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Proof step 5 from proof and-size-one-01 (previous step)

# bits2 = # bits1, ∀ bits, bits1, bits0. # bits < # bits2 → # bits = # bits1 → bits1 = [] ∨ bits0 = [] ∨ bitsand(true ' + bits, bits0 + bits1) = bits0 + bitsand(bits, bits1) ⊦ bits1 = [] ∨ bits0 = [] ∨ bitsand(true ' + bits2, bits0 + bits1) = bits0 + bitsand(bits2, bits1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsand(bits, bits0) = bitsand(bits.butlast, bits0.butlast) + (bits.last ∧ bits0.last)
Proof step 7:
1 < #(true ' + bits2) ∧ 1 < #(bits0 + bits1), # bits2 = # bits1, ∀ bits, bits1, bits0. # bits < # bits2 → # bits = # bits1 → bits1 = [] ∨ bits0 = [] ∨ bitsand(true ' + bits, bits0 + bits1) = bits0 + bitsand(bits, bits1) ⊦ bits1 = [] ∨ bits0 = [] ∨ bitsand((true ' + bits2).butlast, (bits0 + bits1).butlast) + ((true ' + bits2).last ∧ (bits0 + bits1).last) = bits0 + bitsand(bits2, bits1)

Rule argument

name: and-rec rotate: F subst: [bits0, bits] → [bits0 + bits1, true ' + bits2]

Additional information

Used simplifier rules: ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # 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 = [];% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0;% ⊦ # [] = 0;% ⊦ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1;% ⊦ # bits = 1 ↔ bits = bits.first + [];% ⊦ boolvar + bits = boolvar ' + bits;% ⊦ bits + [] = bits;% ⊦ # bits + # bits0 = 1 ∧ bits ≠ [] → # bits = 1 ∧ bits0 = []; (forward) Used terms: bits0 + bits1, true ' + bits2