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
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end
stringlengths
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2.09M
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2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_holds_imp_def_in_env
[1287, 1]
[1320, 29]
apply Exists.intro d
D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ βˆƒ d ∈ E_hd :: E_tl, name = d.name ∧ args.length = d.args.length
D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d ∈ E_hd :: E_tl ∧ name = d.name ∧ args.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_holds_imp_def_in_env
[1287, 1]
[1320, 29]
constructor
D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d ∈ E_hd :: E_tl ∧ name = d.name ∧ args.length = d.args.length
case left D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d ∈ E_hd :: E_tl case right D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ name = d.name ∧ args.length = d.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_holds_imp_def_in_env
[1287, 1]
[1320, 29]
simp
case left D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d ∈ E_hd :: E_tl
case left D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d = E_hd ∨ d ∈ E_tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_holds_imp_def_in_env
[1287, 1]
[1320, 29]
right
case left D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d = E_hd ∨ d ∈ E_tl
case left.h D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d ∈ E_tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_holds_imp_def_in_env
[1287, 1]
[1320, 29]
exact E_ih_1_left
case left.h D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ d ∈ E_tl
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.def_holds_imp_def_in_env
[1287, 1]
[1320, 29]
exact E_ih_1_right
case right D : Type I : Interpretation D V : Valuation D M : MetaValuation D name : DefName args : List VarName E_hd : Definition E_tl : List Definition c1 : Β¬(name = E_hd.name ∧ args.length = E_hd.args.length) h1 : Holds D I V M E_tl (def_ name args) E_ih : βˆƒ d ∈ E_tl, name = d.name ∧ args.length = d.args.length d : Definition E_ih_1_left : d ∈ E_tl E_ih_1_right : name = d.name ∧ args.length = d.args.length ⊒ name = d.name ∧ args.length = d.args.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
induction F generalizing V
D : Type I : Interpretation D V : Valuation D M : MetaValuation D E E' : Env F : Formula h1 : βˆƒ E1, E' = E1 ++ E h2 : IsMetaVarOrAllDefInEnv E F h3 : E'.Nodup ⊒ Holds D I V M E' F ↔ Holds D I V M E F
case meta_var_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup a✝ : MetaVarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (meta_var_ a✝) ⊒ Holds D I V M E' (meta_var_ a✝) ↔ Holds D I V M E (meta_var_ a✝) case pred_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup a✝¹ : PredName a✝ : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (pred_ a✝¹ a✝) ⊒ Holds D I V M E' (pred_ a✝¹ a✝) ↔ Holds D I V M E (pred_ a✝¹ a✝) case eq_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup a✝¹ a✝ : VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (eq_ a✝¹ a✝) ⊒ Holds D I V M E' (eq_ a✝¹ a✝) ↔ Holds D I V M E (eq_ a✝¹ a✝) case true_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup V : Valuation D h2 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I V M E' true_ ↔ Holds D I V M E true_ case not_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup a✝ : Formula a_ih✝ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝ β†’ (Holds D I V M E' a✝ ↔ Holds D I V M E a✝) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E a✝.not_ ⊒ Holds D I V M E' a✝.not_ ↔ Holds D I V M E a✝.not_ case imp_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝¹ β†’ (Holds D I V M E' a✝¹ ↔ Holds D I V M E a✝¹) a_ih✝ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝ β†’ (Holds D I V M E' a✝ ↔ Holds D I V M E a✝) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (a✝¹.imp_ a✝) ⊒ Holds D I V M E' (a✝¹.imp_ a✝) ↔ Holds D I V M E (a✝¹.imp_ a✝) case forall_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝ β†’ (Holds D I V M E' a✝ ↔ Holds D I V M E a✝) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (forall_ a✝¹ a✝) ⊒ Holds D I V M E' (forall_ a✝¹ a✝) ↔ Holds D I V M E (forall_ a✝¹ a✝) case def_ D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup a✝¹ : DefName a✝ : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ a✝¹ a✝) ⊒ Holds D I V M E' (def_ a✝¹ a✝) ↔ Holds D I V M E (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case meta_var_ X => simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : MetaVarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (meta_var_ X) ⊒ Holds D I V M E' (meta_var_ X) ↔ Holds D I V M E (meta_var_ X)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case pred_ X xs => simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : PredName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ Holds D I V M E' (pred_ X xs) ↔ Holds D I V M E (pred_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case eq_ x y => simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x y : VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ Holds D I V M E' (eq_ x y) ↔ Holds D I V M E (eq_ x y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case true_ => simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup V : Valuation D h2 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I V M E' true_ ↔ Holds D I V M E true_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case not_ phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h2 simp only [Holds] congr! 1 exact phi_ih V h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi.not_ ⊒ Holds D I V M E' phi.not_ ↔ Holds D I V M E phi.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case forall_ x phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h2 simp only [Holds] apply forall_congr' intro a exact phi_ih (Function.updateITE V x a) h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (forall_ x phi) ⊒ Holds D I V M E' (forall_ x phi) ↔ Holds D I V M E (forall_ x phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : MetaVarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (meta_var_ X) ⊒ Holds D I V M E' (meta_var_ X) ↔ Holds D I V M E (meta_var_ X)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : PredName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ Holds D I V M E' (pred_ X xs) ↔ Holds D I V M E (pred_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x y : VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ Holds D I V M E' (eq_ x y) ↔ Holds D I V M E (eq_ x y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup V : Valuation D h2 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I V M E' true_ ↔ Holds D I V M E true_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
