Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
start
stringlengths
6
10
end
stringlengths
6
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2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
case _ c1 => cases h2 case inl c2 => subst c2 simp at c1 cases h3 case intro h3_left h3_right => exfalso exact c1 h3_left h3_right case inr c2 => specialize ih h1_right_right c2 simp only [← ih] apply holds_coincide_env D I (Function.updateListITE V d.args (List.map V args)) M tl (hd :: tl) d.F s2 apply def_in_well_formed_env_is_meta_var_or_all_def_in_env tl d h1_right_right c2 exact s1
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
cases h2
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
case inl D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length h✝ : d = hd ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F case inr D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length h✝ : d ∈ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
case inl c2 => subst c2 exact holds_coincide_env D I (Function.updateListITE V d.args (List.map V args)) M tl (d :: tl) d.F s2 h1_right_left s1
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length c2 : d = hd ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
subst c2
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length c2 : d = hd ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name ∧ args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
exact holds_coincide_env D I (Function.updateListITE V d.args (List.map V args)) M tl (d :: tl) d.F s2 h1_right_left s1
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name ∧ args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
cases h3
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length c2 : d ∈ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
case intro D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length c2 : d ∈ tl left✝ : name = d.name right✝ : args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
cases c1
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : name = hd.name ∧ args.length = hd.args.length c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
case intro D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length left✝ : name = hd.name right✝ : args.length = hd.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
exfalso
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
apply h1_left d c2
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ False
case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ hd.name = d.name case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ hd.args.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
simp only [← c1_left]
case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ hd.name = d.name
case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ name = d.name
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
exact h3_left
case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ name = d.name
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
simp only [← c1_right]
case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ hd.args.length = d.args.length
case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ args.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
exact h3_right
case a D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c2 : d ∈ tl h3_left : name = d.name h3_right : args.length = d.args.length c1_left : name = hd.name c1_right : args.length = hd.args.length ⊒ args.length = d.args.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
cases h2
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h2 : d = hd ∨ d ∈ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
case inl D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) h✝ : d = hd ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args) case inr D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) h✝ : d ∈ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
case inl c2 => subst c2 simp at c1 cases h3 case intro h3_left h3_right => exfalso exact c1 h3_left h3_right
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d = hd ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
case inr c2 => specialize ih h1_right_right c2 simp only [← ih] apply holds_coincide_env D I (Function.updateListITE V d.args (List.map V args)) M tl (hd :: tl) d.F s2 apply def_in_well_formed_env_is_meta_var_or_all_def_in_env tl d h1_right_right c2 exact s1
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
subst c2
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d = hd ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : Β¬(name = d.name ∧ args.length = d.args.length) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
