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+(*** Montgomery Multiplication *)
+(** This file implements Montgomery Form, Montgomery Reduction, and
+    Montgomery Multiplication on [ZLikeOps].  We follow [Montgomery/Z.v]. *)
+Require Import Coq.ZArith.ZArith Coq.Lists.List Coq.Classes.Morphisms Coq.micromega.Psatz.
+Require Import Crypto.Arithmetic.MontgomeryReduction.Definition.
+Require Import Crypto.Arithmetic.MontgomeryReduction.Proofs.
+Require Import Crypto.LegacyArithmetic.ZBounded.
+Require Import Crypto.Util.ZUtil.
+Require Import Crypto.Util.Tactics.Test.
+Require Import Crypto.Util.Tactics.Not.
+Require Import Crypto.Util.LetIn.
+Require Import Crypto.Util.Notations.
+
+Local Open Scope small_zlike_scope.
+Local Open Scope large_zlike_scope.
+Local Open Scope Z_scope.
+
+Section montgomery.
+  Context (small_bound modulus : Z) {ops : ZLikeOps small_bound small_bound modulus} {props : ZLikeProperties ops}
+          (modulus' : SmallT)
+          (modulus'_valid : small_valid modulus')
+          (modulus_nonzero : modulus <> 0).
+
+  (** pull out a common subexpression *)
+  Local Ltac cse :=
+    let RHS := match goal with |- _ = ?decode ?RHS /\ _ => RHS end in
+    let v := fresh in
+    match RHS with
+    | context[?e] => not is_var e; set (v := e) at 1 2; test clearbody v
+    end;
+    revert v;
+    match goal with
+    | [ |- let v := ?val in ?LHS = ?decode ?RHS /\ ?P ]
+      => change (LHS = decode (dlet v := val in RHS) /\ P)
+    end.
+
+  Definition partial_reduce : forall v : LargeT,
+      { partial_reduce : SmallT
+      | large_valid v
+        -> decode_small partial_reduce = MontgomeryReduction.Definition.partial_reduce modulus small_bound (decode_small modulus') (decode_large v)
+           /\ small_valid partial_reduce }.
+  Proof.
+    intro T. evar (pr : SmallT); exists pr. intros T_valid.
+    assert (0 <= decode_large T < small_bound * small_bound) by auto using decode_large_valid.
+    assert (0 <= decode_small (Mod_SmallBound T) < small_bound) by auto using decode_small_valid, Mod_SmallBound_valid.
+    assert (0 <= decode_small modulus' < small_bound) by auto using decode_small_valid.
+    assert (0 <= decode_small modulus_digits < small_bound) by auto using decode_small_valid, modulus_digits_valid.
+    assert (0 <= modulus) by apply (modulus_nonneg _).
+    assert (modulus < small_bound) by (rewrite <- modulus_digits_correct; omega).
+    rewrite <- partial_reduce_alt_eq by omega.
+    cbv [MontgomeryReduction.Definition.partial_reduce MontgomeryReduction.Definition.partial_reduce_alt MontgomeryReduction.Definition.prereduce].
+    pull_zlike_decode.
+    cse.
+    subst pr; split; [ reflexivity | exact _ ].
+  Defined.
+
+  Definition reduce_via_partial : forall v : LargeT,
+      { reduce : SmallT
+      | large_valid v
+        -> decode_small reduce = MontgomeryReduction.Definition.reduce_via_partial modulus small_bound (decode_small modulus') (decode_large v)
+           /\ small_valid reduce }.
+  Proof.
+    intro T. evar (pr : SmallT); exists pr. intros T_valid.
+    assert (0 <= decode_large T < small_bound * small_bound) by auto using decode_large_valid.
+    assert (0 <= decode_small (Mod_SmallBound T) < small_bound) by auto using decode_small_valid, Mod_SmallBound_valid.
+    assert (0 <= decode_small modulus' < small_bound) by auto using decode_small_valid.
+    assert (0 <= decode_small modulus_digits < small_bound) by auto using decode_small_valid, modulus_digits_valid.
+    assert (0 <= modulus) by apply (modulus_nonneg _).
+    assert (modulus < small_bound) by (rewrite <- modulus_digits_correct; omega).
+    unfold reduce_via_partial.
+    rewrite <- partial_reduce_alt_eq by omega.
+    cbv [MontgomeryReduction.Definition.partial_reduce MontgomeryReduction.Definition.partial_reduce_alt MontgomeryReduction.Definition.prereduce].
+    pull_zlike_decode.
+    cse.
+    subst pr; split; [ reflexivity | exact _ ].
+  Defined.
+
+  Section correctness.
+    Context (R' : Z)
+            (Hmod : Z.equiv_modulo modulus (small_bound * R') 1)
+            (Hmod' : Z.equiv_modulo small_bound (modulus * (decode_small modulus')) (-1))
+            (v : LargeT)
+            (H : large_valid v)
+            (Hv : 0 <= decode_large v <= small_bound * modulus).
+    Lemma reduce_via_partial_correct'
+      : Z.equiv_modulo modulus
+                       (decode_small (proj1_sig (reduce_via_partial v)))
+                       (decode_large v * R')
+        /\ Z.min 0 (small_bound - modulus) <= (decode_small (proj1_sig (reduce_via_partial v))) < modulus.
+    Proof using H Hmod Hmod' Hv.
+      rewrite (proj1 (proj2_sig (reduce_via_partial v) H)).
+      eauto 6 using reduce_via_partial_correct, reduce_via_partial_in_range, decode_small_valid.
+    Qed.
+
+    Lemma reduce_via_partial_correct''
+      : Z.equiv_modulo modulus
+                       (decode_small (proj1_sig (reduce_via_partial v)))
+                       (decode_large v * R')
+        /\ 0 <= (decode_small (proj1_sig (reduce_via_partial v))) < modulus.
+    Proof using H Hmod Hmod' Hv.
+      pose proof (proj2 (proj2_sig (reduce_via_partial v) H)) as H'.
+      apply decode_small_valid in H'.
+      destruct reduce_via_partial_correct'; split; eauto; omega.
+    Qed.
+
+    Theorem reduce_via_partial_correct
+      : decode_small (proj1_sig (reduce_via_partial v)) = (decode_large v * R') mod modulus.
+    Proof using H Hmod Hmod' Hv.
+      rewrite <- (proj1 reduce_via_partial_correct'').
+      rewrite Z.mod_small by apply reduce_via_partial_correct''.
+      reflexivity.
+    Qed.
+  End correctness.
+End montgomery.