def succ_mul_choose

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/fact.bend:137

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S

137# (n + 1) C(n, k) == C(n + 1, k + 1) (k + 1)   (Mathlib Nat.add_one_mul_choose_eq)
138def succ_mul_choose(+n: Nat, +k: Nat) -> {Nat.mul(1n+n, S.choose(n, k)) == Nat.mul(S.choose(1n+n, 1n+k), 1n+k) : Nat}:
139  match n k:
140    case 0n 0n:
141      {==}
142    case 0n 1n+j:
143      {==}
144    case 1n+ +m 0n:
145      %Equal.sym(Nat, S.choose(1n+m, 0n), 1n, choose_zero_right(1n+m)) : {Nat.mul(2n+m, _) == Nat.mul(Nat.add(S.choose(1n+m, 0n), S.choose(1n+m, 1n)), 1n) : Nat}
146      %Equal.sym(Nat, S.choose(1n+m, 0n), 1n, choose_zero_right(1n+m)) : {Nat.mul(2n+m, 1n) == Nat.mul(Nat.add(_, S.choose(1n+m, 1n)), 1n) : Nat}
147      %Equal.sym(Nat, S.choose(1n+m, 1n), 1n+m, choose_one_right(1n+m)) : {Nat.mul(2n+m, 1n) == Nat.mul(Nat.add(1n, _), 1n) : Nat}
148      {==}
149    case 1n+ +m 1n+ +j:
150      +x = S.choose(1n+m, 1n+j)
151      +y = S.choose(1n+m, 2n+j)
152      +c0 = S.choose(m, j)
153      +c1 = S.choose(m, 1n+j)
154      # (x + y)(2 + j) == x + x(1 + j) + y(2 + j) == x + (1+m) c0 + (1+m) c1 == x + (1+m) x
155      %Equal.sym(Nat, Nat.mul(Nat.add(x, y), 2n+j), Nat.add(Nat.mul(x, 2n+j), Nat.mul(y, 2n+j)), A.mul_add_right(x, y, 2n+j)) : {Nat.mul(2n+m, x) == _ : Nat}
156      %Equal.sym(Nat, Nat.mul(x, 2n+j), Nat.add(x, Nat.mul(x, 1n+j)), A.mul_succ(x, 1n+j)) : {Nat.mul(2n+m, x) == Nat.add(_, Nat.mul(y, 2n+j)) : Nat}
157      %Equal.sym(Nat, Nat.mul(x, 1n+j), Nat.mul(1n+m, c0), Equal.sym(Nat, Nat.mul(1n+m, c0), Nat.mul(x, 1n+j), succ_mul_choose(m, j))) : {Nat.mul(2n+m, x) == Nat.add(Nat.add(x, _), Nat.mul(y, 2n+j)) : Nat}
158      %Equal.sym(Nat, Nat.mul(y, 2n+j), Nat.mul(1n+m, c1), Equal.sym(Nat, Nat.mul(1n+m, c1), Nat.mul(y, 2n+j), succ_mul_choose(m, 1n+j))) : {Nat.mul(2n+m, x) == Nat.add(Nat.add(x, Nat.mul(1n+m, c0)), _) : Nat}
159      %Equal.sym(Nat, Nat.add(Nat.add(x, Nat.mul(1n+m, c0)), Nat.mul(1n+m, c1)), Nat.add(x, Nat.add(Nat.mul(1n+m, c0), Nat.mul(1n+m, c1))), A.add_assoc(x, Nat.mul(1n+m, c0), Nat.mul(1n+m, c1))) : {Nat.mul(2n+m, x) == _ : Nat}
160      %Equal.sym(Nat, Nat.add(Nat.mul(1n+m, c0), Nat.mul(1n+m, c1)), Nat.mul(1n+m, Nat.add(c0, c1)), Equal.sym(Nat, Nat.mul(1n+m, Nat.add(c0, c1)), Nat.add(Nat.mul(1n+m, c0), Nat.mul(1n+m, c1)), A.mul_add_left(1n+m, c0, c1))) : {Nat.mul(2n+m, x) == Nat.add(x, _) : Nat}
161      {==}