From 6a7d93ccb70e16a7841f4d34994c066f2ef10917 Mon Sep 17 00:00:00 2001 From: mrwulf Date: Sat, 11 Jul 2026 17:53:43 +0200 Subject: [PATCH] S5 stage 1: consistency infrastructure (kbelow prefix-split + list surgery) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Theorem 3's binding (steps 1-2) turns on one non-obvious arithmetic fact, isolated and proven here before the main proof: - pow2_exp_unique / kbelow_eq_of_pow2_between: kbelow is pinned by its three defining inequalities (power-of-two, kk recursion branch needs to align MTH(D₁.take n₀) with the fold). - take_take_le, take_drop_prefix: the list-surgery identities relating (D.take n₀) to D.take k and (D.drop k).take (n₀-k). Cones pinned; 21 certs green. Deliberate honest checkpoint — binding + extractCons assembly is the next stage. LTL untouched. Co-Authored-By: Claude Fable 5 --- README.md | 2 +- verification/Proofs/AxiomCheck.lean | 4 ++ verification/Proofs/AxiomCheck.olean | Bin 1632 -> 1744 bytes verification/Proofs/Consistency.lean | 67 ++++++++++++++++++++++++++ verification/Proofs/Consistency.olean | Bin 0 -> 148488 bytes verification/check.sh | 5 +- 6 files changed, 76 insertions(+), 2 deletions(-) create mode 100644 verification/Proofs/Consistency.lean create mode 100644 verification/Proofs/Consistency.olean diff --git a/README.md b/README.md index 7a32906..f808711 100644 --- a/README.md +++ b/README.md @@ -16,7 +16,7 @@ the same discipline as the four `*-ed25519-verified` subject corpora. | L5 | inclusion soundness = EXPLICIT extractor `extractIncl` (Theorem 2) **done, non-vacuous** | | L6a | descent extractor `extractMTH` (Theorem 3 step 3 = Lemma 2, WHOLE-TREE instance) **done, non-vacuous** | | — | Lemma 2, PATH instance (receipt-uniqueness of `Root`): deleted with the vacuous `root_binding` in S3.5 and **not yet restored**; optional, not needed for Theorem 3 | -| L6b | ConsRec binding + Theorem 3 assembly (extractCons) | pending (S5) | +| L6b | ConsRec binding + Theorem 3 assembly (extractCons) | in progress (S5): stage-1 arithmetic/list infrastructure done (kbelow_prefix_eq, take surgery); binding + assembly next | | L6 | pin-store state machine safety (Proposition 1) | pending | ## Discipline (identical to the subject corpora) diff --git a/verification/Proofs/AxiomCheck.lean b/verification/Proofs/AxiomCheck.lean index 8c12f5e..202c3fa 100644 --- a/verification/Proofs/AxiomCheck.lean +++ b/verification/Proofs/AxiomCheck.lean @@ -3,6 +3,7 @@ import Proofs.Basic import Proofs.Completeness import Proofs.Extract import Proofs.Descent +import Proofs.Consistency #print axioms LTLAcc.domsep #print axioms LTLAcc.kbelow_pos #print axioms LTLAcc.kbelow_lt @@ -21,3 +22,6 @@ import Proofs.Descent #print axioms LTLAcc.extractMTH #print axioms LTLAcc.extractMTH_correct #print axioms LTLAcc.extractMTH_nonvacuous +#print axioms LTLAcc.kbelow_prefix_eq +#print axioms LTLAcc.take_take_le +#print axioms LTLAcc.take_drop_prefix diff --git a/verification/Proofs/AxiomCheck.olean b/verification/Proofs/AxiomCheck.olean index 4e325686e7eab2959e079f83a8ce8e9495d40f44..1952ad98a2f4874d42853e8d8f5bb274eafadc9f 100644 GIT binary patch delta 405 zcmaFBbAfk41mld2kzq{rKY17+07i2|`Of)y#hJw=sd>qjP!R?O1qLQY2F3>eX$cGr zE0`ekVbA_kx|q=$;X(q7=KKD$rR66G1-&Z zo^u5=#Ml#)Co;QpeSq==SSCMWc2jgDYWfr-HYDt=<}M^^cO1gM@DP&xoA3sP;tz{(&1b$J2~Is-{PNDlxeA7SkP delta 286 zcmcb>`+#Rc1f#~r$S@|ZZ)^+@z&2Tt*?e*Wa|9#DS1GD?AgiAiDdP8NAi2NswS4_Vwf6QI0?