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/Basic.lean | SetTheory.subset_antisymm | [92, 1] | [93, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z : V
h₁ : x ⊆ y
h₂ : y ⊆ x
z✝ : V
⊢ z✝ ∈ x ↔ z✝ ∈ y | 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 : V
h₁ : x ⊆ y
h₂ : y ⊆ x
z✝ : V
⊢ z✝ ∈ x ↔ z✝ ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_subset | [105, 1] | [109, 8] | unfold russellSet | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ russellSet x ⊆ x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x ⊆ x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ russellSet x ⊆ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_subset | [105, 1] | [109, 8] | intro y hy | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x ⊆ x | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z x y : V
hy : y ∈ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x
⊢ y ∈ x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x ⊆ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_subset | [105, 1] | [109, 8] | aesop | V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z x y : V
hy : y ∈ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x
⊢ y ∈ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y✝ z x y : V
hy : y ∈ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x
⊢ y ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.mem_russellSet_iff | [111, 1] | [113, 7] | unfold russellSet | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈ russellSet x ↔ y ∈ x ∧ ¬y ∈ y | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x ↔ y ∈ 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
⊢ y ∈ russellSet x ↔ y ∈ x ∧ ¬y ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.mem_russellSet_iff | [111, 1] | [113, 7] | simp | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ y ∈ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x ↔ y ∈ x ∧ ¬y ∈ 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
⊢ y ∈ sep (BoundedFormula.not (BoundedFormula.mem (Sum.inr 0) (Sum.inr 0))) (fun i => Empty.elim i) x ↔ y ∈ x ∧ ¬y ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_not_mem | [115, 1] | [122, 22] | intro h | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ ¬russellSet x ∈ x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ ¬russellSet x ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_not_mem | [115, 1] | [122, 22] | by_cases h' : russellSet x ∈ russellSet x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
⊢ False | case pos
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : russellSet x ∈ russellSet x
⊢ False
case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : ¬russellSet x ∈ russellSet x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_not_mem | [115, 1] | [122, 22] | have := mem_russellSet_iff.mp h' | case pos
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : russellSet x ∈ russellSet x
⊢ False | case pos
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : russellSet x ∈ russellSet x
this : russellSet x ∈ x ∧ ¬russellSet x ∈ russellSet x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : russellSet x ∈ russellSet x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_not_mem | [115, 1] | [122, 22] | exact this.2 h' | case pos
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : russellSet x ∈ russellSet x
this : russellSet x ∈ x ∧ ¬russellSet x ∈ russellSet x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : russellSet x ∈ russellSet x
this : russellSet x ∈ x ∧ ¬russellSet x ∈ russellSet x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_not_mem | [115, 1] | [122, 22] | have := h' | case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : ¬russellSet x ∈ russellSet x
⊢ False | case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' this : ¬russellSet x ∈ russellSet x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : ¬russellSet x ∈ russellSet x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_not_mem | [115, 1] | [122, 22] | rw [mem_russellSet_iff, not_and, not_not] at this | case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' this : ¬russellSet x ∈ russellSet x
⊢ False | case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : ¬russellSet x ∈ russellSet x
this : russellSet x ∈ x → russellSet x ∈ russellSet x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' this : ¬russellSet x ∈ russellSet x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.russellSet_not_mem | [115, 1] | [122, 22] | exact h' (this h) | case neg
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : russellSet x ∈ x
h' : ¬russellSet x ∈ russellSet x
this : russellSet x ∈ x → russellSet x ∈ russellSet x
⊢ False | 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 x : V
h : russellSet x ∈ x
h' : ¬russellSet x ∈ russellSet x
this : russellSet x ∈ x → russellSet x ∈ russellSet x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.univ_not_set | [125, 1] | [128, 35] | by_contra h | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ ∃ y, ¬y ∈ x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : ¬∃ y, ¬y ∈ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ ∃ y, ¬y ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.univ_not_set | [125, 1] | [128, 35] | simp only [not_exists, not_not] at h | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : ¬∃ y, ¬y ∈ x
⊢ False | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : ∀ (x_1 : V), x_1 ∈ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : ¬∃ y, ¬y ∈ x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.univ_not_set | [125, 1] | [128, 35] | exact russellSet_not_mem x (h _) | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : ∀ (x_1 : V), x_1 ∈ x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : ∀ (x_1 : V), x_1 ∈ x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.power_not_subset | [131, 1] | [135, 28] | intro h | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ ¬power x ⊆ x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ ¬power x ⊆ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.power_not_subset | [131, 1] | [135, 28] | refine russellSet_not_mem x (h ?_) | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ False | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ russellSet x ∈ power x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.power_not_subset | [131, 1] | [135, 28] | rw [mem_power_iff] | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ russellSet x ∈ power x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ russellSet x ⊆ x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ russellSet x ∈ power x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.power_not_subset | [131, 1] | [135, 28] | exact russellSet_subset x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ russellSet x ⊆ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x ⊆ x
