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Proof step 64 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, 0 < j % 2 ^ i ⊦ 2 ^ i = 2 ^ (n→i(# uint2bits(j % 2 ^ i)) - 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 65:
⊦ 0 ≤ i → int2bits(i) = uint2bits(i)
Proof step 66:
0 ≤ j % 2 ^ i, 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, 0 < j % 2 ^ i ⊦ 2 ^ i = 2 ^ (n→i(# int2bits(j % 2 ^ i)) - 1)

Rule argument

seq: ⊦ 0 ≤ i → int2bits(i) = uint2bits(i) subst: [i] → [j % 2 ^ i] rotate: T spec: int2bits inst: name: int2bits-pos

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ 0 ≤ j ∧ 0 < i → ¬ j % 2 ^ i < 0; Used terms: j % 2 ^ i