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Proof step 12 from proof no-init-violation-left (previous step)

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

User applied rule apply induction

And obtained the following new goals

Proof step 13:
N = N'' + 1 until Boolvar, □ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | Boolvar ::- [PL ¬ P(pm, S)] ∥ false ::- [PL S = s3 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))]], \A ∀ N, R1, pm, S, R2, Boolvar, P, Post2, s, G2, G1, Pre1, Post1, I, Pre2. N < n → □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ P(pm, S)] ∥ 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) ∧ stable(pm, I, R1) ∧ stable(pm, I, R2) ∧ I(pm, S) ∧ Pre2(pm, s) ∧ P-restr(pm, P, I, R1) ∧ P(pm, S) → false, n = N + 1, P(pm, s2), I(pm, s3), I(pm, S), R1(pm, s1, s2), R2(pm, s2, s3), G2(pm, s1, s2), R2(pm, s2, S), R1(pm, s2, S), P(pm, s1), P-restr(pm, P, I, R1), Pre2(pm, s), I(pm, s1), stable(pm, I, R2), stable(pm, I, R1), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), I(pm, s2), R1(pm, s1, S), P(pm, S) ⊦ □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ P(pm, S)] ∥ 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) ∧ stable(pm, I, R1) ∧ stable(pm, I, R2) ∧ I(pm, S) ∧ Pre2(pm, s) ∧ P-restr(pm, P, I, R1) ∧ P(pm, S)

Rule argument

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