def un.shlput

bend-osdev-u32@0.1.0.0 · PROOF.bend:739

import Base
import bend-osdev-u32@0.1.0.0/u32.bend as U

739def un.shlput(n: Nat, c: Bool, w: Word(n), h: {Nat.is_lt(Nat.add(U.U32Nat.bit(c), Nat.double(Word.to_nat(n, w))), U.U32Nat.dbl(n, 1n)) == True{} : Bool}) -> {Word.to_nat(n, Word.shl.put(n, c, w)) == Nat.add(U.U32Nat.bit(c), Nat.double(Word.to_nat(n, w))) : Nat}:
740  match n:
741    case 0n:
742      match c w:
743        case False{} WNil{}:
744          {==}
745        case True{} WNil{}:
746          Empty.absurd({0n == 1n : Nat}, un.ft(h))
747    case 1n+ +p:
748      match c w:
749        case False{} WCon{False{}, +t}:
750          +k = U.U32Nat.dbl(p, 1n)
751          +x = Word.to_nat(p, t)
752          un.double_cong(Word.to_nat(p, Word.shl.put(p, False{}, t)), Nat.double(x), un.shlput(p, False{}, t, Equal.trans(Bool, Nat.is_lt(Nat.double(x), k), Nat.is_lt(Nat.double(Nat.double(x)), Nat.double(k)), True{}, Equal.sym(Bool, Nat.is_lt(Nat.double(Nat.double(x)), Nat.double(k)), Nat.is_lt(Nat.double(x), k), un.lt_dbl(Nat.double(x), k)), h)))
753        case False{} WCon{True{}, +t}:
754          +k = U.U32Nat.dbl(p, 1n)
755          +x = Word.to_nat(p, t)
756          un.double_cong(Word.to_nat(p, Word.shl.put(p, True{}, t)), 1n+Nat.double(x), un.shlput(p, True{}, t, Equal.trans(Bool, Nat.is_lt(1n+Nat.double(x), k), Nat.is_lt(Nat.double(1n+Nat.double(x)), Nat.double(k)), True{}, Equal.sym(Bool, Nat.is_lt(Nat.double(1n+Nat.double(x)), Nat.double(k)), Nat.is_lt(1n+Nat.double(x), k), un.lt_dbl(1n+Nat.double(x), k)), h)))
757        case True{} WCon{False{}, +t}:
758          +k = U.U32Nat.dbl(p, 1n)
759          +x = Word.to_nat(p, t)
760          un.succ_cong(Nat.double(Word.to_nat(p, Word.shl.put(p, False{}, t))), Nat.double(Nat.double(x)), un.double_cong(Word.to_nat(p, Word.shl.put(p, False{}, t)), Nat.double(x), un.shlput(p, False{}, t, Equal.trans(Bool, Nat.is_lt(Nat.double(x), k), Nat.is_lt(1n+Nat.double(Nat.double(x)), Nat.double(k)), True{}, Equal.sym(Bool, Nat.is_lt(1n+Nat.double(Nat.double(x)), Nat.double(k)), Nat.is_lt(Nat.double(x), k), un.lt_dbl1(Nat.double(x), k)), h))))
761        case True{} WCon{True{}, +t}:
762          +k = U.U32Nat.dbl(p, 1n)
763          +x = Word.to_nat(p, t)
764          un.succ_cong(Nat.double(Word.to_nat(p, Word.shl.put(p, True{}, t))), Nat.double(1n+Nat.double(x)), un.double_cong(Word.to_nat(p, Word.shl.put(p, True{}, t)), 1n+Nat.double(x), un.shlput(p, True{}, t, Equal.trans(Bool, Nat.is_lt(1n+Nat.double(x), k), Nat.is_lt(1n+Nat.double(1n+Nat.double(x)), Nat.double(k)), True{}, Equal.sym(Bool, Nat.is_lt(1n+Nat.double(1n+Nat.double(x)), Nat.double(k)), Nat.is_lt(1n+Nat.double(x), k), un.lt_dbl1(1n+Nat.double(x), k)), h))))