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Proof step 159 from proof rg2-lem2-term-unfair (previous step)

□ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | Boolvar ::- [PL N' = N'' + 1 until last ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf false ::- [PL S = s39 ∧ N = n39 ∧ □ ¬ N' < N'' ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S')) ∧ ¬ □ R2(pm, S', S'')] ], □ (N = N' ∧ ¬ N' < N''), N0 = N0'' + 1 until Boolvar, INDHYP-nat(n3) ∧ INDHYP-nat0(N0 + 1), R1(pm, s2, S), I(pm, s3), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s), Pre2(pm, s), I(pm, s2), R1(pm, s3, S), R2(pm, s3, S), N ≤ n3, I(pm, s), G2(pm, s2, s3), R2(pm, s3, s39), n39 ≤ n3, R1(pm, s2, s3), I(pm, S), I(pm, s39) ⊦ ◇ last

User applied rule apply VD induction

And obtained the following new goals

Proof step 160:
□ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | Boolvar ::- [PL N' = N'' + 1 until last ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf false ::- [PL S = s39 ∧ N = n39 ∧ □ ¬ N' < N'' ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S')) ∧ ¬ □ R2(pm, S', S'')] ], □ (N = N' ∧ ¬ N' < N''), N0 = N0'' + 1 until Boolvar, INDHYP-nat(n3) ∧ INDHYP-nat0(N0 + 1), R1(pm, s2, S), I(pm, s3), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s), Pre2(pm, s), I(pm, s2), R1(pm, s3, S), R2(pm, s3, S), N ≤ n3, I(pm, s), G2(pm, s2, s3), R2(pm, s3, s39), n39 ≤ n3, R1(pm, s2, s3), I(pm, S), I(pm, s39) ⊦ [:S | Boolvar ::- [PL N' = N'' + 1 until last ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf false ::- [PL □ ¬ N' < N'' ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S')) ∧ (□ R2(pm, S', S'') → ◇ last)] ] ∧ N0 = N0'' +1 until Boolvar ∧ □ (N = N' ∧ ¬ N' < N'') ∧ □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ I(pm, S) ∧ I(pm, s), ◇ last

Rule argument

subst: [Post1, pm, s, G1, R1, Post2, G2, R2, Pre1, I, Pre2] → [Post1, pm, s, G1, R1, Post2, G2, R2, Pre1, I, Pre2] path: 1, 2, 1, 1, 1, 1, 1, 1

Additional information

Used simplifier rules: lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%anti: ⊦ m ≤ n → (n ≤ m ↔ m = n); from specification nat%s: ⊦ m < n + 1 ↔ ¬ n < m; from specification nat-basic2%ref-1: ⊦ ¬ n < n; from specification nat-basic1%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2 Used terms: Post1, pm, s, G1, R1, Post2, G2, R2, Pre1, I, Pre2