Semiring solver (#50)

Everything/Safe.agda

@@ -49,5 +49,6 @@ open import ClassicalLogic.ClassicalFive
 open import Monoids.Definition
 
 open import Semirings.Definition
+open import Semirings.Solver
 
 module Everything.Safe where

Everything/WithK.agda

@@ -22,4 +22,6 @@ open import Groups.FreeGroups
 open import Groups.Examples.ExampleSheet1
 open import Groups.LectureNotes.Lecture1
 
+open import LectureNotes.NumbersAndSets.Lecture1
+
 module Everything.WithK where

LectureNotes/NumbersAndSets/Lecture1.agda

@@ -0,0 +1,55 @@
+{-# OPTIONS --warning=error --safe #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import LogicalFormulae
+open import Numbers.Integers.Integers
+open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Order
+open import Numbers.Naturals.Exponentiation
+open import Numbers.Primes.PrimeNumbers
+open import Maybe
+open import Semirings.Definition
+open import Semirings.Solver
+
+module NatSolver = Semirings.Solver ℕSemiring multiplicationNIsCommutative
+
+module LectureNotes.NumbersAndSets.Lecture1 where
+
+a-Na : (a : ℕ) → a -N' a ≡ yes 0
+a-Na zero = refl
+a-Na (succ a) = a-Na a
+
+-N''lemma : (a b : ℕ) → (a<b : a ≤N b) → b -N' a ≡ yes (subtractionNResult.result (-N a<b))
+-N''lemma zero (succ b) (inl x) = refl
+-N''lemma (succ a) (succ b) (inl x) = -N''lemma a b (inl (canRemoveSuccFrom<N x))
+-N''lemma a b (inr x) rewrite x | a-Na b = ans
+  where
+    ans : yes 0 ≡ yes (subtractionNResult.result (-N (inr (refl {x = b}))))
+    ans with -N (inr (refl {x = b}))
+    ans | record { result = result ; pr = pr } with result
+    ans | record { result = result ; pr = pr } | zero = refl
+    ans | record { result = result ; pr = pr } | succ bl = exFalso (cannotAddAndEnlarge'' pr)
+
+-N'' : (a b : ℕ) (a<b : a ≤N b) → ℕ
+-N'' a b a<b with -N''lemma a b a<b
+... | bl with b -N' a
+-N'' a b a<b | bl | yes x = x
+
+n3Bigger : (n : ℕ) → (n ≡ 0) || (n ≤N n ^N 3)
+n3Bigger n = exponentiationIncreases n 2
+
+n3Bigger' : (n : ℕ) → n ≤N n ^N 3
+n3Bigger' zero = inr refl
+n3Bigger' (succ n) with n3Bigger (succ n)
+n3Bigger' (succ n) | inr f = f
+
+_!!N_ = NatSolver._!!_
+
+-- How to use the semiring solver
+-- The process is very mechanical; I haven't yet worked out how to do reflection,
+-- so there's quite a bit of transcribing expressions into the Expr form.
+-- The first two arguments to !!N are totally mindless in construction.
+proof : (n : ℕ) → ((n *N n) +N ((2 *N n) +N 1)) ≡ (n +N 1) *N (n +N 1)
+proof n =
+  (plus (times (const n) (const n)) (plus (times (succ (succ zero)) (const n)) (succ zero)) !!N times (plus (const n) (succ zero)) (plus (const n) (succ zero)))
+    (applyEquality (λ i → succ (n *N n) +N (n +N i)) ((const n !!N plus (const n) zero) refl))

Numbers/Integers/Definition.agda

@@ -8,28 +8,3 @@ module Numbers.Integers.Definition where
 data ℤ : Set where
   nonneg : ℕ → ℤ
   negSucc : ℕ → ℤ
-
-data ℤSimple : Set where
-  negativeSucc : (a : ℕ) → ℤSimple
-  positiveSucc : (a : ℕ) → ℤSimple
-  zZero : ℤSimple
-
-convertZ : ℤ → ℤSimple
-convertZ (nonneg zero) = zZero
-convertZ (nonneg (succ x)) = positiveSucc x
-convertZ (negSucc x) = negativeSucc x
-
-convertZ' : ℤSimple → ℤ
-convertZ' (negativeSucc a) = negSucc a
-convertZ' (positiveSucc a) = nonneg (succ a)
-convertZ' zZero = nonneg 0
-
-zIsZ : (a : ℤ) → convertZ' (convertZ a) ≡ a
-zIsZ (nonneg zero) = refl
-zIsZ (nonneg (succ x)) = refl
-zIsZ (negSucc x) = refl
-
-zIsZ' : (a : ℤSimple) → convertZ (convertZ' a) ≡ a
-zIsZ' (negativeSucc a) = refl
-zIsZ' (positiveSucc a) = refl
-zIsZ' zZero = refl

Numbers/Naturals/Naturals.agda

@@ -10,6 +10,7 @@ open import Numbers.Naturals.Addition
 open import Numbers.Naturals.Order
 open import Numbers.Naturals.Multiplication
 open import Numbers.Naturals.Exponentiation
+open import Numbers.Naturals.Subtraction
 open import Semirings.Definition
 open import Monoids.Definition
 
