Semiring solver (#50)

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))