simp at c1
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : Β¬(name = d.name ∧ args.length = d.args.length) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name β†’ Β¬args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
cases h3
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name β†’ Β¬args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
case intro D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name β†’ Β¬args.length = d.args.length left✝ : name = d.name right✝ : args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
case intro h3_left h3_right => exfalso exact c1 h3_left h3_right
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name β†’ Β¬args.length = d.args.length h3_left : name = d.name h3_right : args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
exfalso
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name β†’ Β¬args.length = d.args.length h3_left : name = d.name h3_right : args.length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name β†’ Β¬args.length = d.args.length h3_left : name = d.name h3_right : args.length = d.args.length ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
exact c1 h3_left h3_right
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) h1_right_right : Env.WellFormed tl s1 : Env.Nodup (d :: tl) s2 : βˆƒ E1, d :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, d.name = d'.name β†’ d.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl d.F c1 : name = d.name β†’ Β¬args.length = d.args.length h3_left : name = d.name h3_right : args.length = d.args.length ⊒ False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
specialize ih h1_right_right c2
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition ih : Env.WellFormed tl β†’ d ∈ tl β†’ (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args)) s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
simp only [← ih]
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
apply holds_coincide_env D I (Function.updateListITE V d.args (List.map V args)) M tl (hd :: tl) d.F s2
D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F
case h2 D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ IsMetaVarOrAllDefInEnv tl d.F case h3 D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Env.Nodup (hd :: tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
apply def_in_well_formed_env_is_meta_var_or_all_def_in_env tl d h1_right_right c2
case h2 D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ IsMetaVarOrAllDefInEnv tl d.F case h3 D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Env.Nodup (hd :: tl)
case h3 D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Env.Nodup (hd :: tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.lem_4
[1935, 1]
[1999, 21]
exact s1
case h3 D : Type I : Interpretation D V : Valuation D M : MetaValuation D d : Definition name : DefName args : List VarName h3 : name = d.name ∧ args.length = d.args.length hd : Definition tl : List Definition s1 : Env.Nodup (hd :: tl) s2 : βˆƒ E1, hd :: tl = E1 ++ tl h1_left : βˆ€ d' ∈ tl, hd.name = d'.name β†’ hd.args.length = d'.args.length β†’ False h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F h1_right_right : Env.WellFormed tl c1 : Β¬(name = hd.name ∧ args.length = hd.args.length) c2 : d ∈ tl ih : Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args) ⊒ Env.Nodup (hd :: tl)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
induction h2 generalizing V
D : Type I : Interpretation D V : Valuation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2 : IsConv E F F' ⊒ Holds D I V M E F ↔ Holds D I V M E F'
case conv_refl D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed phi✝ : Formula V : Valuation D ⊒ Holds D I V M E phi✝ ↔ Holds D I V M E phi✝ case conv_symm D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed phi✝ phi'✝ : Formula a✝ : IsConv E phi✝ phi'✝ a_ih✝ : βˆ€ (V : Valuation D), Holds D I V M E phi✝ ↔ Holds D I V M E phi'✝ V : Valuation D ⊒ Holds D I V M E phi'✝ ↔ Holds D I V M E phi✝ case conv_trans D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed phi✝ phi'✝ phi''✝ : Formula a✝¹ : IsConv E phi✝ phi'✝ a✝ : IsConv E phi'✝ phi''✝ a_ih✝¹ : βˆ€ (V : Valuation D), Holds D I V M E phi✝ ↔ Holds D I V M E phi'✝ a_ih✝ : βˆ€ (V : Valuation D), Holds D I V M E phi'✝ ↔ Holds D I V M E phi''✝ V : Valuation D ⊒ Holds D I V M E phi✝ ↔ Holds D I V M E phi''✝ case conv_not D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed phi✝ phi'✝ : Formula a✝ : IsConv E phi✝ phi'✝ a_ih✝ : βˆ€ (V : Valuation D), Holds D I V M E phi✝ ↔ Holds D I V M E phi'✝ V : Valuation D ⊒ Holds D I V M E phi✝.not_ ↔ Holds D I V M E