@@ -20,6 +21,7 @@ module Numbers.Naturals.Naturals where
   open Numbers.Naturals.Multiplication using (_*N_ ; multiplicationNIsCommutative) public
   open Numbers.Naturals.Exponentiation using (_^N_) public
   open Numbers.Naturals.Order using (_<N_ ; le; succPreservesInequality; succIsPositive; addingIncreases; zeroNeverGreater; noIntegersBetweenXAndSuccX; a<SuccA; canRemoveSuccFrom<N) public
+  open Numbers.Naturals.Subtraction using (_-N'_) public
 
   ℕSemiring : Semiring 0 1 _+N_ _*N_
   Monoid.associative (Semiring.monoid ℕSemiring) a b c = equalityCommutative (additionNIsAssociative a b c)

Numbers/Naturals/Subtraction.agda

@@ -0,0 +1,27 @@
+{-# OPTIONS --warning=error --safe --without-K #-}
+
+open import LogicalFormulae
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import WellFoundedInduction
+open import Functions
+open import Orders
+open import Numbers.Naturals.Definition
+open import Numbers.Naturals.Addition
+open import Numbers.Naturals.Order
+open import Numbers.Naturals.Multiplication
+open import Numbers.Naturals.Exponentiation
+open import Semirings.Definition
+open import Monoids.Definition
+open import Maybe
+
+module Numbers.Naturals.Subtraction where
+
+_-N'_ : (a b : ℕ) → Maybe ℕ
+zero -N' zero = yes 0
+zero -N' succ b = no
+succ a -N' zero = yes (succ a)
+succ a -N' succ b = a -N' b
+
+subtractZero : (a : ℕ) → a -N' 0 ≡ yes a
+subtractZero zero = refl
+subtractZero (succ a) = refl

