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Proof step 48 from proof bitsxor-is-orand (previous step)

∀ bits, bits1. # bits1 < #(bits2 + boolvar ') → true → bits = [] ∨ bits1 = [] ∨ bits2int(bitsxor(bits, bits1)) = bits2int(bitsor(bitsand(bitsinvert(bits), bits1), bitsand(bitsinvert(bits1), bits))) ⊦ bits = [], bits2int(bitsxor(bits, bits2 + boolvar ')) = bits2int(bitsor(bitsand(bitsinvert(bits).butlast, bits2) + (bitsinvert(bits).last ∧ boolvar)', bitsand(bitsinvert(bits2) + (¬ boolvar)', bits))), # bits = 1, bits2 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 49:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsand(bits, bits0) = bitsand(bits.butlast, bits0.butlast) + (bits.last ∧ bits0.last)
Proof step 50:
1 < #(bitsinvert(bits2) + (¬ boolvar)') ∧ 1 < # bits, ∀ bits, bits1. # bits1 < #(bits2 + boolvar ') → true → bits = [] ∨ bits1 = [] ∨ bits2int(bitsxor(bits, bits1)) = bits2int(bitsor(bitsand(bitsinvert(bits), bits1), bitsand(bitsinvert(bits1), bits))) ⊦ bits = [], bits2int(bitsxor(bits, bits2 + boolvar ')) = bits2int (bitsor (bitsand(bitsinvert(bits).butlast, bits2) + (bitsinvert(bits).last ∧ boolvar)', bitsand((bitsinvert(bits2) + (¬ boolvar)').butlast, bits.butlast) + ((bitsinvert(bits2) + (¬ boolvar)').last ∧ bits.last))), # bits = 1, bits2 = []

Rule argument

name: and-rec rotate: F subst: [bits0, bits] → [bits, bitsinvert(bits2) + (¬ boolvar)']

Additional information

Used simplifier rules: ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # bitsinvert(bits) = # bits;% ⊦ # boolvar ' = 0 +1;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1);% ⊦ m + 1 ≠ 0;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ 1 = m + 1 ↔ m = 0;% ⊦ # bits = 0 ↔ bits = []; Used terms: bits, bitsinvert(bits2) + (¬ boolvar)'