[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 6 from proof bitsand-len-true (previous step)

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

Rule argument

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

Additional information

Used simplifier rules: and-oner-01: ⊦ bits ≠ [] → bitsand(bits, true ') = bits;% ⊦ 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;% ⊦ bits + [] = bits;% ⊦ m < n ∧ n = n0 → m < n0; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ m < 1 ↔ m = 0; Used terms: true ' + bits1, bits