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internlm
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Lean-Github

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https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases q
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 q : β„• βŠ• Οƒ_0 βŠ• Οƒ_1 p_0 : Οƒ_0 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ q ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), q) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
case inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 val✝ : β„• ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inl val✝ ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inl val✝) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False case inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 val✝ : Οƒ_0 βŠ• Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr val✝ ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr val✝) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => simp intro xs a1 cases a1 case _ left => cases left case _ x a2 => cases a2 case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp case _ right => cases right case _ x a2 => cases a2 case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inl q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inl q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => cases q_0 case _ q_0 => simp sorry case _ q_0 => simp sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inl q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inl q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• ⊒ βˆ€ (x : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1)), ((βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = x) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = x) β†’ Sum.inl q_0 βˆ‰ x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
intro xs a1
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• ⊒ βˆ€ (x : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1)), ((βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = x) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = x) β†’ Sum.inl q_0 βˆ‰ x
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) a1 : (βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = xs) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases a1
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) a1 : (βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = xs) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
case inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) h✝ : βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs case inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) h✝ : βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ left => cases left case _ x a2 => cases a2 case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) left : βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ right => cases right case _ x a2 => cases a2 case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) right : βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases left
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) left : βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
case intro Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) w✝ : EpsilonArrow Οƒ_0 h✝ : w✝ ∈ M_0.epsilon_arrow_list ∧ w✝.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) w✝.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ x a2 => cases a2 case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2 : x ∈ M_0.epsilon_arrow_list ∧ x.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases a2
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2 : x ∈ M_0.epsilon_arrow_list ∧ x.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
case intro Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 left✝ : x ∈ M_0.epsilon_arrow_list right✝ : x.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2_left : x ∈ M_0.epsilon_arrow_list a2_right : x.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases a2_right
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2_left : x ∈ M_0.epsilon_arrow_list a2_right : x.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
case intro Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2_left : x ∈ M_0.epsilon_arrow_list left✝ : x.start_state = p_0 right✝ : List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2_left : x ∈ M_0.epsilon_arrow_list a2_right_left : x.start_state = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp only [← a2_right_right]
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2_left : x ∈ M_0.epsilon_arrow_list a2_right_left : x.start_state = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2_left : x ∈ M_0.epsilon_arrow_list a2_right_left : x.start_state = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ List.map (Sum.inr ∘ Sum.inl) x.stop_state_list
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_0 a2_left : x ∈ M_0.epsilon_arrow_list a2_right_left : x.start_state = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inl) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ List.map (Sum.inr ∘ Sum.inl) x.stop_state_list
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases right
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) right : βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
case intro Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) w✝ : Οƒ_0 h✝ : w✝ ∈ M_0.accepting_state_list ∧ w✝ = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ x a2 => cases a2 case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2 : x ∈ M_0.accepting_state_list ∧ x = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases a2
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2 : x ∈ M_0.accepting_state_list ∧ x = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
case intro Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 left✝ : x ∈ M_0.accepting_state_list right✝ : x = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ a2_left a2_right => cases a2_right case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2_left : x ∈ M_0.accepting_state_list a2_right : x = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases a2_right
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2_left : x ∈ M_0.accepting_state_list a2_right : x = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
case intro Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2_left : x ∈ M_0.accepting_state_list left✝ : x = p_0 right✝ : List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ a2_right_left a2_right_right => simp only [← a2_right_right] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2_left : x ∈ M_0.accepting_state_list a2_right_left : x = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp only [← a2_right_right]
