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/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.eq_of_opair_mem_graph | [81, 1] | [89, 14] | intro ⟨t, ht, ht'⟩ | case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z : V
⊢ (∃ z_1, opair x z_1 ∈ f.graph ∧ z ∈ z_1) → z ∈ y | case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z t : V
ht : opair x t ∈ f.graph
ht' : z ∈ t
⊢ z ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z : V
⊢ (∃ z_1, opair x z_1 ∈ f.graph ∧ z ∈ z_1) → z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.eq_of_opair_mem_graph | [81, 1] | [89, 14] | cases f.isFunction.2 x y t h ht | case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z t : V
ht : opair x t ∈ f.graph
ht' : z ∈ t
⊢ z ∈ y | case h.mpr.refl
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z : V
ht : opair x y ∈ f.graph
ht' : z ∈ y
⊢ z ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z t : V
ht : opair x t ∈ f.graph
ht' : z ∈ t
⊢ z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.eq_of_opair_mem_graph | [81, 1] | [89, 14] | exact ht' | case h.mpr.refl
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z : V
ht : opair x y ∈ f.graph
ht' : z ∈ y
⊢ z ∈ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.refl
V : Type u_1
inst✝ : Zermelo V
r x y z✝ a b c : V
f : a ⟶ b
h : opair x y ∈ f.graph
z : V
ht : opair x y ∈ f.graph
ht' : z ∈ y
⊢ z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph | [91, 1] | [96, 11] | have := mem_dom_of_mem f h | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x (apply f.graph x) ∈ f.graph | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
this : x ∈ dom f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x (apply f.graph x) ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph | [91, 1] | [96, 11] | rw [mem_dom_iff] at this | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
this : x ∈ dom f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
this : ∃ z, opair x z ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
this : x ∈ dom f.graph
⊢ opair x (apply f.graph x) ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph | [91, 1] | [96, 11] | rcases this with ⟨y, hy⟩ | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
this : ∃ z, opair x z ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | case intro
V : Type u_1
inst✝ : Zermelo V
r x y✝ z a b c : V
f : a ⟶ b
h : x ∈ a
y : V
hy : opair x y ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
this : ∃ z, opair x z ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph | [91, 1] | [96, 11] | cases eq_of_opair_mem_graph f hy | case intro
V : Type u_1
inst✝ : Zermelo V
r x y✝ z a b c : V
f : a ⟶ b
h : x ∈ a
y : V
hy : opair x y ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | case intro.refl
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
hy : opair x (apply f.graph x) ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
V : Type u_1
inst✝ : Zermelo V
r x y✝ z a b c : V
f : a ⟶ b
h : x ∈ a
y : V
hy : opair x y ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph | [91, 1] | [96, 11] | exact hy | case intro.refl
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
hy : opair x (apply f.graph x) ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refl
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
hy : opair x (apply f.graph x) ∈ f.graph
⊢ opair x (apply f.graph x) ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph_iff | [98, 1] | [102, 30] | constructor | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x y ∈ f.graph ↔ y = apply f.graph x | case mp
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x y ∈ f.graph → y = apply f.graph x
case mpr
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ y = apply f.graph x → opair x y ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x y ∈ f.graph ↔ y = apply f.graph x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph_iff | [98, 1] | [102, 30] | exact eq_of_opair_mem_graph f | case mp
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x y ∈ f.graph → y = apply f.graph x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x y ∈ f.graph → y = apply f.graph x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph_iff | [98, 1] | [102, 30] | rintro rfl | case mpr
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ y = apply f.graph x → opair x y ∈ f.graph | case mpr
V : Type u_1
inst✝ : Zermelo V
r x z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x (apply f.graph x) ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ y = apply f.graph x → opair x y ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.opair_mem_graph_iff | [98, 1] | [102, 30] | exact opair_mem_graph f h | case mpr
V : Type u_1
inst✝ : Zermelo V
r x z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x (apply f.graph x) ∈ f.graph | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type u_1
inst✝ : Zermelo V
r x z a b c : V
f : a ⟶ b
h : x ∈ a
⊢ opair x (apply f.graph x) ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | suffices f.graph = g.graph from by
cases f
cases this
rfl | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
⊢ f = g | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
⊢ f.graph = g.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
⊢ f = g
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | ext x | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
