url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | apply Exists.intro a | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ ∃ x, (g ∘ f) x = c | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ (g ∘ f) a = c | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ ∃ x, (g ∘ f) x = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | rewrite [comp_def] | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ (g ∘ f) a = c | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ g (f a) = c | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ (g ∘ f) a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | rewrite [←h4] at h3 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ g (f a) = c | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
a : A
h3 : g (f a) = c
h4 : f a = b
⊢ g (f a) = c | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ g (f a) = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | show g (f a) = c from h3 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
a : A
h3 : g (f a) = c
h4 : f a = b
⊢ g (f a) = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
a : A
h3 : g (f a) = c
h4 : f a = b
⊢ g (f a) = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | rewrite [func_from_graph] | A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ ∃ g, graph g = inv (graph f) | A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ is_func_graph (inv (graph f)) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ ∃ g, graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | define | A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ is_func_graph (inv (graph f)) | A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ ∀ (x : B), ∃! y, (x, y) ∈ inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ is_func_graph (inv (graph f))
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | fix b : B | A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ ∀ (x : B), ∃! y, (x, y) ∈ inv (graph f) | A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∃! y, (b, y) ∈ inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
⊢ ∀ (x : B), ∃! y, (x, y) ∈ inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | exists_unique | A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∃! y, (b, y) ∈ inv (graph f) | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∃ y, (b, y) ∈ inv (graph f)
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∀ (y_1 y_2 : A), (b, y_1) ∈ inv (graph f) → (b, y_2) ∈ inv (graph f) → y_1 = y_2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∃! y, (b, y) ∈ inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | define at h2 | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∃ y, (b, y) ∈ inv (graph f) | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
⊢ ∃ y, (b, y) ∈ inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∃ y, (b, y) ∈ inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | obtain (a : A) (h4 : f a = b) from h2 b | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
⊢ ∃ y, (b, y) ∈ inv (graph f) | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ ∃ y, (b, y) ∈ inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
⊢ ∃ y, (b, y) ∈ inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | apply Exists.intro a | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ ∃ y, (b, y) ∈ inv (graph f) | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ (b, a) ∈ inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ ∃ y, (b, y) ∈ inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | define | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ (b, a) ∈ inv (graph f) | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ f a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ (b, a) ∈ inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | show f a = b from h4 | case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ f a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Existence
A B : Type
f : A → B
h1 : one_to_one f
h2 : ∀ (y : B), ∃ x, f x = y
b : B
a : A
h4 : f a = b
⊢ f a = b
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | fix a1 : A | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∀ (y_1 y_2 : A), (b, y_1) ∈ inv (graph f) → (b, y_2) ∈ inv (graph f) → y_1 = y_2 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 : A
⊢ ∀ (y_2 : A), (b, a1) ∈ inv (graph f) → (b, y_2) ∈ inv (graph f) → a1 = y_2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
⊢ ∀ (y_1 y_2 : A), (b, y_1) ∈ inv (graph f) → (b, y_2) ∈ inv (graph f) → y_1 = y_2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | fix a2 : A | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 : A
⊢ ∀ (y_2 : A), (b, a1) ∈ inv (graph f) → (b, y_2) ∈ inv (graph f) → a1 = y_2 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
⊢ (b, a1) ∈ inv (graph f) → (b, a2) ∈ inv (graph f) → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 : A
⊢ ∀ (y_2 : A), (b, a1) ∈ inv (graph f) → (b, y_2) ∈ inv (graph f) → a1 = y_2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | assume h3 : (b, a1) ∈ inv (graph f) | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
⊢ (b, a1) ∈ inv (graph f) → (b, a2) ∈ inv (graph f) → a1 = a2 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : (b, a1) ∈ inv (graph f)
⊢ (b, a2) ∈ inv (graph f) → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
⊢ (b, a1) ∈ inv (graph f) → (b, a2) ∈ inv (graph f) → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | assume h4 : (b, a2) ∈ inv (graph f) | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : (b, a1) ∈ inv (graph f)
⊢ (b, a2) ∈ inv (graph f) → a1 = a2 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : (b, a1) ∈ inv (graph f)
h4 : (b, a2) ∈ inv (graph f)
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : (b, a1) ∈ inv (graph f)
⊢ (b, a2) ∈ inv (graph f) → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | define at h3 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : (b, a1) ∈ inv (graph f)
h4 : (b, a2) ∈ inv (graph f)
⊢ a1 = a2 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = b
h4 : (b, a2) ∈ inv (graph f)
