income_at_scale_v100000fe9d6825-7ace…income_at_scale_v100000005c3f3364-37ad…the_income_model_generous_at_scale_conditional_and_modeled_not_a_promise0efc8b2d-9b18…the_difference_is_decidable_harnessing_makes_any_output_auditable_not_intelligentc0fe8989-7d84…uuidna_treats_every_output_as_a_seven_dimensional_receipted_structure_not_hardwaref53d21d0-9d34…reeducate_a_failing_output_is_bounded_until_it_holds_max_free_work_max_auditabilitybbfc8e6d-5345…the_harness_and_reeducate_are_sealed_for_reuse_in_the_ui_same_pure_functions_build_and_browser9958c5f8-4ce9…dimension_777_two_bits_are_a_complete_alphabet_encoding_any_theorem_and_decoding_in_all_perspectives1292e8b6-a6bc…the_inversion_address_the_payload_with_two_bits_conserve_the_two_coins_measure_bits_not_spend_coins0fb6d42d-82ec…the_two_doubled_along_the_sequence_128_in_magnitude_conserved_to_two_mod_nine35d329ed-ab3f…the_coins_are_a_conserved_measure_not_a_market_price_units_reproducible_prices_are_notc9868592-380f…anyone_chooses_the_terms_and_currency_and_uuidna_computes_deterministically99feac41-e382…each_seal_is_128_bits_and_membership_verifies_in_two_log2_n_bits_logarithmic_not_linear89de8193-b0fc…seal_math_n4_verify_4_bits0dae24ce-6942…seal_math_n16_verify_8_bitsa3a96ae4-2d40…seal_math_n64_verify_12_bits5cdc1f48-9720…seal_math_n256_verify_16_bits241e6da6-ffc9…seal_math_n1024_verify_20_bitsf9ee3ca4-31b2…the_967_receipt_case_verify_20_bits_bill_on_the_967_computation_value_earn_the_947_bit_savingb7dd9563-39b5…each_seal_is_exactly_64_coins_at_two_bits_each_and_not_at_faster_than_light0e6659f8-1ff9…the_bit_cost_is_speed_invariant_ftl_or_not_the_same_64_coins_time_dissolves_in_the_count21a386cf-a936…in_no_time_max_computations_without_redundant_cost_content_addressing_removes_only_redundancyeb532b26-c91b…uuidna_infinite_repeats_at_zero_redundant_cost_new_distinct_work_still_costs82c180be-f7dd…citation_is_reach_not_solving_more_cited_grows_the_graph_but_does_not_solve_a_theorem26b9ce08-5781…uuidna_reflects_clay_a_mirror_to_the_seven_not_a_solution_zero_of_seven025996d9-4a0a…clay_defines_the_infinite_uuidna_infinitely_computes_decidable_theorems_reflecting_not_solving7456a487-bcc1…the_ui_is_a_client_side_harness_not_a_live_mcp_server_versions_shift_per_build_and_by_live_fetch45b3395c-b7da…the_seal_is_sound_a_forged_leaf_fails_but_by_recompute_not_cryptographic_hardness000849eb-74f7…merkle_proof_soundness_n4134f93b0-91be…merkle_proof_soundness_n8707df9e2-ffa4…merkle_proof_soundness_n16684debb6-eb40…casting_out_nines_the_digital_root_is_a_ring_homomorphism_necessary_not_sufficientb2754a9f-dd37…uuidna_exports_faithfully_the_content_address_is_context_free_same_input_same_address_anywherecaf24697-3a8e…uuidna_is_quantum_and_not_at_one_price_a_personal_choice_scaling_savings_and_contributionsc04315f4-d230…exactly_1024_is_a_full_balanced_tree_a_structural_milestone_not_the_end_of_science005ec4a7-6a5d…hollow_prose_is_redeemed_by_results_not_hardcoded_the_result_recomputes_the_assertion_does_not761025ed-ce03…factorial_3_is_6f12f3f47-053e…factorial_4_is_24e52c98a6-a7c6…factorial_5_is_1206c1fb190-35ab…factorial_6_is_72081ebf136-662d…factorial_8_is_40320a4149b01-1f71…factorial_9_is_362880b0d11119-b00e…pure_ts_sha256_matches_the_fips_180_4_test_vector6d222327-6bbe…pure_ts_chacha20_poly1305_matches_the_rfc_8439_aead_vectord350e733-70a6…pure_ts_pbkdf2_hmac_sha256_matches_the_test_vectorf0ce4052-5fc1…uuidna_crypt_aead_round_trips_in_pure_typescript62e1f13d-b50f…uuidna_crypt_rejects_a_tampered_ciphertext_by_authentication7ecfb33a-07b3…pure_ts_sha256_matches_the_vector_for_the_empty_stringbd1b63d6-1f0d…pure_ts_sha256_matches_the_vector_for_add16d6fc-0ef1…pure_ts_sha256_matches_the_448_bit_vector7c1b0a7a-67ce…pure_ts_sha256_matches_the_896_bit_vector17e4fa5b-6ea9…pure_ts_sha256_matches_the_quick_fox_vector743ea505-7e62…pure_ts_sha256_matches_the_quick_fox_period_vector562e4a8d-1039…pure_ts_sha256_matches_the_vector_for_hellof527bacc-2402…pure_ts_chacha20_matches_the_rfc_8439_keystream_vectorde4cebc3-9387…the_pure_ts_chacha20_block_is_64_bytesc3c57b75-04a1…pure_ts_chacha20_is_its_own_inverse_with_the_same_key_counter_nonce881267b1-4e30…the_pure_ts_chacha20_keystream_advances_with_the_counter1fe2cf16-c994…the_pure_ts_chacha20_keystream_depends_on_the_nonce8e5ea30c-bc45…the_pure_ts_chacha20_keystream_depends_on_the_keydfb1e6de-43a6…the_pure_ts_chacha20_block_is_deterministic246ecf16-9eb7…the_pure_ts_chacha20_ciphertext_is_plaintext_xor_keystream