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fips205-slhdsa-verified/verification/Proofs/ForsOuterSpec.lean

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phase 2: FIFTH CERTIFICATE — FORS pk-from-sig (Algorithm 17), inner + outer loops Two theorems, split into two files (METHOD-4 discipline — each proof a clean unit). NB: an early single-file/bare-rfl attempt appeared to "OOM at the clamp", but that memory pressure was a SYMPTOM of the runaway whnf diagnosed below, not a real memory need — the fixed proofs compile in seconds at the default caps. fips205.fors_inner_loop_eq (Proofs/ForsInnerSpec.lean): the extracted inner Merkle auth-path loop for ONE FORS tree (fors_pk_from_sig_free_loop0_loop0) equals the explicit auth-path fold — at level j set tree height j+1, test bit j of THIS tree's leaf index indices[i], hash the current node with auth.tree[j] in the bit order (even: node||auth[j]; odd: auth[j]||node), halving the tree index. Structurally the XMSS auth-path loop, but the bit source is indices[i]>>j and the loop returns the (adrs,node) pair. Cone: kernel-3 + verify_mono.oracle.h. fips205.fors_outer_loop_eq (Proofs/ForsOuterSpec.lean): the extracted outer per-tree loop (fors_pk_from_sig_free_loop0) equals the explicit K-tree fold — for each tree i, compute the leaf with F at tree index (i<<a)+indices[i], run the inner Merkle loop over the A levels, write the result to root[i]. Consumes the inner loop as an opaque sub-call. Cone: kernel-3 + verify_mono.oracle.{f,h} (F per leaf; H transitively through the inner loop). Fidelity review at authorship (three-way, both loops): extracted bodies (gen Funs.lean 893-933 inner, 954-985 outer) == Rust verify_mono.rs fors_pk_from_sig_free (verbatim from upstream fors.rs, hash calls -> oracle) == FIPS 205 Algorithm 17, incl. the even/odd sibling order and the (i<<a)+indices[i] leaf index. Proof: the branched-Merkle recipe (XMSS) for the inner loop (by_cases on the index bit, pair-bind matcher made concrete via bind_congr+rintro then full simp); the HT straight-line recipe for the outer loop, adapted (bind_congr-peeled step lemma + bind_congr x16 induction, both threading the inner-loop sub-call opaquely). loop_unfold_bind / u32_succ / fwd_succ / hnext reused verbatim from ChainSpec. check.sh: PROOFS += ForsInnerSpec, ForsOuterSpec; CERTS += the two fors certs; audit imports both; check.sh settings unchanged (400s/4096MB). ForsOuterSpec compiles in 4.4s / 2.4GB after the fix below. check.sh green over ALL SIX certificates with the axiom audit. README status -> FIVE certificates. DIAGNOSIS NOTE (honesty): ForsOuterSpec's fors_outer_step first closed with a bare `rfl`, which whnf'd the whole 16-bind body INCLUDING the inner-loop `loop` term and hit a DETERMINISTIC 4M-heartbeat timeout (never actually passed — an earlier "green" reading was a misread wrapper exit code; the real error was hidden by check.sh piping per-file output to /dev/null). Fix: peel the 16 binds with bind_congr so the closing rfl only sees the small loop-tail, and close the post-pair-rintro tail with a full simp (the pair `let` won't iota via simp only). This is the HtSpec straight-line recipe adapted for a body that nests a loop. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
2026-07-23 21:56:56 +00:00
/- Proofs/ForsOuterSpec.lean — FORS outer per-tree loop (Algorithm 17).
THEOREM fors_outer_loop_eq: the extracted outer loop
(fors_pk_from_sig_free_loop0) equals the explicit K-tree fold — for each
tree i, compute the leaf with F at tree index (i << a) + indices[i], run
the inner Merkle loop over the A levels, and write the result to root[i].
Straight-line body (like HtSpec) that consumes the inner loop
(fors_pk_from_sig_free_loop0_loop0) as an opaque sub-call — its own
fidelity is fors_inner_loop_eq in ForsInnerSpec. Cone: kernel three +
verify_mono.oracle.{f, h} (F for each leaf; H reached transitively through
the referenced inner loop).
Split from the inner loop (ForsInnerSpec) so each file stays under the
memory ceiling (METHOD-4 file split). Reuses u32_succ / fwd_succ / hnext /
loop_unfold_bind from ChainSpec.