$%d@v zTpOUg3oMh1SlvK&vYK;Ru);L_WOY}Z0OdJ{Sbr=$rlk)uT!Mj#;Q(A!BAa}`2dF_6 bP|*ia8l>8Sft6te6U599IOrcp>Opz{`btf{ diff --git a/verification/Proofs/Consistency.lean b/verification/Proofs/Consistency.lean new file mode 100644 index 0000000..c91af52 --- /dev/null +++ b/verification/Proofs/Consistency.lean @@ -0,0 +1,67 @@ +/- S5 (stage 1) — infrastructure for consistency soundness (paper Theorem 3, + steps 1-2). The binding argument needs one non-obvious arithmetic fact and + two list-surgery facts, isolated and proven here before the main proof. + + Arithmetic: when the pinned prefix of size n₀ spills past the left subtree + (k < n₀ ≤ n, k = kbelow n), the honest tree of `D₁.take n₀` splits at the + SAME point k. That is `kbelow n₀ = k`, and it holds because k is a power of + two with k < n₀ ≤ 2k, which pins kbelow uniquely. -/ +import Proofs.Descent + +namespace LTLAcc + +/-- Two powers of two both lying in `(m/2, m]` (i.e. `2^a < m ≤ 2^(a+1)`) + have the same exponent. -/ +theorem pow2_exp_unique {a j m : Nat} + (ha : 2^a < m) (ha2 : m ≤ 2^(a+1)) (hj : 2^j < m) (hj2 : m ≤ 2^(j+1)) : + a = j := by + rcases Nat.lt_trichotomy a j with h | h | h + · -- a < j ⇒ a+1 ≤ j ⇒ 2^(a+1) ≤ 2^j < m, contradicting m ≤ 2^(a+1) + have : a + 1 ≤ j := h + have hle : 2^(a+1) ≤ 2^j := Nat.pow_le_pow_right (by omega) this + omega + · exact h + · have : j + 1 ≤ a := h + have hle : 2^(j+1) ≤ 2^a := Nat.pow_le_pow_right (by omega) this + omega + +/-- `kbelow` is pinned by its defining inequalities: a power of two `p` + with `p < m ≤ 2p` IS `kbelow m`. -/ +theorem kbelow_eq_of_pow2_between {p m : Nat} (j : Nat) (hp : p = 2^j) + (h1 : p < m) (h2 : m ≤ 2 * p) : kbelow m = p := by + have hm2 : 2 ≤ m := by + have : 1 ≤ p := by rw [hp]; exact Nat.one_le_two_pow + omega + obtain ⟨a, ha⟩ := kbelow_pow2 m + have hlt := kbelow_lt m hm2 + have hle := le_two_kbelow m hm2 + -- kbelow m = 2^a with 2^a < m ≤ 2^(a+1); p = 2^j with 2^j < m ≤ 2^(j+1) + rw [ha] at hlt hle + have hj2 : m ≤ 2^(j+1) := by rw [Nat.pow_succ]; omega + have ha2 : m ≤ 2^(a+1) := by rw [Nat.pow_succ]; omega + have hja : 2^j < m := by omega + have : a = j := pow2_exp_unique hlt ha2 hja hj2 + rw [ha, this, ← hp] + +/-- The specialization used in the binding: with `k = kbelow n`, `2 ≤ n`, + and `k < n₀ ≤ n`, the prefix tree splits at the same `k`. -/ +theorem kbelow_prefix_eq {n n₀ : Nat} (hn : 2 ≤ n) + (hk : k = kbelow n) (hlo : k < n₀) (hhi : n₀ ≤ n) : + kbelow n₀ = k := by + obtain ⟨j, hj⟩ := kbelow_pow2 n + rw [← hk] at hj + have hle := le_two_kbelow n hn + rw [← hk] at hle + exact kbelow_eq_of_pow2_between j hj hlo (by omega) + +/-! ### list surgery -/ + +theorem take_take_le (l : List Bytes) (k n₀ : Nat) (h : k ≤ n₀) : + (l.take n₀).take k = l.take k := by + rw [List.take_take]; congr 1; omega + +theorem take_drop_prefix (l : List Bytes) (k n₀ : Nat) : + (l.take n₀).drop k = (l.drop k).take (n₀ - k) := by + rw [List.drop_take] + +end LTLAcc diff --git a/verification/Proofs/Consistency.olean b/verification/Proofs/Consistency.olean new file mode 100644 index 0000000000000000000000000000000000000000..50cc8997599a384dba8b88891eeed433de96c7eb GIT binary patch literal 148488 zcmbS!d3?;*_x>P>v5!593QD7tvBy?X#i(uUvBwg|l4R_$54FTT6pgj9R6~m>Dy=PQ z3yLV(pepv*Dy4|l@|!#NdB&Mk`~AFr?>`*5&pqd!bMCq4F7G=tkwe0IM`X=XqjGg~ zW%K*P`c^Z%`KjKoZ`I!BprAgr`c>-}6xO?cShbqHYxnI}JFIW*J~e~-)vjK(PrurI z`x*@MkGDRa4 zQic95Nzo`*Liu=?zXn|YdzKxQNd54)by#F%|Ka)%*=`EtC;nBW!k4$Jnn8*>1jc^)ZZolNjAtQjiVl*Hu6CqwogjC zT6SpJNcDnxq+Rjgujg0DZ+pRde=A(sm0$hG^^ltMpN7GG!iGeS?qweq)_-7Budtyg zb`vEx1@baLb0<9c@yo%tDys@gZZ0s8=EUDFi7v&%tJ#uulUSU!8ULzw04jmchE|vPlLf)#bbEho&@VC~KN4-e@p`REC zN?tPLq=S~;F!JrWZRb4wpXF7D!Y(4^1!vQCum0_)0zch6x0v$W0q^>Qv>4dYFJVJ_ 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