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Proof step 74 from proof bitsand-cast (previous step)

sgn(mklist(boolvar, n0 - 1) + boolvar ' + boolvar0 ' + bits1) = boolvar ' + boolvar0 ' + bits1, n = # fillsgn(n, mklist(boolvar, n0 - 1) + boolvar ' + boolvar0 ' + bits1), n0 + 1 + # bits1 < (n0 - 1) + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0), # bits0 ≤ 2 + # bits1, bits0.first, n0 + 1 + # bits1 < n, n ≤ (n0 - 1) + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0), # bitsand(boolvar ' + boolvar0 ' + bits1, bits0) < n ⊦ n0 - 1 = 0, bits0 = [], mklist(bitsand(boolvar ' + boolvar0 ' + bits1, bits0).first, n - # bitsand(boolvar ' + boolvar0 ' + bits1, bits0)) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0) = lastn(n, mklist(boolvar, n0 - 1) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0)), n0 = 0, boolvar ↔ boolvar0, boolvar ∧ boolvar0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 75:
⊦ bits ≠ [] ∧ bits0 ≠ [] → (bitsand(bits, bits0).first ↔ bits.first ∧ bits0.first)
Proof step 76:
boolvar ' + boolvar0 ' + bits1 ≠ [] ∧ bits0 ≠ [], sgn(mklist(boolvar, n0 - 1) + boolvar ' + boolvar0 ' + bits1) = boolvar ' + boolvar0 ' + bits1, n = # fillsgn(n, mklist(boolvar, n0 - 1) + boolvar ' + boolvar0 ' + bits1), n0 + 1 + # bits1 < (n0 - 1) + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0), # bits0 ≤ 2 + # bits1, bits0.first, n0 + 1 + # bits1 < n, n ≤ (n0 - 1) + # bitsand(boolvar ' + boolvar0 ' + bits1, bits0), # bitsand(boolvar ' + boolvar0 ' + bits1, bits0) < n ⊦ n0 - 1 = 0, bits0 = [], mklist((boolvar ' + boolvar0 ' + bits1).first ∧ bits0.first, n - # bitsand(boolvar ' + boolvar0 ' + bits1, bits0)) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0) = lastn(n, mklist(boolvar, n0 - 1) + bitsand(boolvar ' + boolvar0 ' + bits1, bits0)), n0 = 0, boolvar ↔ boolvar0, boolvar ∧ boolvar0

Rule argument

name: and-first rotate: F subst: [bits0, bits] → [bits0, boolvar ' + boolvar0 ' + bits1]

Additional information

Used simplifier rules: ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ boolvar ' ≠ [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1; Used terms: bits0, boolvar ' + boolvar0 ' + bits1