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Proof step 11 from proof rg2-not-pre1-unfair (previous step)

[:S | Boolvar ::- [PL ¬ Pre1(pm, S)] ∥nf false ::- [PL S = s3 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]], N = N'' + 1 until Boolvar, \A ∀ N, Boolvar, Pre1, pm, S, Post2, s, G2, R2, R1, G1, Post1, I, Pre2. N < n → N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ Pre1(pm, S)] ∥nf false ::- [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) ∧ I(pm, s) → (R1(pm, S', S'') ∧ R2(pm, S', S'')) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')), n = N + 1, R1(pm, s1, S), Pre1(pm, s2), I(pm, s2), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s1), Pre2(pm, s), I(pm, s1), I(pm, s), G2(pm, s1, s2), R2(pm, s2, s3), R1(pm, s2, S), R2(pm, s2, S), R1(pm, s1, s2), I(pm, s3), I(pm, S), Pre1(pm, S) ⊦ (R1(pm, S', S'') ∧ R2(pm, S', S'')) -+-> (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 12:
[:S | Boolvar ::- [PL ¬ Pre1(pm, S)] ∥nf false ::- [PL S = s3 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]], N = N'' + 1 until Boolvar, \A ∀ N, Boolvar, Pre1, pm, S, Post2, s, G2, R2, R1, G1, Post1, I, Pre2. N < n → N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ Pre1(pm, S)] ∥nf false ::- [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) ∧ I(pm, s) → (R1(pm, S', S'') ∧ R2(pm, S', S'')) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S')), n = N + 1, R1(pm, s1, S), Pre1(pm, s2), I(pm, s2), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), Pre1(pm, s1), Pre2(pm, s), I(pm, s1), I(pm, s), G2(pm, s1, s2), R2(pm, s2, s3), R1(pm, s2, S), R2(pm, s2, S), R1(pm, s1, s2), I(pm, s3), I(pm, S), Pre1(pm, S) ⊦ N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ Pre1(pm, S)] ∥nf false ::- [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) ∧ I(pm, s), (R1(pm, S', S'') ∧ R2(pm, S', S'')) -+-> (last ⊃ Post1(pm, s, S) ∧ Post2(pm, s, S);G1(pm, S, S') ∨ G2(pm, S, S'))

Rule argument

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

Additional information

Used simplifier rules: 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, Boolvar, Pre1, pm, S, Post2, s, G2, R2, R1, G1, Post1, I, Pre2