⊢ f.graph = g.graph | case h
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ f.graph ↔ x ∈ g.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
⊢ f.graph = g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | constructor | case h
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ f.graph ↔ x ∈ g.graph | case h.mp
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ f.graph → x ∈ g.graph
case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ g.graph → x ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ f.graph ↔ x ∈ g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | cases f | V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
this : f.graph = g.graph
⊢ f = g | case mk
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
g : a ⟶ b
graph✝ : V
isFunction✝ : IsFunction graph✝
dom_eq✝ : dom graph✝ = a
ran_subset✝ : ran graph✝ ⊆ b
h :
∀ (x : V),
x ∈ a →
apply { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph x =
apply g.graph x
this : { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph = g.graph
⊢ { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
this : f.graph = g.graph
⊢ f = g
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | cases this | case mk
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
g : a ⟶ b
graph✝ : V
isFunction✝ : IsFunction graph✝
dom_eq✝ : dom graph✝ = a
ran_subset✝ : ran graph✝ ⊆ b
h :
∀ (x : V),
x ∈ a →
apply { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph x =
apply g.graph x
this : { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph = g.graph
⊢ { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ } = g | case mk.refl
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
g : a ⟶ b
isFunction✝ : IsFunction g.1
dom_eq✝ : dom g.1 = a
ran_subset✝ : ran g.1 ⊆ b
h :
∀ (x : V),
x ∈ a →
apply { graph := g.1, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph x =
apply g.graph x
⊢ { graph := g.1, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ } = g | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
g : a ⟶ b
graph✝ : V
isFunction✝ : IsFunction graph✝
dom_eq✝ : dom graph✝ = a
ran_subset✝ : ran graph✝ ⊆ b
h :
∀ (x : V),
x ∈ a →
apply { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph x =
apply g.graph x
this : { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph = g.graph
⊢ { graph := graph✝, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ } = g
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | rfl | case mk.refl
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
g : a ⟶ b
isFunction✝ : IsFunction g.1
dom_eq✝ : dom g.1 = a
ran_subset✝ : ran g.1 ⊆ b
h :
∀ (x : V),
x ∈ a →
apply { graph := g.1, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph x =
apply g.graph x
⊢ { graph := g.1, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ } = g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.refl
V : Type u_1
inst✝ : Zermelo V
r x y z a b c : V
g : a ⟶ b
isFunction✝ : IsFunction g.1
dom_eq✝ : dom g.1 = a
ran_subset✝ : ran g.1 ⊆ b
h :
∀ (x : V),
x ∈ a →
apply { graph := g.1, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ }.graph x =
apply g.graph x
⊢ { graph := g.1, isFunction := isFunction✝, dom_eq := dom_eq✝, ran_subset := ran_subset✝ } = g
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | intro hx | case h.mp
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ f.graph → x ∈ g.graph | case h.mp
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
hx : x ∈ f.graph
⊢ x ∈ g.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ f.graph → x ∈ g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | obtain ⟨x, y, rfl⟩ := f.isFunction.1 x hx | case h.mp
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
hx : x ∈ f.graph
⊢ x ∈ g.graph | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ opair x y ∈ g.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
hx : x ∈ f.graph
⊢ x ∈ g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | have : x ∈ a := by
refine mem_of_mem_dom f ?_
rw [mem_dom_iff]
exact ⟨y, hx⟩ | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ opair x y ∈ g.graph | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
this : x ∈ a
⊢ opair x y ∈ g.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ opair x y ∈ g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | rw [opair_mem_graph_iff _ this] at hx ⊢ | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
this : x ∈ a
⊢ opair x y ∈ g.graph | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply f.graph x
this : x ∈ a
⊢ y = apply g.graph x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
this : x ∈ a
⊢ opair x y ∈ g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | rw [hx] | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply f.graph x
this : x ∈ a
⊢ y = apply g.graph x | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply f.graph x
this : x ∈ a
⊢ apply f.graph x = apply g.graph x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply f.graph x
this : x ∈ a
⊢ y = apply g.graph x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | exact h _ this | case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply f.graph x
this : x ∈ a
⊢ apply f.graph x = apply g.graph x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply f.graph x
this : x ∈ a
