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Proof step 5 from proof intshiftright-ls-01 (previous step)

j < 32, 0 ≤ j ⊦ bits2int(sublist(0, # int2bits(i), mklist(i < 0, i→n(band(j, 31))) + int2bits(i))) < 0 ↔ ¬ 0 ≤ i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 0 ≤ j ∧ ¬ false ∈ int2bits(i).rest ∧ 0 ≤ i → band(j, i) = j % (i + 1)
Proof step 7:
0 ≤ j ∧ ¬ false ∈ int2bits(31).rest ∧ 0 ≤ 31, j < 32, 0 ≤ j ⊦ bits2int(sublist(0, # int2bits(i), mklist(i < 0, i→n(j % (31 + 1))) + int2bits(i))) < 0 ↔ ¬ 0 ≤ i

Rule argument

name: band-011x-pos-02 rotate: F subst: [i, j] → [31, j]

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false);%0 ≤ i ⊦ int2bits((* i)) = uint2bits(i);%1 < i ⊦ uint2bits((* i)) = uint2bits(i / 2) + (i % 2 = 1);% ⊦ uint2bits(1) = false + true;% ⊦ boolvar + boolvar0 = boolvar ' + boolvar0 ';% ⊦ bits + boolvar = bits + boolvar ';% ⊦ (boolvar ' + bits).rest = bits;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ boolvar ∈ bits + bits0 ↔ ¬ ¬(boolvar ∈ bits ∨ boolvar ∈ bits0);% ⊦ boolvar ∈ boolvar0 ' ↔ boolvar ↔ boolvar0; Used terms: 31, j