Semirings/Solver.agda

@@ -0,0 +1,219 @@
+{-# OPTIONS --warning=error --safe --without-K #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import LogicalFormulae
+open import Semirings.Definition
+open import Maybe
+
+module Semirings.Solver {a : _} {A : Set a} {Zero One : A} {_+_ : A → A → A} {_*_ : A → A → A} (S : Semiring Zero One _+_ _*_) (comm : (a b : A) → a * b ≡ b * a) where
+
+open Semiring S
+
+-- You can see how to use the solver in LectureNotes/NumbersAndSets/Lecture1.agda.
+
+data Expr : Set a where
+  const : A → Expr
+  zero : Expr
+  succ : Expr → Expr
+  plus : Expr → Expr → Expr
+  times : Expr → Expr → Expr
+
+-- TODO: do this in distinct phases. (Work out what those phases are.)
+-- Expose those phases to the user so that they can choose which tactics to use.
+-- Phases:
+--  * In any expression which is all sums, extract all consts out to the front
+--  * In any expression which is all sums, extract all the succs to the front
+
+eval : Expr → A
+eval (const k) = k
+eval zero = Zero
+eval (succ e) = One + (eval e)
+eval (plus x y) = (eval x) + (eval y)
+eval (times x y) = (eval x) * (eval y)
+
+simplePlus : Expr → Expr → Expr
+simplePlus (const x) (const y) = plus (const x) (const y)
+simplePlus (const x) zero = const x
+simplePlus (const x) (succ b) = succ (plus (const x) b)
+simplePlus (const x) (plus a b) = plus (const x) (plus a b)
+simplePlus (const x) (times a b) = plus (const x) (times a b)
+simplePlus zero b = b
+simplePlus (succ a) (const x) = succ (plus (const x) a)
+simplePlus (succ a) zero = succ a
+simplePlus (succ a) (succ b) = succ (succ (plus a b))
+simplePlus (succ a) (plus b c) = succ (plus a (plus b c))
+simplePlus (succ a) (times b c) = succ (plus a (times b c))
+simplePlus (plus a b) (const x) = plus (const x) (plus a b)
+simplePlus (plus a b) zero = plus a b
+simplePlus (plus a b) (succ c) = succ (plus a (plus b c))
+simplePlus (plus a b) (plus c d) = plus a (plus b (plus c d))
+simplePlus (plus a b) (times c d) = plus a (plus b (times c d))
+simplePlus (times a b) (const x) = plus (const x) (times a b)
+simplePlus (times a b) zero = times a b
+simplePlus (times a b) (succ c) = succ (plus (times a b) c)
+simplePlus (times a b) (plus c d) = plus (times a b) (plus c d)
+simplePlus (times a b) (times c d) = plus (times a b) (times c d)
+
+lemma : (a b c : A) → a + (b + c) ≡ b + (a + c)
+lemma a b c rewrite commutative a (b + c) | equalityCommutative (+Associative b c a) = applyEquality (b +_) (commutative _ _)
+
+simplePlusIsPlus : (a b : Expr) → eval (simplePlus a b) ≡ eval (plus a b)
+simplePlusIsPlus (const x) (const x₁) = refl
+simplePlusIsPlus (const x) zero rewrite Semiring.sumZeroRight S x = refl
+simplePlusIsPlus (const x) (succ b) = lemma One x (eval b)
+simplePlusIsPlus (const x) (plus b c) = refl
+simplePlusIsPlus (const x) (times b c) = refl
+simplePlusIsPlus zero b = equalityCommutative (sumZeroLeft (eval b))
+simplePlusIsPlus (succ a) (const x) rewrite lemma One x (eval a) = commutative x _
+simplePlusIsPlus (succ a) zero = equalityCommutative (sumZeroRight _)
+simplePlusIsPlus (succ a) (succ b) rewrite lemma (One + eval a) One (eval b) = applyEquality (One +_) (+Associative One _ _)
+simplePlusIsPlus (succ a) (plus b c) = +Associative _ _ _
+simplePlusIsPlus (succ a) (times b c) = +Associative _ _ _
+simplePlusIsPlus (plus a b) (const x) = commutative x _
+simplePlusIsPlus (plus a b) zero = equalityCommutative (sumZeroRight _)
+simplePlusIsPlus (plus a b) (succ c) rewrite lemma (eval a + eval b) One (eval c) = applyEquality (One +_) (+Associative _ _ _)
+simplePlusIsPlus (plus a b) (plus c d) = +Associative (eval a) _ _
+simplePlusIsPlus (plus a b) (times c d) = +Associative _ _ _
+simplePlusIsPlus (times a b) (const x) = commutative x _
+simplePlusIsPlus (times a b) zero = equalityCommutative (sumZeroRight _)
+simplePlusIsPlus (times a b) (succ c) rewrite lemma One (eval a * eval b) (eval c) = refl
+simplePlusIsPlus (times a b) (plus c d) = refl
+simplePlusIsPlus (times a b) (times c d) = refl
+
+simplify : Expr → Expr
+simplify (const k) = const k
+simplify zero = zero
+simplify (succ e) = succ (simplify e)
+simplify (plus (const k) (const x)) = simplePlus (const k) (const x)
+simplify (plus (const k) zero) = simplify (const k)
+simplify (plus (const k) (succ expr)) = succ (simplify (plus (const k) expr))
+simplify (plus (const k) (plus x y)) = simplePlus (const k) (simplify (plus x y))
+simplify (plus (const k) (times x y)) = simplePlus (const k) (simplify (times x y))
+simplify (plus zero expr2) = simplify expr2
+simplify (plus (succ expr1) expr2) = succ (simplify (plus expr1 expr2))
+simplify (plus (plus expr1 expr3) zero) = simplify (plus expr1 expr3)
+simplify (plus (plus expr1 expr3) (const k)) = simplePlus (const k) (simplify (plus expr1 expr3))
+simplify (plus (plus expr1 expr3) (succ expr2)) = succ (simplePlus (simplify (plus expr1 expr3)) (simplify expr2))
+simplify (plus (plus expr1 expr3) (plus expr2 expr4)) = simplePlus (simplify (plus expr1 expr3)) (simplify (plus expr2 expr4))
