From eb3df7503fede433b7558275aff65b579633848e Mon Sep 17 00:00:00 2001 From: mrwulf Date: Sat, 11 Jul 2026 13:21:17 +0200 Subject: [PATCH] S3/L4-L5: root binding (Lemma 2, Path instance) + Theorem 2, constructive MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The crux layer — the statement whose HAND proof once carried the frontier coverage bug is now kernel-checked. - gen: hash outputs refactored to Hash = {l : List UInt8 // l.length = 32}. MECHANIZATION FINDING: the paper's pair-coincidence step ('equal hnode values of distinct argument pairs are a collision') is load-bearing on FIXED-WIDTH outputs — with unconstrained byte strings x++s = X++Y does not split. hnode_preimage_inj (cone: propext) makes this explicit via List.append_inj on equal-length components. Queued as a half-sentence for the paper's next cycle. - HasCollision := ∃ x y, x ≠ y ∧ sha256 x = sha256 y — appears ONLY as a conclusion, never a hypothesis (no collision-resistance assumed). - hnode_inj_or_collision / hleaf_inj_or_collision: the per-node dichotomy. - root_binding: any accepting reconstruction from (v,P) to the honest root either IS the honest receipt (leaf hash AND full path P = Path m D — case (ii) pinning every consumed sibling) or exhibits a collision. Motive quantifies (v,P); induction on Path; k-fold discipline. - incl_sound (Theorem 2, position binding): accepting a wrong leaf at m yields a collision. Cone [propext, Classical.choice, LTLAcc.sha256, Quot.sound] — the single hash axiom, pinned in check.sh. ALL GREEN. Also: Root n=1 branch changed from list-match to decidable 'if P = []' (well-founded unfolding generated a spurious exhaustiveness obligation); Root_one_cons added. Fable-5 statement-audit passed. LTL untouched. Co-Authored-By: Claude Fable 5 --- README.md | 4 +- verification/Proofs/AxiomCheck.lean | 6 + verification/Proofs/AxiomCheck.olean | Bin 1424 -> 1528 bytes verification/Proofs/Basic.lean | 32 ++-- verification/Proofs/Basic.olean | Bin 483680 -> 492488 bytes verification/Proofs/Binding.lean | 185 +++++++++++++++++++++ verification/Proofs/Binding.olean | Bin 0 -> 288416 bytes verification/Proofs/Completeness.lean | 20 +-- verification/Proofs/Completeness.olean | Bin 443272 -> 412112 bytes verification/check.sh | 7 +- verification/gen/LTLAcc/HashExternal.lean | 24 ++- verification/gen/LTLAcc/HashExternal.olean | Bin 3136 -> 39304 bytes 12 files changed, 246 insertions(+), 32 deletions(-) create mode 100644 verification/Proofs/Binding.lean create mode 100644 verification/Proofs/Binding.olean diff --git a/README.md b/README.md index 97b0e1f..4ab5605 100644 --- a/README.md +++ b/README.md @@ -12,8 +12,8 @@ the same discipline as the four `*-ed25519-verified` subject corpora. | L1 | bytes, hleaf/hnode, domain separation (Lemma 1) | **done** (domsep: axiom-free) | | L2 | MTH, Root, ConsRec definitions + termination | **done** (cones: propext, LTLAcc.sha256, Quot.sound) | | L3 | inclusion