phi'✝.not_ case conv_imp D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed phi✝ phi'✝ psi✝ psi'✝ : Formula a✝¹ : IsConv E phi✝ phi'✝ a✝ : IsConv E psi✝ psi'✝ a_ih✝¹ : βˆ€ (V : Valuation D), Holds D I V M E phi✝ ↔ Holds D I V M E phi'✝ a_ih✝ : βˆ€ (V : Valuation D), Holds D I V M E psi✝ ↔ Holds D I V M E psi'✝ V : Valuation D ⊒ Holds D I V M E (phi✝.imp_ psi✝) ↔ Holds D I V M E (phi'✝.imp_ psi'✝) case conv_forall D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed x✝ : VarName phi✝ phi'✝ : Formula a✝ : IsConv E phi✝ phi'✝ a_ih✝ : βˆ€ (V : Valuation D), Holds D I V M E phi✝ ↔ Holds D I V M E phi'✝ V : Valuation D ⊒ Holds D I V M E (forall_ x✝ phi✝) ↔ Holds D I V M E (forall_ x✝ phi'✝) case conv_unfold D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d✝ : Definition Οƒβœ : Instantiation a✝ : d✝ ∈ E V : Valuation D ⊒ Holds D I V M E (def_ d✝.name (List.map (β†‘Οƒβœ) d✝.args)) ↔ Holds D I V M E (sub Οƒβœ meta_var_ d✝.F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
case conv_refl h2_phi => rfl
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi : Formula V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
case conv_symm h2_phi h2_phi' _ h2_ih => symm exact h2_ih V
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
case conv_trans h2_phi h2_phi' h2_phi'' _ _ h2_ih_1 h2_ih_2 => trans Holds D I V M E h2_phi' exact h2_ih_1 V exact h2_ih_2 V
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi''
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
case conv_not h2_phi h2_phi' _ h2_ih => simp only [Holds] congr! 1 exact h2_ih V
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi.not_ ↔ Holds D I V M E h2_phi'.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
case conv_forall h2_x h2_phi h2_phi' _ h2_ih => simp only [Holds] apply forall_congr' intro a exact h2_ih (Function.updateITE V h2_x a)
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E (forall_ h2_x h2_phi) ↔ Holds D I V M E (forall_ h2_x h2_phi')
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
case conv_unfold d Οƒ h2 => obtain βŸ¨Οƒ', a1⟩ := Οƒ.2 have s1 : IsMetaVarOrAllDefInEnv E d.F := def_in_well_formed_env_is_meta_var_or_all_def_in_env E d h1 h2 simp only [← holds_sub D I V M E Οƒ Οƒ' meta_var_ d.F s1 a1] clear s1 have s2 : d.name = d.name ∧ (List.map Οƒ.val d.args).length = d.args.length simp simp only [← lem_4 D I V M E d d.name (List.map Οƒ.val d.args) h1 h2 s2] clear s2 have s3 : d.F.metaVarSet = βˆ… := no_meta_var_imp_meta_var_set_is_empty d.F d.args d.nf simp only [holds_coincide_meta_var_no_meta_var D I (V ∘ Οƒ.val) (fun (X' : MetaVarName) (V' : Valuation D) => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) M E d.F s3] clear s3 apply holds_coincide_var D I (Function.updateListITE V d.args (List.map V (List.map Οƒ.val d.args))) (V ∘ Οƒ.val) M E d.F d.args d.nf intro v a2 simp exact Function.updateListITE_map_mem V (V ∘ Οƒ.val) d.args v a2
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I V M E (sub Οƒ meta_var_ d.F)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
rfl
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi : Formula V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
symm
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact h2_ih V
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
trans Holds D I V M E h2_phi'
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi''
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi''
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact h2_ih_1 V
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi''
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi''
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact h2_ih_2 V
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_phi'' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_phi' h2_phi'' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi'' V : Valuation D ⊒ Holds D I V M E h2_phi' ↔ Holds D I V M E h2_phi''
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi.not_ ↔ Holds D I V M E h2_phi'.not_
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Β¬Holds D I V M E h2_phi ↔ Β¬Holds D I V M E h2_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
congr! 1
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Β¬Holds D I V M E h2_phi ↔ Β¬Holds D I V M E h2_phi'
case a.h.e'_1.a D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact h2_ih V
case a.h.e'_1.a D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_psi h2_psi' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_psi h2_psi' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi' V : Valuation D ⊒ Holds D I V M E (h2_phi.imp_ h2_psi) ↔ Holds D I V M E (h2_phi'.imp_ h2_psi')