+simplify (plus (plus expr1 expr3) (times expr2 expr4)) = simplePlus (simplify (plus expr1 expr3)) (simplify (times expr2 expr4))
+simplify (plus (times expr1 expr3) (const k)) = simplePlus (const k) (simplify (times expr1 expr3))
+simplify (plus (times expr1 expr3) zero) = times (simplify expr1) (simplify expr3)
+simplify (plus (times expr1 expr3) (succ expr2)) = succ (simplePlus (simplify (times expr1 expr3)) (simplify expr2))
+simplify (plus (times expr1 expr3) (plus expr2 expr4)) = simplePlus (simplify (times expr1 expr3)) (simplify (plus expr2 expr4))
+simplify (plus (times expr1 expr3) (times expr2 expr4)) = simplePlus (simplify (times expr1 expr3)) (simplify (times expr2 expr4))
+simplify (times (const k) (const x)) = times (const k) (const x)
+simplify (times (const k) zero) = zero
+simplify (times (const k) (succ zero)) = const k
+simplify (times (const k) (succ (const y))) = simplePlus (const k) (times (const k) (const y))
+simplify (times (const k) (succ (plus a b))) = simplePlus (const k) (simplify (times (const k) (plus a b)))
+simplify (times (const k) (succ (times a b))) = simplePlus (const k) (simplify (times (const k) (times a b)))
+simplify (times (const k) (succ (succ x))) = simplePlus (const k) (simplify (times (const k) (succ x)))
+simplify (times (const k) (plus expr2 expr3)) = simplePlus (simplify (times (const k) (expr2))) (simplify (times (const k) expr3))
+simplify (times (const k) (times expr2 expr3)) = times (const k) (simplify (times expr2 expr3))
+simplify (times zero expr2) = zero
+simplify (times (succ zero) expr2) = simplify expr2
+simplify (times (succ (succ expr1)) expr2) = simplePlus (simplify expr2) (simplify (times (succ expr1) expr2))
+simplify (times (succ (plus expr1 expr3)) (const k)) = times (const k) (simplify (succ (plus expr1 expr3)))
+simplify (times (succ (plus expr1 expr3)) zero) = zero
+simplify (times (succ (plus expr1 expr3)) (succ expr2)) = succ (simplePlus (simplify (times expr1 expr2)) (simplePlus (simplify (times expr3 expr2)) (simplePlus expr2 (simplify (plus expr1 expr3)))))
+simplify (times (succ (plus expr1 expr3)) (plus expr2 expr4)) = times (simplify (succ (plus expr1 expr3))) (simplify (plus expr2 expr4))
+simplify (times (succ (plus expr1 expr3)) (times expr2 expr4)) = times (simplify (succ (plus expr1 expr3))) (simplify (times expr2 expr4))
+simplify (times (succ (const k)) (const x)) = simplePlus (const x) (times (const k) (const x))
+simplify (times (succ (const k)) zero) = zero
+simplify (times (succ (const k)) (succ (const x))) = succ (simplePlus (const k) (simplePlus (const x) (simplify (times (const k) (const x)))))
+simplify (times (succ (const k)) (succ zero)) = succ (const k)
+simplify (times (succ (const k)) (succ (succ expr2))) = succ (simplePlus (const k) (simplePlus (simplify (succ expr2)) (simplify (times (const k) (succ (expr2))))))
+simplify (times (succ (const k)) (succ (plus expr2 expr3))) = succ (simplePlus (const k) (simplePlus (simplify (plus expr2 expr3)) (simplify (times (const k) (plus expr2 expr3)))))
+simplify (times (succ (const k)) (succ (times expr2 expr3))) = succ (simplePlus (const k) (simplePlus (simplify (times expr2 expr3)) (simplify (times (const k) (times expr2 expr3)))))
+simplify (times (succ (const k)) (plus expr2 expr3)) = simplePlus (simplify (plus expr2 expr3)) (simplify (times (const k) (plus expr2 expr3)))
+simplify (times (succ (const k)) (times expr2 expr3)) = simplePlus (simplify (times expr2 expr3)) (simplify (times (const k) (times expr2 expr3)))
+simplify (times (succ (times expr1 expr3)) (const x)) = times (const x) (simplify (succ (times expr1 expr3)))
+simplify (times (succ (times expr1 expr3)) zero) = zero
+simplify (times (succ (times expr1 expr3)) (succ expr2)) = succ (simplePlus (simplify expr2) (plus (simplify (times expr1 expr3)) (simplify (times (times expr1 expr3) expr2))))
+simplify (times (succ (times expr1 expr3)) (plus expr2 expr4)) = simplePlus (simplify (plus expr2 expr4)) (simplify (times (times expr1 expr3) (plus expr2 expr4)))
+simplify (times (succ (times expr1 expr3)) (times expr2 expr4)) = simplePlus (simplify (times expr2 expr4)) (simplify (times (times expr1 expr3) (times expr2 expr4)))
+simplify (times (plus expr1 expr3) zero) = zero
+simplify (times (plus expr1 expr3) (succ expr2)) = simplePlus (simplify (plus expr1 expr3)) (simplify (times (plus expr1 expr3) expr2))
+simplify (times (plus expr1 expr3) (const x)) = times (const x) (simplify (plus expr1 expr3))
+simplify (times (plus expr1 expr3) (plus expr2 expr4)) = simplePlus (simplify (times expr1 expr2)) (simplePlus (simplify (times expr3 expr2)) (simplePlus (simplify (times expr1 expr4)) (simplify (times expr3 expr4))))
+simplify (times (plus expr1 expr3) (times expr2 expr4)) = simplePlus (simplify (times expr1 (times expr2 expr4))) (simplify (times expr3 (times expr2 expr4)))
+simplify (times (times expr1 expr3) zero) = zero
+simplify (times (times expr1 expr3) (succ expr2)) = simplePlus (simplify (times expr1 expr3)) (simplify (times (times expr1 expr3) expr2))
+simplify (times (times expr1 expr3) (plus expr2 expr4)) = simplePlus (simplify (times (times expr1 expr3) expr2)) (simplify (times (times expr1 expr3) expr4))
