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Proof step 14 from proof intshiftrsign-rec-neg (previous step)

0 ≤ j, 0 < j, i < 0, j < 32 ⊦ bits2int(sublist(0, # int2bits(i), mklist(true, i→n(j)) + int2bits(i))) = bits2int(sublist(0, # int2bits((i - 1) / 2), mklist(true, i→n(band(j - 1, 31))) + int2bits((i - 1) / 2)))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

name: band-011x-pos-02 rotate: F subst: [i, j] → [31, j - 1]

Additional information

Used simplifier rules: band-num: ⊦ band((* i), (* j)) = bits2int(bitsand(int2bits(i), int2bits(j)));%bitsand-oner: ⊦ bitsand(bits + boolvar ', true ') = bits + boolvar;%band-associativity: ⊦ band(i, band(i0, j)) = band(band(i, i0), j);%band-commutativity: ⊦ band(i, j) = band(j, i);% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≤ j → (j ≤ i ↔ i = j);%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;% ⊦ mklist(boolvar, 0) = [];% ⊦ [] + bits = bits;% ⊦ sublist(0, # bits, bits) = bits;% ⊦ bits2int(int2bits(i)) = i;%i < 0 ⊦ int2bits((* i)) = sgn(bitsinvert(uint2bits(abs(i))) +1);% ⊦ bitsinvert(false ' + bits) = true + bitsinvert(bits);% ⊦ bitsinvert(true ') = false ';% ⊦ boolvar + bits = boolvar ' + bits;% ⊦ (bits + false ')+1 = bits + true;% ⊦ sgn(true ' + true ') = true ';% ⊦ bits2int(false ' + bits) = bits2uint(bits);% ⊦ bits2uint(bits + true ') = bits2uint(bits) * 2 + 1;% ⊦ bits2uint(true ') = 1;%n ≠ 0 ⊦ mklist(true, (* n)) = true ' + mklist(true, n -1);% ⊦ bits + [] = bits; Used terms: 31, j - 1