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Proof step 32 from proof bitsand-cast-05 (previous step)

# bits0 = 1 + # bits1, # bits1 < # bits2 + # bits0 ⊦ # bits1 = # bits2 + # bits0, bits2 + bits0 = [], sgn(bitsand(lastn(# bits1, bits2 + bits0), false ' + bits1)) = sgn(bitsand(bits2 + bits0, false ' + bits1)), bits1 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
⊦ # bits1 ≤ # bits0 ∧ bits1 ≠ [] ∧ ¬ bits1.first → bitsand(bits + bits0, bits1) = bitsand(bits0, bits1)
Proof step 34:
#(false ' + bits1) ≤ # bits0 ∧ false ' + bits1 ≠ [] ∧ ¬ (false ' + bits1).first, # bits0 = 1 + # bits1, # bits1 < # bits2 + # bits0 ⊦ # bits1 = # bits2 + # bits0, bits2 + bits0 = [], sgn(bitsand(lastn(# bits1, bits2 + bits0), false ' + bits1)) = sgn(bitsand(bits0, false ' + bits1)), bits1 = []

Rule argument

name: bitsand-app-false rotate: F subst: [bits1, bits0, bits] → [false ' + bits1, bits0, bits2]

Additional information

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