⊢ apply f.graph x = apply g.graph x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | refine mem_of_mem_dom f ?_ | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ x ∈ a | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ x ∈ dom f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ x ∈ a
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | rw [mem_dom_iff] | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ x ∈ dom f.graph | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ ∃ z, opair x z ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ x ∈ dom f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | exact ⟨y, hx⟩ | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ ∃ z, opair x z ∈ f.graph | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ f.graph
⊢ ∃ z, opair x z ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | intro hx | case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ g.graph → x ∈ f.graph | case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
hx : x ∈ g.graph
⊢ x ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
⊢ x ∈ g.graph → x ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | obtain ⟨x, y, rfl⟩ := g.isFunction.1 x hx | case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
hx : x ∈ g.graph
⊢ x ∈ f.graph | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ opair x y ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
V : Type u_1
inst✝ : Zermelo V
r x✝ y z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x : V
hx : x ∈ g.graph
⊢ x ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | have : x ∈ a := by
refine mem_of_mem_dom g ?_
rw [mem_dom_iff]
exact ⟨y, hx⟩ | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ opair x y ∈ f.graph | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
this : x ∈ a
⊢ opair x y ∈ f.graph | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ opair x y ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | rw [opair_mem_graph_iff _ this] at hx ⊢ | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
this : x ∈ a
⊢ opair x y ∈ f.graph | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply g.graph x
this : x ∈ a
⊢ y = apply f.graph x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
this : x ∈ a
⊢ opair x y ∈ f.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | rw [hx] | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply g.graph x
this : x ∈ a
⊢ y = apply f.graph x | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply g.graph x
this : x ∈ a
⊢ apply g.graph x = apply f.graph x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply g.graph x
this : x ∈ a
⊢ y = apply f.graph x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | exact (h _ this).symm | case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply g.graph x
this : x ∈ a
⊢ apply g.graph x = apply f.graph x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : y = apply g.graph x
this : x ∈ a
⊢ apply g.graph x = apply f.graph x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | refine mem_of_mem_dom g ?_ | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ x ∈ a | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ x ∈ dom g.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ x ∈ a
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | rw [mem_dom_iff] | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ x ∈ dom g.graph | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ ∃ z, opair x z ∈ g.graph | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ x ∈ dom g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Function.lean | SetTheory.Function.ext | [106, 1] | [130, 26] | exact ⟨y, hx⟩ | V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ ∃ z, opair x z ∈ g.graph | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x✝ y✝ z a b c : V
f g : a ⟶ b
h : ∀ (x : V), x ∈ a → apply f.graph x = apply g.graph x
x y : V
hx : opair x y ∈ g.graph
⊢ ∃ z, opair x z ∈ g.graph
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_π₂_iff | [31, 1] | [33, 8] | unfold π₂ | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈ π₂ x ↔ ∃ z, y ∈ z ∧ z ∈ ⋃ x ∧ (⋃ x ≠ ⋂ x → ¬z ∈ ⋂ x) | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈
⋃
sep
(BoundedFormula.imp (BoundedFormula.ne (Sum.inl true) (Sum.inl false))
(BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inl false))))
(fun i => if i = true then ⋃ x else ⋂ x) (⋃ x) ↔
∃ z, y ∈ z ∧ z ∈ ⋃ x ∧ (⋃ x ≠ ⋂ x → ¬z ∈ ⋂ x) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈ π₂ x ↔ ∃ z, y ∈ z ∧ z ∈ ⋃ x ∧ (⋃ x ≠ ⋂ x → ¬z ∈ ⋂ x)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_π₂_iff | [31, 1] | [33, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈
⋃
sep
(BoundedFormula.imp (BoundedFormula.ne (Sum.inl true) (Sum.inl false))
(BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inl false))))
(fun i => if i = true then ⋃ x else ⋂ x) (⋃ x) ↔
∃ z, y ∈ z ∧ z ∈ ⋃ x ∧ (⋃ x ≠ ⋂ x → ¬z ∈ ⋂ x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈
⋃
sep
(BoundedFormula.imp (BoundedFormula.ne (Sum.inl true) (Sum.inl false))
(BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inl false))))
(fun i => if i = true then ⋃ x else ⋂ x) (⋃ x) ↔