completeness (Theorem 1) | **done** (incl_complete: propext, Classical.choice, LTLAcc.sha256, Quot.sound) | -| L4 | frontier hash-fold + root binding (Lemma 2) | pending | -| L5 | inclusion/consistency soundness as collision extractors (Theorems 2, 3) | pending | +| L4 | root binding for inclusion (Lemma 2, Path instance) | **done** (root_binding; boundary = single sha256 axiom) | +| L5 | inclusion soundness = collision extractor (Theorem 2) **done**; consistency (Theorem 3) pending | | 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 3e0cd9f..4480d1c 100644 --- a/verification/Proofs/AxiomCheck.lean +++ b/verification/Proofs/AxiomCheck.lean @@ -1,6 +1,7 @@ /- Axiom-cone observation for the audit (Phase 3 of check.sh). -/ import Proofs.Basic import Proofs.Completeness +import Proofs.Binding #print axioms LTLAcc.domsep #print axioms LTLAcc.kbelow_pos #print axioms LTLAcc.kbelow_lt @@ -11,3 +12,8 @@ import Proofs.Completeness #print axioms LTLAcc.ConsRec #print axioms LTLAcc.Path #print axioms LTLAcc.incl_complete +#print axioms LTLAcc.hnode_preimage_inj +#print axioms LTLAcc.hnode_inj_or_collision +#print axioms LTLAcc.hleaf_inj_or_collision +#print axioms LTLAcc.root_binding +#print axioms LTLAcc.incl_sound diff --git a/verification/Proofs/AxiomCheck.olean b/verification/Proofs/AxiomCheck.olean index ad5187d89a51ee537812cff2c4eb1cf629966075..cc18f9838e43150013386b4e8b7faf617afd119d 100644 GIT binary patch delta 388 zcmbQh{eyc#1ml5?k<%FKe{wKD0E}jb@|`mCQZn<>!8`>9CPoIv2LEXZ3=A)zw2jy9 zfPeNcyg&*h7#JA@m>}{%CL92n-&mnM7`=i4qV52gp3KK2#ALuUS&Au+alzzDCVS2c zP;)*^KFH+GCBY0)ZNWTQh}n%Z1Iq1~oX9NBxMOl9vpnYwsMv?e3z^-yen5E=ER#Pm zn{j%uK=fx!He_)J*}!7Xc>yZ+Ve(EEcSQ|WhzjQr>yL%UwDdvFm0)0Eh=7WJn5@Vu gAJ75SBLEexfYKn<77VNm9#B_z;GicUsR!u+07H#ip8x;= delta 298 zcmeytJ%M{d1Y^X;$Z3pR-&hzRfMs$dlR5{8&B(y;ZK9yat(V*Ro3n3g`sa0vz` qh7WL6oviW!3Q*n(C@sJWF$1L9f`OIc1tZj{P#-9u(HcnVL3#iU5lhVg diff --git a/verification/Proofs/Basic.lean b/verification/Proofs/Basic.lean index bce2c64..e63ab36 100644 --- a/verification/Proofs/Basic.lean +++ b/verification/Proofs/Basic.lean @@ -11,10 +11,22 @@ namespace LTLAcc abbrev Bytes := List UInt8 /-- Leaf hash: `H(0x00 ‖ d)` (paper §5.3). -/ -noncomputable def hleaf (d : Bytes) : Bytes := sha256 (0x00 :: d) +noncomputable def hleaf (d : Bytes) : Hash := sha256 (0x00 :: d) -/-- Node hash: `H(0x01 ‖ x ‖ y)` (paper §5.3). -/ -noncomputable def hnode (x y : Bytes) : Bytes := sha256 (0x01 :: (x ++ y)) +/-- Node hash: `H(0x01 ‖ x ‖ y)` (paper §5.3). Arguments are 32-byte + hash values, so the preimage determines the argument pair — the + load-bearing width fact of the paper's Lemma 2. -/ +noncomputable def hnode (x y : Hash) : Hash := sha256 (0x01 :: (x.val ++ y.val)) + +/-- Preimage-level pair injectivity: equal `hnode` PREIMAGES force equal + argument pairs, because both components are exactly 32 bytes. -/ +theorem hnode_preimage_inj {x y X Y : Hash} + (h : (0x01 : UInt8) :: (x.val ++ y.val) = 0x01 :: (X.val ++ Y.val)) : + x = X ∧ y = Y := by + injection h with _ happ + have hlen : x.val.length = X.val.length := by rw [x.property, X.property] + have := List.append_inj happ hlen + exact ⟨Subtype.ext this.1, Subtype.ext this.2⟩ /-- **Lemma 1 (Domain separation), preimage form**: no leaf preimage equals a node preimage as a byte string — the first byte differs. -/ @@ -74,7 +86,7 @@ theorem kbelow_pow2 (n : Nat) : ∃ j, kbelow n = 2 ^ j := by /-- `MTH` (paper §5.3): the RFC 9162 tree head over a leaf-data list. `MTH [] = H(ε)`, `MTH [d] = hleaf d`, and for `n ≥ 2` the split at `k = kbelow n`. -/ -noncomputable def MTH (D : List Bytes) : Bytes := +noncomputable def MTH (D : List Bytes) : Hash := if _h0 : D.length = 0 then sha256 [] else if _h1 : D.length = 1 then hleaf (D.headD []) else @@ -95,11 +107,9 @@ decreasing_by /-- `Root` (paper §5.3 / Appendix B): the consumer's root reconstruction. `none` = rejection on any length mismatch, exactly as deployed. -/ -noncomputable def Root (v : Bytes) (m n : Nat) (P : List Bytes) : Option Bytes := +noncomputable def Root (v : Hash) (m n : Nat) (P : List Hash) : Option Hash := if n = 1 then - match P with - | [] => some v - | _ => none + (if P = [] then some v else none) else if n = 0 then none else match P.getLast? with @@ -125,8 +135,8 @@ decreasing_by the reconstructed pair (old root, new root); `none` = shape mismatch. The flag `b` records whether the size-`n₀` subtree root is carried implicitly (the pinned root `r`) or explicitly in `C`. -/ -noncomputable def ConsRec (n₀ n : Nat) (C : List Bytes) (b : Bool) (r : Bytes) : - Option (Bytes × Bytes) := +noncomputable def ConsRec (n₀ n : Nat) (C : List Hash) (b : Bool) (r : Hash) : + Option (Hash × Hash) := if n₀ = n then if b then match C with @@ -159,7 +169,7 @@ decreasing_by /-- The consumer's acceptance predicate for a consistency proof between pinned head `(n₀, r₀)` and offered head `(n₁, r₁)` (paper §5.3). -/ -def acceptCons (n₀ n₁ : Nat) (r₀ r₁ : Bytes) (C : List Bytes) : Prop := +def acceptCons (n₀ n₁ : Nat) (r₀ r₁ : Hash) (C : List Hash) : Prop := n₀ = 0 ∨ ConsRec n₀ n₁ C true r₀ = some (r₀, r₁) end LTLAcc diff --git a/verification/Proofs/Basic.olean b/verification/Proofs/Basic.olean index 7422fe1bc139de3cbad59d2467535e75a972eb94..0bd38bc095224324519cffd1f5ac728df868379f 100644 GIT binary patch literal 492488 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z9)w>C-Ur85i2h@P_+GS|gW7Y^Ze{I!ouGZj4xH+lq{yrA``A(O3Nfnx^^XfZx1G!E z3cvI5p8H*#Pfd|NjLdd1<0ENdOk?ukKofhqm|xz6tM%PxrNH-?1UTCitDn_^NMDtg9}Di@f@<4(79Vp-0ydBpvkmHa=I# s!