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2_left : x ∈ M_0.accepting_state_list a2_right_left : x = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2_left : x ∈ M_0.accepting_state_list a2_right_left : x = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : Οƒ_0 a2_left : x ∈ M_0.accepting_state_list a2_right_left : x = p_0 a2_right_right : List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = xs ⊒ Sum.inl q_0 βˆ‰ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases q_0
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
case inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 val✝ : Οƒ_0 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inl val✝) ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inl val✝)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False case inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 val✝ : Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inr val✝) ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inr val✝)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => simp sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_0 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inl q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inl q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => simp sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inr q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_0 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inl q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inl q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_0 ⊒ (βˆƒ stop_state_list, ((βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = stop_state_list) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = stop_state_list) ∧ Sum.inr (Sum.inl q_0) ∈ stop_state_list) ↔ βˆƒ stop_state_list, { start_state := p_0, stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q_0 ∈ stop_state_list
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_0 ⊒ (βˆƒ stop_state_list, ((βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = stop_state_list) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = stop_state_list) ∧ Sum.inr (Sum.inl q_0) ∈ stop_state_list) ↔ βˆƒ stop_state_list, { start_state := p_0, stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q_0 ∈ stop_state_list
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inr q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, ((βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = stop_state_list) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = stop_state_list) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) ↔ p_0 ∈ M_0.accepting_state_list ∧ q_0 ∈ M_1.starting_state_list
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, ((βˆƒ a ∈ M_0.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inl) a.stop_state_list = stop_state_list) ∨ βˆƒ a ∈ M_0.accepting_state_list, a = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) M_1.starting_state_list = stop_state_list) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) ↔ p_0 ∈ M_0.accepting_state_list ∧ q_0 ∈ M_1.starting_state_list
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases q
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 q : β„• βŠ• Οƒ_0 βŠ• Οƒ_1 p_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ q ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), q) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
case inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 val✝ : β„• ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inl val✝ ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inl val✝) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False case inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 val✝ : Οƒ_0 βŠ• Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr val✝ ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr val✝) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => simp intro xs x a1 a2 a3 simp only [← a3] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inl q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inl q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => cases q_0 case _ q_0 => simp intro xs x a1 a2 a3 simp only [← a3] simp case _ q_0 => simp sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inl q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inl q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• ⊒ βˆ€ (x : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1)), βˆ€ x_1 ∈ M_1.epsilon_arrow_list, x_1.start_state = p_0 β†’ List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x β†’ Sum.inl q_0 βˆ‰ x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
intro xs x a1 a2 a3
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• ⊒ βˆ€ (x : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1)), βˆ€ x_1 ∈ M_1.epsilon_arrow_list, x_1.start_state = p_0 β†’ List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x β†’ Sum.inl q_0 βˆ‰ x
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp only [← a3]
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ xs
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : β„• xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inl q_0 βˆ‰ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases q_0
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr q_0 ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr q_0) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
case inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 val✝ : Οƒ_0 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inl val✝) ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl val✝)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False case inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 val✝ : Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inr val✝) ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr val✝)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => simp intro xs x a1 a2 a3 simp only [← a3] simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inl q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ q_0 => simp sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inl q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 ⊒ βˆ€ (x : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1)), βˆ€ x_1 ∈ M_1.epsilon_arrow_list, x_1.start_state = p_0 β†’ List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x β†’ Sum.inr (Sum.inl q_0) βˆ‰ x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
intro xs x a1 a2 a3
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 ⊒ βˆ€ (x : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1)), βˆ€ x_1 ∈ M_1.epsilon_arrow_list, x_1.start_state = p_0 β†’ List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x β†’ Sum.inr (Sum.inl q_0) βˆ‰ x
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inr (Sum.inl q_0) βˆ‰ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp only [← a3]
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inr (Sum.inl q_0) βˆ‰ xs