∃ z, y ∈ z ∧ z ∈ ⋃ x ∧ (⋃ x ≠ ⋂ x → ¬z ∈ ⋂ x)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₁_opair | [36, 1] | [39, 8] | unfold π₁ opair | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ π₁ (opair x y) = x | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ⋃ ⋂ pair {x} (pair x y) = x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ π₁ (opair x y) = x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₁_opair | [36, 1] | [39, 8] | ext z | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ⋃ ⋂ pair {x} (pair x y) = x | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ ⋃ ⋂ pair {x} (pair x y) ↔ z ∈ x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ⋃ ⋂ pair {x} (pair x y) = x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₁_opair | [36, 1] | [39, 8] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ ⋃ ⋂ pair {x} (pair x y) ↔ z ∈ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ ⋃ ⋂ pair {x} (pair x y) ↔ z ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | unfold opair | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ π₂ (opair x y) = y | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ π₂ (pair {x} (pair x y)) = y | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ π₂ (opair x y) = y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | ext z | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ π₂ (pair {x} (pair x y)) = y | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ π₂ (pair {x} (pair x y)) ↔ z ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ π₂ (pair {x} (pair x y)) = y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | rw [mem_π₂_iff] | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ π₂ (pair {x} (pair x y)) ↔ z ∈ y | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1,
z ∈ z_1 ∧
z_1 ∈ ⋃ pair {x} (pair x y) ∧ (⋃ pair {x} (pair x y) ≠ ⋂ pair {x} (pair x y) → ¬z_1 ∈ ⋂ pair {x} (pair x y))) ↔
z ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ π₂ (pair {x} (pair x y)) ↔ z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | simp only [sUnion_pair, sInter_pair, ne_eq, union_pair_eq_inter_pair] | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1,
z ∈ z_1 ∧
z_1 ∈ ⋃ pair {x} (pair x y) ∧ (⋃ pair {x} (pair x y) ≠ ⋂ pair {x} (pair x y) → ¬z_1 ∈ ⋂ pair {x} (pair x y))) ↔
z ∈ y | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y)) ↔ z ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1,
z ∈ z_1 ∧
z_1 ∈ ⋃ pair {x} (pair x y) ∧ (⋃ pair {x} (pair x y) ≠ ⋂ pair {x} (pair x y) → ¬z_1 ∈ ⋂ pair {x} (pair x y))) ↔
z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | constructor | case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y)) ↔ z ∈ y | case h.mp
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y)) → z ∈ y
case h.mpr
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ y → ∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y)) ↔ z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | rintro ⟨s, hzs, hsx, hs⟩ | case h.mp
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y)) → z ∈ y | case h.mp.intro.intro.intro
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
⊢ z ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ (∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y)) → z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | by_cases h : s = y | case h.mp.intro.intro.intro
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
⊢ z ∈ y | case pos
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
h : s = y
⊢ z ∈ y
case neg
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
h : ¬s = y
⊢ z ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
⊢ z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | subst h | case pos
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
h : s = y
⊢ z ∈ y | case pos
V : Type u_1
inst✝ : Zermelo V
x z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x s
hs : ¬x = s → ¬s ∈ {x} ∩ pair x s
⊢ z ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
h : s = y
⊢ z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | exact hzs | case pos
V : Type u_1
inst✝ : Zermelo V
x z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x s
hs : ¬x = s → ¬s ∈ {x} ∩ pair x s
⊢ z ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
V : Type u_1
inst✝ : Zermelo V
x z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x s
hs : ¬x = s → ¬s ∈ {x} ∩ pair x s
⊢ z ∈ s
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | aesop | case neg
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
h : ¬s = y
⊢ z ∈ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
V : Type u_1
inst✝ : Zermelo V
x y z✝ z s : V
hzs : z ∈ s
hsx : s ∈ {x} ∪ pair x y
hs : ¬x = y → ¬s ∈ {x} ∩ pair x y
h : ¬s = y
⊢ z ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.π₂_opair | [42, 1] | [53, 10] | aesop | case h.mpr
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ y → ∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
V : Type u_1
inst✝ : Zermelo V
x y z✝ z : V
⊢ z ∈ y → ∃ z_1, z ∈ z_1 ∧ z_1 ∈ {x} ∪ pair x y ∧ (¬x = y → ¬z_1 ∈ {x} ∩ pair x y)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | constructor | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ opair x y = opair z w ↔ x = z ∧ y = w | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ opair x y = opair z w → x = z ∧ y = w
case mpr
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ x = z ∧ y = w → opair x y = opair z w | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ opair x y = opair z w ↔ x = z ∧ y = w
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | intro h | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ opair x y = opair z w → x = z ∧ y = w | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
