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Proof step 5 from proof long-automatic (previous step)

⊦ cast(64, i) = cast(64, j) ↔ (¬ i < 0 ∧ ¬ 9223372036854775807 < i % 18446744073709551616 ⊃ i % 18446744073709551616 ;(¬ i < 0 ∧ ¬ i % 18446744073709551616 ≤ 9223372036854775807 ⊃ i % 18446744073709551616 - 18446744073709551616 ;(¬ 0 ≤ i ∧ ¬ ~9223372036854775808 ≤ i % 18446744073709551616 ⊃ i % 18446744073709551616 + 18446744073709551616 ;(¬ 0 ≤ i ∧ ¬ i % 18446744073709551616 < ~9223372036854775808 ⊃ i % 18446744073709551616;i) ) ) ) = (¬ j < 0 ∧ ¬ 9223372036854775807 < j % 18446744073709551616 ⊃ j % 18446744073709551616 ;(¬ j < 0 ∧ ¬ j % 18446744073709551616 ≤ 9223372036854775807 ⊃ j % 18446744073709551616 - 18446744073709551616 ;(¬ 0 ≤ j ∧ ¬ ~9223372036854775808 ≤ j % 18446744073709551616 ⊃ j % 18446744073709551616 + 18446744073709551616 ;(¬ 0 ≤ j ∧ ¬ j % 18446744073709551616 < ~9223372036854775808 ⊃ j % 18446744073709551616;j) ) ) )

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 0 < i → cast(i, j) = (0 ≤ j ∧ j % 2 ^ i ≤ 2 ^ (i - 1) - 1 ⊃ j % 2 ^ i ;(0 ≤ j ∧ 2 ^ (i - 1) - 1 < j % 2 ^ i ⊃ j % 2 ^ i - 2 ^ i ;(j < 0 ∧ j % 2 ^ i < ~(2 ^ (i - 1)) ⊃ j % 2 ^ i + 2 ^ i;(j < 0 ∧ ~(2 ^ (i - 1)) ≤ j % 2 ^ i ⊃ j % 2 ^ i;j)) ) )
Proof step 7:
0 < 64 ⊦ cast(64, i) = (0 ≤ j ∧ j % 2 ^ 64 ≤ 2 ^ (64 - 1) - 1 ⊃ j % 2 ^ 64 ;(0 ≤ j ∧ 2 ^ (64 - 1) - 1 < j % 2 ^ 64 ⊃ j % 2 ^ 64 - 2 ^ 64 ;(j < 0 ∧ j % 2 ^ 64 < ~(2 ^ (64 - 1)) ⊃ j % 2 ^ 64 + 2 ^ 64;(j < 0 ∧ ~(2 ^ (64 - 1)) ≤ j % 2 ^ 64 ⊃ j % 2 ^ 64;j)) ) ) ↔ (¬ i < 0 ∧ ¬ 9223372036854775807 < i % 18446744073709551616 ⊃ i % 18446744073709551616 ;(¬ i < 0 ∧ ¬ i % 18446744073709551616 ≤ 9223372036854775807 ⊃ i % 18446744073709551616 - 18446744073709551616 ;(¬ 0 ≤ i ∧ ¬ ~9223372036854775808 ≤ i % 18446744073709551616 ⊃ i % 18446744073709551616 + 18446744073709551616 ;(¬ 0 ≤ i ∧ ¬ i % 18446744073709551616 < ~9223372036854775808 ⊃ i % 18446744073709551616;i) ) ) ) = (¬ j < 0 ∧ ¬ 9223372036854775807 < j % 18446744073709551616 ⊃ j % 18446744073709551616 ;(¬ j < 0 ∧ ¬ j % 18446744073709551616 ≤ 9223372036854775807 ⊃ j % 18446744073709551616 - 18446744073709551616 ;(¬ 0 ≤ j ∧ ¬ ~9223372036854775808 ≤ j % 18446744073709551616 ⊃ j % 18446744073709551616 + 18446744073709551616 ;(¬ 0 ≤ j ∧ ¬ j % 18446744073709551616 < ~9223372036854775808 ⊃ j % 18446744073709551616;j) ) ) )

Rule argument

seq: ⊦ 0 < i → cast(i, j) = (0 ≤ j ∧ j % 2 ^ i ≤ 2 ^ (i - 1) - 1 ⊃ j % 2 ^ i ;(0 ≤ j ∧ 2 ^ (i - 1) - 1 < j % 2 ^ i ⊃ j % 2 ^ i - 2 ^ i ;(j < 0 ∧ j % 2 ^ i < ~(2 ^ (i - 1)) ⊃ j % 2 ^ i + 2 ^ i ;(j < 0 ∧ ~(2 ^ (i - 1)) ≤ j % 2 ^ i ⊃ j % 2 ^ i;j) ) ) ) subst: [j, i] → [j, 64] rotate: F spec: intops inst: name: cast-if

Additional information

Used terms: j, 64