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import { Add, Multiply, Succ, _0, _1, _3 } from './nat';
import { Op, Rewrite, ChainRewrites, Sym, Equation, Refl, VerifyEquation, assert, RewriteBase, Subst, ApplyRewrite, Evaluate, ReflR, OpToStr } from './util';
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'>>>,
]
}
}
export namespace equation {
type F<X extends Op> = { op: '<F>', a: _0, b: X };
export type Reflexivity<A extends Op> = Rewrite<Equation<A, A>, 'true'>;
export type Symmetry<A extends Op, B extends Op> = Rewrite<Equation<A, B>, Equation<B, A>>;
export type Congruence<A extends Op, B extends Op> = Rewrite<Equation<A, B>, Equation<F<A>, F<B>>>;
export type SubstituteEq<O extends { op: '=', a: Op, b: Op }> = Rewrite<O['a'], O['b']>
export interface SubstituteEq_Proof<A extends Op, B extends Op> {
type: 'rewrite',
known: {
'a = b': Equation<A, B>,
},
left: Equation<F<A>, F<B>>; // f(a) = f(b)
right: ChainRewrites<[
Sym<Congruence<A, B>>, // a = b
SubstituteEq<this['known']['a = b']>, // b = b
Reflexivity<B>,
], this['left']>;
};
export namespace spec {
export type substitution = [
assert<VerifyEquation<SubstituteEq_Proof<'A', 'B'>>>,
]
}
export interface Transitivity_Proof<A extends Op, B extends Op, C extends Op> {
type: 'rewrite',
known: {
'a = b': Equation<A, B>,
'b = c': Equation<B, C>,
},
left: Equation<Equation<A, B>, Equation<B, C>>; // (a = b) = (b = c)
right: ChainRewrites<[
SubstituteEq<this['known']['a = b']>, // (b = b) = (b = c)
SubstituteEq<ApplyRewrite<this['known']['b = c'], Symmetry<B, C>>>, // (b = b) = (c = b)
Reflexivity<B>, // true = (b = b)
Reflexivity<B>, // true = true
Reflexivity<'true'>,
], this['left']>;
};
export namespace spec {
export type transitivity = [
assert<VerifyEquation<Transitivity_Proof<'A', 'B', 'C'>>>,
]
}
}
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