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Proof step 6 from proof bitsand-zeros-01 (previous step)

1 + # bits1 < # bits, ∀ bits1, bits0. # bits0 < # bits → 1 + # bits1 < # bits0 → bits0 = [] ∨ bitsand(bits0, false ' + bits1) = false ' + mklist(false, # bits1) ∨ true ∈ bits0 ⊦ bitsand(bits, false ' + bits1) = false ' + mklist(false, # bits1), bits = [], true ∈ bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsand(bits, bits0) = bitsand(bits.butlast, bits0.butlast) + (bits.last ∧ bits0.last)
Proof step 8:
1 < # bits ∧ 1 < #(false ' + bits1), 1 + # bits1 < # bits, ∀ bits1, bits0. # bits0 < # bits → 1 + # bits1 < # bits0 → bits0 = [] ∨ bitsand(bits0, false ' + bits1) = false ' + mklist(false, # bits1) ∨ true ∈ bits0 ⊦ bitsand(bits.butlast, (false ' + bits1).butlast) + (bits.last ∧ (false ' + bits1).last) = false ' + mklist(false, # bits1), bits = [], true ∈ bits

Rule argument

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

Additional information

Used simplifier rules: and-zero-03: ⊦ bits ≠ [] → bitsand(bits, false ') = false ';% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1);% ⊦ # bits = 0 ↔ bits = [];% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # boolvar ' = 0 +1;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ m + 1 ≠ 0;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ 1 = m + 1 ↔ m = 0;% ⊦ m < n ∧ n = n0 → m < n0; (forward)% ⊦ ¬ m + n < m;% ⊦ bits + [] = bits;% ⊦ # [] = 0;% ⊦ mklist(boolvar, 0) = [];% ⊦ boolvar ' = boolvar0 ' ↔ boolvar ↔ boolvar0; Used terms: false ' + bits1, bits