⊢ x = z ∧ y = w | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ opair x y = opair z w → x = z ∧ y = w
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | have h₁ := congrArg π₁ h | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
⊢ x = z ∧ y = w | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : π₁ (opair x y) = π₁ (opair z w)
⊢ x = z ∧ y = w | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
⊢ x = z ∧ y = w
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | have h₂ := congrArg π₂ h | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : π₁ (opair x y) = π₁ (opair z w)
⊢ x = z ∧ y = w | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : π₁ (opair x y) = π₁ (opair z w)
h₂ : π₂ (opair x y) = π₂ (opair z w)
⊢ x = z ∧ y = w | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : π₁ (opair x y) = π₁ (opair z w)
⊢ x = z ∧ y = w
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | simp at h₁ h₂ | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : π₁ (opair x y) = π₁ (opair z w)
h₂ : π₂ (opair x y) = π₂ (opair z w)
⊢ x = z ∧ y = w | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : x = z
h₂ : y = w
⊢ x = z ∧ y = w | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : π₁ (opair x y) = π₁ (opair z w)
h₂ : π₂ (opair x y) = π₂ (opair z w)
⊢ x = z ∧ y = w
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | exact ⟨h₁, h₂⟩ | case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : x = z
h₂ : y = w
⊢ x = z ∧ y = w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
h : opair x y = opair z w
h₁ : x = z
h₂ : y = w
⊢ x = z ∧ y = w
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | rintro ⟨rfl, rfl⟩ | case mpr
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ x = z ∧ y = w → opair x y = opair z w | case mpr.intro
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z x y : V
⊢ opair x y = opair x y | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ x y z w : V
⊢ x = z ∧ y = w → opair x y = opair z w
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_injective | [56, 1] | [65, 8] | rfl | case mpr.intro
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z x y : V
⊢ opair x y = opair x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z x y : V
⊢ opair x y = opair x y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.eq_pair_iff | [76, 1] | [78, 7] | rw [ext_iff] | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x = pair y z ↔ ∀ (t : V), t ∈ x ↔ t = y ∨ t = z | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ (∀ (z_1 : V), z_1 ∈ x ↔ z_1 ∈ pair y z) ↔ ∀ (t : V), t ∈ x ↔ t = y ∨ t = z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x = pair y z ↔ ∀ (t : V), t ∈ x ↔ t = y ∨ t = z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.eq_pair_iff | [76, 1] | [78, 7] | simp | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ (∀ (z_1 : V), z_1 ∈ x ↔ z_1 ∈ pair y z) ↔ ∀ (t : V), t ∈ x ↔ t = y ∨ t = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ (∀ (z_1 : V), z_1 ∈ x ↔ z_1 ∈ pair y z) ↔ ∀ (t : V), t ∈ x ↔ t = y ∨ t = z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_eqPair | [81, 1] | [86, 7] | unfold BoundedFormula.eqPair | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqPair x y z) v l ↔
interpretTerm V v l x = pair (interpretTerm V v l y) (interpretTerm V v l z) | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.iff (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.or (BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
interpretTerm V v l x = pair (interpretTerm V v l y) (interpretTerm V v l z) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqPair x y z) v l ↔
interpretTerm V v l x = pair (interpretTerm V v l y) (interpretTerm V v l z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_eqPair | [81, 1] | [86, 7] | rw [eq_pair_iff] | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.iff (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.or (BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
interpretTerm V v l x = pair (interpretTerm V v l y) (interpretTerm V v l z) | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.iff (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.or (BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
∀ (t : V), t ∈ interpretTerm V v l x ↔ t = interpretTerm V v l y ∨ t = interpretTerm V v l z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.iff (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.or (BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
interpretTerm V v l x = pair (interpretTerm V v l y) (interpretTerm V v l z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_eqPair | [81, 1] | [86, 7] | simp | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.iff (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.or (BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
∀ (t : V), t ∈ interpretTerm V v l x ↔ t = interpretTerm V v l y ∨ t = interpretTerm V v l z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.iff (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.or (BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eq (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
