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Proof step 10 from proof int2bits-add-low (previous step)

0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, j * 2 ^ i + j0 < 0, 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0 < 0 ⊦ sgn(int2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ i < ~1 ∧ 0 ≤ j → (2 ^ j + i < 0 ↔ ¬ # int2bits(i) ≤ i→n(j) + 1)
Proof step 12:
j * 2 ^ i + j0 < ~1 ∧ 0 ≤ n→i(# int2bits(~(j * 2 ^ i + j0))), 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, j * 2 ^ i + j0 < 0, ¬ # int2bits(j * 2 ^ i + j0) ≤ i→n(n→i(# int2bits(~(j * 2 ^ i + j0)))) + 1 ⊦ sgn(int2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

name: int2bits-lelen-02 rotate: F subst: [i, j] → [j * 2 ^ i + j0, n→i(# int2bits(~(j * 2 ^ i + j0)))]

Additional information

Used simplifier rules: int2bits-pos-num: 0 ≤ i ⊦ int2bits((* i)) = uint2bits(i);%uint2bits-one: ⊦ uint2bits(1) = false + true;%uint2bits-pos-num: 1 < i ⊦ uint2bits((* i)) = uint2bits(i / 2) + (i % 2 = 1);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1);% ⊦ 0 ≤ n→i(n);% ⊦ j < i ∧ i ≤ 0 → j < 0; (forward)% ⊦ boolvar + boolvar0 = boolvar ' + boolvar0 ';% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ bits + boolvar = bits + boolvar ';% ⊦ (boolvar ' + bits).rest = bits;% ⊦ sgn(true ' + true ') = true '; Used terms: j * 2 ^ i + j0, n→i(# int2bits(~(j * 2 ^ i + j0)))