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_psi h2_psi' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_psi h2_psi' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi' V : Valuation D ⊒ Holds D I V M E h2_phi β†’ Holds D I V M E h2_psi ↔ Holds D I V M E h2_phi' β†’ Holds D I V M E h2_psi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
congr! 1
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_psi h2_psi' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_psi h2_psi' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi' V : Valuation D ⊒ Holds D I V M E h2_phi β†’ Holds D I V M E h2_psi ↔ Holds D I V M E h2_phi' β†’ Holds D I V M E h2_psi'
case a.h.a D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_psi h2_psi' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_psi h2_psi' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' case a.h'.a D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_psi h2_psi' : Formula a✝² : IsConv E h2_phi h2_phi' a✝¹ : IsConv E h2_psi h2_psi' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi' V : Valuation D a✝ : Holds D I V M E h2_phi' ⊒ Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact h2_ih_1 V
case a.h.a D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_psi h2_psi' : Formula a✝¹ : IsConv E h2_phi h2_phi' a✝ : IsConv E h2_psi h2_psi' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi' V : Valuation D ⊒ Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact h2_ih_2 V
case a.h'.a D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_phi h2_phi' h2_psi h2_psi' : Formula a✝² : IsConv E h2_phi h2_phi' a✝¹ : IsConv E h2_psi h2_psi' h2_ih_1 : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' h2_ih_2 : βˆ€ (V : Valuation D), Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi' V : Valuation D a✝ : Holds D I V M E h2_phi' ⊒ Holds D I V M E h2_psi ↔ Holds D I V M E h2_psi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ Holds D I V M E (forall_ h2_x h2_phi) ↔ Holds D I V M E (forall_ h2_x h2_phi')
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE V h2_x d) M E h2_phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V h2_x d) M E h2_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
apply forall_congr'
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE V h2_x d) M E h2_phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V h2_x d) M E h2_phi'
case h D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V h2_x a) M E h2_phi ↔ Holds D I (Function.updateITE V h2_x a) M E h2_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
intro a
case h D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V h2_x a) M E h2_phi ↔ Holds D I (Function.updateITE V h2_x a) M E h2_phi'
case h D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D a : D ⊒ Holds D I (Function.updateITE V h2_x a) M E h2_phi ↔ Holds D I (Function.updateITE V h2_x a) M E h2_phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact h2_ih (Function.updateITE V h2_x a)
case h D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed h2_x : VarName h2_phi h2_phi' : Formula a✝ : IsConv E h2_phi h2_phi' h2_ih : βˆ€ (V : Valuation D), Holds D I V M E h2_phi ↔ Holds D I V M E h2_phi' V : Valuation D a : D ⊒ Holds D I (Function.updateITE V h2_x a) M E h2_phi ↔ Holds D I (Function.updateITE V h2_x a) M E h2_phi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
obtain βŸ¨Οƒ', a1⟩ := Οƒ.2
D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I V M E (sub Οƒ meta_var_ d.F)
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I V M E (sub Οƒ meta_var_ d.F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
have s1 : IsMetaVarOrAllDefInEnv E d.F := def_in_well_formed_env_is_meta_var_or_all_def_in_env E d h1 h2
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I V M E (sub Οƒ meta_var_ d.F)
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s1 : IsMetaVarOrAllDefInEnv E d.F ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I V M E (sub Οƒ meta_var_ d.F)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp only [← holds_sub D I V M E Οƒ Οƒ' meta_var_ d.F s1 a1]
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s1 : IsMetaVarOrAllDefInEnv E d.F ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I V M E (sub Οƒ meta_var_ d.F)
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s1 : IsMetaVarOrAllDefInEnv E d.F ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
clear s1
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s1 : IsMetaVarOrAllDefInEnv E d.F ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
have s2 : d.name = d.name ∧ (List.map Οƒ.val d.args).length = d.args.length
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
