Split off more of ZUtil

16 files changed, 1238 insertions, 0 deletions

diff --git a/src/Util/ZUtil/Div.v b/src/Util/ZUtil/Div.v
new file mode 100644
index 000000000..325818a2c
--- /dev/null
+++ b/src/Util/ZUtil/Div.v
@@ -0,0 +1,36 @@
+Require Import Coq.ZArith.ZArith Coq.micromega.Lia.
+Require Import Crypto.Util.ZUtil.Tactics.CompareToSgn.
+Require Import Crypto.Util.ZUtil.Hints.Core.
+Local Open Scope Z_scope.
+
+Module Z.
+  Lemma div_mul' : forall a b : Z, b <> 0 -> (b * a) / b = a.
+  Proof. intros. rewrite Z.mul_comm. apply Z.div_mul; auto. Qed.
+  Hint Rewrite div_mul' using zutil_arith : zsimplify.
+
+  Local Ltac replace_to_const c :=
+    repeat match goal with
+           | [ H : ?x = ?x |- _ ] => clear H
+           | [ H : ?x = c, H' : context[?x] |- _ ] => rewrite H in H'
+           | [ H : c = ?x, H' : context[?x] |- _ ] => rewrite <- H in H'
+           | [ H : ?x = c |- context[?x] ] => rewrite H
+           | [ H : c = ?x |- context[?x] ] => rewrite <- H
+           end.
+
+  Lemma lt_div_0 n m : n / m < 0 <-> ((n < 0 < m \/ m < 0 < n) /\ 0 < -(n / m)).
+  Proof.
+    Z.compare_to_sgn; rewrite Z.sgn_opp; simpl.
+    pose proof (Zdiv_sgn n m) as H.
+    pose proof (Z.sgn_spec (n / m)) as H'.
+    repeat first [ progress intuition auto
+                 | progress simpl in *
+                 | congruence
+                 | lia
+                 | progress replace_to_const (-1)
+                 | progress replace_to_const 0
+                 | progress replace_to_const 1
+                 | match goal with
+                   | [ x : Z |- _ ] => destruct x
+                   end ].
+  Qed.
+End Z.
diff --git a/src/Util/ZUtil/Tactics.v b/src/Util/ZUtil/Tactics.v
new file mode 100644
index 000000000..3700611df
--- /dev/null
+++ b/src/Util/ZUtil/Tactics.v
@@ -0,0 +1,11 @@
+Require Export Crypto.Util.ZUtil.Tactics.CompareToSgn.
+Require Export Crypto.Util.ZUtil.Tactics.DivideExistsMul.
+Require Export Crypto.Util.ZUtil.Tactics.DivModToQuotRem.
+Require Export Crypto.Util.ZUtil.Tactics.LinearSubstitute.
+Require Export Crypto.Util.ZUtil.Tactics.LtbToLt.
+Require Export Crypto.Util.ZUtil.Tactics.PeelLe.
+Require Export Crypto.Util.ZUtil.Tactics.PrimeBound.
+Require Export Crypto.Util.ZUtil.Tactics.ReplaceNegWithPos.
+Require Export Crypto.Util.ZUtil.Tactics.SimplifyFractionsLe.
+Require Export Crypto.Util.ZUtil.Tactics.ZeroBounds.
+Require Export Crypto.Util.ZUtil.Tactics.Ztestbit.
diff --git a/src/Util/ZUtil/Tactics/CompareToSgn.v b/src/Util/ZUtil/Tactics/CompareToSgn.v
new file mode 100644
index 000000000..31588815b
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/CompareToSgn.v
@@ -0,0 +1,8 @@
+Require Import Coq.ZArith.ZArith.
+Module Z.
+  Ltac compare_to_sgn :=
+    repeat match goal with
+           | [ H : _ |- _ ] => progress rewrite <- ?Z.sgn_neg_iff, <- ?Z.sgn_pos_iff, <- ?Z.sgn_null_iff in H
+           | _ => progress rewrite <- ?Z.sgn_neg_iff, <- ?Z.sgn_pos_iff, <- ?Z.sgn_null_iff
+           end.
+End Z.
diff --git a/src/Util/ZUtil/Tactics/DivModToQuotRem.v b/src/Util/ZUtil/Tactics/DivModToQuotRem.v
new file mode 100644
index 000000000..b37047397
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/DivModToQuotRem.v
@@ -0,0 +1,40 @@
+Require Import Coq.ZArith.ZArith.
+Require Import Crypto.Util.ZUtil.Hints.Core.
+Local Open Scope Z_scope.
+
+Module Z.
+  (** [div_mod_to_quot_rem] replaces [x / y] and [x mod y] with new
+      variables [q] and [r] while simultaneously adding facts that
+      uniquely specify [q] and [r] to the context (roughly, that [y *
+      q + r = x] and that [0 <= r < y]. *)
+  Ltac div_mod_to_quot_rem_inequality_solver :=
+    zutil_arith_more_inequalities.
+  Ltac generalize_div_eucl x y :=
+    let H := fresh in
+    let H' := fresh in
+    assert (H' : y <> 0) by div_mod_to_quot_rem_inequality_solver;
+    generalize (Z.div_mod x y H'); clear H';
+    first [ assert (H' : 0 < y) by div_mod_to_quot_rem_inequality_solver;
+            generalize (Z.mod_pos_bound x y H'); clear H'
+          | assert (H' : y < 0) by div_mod_to_quot_rem_inequality_solver;
+            generalize (Z.mod_neg_bound x y H'); clear H'
+          | assert (H' : y < 0 \/ 0 < y) by (apply Z.neg_pos_cases; div_mod_to_quot_rem_inequality_solver);
+            let H'' := fresh in
+            assert (H'' : y < x mod y <= 0 \/ 0 <= x mod y < y)
+              by (destruct H'; [ left; apply Z.mod_neg_bound; assumption
+                               | right; apply Z.mod_pos_bound; assumption ]);
+            clear H'; revert H'' ];
+    let q := fresh "q" in
+    let r := fresh "r" in
+    set (q := x / y);
+    set (r := x mod y);
+    clearbody q r.
+
+  Ltac div_mod_to_quot_rem_step :=
+    match goal with
+    | [ |- context[?x / ?y] ] => generalize_div_eucl x y
+    | [ |- context[?x mod ?y] ] => generalize_div_eucl x y
+    end.
+
+  Ltac div_mod_to_quot_rem := repeat div_mod_to_quot_rem_step; intros.
+End Z.
diff --git a/src/Util/ZUtil/Tactics/DivideExistsMul.v b/src/Util/ZUtil/Tactics/DivideExistsMul.v
new file mode 100644
index 000000000..07cebc8f8
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/DivideExistsMul.v
@@ -0,0 +1,14 @@
+Require Import Coq.ZArith.ZArith Coq.omega.Omega.
+Local Open Scope Z_scope.
+
+Module Z.
+  Ltac divide_exists_mul := let k := fresh "k" in
+  match goal with
+  | [ H : (?a | ?b) |- _ ] => apply Z.mod_divide in H; try apply Zmod_divides in H;
+                              match type of H with
+                              | ex _ => destruct H as [k H]
+                              | _ => destruct H
+                              end
+  | [ |- (?a | ?b) ] => apply Z.mod_divide; try apply Zmod_divides
+  end; (omega || auto).
+End Z.
diff --git a/src/Util/ZUtil/Tactics/LinearSubstitute.v b/src/Util/ZUtil/Tactics/LinearSubstitute.v
new file mode 100644
index 000000000..d03c9d196
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/LinearSubstitute.v
@@ -0,0 +1,66 @@
+Require Import Coq.omega.Omega Coq.ZArith.ZArith.
+Require Import Crypto.Util.Tactics.Contains.
+Require Import Crypto.Util.Tactics.Not.
+Local Open Scope Z_scope.
+
+Module Z.
+  Lemma move_R_pX x y z : x + y = z -> x = z - y.
+  Proof. omega. Qed.
+  Lemma move_R_mX x y z : x - y = z -> x = z + y.
+  Proof. omega. Qed.
+  Lemma move_R_Xp x y z : y + x = z -> x = z - y.
+  Proof. omega. Qed.
+  Lemma move_R_Xm x y z : y - x = z -> x = y - z.
+  Proof. omega. Qed.
+  Lemma move_L_pX x y z : z = x + y -> z - y = x.
+  Proof. omega. Qed.