∀ (t : V), t ∈ interpretTerm V v l x ↔ t = interpretTerm V v l y ∨ t = interpretTerm V v l z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_eqSingleton | [93, 1] | [97, 7] | unfold BoundedFormula.eqSingleton | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqSingleton x y) v l ↔ interpretTerm V v l x = {interpretTerm V v l y} | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqPair x y y) v l ↔ interpretTerm V v l x = {interpretTerm V v l y} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqSingleton x y) v l ↔ interpretTerm V v l x = {interpretTerm V v l y}
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_eqSingleton | [93, 1] | [97, 7] | simp | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqPair x y y) v l ↔ interpretTerm V v l x = {interpretTerm V v l y} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqPair x y y) v l ↔ interpretTerm V v l x = {interpretTerm V v l y}
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_isPair | [108, 1] | [111, 7] | unfold BoundedFormula.isPair IsPair | V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isPair x) v l ↔ IsPair (interpretTerm V v l x) | V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.eqPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr (Fin.last (n + 1))))))
v l ↔
∃ y z, interpretTerm V v l x = pair y z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isPair x) v l ↔ IsPair (interpretTerm V v l x)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_isPair | [108, 1] | [111, 7] | simp | V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.eqPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr (Fin.last (n + 1))))))
v l ↔
∃ y z, interpretTerm V v l x = pair y z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.eqPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr (Fin.last (n + 1))))))
v l ↔
∃ y z, interpretTerm V v l x = pair y z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_eqOPair | [125, 1] | [129, 8] | unfold BoundedFormula.eqOPair opair | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqOPair x y z) v l ↔
interpretTerm V v l x = opair (interpretTerm V v l y) (interpretTerm V v l z) | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.and
(BoundedFormula.eqSingleton (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc y)))
(BoundedFormula.eqPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc y)) (termSucc (termSucc z))))
(BoundedFormula.eqPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })))))
v l ↔
interpretTerm V v l x = pair {interpretTerm V v l y} (pair (interpretTerm V v l y) (interpretTerm V v l z)) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqOPair x y z) v l ↔
interpretTerm V v l x = opair (interpretTerm V v l y) (interpretTerm V v l z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_eqOPair | [125, 1] | [129, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.and
(BoundedFormula.eqSingleton (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc y)))
(BoundedFormula.eqPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc y)) (termSucc (termSucc z))))
(BoundedFormula.eqPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })))))
v l ↔
interpretTerm V v l x = pair {interpretTerm V v l y} (pair (interpretTerm V v l y) (interpretTerm V v l z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.and
(BoundedFormula.eqSingleton (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc y)))
(BoundedFormula.eqPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc y)) (termSucc (termSucc z))))
(BoundedFormula.eqPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })))))
v l ↔
interpretTerm V v l x = pair {interpretTerm V v l y} (pair (interpretTerm V v l y) (interpretTerm V v l z))
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_isOPair | [140, 1] | [143, 8] | unfold BoundedFormula.isOPair IsOPair opair | V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isOPair x) v l ↔ IsOPair (interpretTerm V v l x) | V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.eqOPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) }))))
v l ↔
∃ y z, interpretTerm V v l x = pair {y} (pair y z) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isOPair x) v l ↔ IsOPair (interpretTerm V v l x)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_isOPair | [140, 1] | [143, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.eqOPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) }))))
v l ↔
∃ y z, interpretTerm V v l x = pair {y} (pair y z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.eqOPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) }))))
v l ↔
∃ y z, interpretTerm V v l x = pair {y} (pair y z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_memProd' | [153, 1] | [157, 8] | unfold BoundedFormula.memProd | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memProd x y z) v l ↔
∃ a, a ∈ interpretTerm V v l y ∧ ∃ b, b ∈ interpretTerm V v l z ∧ interpretTerm V v l x = opair a b | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.and
(BoundedFormula.mem (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc y)))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc z))))
(BoundedFormula.eqOPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })))))
v l ↔
∃ a, a ∈ interpretTerm V v l y ∧ ∃ b, b ∈ interpretTerm V v l z ∧ interpretTerm V v l x = opair a b | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memProd x y z) v l ↔
∃ a, a ∈ interpretTerm V v l y ∧ ∃ b, b ∈ interpretTerm V v l z ∧ interpretTerm V v l x = opair a b
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_memProd' | [153, 1] | [157, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.and