-/
import Proofs.ChainSpec
open Aeneas Aeneas.Std Result ControlFlow
open fips205
set_option maxHeartbeats 4000000
namespace fips205
/-- Outer loop body on a non-empty range, resolved to the successor w. -/
theorem hbody_fo {A K N : Std.Usize} (sig_fors : types.ForsSig A K N) (pk_seed : Slice Std.U8)
(a32 : Std.U32) (indices : Array Std.U32 K) (start stop w : Std.U32)
(adrs : types.Adrs) (root : Array (Array Std.U8 N) K)
(hd : decide (start.val < stop.val) = true) (hwok : start + 1#u32 = ok w) :
verify_mono.fors_pk_from_sig_free_loop0.body sig_fors pk_seed a32 indices
{ start := start, «end» := stop } adrs root
= (do
let i1 ← lift (UScalar.cast .Usize start)
let sk ← Array.index_usize sig_fors.private_key_value i1
let adrs1 ← helpers.Adrs.set_tree_height adrs 0#u32
let i2 ← start <<< a32
let i3 ← lift (UScalar.cast .Usize start)
let i4 ← Array.index_usize indices i3
let i5 ← i2 + i4
let adrs2 ← helpers.Adrs.set_tree_index adrs1 i5
let s ← lift (Array.to_slice sk)
let node_0 ← verify_mono.oracle.f N pk_seed adrs2 s
let i6 ← lift (UScalar.cast .Usize start)
let a ← Array.index_usize sig_fors.auth i6
let auth ← types.Auth.Insts.CoreCloneClone.clone a
let (adrs3, node_01) ←
verify_mono.fors_pk_from_sig_free_loop0_loop0
{ start := 0#u32, «end» := a32 } pk_seed adrs2 indices start node_0 auth
let i7 ← lift (UScalar.cast .Usize start)
let a1 ← Array.update root i7 node_01
ok (cont (({ start := w, «end» := stop } : core.ops.range.Range Std.U32), adrs3, a1))) := by
unfold verify_mono.fors_pk_from_sig_free_loop0.body
rw [hnext hd hwok]
simp
/-- The outer per-tree fold: at tree i, F-leaf then the inner Merkle loop,
writing root[i]. The inner loop is consumed as an opaque sub-call. -/
noncomputable def forsOuterFold {A K N : Std.Usize} (sig_fors : types.ForsSig A K N)
(pk_seed : Slice Std.U8) (a32 : Std.U32) (indices : Array Std.U32 K) :
types.Adrs → Array (Array Std.U8 N) K → Std.U32 → Nat →
Result (types.Adrs × Array (Array Std.U8 N) K)
| adrs, root, _, 0 => ok (adrs, root)
| adrs, root, i, (s+1) => do
let i1 ← lift (UScalar.cast .Usize i)
let sk ← Array.index_usize sig_fors.private_key_value i1
let adrs1 ← helpers.Adrs.set_tree_height adrs 0#u32
let i2 ← i <<< a32
let i3 ← lift (UScalar.cast .Usize i)
let i4 ← Array.index_usize indices i3
let i5 ← i2 + i4
let adrs2 ← helpers.Adrs.set_tree_index adrs1 i5
let s0 ← lift (Array.to_slice sk)
let node_0 ← verify_mono.oracle.f N pk_seed adrs2 s0
let i6 ← lift (UScalar.cast .Usize i)
let a ← Array.index_usize sig_fors.auth i6
let auth ← types.Auth.Insts.CoreCloneClone.clone a
let (adrs3, node_01) ←
verify_mono.fors_pk_from_sig_free_loop0_loop0
{ start := 0#u32, «end» := a32 } pk_seed adrs2 indices i node_0 auth
let i7 ← lift (UScalar.cast .Usize i)
let a1 ← Array.update root i7 node_01
let w ← i + 1#u32
forsOuterFold sig_fors pk_seed a32 indices adrs3 a1 w s
/-- One outer loop step = one fold step (straight-line body). -/
theorem fors_outer_step {A K N : Std.Usize} (sig_fors : types.ForsSig A K N) (pk_seed : Slice Std.U8)
(a32 : Std.U32) (indices : Array Std.U32 K) (start stop : Std.U32)
(adrs : types.Adrs) (root : Array (Array Std.U8 N) K)
(hlt : start.val < stop.val) (hb : start.val + 1 < 2 ^ 32) :
verify_mono.fors_pk_from_sig_free_loop0 { start := start, «end» := stop }
sig_fors pk_seed a32 adrs indices root
= (do
let i1 ← lift (UScalar.cast .Usize start)
let sk ← Array.index_usize sig_fors.private_key_value i1
let adrs1 ← helpers.Adrs.set_tree_height adrs 0#u32
let i2 ← start <<< a32
let i3 ← lift (UScalar.cast .Usize start)
let i4 ← Array.index_usize indices i3
let i5 ← i2 + i4
let adrs2 ← helpers.Adrs.set_tree_index adrs1 i5
let s ← lift (Array.to_slice sk)
let node_0 ← verify_mono.oracle.f N pk_seed adrs2 s
let i6 ← lift (UScalar.cast .Usize start)
