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Proof step 76 from proof uint2bits-mod2pot (previous step)

j % 2 = 0, uint2bits((j / 2) % 2 ^ i0) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest)), mklist(false, # lastn(i→n(i0), uint2bits(j / 2).rest)) = lastn(i→n(i0), uint2bits(j / 2).rest), 0 < (j / 2) % 2 ^ i0, 0 ≤ i0, 0 < j, 2 ^ i0 * 2 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ ~1 = i0, j = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 77:
⊦ n ≤ # bits → # lastn(n, bits) = n
Proof step 78:
¬ i→n(i0) ≤ # uint2bits(j / 2).rest, j % 2 = 0, uint2bits((j / 2) % 2 ^ i0) = sgn(false ' + lastn(i→n(i0), uint2bits(j / 2).rest)), mklist(false, # lastn(i→n(i0), uint2bits(j / 2).rest)) = lastn(i→n(i0), uint2bits(j / 2).rest), 0 < (j / 2) % 2 ^ i0, 0 ≤ i0, 0 < j, 2 ^ i0 * 2 < j, ∀ j0, i. i→n(j0) < i→n(j) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ ~1 = i0, j = 1

Rule argument

seq: ⊦ n ≤ # bits → # lastn(n, bits) = n subst: [bits, n] → [uint2bits(j / 2).rest, i→n(i0)] rotate: F spec: list-del inst: bitlista name: lastN-length

Additional information

Used simplifier rules: uint2bits-zero-01: 0 ≤ i ⊦ uint2bits(i) = false ' ↔ i = 0;% ⊦ sgn(false ' + mklist(false, n)) = false ';% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ i < j → ¬ j < i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward) Used terms: uint2bits(j / 2).rest, i→n(i0)