+simplify (times (times expr1 expr3) (times expr2 expr4)) = times (simplify expr1) (simplify (times expr3 (times expr2 expr4)))
+simplify (times (times expr1 expr3) (const x)) = times (simplify (times expr1 expr3)) (const x)
+
+equalPlus : {a b c d : A} → (a ≡ c) → (b ≡ d) → (a + b) ≡ (c + d)
+equalPlus refl refl = refl
+
+equalTimes : {a b c d : A} → (a ≡ c) → (b ≡ d) → (a * b) ≡ (c * d)
+equalTimes refl refl = refl
+
+otherDist : {a b c : A} → (a * b) + (c * b) ≡ (a + c) * b
+otherDist = transitivity (equalPlus (comm _ _) (comm _ _)) (transitivity (equalityCommutative (+DistributesOver* _ _ _)) (comm _ _))
+
+simplifyPreserves : (a : Expr) → eval (simplify a) ≡ eval a
+simplifyPreserves (const x) = refl
+simplifyPreserves zero = refl
+simplifyPreserves (succ a) = applyEquality (One +_) (simplifyPreserves a)
+simplifyPreserves (plus (const x) (const y)) = refl
+simplifyPreserves (plus (const x) zero) = equalityCommutative (sumZeroRight x)
+simplifyPreserves (plus (const x) (succ b)) = transitivity (transitivity (applyEquality (One +_) (transitivity (simplifyPreserves (plus (const x) b)) (commutative x (eval b)))) (+Associative One (eval b) x)) (commutative (One + eval b) x)
+simplifyPreserves (plus (const x) (plus b c)) = transitivity (simplePlusIsPlus (const x) (simplify (plus b c))) (applyEquality (x +_) (simplifyPreserves (plus b c)))
+simplifyPreserves (plus (const x) (times b c)) = transitivity (simplePlusIsPlus (const x) (simplify (times b c))) (applyEquality (x +_) (simplifyPreserves (times b c)))
+simplifyPreserves (plus zero a) = transitivity (simplifyPreserves a) (equalityCommutative (sumZeroLeft (eval a)))
+simplifyPreserves (plus (succ a) b) = transitivity (applyEquality (One +_) (simplifyPreserves (plus a b))) (+Associative One (eval a) (eval b))
+simplifyPreserves (plus (plus a b) (const x)) = transitivity (simplePlusIsPlus (const x) (simplify (plus a b))) (transitivity (commutative x (eval (simplify (plus a b)))) (applyEquality (_+ x) (simplifyPreserves (plus a b))))
+simplifyPreserves (plus (plus a b) zero) = transitivity (simplifyPreserves (plus a b)) (equalityCommutative (sumZeroRight _))
+simplifyPreserves (plus (plus a b) (succ c)) = transitivity (applyEquality (One +_) (simplePlusIsPlus (simplify (plus a b)) (simplify c))) (transitivity (+Associative One _ _) (transitivity (applyEquality (_+ eval (simplify c)) (commutative One _)) (transitivity (equalityCommutative (+Associative _ _ _)) (transitivity (applyEquality (_+ (One + eval (simplify c))) (simplifyPreserves (plus a b))) (applyEquality (λ i → (eval a + eval b) + (One + i)) (simplifyPreserves c))))))
+simplifyPreserves (plus (plus a b) (plus c d)) = transitivity (simplePlusIsPlus (simplify (plus a b)) (simplify (plus c d))) (transitivity (applyEquality (_+ eval (simplify (plus c d))) (simplifyPreserves (plus a b))) (applyEquality ((eval a + eval b) +_) (simplifyPreserves (plus c d))))
+simplifyPreserves (plus (plus a b) (times c d)) = transitivity (simplePlusIsPlus (simplify (plus a b)) (simplify (times c d))) (transitivity (applyEquality (_+ eval (simplify (times c d))) (simplifyPreserves (plus a b))) (applyEquality ((eval a + eval b) +_) (simplifyPreserves (times c d))))
+simplifyPreserves (plus (times a b) (const x)) = transitivity (simplePlusIsPlus (const x) (simplify (times a b))) (transitivity (commutative x _) (applyEquality (_+ x) (simplifyPreserves (times a b))))
+simplifyPreserves (plus (times a b) zero) = transitivity (transitivity (applyEquality (_* eval (simplify b)) (simplifyPreserves a)) (applyEquality ((eval a) *_) (simplifyPreserves b))) (equalityCommutative (sumZeroRight _))
+simplifyPreserves (plus (times a b) (succ c)) = transitivity (transitivity (applyEquality (One +_) (transitivity (simplePlusIsPlus (simplify (times a b)) (simplify c)) (transitivity (transitivity (applyEquality (eval (simplify (times a b)) +_) (simplifyPreserves c)) (applyEquality (_+ eval c) (simplifyPreserves (times a b)))) (commutative _ (eval c))))) (+Associative One (eval c) _)) (commutative _ (eval a * eval b))
+simplifyPreserves (plus (times a b) (plus c d)) = transitivity (simplePlusIsPlus (simplify (times a b)) (simplify (plus c d))) (transitivity (applyEquality (_+ eval (simplify (plus c d))) (simplifyPreserves (times a b))) (applyEquality ((eval a * eval b) +_) (simplifyPreserves (plus c d))))
+simplifyPreserves (plus (times a b) (times c d)) = transitivity (simplePlusIsPlus (simplify (times a b)) (simplify (times c d))) (transitivity (applyEquality (_+ eval (simplify (times c d))) (simplifyPreserves (times a b))) (applyEquality ((eval a * eval b) +_) (simplifyPreserves (times c d))))
+simplifyPreserves (times (const x) (const y)) = refl
+simplifyPreserves (times (const x) zero) = equalityCommutative (productZeroRight x)
+simplifyPreserves (times (const x) (succ (const y))) = transitivity (applyEquality (_+ (x * y)) (equalityCommutative (productOneRight _))) (equalityCommutative (+DistributesOver* x One y))