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : (b, a1) ∈ inv (graph f)
h4 : (b, a2) ∈ inv (graph f)
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | define at h4 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = b
h4 : (b, a2) ∈ inv (graph f)
⊢ a1 = a2 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = b
h4 : f a2 = b
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = b
h4 : (b, a2) ∈ inv (graph f)
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | rewrite [←h4] at h3 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = b
h4 : f a2 = b
⊢ a1 = a2 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = f a2
h4 : f a2 = b
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = b
h4 : f a2 = b
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | define at h1 | case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = f a2
h4 : f a2 = b
⊢ a1 = a2 | case Uniqueness
A B : Type
f : A → B
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = f a2
h4 : f a2 = b
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : one_to_one f
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = f a2
h4 : f a2 = b
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_1 | [190, 1] | [214, 7] | show a1 = a2 from h1 a1 a2 h3 | case Uniqueness
A B : Type
f : A → B
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = f a2
h4 : f a2 = b
⊢ a1 = a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case Uniqueness
A B : Type
f : A → B
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : onto f
b : B
a1 a2 : A
h3 : f a1 = f a2
h4 : f a2 = b
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | apply funext | A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
⊢ g ∘ f = id | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
⊢ ∀ (x : A), (g ∘ f) x = id x | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
⊢ g ∘ f = id
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | fix a : A | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
⊢ ∀ (x : A), (g ∘ f) x = id x | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (g ∘ f) a = id a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
⊢ ∀ (x : A), (g ∘ f) x = id x
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | have h2 : (f a, a) ∈ graph g := by
rewrite [h1] define rfl
done | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (g ∘ f) a = id a | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
h2 : (f a, a) ∈ graph g
⊢ (g ∘ f) a = id a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (g ∘ f) a = id a
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | define at h2 | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
h2 : (f a, a) ∈ graph g
⊢ (g ∘ f) a = id a | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
h2 : g (f a) = a
⊢ (g ∘ f) a = id a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
h2 : (f a, a) ∈ graph g
⊢ (g ∘ f) a = id a
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | show (g ∘ f) a = id a from h2 | case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
h2 : g (f a) = a
⊢ (g ∘ f) a = id a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
h2 : g (f a) = a
⊢ (g ∘ f) a = id a
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | rewrite [h1] | A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (f a, a) ∈ graph g | A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (f a, a) ∈ inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (f a, a) ∈ graph g
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | define | A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (f a, a) ∈ inv (graph f) | A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ f a = f a | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ (f a, a) ∈ inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_2_1 | [216, 1] | [227, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ f a = f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : graph g = inv (graph f)
a : A
⊢ f a = f a
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | define | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
⊢ one_to_one f | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
⊢ ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
⊢ one_to_one f
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | fix a1 : A | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
⊢ ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 : A
⊢ ∀ (x2 : A), f a1 = f x2 → a1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
⊢ ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | fix a2 : A | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 : A
⊢ ∀ (x2 : A), f a1 = f x2 → a1 = x2 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
⊢ f a1 = f a2 → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 : A
⊢ ∀ (x2 : A), f a1 = f x2 → a1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | assume h2 : f a1 = f a2 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
⊢ f a1 = f a2 → a1 = a2 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
⊢ f a1 = f a2 → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | show a1 = a2 from
calc a1
_ = id a1 := by rfl
_ = (g ∘ f) a1 := by rw [h1]
_ = g (f a1) := by rfl
_ = g (f a2) := by rw [h2]
_ = (g ∘ f) a2 := by rfl
_ = id a2 := by rw [h1]
_ = a2 := by rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ a1 = a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ a1 = id a1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ a1 = id a1
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | rw [h1] | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ id a1 = (g ∘ f) a1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ id a1 = (g ∘ f) a1
