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Proof step 35 from proof rg2-sound-main-unfair (previous step)

[:S | [PL S = s10 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s11 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ], \A ∀ N, Post1, pm, s, S, G1, R1, Post2, G2, R2, Pre1, I, Pre2. N < n → [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∧ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ I(pm, S) → ((R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')), n = N + 1, R1(pm, s2, S), I(pm, s2), I(pm, s3), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s), Pre2(pm, s), I(pm, s), G1(pm, s2, s2), R1(pm, s2, s10), G2(pm, s2, s3), R2(pm, s3, s11), R1(pm, s3, S), R2(pm, s3, S), R2(pm, s2, s2), R1(pm, s2, s3), I(pm, s10), I(pm, s11), I(pm, S), boolvar → s2 = s3 ∧ N + 1 = n0 ⊦ (R1(pm, S', S'') ∧ R2(pm, S', S'') ∧ N = N'' + 1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S'))

User applied rule apply induction

And obtained the following new goals

Proof step 36:
[:S | [PL S = s10 ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL S = s11 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ], \A ∀ N, Post1, pm, s, S, G1, R1, Post2, G2, R2, Pre1, I, Pre2. N < n → [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∧ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ I(pm, S) → ((R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')), n = N + 1, R1(pm, s2, S), I(pm, s2), I(pm, s3), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s), Pre2(pm, s), I(pm, s), G1(pm, s2, s2), R1(pm, s2, s10), G2(pm, s2, s3), R2(pm, s3, s11), R1(pm, s3, S), R2(pm, s3, S), R2(pm, s2, s2), R1(pm, s2, s3), I(pm, s10), I(pm, s11), I(pm, S), boolvar → s2 = s3 ∧ N + 1 = n0 ⊦ [:S | [PL R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥nf [PL R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]] ∧ RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ I(pm, S), (R1(pm, S', S'') ∧ R2(pm, S', S'') ∧ N = N'' + 1) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S'))

Rule argument

subst: [N, Post1, pm, s, S, G1, R1, Post2, G2, R2, Pre1, I, Pre2] → [N, Post1, pm, s, S, G1, R1, Post2, G2, R2, Pre1, I, Pre2]

Additional information

Used simplifier rules: succ: ⊦ m +1 = m + 1; from specification nat-basic2%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: N, Post1, pm, s, S, G1, R1, Post2, G2, R2, Pre1, I, Pre2