case s2 D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s2 : d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp
case s2 D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s2 : d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s2 : d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp only [← lem_4 D I V M E d d.name (List.map Οƒ.val d.args) h1 h2 s2]
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s2 : d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length ⊒ Holds D I V M E (def_ d.name (List.map (↑σ) d.args)) ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s2 : d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
clear s2
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s2 : d.name = d.name ∧ (List.map (↑σ) d.args).length = d.args.length ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
have s3 : d.F.metaVarSet = βˆ… := no_meta_var_imp_meta_var_set_is_empty d.F d.args d.nf
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s3 : d.F.metaVarSet = βˆ… ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp only [holds_coincide_meta_var_no_meta_var D I (V ∘ Οƒ.val) (fun (X' : MetaVarName) (V' : Valuation D) => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) M E d.F s3]
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s3 : d.F.metaVarSet = βˆ… ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (meta_var_ X')) E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s3 : d.F.metaVarSet = βˆ… ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) M E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
clear s3
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id s3 : d.F.metaVarSet = βˆ… ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) M E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) M E d.F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
apply holds_coincide_var D I (Function.updateListITE V d.args (List.map V (List.map Οƒ.val d.args))) (V ∘ Οƒ.val) M E d.F d.args d.nf
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I (Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args))) M E d.F ↔ Holds D I (V ∘ ↑σ) M E d.F
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ βˆ€ v ∈ d.args, Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args)) v = (V ∘ ↑σ) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
intro v a2
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ βˆ€ v ∈ d.args, Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args)) v = (V ∘ ↑σ) v
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id v : VarName a2 : v ∈ d.args ⊒ Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args)) v = (V ∘ ↑σ) v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
simp
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id v : VarName a2 : v ∈ d.args ⊒ Function.updateListITE V d.args (List.map V (List.map (↑σ) d.args)) v = (V ∘ ↑σ) v
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id v : VarName a2 : v ∈ d.args ⊒ Function.updateListITE V d.args (List.map (V ∘ ↑σ) d.args) v = V (↑σ v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_conv
[2002, 1]
[2057, 67]
exact Function.updateListITE_map_mem V (V ∘ Οƒ.val) d.args v a2
case intro D : Type I : Interpretation D M : MetaValuation D E : Env F F' : Formula h1 : E.WellFormed d : Definition Οƒ : Instantiation h2 : d ∈ E V : Valuation D Οƒ' : VarName β†’ VarName a1 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id v : VarName a2 : v ∈ d.args ⊒ Function.updateListITE V d.args (List.map (V ∘ ↑σ) d.args) v = V (↑σ v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
induction h1 generalizing M
D : Type I : Interpretation D M : MetaValuation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h1 : IsProof E Ξ“ Ξ” F h2 : E.WellFormed nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E F
case hyp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ : Formula a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : phi✝ ∈ Ξ”βœ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E phi✝ case mp D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula a✝¹ : IsProof E Ξ“βœ Ξ”βœ phi✝ a✝ : IsProof E Ξ“βœ Ξ”βœ (phi✝.imp_ psi✝) a_ih✝¹ : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E phi✝ a_ih✝ : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (phi✝.imp_ psi✝) M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E psi✝ case prop_1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : IsMetaVarOrAllDefInEnv E psi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (phi✝.imp_ (psi✝.imp_ phi✝)) case prop_2 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ chi✝ : Formula a✝² : IsMetaVarOrAllDefInEnv E phi✝ a✝¹ : IsMetaVarOrAllDefInEnv E