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inr (Sum.inl q_0) βˆ‰ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 q_0 : Οƒ_0 xs : List (β„• βŠ• Οƒ_0 βŠ• Οƒ_1) x : EpsilonArrow Οƒ_1 a1 : x ∈ M_1.epsilon_arrow_list a2 : x.start_state = p_0 a3 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs ⊒ Sum.inr (Sum.inl q_0) βˆ‰ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, { start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } ∈ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inl arrow.start_state, stop_state_list := List.map Sum.inl arrow.stop_state_list }) M_0.epsilon_arrow_list) ++ List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) (List.map (fun arrow => { start_state := Sum.inr arrow.start_state, stop_state_list := List.map Sum.inr arrow.stop_state_list }) M_1.epsilon_arrow_list) ++ List.map (fun accepting_state => { start_state := accepting_state, stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) }) (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) = match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr q_0)) with | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) => βˆƒ stop_state_list, { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list | x => False
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, (βˆƒ a ∈ M_1.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) a.stop_state_list = stop_state_list) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) ↔ βˆƒ stop_state_list, { start_state := p_0, stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q_0 ∈ stop_state_list
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
sorry
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 q_0 : Οƒ_1 ⊒ (βˆƒ stop_state_list, (βˆƒ a ∈ M_1.epsilon_arrow_list, a.start_state = p_0 ∧ List.map (Sum.inr ∘ Sum.inr) a.stop_state_list = stop_state_list) ∧ Sum.inr (Sum.inr q_0) ∈ stop_state_list) ↔ βˆƒ stop_state_list, { start_state := p_0, stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q_0 ∈ stop_state_list
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
constructor
case right.right Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 ⊒ ((fun state => state ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = fun p => match p with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False) ∧ (fun state => state ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = fun p => match p with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
case right.right.left Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 ⊒ (fun state => state ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = fun p => match p with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False case right.right.right Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 ⊒ (fun state => state ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = fun p => match p with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
funext p
case right.right.left Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 ⊒ (fun state => state ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = fun p => match p with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
case right.right.left.h Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p : β„• βŠ• Οƒ_0 βŠ• Οƒ_1 ⊒ (p ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match p with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases p
case right.right.left.h Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p : β„• βŠ• Οƒ_0 βŠ• Οƒ_1 ⊒ (p ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match p with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
case right.right.left.h.inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : β„• ⊒ (Sum.inl val✝ ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inl val✝ with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False case right.right.left.h.inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : Οƒ_0 βŠ• Οƒ_1 ⊒ (Sum.inr val✝ ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr val✝ with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : β„• ⊒ (Sum.inl p_0 ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inl p_0 with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => cases p_0 case _ p_0 => simp case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (Sum.inr p_0 ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr p_0 with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : β„• ⊒ (Sum.inl p_0 ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inl p_0 with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases p_0
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (Sum.inr p_0 ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr p_0 with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
case inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : Οƒ_0 ⊒ (Sum.inr (Sum.inl val✝) ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr (Sum.inl val✝) with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False case inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : Οƒ_1 ⊒ (Sum.inr (Sum.inr val✝) ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr (Sum.inr val✝) with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 ⊒ (Sum.inr (Sum.inl p_0) ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr (Sum.inl p_0) with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 ⊒ (Sum.inr (Sum.inr p_0) ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr (Sum.inr p_0) with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 ⊒ (Sum.inr (Sum.inl p_0) ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr (Sum.inl p_0) with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 ⊒ (Sum.inr (Sum.inr p_0) ∈ List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = match Sum.inr (Sum.inr p_0) with | Sum.inr (Sum.inl p') => p' ∈ M_0.starting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
funext p
case right.right.right Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 ⊒ (fun state => state ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = fun p => match p with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
case right.right.right.h Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p : β„• βŠ• Οƒ_0 βŠ• Οƒ_1 ⊒ (p ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match p with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases p
case right.right.right.h Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p : β„• βŠ• Οƒ_0 βŠ• Οƒ_1 ⊒ (p ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match p with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
case right.right.right.h.inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : β„• ⊒ (Sum.inl val✝ ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inl val✝ with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False case right.right.right.h.inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : Οƒ_0 βŠ• Οƒ_1 ⊒ (Sum.inr val✝ ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr val✝ with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : β„• ⊒ (Sum.inl p_0 ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inl p_0 with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => cases p_0 case _ p_0 => simp case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (Sum.inr p_0 ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr p_0 with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : β„• ⊒ (Sum.inl p_0 ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inl p_0 with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