+  Lemma move_L_mX x y z : z = x - y -> z + y = x.
+  Proof. omega. Qed.
+  Lemma move_L_Xp x y z : z = y + x -> z - y = x.
+  Proof. omega. Qed.
+  Lemma move_L_Xm x y z : z = y - x -> y - z = x.
+  Proof. omega. Qed.
+
+  (** [linear_substitute x] attempts to maipulate equations using only
+      addition and subtraction to put [x] on the left, and then
+      eliminates [x].  Currently, it only handles equations where [x]
+      appears once; it does not yet handle equations like [x - x + x =
+      5]. *)
+  Ltac linear_solve_for_in_step for_var H :=
+    let LHS := match type of H with ?LHS = ?RHS => LHS end in
+    let RHS := match type of H with ?LHS = ?RHS => RHS end in
+    first [ match RHS with
+            | ?a + ?b
+              => first [ contains for_var b; apply move_L_pX in H
+                       | contains for_var a; apply move_L_Xp in H ]
+            | ?a - ?b
+              => first [ contains for_var b; apply move_L_mX in H
+                       | contains for_var a; apply move_L_Xm in H ]
+            | for_var
+              => progress symmetry in H
+            end
+          | match LHS with
+            | ?a + ?b
+              => first [ not contains for_var b; apply move_R_pX in H
+                       | not contains for_var a; apply move_R_Xp in H ]
+            | ?a - ?b
+              => first [ not contains for_var b; apply move_R_mX in H
+                       | not contains for_var a; apply move_R_Xm in H ]
+            end ].
+  Ltac linear_solve_for_in for_var H := repeat linear_solve_for_in_step for_var H.
+  Ltac linear_solve_for for_var :=
+    match goal with
+    | [ H : for_var = _ |- _ ] => idtac
+    | [ H : context[for_var] |- _ ]
+      => linear_solve_for_in for_var H;
+         lazymatch type of H with
+         | for_var = _ => idtac
+         | ?T => fail 0 "Could not fully solve for" for_var "in" T "(hypothesis" H ")"
+         end
+    end.
+  Ltac linear_substitute for_var := linear_solve_for for_var; subst for_var.
+  Ltac linear_substitute_all :=
+    repeat match goal with
+           | [ v : Z |- _ ] => linear_substitute v
+           end.
+End Z.
diff --git a/src/Util/ZUtil/Tactics/LtbToLt.v b/src/Util/ZUtil/Tactics/LtbToLt.v
new file mode 100644
index 000000000..df6eae383
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/LtbToLt.v
@@ -0,0 +1,76 @@
+Require Import Coq.ZArith.ZArith.
+Require Import Crypto.Util.Bool.
+Local Open Scope Z_scope.
+
+Module Z.
+  Lemma eqb_cases x y : if x =? y then x = y else x <> y.
+  Proof.
+    pose proof (Z.eqb_spec x y) as H.
+    inversion H; trivial.
+  Qed.
+
+  Lemma geb_spec0 : forall x y : Z, Bool.reflect (x >= y) (x >=? y).
+  Proof.
+    intros x y; pose proof (Zge_cases x y) as H; destruct (Z.geb x y); constructor; omega.
+  Qed.
+  Lemma gtb_spec0 : forall x y : Z, Bool.reflect (x > y) (x >? y).
+  Proof.
+    intros x y; pose proof (Zgt_cases x y) as H; destruct (Z.gtb x y); constructor; omega.
+  Qed.
+
+  Ltac ltb_to_lt_with_hyp H lem :=
+    let H' := fresh in
+    rename H into H';
+    pose proof lem as H;
+    rewrite H' in H;
+    clear H'.
+
+  Ltac ltb_to_lt_in_goal b' lem :=
+    refine (proj1 (@reflect_iff_gen _ _ lem b') _);
+    cbv beta iota.
+
+  Ltac ltb_to_lt_hyps_step :=
+    match goal with
+    | [ H : (?x <? ?y) = ?b |- _ ]
+      => ltb_to_lt_with_hyp H (Zlt_cases x y)
+    | [ H : (?x <=? ?y) = ?b |- _ ]
+      => ltb_to_lt_with_hyp H (Zle_cases x y)
+    | [ H : (?x >? ?y) = ?b |- _ ]
+      => ltb_to_lt_with_hyp H (Zgt_cases x y)
+    | [ H : (?x >=? ?y) = ?b |- _ ]
+      => ltb_to_lt_with_hyp H (Zge_cases x y)
+    | [ H : (?x =? ?y) = ?b |- _ ]
+      => ltb_to_lt_with_hyp H (eqb_cases x y)
+    end.
+  Ltac ltb_to_lt_goal_step :=
+    match goal with
+    | [ |- (?x <? ?y) = ?b ]
+      => ltb_to_lt_in_goal b (Z.ltb_spec0 x y)
+    | [ |- (?x <=? ?y) = ?b ]
+      => ltb_to_lt_in_goal b (Z.leb_spec0 x y)
+    | [ |- (?x >? ?y) = ?b ]
+      => ltb_to_lt_in_goal b (Z.gtb_spec0 x y)
+    | [ |- (?x >=? ?y) = ?b ]
+      => ltb_to_lt_in_goal b (Z.geb_spec0 x y)
+    | [ |- (?x =? ?y) = ?b ]
+      => ltb_to_lt_in_goal b (Z.eqb_spec x y)
+    end.
+  Ltac ltb_to_lt_step :=
+    first [ ltb_to_lt_hyps_step
+          | ltb_to_lt_goal_step ].
+  Ltac ltb_to_lt := repeat ltb_to_lt_step.
+
+  Section R_Rb.
+    Local Ltac t := intros ? ? []; split; intro; ltb_to_lt; omega.
+    Local Notation R_Rb Rb R nR := (forall x y b, Rb x y = b <-> if b then R x y else nR x y).
+    Lemma ltb_lt_iff : R_Rb Z.ltb Z.lt Z.ge. Proof. t. Qed.
+    Lemma leb_le_iff : R_Rb Z.leb Z.le Z.gt. Proof. t. Qed.
+    Lemma gtb_gt_iff : R_Rb Z.gtb Z.gt Z.le. Proof. t. Qed.
+    Lemma geb_ge_iff : R_Rb Z.geb Z.ge Z.lt. Proof. t. Qed.
+    Lemma eqb_eq_iff : R_Rb Z.eqb (@Logic.eq Z) (fun x y => x <> y). Proof. t. Qed.
+  End R_Rb.
+  Hint Rewrite ltb_lt_iff leb_le_iff gtb_gt_iff geb_ge_iff eqb_eq_iff : ltb_to_lt.
+  Ltac ltb_to_lt_in_context :=
+    repeat autorewrite with ltb_to_lt in *;
+    cbv beta iota in *.
+End Z.
diff --git a/src/Util/ZUtil/Tactics/PrimeBound.v b/src/Util/ZUtil/Tactics/PrimeBound.v
new file mode 100644
index 000000000..f914ed7c8
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/PrimeBound.v
@@ -0,0 +1,7 @@
+Require Import Coq.omega.Omega Coq.ZArith.Znumtheory.
+
+Module Z.
+  Ltac prime_bound := match goal with
+  | [ H : prime ?p |- _ ] => pose proof (prime_ge_2 p H); try omega
+  end.
+End Z.
diff --git a/src/Util/ZUtil/Tactics/ReplaceNegWithPos.v b/src/Util/ZUtil/Tactics/ReplaceNegWithPos.v
new file mode 100644
index 000000000..67b5397aa
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/ReplaceNegWithPos.v
@@ -0,0 +1,34 @@
+Require Import Coq.omega.Omega Coq.ZArith.ZArith.
+Local Open Scope Z_scope.
+
+Module Z.