{-8ey6=Q@XD-f*PS`DEc4O6O9r&Etj{CMcqs(JAq;5=ceTbO rw [Root]; simp [h0, h1, hm, hR] - exact fun hh => absurd hh hne | some x => rw [Root]; simp [h0, h1, hm, hR] - exact fun hh => absurd hh hne /-- `Root` at a composite size, right branch (`m ≥ k`). -/ -theorem Root_right (v : Bytes) (m n : Nat) (P : List Bytes) (s : Bytes) +theorem Root_right (v : Hash) (m n : Nat) (P : List Hash) (s : Hash) (hn : 2 ≤ n) (hm : ¬ m < kbelow n) : Root v m n (P ++ [s]) = (Root v (m - kbelow n) (n - kbelow n) P).map (hnode s ·) := by have h1 : ¬ n = 1 := by omega have h0 : ¬ n = 0 := by omega - have hne : ¬ (P ++ [s] = []) := by simp cases hR : Root v (m - kbelow n) (n - kbelow n) P with | none => rw [Root]; simp [h0, h1, hm, hR] - exact fun hh => absurd hh hne | some x => rw [Root]; simp [h0, h1, hm, hR] - exact fun hh => absurd hh hne /-! ### 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hh + calc a :: b :: u = a :: (b :: u) := rfl + _ = a :: ((b :: u).dropLast ++ [s]) := by rw [← this] + _ = (a :: b :: u).dropLast ++ [s] := by simp + +/-- **Root binding** (the Path instance of the paper's Lemma 2): if any + reconstruction from `(v, P)` hits the honest root, then either + `(v, P)` IS the honest receipt — leaf hash and every consumed + sibling — or a collision is exhibited. -/ +theorem root_binding (m : Nat) (D : List Bytes) : + ∀ (v : Hash) (P : List Hash), m < D.length → + Root v m D.length P = some (MTH D) → + (v = hleaf (D.getD m []) ∧ P = Path m D) ∨ HasCollision := by + induction m, D using Path.induct with + | case1 m D hle => + intro v P hm h + have h1 : D.length = 1 := by omega + obtain ⟨d, rfl⟩ := exists_singleton_of_length_one D h1 + have hm0 : m = 0 := by simpa using hm + subst hm0 + have hlen : ([d] : List Bytes).length = 1 := rfl + rw [hlen] at h + cases P with + | nil => + rw [Root_one, MTH_single] at h + simp only [Option.some.injEq] at h + left + refine ⟨?_, by rw [Path]; simp⟩ + have hg : ([d] : List Bytes).getD 0 [] = d := rfl + rw [hg]; exact h + | cons p q => + rw [Root_one_cons] at h + simp at h + | case2 m D hgt k hmk ih => + intro v P hm h + have h2 : 2 ≤ D.length := by omega + have hkeq : k = kbelow D.length := rfl + have hkl : k < D.length := by rw [hkeq]; exact kbelow_lt D.length h2 + have hmk' : m < kbelow D.length := by rw [← hkeq]; exact hmk + have htklen : (D.take k).length = k := by + simp [List.length_take]; omega + cases hP : P.getLast? with + | none => + have hPnil : P = [] := by + cases P with + | nil => rfl + | cons a t => simp at hP + subst hPnil + rw [Root] at h + have hn1 : ¬ D.length = 1 := by omega + have hn0 : ¬ D.length = 0 := by omega + simp [hn1, hn0] at h + | some s => + obtain hsplit := eq_dropLast_append_of_getLast? P s hP + rw [hsplit] at h + rw [Root_left _ _ _ _ _ h2 hmk', ← hkeq] at h + cases hR : Root v m k P.dropLast with + | none => rw [hR] at h; simp at h + | some x => + rw [hR] at h + simp only [Option.map_some, Option.some.injEq] at h + rw [MTH_split D h2, ← hkeq] at h + rcases hnode_inj_or_collision h with ⟨hx, hs⟩ | hc + · have ihm : m < (D.take k).length := by omega + have hR' : Root v m (D.take k).length P.dropLast + = some (MTH (D.take k)) := by rw [htklen, hR, hx] + rcases ih v P.dropLast ihm hR' with ⟨hv, hPd⟩ | hc + · left + refine ⟨?_, ?