psi✝ a✝ : IsMetaVarOrAllDefInEnv E chi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((phi✝.imp_ (psi✝.imp_ chi✝)).imp_ ((phi✝.imp_ psi✝).imp_ (phi✝.imp_ chi✝))) case prop_3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : IsMetaVarOrAllDefInEnv E psi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((phi✝.not_.imp_ psi✝.not_).imp_ (psi✝.imp_ phi✝)) case gen D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ : Formula x✝ : VarName a✝ : IsProof E Ξ“βœ Ξ”βœ phi✝ a_ih✝ : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E phi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (forall_ x✝ phi✝) case pred_1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ psi✝ : Formula x✝ : VarName a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : IsMetaVarOrAllDefInEnv E psi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((forall_ x✝ (phi✝.imp_ psi✝)).imp_ ((forall_ x✝ phi✝).imp_ (forall_ x✝ psi✝))) case pred_2 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ : Formula x✝ : VarName a✝¹ : IsMetaVarOrAllDefInEnv E phi✝ a✝ : NotFree Ξ“βœ x✝ phi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (phi✝.imp_ (forall_ x✝ phi✝)) case eq_1 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula x✝ y✝ : VarName a✝ : Β¬y✝ = x✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (exists_ x✝ (eq_ x✝ y✝)) case eq_2 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula x✝ y✝ z✝ : VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((eq_ x✝ y✝).imp_ ((eq_ x✝ z✝).imp_ (eq_ y✝ z✝))) case eq_3 D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula n✝ : β„• name✝ : PredName xs✝ ys✝ : Fin n✝ β†’ VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (eqSubPred name✝ n✝ xs✝ ys✝) case thm D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ Ξ“'✝ : List (VarName Γ— MetaVarName) Ξ”βœ Ξ”'✝ : List Formula phi✝ : Formula Οƒβœ : Instantiation Ο„βœ : MetaInstantiation a✝³ : βˆ€ X ∈ phi✝.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο„βœ X) a✝² : βˆ€ (x : VarName) (X : MetaVarName), (x, X) ∈ Ξ“βœ β†’ NotFree Ξ“'✝ (β†‘Οƒβœ x) (Ο„βœ X) a✝¹ : βˆ€ psi ∈ Ξ”βœ, IsProof E Ξ“'✝ Ξ”'✝ (sub Οƒβœ Ο„βœ psi) a✝ : IsProof E Ξ“βœ Ξ”βœ phi✝ a_ih✝¹ : βˆ€ psi ∈ Ξ”βœ, βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“'✝ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”'✝ β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (sub Οƒβœ Ο„βœ psi) a_ih✝ : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E phi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“'✝ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”'✝ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (sub Οƒβœ Ο„βœ phi✝) case conv D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed Ξ“βœ : List (VarName Γ— MetaVarName) Ξ”βœ : List Formula phi✝ phi'✝ : Formula a✝² : IsMetaVarOrAllDefInEnv E phi'✝ a✝¹ : IsProof E Ξ“βœ Ξ”βœ phi✝ a✝ : IsConv E phi✝ phi'✝ a_ih✝ : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E phi✝ M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ Ξ“βœ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ Ξ”βœ β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E phi'✝
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case hyp h1_Ξ“ h1_Ξ” h1_phi h1_1 h1_2 => intro V exact hyp h1_phi V h1_2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : h1_phi ∈ h1_Ξ” M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case mp h1_Ξ“ h1_Ξ” h1_phi h1_psi h1_1 h1_2 h1_ih_1 h1_ih_2 => simp only [Holds] at h1_ih_2 intro V exact h1_ih_2 M nf hyp V (h1_ih_1 M nf hyp V)
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_2 : IsProof E h1_Ξ“ h1_Ξ” (h1_phi.imp_ h1_psi) h1_ih_1 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ h1_psi) M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case prop_1 h1_Ξ“ h1_Ξ” h1_phi h1_psi h1_1 h1_2 => simp only [Holds] intro V a1 a2 exact a1
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (h1_psi.imp_ h1_phi))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case prop_2 h1_Ξ“ h1_Ξ” h1_phi h1_psi h1_chi h1_1 h1_2 h1_3 => simp only [Holds] intro V a1 a2 a3 apply a1 a3 exact a2 a3
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case prop_3 h1_Ξ“ h1_Ξ” h1_phi h1_psi h1_1 h1_2 => simp only [Holds] intro V a1 a2 by_contra contra exact a1 contra a2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case gen h1_Ξ“ h1_Ξ” h1_phi h1_x h1_1 h1_ih => simp only [Holds] intro V d exact h1_ih M nf hyp (Function.updateITE V h1_x d)
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (forall_ h1_x h1_phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case pred_1 h1_Ξ“ h1_Ξ” h1_phi h1_psi h1_x h1_1 h1_2 => simp only [Holds] intro V a1 a2 d apply a1 d exact a2 d
