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Proof step 59 from proof bytes2bytes-mone (previous step)

b2i(bytes0.first) = ~1, 0 ≤ b2i((bytes0.rest + by ').first), false ∈ bytes2bits(bytes0), true ∈ bytes2bits(bytes0), ∀ bytes1. # bytes1 < #(bytes0 + by ') → b2i(bytes1.first) = ~1 ∧ 0 ≤ b2i(bytes1.rest.first) → bits2bytes(sgn(bytes2bits(bytes1))) = bytes1 ∨ # bytes1 = 1 ∨ bytes1 = [] ⊦ bits2bytes(sgn(bytes2bits(bytes0)) + byte2bits(by)) = bytes0 + by ', bytes0 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 60:
⊦ 8 < # bits → bits2bytes(bits) = bits2bytes(firstn(# bits - 8, bits)) + i2b(bits2int(lastn(8, bits)))
Proof step 61:
8 < #(sgn(bytes2bits(bytes0)) + byte2bits(by)), b2i(bytes0.first) = ~1, 0 ≤ b2i((bytes0.rest + by ').first), false ∈ bytes2bits(bytes0), true ∈ bytes2bits(bytes0), ∀ bytes1. # bytes1 < #(bytes0 + by ') → b2i(bytes1.first) = ~1 ∧ 0 ≤ b2i(bytes1.rest.first) → bits2bytes(sgn(bytes2bits(bytes1))) = bytes1 ∨ # bytes1 = 1 ∨ bytes1 = [] ⊦ bits2bytes(firstn(#(sgn(bytes2bits(bytes0)) + byte2bits(by)) - 8, sgn(bytes2bits(bytes0)) + byte2bits(by))) + i2b(bits2int(lastn(8, sgn(bytes2bits(bytes0)) + byte2bits(by)))) = bytes0 + by ', bytes0 = []

Rule argument

name: bits2bytes-rec rotate: F subst: [bits] → [sgn(bytes2bits(bytes0)) + byte2bits(by)]

Additional information

Used simplifier rules: byte2bits-length: ⊦ # byte2bits(by) = 8;%bytes2bits-empty-01: ⊦ bytes2bits(bytes) = [] ↔ bytes = [];% ⊦ #(bits + bits0) = # bits + # bits0;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ ¬ m < n ↔ n ≤ m;% ⊦ m + n ≤ m ↔ n = 0;% ⊦ # bits = 0 ↔ bits = [];% ⊦ sgn(bits) = [] ↔ bits = [];% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m; Used terms: sgn(bytes2bits(bytes0)) + byte2bits(by)