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ (g ∘ f) a1 = g (f a1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ (g ∘ f) a1 = g (f a1)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | rw [h2] | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ g (f a1) = g (f a2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ g (f a1) = g (f a2)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ g (f a2) = (g ∘ f) a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ g (f a2) = (g ∘ f) a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | rw [h1] | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ (g ∘ f) a2 = id a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ (g ∘ f) a2 = id a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_3_1 | [232, 1] | [246, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ id a2 = a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
a1 a2 : A
h2 : f a1 = f a2
⊢ id a2 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | have h3 : one_to_one f := Theorem_5_3_3_1 f g h1 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
⊢ graph g = inv (graph f) | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
⊢ graph g = inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
⊢ graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | have h4 : onto f := Theorem_5_3_3_2 f g h2 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
⊢ graph g = inv (graph f) | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
⊢ graph g = inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
⊢ graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | obtain (g' : B → A) (h5 : graph g' = inv (graph f))
from Theorem_5_3_1 f h3 h4 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
⊢ graph g = inv (graph f) | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
⊢ graph g = inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
⊢ graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | have h6 : g' ∘ f = id := Theorem_5_3_2_1 f g' h5 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
⊢ graph g = inv (graph f) | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ graph g = inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
⊢ graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | have h7 : g = g' :=
calc g
_ = id ∘ g := by rfl
_ = (g' ∘ f) ∘ g := by rw [h6]
_ = g' ∘ (f ∘ g) := by rfl
_ = g' ∘ id := by rw [h2]
_ = g' := by rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ graph g = inv (graph f) | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
h7 : g = g'
⊢ graph g = inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | rewrite [←h7] at h5 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
h7 : g = g'
⊢ graph g = inv (graph f) | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g = inv (graph f)
h6 : g' ∘ f = id
h7 : g = g'
⊢ graph g = inv (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
h7 : g = g'
⊢ graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | show graph g = inv (graph f) from h5 | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g = inv (graph f)
h6 : g' ∘ f = id
h7 : g = g'
⊢ graph g = inv (graph f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g = inv (graph f)
h6 : g' ∘ f = id
h7 : g = g'
⊢ graph g = inv (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ g = id ∘ g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ g = id ∘ g
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | rw [h6] | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ id ∘ g = (g' ∘ f) ∘ g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ id ∘ g = (g' ∘ f) ∘ g
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ (g' ∘ f) ∘ g = g' ∘ f ∘ g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ (g' ∘ f) ∘ g = g' ∘ f ∘ g
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | rw [h2] | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ g' ∘ f ∘ g = g' ∘ id | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ g' ∘ f ∘ g = g' ∘ id
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_3_5 | [251, 1] | [267, 7] | rfl | A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ g' ∘ id = g' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
g : B → A
h1 : g ∘ f = id
h2 : f ∘ g = id
h3 : one_to_one f
h4 : onto f
g' : B → A
h5 : graph g' = inv (graph f)
h6 : g' ∘ f = id
⊢ g' ∘ id = g'
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | set F : Set (Set A) := {D : Set A | B ⊆ D ∧ closed f D} | A : Type
f : A → A
B : Set A
⊢ ∃ C, closure f B C | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
⊢ ∃ C, closure f B C | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : A → A
B : Set A
⊢ ∃ C, closure f B C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | set C : Set A := ⋂₀ F | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
⊢ ∃ C, closure f B C | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∃ C, closure f B C | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
⊢ ∃ C, closure f B C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | apply Exists.intro C | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∃ C, closure f B C | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ closure f B C | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∃ C, closure f B C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ closure f B C | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ C ∈ {D | B ⊆ D ∧ closed f D} ∧ ∀ x ∈ {D | B ⊆ D ∧ closed f D}, sub A C x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ closure f B C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | apply And.intro | A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ C ∈ {D | B ⊆ D ∧ closed f D} ∧ ∀ x ∈ {D | B ⊆ D ∧ closed f D}, sub A C x | case left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ C ∈ {D | B ⊆ D ∧ closed f D}
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∀ x ∈ {D | B ⊆ D ∧ closed f D}, sub A C x | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ C ∈ {D | B ⊆ D ∧ closed f D} ∧ ∀ x ∈ {D | B ⊆ D ∧ closed f D}, sub A C x
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define | case left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ C ∈ {D | B ⊆ D ∧ closed f D} | case left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ B ⊆ C ∧ closed f C | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ C ∈ {D | B ⊆ D ∧ closed f D}