+simplifyPreserves (times (const x) (succ zero)) = equalityCommutative (transitivity (applyEquality (x *_) (sumZeroRight One)) (productOneRight x))
+simplifyPreserves (times (const x) (succ (succ b))) = transitivity (simplePlusIsPlus (const x) (simplify (times (const x) (succ b)))) (transitivity (equalPlus (equalityCommutative (productOneRight _)) (simplifyPreserves (times (const x) (succ b)))) (equalityCommutative (+DistributesOver* x One _)))
+simplifyPreserves (times (const x) (succ (plus b c))) = transitivity (simplePlusIsPlus (const x) (simplePlus (simplify (times (const x) b)) (simplify (times (const x) c)))) (transitivity (equalPlus (equalityCommutative (productOneRight _)) (transitivity (simplePlusIsPlus (simplify (times (const x) b)) (simplify (times (const x) c))) (transitivity (equalPlus (simplifyPreserves (times (const x) b)) (simplifyPreserves (times (const x) c))) (equalityCommutative (+DistributesOver* x (eval b) (eval c)))))) (equalityCommutative (+DistributesOver* x One _)))
+simplifyPreserves (times (const x) (succ (times b c))) = transitivity (equalPlus (equalityCommutative (productOneRight x)) (applyEquality (x *_) (simplifyPreserves (times b c)))) (equalityCommutative (+DistributesOver* x One _))
+simplifyPreserves (times (const x) (plus b c)) = transitivity (simplePlusIsPlus (simplify (times (const x) b)) (simplify (times (const x) c))) (transitivity (transitivity (applyEquality (eval (simplify (times (const x) b)) +_) (simplifyPreserves (times (const x) c))) (applyEquality (_+ (x * eval c)) (simplifyPreserves (times (const x) b)))) (equalityCommutative (+DistributesOver* x (eval b) (eval c))))
+simplifyPreserves (times (const x) (times b c)) = applyEquality (x *_) (simplifyPreserves (times b c))
+simplifyPreserves (times zero b) = equalityCommutative (productZeroLeft (eval b))
+simplifyPreserves (times (succ (const x)) (const y)) = transitivity (transitivity (equalPlus (equalityCommutative (productOneRight y)) (comm x y)) (equalityCommutative (+DistributesOver* y One x))) (comm y _)
+simplifyPreserves (times (succ (const x)) zero) = equalityCommutative (productZeroRight _)
+simplifyPreserves (times (succ (const x)) (succ (const y))) = transitivity (transitivity (transitivity (+Associative One x _) (applyEquality ((One + x) +_) (transitivity (equalPlus (equalityCommutative (productOneRight y)) (comm x y)) (equalityCommutative (+DistributesOver* y One x))))) (equalPlus (equalityCommutative (productOneRight (One + x))) (comm _ _))) (equalityCommutative (+DistributesOver* _ _ _))
+simplifyPreserves (times (succ (const x)) (succ zero)) = equalityCommutative (transitivity (applyEquality ((One + x) *_) (sumZeroRight One)) (productOneRight _))
+simplifyPreserves (times (succ (const x)) (succ (succ b))) = transitivity (applyEquality (One +_) (simplePlusIsPlus (const x) (simplePlus (succ (simplify b)) (simplify (times (const x) (succ b)))))) (transitivity (applyEquality (λ i → One + (x + i)) (simplePlusIsPlus (succ (simplify b)) (simplify (times (const x) (succ b))))) (transitivity (applyEquality (λ i → One + (x + ((One + i) + eval (simplify (times (const x) (succ b)))))) (simplifyPreserves b)) (transitivity (applyEquality (λ i → One + (x + ((One + eval b) + i))) (simplifyPreserves (times (const x) (succ b)))) (transitivity (transitivity (+Associative One x _) (equalPlus (equalityCommutative (productOneRight _)) (transitivity (transitivity (transitivity (applyEquality ((One + eval b) +_) (+DistributesOver* x One (eval b))) (transitivity (transitivity (equalityCommutative (+Associative One (eval b) _)) (transitivity (applyEquality (One +_) (transitivity (+Associative (eval b) _ _) (transitivity (equalPlus (transitivity (commutative _ _) (equalPlus (productOneRight x) (equalityCommutative (productOneRight _)))) (comm x _)) (equalityCommutative (+Associative x _ _))))) (+Associative One x _))) (applyEquality ((One + x) +_) (equalityCommutative (+DistributesOver* (eval b) One x))))) (equalPlus (equalityCommutative (productOneRight _)) (comm (eval b) _))) (equalityCommutative (+DistributesOver* (One + x) One _))))) (equalityCommutative (+DistributesOver* (One + x) One _))))))
+simplifyPreserves (times (succ (const x)) (succ (plus b c))) = transitivity (transitivity (transitivity (applyEquality (One +_) (simplePlusIsPlus (const x) (simplePlus (simplify (plus b c)) (simplePlus (simplify (times (const x) b)) (simplify (times (const x) c)))))) (transitivity (+Associative One x _) (applyEquality ((One + x) +_) (transitivity (simplePlusIsPlus (simplify (plus b c)) (simplePlus (simplify (times (const x) b)) (simplify (times (const x) c)))) (transitivity (equalPlus (simplifyPreserves (plus b c)) (simplePlusIsPlus (simplify (times (const x) b)) (simplify (times (const x) c)))) (transitivity (transitivity (transitivity (transitivity (equalityCommutative (+Associative _ _ _ )) (transitivity (applyEquality (eval b +_) (transitivity (applyEquality (eval c +_) (equalPlus (simplifyPreserves (times (const x) b)) (simplifyPreserves (times (const x) c)))) (transitivity (+Associative (eval c) _ _) (transitivity (applyEquality (_+ (x * eval c)) (commutative _ _)) (equalityCommutative (+Associative _ _ _)))))) (+Associative _ _ _))) (equalPlus (equalPlus (equalityCommutative (productOneRight (eval b))) (comm _ _)) (equalPlus (equalityCommutative (productOneRight (eval c))) (comm x _)))) (equalPlus (equalityCommutative (+DistributesOver* (eval b) One x)) (equalityCommutative (+DistributesOver* (eval c) One x)))) (equalPlus (comm _ _) (comm _ _)))))))) (equalPlus (equalityCommutative (productOneRight _)) (equalityCommutative (+DistributesOver* (One + x) _ _)))) (equalityCommutative (+DistributesOver* (One + x) _ _))