⊢ russellSet x ⊆ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.power_ne | [137, 1] | [139, 46] | intro h | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ power x ≠ x | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x = x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
⊢ power x ≠ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.power_ne | [137, 1] | [139, 46] | exact (power_not_subset x) (subset_of_eq h) | V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x = x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x✝ y z x : V
h : power x = x
⊢ False
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_comm | [141, 1] | [142, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ y = y ∩ x | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ y ↔ z✝ ∈ y ∩ x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ y = y ∩ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_comm | [141, 1] | [142, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ y ↔ z✝ ∈ y ∩ 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✝ ∈ x ∩ y ↔ z✝ ∈ y ∩ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_comm | [144, 1] | [145, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ y = y ∪ x | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ y ↔ z✝ ∈ y ∪ x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ y = y ∪ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_comm | [144, 1] | [145, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ y ↔ z✝ ∈ y ∪ 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✝ ∈ x ∪ y ↔ z✝ ∈ y ∪ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_assoc | [147, 1] | [148, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ y ∩ z = x ∩ (y ∩ z) | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ y ∩ z ↔ z✝ ∈ x ∩ (y ∩ z) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ y ∩ z = x ∩ (y ∩ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_assoc | [147, 1] | [148, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ y ∩ z ↔ z✝ ∈ x ∩ (y ∩ z) | 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✝ ∈ x ∩ y ∩ z ↔ z✝ ∈ x ∩ (y ∩ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_assoc | [150, 1] | [151, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ y ∪ z = x ∪ (y ∪ z) | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ y ∪ z ↔ z✝ ∈ x ∪ (y ∪ z) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ y ∪ z = x ∪ (y ∪ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_assoc | [150, 1] | [151, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ y ∪ z ↔ z✝ ∈ x ∪ (y ∪ z) | 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✝ ∈ x ∪ y ∪ z ↔ z✝ ∈ x ∪ (y ∪ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_distrib_left | [153, 1] | [154, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ (y ∪ z) = x ∩ y ∪ x ∩ z | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ (y ∪ z) ↔ z✝ ∈ x ∩ y ∪ x ∩ z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ (y ∪ z) = x ∩ y ∪ x ∩ z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_distrib_left | [153, 1] | [154, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ (y ∪ z) ↔ z✝ ∈ x ∩ y ∪ x ∩ z | 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✝ ∈ x ∩ (y ∪ z) ↔ z✝ ∈ x ∩ y ∪ x ∩ z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_distrib_right | [156, 1] | [157, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ (x ∪ y) ∩ z = x ∩ z ∪ y ∩ z | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ (x ∪ y) ∩ z ↔ z✝ ∈ x ∩ z ∪ y ∩ z | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ (x ∪ y) ∩ z = x ∩ z ∪ y ∩ z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_distrib_right | [156, 1] | [157, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ (x ∪ y) ∩ z ↔ z✝ ∈ x ∩ z ∪ y ∩ z | 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✝ ∈ (x ∪ y) ∩ z ↔ z✝ ∈ x ∩ z ∪ y ∩ z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_distrib_left | [159, 1] | [160, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ y ∩ z = (x ∪ y) ∩ (x ∪ z) | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ y ∩ z ↔ z✝ ∈ (x ∪ y) ∩ (x ∪ z) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ y ∩ z = (x ∪ y) ∩ (x ∪ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_distrib_left | [159, 1] | [160, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ y ∩ z ↔ z✝ ∈ (x ∪ y) ∩ (x ∪ z) | 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✝ ∈ x ∪ y ∩ z ↔ z✝ ∈ (x ∪ y) ∩ (x ∪ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_distrib_right | [162, 1] | [163, 16] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ y ∪ z = (x ∪ z) ∩ (y ∪ z) | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ y ∪ z ↔ z✝ ∈ (x ∪ z) ∩ (y ∪ z) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ y ∪ z = (x ∪ z) ∩ (y ∪ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_distrib_right | [162, 1] | [163, 16] | aesop | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ y ∪ z ↔ z✝ ∈ (x ∪ z) ∩ (y ∪ z) | 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✝ ∈ x ∩ y ∪ z ↔ z✝ ∈ (x ∪ z) ∩ (y ∪ z)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_empty | [166, 1] | [167, 15] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ ∅ = ∅ | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ ∅ ↔ z✝ ∈ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∩ ∅ = ∅
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.inter_empty | [166, 1] | [167, 15] | simp | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∩ ∅ ↔ z✝ ∈ ∅ | 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✝ ∈ x ∩ ∅ ↔ z✝ ∈ ∅
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.empty_inter | [170, 1] | [171, 15] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ∅ ∩ x = ∅ | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ ∅ ∩ x ↔ z✝ ∈ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ∅ ∩ x = ∅
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.empty_inter | [170, 1] | [171, 15] | simp | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ ∅ ∩ x ↔ z✝ ∈ ∅ | 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✝ ∈ ∅ ∩ x ↔ z✝ ∈ ∅
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_empty | [174, 1] | [175, 15] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ x ∪ ∅ = x | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ 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