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | apply And.intro | case left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ B ⊆ C ∧ closed f C | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ B ⊆ C
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ closed f C | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ B ⊆ C ∧ closed f C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | fix a : A | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ B ⊆ C | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
⊢ a ∈ B → a ∈ C | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ B ⊆ C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | assume h1 : a ∈ B | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
⊢ a ∈ B → a ∈ C | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
⊢ a ∈ C | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
⊢ a ∈ B → a ∈ C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
⊢ a ∈ C | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
⊢ ∀ t ∈ F, a ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
⊢ a ∈ C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | fix D : Set A | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
⊢ ∀ t ∈ F, a ∈ t | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
⊢ D ∈ F → a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
⊢ ∀ t ∈ F, a ∈ t
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | assume h2 : D ∈ F | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
⊢ D ∈ F → a ∈ D | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
h2 : D ∈ F
⊢ a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
⊢ D ∈ F → a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define at h2 | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
h2 : D ∈ F
⊢ a ∈ D | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
h2 : B ⊆ D ∧ closed f D
⊢ a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
h2 : D ∈ F
⊢ a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | show a ∈ D from h2.left h1 | case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
h2 : B ⊆ D ∧ closed f D
⊢ a ∈ D | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ B
D : Set A
h2 : B ⊆ D ∧ closed f D
⊢ a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ closed f C | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∀ x ∈ C, f x ∈ C | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ closed f C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | fix a : A | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∀ x ∈ C, f x ∈ C | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
⊢ a ∈ C → f a ∈ C | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∀ x ∈ C, f x ∈ C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | assume h1 : a ∈ C | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
⊢ a ∈ C → f a ∈ C | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
⊢ f a ∈ C | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
⊢ a ∈ C → f a ∈ C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
⊢ f a ∈ C | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
⊢ ∀ t ∈ F, f a ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
⊢ f a ∈ C
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | fix D : Set A | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
⊢ ∀ t ∈ F, f a ∈ t | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
D : Set A
⊢ D ∈ F → f a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
⊢ ∀ t ∈ F, f a ∈ t
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | assume h2 : D ∈ F | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
D : Set A
⊢ D ∈ F → f a ∈ D | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
D : Set A
h2 : D ∈ F
⊢ f a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
D : Set A
⊢ D ∈ F → f a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define at h1 | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
D : Set A
h2 : D ∈ F
⊢ f a ∈ D | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : D ∈ F
⊢ f a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : a ∈ C
D : Set A
h2 : D ∈ F
⊢ f a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | have h3 : a ∈ D := h1 D h2 | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : D ∈ F
⊢ f a ∈ D | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : D ∈ F
h3 : a ∈ D
⊢ f a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : D ∈ F
⊢ f a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define at h2 | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : D ∈ F
h3 : a ∈ D
⊢ f a ∈ D | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
⊢ f a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : D ∈ F
h3 : a ∈ D
⊢ f a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | have h4 : closed f D := h2.right | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
⊢ f a ∈ D | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
h4 : closed f D
⊢ f a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
⊢ f a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define at h4 | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
h4 : closed f D
⊢ f a ∈ D | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
h4 : ∀ x ∈ D, f x ∈ D
⊢ f a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
h4 : closed f D
⊢ f a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | show f a ∈ D from h4 a h3 | case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
h4 : ∀ x ∈ D, f x ∈ D
⊢ f a ∈ D | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
a : A
h1 : ∀ t ∈ F, a ∈ t
D : Set A
h2 : B ⊆ D ∧ closed f D
h3 : a ∈ D
h4 : ∀ x ∈ D, f x ∈ D
⊢ f a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | fix D : Set A | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∀ x ∈ {D | B ⊆ D ∧ closed f D}, sub A C x | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
⊢ D ∈ {D | B ⊆ D ∧ closed f D} → sub A C D | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
⊢ ∀ x ∈ {D | B ⊆ D ∧ closed f D}, sub A C x
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | assume h1 : D ∈ F | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
⊢ D ∈ {D | B ⊆ D ∧ closed f D} → sub A C D | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
