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Diffstat (limited to 'src/natural-numbers.ts')
| -rw-r--r-- | src/natural-numbers.ts | 186 |
1 files changed, 186 insertions, 0 deletions
diff --git a/src/natural-numbers.ts b/src/natural-numbers.ts new file mode 100644 index 0000000..fa8a3ce --- /dev/null +++ b/src/natural-numbers.ts @@ -0,0 +1,186 @@ +import { Add, Multiply, Succ, _0, _1, _3 } from './utils/nat'; +import { Op, Rewrite, ChainRewrites, Sym, Equation, Refl, VerifyEquation, assert, Subst, ApplyRewrite, Evaluate, OpToStr } from './utils/theorem'; + +export namespace addition { + // Identity + export type Identity<A extends Op> = Rewrite<Add<A, _0>, A>; + export type IdentityR<A extends Op> = Rewrite<Add<_0, A>, A>; + + export type SuccLemma<A extends Op, B extends Op> = Rewrite<Add<Succ<A>, B>, Succ<Add<A, B>>> + export type SuccLemmaR<A extends Op, B extends Op> = Rewrite<Add<A, Succ<B>>, Succ<Add<A, B>>> + + // Commutativity + export type Commutativity<A extends Op, B extends Op> = Rewrite<Add<A, B>, Add<B, A>>; + export interface Comm_Base_Proof<B extends Op> { + type: 'rewrite', + left: Equation<Add<_0, B>, Add<B, _0>>; // 0 + b = b + 0 + right: ChainRewrites<[ + IdentityR<B>, // b = b + 0 + Sym<Identity<B>>, // b + 0 = b + 0 + Refl, + ], this['left']>; + }; + export interface Comm_Inductive_Proof<A extends Op, B extends Op> { + type: 'rewrite', + left: Equation<Add<Succ<A>, B>, Add<B, Succ<A>>>; // succ(a) + b = b + succ(a) + right: ChainRewrites<[ + SuccLemma<A, B>, // succ(a + b) = b + succ(a) + Commutativity<A, B>, // succ(b + a) = b + succ(a) + SuccLemmaR<B, A>, // succ(b + a) = succ(b + a) + Refl, + ], this['left']>; + }; + export namespace spec { + export type commutativity = [ + assert<VerifyEquation<Comm_Base_Proof<'B'>>>, + assert<VerifyEquation<Comm_Inductive_Proof<'A', 'B'>>>, + ] + } + + // Associativity + export type Associativity<A extends Op, B extends Op, C extends Op> = Rewrite<Add<Add<A, B>, C>, Add<A, Add<B, C>>>; + export interface Assoc_Base_Proof<B extends Op, C extends Op> { + type: 'rewrite', + left: Equation<Add<Add<_0, B>, C>, Add<_0, Add<B, C>>>; // (0 + b) + c = 0 + (b + c) + right: ChainRewrites<[ + Commutativity<_0, B>, // (b + 0) + c = 0 + (b + c) + Identity<B>, // b + c = 0 + (b + c) + IdentityR<Add<B, C>>, // b + c = b + c + Refl, + ], this['left']>; + }; + export interface Assoc_Inductive_Proof<A extends Op, B extends Op, C extends Op> { + type: 'rewrite', + left: Equation<Add<Add<Succ<A>, B>, C>, Add<Succ<A>, Add<B, C>>>; // (succ(a) + b) + c = succ(a) + (b + c) + right: ChainRewrites<[ + SuccLemma<A, B>, // succ(a + b) + c = succ(a) + (b + c) + SuccLemma<A, Add<B, C>>, // succ(a + b) + c = succ(a + (b + c)) + SuccLemma<Add<A, B>, C>, // succ((a + b) + c) = succ(a + (b + c)) + Associativity<A, B, C>, // succ(a + (b + c)) = succ(a + (b + c)) + Refl, + ], this['left']>; + }; + + export namespace spec { + export type associativity = [ + assert<VerifyEquation<Assoc_Base_Proof<'A', 'B'>>>, + assert<VerifyEquation<Assoc_Inductive_Proof<'A', 'B', 'C'>>>, + ] + } + + // Rearrange + export interface Rearrange_Proof<A extends Op, B extends Op, C extends Op, D extends Op> { + type: 'rewrite', + left: Equation<Add<Add<A, B>, Add<C, D>>, Add<A, Add<Add<B, C>, D>>>; // (a + b) + (c + d) = a + ((b + c) + d) + right: ChainRewrites<[ + Associativity<A, B, Add<C, D>>, // a + (b + (c + d)) = a + ((b + c) + d) + Associativity<B, C, D>, // a + (b + (c + d)) = a + (b + (c + d)) + Refl, + ], this['left']>; + }; + export namespace spec { + export type rearrange = [ + assert<VerifyEquation<Rearrange_Proof<'A', 'B', 'C', 'D'>>>, + ] + } +} + +export namespace multiplication { + // Identity + export type Identity<A extends Op> = Rewrite<Multiply<A, _1>, A>; + export type IdentityR<A