(BoundedFormula.mem (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc y)))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc z))))
(BoundedFormula.eqOPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })))))
v l ↔
∃ a, a ∈ interpretTerm V v l y ∧ ∃ b, b ∈ interpretTerm V v l z ∧ interpretTerm V v l x = opair a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.and
(BoundedFormula.mem (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc y)))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc z))))
(BoundedFormula.eqOPair (termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) })
(Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })))))
v l ↔
∃ a, a ∈ interpretTerm V v l y ∧ ∃ b, b ∈ interpretTerm V v l z ∧ interpretTerm V v l x = opair a b
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | unfold prod | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈ prod x y ↔ ∃ a, a ∈ x ∧ ∃ b, b ∈ y ∧ z = opair a b | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈
sep (BoundedFormula.memProd (Sum.inr 0) (Sum.inl true) (Sum.inl false)) (fun i => if i = true then x else y)
(power (power (x ∪ y))) ↔
∃ a, a ∈ x ∧ ∃ b, b ∈ y ∧ z = opair a b | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈ prod x y ↔ ∃ a, a ∈ x ∧ ∃ b, b ∈ y ∧ z = opair a b
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | simp only [mem_sep_iff, mem_power_iff, interpret_memProd', interpret_inl, ite_true,
ite_false, interpret_inr, and_iff_right_iff_imp, forall_exists_index, and_imp] | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈
sep (BoundedFormula.memProd (Sum.inr 0) (Sum.inl true) (Sum.inl false)) (fun i => if i = true then x else y)
(power (power (x ∪ y))) ↔
∃ a, a ∈ x ∧ ∃ b, b ∈ y ∧ z = opair a b | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ∀ (x_1 : V), x_1 ∈ x → ∀ (x_2 : V), x_2 ∈ y → z = opair x_1 x_2 → z ⊆ power (x ∪ y) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ z ∈
sep (BoundedFormula.memProd (Sum.inr 0) (Sum.inl true) (Sum.inl false)) (fun i => if i = true then x else y)
(power (power (x ∪ y))) ↔
∃ a, a ∈ x ∧ ∃ b, b ∈ y ∧ z = opair a b
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | rintro a ha b hb rfl | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ∀ (x_1 : V), x_1 ∈ x → ∀ (x_2 : V), x_2 ∈ y → z = opair x_1 x_2 → z ⊆ power (x ∪ y) | V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ opair a b ⊆ power (x ∪ y) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ∀ (x_1 : V), x_1 ∈ x → ∀ (x_2 : V), x_2 ∈ y → z = opair x_1 x_2 → z ⊆ power (x ∪ y)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | intro z hz | V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ opair a b ⊆ power (x ∪ y) | V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
z : V
hz : z ∈ opair a b
⊢ z ∈ power (x ∪ y) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ opair a b ⊆ power (x ∪ y)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | rw [opair, mem_pair_iff] at hz | V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
z : V
hz : z ∈ opair a b
⊢ z ∈ power (x ∪ y) | V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
z : V
hz : z = {a} ∨ z = pair a b
⊢ z ∈ power (x ∪ y) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
z : V
hz : z ∈ opair a b
⊢ z ∈ power (x ∪ y)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | rw [mem_power_iff] | case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ pair a b ∈ power (x ∪ y) | case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ pair a b ⊆ x ∪ y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ pair a b ∈ power (x ∪ y)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | intro t ht | case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ pair a b ⊆ x ∪ y | case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
t : V
ht : t ∈ pair a b
⊢ t ∈ x ∪ y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
⊢ pair a b ⊆ x ∪ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.mem_prod_iff | [165, 1] | [175, 10] | aesop | case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
t : V
ht : t ∈ pair a b
⊢ t ∈ x ∪ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
V : Type u_1
inst✝ : Zermelo V
x y a : V
ha : a ∈ x
b : V
hb : b ∈ y
t : V
ht : t ∈ pair a b
⊢ t ∈ x ∪ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_mem_prod_iff | [178, 1] | [180, 8] | rw [mem_prod_iff] | V : Type u_1
inst✝ : Zermelo V
x y z a b : V
⊢ opair a b ∈ prod x y ↔ a ∈ x ∧ b ∈ y | V : Type u_1
inst✝ : Zermelo V
x y z a b : V
⊢ (∃ a_1, a_1 ∈ x ∧ ∃ b_1, b_1 ∈ y ∧ opair a b = opair a_1 b_1) ↔ a ∈ x ∧ b ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z a b : V
⊢ opair a b ∈ prod x y ↔ a ∈ x ∧ b ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.opair_mem_prod_iff | [178, 1] | [180, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x y z a b : V
⊢ (∃ a_1, a_1 ∈ x ∧ ∃ b_1, b_1 ∈ y ∧ opair a b = opair a_1 b_1) ↔ a ∈ x ∧ b ∈ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z a b : V
⊢ (∃ a_1, a_1 ∈ x ∧ ∃ b_1, b_1 ∈ y ∧ opair a b = opair a_1 b_1) ↔ a ∈ x ∧ b ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/OPair.lean | SetTheory.interpret_isOPairIn | [183, 1] | [186, 40] | rw [mem_prod_iff, interpret_memProd'] | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memProd x y z) v l ↔
interpretTerm V v l x ∈ prod (interpretTerm V v l y) (interpretTerm V v l z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z✝ : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memProd x y z) v l ↔
