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Proof step 860 from proof bitsand-cast (previous step)

sgn(mklist(boolvar, m) + boolvar ' + boolvar0 ' + bits) = boolvar ' + boolvar0 ' + bits, # bits0 ≤ 2 + # bits + m, # bits0 ≤ # bitsand(boolvar ' + boolvar0 ' + bits, bits0), 2 + # bits ≤ # bitsand(boolvar ' + boolvar0 ' + bits, bits0), # bits0 ≤ (2 + # bits) + # bits2, # bits0 ≤ 2 + # bits, bits0.first, # bitsand(boolvar ' + boolvar0 ' + bits, bits0) ≤ 2 + # bits + m ⊦ bits0 = [], lastn (2 + # bits + m, mklist(bitsand(boolvar ' + boolvar0 ' + bits, bits0).first, 2 + # bits + m - # bitsand(boolvar ' + boolvar0 ' + bits, bits0)) + bitsand(boolvar ' + boolvar0 ' + bits, bits0)) = lastn(2 + # bits + m, mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits, bits0)), boolvar ↔ boolvar0, boolvar ∧ boolvar0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 861:
⊦ bits0.first ∧ # bits0 ≤ # bits ∧ bits0 ≠ [] → # bitsand(bits, bits0) = # bits
Proof step 862:
bits0.first ∧ # bits0 ≤ #(boolvar ' + boolvar0 ' + bits) ∧ bits0 ≠ [], sgn(mklist(boolvar, m) + boolvar ' + boolvar0 ' + bits) = boolvar ' + boolvar0 ' + bits, # bits0 ≤ 2 + # bits + m, # bits0 ≤ #(boolvar ' + boolvar0 ' + bits), 2 + # bits ≤ #(boolvar ' + boolvar0 ' + bits), # bits0 ≤ (2 + # bits) + # bits2, # bits0 ≤ 2 + # bits, bits0.first, #(boolvar ' + boolvar0 ' + bits) ≤ 2 + # bits + m ⊦ bits0 = [], lastn (2 + # bits + m, mklist(bitsand(boolvar ' + boolvar0 ' + bits, bits0).first, 2 + # bits + m - #(boolvar ' + boolvar0 ' + bits)) + bitsand(boolvar ' + boolvar0 ' + bits, bits0)) = lastn(2 + # bits + m, mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits, bits0)), boolvar ↔ boolvar0, boolvar ∧ boolvar0

Rule argument

name: bitsand-len-true rotate: F subst: [bits0, bits] → [bits0, boolvar ' + boolvar0 ' + bits]

Additional information

Used simplifier rules: ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ m + n = n + m;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ m ≤ n → (n < m ↔ false); Used terms: bits0, boolvar ' + boolvar0 ' + bits