+  Ltac clean_neg :=
+    repeat match goal with
+           | [ H : (-?x) < 0 |- _ ] => assert (0 < x) by omega; clear H
+           | [ H : 0 > (-?x) |- _ ] => assert (0 < x) by omega; clear H
+           | [ H : (-?x) <= 0 |- _ ] => assert (0 <= x) by omega; clear H
+           | [ H : 0 >= (-?x) |- _ ] => assert (0 <= x) by omega; clear H
+           | [ H : -?x <= -?y |- _ ] => apply Z.opp_le_mono in H
+           | [ |- -?x <= -?y ] => apply Z.opp_le_mono
+           | _ => progress rewrite <- Z.opp_le_mono in *
+           | [ H : 0 <= ?x, H' : 0 <= ?y, H'' : -?x <= ?y |- _ ] => clear H''
+           | [ H : 0 < ?x, H' : 0 <= ?y, H'' : -?x <= ?y |- _ ] => clear H''
+           | [ H : 0 <= ?x, H' : 0 < ?y, H'' : -?x <= ?y |- _ ] => clear H''
+           | [ H : 0 < ?x, H' : 0 < ?y, H'' : -?x <= ?y |- _ ] => clear H''
+           | [ H : 0 < ?x, H' : 0 <= ?y, H'' : -?x < ?y |- _ ] => clear H''
+           | [ H : 0 <= ?x, H' : 0 < ?y, H'' : -?x < ?y |- _ ] => clear H''
+           | [ H : 0 < ?x, H' : 0 < ?y, H'' : -?x < ?y |- _ ] => clear H''
+           end.
+  Ltac replace_with_neg x :=
+    assert (x = -(-x)) by (symmetry; apply Z.opp_involutive); generalize dependent (-x);
+    let x' := fresh in
+    rename x into x'; intro x; intros; subst x';
+    clean_neg.
+  Ltac replace_all_neg_with_pos :=
+    repeat match goal with
+           | [ H : ?x < 0 |- _ ] => replace_with_neg x
+           | [ H : 0 > ?x |- _ ] => replace_with_neg x
+           | [ H : ?x <= 0 |- _ ] => replace_with_neg x
+           | [ H : 0 >= ?x |- _ ] => replace_with_neg x
+           end.
+End Z.
diff --git a/src/Util/ZUtil/Tactics/SimplifyFractionsLe.v b/src/Util/ZUtil/Tactics/SimplifyFractionsLe.v
new file mode 100644
index 000000000..c5b024eca
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/SimplifyFractionsLe.v
@@ -0,0 +1,133 @@
+Require Import Coq.ZArith.ZArith.
+Require Import Coq.micromega.Lia.
+Require Import Crypto.Util.ZUtil.Tactics.ZeroBounds.
+Require Import Crypto.Util.ZUtil.Div.
+Local Open Scope Z_scope.
+
+Module Z.
+  (** * [Z.simplify_fractions_le] *)
+  (** The culmination of this series of tactics,
+      [Z.simplify_fractions_le], will use the fact that [a * (b / c) <=
+      (a * b) / c], and do some reasoning modulo associativity and
+      commutativity in [Z] to perform such a reduction.  It may leave
+      over goals if it cannot prove that some denominators are non-zero.
+      If the rewrite [a * (b / c)] → [(a * b) / c] is safe to do on the
+      LHS of the goal, this tactic should not turn a solvable goal into
+      an unsolvable one.
+
+      After running, the tactic does some basic rewriting to simplify
+      fractions, e.g., that [a * b / b = a]. *)
+  Ltac split_sums_step :=
+    match goal with
+    | [ |- _ + _ <= _ ]
+      => etransitivity; [ eapply Z.add_le_mono | ]
+    | [ |- _ - _ <= _ ]
+      => etransitivity; [ eapply Z.sub_le_mono | ]
+    end.
+  Ltac split_sums :=
+    try (split_sums_step; [ split_sums.. | ]).
+  Ltac pre_reorder_fractions_step :=
+    match goal with
+    | [ |- context[?x / ?y * ?z] ]
+      => lazymatch z with
+         | context[_ / _] => fail
+         | _ => idtac
+         end;
+         rewrite (Z.mul_comm (x / y) z)
+    | _ => let LHS := match goal with |- ?LHS <= ?RHS => LHS end in
+           match LHS with
+           | context G[?x * (?y / ?z)]
+             => let G' := context G[(x * y) / z] in
+                transitivity G'
+           end
+    end.
+  Ltac pre_reorder_fractions :=
+    try first [ split_sums_step; [ pre_reorder_fractions.. | ]
+              | pre_reorder_fractions_step; [ .. | pre_reorder_fractions ] ].
+  Ltac split_comparison :=
+    match goal with
+    | [ |- ?x <= ?x ] => reflexivity
+    | [ H : _ >= _ |- _ ]
+      => apply Z.ge_le_iff in H
+    | [ |- ?x * ?y <= ?z * ?w ]
+      => lazymatch goal with
+         | [ H : 0 <= x |- _ ] => idtac
+         | [ H : x < 0 |- _ ] => fail
+         | _ => destruct (Z_lt_le_dec x 0)
+         end;
+         [ ..
+         | lazymatch goal with
+           | [ H : 0 <= y |- _ ] => idtac
+           | [ H : y < 0 |- _ ] => fail
+           | _ => destruct (Z_lt_le_dec y 0)
+           end;
+           [ ..
+           | apply Zmult_le_compat; [ | | assumption | assumption ] ] ]
+    | [ |- ?x / ?y <= ?z / ?y ]
+      => lazymatch goal with
+         | [ H : 0 < y |- _ ] => idtac
+         | [ H : y <= 0 |- _ ] => fail
+         | _ => destruct (Z_lt_le_dec 0 y)
+         end;
+         [ apply Z_div_le; [ apply Z.gt_lt_iff; assumption | ]
+         | .. ]
+    | [ |- ?x / ?y <= ?x / ?z ]
+      => lazymatch goal with
+         | [ H : 0 <= x |- _ ] => idtac
+         | [ H : x < 0 |- _ ] => fail
+         | _ => destruct (Z_lt_le_dec x 0)
+         end;
+         [ ..
+         | lazymatch goal with
+           | [ H : 0 < z |- _ ] => idtac
+           | [ H : z <= 0 |- _ ] => fail
+           | _ => destruct (Z_lt_le_dec 0 z)
+           end;
+           [ apply Z.div_le_compat_l; [ assumption | split; [ assumption | ] ]
+           | .. ] ]
+    | [ |- _ + _ <= _ + _ ]
+      => apply Z.add_le_mono
+    | [ |- _ - _ <= _ - _ ]
+      => apply Z.sub_le_mono
+    | [ |- ?x * (?y / ?z) <= (?x * ?y) / ?z ]
+      => apply Z.div_mul_le
+    end.
+  Ltac split_comparison_fin_step :=
+    match goal with
+    | _ => assumption
+    | _ => lia
+    | _ => progress subst
+    | [ H : ?n * ?m < 0 |- _ ]
+      => apply (proj1 (Z.lt_mul_0 n m)) in H; destruct H as [ [??]|[??] ]
+    | [ H : ?n / ?m < 0 |- _ ]
+      => apply (proj1 (Z.lt_div_0 n m)) in H; destruct H as [ [ [??]|[??] ] ? ]
+    | [ H : (?x^?y) <= ?n < _, H' : ?n < 0 |- _ ]
+      => assert (0 <= x^y) by Z.zero_bounds; lia
+    | [ H : (?x^?y) < 0 |- _ ]
+      => assert (0 <= x^y) by Z.zero_bounds; lia
+    | [ H : (?x^?y) <= 0 |- _ ]
+      => let H' := fresh in
+         assert (H' : 0 <= x^y) by Z.zero_bounds;
+         assert (x^y = 0) by lia;
+         clear H H'
+    | [ H : _^_ = 0 |- _ ]
+      => apply Z.pow_eq_0_iff in H; destruct H as [ ?|[??] ]
+    | [ H : 0 <= ?x, H' : ?x - 1 < 0 |- _ ]
+      => assert (x = 0) by lia; clear H H'
+    | [ |- ?x <= ?y ] => is_evar x; reflexivity
+    | [ |- ?x <= ?y ] => is_evar y; reflexivity
+    end.
+  Ltac split_comparison_fin := repeat split_comparison_fin_step.
+  Ltac simplify_fractions_step :=
+    match goal with
+    | _ => rewrite Z.div_mul by (try apply Z.pow_nonzero; Z.zero_bounds)
+    | [ |- context[?x * ?y / ?x] ]
+      => rewrite (Z.mul_comm x y)
+    | [ |- ?x <= ?x ] => reflexivity
+    end.
+  Ltac simplify_fractions := repeat simplify_fractions_step.
+  Ltac simplify_fractions_le :=
+    pre_reorder_fractions;
+    [ repeat split_comparison; split_comparison_fin; Z.zero_bounds..
+    | simplify_fractions ].