unfold IsMetaVarOrAllDefInEnv at h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi.not_ ⊒ Holds D I V M E' phi.not_ ↔ Holds D I V M E phi.not_
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I V M E' phi.not_ ↔ Holds D I V M E phi.not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I V M E' phi.not_ ↔ Holds D I V M E phi.not_
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Β¬Holds D I V M E' phi ↔ Β¬Holds D I V M E phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
congr! 1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Β¬Holds D I V M E' phi ↔ Β¬Holds D I V M E phi
case a.h.e'_1.a D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I V M E' phi ↔ Holds D I V M E phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
exact phi_ih V h2
case a.h.e'_1.a D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I V M E' phi ↔ Holds D I V M E phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
unfold IsMetaVarOrAllDefInEnv at h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (phi.imp_ psi) ⊒ Holds D I V M E' (phi.imp_ psi) ↔ Holds D I V M E (phi.imp_ psi)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' (phi.imp_ psi) ↔ Holds D I V M E (phi.imp_ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
cases h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' (phi.imp_ psi) ↔ Holds D I V M E (phi.imp_ psi)
case intro D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D left✝ : IsMetaVarOrAllDefInEnv E phi right✝ : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' (phi.imp_ psi) ↔ Holds D I V M E (phi.imp_ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2_left : IsMetaVarOrAllDefInEnv E phi h2_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' (phi.imp_ psi) ↔ Holds D I V M E (phi.imp_ psi)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2_left : IsMetaVarOrAllDefInEnv E phi h2_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' phi β†’ Holds D I V M E' psi ↔ Holds D I V M E phi β†’ Holds D I V M E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
congr! 1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2_left : IsMetaVarOrAllDefInEnv E phi h2_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' phi β†’ Holds D I V M E' psi ↔ Holds D I V M E phi β†’ Holds D I V M E psi
case a.h.a D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2_left : IsMetaVarOrAllDefInEnv E phi h2_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' phi ↔ Holds D I V M E phi case a.h'.a D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2_left : IsMetaVarOrAllDefInEnv E phi h2_right : IsMetaVarOrAllDefInEnv E psi a✝ : Holds D I V M E phi ⊒ Holds D I V M E' psi ↔ Holds D I V M E psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
exact phi_ih V h2_left
case a.h.a D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2_left : IsMetaVarOrAllDefInEnv E phi h2_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I V M E' phi ↔ Holds D I V M E phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
exact psi_ih V h2_right
case a.h'.a D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I V M E' psi ↔ Holds D I V M E psi) V : Valuation D h2_left : IsMetaVarOrAllDefInEnv E phi h2_right : IsMetaVarOrAllDefInEnv E psi a✝ : Holds D I V M E phi ⊒ Holds D I V M E' psi ↔ Holds D I V M E psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
unfold IsMetaVarOrAllDefInEnv at h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (forall_ x phi) ⊒ Holds D I V M E' (forall_ x phi) ↔ Holds D I V M E (forall_ x phi)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I V M E' (forall_ x phi) ↔ Holds D I V M E (forall_ x phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I V M E' (forall_ x phi) ↔ Holds D I V M E (forall_ x phi)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE V x d) M E' phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V x d) M E phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
apply forall_congr'
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE V x d) M E' phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V x d) M E phi
case h D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V x a) M E' phi ↔ Holds D I (Function.updateITE V x a) M E phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
intro a
case h D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V x a) M E' phi ↔ Holds D I (Function.updateITE V x a) M E phi
case h D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi a : D ⊒ Holds D I (Function.updateITE V x a) M E' phi ↔ Holds D I (Function.updateITE V x a) M E phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
exact phi_ih (Function.updateITE V x a) h2
case h D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I V M E' phi ↔ Holds D I V M E phi) V : Valuation D h2 : IsMetaVarOrAllDefInEnv E phi a : D ⊒ Holds D I (Function.updateITE V x a) M E' phi ↔ Holds D I (Function.updateITE V x a) M E phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
apply Exists.elim h1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ X xs) ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ X xs) ⊒ βˆ€ (a : Env), E' = a ++ E β†’ (Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
