From 801ae08fe805ff2c90cb13c8ae8d3bde38f44353 Mon Sep 17 00:00:00 2001 From: mrwulf Date: Sat, 11 Jul 2026 12:06:20 +0200 Subject: [PATCH] L3: Theorem 1 (inclusion completeness) kernel-checked Path (prover-side inclusion path, paper SS5.3) with termination via the kbelow bounds; self-contained list lemmas (getD_take, getD_drop - no stdlib-name dependence); equation lemmas MTH_single/MTH_split/Root_one/ Root_left/Root_right (Option.map form; matcher side conditions closed explicitly); Theorem 1 by functional induction on Path with a k-fold discipline against the let-bound split point. incl_complete cone: [propext, Classical.choice, LTLAcc.sha256, Quot.sound] - pinned exactly in check.sh alongside Path. THE button green end to end. LTL untouched. Co-Authored-By: Claude Fable 5 --- README.md | 2 +- verification/Proofs/AxiomCheck.lean | 3 + verification/Proofs/AxiomCheck.olean | Bin 1312 -> 1424 bytes verification/Proofs/Completeness.lean | 166 +++++++++++++++++++++++++ verification/Proofs/Completeness.olean | Bin 0 -> 443272 bytes verification/check.sh | 4 +- 6 files changed, 173 insertions(+), 2 deletions(-) create mode 100644 verification/Proofs/Completeness.lean create mode 100644 verification/Proofs/Completeness.olean diff --git a/README.md b/README.md index b15c5b9..97b0e1f 100644 --- a/README.md +++ b/README.md @@ -11,7 +11,7 @@ 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) | pending | +| 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 | | L6 | pin-store state machine safety (Proposition 1) | pending | diff --git a/verification/Proofs/AxiomCheck.lean b/verification/Proofs/AxiomCheck.lean index 669e2f3..8c0d204 100644 --- a/verification/Proofs/AxiomCheck.lean +++ b/verification/Proofs/AxiomCheck.lean @@ -1,5 +1,6 @@ /- Axiom-cone observation for the audit (Phase 3 of check.sh). -/ import Proofs.Basic +import Proofs.Completeness #print axioms LTLAcc.domsep #print axioms LTLAcc.kbelow_pos #print axioms LTLAcc.kbelow_lt @@ -7,3 +8,5 @@ import Proofs.Basic #print axioms LTLAcc.MTH #print axioms LTLAcc.Root #print axioms LTLAcc.ConsRec +#print axioms LTLAcc.Path +#print axioms LTLAcc.incl_complete diff --git a/verification/Proofs/AxiomCheck.olean b/verification/Proofs/AxiomCheck.olean index c21ff1c941b6124d6aeec55e1d4e3e94571e8c7b..ad5187d89a51ee537812cff2c4eb1cf629966075 100644 GIT binary patch delta 359 zcmZ3$HGz9V1XBd-#wcCJpr5=95CEfjpnT{2+=86clGMD^;$o-}g8~B+BLicD|Fi@K zh6X4-InL?b(fxYSAT<&Uj0_tl7cz=-vOu-4OrFRn&v;?-O2&953#Q3jOm>_ZOb`P) zCI>RPb8dk0E=*p?la-0QF>5j>U%6(*4{))KEb;*gP~HkC cEx-!V0kYbHftBF})V~T)2cy#(Na{g)0P#;$w*UYD delta 271 zcmbQhy?|>%1k($ajZwOcT;G@(Ab@%DPe%F43QQ49Cm1JJG1+ncU~KT8mcYOuG5H{q zJEsE^gqJW`kl9^n0+hD_N_(G?mV3H-_A`)C3Jgq4511xbG0Sl>FvE;k$n4Ik0OdJM z{>W_3RRHBpV4iHm;>L7>d2$tt8RrkUj-4#-i~*AwS;aXUSYW0ovdRZ+fU2;7iY|cC WAge7HSQ#3i&fb87-hreZqz3>%xkoGj diff --git a/verification/Proofs/Completeness.lean b/verification/Proofs/Completeness.lean new file mode 100644 index 0000000..24ae35e --- /dev/null +++ b/verification/Proofs/Completeness.lean @@ -0,0 +1,166 @@ +/- L3 of the accumulator pyramid: the prover-side Path function and + **Theorem 1 (inclusion completeness)** — honest receipts always verify: + + Root (hleaf D[m]) m |D| (Path m D) = some (MTH D) + + No hash property is used anywhere (the theorem is about shapes). -/ +import Proofs.Basic + +namespace LTLAcc + +/-- `Path` (paper §5.3): the operator's inclusion path for index `m`. + Sibling subtree heads, leaf-to-root order (the paper's `‖ [s]`). -/ +noncomputable def Path (m : Nat) (D : List Bytes) : List Bytes := + if D.length ≤ 1 then [] + else + let k := kbelow D.length + if m < k then Path m (D.take k) ++ [MTH (D.drop k)] + else Path (m - k) (D.drop k) ++ [MTH (D.take k)] +termination_by D.length +decreasing_by + · simp only [List.length_take] + have h2 : 2 ≤ D.length := by omega + have hk := kbelow_lt D.length h2 + omega + · simp only [List.length_drop] + have hp := kbelow_pos D.length + omega + +/-! ### List helper lemmas (self-contained; no stdlib-name dependence) -/ + +theorem getD_take (l : List Bytes) (k m : Nat) (h : m < k) : + (l.take k).getD m [] = l.getD m [] := by + induction l generalizing k m with + | nil => simp + | cons a t ih => + cases k with + | zero => omega + | succ k' => + cases m with + | zero => simp + | succ m' => + simp only [List.take_succ_cons, List.getD_cons_succ] + exact ih k' m' (by omega) + +theorem getD_drop (l : List