+End Z.
diff --git a/src/Util/ZUtil/Tactics/ZeroBounds.v b/src/Util/ZUtil/Tactics/ZeroBounds.v
new file mode 100644
index 000000000..d10b6714c
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/ZeroBounds.v
@@ -0,0 +1,27 @@
+Require Import Coq.ZArith.ZArith Coq.omega.Omega.
+Require Import Crypto.Util.ZUtil.Tactics.PrimeBound.
+Local Open Scope Z_scope.
+
+Module Z.
+  (* prove that combinations of known positive/nonnegative numbers are positive/nonnegative *)
+  Ltac zero_bounds' :=
+    repeat match goal with
+    | [ |- 0 <= _ + _] => apply Z.add_nonneg_nonneg
+    | [ |- 0 <= _ - _] => apply Z.le_0_sub
+    | [ |- 0 <= _ * _] => apply Z.mul_nonneg_nonneg
+    | [ |- 0 <= _ / _] => apply Z.div_pos
+    | [ |- 0 <= _ ^ _ ] => apply Z.pow_nonneg
+    | [ |- 0 <= Z.shiftr _ _] => apply Z.shiftr_nonneg
+    | [ |- 0 <= _ mod _] => apply Z.mod_pos_bound
+    | [ |- 0 < _ + _] => try solve [apply Z.add_pos_nonneg; zero_bounds'];
+                         try solve [apply Z.add_nonneg_pos; zero_bounds']
+    | [ |- 0 < _ - _] => apply Z.lt_0_sub
+    | [ |- 0 < _ * _] => apply Z.lt_0_mul; left; split
+    | [ |- 0 < _ / _] => apply Z.div_str_pos
+    | [ |- 0 < _ ^ _ ] => apply Z.pow_pos_nonneg
+    end; try omega; try Z.prime_bound; auto.
+
+  Ltac zero_bounds := try omega; try Z.prime_bound; zero_bounds'.
+
+  Hint Extern 1 => progress zero_bounds : zero_bounds.
+End Z.
diff --git a/src/Util/ZUtil/Tactics/Ztestbit.v b/src/Util/ZUtil/Tactics/Ztestbit.v
new file mode 100644
index 000000000..d12de5330
--- /dev/null
+++ b/src/Util/ZUtil/Tactics/Ztestbit.v
@@ -0,0 +1,22 @@
+Require Import Coq.ZArith.ZArith.
+Require Import Crypto.Util.ZUtil.Testbit.
+Require Import Crypto.Util.ZUtil.Hints.Core.
+
+Ltac Ztestbit_full_step :=
+  match goal with
+  | _ => progress autorewrite with Ztestbit_full
+  | [ |- context[Z.testbit ?x ?y] ]
+    => rewrite (Z.testbit_neg_r x y) by zutil_arith
+  | [ |- context[Z.testbit ?x ?y] ]
+    => rewrite (Z.bits_above_pow2 x y) by zutil_arith
+  | [ |- context[Z.testbit ?x ?y] ]
+    => rewrite (Z.bits_above_log2 x y) by zutil_arith
+  end.
+Ltac Ztestbit_full := repeat Ztestbit_full_step.
+
+Ltac Ztestbit_step :=
+  match goal with
+  | _ => progress autorewrite with Ztestbit
+  | _ => progress Ztestbit_full_step
+  end.
+Ltac Ztestbit := repeat Ztestbit_step.
diff --git a/src/Util/ZUtil/Testbit.v b/src/Util/ZUtil/Testbit.v
new file mode 100644
index 000000000..a315f7e4b
--- /dev/null
+++ b/src/Util/ZUtil/Testbit.v
@@ -0,0 +1,90 @@
+Require Import Coq.ZArith.ZArith.
+Require Import Crypto.Util.ZUtil.Definitions.
+Require Import Crypto.Util.ZUtil.Hints.
+Require Import Crypto.Util.ZUtil.Notations.
+Require Import Crypto.Util.Tactics.BreakMatch.
+Local Open Scope Z_scope.
+
+Module Z.
+  Lemma ones_spec : forall n m, 0 <= n -> 0 <= m -> Z.testbit (Z.ones n) m = if Z_lt_dec m n then true else false.
+  Proof.
+    intros.
+    break_match.
+    + apply Z.ones_spec_low. omega.
+    + apply Z.ones_spec_high. omega.
+  Qed.
+  Hint Rewrite ones_spec using zutil_arith : Ztestbit.
+
+  Lemma ones_spec_full : forall n m, Z.testbit (Z.ones n) m
+                                     = if Z_lt_dec m 0
+                                       then false
+                                       else if Z_lt_dec n 0
+                                            then true
+                                            else if Z_lt_dec m n then true else false.
+  Proof.
+    intros.
+    repeat (break_match || autorewrite with Ztestbit); try reflexivity; try omega.
+    unfold Z.ones.
+    rewrite <- Z.shiftr_opp_r, Z.shiftr_eq_0 by (simpl; omega); simpl.
+    destruct m; simpl in *; try reflexivity.
+    exfalso; auto using Zlt_neg_0.
+  Qed.
+  Hint Rewrite ones_spec_full : Ztestbit_full.
+
+  Lemma testbit_pow2_mod : forall a n i, 0 <= n ->
+  Z.testbit (Z.pow2_mod a n) i = if Z_lt_dec i n then Z.testbit a i else false.
+  Proof.
+  cbv [Z.pow2_mod]; intros; destruct (Z_le_dec 0 i);
+      repeat match goal with
+          | |- _ => rewrite Z.testbit_neg_r by omega
+          | |- _ => break_innermost_match_step
+          | |- _ => omega
+          | |- _ => reflexivity
+          | |- _ => progress autorewrite with Ztestbit
+          end.
+  Qed.
+  Hint Rewrite testbit_pow2_mod using zutil_arith : Ztestbit.
+
+  Lemma testbit_pow2_mod_full : forall a n i,
+      Z.testbit (Z.pow2_mod a n) i = if Z_lt_dec n 0
+                                     then if Z_lt_dec i 0 then false else Z.testbit a i
+                                     else if Z_lt_dec i n then Z.testbit a i else false.
+  Proof.
+    intros; destruct (Z_lt_dec n 0); [ | apply testbit_pow2_mod; omega ].
+    unfold Z.pow2_mod.
+    autorewrite with Ztestbit_full;
+      repeat break_match;
+      autorewrite with Ztestbit;
+      reflexivity.
+  Qed.
+  Hint Rewrite testbit_pow2_mod_full : Ztestbit_full.
+
+  Lemma bits_above_pow2 a n : 0 <= a < 2^n -> Z.testbit a n = false.
+  Proof.
+    intros.
+    destruct (Z_zerop a); subst; autorewrite with Ztestbit; trivial.
+    apply Z.bits_above_log2; auto with zarith concl_log2.
+  Qed.
+  Hint Rewrite bits_above_pow2 using zutil_arith : Ztestbit.
+
+  Lemma testbit_low : forall n x i, (0 <= i < n) ->
+    Z.testbit x i = Z.testbit (Z.land x (Z.ones n)) i.
+  Proof.
+    intros.
+    rewrite Z.land_ones by omega.
+    symmetry.
+    apply Z.mod_pow2_bits_low.
+    omega.
+  Qed.
+
+  Lemma testbit_add_shiftl_low : forall i, (0 <= i) -> forall a b n, (i < n) ->
+    Z.testbit (a + Z.shiftl b n) i = Z.testbit a i.
+  Proof.
+    intros.
+    erewrite Z.testbit_low; eauto.
+    rewrite Z.land_ones, Z.shiftl_mul_pow2 by omega.
+    rewrite Z.mod_add by (pose proof (Z.pow_pos_nonneg 2 n); omega).
+    auto using Z.mod_pow2_bits_low.
+  Qed.
+  Hint Rewrite testbit_add_shiftl_low using zutil_arith : Ztestbit.
+End Z.
diff --git a/src/Util/ZUtil/ZSimplify.v b/src/Util/ZUtil/ZSimplify.v
new file mode 100644
index 000000000..b27a92a0c
--- /dev/null
+++ b/src/Util/ZUtil/ZSimplify.v
@@ -0,0 +1,2 @@
+Require Export Crypto.Util.ZUtil.ZSimplify.Autogenerated.
+Require Export Crypto.Util.ZUtil.ZSimplify.Simple.
diff --git a/src/Util/ZUtil/ZSimplify/Autogenerated.v b/src/Util/ZUtil/ZSimplify/Autogenerated.v
new file mode 100644
index 000000000..b3b6d36b6
--- /dev/null
+++ b/src/Util/ZUtil/ZSimplify/Autogenerated.v
@@ -0,0 +1,590 @@
+Require Import Coq.ZArith.ZArith Coq.micromega.Lia.