intro E1 h1_1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ X xs) ⊒ βˆ€ (a : Env), E' = a ++ E β†’ (Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs))
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ X xs) E1 : Env h1_1 : E' = E1 ++ E ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
clear h1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h1 : βˆƒ E1, E' = E1 ++ E h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ X xs) E1 : Env h1_1 : E' = E1 ++ E ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ X xs) E1 : Env h1_1 : E' = E1 ++ E ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
unfold IsMetaVarOrAllDefInEnv at h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D h2 : IsMetaVarOrAllDefInEnv E (def_ X xs) E1 : Env h1_1 : E' = E1 ++ E ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E h2 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
apply Exists.elim h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E h2 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E h2 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ βˆ€ (a : Definition), a ∈ E ∧ X = a.name ∧ xs.length = a.args.length β†’ (Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
intro a h2_1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E h2 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length ⊒ βˆ€ (a : Definition), a ∈ E ∧ X = a.name ∧ xs.length = a.args.length β†’ (Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs))
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E h2 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length a : Definition h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
clear h2
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E h2 : βˆƒ d ∈ E, X = d.name ∧ xs.length = d.args.length a : Definition h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
unfold Env.Nodup at h3
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : E'.Nodup X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) E' X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
cases h2_1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) E' X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
case intro D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) E' X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition left✝ : a ∈ E right✝ : X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
cases h2_1_right
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) E' X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition h2_1_left : a ∈ E h2_1_right : X = a.name ∧ xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
case intro D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) E' X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition h2_1_left : a ∈ E left✝ : X = a.name right✝ : xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
subst h1_1
D : Type I : Interpretation D M : MetaValuation D E E' : Env h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) E' X : DefName xs : List VarName V : Valuation D E1 : Env h1_1 : E' = E1 ++ E a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length ⊒ Holds D I V M E' (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D E1 : Env a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1 ++ E) ⊒ Holds D I V M (E1 ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
induction E1
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D E1 : Env a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1 ++ E) ⊒ Holds D I V M (E1 ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
case nil D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) ([] ++ E) ⊒ Holds D I V M ([] ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs) case cons D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length head✝ : Definition tail✝ : List Definition tail_ih✝ : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (tail✝ ++ E) β†’ (Holds D I V M (tail✝ ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (head✝ :: tail✝ ++ E) ⊒ Holds D I V M (head✝ :: tail✝ ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case nil => simp
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) ([] ++ E) ⊒ Holds D I V M ([] ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) ([] ++ E) ⊒ Holds D I V M ([] ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp at h3
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_hd :: E1_tl ++ E) ⊒ Holds D I V M (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3 : (βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length) ∧ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ Holds D I V M (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
cases h3
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3 : (βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length) ∧ List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ Holds D I V M (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
case intro D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) left✝ : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length right✝ : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ Holds D I V M (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ Holds D I V M (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ Holds D I V M (E1_hd :: (E1_tl ++ E)) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ Holds D I V M (E1_hd :: (E1_tl ++ E)) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ (if X = E1_hd.name ∧ xs.length = E1_hd.args.length then Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) M (E1_tl ++ E) E1_hd.F else Holds D I V M (E1_tl ++ E) (def_ X xs)) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
split_ifs
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) ⊒ (if X = E1_hd.name ∧ xs.length = E1_hd.args.length then Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) M (E1_tl ++ E) E1_hd.F else Holds D I V M (E1_tl ++ E) (def_ X xs)) ↔ Holds D I V M E (def_ X xs)