+simplifyPreserves (times (succ (const x)) (succ (times b c))) = transitivity (transitivity (applyEquality (One +_) (simplePlusIsPlus (const x) (simplePlus (simplify (times b c)) (times (const x) (simplify (times b c)))))) (transitivity (+Associative One x _) (equalPlus (equalityCommutative (productOneRight _)) (transitivity (simplePlusIsPlus (simplify (times b c)) (times (const x) (simplify (times b c)))) (transitivity (transitivity (equalPlus (transitivity (simplifyPreserves (times b c)) (equalityCommutative (productOneRight _))) (transitivity (comm _ _) (applyEquality (_* x) (simplifyPreserves (times b c))))) (equalityCommutative (+DistributesOver* _ One x))) (comm _ _)))))) (equalityCommutative (+DistributesOver* _ _ _))
+simplifyPreserves (times (succ (const x)) (plus b c)) = transitivity (simplePlusIsPlus (simplify (plus b c)) (simplePlus (simplify (times (const x) b)) (simplify (times (const x) c)))) (transitivity (transitivity (equalPlus (transitivity (simplifyPreserves (plus b c)) (equalityCommutative (productOneRight _))) (transitivity (simplePlusIsPlus (simplify (times (const x) b)) (simplify (times (const x) c))) (transitivity (transitivity (equalPlus (simplifyPreserves (times (const x) b)) (simplifyPreserves (times (const x) c))) (equalityCommutative (+DistributesOver* x _ _))) (comm x _)))) (equalityCommutative (+DistributesOver* _ One x))) (comm _ _))
+simplifyPreserves (times (succ (const x)) (times b c)) = transitivity (simplePlusIsPlus (simplify (times b c)) (times (const x) (simplify (times b c)))) (transitivity (transitivity (equalPlus (transitivity (simplifyPreserves (times b c)) (equalityCommutative (productOneRight _))) (transitivity (comm x _) (applyEquality (_* x) (simplifyPreserves (times b c))))) (equalityCommutative (+DistributesOver* _ One x))) (comm _ _))
+simplifyPreserves (times (succ zero) b) = transitivity (simplifyPreserves b) (transitivity (equalityCommutative (productOneLeft (eval b))) (applyEquality (_* eval b) (equalityCommutative (sumZeroRight One))))
+simplifyPreserves (times (succ (succ a)) b) = transitivity (simplePlusIsPlus (simplify b) (simplify (times (succ a) b))) (transitivity (equalPlus (simplifyPreserves b) (simplifyPreserves (times (succ a) b))) (transitivity (transitivity (equalPlus (equalityCommutative (productOneRight _)) (comm _ _)) (equalityCommutative (+DistributesOver* (eval b) One _))) (comm (eval b) _)))
+simplifyPreserves (times (succ (plus a b)) (const x)) = transitivity (comm x _) (applyEquality (λ i → (One + i) * x) (simplifyPreserves (plus a b)))
+simplifyPreserves (times (succ (plus a b)) zero) = equalityCommutative (productZeroRight _)
+simplifyPreserves (times (succ (plus a b)) (succ c)) = transitivity (transitivity (transitivity (applyEquality (One +_) (transitivity (simplePlusIsPlus (simplify (times a c)) (simplePlus (simplify (times b c)) (simplePlus c (simplify (plus a b))))) (transitivity (transitivity (equalPlus (simplifyPreserves (times a c)) (simplePlusIsPlus (simplify (times b c)) (simplePlus c (simplify (plus a b))))) (transitivity (+Associative _ _ _) (transitivity (transitivity (equalPlus (transitivity (equalPlus (comm _ _) (transitivity (simplifyPreserves (times b c)) (comm _ _))) (equalityCommutative (+DistributesOver* _ _ _))) (transitivity (simplePlusIsPlus c (simplify (plus a b))) (transitivity (commutative _ _) (equalPlus (simplifyPreserves (plus a b)) (equalityCommutative (productOneRight _)))))) (commutative _ _)) (equalityCommutative (+Associative _ _ _))))) (applyEquality ((eval a + eval b) +_) (equalityCommutative (+DistributesOver* _ One _)))))) (+Associative One _ _)) (equalPlus (equalityCommutative (productOneRight _)) (comm _ _))) (equalityCommutative (+DistributesOver* _ _ _))
+simplifyPreserves (times (succ (plus a b)) (plus c d)) = equalTimes (applyEquality (One +_) (simplifyPreserves (plus a b))) (simplifyPreserves (plus c d))
+simplifyPreserves (times (succ (plus a b)) (times c d)) = equalTimes (applyEquality (One +_) (simplifyPreserves (plus a b))) (simplifyPreserves (times c d))
+simplifyPreserves (times (succ (times a b)) (const x)) = transitivity (comm x _) (applyEquality (λ i → (One + i) * x) (simplifyPreserves (times a b)))
+simplifyPreserves (times (succ (times a b)) zero) = equalityCommutative (productZeroRight _)
