def lt_sub2

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/roots.bend:108

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L

108# a < b and c <= a give a - c < b - c
109def lt_sub2(+a: Nat, +b: Nat, +c: Nat, +h1: {Nat.is_lt(a, b) == True{} : Bool}, +h2: {Nat.is_le(c, a) == True{} : Bool}) -> {Nat.is_lt(Nat.sub(a, c), Nat.sub(b, c)) == True{} : Bool}:
110  match a b c:
111    case 0n 0n c0:
112      Empty.absurd({Nat.is_lt(Nat.sub(0n, c0), Nat.sub(0n, c0)) == True{} : Bool}, L.false_true(h1))
113    case 0n 1n+bp 0n:
114      {==}
115    case 0n 1n+bp 1n+cp:
116      Empty.absurd({Nat.is_lt(Nat.sub(0n, 1n+cp), Nat.sub(1n+bp, 1n+cp)) == True{} : Bool}, L.false_true(h2))
117    case 1n+ap 0n c0:
118      Empty.absurd({Nat.is_lt(Nat.sub(1n+ap, c0), Nat.sub(0n, c0)) == True{} : Bool}, L.false_true(h1))
119    case 1n+ap 1n+bp 0n:
120      h1
121    case 1n+ +ap 1n+ +bp 1n+ +cp:
122      lt_sub2(ap, bp, cp, h1, h2)