case pos D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) h✝ : X = E1_hd.name ∧ xs.length = E1_hd.args.length ⊒ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) M (E1_tl ++ E) E1_hd.F ↔ Holds D I V M E (def_ X xs) case neg D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) h✝ : Β¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length) ⊒ Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
case _ c1 => exact E1_ih h3_right
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1 : Β¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length) ⊒ Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
cases c1
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1 : X = E1_hd.name ∧ xs.length = E1_hd.args.length ⊒ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) M (E1_tl ++ E) E1_hd.F ↔ Holds D I V M E (def_ X xs)
case intro D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) left✝ : X = E1_hd.name right✝ : xs.length = E1_hd.args.length ⊒ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) M (E1_tl ++ E) E1_hd.F ↔ Holds D I V M E (def_ X xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
exfalso
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) M (E1_tl ++ E) E1_hd.F ↔ Holds D I V M E (def_ X xs)
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ False
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
apply h3_left a
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ False
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ a ∈ E1_tl ∨ a ∈ E case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.name = a.name case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.args.length = a.args.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
right
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ a ∈ E1_tl ∨ a ∈ E
case a.h D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ a ∈ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
exact h2_1_left
case a.h D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ a ∈ E
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [← h2_1_right_left]
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.name = a.name
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.name = X
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [c1_left]
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.name = X
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [← h2_1_right_right]
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.args.length = a.args.length
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.args.length = xs.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
simp only [c1_right]
case a D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1_left : X = E1_hd.name c1_right : xs.length = E1_hd.args.length ⊒ E1_hd.args.length = xs.length
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_coincide_env
[1379, 1]
[1462, 35]
exact E1_ih h3_right
D : Type I : Interpretation D M : MetaValuation D E : Env X : DefName xs : List VarName V : Valuation D a : Definition h2_1_left : a ∈ E h2_1_right_left : X = a.name h2_1_right_right : xs.length = a.args.length E1_hd : Definition E1_tl : List Definition E1_ih : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ d1.args.length = d2.args.length β†’ False) (E1_tl ++ E) β†’ (Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)) h3_left : βˆ€ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E β†’ E1_hd.name = a'.name β†’ Β¬E1_hd.args.length = a'.args.length h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name β†’ Β¬d1.args.length = d2.args.length) (E1_tl ++ E) c1 : Β¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length) ⊒ Holds D I V M (E1_tl ++ E) (def_ X xs) ↔ Holds D I V M E (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
induction F generalizing V
D : Type I : Interpretation D V : Valuation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation F : Formula h1 : IsMetaVarOrAllDefInEnv E F h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E F ↔ Holds D I V M E (sub Οƒ Ο„ F)
case meta_var_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id a✝ : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ a✝) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ a✝) ↔ Holds D I V M E (sub Οƒ Ο„ (meta_var_ a✝)) case pred_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id a✝¹ : PredName a✝ : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (pred_ a✝¹ a✝) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (pred_ a✝¹ a✝) ↔ Holds D I V M E (sub Οƒ Ο„ (pred_ a✝¹ a✝)) case eq_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id a✝¹ a✝ : VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (eq_ a✝¹ a✝) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (eq_ a✝¹ a✝) ↔ Holds D I V M E (sub Οƒ Ο„ (eq_ a✝¹ a✝)) case true_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id V : Valuation D h1 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E true_ ↔ Holds D I V M E (sub Οƒ Ο„ true_) case not_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id a✝ : Formula a_ih✝ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝ β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E a✝ ↔ Holds D I V M E (sub Οƒ Ο„ a✝)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E a✝.not_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E a✝.not_ ↔ Holds D I V M E (sub Οƒ Ο„ a✝.not_) case imp_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id a✝¹ a✝ : Formula a_ih✝¹ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝¹ β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E a✝¹ ↔ Holds D I V M E (sub Οƒ Ο„ a✝¹)) a_ih✝ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝ β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E a✝ ↔ Holds D I V M E (sub Οƒ Ο„ a✝)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (a✝¹.imp_ a✝) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (a✝¹.imp_ a✝) ↔ Holds D I V M E (sub Οƒ Ο„ (a✝¹.imp_ a✝)) case forall_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id a✝¹ : VarName a✝ : Formula a_ih✝ : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E a✝ β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E a✝ ↔ Holds D I V M E (sub Οƒ Ο„ a✝)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (forall_ a✝¹ a✝) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ a✝¹ a✝) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ a✝¹ a✝)) case def_ D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id a✝¹ : DefName a✝ : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (def_ a✝¹ a✝) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (def_ a✝¹ a✝) ↔ Holds D I V M E (sub Οƒ Ο„ (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case meta_var_ X => cases h2 case intro h2_left h2_right => unfold sub simp only [Holds] simp only [Function.comp.assoc] simp only [h2_left] simp only [Function.comp_id]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ X) ↔ Holds D I V M E (sub Οƒ Ο„ (meta_var_ X))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case pred_ X xs => unfold sub simp only [Holds] simp