let a ← Array.index_usize sig_fors.auth i6
let auth ← types.Auth.Insts.CoreCloneClone.clone a
let (adrs3, node_01) ←
verify_mono.fors_pk_from_sig_free_loop0_loop0
{ start := 0#u32, «end» := a32 } pk_seed adrs2 indices start node_0 auth
let i7 ← lift (UScalar.cast .Usize start)
let a1 ← Array.update root i7 node_01
let w ← start + 1#u32
verify_mono.fors_pk_from_sig_free_loop0 { start := w, «end» := stop }
sig_fors pk_seed a32 adrs3 indices a1) := by
obtain ⟨w, hwok, _⟩ := u32_succ hb
have hd : decide (start.val < stop.val) = true := by simp [hlt]
conv_lhs => rw [verify_mono.fors_pk_from_sig_free_loop0, loop_unfold_bind]
dsimp only
rw [hbody_fo sig_fors pk_seed a32 indices start stop w adrs root hd hwok]
simp only [bind_assoc, bind_ok]
conv_rhs => rw [show (start + 1#u32) = ok w from hwok]
simp only [bind_tc_ok, bind_ok]
-- Peel all 16 binds with bind_congr so the closing `rfl` only whnf's the
-- small loop-tail, not the whole body threading the inner `loop` term
-- (a bare `rfl` here blows the 4M-heartbeat whnf budget — the HtSpec body had
-- no nested loop, so its rfl was cheap; the FORS outer body does).
apply bind_congr; intro i1
apply bind_congr; intro sk
apply bind_congr; intro adrs1
apply bind_congr; intro i2
apply bind_congr; intro i3
apply bind_congr; intro i4
apply bind_congr; intro i5
apply bind_congr; intro adrs2
apply bind_congr; intro s0
apply bind_congr; intro node_0
apply bind_congr; intro i6
apply bind_congr; intro a
apply bind_congr; intro auth
apply bind_congr; rintro ⟨adrs3, node_01⟩
-- the pair `let` blocks further bind_congr (won't iota via simp only); the
-- remaining tail (i7, a1, loop recursion) is small, so a full simp closes it
-- cheaply — no whnf over the body, so no heartbeat blowup
simp [bind_assoc, bind_ok, verify_mono.fors_pk_from_sig_free_loop0]
/-- **Algorithm 17 outer fidelity.** The extracted per-tree loop over
[start, start+s) equals the explicit K-tree fold. -/
theorem fors_outer_loop_eq {A K N : Std.Usize} (sig_fors : types.ForsSig A K N)
(pk_seed : Slice Std.U8) (a32 : Std.U32) (indices : Array Std.U32 K) (s : Nat) :
∀ (start : Std.U32) (adrs : types.Adrs) (root : Array (Array Std.U8 N) K),
start.val + s < 2 ^ 32 →
∀ (stop : Std.U32), stop.val = start.val + s →
verify_mono.fors_pk_from_sig_free_loop0 { start := start, «end» := stop }
sig_fors pk_seed a32 adrs indices root
= forsOuterFold sig_fors pk_seed a32 indices adrs root start s := by
induction s with
| zero =>
intro start adrs root _ stop hstop
have hse : start = stop := by apply Std.UScalar.eq_of_val_eq; omega
subst hse
unfold verify_mono.fors_pk_from_sig_free_loop0 forsOuterFold
rw [loop.eq_1]
unfold verify_mono.fors_pk_from_sig_free_loop0.body core.iter.range.IteratorRange.next
simp [core.cmp.impls.PartialOrdU32.lt]
| succ k ih =>
intro start adrs root hb stop hstop
have hlt : start.val < stop.val := by omega
have hb1 : start.val + 1 < 2 ^ 32 := by omega
obtain ⟨w, hwok, hwv⟩ := u32_succ hb1
rw [fors_outer_step sig_fors pk_seed a32 indices start stop adrs root hlt hb1]
unfold forsOuterFold
rw [hwok]
simp only [bind_tc_ok, bind_ok]
have hbound : w.val + k < 2 ^ 32 := by omega
have hstop' : stop.val = w.val + k := by omega
apply bind_congr; intro i1
apply bind_congr; intro sk
apply bind_congr; intro adrs1
apply bind_congr; intro i2
apply bind_congr; intro i3
apply bind_congr; intro i4
apply bind_congr; intro i5
apply bind_congr; intro adrs2
apply bind_congr; intro s0
apply bind_congr; intro node_0
apply bind_congr; intro i6
apply bind_congr; intro a
apply bind_congr; intro auth
apply bind_congr; rintro ⟨adrs3, node_01⟩
apply bind_congr; intro i7
apply bind_congr; intro a1
exact ih w adrs3 a1 hbound stop hstop'
end fips205
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