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stringclasses 147
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stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
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stringlengths 1
11.2k
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stringlengths 3
2.09M
| state_after
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2.09M
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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply Forall_spec_id' (List.ofFn args_u) | case a.a.a.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))) | case a.a.a.h1.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.axiom_ | case a.a.a.h1.h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))) | case a.a.a.h1.h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsAxiom
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact IsAxiom.eq_2_pred_const_ name n args_u args_v | case a.a.a.h1.h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsAxiom
(Forall_ (List.ofFn args_u)
(Forall_ (List.ofFn args_v)
((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_
((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | clear h2 | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u))
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | clear h3 | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [And_] | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | induction n | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
args_u args_v : Fin n → VarName
h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) | case a.a.zero
P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
case a.a.succ
P_r P_s : Formula
r s : VarName
name : PredName
n✝ : ℕ
a✝ :
∀ (args_u args_v : Fin n✝ → VarName),
(∀ (i : Fin n✝), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n✝ + 1) → VarName
h1_1 : ∀ (i : Fin (n✝ + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | case _ =>
simp
SC | P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp | P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) | P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ true_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | P_r P_s : Formula
r s : VarName
name : PredName
args_u args_v : Fin 0 → VarName
h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ true_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply
IsDeduct.mp_
((eq_ r s).imp_
(List.foldr and_ true_
(List.ofFn fun (i : Fin n) => eq_ (args_u i.succ) (args_v i.succ)))) | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))) | case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))))
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))))
case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_and_] | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).and_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).imp_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅
(((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_
(((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_
((eq_ r s).imp_
((eq_ (args_u 0) (args_v 0)).imp_
(List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | specialize h1_1 0 | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : args_u 0 = args_v 0 ∨ args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h1_1 | case a.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : args_u 0 = args_v 0 ∨ args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | case a.a.inl
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h✝ : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
case a.a.inr
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h✝ : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | case _ c1 =>
apply IsDeduct.mp_ (eq_ (args_u 0) (args_v 0))
case _ =>
SC
case _ =>
simp only [c1]
apply specId (args_v 0)
apply IsDeduct.axiom_
apply IsAxiom.eq_1_ | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | case _ c1 =>
cases c1
case _ c1_left c1_right =>
subst c1_left
subst c1_right
SC | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ (eq_ (args_u 0) (args_v 0)) | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | case _ =>
SC | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | case _ =>
simp only [c1]
apply specId (args_v 0)
apply IsDeduct.axiom_
apply IsAxiom.eq_1_ | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [c1] | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0)) | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_v 0) (args_v 0)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply specId (args_v 0) | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (eq_ (args_v 0) (args_v 0)) | case h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.axiom_ | case h1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsDeduct ∅ (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0))) | case h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsAxiom (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsAxiom.eq_1_ | case h1.a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = args_v 0
⊢ IsAxiom (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases c1 | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1 : args_u 0 = r ∧ args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | case intro
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
left✝ : args_u 0 = r
right✝ : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | case _ c1_left c1_right =>
subst c1_left
subst c1_right
SC | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1_left : args_u 0 = r
c1_right : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | subst c1_left | P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
c1_left : args_u 0 = r
c1_right : args_v 0 = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) | P_r P_s : Formula
s : VarName
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
c1_right : args_v 0 = s
ih :
∀ (args_u_1 args_v : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v i ∨ args_u_1 i = args_u 0 ∧ args_v i = s) →
IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (eq_ (args_u 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | subst c1_right | P_r P_s : Formula
s : VarName
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
c1_right : args_v 0 = s
ih :
∀ (args_u_1 args_v : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v i ∨ args_u_1 i = args_u 0 ∧ args_v i = s) →
IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (eq_ (args_u 0) (args_v 0))) | P_r P_s : Formula
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
ih :
∀ (args_u_1 args_v_1 : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v_1 i ∨ args_u_1 i = args_u 0 ∧ args_v_1 i = args_v 0) →
IsDeduct ∅
((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v_1 i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (eq_ (args_u 0) (args_v 0))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | P_r P_s : Formula
name : PredName
n : ℕ
args_u args_v : Fin (n + 1) → VarName
ih :
∀ (args_u_1 args_v_1 : Fin n → VarName),
(∀ (i : Fin n), args_u_1 i = args_v_1 i ∨ args_u_1 i = args_u 0 ∧ args_v_1 i = args_v 0) →
IsDeduct ∅
((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v_1 i))))
⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (eq_ (args_u 0) (args_v 0))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply ih | case a
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))) | case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ ∀ (i : Fin n), args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | intro i | case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
⊢ ∀ (i : Fin n), args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s | case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
i : Fin n
⊢ args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply h1_1 | case a.h1_1
P_r P_s : Formula
r s : VarName
name : PredName
n : ℕ
ih :
∀ (args_u args_v : Fin n → VarName),
(∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) →
IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
args_u args_v : Fin (n + 1) → VarName
h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s
i : Fin n
⊢ args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h2 | P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u.not_
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)) | P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h3 | P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u.not_
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)) | P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | specialize h1_ih h2 h3 | P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)) | P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ ((eq_ r s).imp_ (P_u.iff_ P_v)) | P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)) | case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (P_u.iff_ P_v)).imp_ ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)))
case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_iff_] | case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (P_u.iff_ P_v)).imp_ ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))) | case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).and_ (P_v.not_.imp_ P_u.not_)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_and_] | case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).and_ (P_v.not_.imp_ P_u.not_)))) | case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).imp_ (P_v.not_.imp_ P_u.not_).not_).not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).imp_ (P_v.not_.imp_ P_u.not_).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h1_ih | case a
P_r P_s : Formula
r s : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h2 : ¬isBoundIn r P_u
h3 : ¬isBoundIn s P_u
h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h2 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬isBoundIn r (P_u.imp_ Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬(isBoundIn r P_u ∨ isBoundIn r Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h2 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬(isBoundIn r P_u ∨ isBoundIn r Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2 : ¬isBoundIn r P_u ∧ ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h2 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2 : ¬isBoundIn r P_u ∧ ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | case intro
