def minimum_choice

stelliferous@0.0.2.0 · proofs/nat_extrema.bend:13

import Base

13def minimum_choice(+left: Nat,+right: Nat) -> {Nat.min(left,right) == Bool.pick(Nat,Nat.is_lt(left,right),left,right) : Nat}:
14  match left right:
15    case 0n 0n: {==}
16    case 0n 1n+right: {==}
17    case 1n+left 0n: {==}
18    case 1n+ +left 1n+ +right:
19      Equal.trans(Nat,1n+Nat.min(left,right),1n+Bool.pick(Nat,Nat.is_lt(left,right),left,right),Bool.pick(Nat,Nat.is_lt(left,right),1n+left,1n+right),
20        Equal.cong(Nat,Nat,n => 1n+n,Nat.min(left,right),Bool.pick(Nat,Nat.is_lt(left,right),left,right),minimum_choice(left,right)),successor_pick(Nat.is_lt(left,right),left,right))