Split off more of ZUtil

2 files changed, 672 insertions, 0 deletions

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.