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
left✝ : ¬isBoundIn r P_u
right✝ : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h3 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬(isBoundIn s P_u ∨ isBoundIn s Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h3 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬(isBoundIn s P_u ∨ isBoundIn s Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3 : ¬isBoundIn s P_u ∧ ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h3 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3 : ¬isBoundIn s P_u ∧ ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | case intro
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
left✝ : ¬isBoundIn s P_u
right✝ : ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | specialize h1_ih_1 h2_left h3_left | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | specialize h1_ih_2 h2_right h3_right | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ ((eq_ r s).imp_ (Q_u.iff_ Q_v)) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))))
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (Q_u.iff_ Q_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ ((eq_ r s).imp_ (P_u.iff_ P_v)) | case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))) | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (P_u.iff_ P_v)).imp_
(((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))))
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_iff_] | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (P_u.iff_ P_v)).imp_
(((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))))) | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).and_ (Q_v.imp_ Q_u))).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).and_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_and_] | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).and_ (Q_v.imp_ Q_u))).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).and_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)))))) | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).imp_ (Q_v.imp_ Q_u).not_).not_).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).imp_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).imp_ (Q_v.imp_ Q_u).not_).not_).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).imp_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h1_ih_1 | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h1_ih_2 | case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (Q_u.iff_ Q_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h2 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r (forall_ x P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬(r = x ∨ isBoundIn r P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h2 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬(r = x ∨ isBoundIn r P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2 : r ≠ x ∧ ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h2 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2 : r ≠ x ∧ ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case intro
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
left✝ : r ≠ x
right✝ : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h3 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬(s = x ∨ isBoundIn s P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h3 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬(s = x ∨ isBoundIn s P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3 : s ≠ x ∧ ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h3 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3 : s ≠ x ∧ ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case intro
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
left✝ : s ≠ x
right✝ : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply deduction_theorem | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct (∅ ∪ {eq_ r s}) ((forall_ x P_u).iff_ (forall_ x P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct (∅ ∪ {eq_ r s}) ((forall_ x P_u).iff_ (forall_ x P_v)) | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x P_u).iff_ (forall_ x P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ (forall_ x (P_u.iff_ P_v)) | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x P_u).iff_ (forall_ x P_v)) | case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (forall_ x (P_u.iff_ P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply proof_imp_deduct | case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply T_18_1 | case h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ (eq_ r s) | case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (forall_ x (P_u.iff_ P_v)) | case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (eq_ r s) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply proof_imp_deduct | case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))) | case h1.a.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))) | case h1.a.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))) | case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))).imp_ ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))))
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply T_19_TS_21_left | case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))).imp_ ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬isFreeIn x (eq_ r s) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isFreeIn] | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬isFreeIn x (eq_ r s) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬(x = r ∨ x = s) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬(x = r ∨ x = s) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r ∧ x ≠ s |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | constructor | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r ∧ x ≠ s | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ s |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [ne_comm] | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ r ≠ x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h2_left | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ r ≠ x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [ne_comm] | case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ s | case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ s ≠ x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h3_left | case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ s ≠ x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply generalization | case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ∀ H ∈ ∅, ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h1_ih h2_right h3_right | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | intro H a1 | case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ∀ H ∈ ∅, ¬isFreeIn x H | case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp at a1 | case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.assume_ | case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (eq_ r s) | case h1.a.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ eq_ r s ∈ {eq_ r s} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp | case h1.a.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ eq_ r s ∈ {eq_ r s} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | sorry | case exists_
P_r P_s : Formula
r s x✝ : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r (exists_ x✝ P_u✝)
h3 : ¬isBoundIn s (exists_ x✝ P_u✝)
⊢ IsProof ((eq_ r s).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.left_id_left_inverse | [74, 1] | [83, 22] | simp only [Function.LeftInverse] | α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ LeftInverse g f | α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.left_id_left_inverse | [74, 1] | [83, 22] | intro x | α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x | α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.left_id_left_inverse | [74, 1] | [83, 22] | exact congrFun h1 x | α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.right_id_right_inverse | [86, 1] | [94, 45] | simp only [Function.RightInverse] | α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ RightInverse g f | α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.right_id_right_inverse | [86, 1] | [94, 45] | exact Function.left_id_left_inverse g f h1 | α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | funext x | α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
⊢ updateITE f a b = Function.updateITE' f a b | case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | simp only [Function.updateITE] | case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x | case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | simp only [Function.updateITE'] | case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x | case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | split_ifs | case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x | case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : x = a
h✝ : a = x
⊢ b = b
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : x = a
h✝ : ¬a = x
⊢ b = f x
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : ¬x = a
h✝ : a = x
⊢ f x = b
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : ¬x = a
h✝ : ¬a = x
⊢ f x = f x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | case _ c1 c2 =>
rfl | α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : a = x
⊢ b = b | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | case _ c1 c2 =>
subst c1
contradiction | α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : ¬a = x
⊢ b = f x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | case _ c1 c2 =>
subst c2
contradiction | α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : a = x
⊢ f x = b | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateITE_eq_Function.updateITE' | [99, 1] | [121, 8] | case _ c1 c2 =>
rfl | α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : ¬a = x
⊢ f x = f x | no goals |
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