cases p_0
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 βŠ• Οƒ_1 ⊒ (Sum.inr p_0 ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr p_0 with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
case inl Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : Οƒ_0 ⊒ (Sum.inr (Sum.inl val✝) ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr (Sum.inl val✝) with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False case inr Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 val✝ : Οƒ_1 ⊒ (Sum.inr (Sum.inr val✝) ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr (Sum.inr val✝) with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 ⊒ (Sum.inr (Sum.inl p_0) ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr (Sum.inl p_0) with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
case _ p_0 => simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 ⊒ (Sum.inr (Sum.inr p_0) ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr (Sum.inr p_0) with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_0 ⊒ (Sum.inr (Sum.inl p_0) ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr (Sum.inl p_0) with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/RegExpToEpsilonNFA.lean
match_concat_EpsilonNFA_toAbstract
[557, 1]
[690, 19]
simp
Ξ± : Type inst✝ : DecidableEq Ξ± Οƒ_0 Οƒ_1 : Type M_0 : EpsilonNFA Ξ± Οƒ_0 M_1 : EpsilonNFA Ξ± Οƒ_1 p_0 : Οƒ_1 ⊒ (Sum.inr (Sum.inr p_0) ∈ List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = match Sum.inr (Sum.inr p_0) with | Sum.inr (Sum.inr p') => p' ∈ M_1.accepting_state_list | x => False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
induction h1 generalizing V
D : Type I J : Interpretation D V : VarAssignment D E : Env A : Formula P : PredName zs : List VarName H B : Formula h1 : IsSub P zs H A B h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) ⊒ Holds D I V E B ↔ Holds D J V E A
case pred_const_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : PredName xs✝ : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_const_ X✝ xs✝) ↔ Holds D J V E (pred_const_ X✝ xs✝) case pred_not_occurs_in D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : PredName xs✝ : List VarName a✝ : Β¬(X✝ = P ∧ xs✝.length = zs.length) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_var_ X✝ xs✝) ↔ Holds D J V E (pred_var_ X✝ xs✝) case pred_occurs_in D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : PredName ts✝ : List VarName a✝¹ : X✝ = P ∧ ts✝.length = zs.length a✝ : Var.All.Rec.admits (Function.updateListITE id zs ts✝) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs ts✝) H) ↔ Holds D J V E (pred_var_ P ts✝) case eq_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) x✝ y✝ : VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (eq_ x✝ y✝) ↔ Holds D J V E (eq_ x✝ y✝) case true_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E true_ ↔ Holds D J V E true_ case false_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E false_ ↔ Holds D J V E false_ case not_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ phi'✝ : Formula a✝ : IsSub P zs H phi✝ phi'✝ a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E phi'✝.not_ ↔ Holds D J V E phi✝.not_ case imp_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.imp_ psi'✝) ↔ Holds D J V E (phi✝.imp_ psi✝) case and_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.and_ psi'✝) ↔ Holds D J V E (phi✝.and_ psi✝) case or_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.or_ psi'✝) ↔ Holds D J V E (phi✝.or_ psi✝) case iff_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSub P zs H phi✝ phi'✝ a✝ : IsSub P zs H psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E psi'✝ ↔ Holds D J V E psi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (phi'✝.iff_ psi'✝) ↔ Holds D J V E (phi✝.iff_ psi✝) case forall_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) x✝ : VarName phi✝ phi'✝ : Formula a✝¹ : Β¬isFreeIn x✝ H a✝ : IsSub P zs H phi✝ phi'✝ a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (forall_ x✝ phi'✝) ↔ Holds D J V E (forall_ x✝ phi✝) case exists_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) x✝ : VarName phi✝ phi'✝ : Formula a✝¹ : Β¬isFreeIn x✝ H a✝ : IsSub P zs H phi✝ phi'✝ a_ih✝ : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E phi'✝ ↔ Holds D J V E phi✝) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (exists_ x✝ phi'✝) ↔ Holds D J V E (exists_ x✝ phi✝) case def_ D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) X✝ : DefName xs✝ : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (def_ X✝ xs✝) ↔ Holds D J V E (def_ X✝ xs✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case pred_const_ h1_X h1_ts => simp only [Holds] simp only [h3_const]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_const_ h1_X h1_ts) ↔ Holds D J V E (pred_const_ h1_X h1_ts)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case pred_occurs_in h1_X h1_ts h1_1 h1_2 => obtain s1 := Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id zs h1_ts) H h1_2 obtain s2 := Function.updateListITE_comp id V zs h1_ts simp only [s2] at s1 simp at s1 specialize h2 h1_X (List.map V h1_ts) simp only [s1] at h2 simp only [Holds] apply h2 simp exact h1_1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case eq_ h1_x h1_y => simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x h1_y : VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (eq_ h1_x h1_y) ↔ Holds D J V E (eq_ h1_x h1_y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case true_ | false_ => simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E false_ ↔ Holds D J V E false_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case not_ h1_phi h1_phi' _ h1_ih => simp only [Holds] congr! 1 exact h1_ih V h2
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_phi h1_phi' : Formula a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E h1_phi'.not_ ↔ Holds D J V E h1_phi.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
case forall_ h1_x h1_phi h1_phi' h1_1 _ h1_ih | exists_ h1_x h1_phi h1_phi' h1_1 _ h1_ih => simp only [Holds] first | apply forall_congr' | apply exists_congr intro d apply h1_ih intro Q ds a1 specialize h2 Q ds a1 have s1 : Holds D I (Function.updateListITE (Function.updateITE V h1_x d) zs ds) E H ↔ Holds D I (Function.updateListITE V zs ds) E H := by apply Holds_coincide_Var intro v a1 apply Function.updateListITE_updateIte intro contra subst contra contradiction simp only [h2] at s1 exact s1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_x : VarName h1_phi h1_phi' : Formula h1_1 : Β¬isFreeIn h1_x H a✝ : IsSub P zs H h1_phi h1_phi' h1_ih : βˆ€ (V : VarAssignment D), (βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) β†’ (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (exists_ h1_x h1_phi') ↔ Holds D J V E (exists_ h1_x h1_phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [Holds]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_const_ h1_X h1_ts) ↔ Holds D J V E (pred_const_ h1_X h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ I.pred_const_ h1_X (List.map V h1_ts) ↔ J.pred_const_ h1_X (List.map V h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [h3_const]
