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Proof step 45 from proof cast-mod-pos-high (previous step)

i→n(i) + 1 < # int2bits(j % 2 ^ i), i→n(i) ≤ (# fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1) + # int2bits(j / 2 ^ i), 2 ^ (i - 1) ≤ j % 2 ^ i, 0 < i, 0 ≤ j, 2 ^ (i - 1) ≤ j, 0 ≤ 2 ^ (i - 1) + j, 2 ^ i ≤ j ⊦ i→n(i) = # fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) - 1, fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)) = [], j % 2 ^ i = 2 ^ i + bits2int(lastn(i→n(i), int2bits(j / 2 ^ i) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ 0 < i → (m < # int2bits(i) ↔ ¬ i < 2 ^ (n→i(m) - 1)) subst: [i, m] → [j % 2 ^ i, i→n(i) + 1] rotate: F spec: int2bits inst: name: int2bits-lslen-06

Additional information

Used simplifier rules: ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 ≤ j ∧ 0 < i → (j % 2 ^ i ≤ 0 ↔ j % 2 ^ i = 0);% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ m < n → n ≠ 0; (forward)% ⊦ int2bits(0) = false ';% ⊦ # boolvar ' = 0 +1;% ⊦ fillsgn(n + 1, boolvar ') = mklist(boolvar, n + 1);% ⊦ mklist(boolvar, m + n) = mklist(boolvar, m) + mklist(boolvar, n);%n ≠ 0 ⊦ mklist(false, (* n)) = false ' + mklist(false, n -1);% ⊦ mklist(boolvar, 0) = [];% ⊦ bits + [] = bits;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ mklist(boolvar, n) + boolvar ' = boolvar ' + mklist(boolvar, n);% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ (boolvar ' + bits).rest = bits;% ⊦ lastn(n, bits + mklist(boolvar, n)) = mklist(boolvar, n);% ⊦ boolvar ' + bits ≠ [];% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ # mklist(boolvar, n) = n;% ⊦ n ≤ m → m + n0 - n = (m - n) + n0;% ⊦ 0 + m = m;% ⊦ i→n(i) = i→n(j) ↔ abs(i) = abs(j); Used terms: j % 2 ^ i, i→n(i) + 1