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((forall_ h1_x (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_x h1_phi).imp_ (forall_ h1_x h1_psi)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case pred_2 h1_Ξ“ h1_Ξ” h1_phi h1_x h1_1 h1_2 => have s1 : IsNotFree D I M E h1_phi h1_x apply not_free_imp_is_not_free D I M E h1_phi h1_Ξ“ h1_x h1_2 exact nf h1_x simp only [Holds] intro V a1 a unfold IsNotFree at s1 simp only [← s1 V a] exact a1
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : NotFree h1_Ξ“ h1_x h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case eq_1 h1_Ξ“ h1_Ξ” h1_x h1_y h1_1 => unfold exists_ simp only [Holds] simp intro V apply Exists.intro (V h1_y) unfold Function.updateITE simp
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_x h1_y : VarName h1_1 : Β¬h1_y = h1_x M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (exists_ h1_x (eq_ h1_x h1_y))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case eq_3 h1_Ξ“ h1_Ξ” h1_n h1_name h1_xs h1_ys => sorry
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_n : β„• h1_name : PredName h1_xs h1_ys : Fin h1_n β†’ VarName M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (eqSubPred h1_name h1_n h1_xs h1_ys)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
case conv h1_Ξ“ h1_Ξ” h1_phi h1_phi' h1_1 h1_2 h1_3 h1_ih => intro V have s1 : Holds D I V M E h1_phi := h1_ih M nf hyp V simp only [← holds_conv D I V M E h1_phi h1_phi' h2 h1_3] exact s1
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_phi' : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi' h1_2 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_3 : IsConv E h1_phi h1_phi' h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : h1_phi ∈ h1_Ξ” M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : h1_phi ∈ h1_Ξ” M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ Holds D I V M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact hyp h1_phi V h1_2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : h1_phi ∈ h1_Ξ” M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D ⊒ Holds D I V M E h1_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds] at h1_ih_2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_2 : IsProof E h1_Ξ“ h1_Ξ” (h1_phi.imp_ h1_psi) h1_ih_1 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ h1_psi) M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_psi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_2 : IsProof E h1_Ξ“ h1_Ξ” (h1_phi.imp_ h1_psi) h1_ih_1 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_2 : IsProof E h1_Ξ“ h1_Ξ” (h1_phi.imp_ h1_psi) h1_ih_1 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_psi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_2 : IsProof E h1_Ξ“ h1_Ξ” (h1_phi.imp_ h1_psi) h1_ih_1 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi V : Valuation D ⊒ Holds D I V M E h1_psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact h1_ih_2 M nf hyp V (h1_ih_1 M nf hyp V)
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_2 : IsProof E h1_Ξ“ h1_Ξ” (h1_phi.imp_ h1_psi) h1_ih_1 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F h1_ih_2 : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi V : Valuation D ⊒ Holds D I V M E h1_psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (h1_psi.imp_ h1_phi))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V a1 a2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Holds D I V M E h1_phi a2 : Holds D I V M E h1_psi ⊒ Holds D I V M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact a1
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Holds D I V M E h1_phi a2 : Holds D I V M E h1_psi ⊒ Holds D I V M E h1_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi)))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), (Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_chi) β†’ (Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi) β†’ Holds D I V M E h1_phi β†’ Holds D I V M E h1_chi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V a1 a2 a3
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), (Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_chi) β†’ (Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi) β†’ Holds D I V M E h1_phi β†’ Holds D I V M E h1_chi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_chi a2 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi a3 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_chi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
apply a1 a3
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_chi a2 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi a3 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_chi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_chi a2 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi a3 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact a2 a3