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : PredName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (pred_ X xs) ↔ Holds D I V M E (sub Οƒ Ο„ (pred_ X xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case eq_ x y => unfold sub simp only [Holds] simp
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x y : VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (eq_ x y) ↔ Holds D I V M E (sub Οƒ Ο„ (eq_ x y))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case true_ => unfold sub simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id V : Valuation D h1 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E true_ ↔ Holds D I V M E (sub Οƒ Ο„ true_)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case not_ phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h1 unfold sub simp only [Holds] congr! 1 exact phi_ih V h1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi.not_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi.not_ ↔ Holds D I V M E (sub Οƒ Ο„ phi.not_)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case forall_ x phi phi_ih => unfold IsMetaVarOrAllDefInEnv at h1 cases h2 case intro h2_left h2_right => unfold sub simp only [Holds] apply forall_congr' intro a have s1 : Function.updateITE V (Οƒ.val x) a ∘ Οƒ.val = Function.updateITE (V ∘ Οƒ.val) x a apply Function.updateITE_comp_right_injective apply Instantiation.is_injective simp only [← s1] exact phi_ih (Function.updateITE V (Οƒ.val x) a) h1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ x phi))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
cases h2
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ X) ↔ Holds D I V M E (sub Οƒ Ο„ (meta_var_ X))
case intro D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) left✝ : ↑σ ∘ Οƒ' = id right✝ : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ X) ↔ Holds D I V M E (sub Οƒ Ο„ (meta_var_ X))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case intro h2_left h2_right => unfold sub simp only [Holds] simp only [Function.comp.assoc] simp only [h2_left] simp only [Function.comp_id]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ X) ↔ Holds D I V M E (sub Οƒ Ο„ (meta_var_ X))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold sub
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ X) ↔ Holds D I V M E (sub Οƒ Ο„ (meta_var_ X))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ X) ↔ Holds D I V M E (Ο„ X)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (meta_var_ X) ↔ Holds D I V M E (Ο„ X)
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I ((V ∘ ↑σ) ∘ Οƒ') M E (Ο„ X) ↔ Holds D I V M E (Ο„ X)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Function.comp.assoc]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I ((V ∘ ↑σ) ∘ Οƒ') M E (Ο„ X) ↔ Holds D I V M E (Ο„ X)
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ ∘ Οƒ') M E (Ο„ X) ↔ Holds D I V M E (Ο„ X)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [h2_left]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ ∘ Οƒ') M E (Ο„ X) ↔ Holds D I V M E (Ο„ X)
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ id) M E (Ο„ X) ↔ Holds D I V M E (Ο„ X)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Function.comp_id]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation X : MetaVarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X) h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ id) M E (Ο„ X) ↔ Holds D I V M E (Ο„ X)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold sub
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : PredName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (pred_ X xs) ↔ Holds D I V M E (sub Οƒ Ο„ (pred_ X xs))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : PredName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (pred_ X xs) ↔ Holds D I V M E (pred_ X (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : PredName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (pred_ X xs) ↔ Holds D I V M E (pred_ X (List.map (↑σ) xs))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : PredName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ I X (List.map (V ∘ ↑σ) xs) ↔ I X (List.map V (List.map (↑σ) xs))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id X : PredName xs : List VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs) ⊒ I X (List.map (V ∘ ↑σ) xs) ↔ I X (List.map V (List.map (↑σ) xs))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold sub
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x y : VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (eq_ x y) ↔ Holds D I V M E (sub Οƒ Ο„ (eq_ x y))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x y : VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (eq_ x y) ↔ Holds D I V M E (eq_ (↑σ x) (↑σ y))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x y : VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (eq_ x y) ↔ Holds D I V M E (eq_ (↑σ x) (↑σ y))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x y : VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ (V ∘ ↑σ) x = (V ∘ ↑σ) y ↔ V (↑σ x) = V (↑σ y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x y : VarName V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (eq_ x y) ⊒ (V ∘ ↑σ) x = (V ∘ ↑σ) y ↔ V (↑σ x) = V (↑σ y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold sub
