def dn

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/w64sqrt.bend:78

import Base
import bend-collections-laws-math@1.0.0.0/src/math/w64.bend as X
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
import bend-collections-laws-math@1.0.0.0/proofs/lib/u32.bend as U
import bend-collections-laws-math@1.0.0.0/proofs/lib/word.bend as WD

78def dn(+fu: Nat, +one: Nat, +h1: {one == 1n : Nat}, +x: U32, +r: U32, +hf: {Nat.is_lt(v(r), fu) == True{} : Bool}, +hb: {Nat.is_lt(v(r), WD.sc(16n, one)) == True{} : Bool}, +cb: Bool, +hc: {U32.is_lt(x, U32.mul(r, r)) == cb : Bool}, +nval: Nat, +hnr: {v(r) == nval : Nat}) -> {Nat.is_le(sq(v(X.down32(fu, x, r, cb))), v(x)) == True{} : Bool} & {Nat.is_lt(v(X.down32(fu, x, r, cb)), WD.sc(16n, one)) == True{} : Bool}:
79  match fu:
80    case 0n:
81      Empty.absurd({Nat.is_le(sq(v(X.down32(0n, x, r, cb))), v(x)) == True{} : Bool} & {Nat.is_lt(v(X.down32(0n, x, r, cb)), WD.sc(16n, one)) == True{} : Bool}, N.lt_zero_absurd(v(r), hf))
82    case 1n+ +g:
83      match cb:
84        case False{}:
85          +hc2 = Equal.trans(Bool, Nat.is_lt(v(x), sq(v(r))), U32.is_lt(x, U32.mul(r, r)), False{}, Equal.sym(Bool, U32.is_lt(x, U32.mul(r, r)), Nat.is_lt(v(x), sq(v(r))), lt_sq(one, h1, x, r, hb)), hc)
86          (N.not_lt_le(v(x), sq(v(r)), hc2), hb)
87        case True{}:
88          match nval:
89            case 0n:
90              +hc2 = Equal.trans(Bool, Nat.is_lt(v(x), sq(v(r))), U32.is_lt(x, U32.mul(r, r)), True{}, Equal.sym(Bool, U32.is_lt(x, U32.mul(r, r)), Nat.is_lt(v(x), sq(v(r))), lt_sq(one, h1, x, r, hb)), hc)
91              +h0 = L.subst(Nat, z => {Nat.is_lt(v(x), sq(z)) == True{} : Bool}, v(r), 0n, hnr, hc2)
92              Empty.absurd({Nat.is_le(sq(v(X.down32(1n+g, x, r, True{}))), v(x)) == True{} : Bool} & {Nat.is_lt(v(X.down32(1n+g, x, r, True{})), WD.sc(16n, one)) == True{} : Bool}, N.lt_zero_absurd(v(x), h0))
93            case 1n+ +rp:
94              +r1 = U32.sub(r, 1)
95              +e1 = Equal.trans(Nat, v(r1), Nat.sub(v(r), 1n), rp, U.sub_nat(r, 1, L.subst(Nat, z => {Nat.is_le(1n, z) == True{} : Bool}, 1n+rp, v(r), Equal.sym(Nat, v(r), 1n+rp, hnr), N.zero_le(rp))), Equal.trans(Nat, Nat.sub(v(r), 1n), Nat.sub(1n+rp, 1n), rp, Equal.cong(Nat, Nat, t => Nat.sub(t, 1n), v(r), 1n+rp, hnr), N.sub_zero(rp)))
96              +hf1 = L.subst(Nat, z => {Nat.is_lt(z, g) == True{} : Bool}, rp, v(r1), Equal.sym(Nat, v(r1), rp, e1), L.subst(Nat, z => {Nat.is_lt(z, 1n+g) == True{} : Bool}, v(r), 1n+rp, hnr, hf))
97              +hb1 = L.subst(Nat, z => {Nat.is_lt(z, WD.sc(16n, one)) == True{} : Bool}, rp, v(r1), Equal.sym(Nat, v(r1), rp, e1), N.lt_trans(rp, 1n+rp, WD.sc(16n, one), N.lt_succ(rp), L.subst(Nat, z => {Nat.is_lt(z, WD.sc(16n, one)) == True{} : Bool}, v(r), 1n+rp, hnr, hb)))
98              dn(g, one, h1, x, r1, hf1, hb1, U32.is_lt(x, U32.mul(r1, r1)), {==}, rp, e1)