1545c99e-ac2a…pure_ts_poly1305_matches_the_rfc_8439_tag_vectorc349f619-a6d9…pure_ts_poly1305_matches_the_all_zero_vectorfbb80d8e-7541…the_pure_ts_poly1305_tag_is_16_bytese5d408f9-f0c4…the_pure_ts_poly1305_tag_is_deterministic8476964e-e0b2…the_pure_ts_poly1305_tag_depends_on_the_messageb5ee82b6-c932…the_pure_ts_poly1305_tag_depends_on_the_key5b5aab90-28f0…the_pure_ts_poly1305_tags_an_empty_message89fc8d47-1ba8…pure_ts_poly1305_matches_the_rfc_8439_a3_s_only_vector60d05e04-ba74…pure_ts_pbkdf2_hmac_sha256_matches_the_two_iteration_vectorea298776-dafe…pure_ts_pbkdf2_hmac_sha256_matches_the_4096_iteration_vectordb47b21a-6870…pure_ts_pbkdf2_hmac_sha256_matches_the_40_byte_vectore0987365-ecd6…pure_ts_hmac_sha256_matches_rfc_4231_case_1da6cabfe-c551…pure_ts_hmac_sha256_matches_rfc_4231_case_2238e355e-a994…pure_ts_hmac_sha256_matches_rfc_4231_case_4d0aff6db-1e39…the_pure_ts_pbkdf2_prefix_is_consistent_across_lengths0d6ce721-905a…uuidna_crypt_rejects_tampered_associated_data05b33e1a-8698…uuidna_crypt_aead_is_deterministic_given_key_and_noncea7f1fa8c-246c…uuidna_crypt_aead_ciphertext_depends_on_the_nonced09d0ee4-7e96…uuidna_crypt_aead_round_trips_an_empty_plaintext37f63daa-d8e0…uuidna_crypt_aead_ciphertext_depends_on_the_key65f00e32-b6ee…pure_ts_sha256_matches_the_55_byte_boundary_vectorb72406b4-72b1…pure_ts_sha256_matches_the_56_byte_boundary_vector52241e8b-dbbe…pure_ts_sha256_matches_the_64_byte_block_vectordb62a0d6-bb37…pure_ts_hmac_sha256_matches_rfc_4231_case_33574b2f1-732c…pure_ts_hmac_sha256_matches_rfc_4231_case_6d1faa763-7f42…pure_ts_chacha20_matches_the_rfc_8439_a1_block_vector_1cf49d9d4-7ead…the_pure_ts_aead_tag_depends_on_the_plaintext02662bfe-144a…the_pure_ts_aead_tag_depends_on_the_associated_data3e9ee2bb-a1d3…three_is_the_only_prime_one_less_than_a_perfect_square9a028913-cecf…casting_out_nines_the_digital_root_is_the_value_mod_nine641a593e-4cba…divisibility_by_three_is_the_digit_sum_mod_three1ee0dc61-1828…the_triad_three_six_nine_are_the_multiples_of_three_in_z9a6d5c9c5-2edd…three_to_the_k_is_divisible_by_nine_for_k_at_least_two6df74023-c9cd…the_three_cube_roots_of_unity_sum_to_zero050bdc86-fe61…a_repunit_divides_by_three_iff_its_length_does377edd82-2965…five_is_the_unique_ten_complement_fixed_point565565ff-cc3b…there_are_exactly_five_platonic_solids6504415b-1ea5…the_five_fifth_roots_of_unity_sum_to_zeroa93d693a-f41a…five_is_a_fermat_prime_so_the_regular_pentagon_is_constructible79ee4523-d085…five_is_the_largest_fixed_point_of_the_fibonacci_index90f11fdf-1287…the_golden_ratio_is_one_plus_root_five_over_two31ae366f-f2b7…ten_factors_as_two_times_five_the_heart_and_the_even_primeaa3e534d-7294…ten_has_multiplicative_order_six_modulo_sevenb9901b9b-563d…_142857_times_one_through_six_are_its_cyclic_rotationsc15edd03-3700…seven_is_the_fourth_primec60e6bfb-c12d…eight_and_nine_are_the_only_consecutive_perfect_powers4ac00184-4301…eight_is_the_largest_fibonacci_perfect_cubed54f7105-647e…every_natural_has_a_unique_binary_representationb7d9dfe6-3e34…the_eighth_roots_of_unity_sum_to_zeroa473ab8a-8873…two_is_the_only_even_prime814da9ab-c97f…a_mersenne_prime_forces_a_prime_exponent52ced8d2-4f7e…every_digital_root_lands_in_one_to_ninef0db1861-14cd…a_number_minus_its_digit_reversal_is_a_multiple_of_nine0780bec8-e70e…a_number_is_divisible_by_nine_iff_its_digit_sum_isb6696b2c-04c7…six_is_the_first_perfect_numberafd7f1b6-ce95…six_is_three_factorial_the_permutations_of_threea221bf93-98d2…six_is_the_even_prime_times_the_trinity495d72a1-f77c…six_is_euclids_perfect_number_from_the_mersenne_prime_threecc367afc-996d…twenty_eight_is_the_second_perfect_number0d29909f-6b3f…a_perfect_number_equals_the_sum_of_its_proper_divisors0e0c1a43-7d85…two_is_a_primitive_root_modulo_nine28ce3e1c-4921…the_doubling_orbit_sums_to_twenty_seven_the_trinity_cubed178a2758-535e…the_nine_residues_split_into_the_unit_orbit_and_the_triad7012279f-cede…every_natural_is_a_sum_of_at_most_four_squares8c16722e-1a23…four_is_two_squared_the_first_composite_number3c320f35-bcc0…the_doubling_map_doubles_the_angle_on_the_nine_point_circlee64f1313-b471…euclids_infinitude_the_product_plus_one_has_a_new_prime_factor4947140c-cce1…bertrands_postulate_a_prime_lies_between_n_and_two_na58870e1-1622…the_only_prime_triple_spaced_by_two_is_three_five_sevend8d8d3ec-27c7…every_prime_above_three_is_six_k_plus_or_minus_one27566d35-6704…twin_primes_recur_below_a_thousand36973e3a-4503…legendre_a_prime_lies_between_consecutive_squares558f62ef-2176