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id V : Valuation D h1 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E true_ ↔ Holds D I V M E (sub Οƒ Ο„ true_)
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id V : Valuation D h1 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E true_ ↔ Holds D I V M E true_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id V : Valuation D h1 : IsMetaVarOrAllDefInEnv E true_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E true_ ↔ Holds D I V M E true_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold IsMetaVarOrAllDefInEnv at h1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi.not_ ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi.not_ ↔ Holds D I V M E (sub Οƒ Ο„ phi.not_)
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi.not_ ↔ Holds D I V M E (sub Οƒ Ο„ phi.not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold sub
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi.not_ ↔ Holds D I V M E (sub Οƒ Ο„ phi.not_)
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi.not_ ↔ Holds D I V M E (sub Οƒ Ο„ phi).not_
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi.not_ ↔ Holds D I V M E (sub Οƒ Ο„ phi).not_
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Β¬Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Β¬Holds D I V M E (sub Οƒ Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
congr! 1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Β¬Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Β¬Holds D I V M E (sub Οƒ Ο„ phi)
case a.h.e'_1.a D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
exact phi_ih V h1
case a.h.e'_1.a D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold IsMetaVarOrAllDefInEnv at h1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (phi.imp_ psi) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (phi.imp_ psi) ↔ Holds D I V M E (sub Οƒ Ο„ (phi.imp_ psi))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (phi.imp_ psi) ↔ Holds D I V M E (sub Οƒ Ο„ (phi.imp_ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
cases h1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (phi.imp_ psi) ↔ Holds D I V M E (sub Οƒ Ο„ (phi.imp_ psi))
case intro D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D left✝ : IsMetaVarOrAllDefInEnv E phi right✝ : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (phi.imp_ psi) ↔ Holds D I V M E (sub Οƒ Ο„ (phi.imp_ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold sub
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (phi.imp_ psi) ↔ Holds D I V M E (sub Οƒ Ο„ (phi.imp_ psi))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (phi.imp_ psi) ↔ Holds D I V M E ((sub Οƒ Ο„ phi).imp_ (sub Οƒ Ο„ psi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
simp only [Holds]
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (phi.imp_ psi) ↔ Holds D I V M E ((sub Οƒ Ο„ phi).imp_ (sub Οƒ Ο„ psi))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi β†’ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ phi) β†’ Holds D I V M E (sub Οƒ Ο„ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
congr! 1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi β†’ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ phi) β†’ Holds D I V M E (sub Οƒ Ο„ psi)
case a.h.a D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi) case a.h'.a D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi a✝ : Holds D I V M E (sub Οƒ Ο„ phi) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
exact phi_ih V h1_left
case a.h.a D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
exact psi_ih V h1_right
case a.h'.a D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id phi psi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) psi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E psi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)) V : Valuation D h1_left : IsMetaVarOrAllDefInEnv E phi h1_right : IsMetaVarOrAllDefInEnv E psi a✝ : Holds D I V M E (sub Οƒ Ο„ phi) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E psi ↔ Holds D I V M E (sub Οƒ Ο„ psi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold IsMetaVarOrAllDefInEnv at h1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi) ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ x phi))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ x phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
cases h2
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation h2 : ↑σ ∘ Οƒ' = id ∧ Οƒ' ∘ ↑σ = id x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ x phi))
case intro D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi left✝ : ↑σ ∘ Οƒ' = id right✝ : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ x phi))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
case intro h2_left h2_right => unfold sub simp only [Holds] apply forall_congr' intro a have s1 : Function.updateITE V (Οƒ.val x) a ∘ Οƒ.val = Function.updateITE (V ∘ Οƒ.val) x a apply Function.updateITE_comp_right_injective apply Instantiation.is_injective simp only [← s1] exact phi_ih (Function.updateITE V (Οƒ.val x) a) h1
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ x phi))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/MM0/MM0.lean
MM0.holds_sub
[1465, 1]
[1577, 15]
unfold sub
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (sub Οƒ Ο„ (forall_ x phi))
D : Type I : Interpretation D M : MetaValuation D E : Env Οƒ : Instantiation Οƒ' : VarName β†’ VarName Ο„ : MetaInstantiation x : VarName phi : Formula phi_ih : βˆ€ (V : Valuation D), IsMetaVarOrAllDefInEnv E phi β†’ (Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E phi ↔ Holds D I V M E (sub Οƒ Ο„ phi)) V : Valuation D h1 : IsMetaVarOrAllDefInEnv E phi h2_left : ↑σ ∘ Οƒ' = id h2_right : Οƒ' ∘ ↑σ = id ⊒ Holds D I (V ∘ ↑σ) (fun X' V' => Holds D I (V' ∘ Οƒ') M E (Ο„ X')) E (forall_ x phi) ↔ Holds D I V M E (forall_ (↑σ x) (sub Οƒ Ο„ phi))