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Proof step 2 from proof getsetShort-theorem (previous step)

~32768 ≤ j, j ≤ 32767 ⊦ bor(b2i(i2b(j i>> 8)) i<< 8, band(255, b2i(i2b(j)))) = j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ j0 < 2 ^ i ∧ 0 ≤ j0 ∧ 0 ≤ i ∧ i < 32 → bor(j i<< i, j0) = j * 2 ^ i + j0
Proof step 4:
band(255, b2i(i2b(j))) < 2 ^ 8 ∧ 0 ≤ band(255, b2i(i2b(j))) ∧ 0 ≤ 8 ∧ 8 < 32, ~32768 ≤ j, j ≤ 32767 ⊦ b2i(i2b(j i>> 8)) * 2 ^ 8 + band(255, b2i(i2b(j))) = j

Rule argument

spec: intops inst: name: bor-shift-plus seq: ⊦ j0 < 2 ^ i ∧ 0 ≤ j0 ∧ 0 ≤ i ∧ i < 32 → bor(j i<< i, j0) = j * 2 ^ i + j0 subst: [j0, i, j] → [band(255, b2i(i2b(j))), 8, b2i(i2b(j i>> 8))] flags: rewrite

Additional information

Used simplifier rules: lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%band-max-04: ⊦ ¬ 256 ≤ band(i, 255); from specification intops%band-associativity: ⊦ band(i, band(i0, j)) = band(band(i, i0), j); from specification intops%band-commutativity: ⊦ band(i, j) = band(j, i); from specification intops%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%band-notneg: ⊦ 0 ≤ i → ¬ band(i, j) < 0; from specification intops Used terms: band(255, b2i(i2b(j))), 8, b2i(i2b(j i>> 8))