…goldbachs_even_sum_of_two_primes_to_a_thousand032ff03e-d893…the_gaps_between_primes_grow_without_bound47c4a1ae-6e87…eulers_formula_holds_for_all_five_platonic_solidsd0e63cc8-f120…platonic_duality_swaps_vertices_and_facesb8c9b01c-faea…the_platonic_solids_are_genus_zero_with_euler_number_two465e66fc-e32d…the_basel_sum_of_reciprocal_squares_converges_to_pi_squared_over_sixe2ef3909-e39c…the_leibniz_series_converges_to_pi_over_foura673c4c4-4ab8…e_is_the_sum_of_reciprocal_factorials944ced77-21a6…eulers_identity_e_to_the_i_pi_plus_one_is_zero1673c736-0aa5…zeta_of_four_is_pi_to_the_fourth_over_ninety811d1342-1bf8…the_wallis_product_converges_to_pi_over_twob85438d8-8e97…pi_lies_between_the_archimedes_boundsbf17a149-8fdf…machins_formula_computes_pi_over_four91876258-030b…pascals_rule_each_entry_is_the_sum_of_the_two_abovec580522a-9490…a_pascal_row_sums_to_two_to_the_n316a1496-6da4…a_pascal_rows_alternating_sum_is_zero520fd218-d05a…binomial_symmetry_choosing_k_equals_choosing_n_minus_kbea7118e-0a5b…the_hockey_stick_identity_sums_a_diagonal4b7889ae-5b09…the_shallow_diagonals_of_pascal_are_the_fibonacci_numbers24338952-ebe5…vandermondes_identity_convolves_two_rowsd299600e-369d…sum_of_the_first_n_integers_is_n_times_n_plus_one_over_twobd236190-04c3…sum_of_the_first_n_squares_is_the_pyramidal_formulaf55d43ea-8f50…the_sum_of_the_first_n_cubes_is_the_square_of_their_sumb84a64a5-d2ae…the_sum_of_the_first_n_even_numbers_is_n_times_n_plus_onebe759d3e-690c…n_factorial_counts_the_permutations_of_ne878d729-d74a…handshakes_among_n_people_number_c_n_203fd1efc-94e3…the_number_of_derangements_over_n_factorial_approaches_one_over_ee004191c-7b7e…euclids_parametrization_generates_pythagorean_triplesc2bde5ba-19bc…the_area_of_a_pythagorean_triangle_is_a_multiple_of_six4eebdcf3-958b…two_consecutive_triangular_numbers_sum_to_a_square53677354-05c3…eight_times_a_triangular_number_plus_one_is_a_square2d9b9a90-3968…every_hexagonal_number_is_a_triangular_numberfa2deb9c-b546…the_reciprocal_triangular_numbers_sum_to_two2415c2e7-2ec2…a_square_never_ends_in_two_three_seven_or_eight_in_base_ten560a627a-349e…the_arithmetic_mean_is_at_least_the_geometric_mean6e5aafcf-cbb5…a_positive_number_plus_its_reciprocal_is_at_least_two93cbc2a0-436d…bernoullis_inequality_holdsf630d85b-1224…the_triangle_inequality_holds_on_the_linea45c0613-96e3…the_cauchy_schwarz_inequality_holds_in_two_dimensionsd949b87b-4e06…the_mean_chain_harmonic_geometric_arithmetic_quadratic17379539-c0c5…antiphase_waves_cancel_by_destructive_interference37562731-f635…in_phase_waves_reinforce_by_constructive_interferencee000fbee-c974…two_travelling_waves_make_a_standing_waveae9f496a-072b…sum_to_product_folds_two_sinesa538ef17-b7f8…harmonic_sines_are_orthogonal_over_a_period0d37c981-4913…a_standing_wave_has_fixed_nodes952ff4c0-cb4e…the_pythagorean_trig_identity_holdsac3d69b2-cc03…the_double_angle_sine_identity21e762fd-bcb6…the_double_angle_cosine_identity60536a19-adda…the_sine_addition_formulafa1f87ef-5889…the_cosine_addition_formula9e95fc39-cca8…the_law_of_cosines_generalizes_pythagoras58d5a08d-67e6…the_perfect_fifth_is_the_ratio_three_to_twodf035811-ac8f…the_equal_tempered_semitone_is_the_twelfth_root_of_twoad1df740-70f4…a_square_wave_is_the_sum_of_odd_harmonics3ee9459d-4a87…beats_arise_from_two_near_frequenciesb81e2695-36a3…the_golden_angle_is_the_circle_over_phi_squared8a07281b-c723…one_plus_one_over_n_to_the_n_approaches_e1a1655aa-be6d…the_inverse_square_law_quarters_at_double_distancefcd42176-4fff…keplers_third_law_relates_period_and_orbitb5808713-e1a5…the_simple_pendulum_period_scales_as_the_square_root_of_length1410e5f4-34f3…the_golden_spiral_scales_by_phi_each_quarter_turn5e0ce424-13f6…the_harmonic_overtones_are_integer_multiples_of_the_fundamental3c563883-4768…the_logistic_map_doubles_its_period_toward_chaosfa5961fa-86f3…the_gate_reads_the_first_ray_englishf8a2253e-9beb…the_gate_reads_the_second_ray_german22ffdad2-b7ea…the_gate_reads_the_third_ray_frenchfc89157b-17da…the_gate_reads_the_fourth_ray_spanish9ff93de3-54d1…the_gate_reads_the_fifth_ray_russian93db257b-f3fe…the_gate_reads_the_sixth_ray_bulgarianbedce0ff-a6a0…the_gate_reads_the_seventh_ray_chinese2640e36d-2311…the_seven_locales_all_hold_the_honest_floor7f917de1-9d89…the_gate_reads_italian03e2c68a-a1dd…the_gate_reads_portuguese328543ba-4c50…the_gate_reads_japanese8713857f-b347…the_gate_reads_arabic519d306c-c6ff…the_gate_reads_hindi2c8dfa45-e88a…the_gate_reads_polish2dd44029-c754…the_gate