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Proof step 61 from proof cast-mod-pos-high (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ 0 < i → bits2int(int2bits(i).rest) = i - 2 ^ (n→i(# uint2bits(i)) - 1) subst: [i] → [j % 2 ^ i] rotate: F spec: int2bits inst: name: int2int-rest

Additional information

Used simplifier rules: ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 ≤ j ∧ 0 < i → (j % 2 ^ i ≤ 0 ↔ j % 2 ^ i = 0);% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ m < n ∧ m = n0 → n0 < n; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ int2bits(0) = false ';% ⊦ boolvar '.rest = [];% ⊦ # [] = 0;% ⊦ i→n(i) = 0 ↔ i = 0;% ⊦ # boolvar ' = 0 +1;% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ 2 ^ i ≤ 0 ↔ i < 0;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i < 1 ↔ i ≤ 0;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: j % 2 ^ i