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Proof step 51 from proof no-progress-left (previous step)

□ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | Boolvar ::- [PL S = s20 ∧ ¬ Boolvar until □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ false ::- [PL S = s21 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ], N = N'' + 1 until Boolvar, \A ∀ N, R1, pm, S, R2, Boolvar, P, Post1, s, G1, Post2, G2, Pre1, I, Pre2. N < n → □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ Boolvar until □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, 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) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ P-restr(pm, P, I, R1) → ◇ □ (¬ P(pm, S) ∧ ¬ last), n = N + 1, R1(pm, s2, S), I(pm, s3), I(pm, s2), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), stable(pm, I, R1), stable(pm, I, R2), Pre1(pm, s), Pre2(pm, s), P-restr(pm, P, I, R1), R1(pm, s3, S), R2(pm, s3, S), G1(pm, s2, s2), R1(pm, s2, s20), G2(pm, s2, s3), R2(pm, s3, s21), R2(pm, s2, s2), R1(pm, s2, s3), I(pm, s20), I(pm, S), I(pm, s21), boolvar1 → s2 = s3 ∧ ¬ boolvar0 ∧ N + 1 = n0 ⊦ ¬ P(pm, s2) ∧ □ (¬ P(pm, S) ∧ ¬ last), ◇ □ (¬ P(pm, S) ∧ ¬ last)

User applied rule apply induction

And obtained the following new goals

Proof step 52:
□ (R1(pm, S', S'') ∧ R2(pm, S', S'')), [:S | Boolvar ::- [PL S = s20 ∧ ¬ Boolvar until □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, S'))] ∥ false ::- [PL S = s21 ∧ R2(pm, S', S'') -+-> (last ⊃ Post2(pm, s, S);G2(pm, S, S'))] ], N = N'' + 1 until Boolvar, \A ∀ N, R1, pm, S, R2, Boolvar, P, Post1, s, G1, Post2, G2, Pre1, I, Pre2. N < n → □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ Boolvar until □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, 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) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ P-restr(pm, P, I, R1) → ◇ □ (¬ P(pm, S) ∧ ¬ last), n = N + 1, R1(pm, s2, S), I(pm, s3), I(pm, s2), RG-restr(pm, R1, G1, Pre1, Post1, I, R2, G2, Pre2, Post2), stable(pm, I, R1), stable(pm, I, R2), Pre1(pm, s), Pre2(pm, s), P-restr(pm, P, I, R1), R1(pm, s3, S), R2(pm, s3, S), G1(pm, s2, s2), R1(pm, s2, s20), G2(pm, s2, s3), R2(pm, s3, s21), R2(pm, s2, s2), R1(pm, s2, s3), I(pm, s20), I(pm, S), I(pm, s21), boolvar1 → s2 = s3 ∧ ¬ boolvar0 ∧ N + 1 = n0 ⊦ □ (R1(pm, S', S'') ∧ R2(pm, S', S'')) ∧ N = N'' +1 until Boolvar ∧ [:S | Boolvar ::- [PL ¬ Boolvar until □ (¬ P(pm, S) ∧ ¬ last) ∧ R1(pm, S', S'') -+-> (last ⊃ Post1(pm, s, S);G1(pm, S, 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) ∧ Pre1(pm, s) ∧ Pre2(pm, s) ∧ P-restr(pm, P, I, R1), ¬ P(pm, s2) ∧ □ (¬ P(pm, S) ∧ ¬ last), ◇ □ (¬ P(pm, S) ∧ ¬ last)

Rule argument

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