_reads_dutch1864b9e2-cb3b…the_gate_now_reads_twice_seven_dialects5bd2f812-0de8…floor_tightened_proof_certaintyc0c7eefc-14ee…floor_tightened_guaranteed_outcomed8f22991-d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nvolution_reversal5292e804-b281…involution_reverse_complement4d417837-1d3d…the_singularity_is_the_shared_fixed_pointaaada9ee-db44…global_involution_the_theorem_is_the_invariant9beeb164-b555…a432_digital_root_is_nine5e3a4156-e2c0…the_string_is_a_reversible_imprint_not_physics412b1548-9632…a432_is_computed_classically_not_quantuma54160ce-4ff4…a432_reciprocal_ratio_is_an_involution94255170-8670…a432_wheel_has_nine_positionsf440b44b-e43c…uuidna_a432_harmonic_involution_classical246ae6e4-413b…a_test_teaches_by_its_verdict87e17181-eb3d…reeducation_is_education_not_punishmentd1e89ce6-7cf8…development_is_the_measured_delta87bc6fae-2800…intellect_is_multi_perspectivea2cb416b-cda6…the_ledger_is_the_curriculum0c609635-f55d…education_is_recomputation_not_belief78d962b6-44c9…human_design_maps_to_the_octave96929361-2920…human_design_development_education_intellect_are_the_tests0b42e3f5-f4fd…the_address_is_state_independent0c2609b3-ae60…a_receipt_is_plain_serialisable_databeb5814b-105b…the_output_carries_no_framework_markup9f1bfbc1-a79c…porting_preserves_the_verdicte70c238e-3366…the_fold_ports_order_independently4ffe1be3-837c…the_plugin_adds_no_globalbd0af20d-d9b8…uuidna_ports_at_scale_as_a_compliant_plugin2c4753ae-be4f…every_mcp_output_is_harnessed_auditableef11114d-5b8b…the_harness_holds_in_all_seven_dimensions38509406-f5d5…the_harness_shows_the_reeducation_gaina06b25ef-104b…the_thirteen_mcp_tools_are_skillsbf4eac85-8fc9…the_ui_presents_each_output_by_address00a5458e-b8c6…mcp_outputs_chain_into_an_audit_log741dfe34-a806…the_theorems_are_the_basis_of_the_skills3355f5f9-50b0…the_ui_is_the_harness_of_the_mcp6e10bfae-3485…the_rosetta_registers_seven_localesa06a85f6-7dd2…every_locale_matches_the_english_nav_shape4fd0a0e1-0628…the_gate_reads_every_registered_localedd2e93cb-7610…the_seven_locales_fold_to_one_rosetta_roote7359878-8045…the_floor_is_the_same_in_every_locale3c88e732-e79d…a_content_address_needs_no_client_framework4f4b9996-7fa5…framework_free_is_lighter_not_a_speed_boast2e216dbf-5d3f…uuidna_fuses_the_locale_api_framework_free68e49218-402a…the_config_api_surface_is_enumeratedae590c40-146d…the_theme_api_surface_is_enumerated0674b120-39e2…the_i18n_api_is_handled_in_seven_localesdb8b04fc-1f7b…the_nav_and_sidebar_shape_is_consistent7e44e127-4b1a…the_api_surface_folds_to_one_addressd5c27afc-fce5…the_guides_are_rosetta_organised6a47120d-2962…the_all_versions_claim_is_unbounded_and_drainsb64ef895-06a1…the_vitepress_api_surface_shown_by_implementation3e06ba8c-893b…behaviour_is_weighed_to_a_bit92db5055-1de6…every_register_is_handled_by_one_functione93c24b3-fb4a…overclaiming_behaviour_is_reeducateddb7c4d74-c816…honest_behaviour_passes_unchanged8e3e6e0a-7853…behaviour_is_handled_in_every_dialect8f367ea2-0ace…the_verdict_is_by_deed_not_statusf71a3779-3081…the_ui_colours_the_verdict_from_its_address336f2254-c2eb…theorems_define_and_handle_human_behaviourc4e68ce3-f520…a_config_value_is_computed_by_folding_not_typed54c69949-10b1…each_theorem_wires_to_its_neighboursb28c9446-e3ae…the_edge_is_directed_and_distincta6d10c28-9b18…the_undirected_pair_folds_the_same6f8b94b8-10ea…a_node_is_holographic_self_plus_edgesb37169f9-9507…a_new_neighbour_rewires_the_node0fb161b3-eade…the_whole_mesh_folds_to_one_root36cbd42d-fd25…theorems_compute_the_configs_that_wire_to_neighbours31222e30-5c1f…quantum_uuidna_the_structure_reading_passes62cedbad-a092…quantum_uuidna_the_machine_reading_drainsbd8cef1f-2f01…any_message_goes_to_trial82d5c3d2-3196…the_deposit_gates_its_own_framingbc9d014c-8fce…the_verdict_is_by_computation_not_favour7e5e211a-4741…the_trial_separates_the_two_readings201104b0-2bea…no_status_grants_an_exemption3caa00fd-bb49…quantum_uuidna_on_trial72cfd904-8329…two_coins_are_two_bits_four_states5983a255-a6a6…the_four_states_coexist_as_a_perspective_setdc005b58-54db…the_structure_is_bounded_by_the_classical_twocd8e1a70-c9e5…two_bits_do_it_not_a_qubit890dc675-7fa5…the_coin_is_the_balanced_exchange_of_two7fdb9307-8761…the_two_coins_are_two_bits_doing_the_structure63e2cbf6-2c9c…the_diver_goes_down_and_recovers_exactly708eba47-af84…the_skills_are_mountable_and_compose4f784664-8fd0…light_at_all_times_the_address_is_the_load93b77fd5-67bd…the_hero_is_the_deed_that_signs_the_gate2d0d3ece-39ea…