Split out much more structure (#37)

2025-10-18 17:48:40 +00:00 · 2019-08-18 14:57:41 +01:00
parent 69389bb889
commit 1f26064502
52 changed files with 2137 additions and 2014 deletions
--- a/Numbers/Naturals/Multiplication.agda
+++ b/Numbers/Naturals/Multiplication.agda
@@ -0,0 +1,82 @@
+{-# OPTIONS --warning=error --safe --without-K #-}
+
+open import LogicalFormulae
+open import Numbers.Naturals.Definition
+open import Numbers.Naturals.Addition
+
+module Numbers.Naturals.Multiplication where
+
+infix 25 _*N_
+_*N_ : ℕ → ℕ → ℕ
+zero *N y = zero
+(succ x) *N y = y +N (x *N y)
+
+productZeroIsZeroLeft : (a : ℕ) → (zero *N a ≡ zero)
+productZeroIsZeroLeft a = refl
+
+productZeroIsZeroRight : (a : ℕ) → (a *N zero ≡ zero)
+productZeroIsZeroRight zero = refl
+productZeroIsZeroRight (succ a) = productZeroIsZeroRight a
+
+productWithOneLeft : (a : ℕ) → ((succ zero) *N a) ≡ a
+productWithOneLeft a = transitivity refl (transitivity (applyEquality (λ { m -> a +N m }) refl) (additionNIsCommutative a zero))
+
+productWithOneRight : (a : ℕ) → a *N succ zero ≡ a
+productWithOneRight zero = refl
+productWithOneRight (succ a) = transitivity refl (addingPreservesEqualityLeft (succ zero) (productWithOneRight a))
+
+productDistributes : (a b c : ℕ) → (a *N (b +N c)) ≡ a *N b +N a *N c
+productDistributes zero b c = refl
+productDistributes (succ a) b c = transitivity refl
+  (transitivity (addingPreservesEqualityLeft (b +N c) (productDistributes a b c))
+    (transitivity (equalityCommutative (additionNIsAssociative (b +N c) (a *N b) (a *N c)))
+      (transitivity (addingPreservesEqualityRight (a *N c) (additionNIsCommutative (b +N c) (a *N b)))
+        (transitivity (addingPreservesEqualityRight (a *N c) (equalityCommutative (additionNIsAssociative (a *N b) b c)))
+          (transitivity (addingPreservesEqualityRight (a *N c) (addingPreservesEqualityRight c (additionNIsCommutative (a *N b) b)))
+            (transitivity (addingPreservesEqualityRight (a *N c) (addingPreservesEqualityRight {b +N a *N b} {(succ a) *N b} c (refl)))
+              (transitivity (additionNIsAssociative ((succ a) *N b) c (a *N c))
+                (transitivity (addingPreservesEqualityLeft (succ a *N b) refl)
+                  refl)
+            )
+        ))))))
+
+productPreservesEqualityLeft : (a : ℕ) → {b c : ℕ} → b ≡ c → a *N b ≡ a *N c
+productPreservesEqualityLeft a {b} {.b} refl = refl
+
+aSucB : (a b : ℕ) → a *N succ b ≡ a *N b +N a
+aSucB a b = transitivity {_} {ℕ} {a *N succ b} {a *N (b +N succ zero)} (productPreservesEqualityLeft a (succIsAddOne b)) (transitivity {_} {ℕ} {a *N (b +N succ zero)} {a *N b +N a *N succ zero} (productDistributes a b (succ zero)) (addingPreservesEqualityLeft (a *N b) (productWithOneRight a)))
+
+aSucBRight : (a b : ℕ) → (succ a) *N b ≡ a *N b +N b
+aSucBRight a b = additionNIsCommutative b (a *N b)
+
+multiplicationNIsCommutative : (a b : ℕ) → (a *N b) ≡ (b *N a)
+multiplicationNIsCommutative zero b = transitivity (productZeroIsZeroLeft b) (equalityCommutative (productZeroIsZeroRight b))
+multiplicationNIsCommutative (succ a) zero = multiplicationNIsCommutative a zero
+multiplicationNIsCommutative (succ a) (succ b) = transitivity refl
+  (transitivity (addingPreservesEqualityLeft (succ b) (aSucB a b))
+    (transitivity (additionNIsCommutative (succ b) (a *N b +N a))
+      (transitivity (additionNIsAssociative (a *N b) a (succ b))
+        (transitivity (addingPreservesEqualityLeft (a *N b) (succCanMove a b))
+          (transitivity (addingPreservesEqualityLeft (a *N b) (additionNIsCommutative (succ a) b))
+            (transitivity (equalityCommutative (additionNIsAssociative (a *N b) b (succ a)))
+              (transitivity (addingPreservesEqualityRight (succ a) (equalityCommutative (aSucBRight a b)))
+                (transitivity (addingPreservesEqualityRight (succ a) (multiplicationNIsCommutative (succ a) b))
+                  (transitivity (additionNIsCommutative (b *N (succ a)) (succ a))
+                    refl
+                )))))))))
+
+productDistributes' : (a b c : ℕ) → (a +N b) *N c ≡ a *N c +N b *N c
+productDistributes' a b c rewrite multiplicationNIsCommutative (a +N b) c | productDistributes c a b | multiplicationNIsCommutative c a | multiplicationNIsCommutative c b = refl
+
+flipProductsWithinSum : (a b c : ℕ) → (c *N a +N c *N b ≡ a *N c +N b *N c)
+flipProductsWithinSum a b c = transitivity (addingPreservesEqualityRight (c *N b) (multiplicationNIsCommutative c a)) ((addingPreservesEqualityLeft (a *N c) (multiplicationNIsCommutative c b)))
+
+productDistributesRight : (a b c : ℕ) → (a +N b) *N c ≡ a *N c +N b *N c
+productDistributesRight a b c = transitivity (multiplicationNIsCommutative (a +N b) c) (transitivity (productDistributes c a b) (flipProductsWithinSum a b c))
+
+multiplicationNIsAssociative : (a b c : ℕ) → (a *N (b *N c)) ≡ ((a *N b) *N c)
+multiplicationNIsAssociative zero b c = refl
+multiplicationNIsAssociative (succ a) b c =
+    transitivity refl
+      (transitivity refl
+        (transitivity (applyEquality ((λ x → b *N c +N x)) (multiplicationNIsAssociative a b c)) (transitivity (equalityCommutative (productDistributesRight b (a *N b) c)) refl)))