interpretTerm V v l x ∈ prod (interpretTerm V v l y) (interpretTerm V v l z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | not_not | [104, 1] | [105, 23] | by_cases p <;> aesop | V : Type ?u.5243
inst✝ : SetTheory V
p : Prop
⊢ ¬¬p ↔ p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type ?u.5243
inst✝ : SetTheory V
p : Prop
⊢ ¬¬p ↔ p
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | not_forall | [108, 1] | [112, 10] | constructor | V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (¬∀ (x : α), p x) ↔ ∃ x, ¬p x | case mp
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (¬∀ (x : α), p x) → ∃ x, ¬p x
case mpr
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (∃ x, ¬p x) → ¬∀ (x : α), p x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (¬∀ (x : α), p x) ↔ ∃ x, ¬p x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | not_forall | [108, 1] | [112, 10] | by_contra | case mp
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (¬∀ (x : α), p x) → ∃ x, ¬p x | case mp
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
x✝ : ¬((¬∀ (x : α), p x) → ∃ x, ¬p x)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (¬∀ (x : α), p x) → ∃ x, ¬p x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | not_forall | [108, 1] | [112, 10] | aesop | case mp
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
x✝ : ¬((¬∀ (x : α), p x) → ∃ x, ¬p x)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
x✝ : ¬((¬∀ (x : α), p x) → ∃ x, ¬p x)
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | not_forall | [108, 1] | [112, 10] | aesop | case mpr
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (∃ x, ¬p x) → ¬∀ (x : α), p x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type ?u.5314
inst✝ : SetTheory V
α : Sort u_1
p : α → Prop
⊢ (∃ x, ¬p x) → ¬∀ (x : α), p x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | or_iff | [114, 1] | [115, 23] | by_cases p <;> aesop | V : Type ?u.6497
inst✝ : SetTheory V
p q : Prop
⊢ p ∨ q ↔ ¬p → q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type ?u.6497
inst✝ : SetTheory V
p q : Prop
⊢ p ∨ q ↔ ¬p → q
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | and_iff | [117, 1] | [118, 23] | by_cases p <;> aesop | V : Type ?u.6701
inst✝ : SetTheory V
p q : Prop
⊢ p ∧ q ↔ ¬(p → ¬q) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type ?u.6701
inst✝ : SetTheory V
p q : Prop
⊢ p ∧ q ↔ ¬(p → ¬q)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | Fin.snoc_castSucc | [132, 1] | [135, 6] | rw [Fin.snoc, dif_pos] | V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ snoc l x (castSucc k) = l k | V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ l { val := (castSucc k).val, isLt := ?hc } = l k
case hc
V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ (castSucc k).val < n | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ snoc l x (castSucc k) = l k
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | Fin.snoc_castSucc | [132, 1] | [135, 6] | rfl | V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ l { val := (castSucc k).val, isLt := ?hc } = l k
case hc
V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ (castSucc k).val < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ l { val := (castSucc k).val, isLt := ?hc } = l k
case hc
V : Type ?u.7418
inst✝ : SetTheory V
n : Nat
α : Sort u_1
l : Fin n → α
x : α
k : Fin n
⊢ (castSucc k).val < n
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | Nat.lt_add | [138, 1] | [145, 28] | induction m with
| zero =>
rw [Nat.lt_succ]
exact Nat.le_refl n
| succ m ih =>
rw [← Nat.add_assoc, Nat.lt_succ]
exact Nat.le_of_lt ih | V : Type ?u.7600
inst✝ : SetTheory V
n m : Nat
⊢ n < n + (m + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type ?u.7600
inst✝ : SetTheory V
n m : Nat
⊢ n < n + (m + 1)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | Nat.lt_add | [138, 1] | [145, 28] | rw [Nat.lt_succ] | case zero
V : Type ?u.7600
inst✝ : SetTheory V
n : Nat
⊢ n < n + (zero + 1) | case zero
V : Type ?u.7600
inst✝ : SetTheory V
n : Nat
⊢ n ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
V : Type ?u.7600
inst✝ : SetTheory V
n : Nat
⊢ n < n + (zero + 1)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | Nat.lt_add | [138, 1] | [145, 28] | exact Nat.le_refl n | case zero
V : Type ?u.7600
inst✝ : SetTheory V
n : Nat
⊢ n ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
V : Type ?u.7600
inst✝ : SetTheory V
n : Nat
⊢ n ≤ n
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | Nat.lt_add | [138, 1] | [145, 28] | rw [← Nat.add_assoc, Nat.lt_succ] | case succ
V : Type ?u.7600
inst✝ : SetTheory V
n m : Nat
ih : n < n + (m + 1)
⊢ n < n + (succ m + 1) | case succ
V : Type ?u.7600
inst✝ : SetTheory V
n m : Nat
ih : n < n + (m + 1)
⊢ n ≤ n + succ m | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
V : Type ?u.7600
inst✝ : SetTheory V
n m : Nat
ih : n < n + (m + 1)
⊢ n < n + (succ m + 1)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Formula.lean | Nat.lt_add | [138, 1] | [145, 28] | exact Nat.le_of_lt ih | case succ
V : Type ?u.7600
inst✝ : SetTheory V
n m : Nat
ih : n < n + (m + 1)
⊢ n ≤ n + succ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
V : Type ?u.7600
inst✝ : SetTheory V
n m : Nat
ih : n < n + (m + 1)
⊢ n ≤ n + succ m
TACTIC:
|
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