+Require Import Crypto.Util.ZUtil.Hints.Core.
+Require Import Crypto.Util.ZUtil.Tactics.DivModToQuotRem.
+Local Open Scope Z_scope.
+
+Module Z.
+  Local Ltac simplify_div_tac :=
+    intros; Z.div_mod_to_quot_rem; nia.
+  (* Naming Convention: [X] for thing being divided by, [p] for plus,
+     [m] for minus, [d] for div, and [_] to separate parentheses and
+     multiplication. *)
+  (* Mathematica code to generate these hints:
+<<
+ClearAll[minus, plus, div, mul, combine, parens, ExprToString,
+  ExprToExpr, ExprToName, SymbolsIn, Chars, RestFrom, a, b, c, d, e,
+  f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, X];
+Chars = StringSplit["abcdefghijklmnopqrstuvwxyz", ""];
+RestFrom[i_, len_] :=
+ Join[{mul[Chars[[i]], "X"]}, Take[Drop[Chars, i], len]]
+Exprs = Flatten[
+   Map[{#1, #1 /. mul[a_, "X", b___] :> mul["X", a, b]} &, Flatten[{
+      Table[
+       Table[div[
+         combine @@
+          Join[Take[Chars, start - 1], RestFrom[start, len]],
+         "X"], {len, 0, 10 - start}], {start, 1, 2}],
+      Table[
+       Table[div[
+         combine["a",
+          parens @@
+           Join[Take[Drop[Chars, 1], start - 1],
+            RestFrom[1 + start, len]]], "X"], {len, 0,
+         10 - start}], {start, 1, 2}],
+      div[combine["a", parens["b", parens["c", mul["d", "X"]], "e"]],
+       "X"],
+      div[combine["a", "b", parens["c", mul["d", "X"]], "e"], "X"],
+      div[combine["a", "b", mul["c", "X", "d"], "e", "f"], "X"],
+      div[combine["a", mul["b", "X", "c"], "d", "e"], "X"],
+      div[
+       combine["a",
+        parens["b", parens["c", mul["d", "X", "e"]], "f"]], "X"],
+      div[combine["a", parens["b", mul["c", "X", "d"]], "e"], "X"]}]]];
+ExprToString[div[x_, y_], withparen_: False] :=
+ With[{v := ExprToString[x, True] <> " / " <> ExprToString[y, True]},
+  If[withparen, "(" <> v <> ")", v]]
+ExprToString[combine[x_], withparen_: False] :=
+ ExprToString[x, withparen]
+ExprToString[combine[x_, minus, y__], withparen_: False] :=
+ With[{v :=
+    ExprToString[x, False] <> " - " <>
+     ExprToString[combine[y], False]},
+  If[withparen, "(" <> v <> ")", v]]
+ExprToString[combine[minus, y__], withparen_: False] :=
+ With[{v := "-" <> ExprToString[combine[y], False]},
+  If[withparen, "(" <> v <> ")", v]]
+ExprToString[combine[x_, y__], withparen_: False] :=
+ With[{v :=
+    ExprToString[x, False] <> " + " <>
+     ExprToString[combine[y], False]},
+  If[withparen, "(" <> v <> ")", v]]
+ExprToString[mul[x_], withparen_: False] := ExprToString[x]
+ExprToString[mul[x_, y__], withparen_: False] :=
+ With[{v :=
+    ExprToString[x, False] <> " * " <> ExprToString[mul[y], False]},
+  If[withparen, "(" <> v <> ")", v]]
+ExprToString[parens[x__], withparen_: False] :=
+ "(" <> ExprToString[combine[x]] <> ")"
+ExprToString[x_String, withparen_: False] := x
+ExprToExpr[div[x__]] := Divide @@ Map[ExprToExpr, {x}]
+ExprToExpr[mul[x__]] := Times @@ Map[ExprToExpr, {x}]
+ExprToExpr[combine[]] := 0
+ExprToExpr[combine[minus, y_, z___]] := -ExprToExpr[y] +
+  ExprToExpr[combine[z]]
+ExprToExpr[combine[x_, y___]] := ExprToExpr[x] + ExprToExpr[combine[y]]
+ExprToExpr[parens[x__]] := ExprToExpr[combine[x]]
+ExprToExpr[x_String] := Symbol[x]
+ExprToName["X", ispos_: True] := If[ispos, "X", "mX"]
+ExprToName[x_String, ispos_: True] := If[ispos, "p", "m"]
+ExprToName[div[x_, y_], ispos_: True] :=
+ If[ispos, "p", "m"] <> ExprToName[x] <> "d" <> ExprToName[y]
+ExprToName[mul[x_], ispos_: True] :=
+ If[ispos, "", "m_"] <> ExprToName[x] <> "_"
+ExprToName[mul[x_, y__], ispos_: True] :=
+ If[ispos, "", "m_"] <> ExprToName[x] <> ExprToName[mul[y]]
+ExprToName[combine[], ispos_: True] := ""
+ExprToName[combine[x_], ispos_: True] := ExprToName[x, ispos]
+ExprToName[combine[x_, minus, mul[y__], z___], ispos_: True] :=
+ ExprToName[x, ispos] <> "m_" <> ExprToName[mul[y], True] <>
+  ExprToName[combine[z], True]
+ExprToName[combine[x_, mul[y__], z___], ispos_: True] :=
+ ExprToName[x, ispos] <> "p_" <> ExprToName[mul[y], True] <>
+  ExprToName[combine[z], True]
+ExprToName[combine[x_, y__], ispos_: True] :=
+ ExprToName[x, ispos] <> ExprToName[combine[y], True]
+ExprToName[combine[x_, minus, y__], ispos_: True] :=
+ ExprToName[x, ispos] <> ExprToName[combine[y], True]
+ExprToName[combine[x_, y__], ispos_: True] :=
+ ExprToName[x, ispos] <> ExprToName[combine[y], True]
+ExprToName[parens[x__], ispos_: True] :=
+ "_o_" <> ExprToName[combine[x], ispos] <> "_c_"
+SymbolsIn[x_String] := {x <> " "}
+SymbolsIn[f_[y___]] := Join @@ Map[SymbolsIn, {y}]
+StringJoin @@
+ Map[{"  Lemma simplify_div_" <> ExprToName[#1] <> " " <>
+     StringJoin @@ Sort[DeleteDuplicates[SymbolsIn[#1]]] <>
+     ": X <> 0 -> " <> ExprToString[#1] <> " = " <>
+     StringReplace[(FullSimplify[ExprToExpr[#1]] // InputForm //
+        ToString), "/" -> " / "] <> "." <>
+     "\n  Proof. simplify_div_tac. Qed.\n  Hint Rewrite \
+simplify_div_" <> ExprToName[#1] <>
+     " using zutil_arith : zsimplify.\n"} &, Exprs]
+>> *)
+  Lemma simplify_div_ppX_dX a X : X <> 0 -> (a * X) / X = a.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_dX a X : X <> 0 -> (X * a) / X = a.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_pdX a b X : X <> 0 -> (a * X + b) / X = a + b / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_pdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_pdX a b X : X <> 0 -> (X * a + b) / X = a + b / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_pdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_ppdX a b c X : X <> 0 -> (a * X + b + c) / X = a + (b + c) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_ppdX a b c X : X <> 0 -> (X * a + b + c) / X = a + (b + c) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_pppdX a b c d X : X <> 0 -> (a * X + b + c + d) / X = a + (b + c + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_pppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_pppdX a b c d X : X <> 0 -> (X * a + b + c + d) / X = a + (b + c + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_pppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_ppppdX a b c d e X : X <> 0 -> (a * X + b + c + d + e) / X = a + (b + c + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_ppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_ppppdX a b c d e X : X <> 0 -> (X * a + b + c + d + e) / X = a + (b + c + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_ppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_pppppdX a b c d e f X : X <> 0 -> (a * X + b + c + d + e + f) / X = a + (b + c + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_pppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_pppppdX a b c d e f X : X <> 0 -> (X * a + b + c + d + e + f) / X = a + (b + c + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_pppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_ppppppdX a b c d e f g X : X <> 0 -> (a * X + b + c + d + e + f + g) / X = a + (b + c + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_ppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_ppppppdX a b c d e f g X : X <> 0 -> (X * a + b + c + d + e + f + g) / X = a + (b + c + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_ppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_pppppppdX a b c d e f g h X : X <> 0 -> (a * X + b + c + d + e + f + g + h) / X = a + (b + c + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_pppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_pppppppdX a b c d e f g h X : X <> 0 -> (X * a + b + c + d + e + f + g + h) / X = a + (b + c + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_pppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_ppppppppdX a b c d e f g h i X : X <> 0 -> (a * X + b + c + d + e + f + g + h + i) / X = a + (b + c + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_ppppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_ppppppppdX a b c d e f g h i X : X <> 0 -> (X * a + b + c + d + e + f + g + h + i) / X = a + (b + c + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_ppppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppX_pppppppppdX a b c d e f g h i j X : X <> 0 -> (a * X + b + c + d + e + f + g + h + i + j) / X = a + (b + c + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppX_pppppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pXp_pppppppppdX a b c d e f g h i j X : X <> 0 -> (X * a + b + c + d + e + f + g + h + i + j) / X = a + (b + c + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pXp_pppppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_dX a b X : X <> 0 -> (a + b * X) / X = b + a / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_dX a b X : X <> 0 -> (a + X * b) / X = b + a / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_pdX a b c X : X <> 0 -> (a + b * X + c) / X = b + (a + c) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_pdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_pdX a b c X : X <> 0 -> (a + X * b + c) / X = b + (a + c) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_pdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_ppdX a b c d X : X <> 0 -> (a + b * X + c + d) / X = b + (a + c + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_ppdX a b c d X : X <> 0 -> (a + X * b + c + d) / X = b + (a + c + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_pppdX a b c d e X : X <> 0 -> (a + b * X + c + d + e) / X = b + (a + c + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_pppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_pppdX a b c d e X : X <> 0 -> (a + X * b + c + d + e) / X = b + (a + c + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_pppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_ppppdX a b c d e f X : X <> 0 -> (a + b * X + c + d + e + f) / X = b + (a + c + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_ppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_ppppdX a b c d e f X : X <> 0 -> (a + X * b + c + d + e + f) / X = b + (a + c + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_ppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_pppppdX a b c d e f g X : X <> 0 -> (a + b * X + c + d + e + f + g) / X = b + (a + c + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_pppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_pppppdX a b c d e f g X : X <> 0 -> (a + X * b + c + d + e + f + g) / X = b + (a + c + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_pppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_ppppppdX a b c d e f g h X : X <> 0 -> (a + b * X + c + d + e + f + g + h) / X = b + (a + c + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_ppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_ppppppdX a b c d e f g h X : X <> 0 -> (a + X * b + c + d + e + f + g + h) / X = b + (a + c + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_ppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_pppppppdX a b c d e f g h i X : X <> 0 -> (a + b * X + c + d + e + f + g + h + i) / X = b + (a + c + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_pppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_pppppppdX a b c d e f g h i X : X <> 0 -> (a + X * b + c + d + e + f + g + h + i) / X = b + (a + c + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_pppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pX_ppppppppdX a b c d e f g h i j X : X <> 0 -> (a + b * X + c + d + e + f + g + h + i + j) / X = b + (a + c + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pX_ppppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xp_ppppppppdX a b c d e f g h i j X : X <> 0 -> (a + X * b + c + d + e + f + g + h + i + j) / X = b + (a + c + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xp_ppppppppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX__c_dX a b X : X <> 0 -> (a + (b * X)) / X = b + a / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX__c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp__c_dX a b X : X <> 0 -> (a + (X * b)) / X = b + a / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp__c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_p_c_dX a b c X : X <> 0 -> (a + (b * X + c)) / X = b + (a + c) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_p_c_dX a b c X : X <> 0 -> (a + (X * b + c)) / X = b + (a + c) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_pp_c_dX a b c d X : X <> 0 -> (a + (b * X + c + d)) / X = b + (a + c + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_pp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_pp_c_dX a b c d X : X <> 0 -> (a + (X * b + c + d)) / X = b + (a + c + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_pp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_ppp_c_dX a b c d e X : X <> 0 -> (a + (b * X + c + d + e)) / X = b + (a + c + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_ppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_ppp_c_dX a b c d e X : X <> 0 -> (a + (X * b + c + d + e)) / X = b + (a + c + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_ppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_pppp_c_dX a b c d e f X : X <> 0 -> (a + (b * X + c + d + e + f)) / X = b + (a + c + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_pppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_pppp_c_dX a b c d e f X : X <> 0 -> (a + (X * b + c + d + e + f)) / X = b + (a + c + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_pppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_ppppp_c_dX a b c d e f g X : X <> 0 -> (a + (b * X + c + d + e + f + g)) / X = b + (a + c + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_ppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_ppppp_c_dX a b c d e f g X : X <> 0 -> (a + (X * b + c + d + e + f + g)) / X = b + (a + c + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_ppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_pppppp_c_dX a b c d e f g h X : X <> 0 -> (a + (b * X + c + d + e + f + g + h)) / X = b + (a + c + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_pppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_pppppp_c_dX a b c d e f g h X : X <> 0 -> (a + (X * b + c + d + e + f + g + h)) / X = b + (a + c + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_pppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_ppppppp_c_dX a b c d e f g h i X : X <> 0 -> (a + (b * X + c + d + e + f + g + h + i)) / X = b + (a + c + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_ppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_ppppppp_c_dX a b c d e f g h i X : X <> 0 -> (a + (X * b + c + d + e + f + g + h + i)) / X = b + (a + c + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_ppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_pppppppp_c_dX a b c d e f g h i j X : X <> 0 -> (a + (b * X + c + d + e + f + g + h + i + j)) / X = b + (a + c + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_pppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_pppppppp_c_dX a b c d e f g h i j X : X <> 0 -> (a + (X * b + c + d + e + f + g + h + i + j)) / X = b + (a + c + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_pppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pX_ppppppppp_c_dX a b c d e f g h i j k X : X <> 0 -> (a + (b * X + c + d + e + f + g + h + i + j + k)) / X = b + (a + c + d + e + f + g + h + i + j + k) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pX_ppppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_Xp_ppppppppp_c_dX a b c d e f g h i j k X : X <> 0 -> (a + (X * b + c + d + e + f + g + h + i + j + k)) / X = b + (a + c + d + e + f + g + h + i + j + k) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_Xp_ppppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX__c_dX a b c X : X <> 0 -> (a + (b + c * X)) / X = c + (a + b) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX__c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp__c_dX a b c X : X <> 0 -> (a + (b + X * c)) / X = c + (a + b) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp__c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_p_c_dX a b c d X : X <> 0 -> (a + (b + c * X + d)) / X = c + (a + b + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_p_c_dX a b c d X : X <> 0 -> (a + (b + X * c + d)) / X = c + (a + b + d) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_pp_c_dX a b c d e X : X <> 0 -> (a + (b + c * X + d + e)) / X = c + (a + b + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_pp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_pp_c_dX a b c d e X : X <> 0 -> (a + (b + X * c + d + e)) / X = c + (a + b + d + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_pp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_ppp_c_dX a b c d e f X : X <> 0 -> (a + (b + c * X + d + e + f)) / X = c + (a + b + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_ppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_ppp_c_dX a b c d e f X : X <> 0 -> (a + (b + X * c + d + e + f)) / X = c + (a + b + d + e + f) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_ppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_pppp_c_dX a b c d e f g X : X <> 0 -> (a + (b + c * X + d + e + f + g)) / X = c + (a + b + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_pppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_pppp_c_dX a b c d e f g X : X <> 0 -> (a + (b + X * c + d + e + f + g)) / X = c + (a + b + d + e + f + g) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_pppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_ppppp_c_dX a b c d e f g h X : X <> 0 -> (a + (b + c * X + d + e + f + g + h)) / X = c + (a + b + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_ppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_ppppp_c_dX