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Proof step 380 from proof bitsand-cast (previous step)

¬(# bits0 ≤ #(boolvar ' + boolvar0 ' + bits) ∧ bits0 ≠ [] ∧ ¬ bits0.first), sgn(mklist(boolvar, n - 1) + boolvar ' + boolvar0 ' + bits) = boolvar ' + boolvar0 ' + bits ⊦ boolvar ↔ boolvar0, bits0 = [], n = 0, lastn(n + 1 + # bits, fillsgn(n + 1 + # bits, bitsand(boolvar ' + boolvar0 ' + bits, bits0))) = lastn(n + 1 + # bits, fillsgn(n + 1 + # bits, bitsand((bits2 + mklist(boolvar, n - 1)) + boolvar ' + boolvar0 ' + bits, bits0))), boolvar ∧ boolvar0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 381:
⊦ # bits1 ≤ # bits0 ∧ bits1 ≠ [] ∧ bits1.first → bitsand(bits + bits0, bits1) = bits + bitsand(bits0, bits1)
Proof step 382:
¬(# bits0 ≤ #(boolvar ' + boolvar0 ' + bits) ∧ bits0 ≠ [] ∧ bits0.first), ¬(# bits0 ≤ #(boolvar ' + boolvar0 ' + bits) ∧ bits0 ≠ [] ∧ ¬ bits0.first), sgn(mklist(boolvar, n - 1) + boolvar ' + boolvar0 ' + bits) = boolvar ' + boolvar0 ' + bits ⊦ boolvar ↔ boolvar0, bits0 = [], n = 0, lastn(n + 1 + # bits, fillsgn(n + 1 + # bits, bitsand(boolvar ' + boolvar0 ' + bits, bits0))) = lastn(n + 1 + # bits, fillsgn(n + 1 + # bits, bitsand((bits2 + mklist(boolvar, n - 1)) + boolvar ' + boolvar0 ' + bits, bits0))), boolvar ∧ boolvar0
Proof step 834:
# bits0 ≤ #(boolvar ' + boolvar0 ' + bits) ∧ bits0 ≠ [] ∧ bits0.first, ¬(# bits0 ≤ #(boolvar ' + boolvar0 ' + bits) ∧ bits0 ≠ [] ∧ ¬ bits0.first), sgn(mklist(boolvar, n - 1) + boolvar ' + boolvar0 ' + bits) = boolvar ' + boolvar0 ' + bits ⊦ boolvar ↔ boolvar0, bits0 = [], n = 0, lastn(n + 1 + # bits, fillsgn(n + 1 + # bits, bitsand(boolvar ' + boolvar0 ' + bits, bits0))) = lastn(n + 1 + # bits, fillsgn(n + 1 + # bits, (bits2 + mklist(boolvar, n - 1)) + bitsand(boolvar ' + boolvar0 ' + bits, bits0))), boolvar ∧ boolvar0

Rule argument

name: bitsand-app-true rotate: F subst: [bits1, bits0, bits] → [bits0, boolvar ' + boolvar0 ' + bits, bits2 + mklist(boolvar, n - 1)]

Additional information

Used terms: bits0, boolvar ' + boolvar0 ' + bits, bits2 + mklist(boolvar, n - 1)