⊢ sub A C D | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
⊢ D ∈ {D | B ⊆ D ∧ closed f D} → sub A C D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
⊢ sub A C D | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
⊢ ∀ ⦃a : A⦄, a ∈ C → a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
⊢ sub A C D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | fix a : A | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
⊢ ∀ ⦃a : A⦄, a ∈ C → a ∈ D | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
⊢ a ∈ C → a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
⊢ ∀ ⦃a : A⦄, a ∈ C → a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | assume h2 : a ∈ C | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
⊢ a ∈ C → a ∈ D | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
h2 : a ∈ C
⊢ a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
⊢ a ∈ C → a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | define at h2 | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
h2 : a ∈ C
⊢ a ∈ D | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
h2 : ∀ t ∈ F, a ∈ t
⊢ a ∈ D | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
h2 : a ∈ C
⊢ a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_4_5 | [270, 1] | [313, 7] | show a ∈ D from h2 D h1 | case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
h2 : ∀ t ∈ F, a ∈ t
⊢ a ∈ D | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type
f : A → A
B : Set A
F : Set (Set A) := {D | B ⊆ D ∧ closed f D}
C : Set A := ⋂₀ F
D : Set A
h1 : D ∈ F
a : A
h2 : ∀ t ∈ F, a ∈ t
⊢ a ∈ D
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.image_def | [331, 1] | [332, 47] | rfl | A B : Type
f : A → B
X : Set A
b : B
⊢ b ∈ image f X ↔ ∃ x ∈ X, f x = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
X : Set A
b : B
⊢ b ∈ image f X ↔ ∃ x ∈ X, f x = b
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.inverse_image_def | [334, 1] | [335, 46] | rfl | A B : Type
f : A → B
Y : Set B
a : A
⊢ a ∈ inverse_image f Y ↔ f a ∈ Y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
Y : Set B
a : A
⊢ a ∈ inverse_image f Y ↔ f a ∈ Y
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | fix y : B | A B : Type
f : A → B
W X : Set A
⊢ image f (W ∩ X) ⊆ image f W ∩ image f X | A B : Type
f : A → B
W X : Set A
y : B
⊢ y ∈ image f (W ∩ X) → y ∈ image f W ∩ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
W X : Set A
⊢ image f (W ∩ X) ⊆ image f W ∩ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | assume h1 : y ∈ image f (W ∩ X) | A B : Type
f : A → B
W X : Set A
y : B
⊢ y ∈ image f (W ∩ X) → y ∈ image f W ∩ image f X | A B : Type
f : A → B
W X : Set A
y : B
h1 : y ∈ image f (W ∩ X)
⊢ y ∈ image f W ∩ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
W X : Set A
y : B
⊢ y ∈ image f (W ∩ X) → y ∈ image f W ∩ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | define at h1 | A B : Type
f : A → B
W X : Set A
y : B
h1 : y ∈ image f (W ∩ X)
⊢ y ∈ image f W ∩ image f X | A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
⊢ y ∈ image f W ∩ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
W X : Set A
y : B
h1 : y ∈ image f (W ∩ X)
⊢ y ∈ image f W ∩ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | obtain (x : A) (h2 : x ∈ W ∩ X ∧ f x = y) from h1 | A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
⊢ y ∈ image f W ∩ image f X | A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : x ∈ W ∩ X ∧ f x = y
⊢ y ∈ image f W ∩ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
⊢ y ∈ image f W ∩ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | define : x ∈ W ∩ X at h2 | A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : x ∈ W ∩ X ∧ f x = y
⊢ y ∈ image f W ∩ image f X | A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f W ∩ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : x ∈ W ∩ X ∧ f x = y
⊢ y ∈ image f W ∩ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | apply And.intro | A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f W ∩ image f X | case left
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f W
case right
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f W ∩ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | define | case left
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f W | case left
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ ∃ x ∈ W, f x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f W
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | show ∃ (x : A), x ∈ W ∧ f x = y from
Exists.intro x (And.intro h2.left.left h2.right) | case left
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ ∃ x ∈ W, f x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ ∃ x ∈ W, f x = y
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_1 | [337, 1] | [354, 7] | show y ∈ image f X from
Exists.intro x (And.intro h2.left.right h2.right) | case right
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f X | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A B : Type
f : A → B
W X : Set A
y : B
h1 : ∃ x ∈ W ∩ X, f x = y
x : A
h2 : (x ∈ W ∧ x ∈ X) ∧ f x = y
⊢ y ∈ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_2 | [356, 1] | [380, 7] | apply Set.ext | A B : Type
f : A → B
W X : Set A
h1 : one_to_one f
⊢ image f (W ∩ X) = image f W ∩ image f X | case h
A B : Type
f : A → B
W X : Set A
h1 : one_to_one f
⊢ ∀ (x : B), x ∈ image f (W ∩ X) ↔ x ∈ image f W ∩ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
W X : Set A
h1 : one_to_one f
⊢ image f (W ∩ X) = image f W ∩ image f X
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_5_2_2 | [356, 1] | [380, 7] | fix y : B | case h
A B : Type
f : A → B
W X : Set A
h1 : one_to_one f
⊢ ∀ (x : B), x ∈ image f (W ∩ X) ↔ x ∈ image f W ∩ image f X | case h
A B : Type
f : A → B
W X : Set A
h1 : one_to_one f
y : B
⊢ y ∈ image f (W ∩ X) ↔ y ∈ image f W ∩ image f X | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
f : A → B
W X : Set A
h1 : one_to_one f
⊢ ∀ (x : B), x ∈ image f (W ∩ X) ↔ x ∈ image f W ∩ image f X
TACTIC:
|
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