a b c d e f g h X : X <> 0 -> (a + (b + X * c + d + e + f + g + h)) / X = c + (a + b + d + e + f + g + h) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_ppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_pppppp_c_dX a b c d e f g h i X : X <> 0 -> (a + (b + c * X + d + e + f + g + h + i)) / X = c + (a + b + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_pppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_pppppp_c_dX a b c d e f g h i X : X <> 0 -> (a + (b + X * c + d + e + f + g + h + i)) / X = c + (a + b + d + e + f + g + h + i) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_pppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_ppppppp_c_dX a b c d e f g h i j X : X <> 0 -> (a + (b + c * X + d + e + f + g + h + i + j)) / X = c + (a + b + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_ppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_ppppppp_c_dX a b c d e f g h i j X : X <> 0 -> (a + (b + X * c + d + e + f + g + h + i + j)) / X = c + (a + b + d + e + f + g + h + i + j) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_ppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pX_pppppppp_c_dX a b c d e f g h i j k X : X <> 0 -> (a + (b + c * X + d + e + f + g + h + i + j + k)) / X = c + (a + b + d + e + f + g + h + i + j + k) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pX_pppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xp_pppppppp_c_dX a b c d e f g h i j k X : X <> 0 -> (a + (b + X * c + d + e + f + g + h + i + j + k)) / X = c + (a + b + d + e + f + g + h + i + j + k) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xp_pppppppp_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_p_o_pp_pX__c_p_c_dX a b c d e X : X <> 0 -> (a + (b + (c + d * X) + e)) / X = d + (a + b + c + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_p_o_pp_pX__c_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_p_o_pp_Xp__c_p_c_dX a b c d e X : X <> 0 -> (a + (b + (c + X * d) + e)) / X = d + (a + b + c + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_p_o_pp_Xp__c_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_o_pp_pX__c_pdX a b c d e X : X <> 0 -> (a + b + (c + d * X) + e) / X = d + (a + b + c + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_o_pp_pX__c_pdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_o_pp_Xp__c_pdX a b c d e X : X <> 0 -> (a + b + (c + X * d) + e) / X = d + (a + b + c + e) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_o_pp_Xp__c_pdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pppp_pXp_ppdX a b c d e f X : X <> 0 -> (a + b + c * X * d + e + f) / X = (a + b + e + f + c*d*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pppp_pXp_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pppp_Xpp_ppdX a b c d e f X : X <> 0 -> (a + b + X * c * d + e + f) / X = (a + b + e + f + c*d*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pppp_Xpp_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_pXp_ppdX a b c d e X : X <> 0 -> (a + b * X * c + d + e) / X = (a + d + e + b*c*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_pXp_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_ppp_Xpp_ppdX a b c d e X : X <> 0 -> (a + X * b * c + d + e) / X = (a + d + e + b*c*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_ppp_Xpp_ppdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_p_o_pp_pXp__c_p_c_dX a b c d e f X : X <> 0 -> (a + (b + (c + d * X * e) + f)) / X = (a + b + c + f + d*e*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_p_o_pp_pXp__c_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_p_o_pp_Xpp__c_p_c_dX a b c d e f X : X <> 0 -> (a + (b + (c + X * d * e) + f)) / X = (a + b + c + f + d*e*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_p_o_pp_Xpp__c_p_c_dX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_pXp__c_pdX a b c d e X : X <> 0 -> (a + (b + c * X * d) + e) / X = (a + b + e + c*d*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_pXp__c_pdX using zutil_arith : zsimplify.
+  Lemma simplify_div_pp_o_pp_Xpp__c_pdX a b c d e X : X <> 0 -> (a + (b + X * c * d) + e) / X = (a + b + e + c*d*X) / X.
+  Proof. simplify_div_tac. Qed.
+  Hint Rewrite simplify_div_pp_o_pp_Xpp__c_pdX using zutil_arith : zsimplify.
+
+
+  (* Naming convention: [X] for thing being aggregated, [p] for plus,
+     [m] for minus, [_] for parentheses *)
+  (* Python code to generate these hints:
+<<
+#!/usr/bin/env python
+
+def sgn(v):
+    if v < 0:
+        return -1
+    elif v == 0:
+        return 0
+    elif v > 0:
+        return 1
+
+def to_eqn(name):
+    vars_left = list('abcdefghijklmnopqrstuvwxyz')
+    ret = ''
+    close = ''
+    while name:
+        if name[0] == 'X':
+            ret += ' X'
+            name = name[1:]
+        elif not name[0].isdigit():
+            ret += ' ' + vars_left[0]
+            vars_left = vars_left[1:]
+        if name:
+            if name[0] == 'm': ret += ' -'
+            elif name[0] == 'p': ret += ' +'
+            elif name[0].isdigit(): ret += ' %s *' % name[0]
+            name = name[1:]
+        if name and name[0] == '_':
+            ret += ' ('
+            close += ')'
+            name = name[1:]
+    if ret[-1] != 'X':
+        ret += ' ' + vars_left[0]
+    return (ret + close).strip().replace('( ', '(')
+
+def simplify(eqn):
+    counts = {}
+    sign_stack = [1, 1]
+    for i in eqn:
+        if i == ' ': continue
+        elif i == '(': sign_stack.append(sign_stack[-1])
+        elif i == ')': sign_stack.pop()
+        elif i == '+': sign_stack.append(sgn(sign_stack[-1]))
+        elif i == '-': sign_stack.append(-sgn(sign_stack[-1]))
+        elif i == '*': continue
+        elif i.isdigit(): sign_stack[-1] *= int(i)
+        else:
+            counts[i] = counts.get(i,0) + sign_stack.pop()
+    ret = ''
+    for k in sorted(counts.keys()):
+        if counts[k] == 1: ret += ' + %s' % k
+        elif counts[k] == -1: ret += ' - %s' % k
+        elif counts[k] < 0: ret += ' - %d * %s' % (abs(counts[k]), k)
+        elif counts[k] > 0: ret += ' + %d * %s' % (abs(counts[k]), k)
+    if ret == '': ret = '0'
+    return ret.strip(' +')
+
+
+def to_def(name):
+    eqn = to_eqn(name)
+    return r'''  Lemma simplify_%s %s X : %s = %s.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_%s : zsimplify.''' % (name, ' '.join(sorted(set(eqn) - set('*+-() 0123456789X'))), eqn, simplify(eqn), name)
+
+names = []
+names += ['%sX%s%sX' % (a, b, c) for a in 'mp' for b in 'mp' for c in 'mp']
+names += ['%sX%s_X%s' % (a, b, c) for a in 'mp' for b in 'mp' for c in 'mp']
+names += ['X%s%s_X%s' % (a, b, c) for a in 'mp' for b in 'mp' for c in 'mp']
+names += ['%sX%s_%sX' % (a, b, c) for a in 'mp' for b in 'mp' for c in 'mp']
+names += ['X%s%s_%sX' % (a, b, c) for a in 'mp' for b in 'mp' for c in 'mp']
+names += ['m2XpX', 'm2XpXpX']
+
+print(r'''  (* Python code to generate these hints:
+<<''')
+print(open(__file__).read())
+print(r'''
+>> *)''')
+for name in names:
+    print(to_def(name))
+
+
+>> *)
+  Lemma simplify_mXmmX a b X : a - X - b - X = - 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXmmX : zsimplify.
+  Lemma simplify_mXmpX a b X : a - X - b + X = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXmpX : zsimplify.
+  Lemma simplify_mXpmX a b X : a - X + b - X = - 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXpmX : zsimplify.
+  Lemma simplify_mXppX a b X : a - X + b + X = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXppX : zsimplify.
+  Lemma simplify_pXmmX a b X : a + X - b - X = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXmmX : zsimplify.
+  Lemma simplify_pXmpX a b X : a + X - b + X = 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXmpX : zsimplify.
+  Lemma simplify_pXpmX a b X : a + X + b - X = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXpmX : zsimplify.
+  Lemma simplify_pXppX a b X : a + X + b + X = 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXppX : zsimplify.
+  Lemma simplify_mXm_Xm a b X : a - X - (X - b) = - 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXm_Xm : zsimplify.
+  Lemma simplify_mXm_Xp a b X : a - X - (X + b) = - 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXm_Xp : zsimplify.
+  Lemma simplify_mXp_Xm a b X : a - X + (X - b) = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXp_Xm : zsimplify.
+  Lemma simplify_mXp_Xp a b X : a - X + (X + b) = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXp_Xp : zsimplify.
+  Lemma simplify_pXm_Xm a b X : a + X - (X - b) = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXm_Xm : zsimplify.
+  Lemma simplify_pXm_Xp a b X : a + X - (X + b) = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXm_Xp : zsimplify.
+  Lemma simplify_pXp_Xm a b X : a + X + (X - b) = 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXp_Xm : zsimplify.