+simplifyPreserves (times (succ (times a b)) (succ c)) = transitivity (transitivity (transitivity (applyEquality (One +_) (transitivity (simplePlusIsPlus (simplify c) (plus (simplify (times a b)) (simplify (times (times a b) c)))) (transitivity (transitivity (transitivity (equalPlus (transitivity (simplifyPreserves c) (equalityCommutative (productOneRight _))) (transitivity (commutative _ _) (equalPlus (transitivity (simplifyPreserves (times (times a b) c)) (comm _ _)) (simplifyPreserves (times a b))))) (+Associative _ _ _)) (applyEquality (_+ (eval a * eval b)) (equalityCommutative (+DistributesOver* _ One _)))) (commutative _ _)))) (+Associative One _ _)) (equalPlus (equalityCommutative (productOneRight _)) (comm _ _))) (equalityCommutative (+DistributesOver* _ One _))
+simplifyPreserves (times (succ (times a b)) (plus c d)) = transitivity (simplePlusIsPlus (simplify (plus c d)) (simplePlus (simplify (times (times a b) c)) (simplify (times (times a b) d)))) (transitivity (equalPlus (simplifyPreserves (plus c d)) (simplePlusIsPlus (simplify (times (times a b) c)) (simplify (times (times a b) d)))) (transitivity (transitivity (equalPlus (equalityCommutative (productOneRight _)) (transitivity (transitivity (equalPlus (simplifyPreserves (times (times a b) c)) (simplifyPreserves (times (times a b) d))) (equalityCommutative (+DistributesOver* _ _ _))) (comm _ _))) (equalityCommutative (+DistributesOver* _ One _))) (comm _ _)))
+simplifyPreserves (times (succ (times a b)) (times c d)) = transitivity (simplePlusIsPlus (simplify (times c d)) (times (simplify a) (simplify (times b (times c d))))) (transitivity (equalPlus (simplifyPreserves (times c d)) (equalTimes (simplifyPreserves a) (simplifyPreserves (times b (times c d))))) (transitivity (transitivity (equalPlus (equalityCommutative (productOneRight _)) (transitivity (*Associative (eval a) _ _) (comm _ _))) (equalityCommutative (+DistributesOver* _ One _))) (comm _ _)))
+simplifyPreserves (times (plus a b) (const x)) = transitivity (comm x _) (applyEquality (_* x) (simplifyPreserves (plus a b)))
+simplifyPreserves (times (plus a b) zero) = equalityCommutative (productZeroRight _)
+simplifyPreserves (times (plus a b) (succ c)) = transitivity (simplePlusIsPlus (simplify (plus a b)) (simplify (times (plus a b) c))) (transitivity (applyEquality ((eval (simplify (plus a b))) +_) (simplifyPreserves (times (plus a b) c))) (transitivity (applyEquality (_+ ((eval a + eval b) * eval c)) (simplifyPreserves (plus a b))) (transitivity (equalPlus (equalityCommutative (productOneRight _)) refl) (equalityCommutative (+DistributesOver* _ One (eval c))))))
+simplifyPreserves (times (plus a b) (plus c d)) = transitivity (simplePlusIsPlus (simplify (times a c)) (simplePlus (simplify (times b c)) (simplePlus (simplify (times a d)) (simplify (times b d))))) (transitivity (transitivity (transitivity (transitivity (equalPlus (transitivity (simplifyPreserves (times a c)) (comm _ _)) (transitivity (simplePlusIsPlus (simplify (times b c)) (simplePlus (simplify (times a d)) (simplify (times b d)))) (equalPlus (transitivity (simplifyPreserves (times b c)) (comm _ _)) (transitivity (simplePlusIsPlus (simplify (times a d)) (simplify (times b d))) (equalPlus (transitivity (simplifyPreserves (times a d)) (comm _ _)) (transitivity (simplifyPreserves (times b d)) (comm _ _))))))) (+Associative _ _ _)) (equalPlus (equalityCommutative (+DistributesOver* (eval c) (eval a) _)) (equalityCommutative (+DistributesOver* (eval d) (eval a) _)))) (equalPlus (comm (eval c) _) (comm (eval d) _))) (equalityCommutative (+DistributesOver* _ (eval c) (eval d))))
+simplifyPreserves (times (plus a b) (times c d)) = transitivity (simplePlusIsPlus (simplify (times a (times c d))) (simplify (times b (times c d)))) (transitivity (equalPlus (simplifyPreserves (times a (times c d))) (simplifyPreserves (times b (times c d)))) (transitivity (equalPlus (comm (eval a) _) (comm (eval b) _)) (transitivity (equalityCommutative (+DistributesOver* _ (eval a) (eval b))) (comm _ _))))
+simplifyPreserves (times (times a b) (const x)) = applyEquality (_* x) (simplifyPreserves (times a b))
+simplifyPreserves (times (times a b) zero) = equalityCommutative (productZeroRight _)
+simplifyPreserves (times (times a b) (succ c)) = transitivity (simplePlusIsPlus (simplify (times a b)) (simplify (times (times a b) c))) (transitivity (equalPlus (simplifyPreserves (times a b)) (simplifyPreserves (times (times a b) c))) (transitivity (applyEquality (_+ ((eval a * eval b) * eval c)) (equalityCommutative (productOneRight _))) (equalityCommutative (+DistributesOver* _ One (eval c)))))
+simplifyPreserves (times (times a b) (plus c d)) = transitivity (simplePlusIsPlus (simplify (times (times a b) c)) (simplify (times (times a b) d))) (transitivity (equalPlus (simplifyPreserves (times (times a b) c)) (simplifyPreserves (times (times a b) d))) (equalityCommutative (+DistributesOver* _ _ _)))
+simplifyPreserves (times (times a b) (times c d)) = transitivity (equalTimes (simplifyPreserves a) (simplifyPreserves (times b (times c d)))) (*Associative (eval a) (eval b) _)
+
+_!!_ : (x y : Expr) (pr : eval (simplify x) ≡ eval (simplify y)) → eval x ≡ eval y
+_!!_ = λ a b c → transitivity (equalityCommutative (simplifyPreserves a)) (transitivity c (simplifyPreserves b))