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Proof step 65 from proof bitsand-cast (previous step)

sgn(mklist(boolvar, m) + boolvar ' + boolvar0 ' + bits1) = boolvar ' + boolvar0 ' + bits1, n = # fillsgn(n, mklist(boolvar, m) + boolvar ' + boolvar0 ' + bits1), # bits0 ≤ 2 + # bits1, bits0.first, m + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0) < n, # bitsand(boolvar ' + boolvar0 ' + bits1, bits0) < n, 2 + # bits1 + m < n ⊦ (mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0)).first ↔ boolvar, mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0) = [], boolvar ↔ boolvar0, bits0 = [], mklist(boolvar, n - # bitsand(boolvar ' + boolvar0 ' + bits1, bits0)) = mklist((mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0)).first, n - m + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0)) + mklist(boolvar, m), boolvar ∧ boolvar0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 66:
⊦ bits ≠ [] → ((bits + bits0).first ↔ bits.first)
Proof step 67:
¬ mklist(boolvar, m) ≠ [], sgn(mklist(boolvar, m) + boolvar ' + boolvar0 ' + bits1) = boolvar ' + boolvar0 ' + bits1, n = # fillsgn(n, mklist(boolvar, m) + boolvar ' + boolvar0 ' + bits1), # bits0 ≤ 2 + # bits1, bits0.first, m + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0) < n, # bitsand(boolvar ' + boolvar0 ' + bits1, bits0) < n, 2 + # bits1 + m < n ⊦ (mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0)).first ↔ boolvar, mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0) = [], boolvar ↔ boolvar0, bits0 = [], mklist(boolvar, n - # bitsand(boolvar ' + boolvar0 ' + bits1, bits0)) = mklist((mklist(boolvar, m) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0)).first, n - m + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0)) + mklist(boolvar, m), boolvar ∧ boolvar0

Rule argument

seq: ⊦ bits ≠ [] → ((bits + bits0).first ↔ bits.first) subst: [bits0, bits] → [bitsand(boolvar ' + boolvar0 ' + bits1, bits0), mklist(boolvar, m)] rotate: F spec: list inst: bitlista name: first-01

Additional information

Used simplifier rules: ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ m ≠ 0 → (mklist(boolvar, m).first ↔ boolvar); Used terms: bitsand(boolvar ' + boolvar0 ' + bits1, bits0), mklist(boolvar, m)