Bytes) (k i : Nat) : + (l.drop k).getD i [] = l.getD (k + i) [] := by + induction l generalizing k with + | nil => simp + | cons a t ih => + cases k with + | zero => simp + | succ k' => + have hidx : k' + 1 + i = (k' + i) + 1 := by omega + simp only [List.drop_succ_cons, hidx, List.getD_cons_succ] + exact ih k' + +theorem exists_singleton_of_length_one (l : List Bytes) + (h : l.length = 1) : ∃ d, l = [d] := by + cases l with + | nil => simp at h + | cons a t => + cases t with + | nil => exact ⟨a, rfl⟩ + | cons b u => simp at h + +/-! ### Equation lemmas -/ + +theorem MTH_single (d : Bytes) : MTH [d] = hleaf d := by + rw [MTH]; rfl + +theorem MTH_split (D : List Bytes) (h : 2 ≤ D.length) : + MTH D = hnode (MTH (D.take (kbelow D.length))) + (MTH (D.drop (kbelow D.length))) := by + rw [MTH] + have h0 : ¬ D.length = 0 := by omega + have h1 : ¬ D.length = 1 := by omega + simp only [h0, h1, dite_false] + +theorem Root_one (v : Bytes) (m : Nat) : Root v m 1 [] = some v := by + rw [Root]; rfl + +/-- `Root` at a composite size, left branch (`m < k`). -/ +theorem Root_left (v : Bytes) (m n : Nat) (P : List Bytes) (s : Bytes) + (hn : 2 ≤ n) (hm : m < kbelow n) : + Root v m n (P ++ [s]) = (Root v m (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) 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 + +/-- `Root` at a composite size, right branch (`m ≥ k`). -/ +theorem Root_right (v : Bytes) (m n : Nat) (P : List Bytes) (s : Bytes) + (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 + +/-! ### Theorem 1 -/ + +/-- **Theorem 1 (Inclusion completeness)**, paper §6: for every non-empty + leaf list `D` and every `m < |D|`, the honestly produced receipt + verifies to the honest root. -/ +theorem incl_complete (m : Nat) (D : List Bytes) (hm : m < D.length) : + Root (hleaf (D.getD m [])) m D.length (Path m D) = some (MTH D) := by + induction m, D using Path.induct with + | case1 m D hle => + -- |D| ≤ 1 and m < |D| force D = [d], m = 0 + 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 + rw [Path] + simp only [List.length_singleton, if_pos (by omega : (1:Nat) ≤ 1)] + rw [MTH_single] + simpa using Root_one (hleaf d) 0 + | case2 m D hgt k hmk ih => + 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 + rw [Path] + simp only [if_neg hgt, ← hkeq, if_pos hmk] + rw [Root_left _ _ _ _ _ h2 hmk', ← hkeq] + have htklen : (D.take k).length = k := by + simp [List.length_take]; omega + have ihm : m < (D.take k).length := by omega + have hrec := ih ihm + rw [getD_take D k m hmk, htklen] at hrec + rw [hrec, MTH_split D h2, ← hkeq] + rfl + | case3 m D hgt k hmk ih => + 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 + rw [Path] + simp only [if_neg hgt, ← hkeq, if_neg hmk] + rw [Root_right _ _ _ _ _ h2 hmk', ← hkeq] + have hidx : k + (m - k) = m := by omega + have hget : (D.drop k).getD (m - k) [] = D.getD m [] := by + rw [getD_drop, hidx] + have hdplen : (D.drop k).length = D.length - k := by + simp [List.length_drop] + have ihm : m - k < (D.drop k).length := by omega + have hrec := ih ihm + rw [hget, hdplen] at hrec + rw [hrec, MTH_split D h2, ← hkeq] + rfl + +end LTLAcc diff --git a/verification/Proofs/Completeness.olean b/verification/Proofs/Completeness.olean new file mode 100644 index 0000000000000000000000000000000000000000..95ee8a96f191dac1cdc73f0ef18b08140d71481b GIT binary patch literal 443272 zcmbTfd0>p!_Xj*;5)*5zjeYP%Xld*zT4Rr`#vY1H5XP2-P@%<8OYEVR)Yi1N*kh|G zYV1);EUl=j)Eay2q15}_c|M zwF0XLRvX^8rs1EjTK)Rg2n!4f>QlF0&3-}QVg18v)(NZEw_m;RzV-Ul3F=p`R*gRW z>h3WDo=(?KcG029?Fr@ zk0xQ!12v~zm(pX%q5E$J{j5-?|E4+@%PR(DXE55O{hIz~Idv5V(nQ(;YNwpW_g{Xs z?cBDp8U=Mpzmg!|z^91M_QDPRRJhWUTm477NPYDm4FkeQGz%ROHaHwVE28wJK~ENF z-_u{ts(q&9MujVVdDMTjintKoX{oIKq_mvK%FTeVa 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GEN_MODULES=( LTLAcc/HashExternal ) -PROOFS=( Basic ) +PROOFS=( Basic Completeness ) # Certificates and their exact expected cones (observed at first green # compile, 2026-07-10; any drift in EITHER direction is a failure). @@ -29,6 +29,8 @@ declare -A CONES=( [LTLAcc.MTH]="propext, LTLAcc.sha256, Quot.sound" [LTLAcc.Root]="propext, LTLAcc.sha256, Quot.sound" [LTLAcc.ConsRec]="propext, LTLAcc.sha256, Quot.sound" + [LTLAcc.Path]="propext, LTLAcc.sha256, Quot.sound" + [LTLAcc.incl_complete]="propext, Classical.choice, LTLAcc.sha256, Quot.sound" ) free -m | awk '/Mem:/{if($7<2048){print "FATAL: <2GB RAM available — refusing to compile"; exit 1}}'