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi h1_chi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi h1_3 : IsMetaVarOrAllDefInEnv E h1_chi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_chi a2 : Holds D I V M E h1_phi β†’ Holds D I V M E h1_psi a3 : Holds D I V M E h1_phi ⊒ Holds D I V M E h1_psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), (Β¬Holds D I V M E h1_phi β†’ Β¬Holds D I V M E h1_psi) β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V a1 a2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), (Β¬Holds D I V M E h1_phi β†’ Β¬Holds D I V M E h1_psi) β†’ Holds D I V M E h1_psi β†’ Holds D I V M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Β¬Holds D I V M E h1_phi β†’ Β¬Holds D I V M E h1_psi a2 : Holds D I V M E h1_psi ⊒ Holds D I V M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
by_contra contra
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Β¬Holds D I V M E h1_phi β†’ Β¬Holds D I V M E h1_psi a2 : Holds D I V M E h1_psi ⊒ Holds D I V M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Β¬Holds D I V M E h1_phi β†’ Β¬Holds D I V M E h1_psi a2 : Holds D I V M E h1_psi contra : Β¬Holds D I V M E h1_phi ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact a1 contra a2
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : Β¬Holds D I V M E h1_phi β†’ Β¬Holds D I V M E h1_psi a2 : Holds D I V M E h1_psi contra : Β¬Holds D I V M E h1_phi ⊒ False
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E (forall_ h1_x h1_phi)
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D) (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V d
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D) (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D d : D ⊒ Holds D I (Function.updateITE V h1_x d) M E h1_phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
exact h1_ih M nf hyp (Function.updateITE V h1_x d)
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi : Formula h1_x : VarName h1_1 : IsProof E h1_Ξ“ h1_Ξ” h1_phi h1_ih : βˆ€ (M : MetaValuation D), (βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v) β†’ (βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F) β†’ βˆ€ (V : Valuation D), Holds D I V M E h1_phi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D d : D ⊒ Holds D I (Function.updateITE V h1_x d) M E h1_phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
simp only [Holds]
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), Holds D I V M E ((forall_ h1_x (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_x h1_phi).imp_ (forall_ h1_x h1_psi)))
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), (βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi β†’ Holds D I (Function.updateITE V h1_x d) M E h1_psi) β†’ (βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi) β†’ βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
intro V a1 a2 d
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F ⊒ βˆ€ (V : Valuation D), (βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi β†’ Holds D I (Function.updateITE V h1_x d) M E h1_psi) β†’ (βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi) β†’ βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_psi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi β†’ Holds D I (Function.updateITE V h1_x d) M E h1_psi a2 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi d : D ⊒ Holds D I (Function.updateITE V h1_x d) M E h1_psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_is_proof
[2060, 1]
[2162, 13]
apply a1 d
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi β†’ Holds D I (Function.updateITE V h1_x d) M E h1_psi a2 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi d : D ⊒ Holds D I (Function.updateITE V h1_x d) M E h1_psi
D : Type I : Interpretation D E : Env Ξ“ : List (VarName Γ— MetaVarName) Ξ” : List Formula F : Formula h2 : E.WellFormed h1_Ξ“ : List (VarName Γ— MetaVarName) h1_Ξ” : List Formula h1_phi h1_psi : Formula h1_x : VarName h1_1 : IsMetaVarOrAllDefInEnv E h1_phi h1_2 : IsMetaVarOrAllDefInEnv E h1_psi M : MetaValuation D nf : βˆ€ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Ξ“ β†’ IsNotFree D I M E (meta_var_ X) v hyp : βˆ€ (F : Formula) (V : Valuation D), F ∈ h1_Ξ” β†’ Holds D I V M E F V : Valuation D a1 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi β†’ Holds D I (Function.updateITE V h1_x d) M E h1_psi a2 : βˆ€ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi d : D ⊒ Holds D I (Function.updateITE V h1_x d) M E h1_phi