extends Op> = Rewrite<Multiply<_1, A>, A>; + + export type SuccLemma<A extends Op, B extends Op> = Rewrite<Multiply<Succ<A>, B>, Add<B, Multiply<A, B>>>; + export type SuccLemmaR<A extends Op, B extends Op> = Rewrite<Multiply<A, Succ<B>>, Add<A, Multiply<A, B>>>; + + // Commutativity + export type Commutativity<A extends Op, B extends Op> = Rewrite<Multiply<A, B>, Multiply<B, A>>; + export interface Comm_Base_Proof<B extends Op> { + type: 'rewrite', + left: Equation<Multiply<_1, B>, Multiply<B, _1>>; // 1 * b = b * 1 + right: ChainRewrites<[ + IdentityR<B>, // b = b * 1 + Identity<B>, // b = b + Refl, + ], this['left']>; + }; + export interface Comm_Inductive_Proof<A extends Op, B extends Op> { + type: 'rewrite', + left: Equation<Multiply<Succ<A>, B>, Multiply<B, Succ<A>>>; // succ(a)*b = b*succ(a) + right: ChainRewrites<[ + SuccLemma<A, B>, // b + a*b = b*succ(a) + SuccLemmaR<B, A>, // b + a*b = b + b*a + Commutativity<B, A>, // b + a*b = b + a*b + Refl, + ], this['left']>; + }; + export namespace spec { + export type commutativity = [ + assert<VerifyEquation<Comm_Base_Proof<'B'>>>, + assert<VerifyEquation<Comm_Inductive_Proof<'A', 'B'>>>, + ] + } + + // Distributivity + export type Distributivity<A extends Op, B extends Op, C extends Op> = + Rewrite<Multiply<Add<A, B>, C>, Add<Multiply<A, C>, Multiply<B, C>>>; + export interface Dist_Base_Proof<B extends Op, C extends Op> { + type: 'rewrite', + left: Equation<Multiply<Add<_1, B>, C>, Add<Multiply<_1, C>, Multiply<B, C>>>; // (1 + b)*c = 1*c + b*c + right: ChainRewrites<[ + IdentityR<C>, // (1 + b)*c = c + b*c + addition.SuccLemma<_0, B>, // succ(0 + b)*c = c + b*c + addition.IdentityR<B>, // succ(b)*c = c + b*c + SuccLemma<B, C>, // c + b*c = c + b*c + Refl, + ], this['left']>; + }; + export interface Dist_Inductive_Proof<A extends Op, B extends Op, C extends Op> { + type: 'rewrite', + left: Equation<Multiply<Add<Succ<A>, B>, C>, Add<Multiply<Succ<A>, C>, Multiply<B, C>>>; // (succ(a) + b)*c = succ(a)*c + b*c + right: ChainRewrites<[ + addition.SuccLemma<A, B>, // succ(a + b)*c = succ(a)*c + b*c + SuccLemma<Add<A, B>, C>, // c + (a + b)*c = succ(a)*c + b*c + SuccLemma<A, C>, // c + (a + b)*c = (c + a*c) + b*c + Distributivity<A, B, C>, // c + (a*c + b*c) = (c + a*c) + b*c + addition.Associativity<C, Multiply<A, C>, Multiply<B, C>>, // (c + a*c) + b*c = (c + a*c) + b*c + Refl, + ], this['left']>; + }; + export namespace spec { + export type distributivity = [ + assert<VerifyEquation<Dist_Base_Proof<'B', 'C'>>>, + assert<VerifyEquation<Dist_Inductive_Proof<'A', 'B', 'C'>>>, + ] + } + + // Associativity + export type Associativity<A extends Op, B extends Op, C extends Op> = + Rewrite<Multiply<Multiply<A, B>, C>, Multiply<A, Multiply<B, C>>>; + export interface Assoc_Base_Proof<B extends Op, C extends Op> { + type: 'rewrite', + left: Equation<Multiply<Multiply<_1, B>, C>, Multiply<_1, Multiply<B, C>>>; // (1*b)*c = 1*(b*c) + right: ChainRewrites<[ + IdentityR<B>, // b*c = 1*(b*c) + IdentityR<Multiply<B, C>>, // b*c = b*c + Refl, + ], this['left']>; + }; + export interface Assoc_Inductive_Proof<A extends Op, B extends Op, C extends Op> { + type: 'rewrite', + left: Equation<Multiply<Multiply<Succ<A>, B>, C>, Multiply<Succ<A>, Multiply<B, C>>>; // (succ(a)*b)*c = succ(a)*(b*c) + right: ChainRewrites<[ + SuccLemma<A, B>, // (b + a*b)*c = succ(a) * (b*c) + Distributivity<B, Multiply<A, B>, C>, // b*c + (a*b)*c = succ(a) * (b*c) + SuccLemma<A, Multiply<B, C>>, // b*c + (a*b)*c = b*c + a*(b*c) + Associativity<A, B, C>, // b*c + a*(b*c) = b*c + a*(b*c) + Refl, + ], this['left']>; + }; + export namespace spec { + export type associativity = [ + assert<VerifyEquation<Assoc_Base_Proof<'B', 'C'>>>, + assert<VerifyEquation<Assoc_Inductive_Proof<'A', 'B', 'C'>>>, + ] + } +} |