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Proof step 36 from proof cast-mod-pos-low (previous step)

j % 2 ^ i < i, j % 2 ^ i ≤ 0, 2 ^ i ≤ j, 0 ≤ 2 ^ (i - 1) + j, j % 2 ^ i < j, 2 ^ (i - 1) ≤ j, 0 ≤ j, 0 < i, j % 2 ^ i < 2 ^ (i - 1), i→n(i) ≤ (# int2bits(j % 2 ^ i) - 1) + # int2bits(j / 2 ^ i), i→n(i) + 1 < # int2bits(j % 2 ^ i), i→n(i) < # int2bits(j % 2 ^ i) ⊦ bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + int2bits(j % 2 ^ i).rest)) = j % 2 ^ i, i→n(i) = # int2bits(j % 2 ^ i) - 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 37:
⊦ 0 ≤ i ∧ 0 < j → (i % j ≤ 0 ↔ i % j = 0)

Rule argument

seq: ⊦ 0 ≤ i ∧ 0 < j → (i % j ≤ 0 ↔ i % j = 0) subst: [j, i] → [2 ^ i, j] rotate: F spec: int-div inst: name: mod-zero-02

Additional information

Used simplifier rules: ⊦ i < j → ¬ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward)% ⊦ int2bits(0) = false ';% ⊦ # boolvar ' = 0 +1;% ⊦ i→n(i) = 0 ↔ i = 0;% ⊦ boolvar '.rest = [];% ⊦ bits + [] = bits;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ m < 1 ↔ m = 0; Used terms: 2 ^ i, j