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Proof step 8 from proof fill-lastn-04 (previous step)

# bits0 ≤ m, # fillsgn(m + 1, bits0) ≤ m ⊦ lastn(m, bits + fillsgn(m + 1, bits0).rest) = fillsgn(m, bits0), fillsgn(m + 1, bits0) = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ # bits ≤ n ∧ bits ≠ [] → fillsgn(n, bits) = mklist(bits.first, n - # bits) + bits

Rule argument

name: fill-fill rotate: F subst: [bits, n] → [bits0, m + 1]

Additional information

Used simplifier rules: fillsgn-empty: ⊦ fillsgn(n, []) = mklist(false, n);%fill-length: ⊦ # bits ≤ n → # fillsgn(n, bits) = n;% ⊦ m + n ≤ m + n0 ↔ n ≤ n0;%n ≤ m ⊦ (m - n) + m0 ≤ n0 ↔ m + m0 ≤ n0 + n;% ⊦ # mklist(boolvar, n) = n;% ⊦ # boolvar ' = 0 +1;% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ n ≤ m → m + n0 - n = (m - n) + n0;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ mklist(boolvar, m + n) = mklist(boolvar, m) + mklist(boolvar, n);% ⊦ mklist(boolvar, 1) = boolvar ';% ⊦ mklist(boolvar, n) + boolvar ' = boolvar ' + mklist(boolvar, n);% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ boolvar ' + bits ≠ [];% ⊦ (boolvar ' + bits).rest = bits;% ⊦ lastn(n, bits + mklist(boolvar, n)) = mklist(boolvar, n);% ⊦ mklist(boolvar, m) = mklist(boolvar, n) ↔ m = n;% ⊦ ¬ m < n ↔ n ≤ m;% ⊦ m + n ≤ m ↔ n = 0;% ⊦ m < n ∧ n ≤ n0 → m < n0; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ # bits = 0 ↔ bits = [];% ⊦ ¬ m + n < m; Used terms: bits0, m + 1