internlm/Lean-Github huggingface dataset:
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : name = hd.name ∧ args.length = hd.args.length
c2 : d ∈ tl
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔
    Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
Tactic:
cases h3
Proof state after:
case intro
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : name = hd.name ∧ args.length = hd.args.length
c2 : d ∈ tl
left✝ : name = d.name
right✝ : args.length = d.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔
    Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : name = hd.name ∧ args.length = hd.args.length
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔
    Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
Tactic:
cases c1
Proof state after:
case intro
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
left✝ : name = hd.name
right✝ : args.length = hd.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔
    Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔
    Holds D I (Function.updateListITE V hd.args (List.map V args)) M tl hd.F
Tactic:
exfalso
Proof state after:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ False
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ False
Tactic:
apply h1_left d c2
Proof state after:
case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ hd.name = d.name

case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ hd.args.length = d.args.length
Theorem: MM0.lem_4
Proof state before:
case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ hd.name = d.name
Tactic:
simp only [← c1_left]
Proof state after:
case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ name = d.name
Theorem: MM0.lem_4
Proof state before:
case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ name = d.name
Tactic:
exact h3_left
Proof state after:
no goals
Theorem: MM0.lem_4
Proof state before:
case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ hd.args.length = d.args.length
Tactic:
simp only [← c1_right]
Proof state after:
case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ args.length = d.args.length
Theorem: MM0.lem_4
Proof state before:
case a
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c2 : d ∈ tl
h3_left : name = d.name
h3_right : args.length = d.args.length
c1_left : name = hd.name
c1_right : args.length = hd.args.length
⊢ args.length = d.args.length
Tactic:
exact h3_right
Proof state after:
no goals
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h2 : d = hd ∨ d ∈ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : ¬(name = hd.name ∧ args.length = hd.args.length)
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
cases h2
Proof state after:
case inl
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : ¬(name = hd.name ∧ args.length = hd.args.length)
h✝ : d = hd
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)

case inr
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : ¬(name = hd.name ∧ args.length = hd.args.length)
h✝ : d ∈ tl
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : ¬(name = hd.name ∧ args.length = hd.args.length)
c2 : d = hd
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
case inl c2 =>
  subst c2
  simp at c1
  cases h3
  case intro h3_left h3_right =>
    exfalso
    exact c1 h3_left h3_right
Proof state after:
no goals
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : ¬(name = hd.name ∧ args.length = hd.args.length)
c2 : d ∈ tl
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
case inr c2 =>
  specialize ih h1_right_right c2
  simp only [← ih]
  apply holds_coincide_env D I (Function.updateListITE V d.args (List.map V args)) M tl (hd :: tl) d.F s2
  apply def_in_well_formed_env_is_meta_var_or_all_def_in_env tl d h1_right_right c2
  exact s1
Proof state after:
no goals
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
hd : Definition
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
s1 : Env.Nodup (hd :: tl)
s2 : ∃ E1, hd :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, hd.name = d'.name → hd.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl hd.F
h1_right_right : Env.WellFormed tl
c1 : ¬(name = hd.name ∧ args.length = hd.args.length)
c2 : d = hd
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (hd :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
subst c2
Proof state after:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : ¬(name = d.name ∧ args.length = d.args.length)
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : ¬(name = d.name ∧ args.length = d.args.length)
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
simp at c1
Proof state after:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : name = d.name → ¬args.length = d.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
h3 : name = d.name ∧ args.length = d.args.length
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : name = d.name → ¬args.length = d.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
cases h3
Proof state after:
case intro
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : name = d.name → ¬args.length = d.args.length
left✝ : name = d.name
right✝ : args.length = d.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : name = d.name → ¬args.length = d.args.length
h3_left : name = d.name
h3_right : args.length = d.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
case intro h3_left h3_right =>
  exfalso
  exact c1 h3_left h3_right
Proof state after:
no goals
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : name = d.name → ¬args.length = d.args.length
h3_left : name = d.name
h3_right : args.length = d.args.length
⊢ Holds D I (Function.updateListITE V d.args (List.map V args)) M (d :: tl) d.F ↔ Holds D I V M tl (def_ name args)
Tactic:
exfalso
Proof state after:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : name = d.name → ¬args.length = d.args.length
h3_left : name = d.name
h3_right : args.length = d.args.length
⊢ False
Theorem: MM0.lem_4
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M : MetaValuation D
d : Definition
name : DefName
args : List VarName
tl : List Definition
ih :
  Env.WellFormed tl →
    d ∈ tl →
      (Holds D I (Function.updateListITE V d.args (List.map V args)) M tl d.F ↔ Holds D I V M tl (def_ name args))
h1_right_right : Env.WellFormed tl
s1 : Env.Nodup (d :: tl)
s2 : ∃ E1, d :: tl = E1 ++ tl
h1_left : ∀ d' ∈ tl, d.name = d'.name → d.args.length = d'.args.length → False
h1_right_left : IsMetaVarOrAllDefInEnv tl d.F
c1 : name = d.name → ¬args.length = d.args.length
h3_left : name = d.name
h3_right : args.length = d.args.length
⊢ False
Tactic:
exact c1 h3_left h3_right
Proof state after:
no goals