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ I.pred_const_ h1_X (List.map V h1_ts) ↔ J.pred_const_ h1_X (List.map V h1_ts)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp at h1_1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : Β¬(h1_X = P ∧ h1_ts.length = zs.length) V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (pred_var_ h1_X h1_ts) ↔ Holds D J V E (pred_var_ h1_X h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ Holds D I V E (pred_var_ h1_X h1_ts) ↔ Holds D J V E (pred_var_ h1_X h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
apply Holds_coincide_PredVar
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ Holds D I V E (pred_var_ h1_X h1_ts) ↔ Holds D J V E (pred_var_ h1_X h1_ts)
case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_const_ = J.pred_const_ case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (pred_var_ h1_X h1_ts) β†’ (I.pred_var_ P ds ↔ J.pred_var_ P ds)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact h3_const
case h1 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_const_ = J.pred_const_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
intro X ds a1
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length ⊒ βˆ€ (P : PredName) (ds : List D), predVarOccursIn P ds.length (pred_var_ h1_X h1_ts) β†’ (I.pred_var_ P ds ↔ J.pred_var_ P ds)
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : predVarOccursIn X ds.length (pred_var_ h1_X h1_ts) ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [predVarOccursIn] at a1
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : predVarOccursIn X ds.length (pred_var_ h1_X h1_ts) ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : X = h1_X ∧ ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
cases a1
case h2 D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1 : X = h1_X ∧ ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
case h2.intro D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D left✝ : X = h1_X right✝ : ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
subst a1_left
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) h1_1 : h1_X = P β†’ Β¬h1_ts.length = zs.length X : PredName ds : List D a1_left : X = h1_X a1_right : ds.length = h1_ts.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
apply h3_var
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ I.pred_var_ X ds ↔ J.pred_var_ X ds
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ Β¬(X = P ∧ ds.length = zs.length)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ Β¬(X = P ∧ ds.length = zs.length)
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ X = P β†’ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
intro a2
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length ⊒ X = P β†’ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length a2 : X = P ⊒ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
subst a2
case a D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_ts : List VarName V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) X : PredName ds : List D a1_right : ds.length = h1_ts.length h1_1 : X = P β†’ Β¬h1_ts.length = zs.length a2 : X = P ⊒ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : X = X β†’ Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp at h1_1
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : X = X β†’ Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
intro contra
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length ⊒ Β¬ds.length = zs.length
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
apply h1_1
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ False
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
trans List.length ds
case a D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = zs.length
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = ds.length D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = zs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [eq_comm]
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ h1_ts.length = ds.length
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = h1_ts.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact a1_right
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = h1_ts.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
exact contra
D : Type I J : Interpretation D E : Env A : Formula zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h1_ts : List VarName V : VarAssignment D X : PredName ds : List D a1_right : ds.length = h1_ts.length h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = X ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h2 : βˆ€ (Q : PredName) (ds : List D), Q = X ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ X ds) h1_1 : Β¬h1_ts.length = zs.length contra : ds.length = zs.length ⊒ ds.length = zs.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
obtain s1 := Sub.Var.All.Rec.substitution_theorem D I V E (Function.updateListITE id zs h1_ts) H h1_2
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
obtain s2 := Function.updateListITE_comp id V zs h1_ts
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp only [s2] at s1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s1 : Holds D I (V ∘ Function.updateListITE id zs h1_ts) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE (V ∘ id) zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
simp at s1
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE (V ∘ id) zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Pred/One/Ind/Sub.lean
FOL.NV.Sub.Pred.One.Ind.substitution_theorem
[131, 1]
[245, 15]
specialize h2 h1_X (List.map V h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D h2 : βˆ€ (Q : PredName) (ds : List D), Q = P ∧ ds.length = zs.length β†’ (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds) s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)
D : Type I J : Interpretation D E : Env A : Formula P : PredName zs : List VarName H B : Formula h3_const : I.pred_const_ = J.pred_const_ h3_var : βˆ€ (Q : PredName) (ds : List D), Β¬(Q = P ∧ ds.length = zs.length) β†’ (I.pred_var_ Q ds ↔ J.pred_var_ Q ds) h1_X : PredName h1_ts : List VarName h1_1 : h1_X = P ∧ h1_ts.length = zs.length h1_2 : Var.All.Rec.admits (Function.updateListITE id zs h1_ts) H V : VarAssignment D s2 : V ∘ Function.updateListITE id zs h1_ts = Function.updateListITE (V ∘ id) zs (List.map V h1_ts) s1 : Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) h2 : h1_X = P ∧ (List.map V h1_ts).length = zs.length β†’ (Holds D I (Function.updateListITE V zs (List.map V h1_ts)) E H ↔ J.pred_var_ P (List.map V h1_ts)) ⊒ Holds D I V E (Var.All.Rec.fastReplaceFree (Function.updateListITE id zs h1_ts) H) ↔ Holds D J V E (pred_var_ P h1_ts)