/- 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