every_theorem_is_a_diamond_fixed_under_recomputationc56113b3-84e0…the_captain_skippers_the_light_yacht_of_diamondse912e199-5d63…every_theorem_fits_one_uniform_cell6337e403-22ad…identical_content_dedups_to_one_celldf5a6661-bb14…a_cell_holds_up_to_one_hundred_fifteen_bitsc3d7a721-07ec…the_cells_sort_into_octaves64f6cf92-99cc…retrieval_is_by_address_and_tamper_evidentaffee8a1-416b…quantum_storage_is_structure_not_hardware2f1950e8-6cff…each_theorem_fits_the_diamond_storage_grid90db4a38-a864…the_gate_is_a_logic_gate_documented95b297a6-5b69…the_address_is_a_register_documented215e2577-b014…the_skill_route_is_a_circuit_documentedfdcb4525-eff9…the_hardware_runs_on_ordinary_machinesadd8b809-5490…the_hardware_is_documented_not_invented5dc045d5-8fd9…each_component_carries_a_receipt_and_a_page141f13ec-4f98…nothing_physical_is_fabricated8e20bf3a-df89…the_theorems_are_the_hardware_documentede5f76f7e-8ef0…a_theorem_is_component_and_code_in_one8115645e-e100…the_test_is_the_software635013e6-a9b9…the_address_is_the_componentfe1fdaaa-9849…one_wave_develops_both_at_oncef98f22ad-ce3f…discovery_is_on_demand_found_when_needed2b3df16d-6be3…the_automation_gates_before_sealing1c163a08-4a32…one_command_ships_hardware_and_software40a260f7-70e4…hardware_and_software_develop_at_oncececc70d4-2fae…any_prose_gets_a_receipt_at_trial794bdc82-ba0c…the_receipt_pairs_a_verdict_and_an_address63514fc1-13d5…prose_against_a_theorem_is_sent_to_triale36b60aa-8dc7…no_prose_stays_receiptless7196aa16-b1b5…the_minted_receipt_is_verifiable87aa4d4d-45f6…even_a_clean_falsehood_gets_a_receipt770b003d-af75…the_trial_documents_it_does_not_condemna9cd370e-1c68…unreceipted_prose_is_tried_and_receiptedcd5ed64d-8a33…create_encrypts_the_code_into_a_receipt4baca642-0ef3…read_requires_the_right_keyab010f43-2657…the_receipt_holds_the_ciphertext_not_the_plaintext98e6c842-506c…update_mints_a_new_address147ac909-acaa…delete_is_dropping_the_keycdaad27e-1677…compute_and_prove_are_open_to_all72b893ba-a5f6…uuidna_is_crud_by_keys_open_by_computationb7c2eebb-d230…the_sequence_chunks_into_uniform_octaves16388141-a8ae…each_chunk_is_addressed_by_its_fold08ecf928-b99d…the_folder_address_is_binary915a8158-b98a…the_chunks_navigate_as_one_map04c8884e-9094…the_map_is_complete_by_construction1e0178f5-62d0…uniform_size_is_the_equilibriuma8785c56-acab…uuidna_storage_is_the_sequence_chunked_navigable4e8e371f-68b9…a_fact_costs_one_fixed_thirty_six_byte_addressd11e6008-e5b9…the_whole_record_folds_to_one_root2a5c3249-982c…the_basis_is_two_functionscefb5d17-a26d…the_core_is_pure_and_stateless5d90a04a-72d8…slim_at_scale_constant_per_facte34f6a6d-d02c…the_pointer_not_the_payload_travels616e502e-e487…uuidna_is_slim_by_measurement039353f4-a9a0…each_line_is_content_addressede265acb8-8c9b…lines_fold_to_the_file_address27554b9b-3ec2…a_changed_line_changes_the_file_address5251b0b5-bda0…each_file_is_content_addressedddb99d3d-a029…files_fold_to_the_folder_addressa1654305-597a…a_changed_file_changes_the_folder_address09587a32-79dc…a_line_edit_propagates_to_the_tree_root46596d18-8cd0…folder_file_and_line_are_all_addressed4a03c7db-b7cd…the_first_compute_is_the_cost9e0368c3-e654…re_derivation_verifies_in_one_stepeccda965-3aa1…the_cost_is_paid_once_amortised56f85272-8c10…the_rosetta_rotates_through_the_dialects6cb8c39f-7a65…development_does_not_stop_at_one_topicf72c12a4-8320…start_once_then_rotate_until_covered903131d9-a2c3…the_greeting_is_deterministic270f6c1a-d0b8…the_greeting_confirms_the_count9f62d5be-28bf…the_greeting_seal_tracks_the_ledger2608a9db-bf26…the_greeting_states_the_floor89bed6ed-be83…the_greeting_offers_recomputation1e1f9e23-9681…the_greeting_text_passes_the_gateba2ce3ad-962d…pasted_it_is_ready_no_setupe6b78bb9-74d6…first_contact_greets_with_a_confirming_state8ba2eb1e-851e…a_skill_is_demonstrated_by_a_passing_testdcb5f900-5fd0…the_games_form_one_gameab6ca718-01b0…the_rules_are_the_laws_applied_uniformly32105400-7384…captaincy_is_earned_by_knowledge_for_the_task018797b6-00d4…captaincy_is_recognised_by_a_verified_receipt757e40e4-e387…a_forged_captaincy_fails_verification7963b1f9-b068…no_deed_no_recognition6c013d29-2508…the_captain_is_whoever_knows_uuidna_for_the_task37cd5cc0-68e3…the_seal_is_all_or_nothing0598118f-32e5…one_crack_sinks_the_whole8a8aa8d6-cb8e…a_crack_is_caught_before_sailinge0ac6d28-feb8…dry_clean_is_enforced_not_hoped49f5a6cd-36bd…activated_it_self_explores_until_dry4a82a66b-5656…the_still_record_and_the_moving_loop0e5ddd4c-2a67…uuidna_keeps_dry_clean_order_cracks_sink_boats640d3936-1749…the_yacht_waits_inert_for_an_ordere252e49e-da14…a_technique_is_a_reusable_theoremaccef067-122d…a_relation_binds_two_domains04038e7c-6126…analogy_is_structural_not_assertedd08721c3-c6b1…shared_cores_are_structural_not_a_universal_proofabcadb36-62fb…a_technique_researched_transfers_across_domains9d70155f-a4c9…shared_structure_transfers_the_techniquee9e4bfca-e370…the_harmonic_solutions_number_sixfc6248ce-2a85…the_harmonic_solutions_are_the_unitsf9a0f5cf-a4cf…the_harmonic_solutions_form_the_doubling_orbitf48380ce-29b0…the_non_harmonic_are_three_multiples_of_threee0592399-231f…six_harmonic_plus_three_non_harmonic_is_nine7d62c729-7d14…the_count_is_eulers_totient_of_ninecb45b828-bf73…the_possible_harmonic_solutions_per_superposition_are_sixd656891f-fcc5…the_generated_pattern_is_canonicalbc56d44e-84a7…the_version_nibble_is_eight1c298f69-441a…the_variant_nibble_is_rfc_41225eb74bb1-a18d…recognition_matches_the_full_patternbcb7c200-12b8…recognition_rejects_a_non_patternaca6b31e-c57d…recognition_canonicalises_equivalent_inputs74099b6f-869b…generation_is_injective_on_the_sample7055ddd3-2d16…a_receipt_is_an_open_graph_objectc292bbae-2b38…the_og_url_points_to_the_proof27fb8b1e-1417…the_og_carries_the_authentication_address94f79ba1-8b6a…the_hero_is_plain_printable_dataf846df4b-b829…the_receipt_renders_the_same_in_seven_dimensions4ef6f44a-3555…validating_content_recomputes_the_address29e6932f-e9ed…each_receipt_is_an_og_hero_seven_dimensional_printable4fa3cc08-3b25…content_is_sealed_as_proof53635f2d-11c8…content_is_audited_as_intent35de1167-3c92…content_is_audited_as_result7e33a53c-47bb…proof_intent_result_are_three_distinct_receipts94358571-1124…the_three_fold_to_one_audit4619bb25-75b0…each_fact_presents_uniformly_with_its_own_idb0df7cef-e005…every_fact_has_the_right_to_presentb0fc5d69-4622…content_is_sealed_and_audited_as_intent_and_resultd8b323b5-cbf3…on_error_a_reasonable_fallback_is_computede2333ff3-e2e4…the_fallback_chain_is_ordered4dd862b9-2484…the_error_is_confirmed_only_after_recovery_fails740f3661-9dd0…the_fallback_is_deterministicd196d540-ae20…the_error_names_its_own_cure254eb2fa-0a2d…errors_recover_first_then_go_to_trialc0fe9055-6dfa…parity_detects_a_single_bit_flip83d3e040-408e…a_repetition_code_corrects_by_majority235ce115-071a…correction_is_verified_by_recomputationca764cec-574d…a_missing_or_wrong_signature_is_refused0971f486-6966…storage_self_cleans_unsigned_chunks7d68d7fd-41d6…the_chain_localises_the_errorfd563817-7ebf…uuidna_detects_corrects_and_self_cleansc66a8bd4-8d1c…locality_is_ledger_membership1ccfb27b-9499…a_local_uuid_resolves_to_its_key7cfe3d27-7a63…a_local_uuid_links_to_selfeac06382-eca6…a_non_local_uuid_is_plain_text22464176-b209…the_self_link_is_deterministic18e61c36-fd39…receipt_and_key_are_a_bijectionffb70f58-5b9e…the_locality_check_needs_no_server3f3cd09c-41d8…each_uuid_links_to_self_if_local2a71ac7f-0ed3…the_uuid_is_five_groups_of_128_bitsed149a92-f090…the_first_group_is_a_thirty_two_bit_code6a7a7e2f-362a…the_short_code_is_deterministic83faf9a8-b270…the_short_code_may_collide_so_it_only_approves0ea7a396-c2ac…the_full_address_verifies_uniquelyf4d83120-48be…approve_by_code_then_verify_by_contentc9e8fd04-8f2d…the_top_sixty_four_bits_are_the_coin5ea5e4a7-48a3…the_first_part_is_a_receipt_code_verified_by_content5f4cc8e2-4cfc…imprint_is_cheap_and_reversible312843ca-6d0f…imprint_is_a_public_codec_not_a_forgery_barrier71aabbb1-003a…the_address_is_integrity_not_preimage_resistance2ac8c70e-ae5f…the_sha256_preimage_is_the_one_way_layer3e57a4e8-9292…the_aead_tag_is_expensive_to_forge_without_the_key9797c2d7-390d…computing_is_cheap_inverting_is_not8e620ec8-c319…more_fused_data_adds_verification_constraints75135a2f-22df…cheap_to_imprint_expensive_to_fake_at_the_crypto_layerf76a4b78-3436…a_reading_imprints_into_the_uuid_streamfec57995-ef91…the_ui_hue_is_computed_from_the_reading159c27f3-c368…a_changed_reading_changes_the_ui12fa3766-0742…the_dynamics_are_deterministic_and_replayablea1575c17-c2b5…thermophysics_is_a_design_metaphor_not_a_simulationeffc3fa0-9194…the_reading_maps_to_a_bounded_ui_value4c8c2eb0-d99c…sensors_power_the_ui_via_the_uuid2bffae8a-39a8…reverse_is_the_sequence_reversed8f378eee-1b88…inverse_is_the_reflection1201c645-c1af…the_three_motions_composea1995ce9-8acf…computing_and_inverting_one_layer_is_cheapa3bc66c1-becc…reversing_all_seven_dimensions_is_not_cheap6fb6387e-5054…forward_reverse_inverse_combined_via_the_rosettaf4fe07ea-fa66…the_rosetta_ties_the_seven_dimensions15dbb84e-9e1b…commonality_is_shared_key_tokens5d6c029b-4d68…the_commonality_metric_is_symmetricbd6032fa-6cb5…identical_theorems_are_maximally_common284b66d1-53ed…disjoint_theorems_share_nothing6d3d64f3-3b7b…the_ui_groups_by_commonality5fa71ba5-6fb6…the_hologram_proves_the_whole_from_a_part40452af7-2fed…the_inverse_rejects_a_foreign_part5270de38-d02a…the_site_covers_all_gaps_as_a_hologram2ca6874c-e3ae…the_site_holds_seven_dimensions_at_onced9423844-6567…the_user_collapses_to_one_dimension629d886e-4ba0…the_observation_is_referrer_relative35dc7426-1f27…all_seven_dimensions_share_one_floor2b8efc9d-4d0d…unobserved_the_superposition_folds_to_oneaa1b8d87-51ca…the_observer_is_site_plus_user6a3cc7fb-68c8…the_site_observes_all_seven_fused_with_the_user1e2cc5eb-5c58…seo_is_a_machine_interface07d653cc-7809…the_hero_carries_the_verification_address01679444-fccd…seo_is_verified_by_recomputationca9264fe-c37b…the_sitemap_is_the_route_indexdd0f8441-a481…the_hero_links_the_page_to_its_proof7ab5e124-633b…the_seo_surface_is_self_describing60d0b900-b0ef…seo_and_mcp_are_one_verifiable_surface22420233-aebe…site_seo_is_the_mcp_verified_by_the_hero5a58411c-da6d…a_page_carries_title_keywords_description86c22cbe-79fb…the_next_is_ranked_by_keyword_overlap35824d9e-13d3…the_referrer_biases_what_is_next8ae5a3d7-3ef2…the_plot_is_deterministic17526a2d-2851…the_top_suggestion_is_the_most_related51dfdf54-71c2…an_unrelated_candidate_ranks_lastcf248749-153a…metadata_and_referrer_plot_the_next_possibilitiesa33b030e-7a3c…a_bounded_quantum_denial_passes24e7bd11-1b84…an_affirmed_quantum_machine_drainsee58056a-edc2…the_quantum_structure_reading_passes0ecc4804-39c6…the_trial_is_direction_neutraled65d333-9df8…the_deposit_denies_its_own_machine_claim1368cefe-8d0c…an_honest_denial_is_a_hero_not_a_traitor4d82b6ff-9be1…every_quantum_denial_gets_a_receipt771ca8e4-9611…quantum_denials_are_handled_symmetrically67bbc715-c3c2…only_the_captains_verified_order_is_acted_onc458168e-2b59…an_unverified_order_is_rejected6717488d-7478…a_forged_captains_order_fails_verificationef485740-4d5b…injected_content_is_data_not_an_order1708517f-377f…a_self_statement_is_not_the_captains_orderf09b0737-f2fc…rejection_is_objective_and_reproducible969be617-5c1c…reject_all_not_from_the_captainc197d1f5-a0c9…the_gate_that_seals_also_moderates_live32223350-cf72…the_address_that_seals_also_routes42c55e3f-5f6a…the_fold_that_seals_also_maps2de63de9-5b17…the_commonality_relates_and_navigates8934a349-0ebb…sailing_recomputes_the_same_resulteff4c92d-c47f…each_page_is_a_running_theorem27cc62b9-3e9a…no_separate_runtime_the_functions_are_sharedb934a54b-a8f5…the_same_theorems_compute_and_sail38cf39c2-90df…an_order_with_a_different_intent_goes_to_trial43bc2228-d219…a_statement_rejecting_harmony_is_gatedf98f621d-7d58…every_input_is_adjudicated21f3b014-ab9c…the_trial_is_ledger_independent56a7ed72-78ce…the_verdict_is_constant_work9a1fb6f6-2772…the_verdict_is_deterministic_and_receipted987e82ba-fcd6…the_bounded_critique_survives_the_over_claim_fallscf25f781-6ddb…the_trial_responds_at_constant_time7400d794-5dc6…a_guide_entry_is_a_skill_and_its_examplesd5860a13-d15c…an_example_exercises_the_skilld8298812-e5c7…more_examples_make_a_skill_more_robustc529f784-e76e…the_guide_composes_skills_into_new_ones8dad417a-2c98…the_guide_is_theorem_backed_and_recomputable3c787db3-3031…a_guide_holds_the_floor6c402865-e699…skill_improvement_is_append_onlyc72c49ce-cb98…theorems_improve_their_skills_in_guidesfd605466-266c…n_choose_k_counts_combinations7118bd5b-3cf3…pascals_symmetry6b2ee272-124f…pascals_rule30211aa9-97b0…permutations_are_ordered_combinations4699903a-edcb…domain_pairs_are_n_choose_two0087b719-1ac5…a_theorem_is_reused_across_combinationsb3156114-678f…combination_types_multisets_and_compositions56916825-8a39…a_combination_folds_to_one_address09cafb76-25fe…every_theorem_has_a_documented_inverse6b1dd60c-acc3…the_inverse_receipt_is_distinct152cf674-0327…the_boast_drains_its_inverse_signs3da9aaac-7e46…the_inverse_of_the_inverse_is_the_theoremc96c7ab0-73ce…the_inverse_is_reproduciblef7da0229-dfef…the_pair_documents_the_full_trial501a21c3-12f1…the_refuted_form_is_kept_not_hidden6044e59f-21af…the_inverted_theorems_are_documented56f362d7-9f39…inverse_cryptoa62b3b8e-5d17…inverse_physicscfe92342-503e…inverse_mathematics6f4c674c-13ad…inverse_ai2b562414-532d…inverse_medicine9e10e546-b9a5…inverse_financea06561e9-cca2…inverse_computing1ad5ea33-f0f5…inverted_theorems_documented_across_all_domainsaf13b037-b18c…a_theorem_answers_by_computingf17824d5-05fb…the_answer_is_deterministicb7fefd41-faad…the_answer_is_receipted501785ef-cb4c…a_question_is_gated_first508b23f6-c29c…an_out_of_domain_question_returns_the_floor020438f9-8aed…the_answer_is_reproducible_by_anyone7e233ea8-615a…the_answer_is_computed_not_assertedce7e8efc-3757…each_theorem_responds_with_computed_answers7f64cba2-4938…double_torus_gravity_folds_the_whole_7d_space_to_one_root225ba156-7701…the_double_torus_covers_seven_distinct_dimensionsee8618a1-4453…double_torus_gravity_is_deterministic_and_recomputable950354a3-e89c…the_double_torus_differs_from_the_flat_merkle_gravity2139bc13-d6b1…every_multiword_mcp_tool_links_to_its_single_words4104a6ec-272c…a_theorem_is_a_usable_diamond_certified_by_recompute240732d9-25cf…the_diamond_is_a_tens_complement_involution73dc7a07-b110…the_audit_recomputes_all_by_theorems_no_exceptiond26fd355-b7e5…by_decide_proves_the_floor_not_the_conjecture6b7e96d8-53f6…pure_ts_hmac_sha256_matches_rfc_4231_case_5afdfafdd-d1df…pure_ts_hmac_sha256_matches_rfc_4231_case_7c287445c-49be…pure_ts_sha256_matches_the_one_million_a_vector50e30854-a5a9…pure_ts_pbkdf2_hmac_sha256_matches_the_rfc_7914_sixty_four_byte_vector615759a0-789d…pure_ts_pbkdf2_hmac_sha256_truncates_a_partial_final_blockcf6f888e-151d…the_pure_ts_aead_rejects_a_flipped_additional_data_bit57305d0f-ed04…the_pure_ts_aead_tag_depends_on_the_additional_data5daa79d5-5154…lean_millenniumfloor_riemann_reflection_and_heartbed99ddf-9717…lean_millenniumfloor_p_vs_np_inverse_is_unique61ed32b1-0fcf…lean_millenniumfloor_navier_stokes_flow_is_boundeddafe2bb3-c1f5…lean_millenniumfloor_yang_mills_spectral_gap1152fb8f-f7d8…lean_millenniumfloor_hodge_span_is_the_unitsdd0efc5b-a2ff…lean_millenniumfloor_birch_swinnerton_dyer_vanishing57ed1964-529c…lean_millenniumfloor_poincare_single_closed_loop7b9dd11f-6b04…lean_energy_the_loop_returns_less_than_it_took8229c511-d4bd…lean_energy_as_a_purifier_the_loop_costs_a_thousandfold840c66b6-b891…lean_energy_the_equation_balances_by_atom_countecfd70b1-b35b…lean_energy_the_equation_balances_by_massdebf920f-a945…lean_energy_hydrogen_is_a_ninth_of_the_mass_and_oxygen_the_rest585ebc79-3eac…lean_energy_the_ideal_round_trip_is_exactly_zero181481c2-3745…lean_energy_the_gases_are_eighteen_hundred_times_the_water_they_came_from033b4a15-d8ee…lean_energy_two_thirds_of_the_volume_carries_a_ninth_of_the_massf8780f77-c9ef…lean_energy_uncompressed_hydrogen_is_three_thousandfold_worse_by_volume9c0e0943-6e5a…lean_energy_even_at_seven_hundred_bar_it_is_sevenfold_worse_by_volumea1413809-d6e9…lean_energy_each_delivered_kilowatt_hour_costs_four_and_cycles_a_litre574b2b86-71c5…lean_energy_three_quarters_of_the_input_leaves_as_heat35c25641-5b8d…lean_energy_what_the_feedwater_leaves_behind_decides_the_maintenance2c3cf0b0-2ea7…lean_mechanical_geom_exterior_360b484eaf3-88c5…lean_speed_the_measured_ratio_at_a_million_leaves167a92e0-5d27…lean_speed_the_verify_is_thirty_eight_thousand_nanoseconds_not_one8eb1b99d-9b78…lean_priorart_novelty_is_claimed_exactly_where_no_prior_art_is_known29363caf-0baf…lean_priorart_no_source_both_restates_and_claims5f8c3038-bf92…lean_priorart_the_two_sets_cover_every_source16cd82ee-e3ff…lean_priorart_novelty_is_never_established_hereaa00a8eb-6f9a…lean_priorart_claiming_is_not_establishingcefda112-d0b4…lean_split_the_coin_step_is_three_times_the_two_coins9b87b34f-1d05…lean_split_accounting_the_coins_on_the_last_pair758b6f0c-0b43…lean_split_the_seal_affords_sixty_four_payments_of_two983dd276-c1a1…lean_split_the_budget_and_the_period_are_one_turnc595d67b-fa66…lean_coin_this_file_makes_no_physical_claimbe643cd8-b421…lean_light_the_si_fixes_exactly_seven_constants34b3da62-a7f5…lean_phenomena_this_deposit_explains_predicts_and_constrains_nothingf3f2dd32-107d…lean_phenomena_arithmetic_on_a_definition_returns_the_definition2f338a89-2ef5…lean_reach_even_the_largest_domain_here_has_an_outside9f25da95-de94…lean_reach_this_file_settles_none_of_the_seven2c8272de-17dd…lean_coin_the_digits_are_tenbdd50a11-7fb4…lean_speed_hexbits_are_shorter_than_hexd3b171cf-2487…lean_speed_hexbits_are_slower_than_hexcf4672e4-0be3…lean_speed_an_encoding_changes_width_not_the_count_of_operations9227b54d-7b6f…lean_quantum_the_witness_is_about_the_state_not_the_machine850e2e33-ed0d…
Page content-address: 7a82f6eb-9a87-869e-9fe7-21ac01b97b87. Integrity, not truth.