+  Lemma simplify_pXp_Xp a b X : a + X + (X + b) = 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXp_Xp : zsimplify.
+  Lemma simplify_Xmm_Xm a b X : X - a - (X - b) = - a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmm_Xm : zsimplify.
+  Lemma simplify_Xmm_Xp a b X : X - a - (X + b) = - a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmm_Xp : zsimplify.
+  Lemma simplify_Xmp_Xm a b X : X - a + (X - b) = 2 * X - a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmp_Xm : zsimplify.
+  Lemma simplify_Xmp_Xp a b X : X - a + (X + b) = 2 * X - a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmp_Xp : zsimplify.
+  Lemma simplify_Xpm_Xm a b X : X + a - (X - b) = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpm_Xm : zsimplify.
+  Lemma simplify_Xpm_Xp a b X : X + a - (X + b) = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpm_Xp : zsimplify.
+  Lemma simplify_Xpp_Xm a b X : X + a + (X - b) = 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpp_Xm : zsimplify.
+  Lemma simplify_Xpp_Xp a b X : X + a + (X + b) = 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpp_Xp : zsimplify.
+  Lemma simplify_mXm_mX a b X : a - X - (b - X) = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXm_mX : zsimplify.
+  Lemma simplify_mXm_pX a b X : a - X - (b + X) = - 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXm_pX : zsimplify.
+  Lemma simplify_mXp_mX a b X : a - X + (b - X) = - 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXp_mX : zsimplify.
+  Lemma simplify_mXp_pX a b X : a - X + (b + X) = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_mXp_pX : zsimplify.
+  Lemma simplify_pXm_mX a b X : a + X - (b - X) = 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXm_mX : zsimplify.
+  Lemma simplify_pXm_pX a b X : a + X - (b + X) = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXm_pX : zsimplify.
+  Lemma simplify_pXp_mX a b X : a + X + (b - X) = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXp_mX : zsimplify.
+  Lemma simplify_pXp_pX a b X : a + X + (b + X) = 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_pXp_pX : zsimplify.
+  Lemma simplify_Xmm_mX a b X : X - a - (b - X) = 2 * X - a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmm_mX : zsimplify.
+  Lemma simplify_Xmm_pX a b X : X - a - (b + X) = - a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmm_pX : zsimplify.
+  Lemma simplify_Xmp_mX a b X : X - a + (b - X) = - a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmp_mX : zsimplify.
+  Lemma simplify_Xmp_pX a b X : X - a + (b + X) = 2 * X - a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xmp_pX : zsimplify.
+  Lemma simplify_Xpm_mX a b X : X + a - (b - X) = 2 * X + a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpm_mX : zsimplify.
+  Lemma simplify_Xpm_pX a b X : X + a - (b + X) = a - b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpm_pX : zsimplify.
+  Lemma simplify_Xpp_mX a b X : X + a + (b - X) = a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpp_mX : zsimplify.
+  Lemma simplify_Xpp_pX a b X : X + a + (b + X) = 2 * X + a + b.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_Xpp_pX : zsimplify.
+  Lemma simplify_m2XpX a X : a - 2 * X + X = - X + a.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_m2XpX : zsimplify.
+  Lemma simplify_m2XpXpX a X : a - 2 * X + X + X = a.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_m2XpXpX : zsimplify.
+End Z.
diff --git a/src/Util/ZUtil/ZSimplify/Simple.v b/src/Util/ZUtil/ZSimplify/Simple.v
new file mode 100644
index 000000000..9b5e0e9c8
--- /dev/null
+++ b/src/Util/ZUtil/ZSimplify/Simple.v
@@ -0,0 +1,82 @@
+Require Import Coq.ZArith.ZArith Coq.micromega.Lia Coq.omega.Omega.
+Require Import Crypto.Util.ZUtil.Hints.Core.
+Local Open Scope Z_scope.
+
+Module Z.
+  Lemma sub_same_minus (x y : Z) : x - (x - y) = y.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_minus : zsimplify.
+  Lemma sub_same_plus (x y : Z) : x - (x + y) = -y.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_plus : zsimplify.
+  Lemma sub_same_minus_plus (x y z : Z) : x - (x - y + z) = y - z.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_minus_plus : zsimplify.
+  Lemma sub_same_plus_plus (x y z : Z) : x - (x + y + z) = -y - z.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_plus_plus : zsimplify.
+  Lemma sub_same_minus_minus (x y z : Z) : x - (x - y - z) = y + z.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_minus_minus : zsimplify.
+  Lemma sub_same_plus_minus (x y z : Z) : x - (x + y - z) = z - y.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_plus_minus : zsimplify.
+  Lemma sub_same_minus_then_plus (x y w : Z) : x - (x - y) + w = y + w.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_minus_then_plus : zsimplify.
+  Lemma sub_same_plus_then_plus (x y w : Z) : x - (x + y) + w = w - y.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_plus_then_plus : zsimplify.
+  Lemma sub_same_minus_plus_then_plus (x y z w : Z) : x - (x - y + z) + w = y - z + w.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_minus_plus_then_plus : zsimplify.
+  Lemma sub_same_plus_plus_then_plus (x y z w : Z) : x - (x + y + z) + w = w - y - z.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_plus_plus_then_plus : zsimplify.
+  Lemma sub_same_minus_minus_then_plus (x y z w : Z) : x - (x - y - z) + w = y + z + w.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_minus_minus : zsimplify.
+  Lemma sub_same_plus_minus_then_plus (x y z w : Z) : x - (x + y - z) + w = z - y + w.
+  Proof. lia. Qed.
+  Hint Rewrite sub_same_plus_minus_then_plus : zsimplify.
+
+  Lemma simplify_twice_sub_sub x y : 2 * x - (x - y) = x + y.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_twice_sub_sub : zsimplify.
+
+  Lemma simplify_twice_sub_add x y : 2 * x - (x + y) = x - y.
+  Proof. lia. Qed.
+  Hint Rewrite simplify_twice_sub_add : zsimplify.
+
+  Lemma simplify_2XmX X : 2 * X - X = X.
+  Proof. omega. Qed.
+  Hint Rewrite simplify_2XmX : zsimplify.
+
+  Lemma simplify_add_pos x y : Z.pos x + Z.pos y = Z.pos (x + y).
+  Proof. reflexivity. Qed.
+  Hint Rewrite simplify_add_pos : zsimplify_Z_to_pos.
+  Lemma simplify_add_pos10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
+    : Z.pos x0 + (Z.pos x1 + (Z.pos x2 + (Z.pos x3 + (Z.pos x4 + (Z.pos x5 + (Z.pos x6 + (Z.pos x7 + (Z.pos x8 + Z.pos x9))))))))
+      = Z.pos (x0 + (x1 + (x2 + (x3 + (x4 + (x5 + (x6 + (x7 + (x8 + x9))))))))).
+  Proof. reflexivity. Qed.
+  Hint Rewrite simplify_add_pos10 : zsimplify_Z_to_pos.
+  Lemma simplify_mul_pos x y : Z.pos x * Z.pos y = Z.pos (x * y).
+  Proof. reflexivity. Qed.
+  Hint Rewrite simplify_mul_pos : zsimplify_Z_to_pos.
+  Lemma simplify_mul_pos10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
+    : Z.pos x0 * (Z.pos x1 * (Z.pos x2 * (Z.pos x3 * (Z.pos x4 * (Z.pos x5 * (Z.pos x6 * (Z.pos x7 * (Z.pos x8 * Z.pos x9))))))))
+      = Z.pos (x0 * (x1 * (x2 * (x3 * (x4 * (x5 * (x6 * (x7 * (x8 * x9))))))))).
+  Proof. reflexivity. Qed.
+  Hint Rewrite simplify_mul_pos10 : zsimplify_Z_to_pos.
+  Lemma simplify_sub_pos x y : Z.pos x - Z.pos y = Z.pos_sub x y.
+  Proof. reflexivity. Qed.
+  Hint Rewrite simplify_sub_pos : zsimplify_Z_to_pos.
+
+  Lemma two_sub_sub_inner_sub x y z : 2 * x - y - (x - z) = x - y + z.
+  Proof. clear; lia. Qed.
+  Hint Rewrite two_sub_sub_inner_sub : zsimplify.
+
+  Lemma minus_minus_one : - -1 = 1.
+  Proof. reflexivity. Qed.
+  Hint Rewrite minus_minus_one : zsimplify.
+End Z.