_⟩ + · rw [hv]; exact congrArg hleaf (getD_take D k m hmk) + · have hRHS : Path m D = Path m (D.take k) ++ [MTH (D.drop k)] := by + rw [Path]; simp only [if_neg hgt, ← hkeq, if_pos hmk] + rw [hRHS, hsplit, hPd, hs] + · exact Or.inr hc + · exact Or.inr hc + | case3 m D hgt k hmk ih => + intro v P hm h + have h2 : 2 ≤ D.length := by omega + have hkeq : k = kbelow D.length := rfl + have hkl : k < D.length := by rw [hkeq]; exact kbelow_lt D.length h2 + have hkp : 0 < k := by rw [hkeq]; exact kbelow_pos D.length + have hmk' : ¬ m < kbelow D.length := by rw [← hkeq]; exact hmk + have hdplen : (D.drop k).length = D.length - k := by + simp [List.length_drop] + cases hP : P.getLast? with + | none => + have hPnil : P = [] := by + cases P with + | nil => rfl + | cons a t => simp at hP + subst hPnil + rw [Root] at h + have hn1 : ¬ D.length = 1 := by omega + have hn0 : ¬ D.length = 0 := by omega + simp [hn1, hn0] at h + | some s => + obtain hsplit := eq_dropLast_append_of_getLast? P s hP + rw [hsplit] at h + rw [Root_right _ _ _ _ _ h2 hmk', ← hkeq] at h + cases hR : Root v (m - k) (D.length - k) P.dropLast with + | none => rw [hR] at h; simp at h + | some x => + rw [hR] at h + simp only [Option.map_some, Option.some.injEq] at h + rw [MTH_split D h2, ← hkeq] at h + rcases hnode_inj_or_collision h with ⟨hs, hx⟩ | hc + · have ihm : m - k < (D.drop k).length := by omega + have hR' : Root v (m - k) (D.drop k).length P.dropLast + = some (MTH (D.drop k)) := by rw [hdplen, hR, hx] + rcases ih v P.dropLast ihm hR' with ⟨hv, hPd⟩ | hc + · left + have hidx : k + (m - k) = m := by omega + refine ⟨?_, ?_⟩ + · rw [hv] + have hh : (D.drop k).getD (m - k) [] = D.getD m [] := by + rw [getD_drop, hidx] + exact congrArg hleaf hh + · have hRHS : Path m D = Path (m - k) (D.drop k) ++ [MTH (D.take k)] := by + rw [Path]; simp only [if_neg hgt, ← hkeq, if_neg hmk] + rw [hRHS, hsplit, hPd, hs] + · exact Or.inr hc + · exact Or.inr hc + +/-- **Theorem 2 (Inclusion soundness: position binding)**, paper §6, + constructive form: an accepting receipt whose leaf differs from the + honest leaf at position `m` exhibits a SHA-256 collision. -/ +theorem incl_sound (m : Nat) (D : List Bytes) (hm : m < D.length) + (d : Bytes) (P : List Hash) + (h : Root (hleaf d) m D.length P = some (MTH D)) : + d = D.getD m [] ∨ HasCollision := by + rcases root_binding m D (hleaf d) P hm h with ⟨hv, _⟩ | hc + · exact hleaf_inj_or_collision hv + · exact Or.inr hc + +end LTLAcc diff --git a/verification/Proofs/Binding.olean b/verification/Proofs/Binding.olean new file mode 100644 index 0000000000000000000000000000000000000000..fc0eadef1e2b524493156c2d978c684fc040e779 GIT binary patch literal 288416 zcmbTfd3?;*_dh-)6JsCy*kg}U#!{x1iaqwRl-LtVNG!<^-UhA4RifV=bU@ax##Y$c}>j7h|uWVxf<4K zV6J0+drVk;-M@btgoo7&HCrr08im&nw?u>vji~=lXydT(#t~tShrDA6Z``2XknqM~ zVLIJj>&xakUF6XHZmP89(&?V+p_bGae|LzE8jJtx{B^l=xp>{55a@9XEwD4~*9v7+ zp~&<*7R?MD?Rj!l>WKGGEl>*LpiX~*$ 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