def quot_lt_g

bend-collections-laws-math@1.0.0.0 · proofs/lib/arith.bend:81

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

81def quot_lt_g(+k: Nat, +q: Nat, +d: Nat, +r: Nat, +c: Bool, +hc: {Nat.is_lt(q, sc(k, 1n)) == c : Bool}, +h: {Nat.is_lt(Nat.add(Nat.mul(q, d), r), sc(k, d)) == True{} : Bool}) -> {Nat.is_lt(q, sc(k, 1n)) == True{} : Bool}:
82  match c:
83    case True{}:
84      hc
85    case False{}:
86      +l1 = mul_le(sc(k, 1n), q, d, N.not_lt_le(q, sc(k, 1n), hc))
87      +e1 = Equal.trans(Nat, Nat.mul(sc(k, 1n), d), sc(k, Nat.mul(1n, d)), sc(k, d), Equal.sym(Nat, sc(k, Nat.mul(1n, d)), Nat.mul(sc(k, 1n), d), sc_mul(k, 1n, d)), Equal.cong(Nat, Nat, z => sc(k, z), Nat.mul(1n, d), d, one_mul(d)))
88      +l2 = L.subst(Nat, z => {Nat.is_le(z, Nat.mul(q, d)) == True{} : Bool}, Nat.mul(sc(k, 1n), d), sc(k, d), e1, l1)
89      +l3 = N.le_trans(sc(k, d), Nat.mul(q, d), Nat.add(Nat.mul(q, d), r), l2, N.le_add_right(Nat.mul(q, d), r))
90      Empty.absurd({Nat.is_lt(q, sc(k, 1n)) == True{} : Bool}, L.true_false(Equal.trans(Bool, True{}, Nat.is_le(sc(k, d), Nat.add(Nat.mul(q, d), r)), False{}, Equal.sym(Bool, Nat.is_le(sc(k, d), Nat.add(Nat.mul(q, d), r)), True{}, l3), N.lt_not_le(Nat.add(Nat.mul(q, d), r), sc(k, d), h))))