0specs/CSP-DT-ext/proofs/3377954148genfailure-def ⊦ f'(p)(el, E) ↔ (∃ st, st', il. p(st) ∧ el = il |E ∧ OP(il)(st, st') ∧ st' ↓ ∧ refusal(st', E))00genfailure-def-proofgenfailure-def-proof-infolocalsimpINIT-emptyPROG ⊦ Init^ ⊗ Op^([]) = Init^15INIT-emptyPROG-proofINIT-emptyPROG-proof-infoLemma-1¬ div([]) ⊦ SEM^(el)(E ', ω) ↔ div(el)init-divinit-taustar-divdiv-lemmaassoc-INIT-OP-FIN1024Lemma-1-proofLemma-1-proof-infoLemma-2¬ div([]), ¬ div(el) ⊦ SEM^(el)(E ', E' ') ↔ failure(el, E')genfailure-defprog-omega-divinit-taustar-divrefusal-lemmaassoc-INIT-OP-FIN1849Lemma-2-proofLemma-2-proof-infoLemma-3∀ el0. el0 ≠ el ∧ el0 ⊑ el → ¬ div(el0) ⊦ SEM^(el)(E ', ⊥) ↔ (∃ el1, ei. el1 + ei ' ⊑ el ∧ SEM^(el1)(E ', (λ ei0. ei0 = ei) '))optau-PROGtaustar-idprog-taustarinit-taustar-divprog-omegaprog-omega-divassoc-OP-OP-FINassoc-INIT-OP-FIN73186Lemma-3-proofLemma-3-proof-infoLemma-3-old¬ div(el), ¬ div([]) ⊦ SEM^(el)(E ', ⊥) ↔ (∃ el1, ei. el1 + ei ' ⊑ el ∧ SEM^(el1)(E ', (λ ei0. ei0 = ei) '))prog-taustarPROG-optauoptau-PROGtaustar-idinit-taustar-divprog-omegaprog-omega-divassoc-OP-OP-FINassoc-INIT-OP-FIN58150Lemma-3-old-proofLemma-3-old-proof-infoLemma-3a ⊦ ¬ div([]) ∧ div(el0) ∧ el0 ≠ el ∧ el0 ⊑ el → (¬ SEM^(el)(E ', ⊥) ↔ false)Lemma-1assoc-OP-OP-FINprog-omega-divassoc-INIT-OP-FIN716Lemma-3a-proofLemma-3a-proof-infolocalsimpsimpPROG-optau ⊦ il |E = ei ' ∧ OP(il)(st, st') → (¬ Opτ(ei)(st, st') ↔ false)38PROG-optau-proofPROG-optau-proof-infosimplocalsimpPROG-optau-taustar ⊦ il |E = ei ' ∧ ¬ e∈(il0) ∧ OP(il)(st, st0) ∧ OP(il0)(st0, st') → (¬ Opτ(ei)(st, st') ↔ false)PROG-optau14PROG-optau-taustar-proofPROG-optau-taustar-proof-infosimplocalsimpPROG-taustar-optau ⊦ il |E = ei ' ∧ ¬ e∈(il0) ∧ OP(il0)(st, st0) ∧ OP(il)(st0, st') → (¬ Opτ(ei)(st, st') ↔ false)PROG-optau14PROG-taustar-optau-proofPROG-taustar-optau-proof-infosimplocalsimpPROG-taustar-optau-taustar ⊦ il |E = ei ' ∧ ¬ e∈(il0) ∧ ¬ e∈(il1) ∧ OP(il0)(st, st0) ∧ OP(il)(st0, st1) ∧ OP(il1)(st1, st') → (¬ Opτ(ei)(st, st') ↔ false)PROG-optau26PROG-taustar-optau-taustar-proofPROG-taustar-optau-taustar-proof-infosimplocalsimpSEM-div-full ⊦ ¬ div([]) ∧ div(el) → (¬ SEM^(el)(g⊥ω, E ') ↔ false)prog-omega-divassoc-INIT-OP-FINLemma-1412SEM-div-full-proofSEM-div-full-proof-infosimplocalsimpSEM-no-omega ⊦ ¬ div([]) → (SEM^(el)(g⊥ω, no) ↔ SEM^(el)(g⊥ω, ω))init-taustar-divprog-omegaprog-omega-divassoc-INIT-OP-FIN1640SEM-no-omega-proofSEM-no-omega-proof-infosimplocalsimpSEM-omega-full ⊦ ¬ div([]) ∧ SEM^(el)(g⊥ω, ω) → (¬ SEM^(el)(g⊥ω, E ') ↔ false)prog-omega-divassoc-INIT-OP-FIN310SEM-omega-full-proofSEM-omega-full-proof-infosimplocalsimpassoc-INIT-OP-FIN ⊦ (init⊥ω ⊗ op⊥ω) ⊗ fin⊥ω = init⊥ω ⊗ (op⊥ω ⊗ fin⊥ω)111assoc-INIT-OP-FIN-proofassoc-INIT-OP-FIN-proof-infolocalsimpassoc-INIT-OP-OP ⊦ (init⊥ω ⊗ op⊥ω) ⊗ op⊥ω0 = init⊥ω ⊗ (op⊥ω ⊗ op⊥ω0)111assoc-INIT-OP-OP-proofassoc-INIT-OP-OP-proof-infolocalsimpassoc-OP-OP-FIN ⊦ (op⊥ω ⊗ op⊥ω0) ⊗ fin⊥ω = op⊥ω ⊗ op⊥ω0 ⊗ fin⊥ω111assoc-OP-OP-FIN-proofassoc-OP-OP-FIN-proof-infolocalsimpdiv-lemma¬ st ↑ ⊦ Op^(el)(st ', ω) ↔ (∃ st', il. il |E ⊑ el ∧ OP(il)(st, st') ∧ st' ↑)PROG-optauoptau-PROG1640div-lemma-proofdiv-lemma-proof-infoinit-div ⊦ INITS(st) ∧ st ↑ → (¬ div([]) ↔ false)14init-div-proofinit-div-proof-infosimplocalsimpinit-taustar-div ⊦ INITS(st) ∧ τ*(st, st0) ∧ st0 ↑ → (¬ div([]) ↔ false)36init-taustar-div-proofinit-taustar-div-proof-infosimplocalsimpoptau-PROG ⊦ Opτ(ei)(st, st') → (∃ il. il |E = ei ' ∧ OP(il)(st, st'))512optau-PROG-proofoptau-PROG-proof-infolocalforwardprog-omega ⊦ el ≠ [] ∧ Op^(el)(st ', st' ') ∧ st' ↑ → (¬ Op^(el)(st ', ω) ↔ false)2150prog-omega-proofprog-omega-proof-infolocalsimpprog-omega-div ⊦ INITS(st) ∧ τ*(st, st1) ∧ Op^(el0)(st1 ', ω) → div(el0)assoc-INIT-OP-FINLemma-138prog-omega-div-proofprog-omega-div-proof-infolocalforwardprog-taustar ⊦ el ≠ [] ∧ Op^(el)(st ', st0 ') ∧ τ*(st0, st') → (¬ Op^(el)(st ', st' ') ↔ false)1843prog-taustar-proofprog-taustar-proof-infolocalsimprefusal-lemma¬ ran((λ s⊥ω. s⊥ω ≠ ⊥ ∧ s⊥ω ≠ ω ∧ p(s⊥ω .s)) ≪ Op^(el))(ω), ∀ st. p(st) → ¬ st ↑ ⊦ f'(p)(el, E) ↔ ran(((λ s⊥ω. s⊥ω ≠ ⊥ ∧ s⊥ω ≠ ω ∧ p(s⊥ω .s)) ≪ Op^(el)) ⊗ Fin^)(E ')PROG-optauoptau-PROGgenfailure-def54138refusal-lemma-proofrefusal-lemma-proof-infotaustar-id ⊦ (∀ st0. ¬ τ(st, st0)) ∧ τ*(st, st') → st = st'38taustar-id-prooftaustar-id-proof-info