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Proof step 81 from proof uint2bits-mod2pot (previous step)

uint2bits(i1 % 2 ^ i0) = sgn(false ' + lastn(i→n(i0), uint2bits(i1).rest)), mklist(false, # lastn(i→n(i0), uint2bits(i1).rest)) = lastn(i→n(i0), uint2bits(i1).rest), # uint2bits(i1).rest < i→n(i0), 0 < i1 % 2 ^ i0, 0 ≤ i1, 0 < i0, 0 < i1, 2 ^ i0 < i1, ∀ j0, i. i→n(j0) < i→n(i1 * 2 + 0) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ i0 = 0, ~1 = i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 82:
⊦ bits ≠ [] → # bits.rest = (# bits)-1
Proof step 83:
uint2bits(i1) ≠ [], uint2bits(i1 % 2 ^ i0) = sgn(false ' + lastn(i→n(i0), uint2bits(i1).rest)), mklist(false, # lastn(i→n(i0), uint2bits(i1).rest)) = lastn(i→n(i0), uint2bits(i1).rest), (# uint2bits(i1))-1 < i→n(i0), 0 < i1 % 2 ^ i0, 0 ≤ i1, 0 < i0, 0 < i1, 2 ^ i0 < i1, ∀ j0, i. i→n(j0) < i→n(i1 * 2 + 0) → 0 ≤ i ∧ 2 ^ i < j0 → uint2bits(j0 % 2 ^ i) = sgn(false ' + lastn(i→n(i), uint2bits(j0).rest)) ⊦ i0 = 0, ~1 = i0

Rule argument

seq: ⊦ bits ≠ [] → # bits.rest = (# bits)-1 subst: [bits] → [uint2bits(i1)] rotate: F spec: list-data inst: bitlista name: length-rest

Additional information

Used simplifier rules: uint2bits-empty: 0 ≤ i ⊦ uint2bits(i) ≠ []; Used terms: uint2bits(i1)