⊢ x ∪ ∅ = x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.union_empty | [174, 1] | [175, 15] | simp | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ x ∪ ∅ ↔ 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✝ ∈ x ∪ ∅ ↔ z✝ ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.empty_union | [178, 1] | [179, 15] | ext | V : Type u_1
inst✝ : Zermelo V
x y z : V
⊢ ∅ ∪ x = x | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ ∅ ∪ 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
⊢ ∅ ∪ x = x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Basic.lean | SetTheory.empty_union | [178, 1] | [179, 15] | simp | case h
V : Type u_1
inst✝ : Zermelo V
x y z z✝ : V
⊢ z✝ ∈ ∅ ∪ x ↔ 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✝ ∈ ∅ ∪ x ↔ z✝ ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqUnion | [16, 1] | [21, 7] | unfold BoundedFormula.eqUnion | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqUnion x y z) v l ↔ interpretTerm V v l x = interpretTerm V v l y ∪ interpretTerm V v l z | V : Type u_1
inst✝ : Zermelo 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.mem (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
interpretTerm V v l x = 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
α : Type
n : Nat
v : α → V
l : Fin n → V
x y z : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqUnion x y z) v l ↔ interpretTerm V v l x = interpretTerm V v l y ∪ interpretTerm V v l z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqUnion | [16, 1] | [21, 7] | rw [ext_iff] | V : Type u_1
inst✝ : Zermelo 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.mem (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
interpretTerm V v l x = interpretTerm V v l y ∪ interpretTerm V v l z | V : Type u_1
inst✝ : Zermelo 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.mem (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
∀ (z_1 : V), z_1 ∈ interpretTerm V v l x ↔ z_1 ∈ 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
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.mem (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
interpretTerm V v l x = interpretTerm V v l y ∪ interpretTerm V v l z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqUnion | [16, 1] | [21, 7] | simp | V : Type u_1
inst✝ : Zermelo 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.mem (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
∀ (z_1 : V), z_1 ∈ interpretTerm V v l x ↔ z_1 ∈ 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
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.mem (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc z)))))
v l ↔
∀ (z_1 : V), z_1 ∈ interpretTerm V v l x ↔ z_1 ∈ interpretTerm V v l y ∪ interpretTerm V v l z
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqSucc | [34, 1] | [38, 7] | unfold BoundedFormula.eqSucc succ | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqSucc x y) v l ↔ interpretTerm V v l x = succ (interpretTerm V v l y) | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqSingleton (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eqUnion (termSucc x) (termSucc y) (Sum.inr (Fin.last n)))))
v l ↔
interpretTerm V v l x = interpretTerm V v l y ∪ {interpretTerm V v l y} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqSucc x y) v l ↔ interpretTerm V v l x = succ (interpretTerm V v l y)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqSucc | [34, 1] | [38, 7] | simp | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqSingleton (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eqUnion (termSucc x) (termSucc y) (Sum.inr (Fin.last n)))))
v l ↔
interpretTerm V v l x = interpretTerm V v l y ∪ {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
α : Type
n : Nat
v : α → V
l : Fin n → V
x y : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqSingleton (Sum.inr (Fin.last n)) (termSucc y))
(BoundedFormula.eqUnion (termSucc x) (termSucc y) (Sum.inr (Fin.last n)))))
v l ↔
interpretTerm V v l x = interpretTerm V v l y ∪ {interpretTerm V v l y}
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqEmpty | [45, 1] | [49, 7] | unfold BoundedFormula.eqEmpty | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqEmpty x) v l ↔ interpretTerm V v l x = ∅ | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.all (BoundedFormula.not (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x)))) v l ↔
interpretTerm V v l x = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.eqEmpty x) v l ↔ interpretTerm V v l x = ∅
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqEmpty | [45, 1] | [49, 7] | rw [eq_empty_iff_forall_not_mem] | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.all (BoundedFormula.not (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x)))) v l ↔
interpretTerm V v l x = ∅ | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.all (BoundedFormula.not (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x)))) v l ↔
∀ (y : V), ¬y ∈ interpretTerm V v l x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.all (BoundedFormula.not (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x)))) v l ↔
interpretTerm V v l x = ∅
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_eqEmpty | [45, 1] | [49, 7] | simp | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.all (BoundedFormula.not (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x)))) v l ↔
∀ (y : V), ¬y ∈ interpretTerm V v l x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.all (BoundedFormula.not (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x)))) v l ↔
∀ (y : V), ¬y ∈ interpretTerm V v l x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_succClosed | [65, 1] | [69, 7] | unfold BoundedFormula.succClosed | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.succClosed x) v l ↔ ∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqSucc (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc x)))))))
v l ↔
∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.succClosed x) v l ↔ ∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_succClosed | [65, 1] | [69, 7] | simp | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqSucc (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc x)))))))
v l ↔
∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqSucc (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc x)))))))
v l ↔
∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.inductive_iff | [71, 1] | [76, 21] | constructor | V : Type u_1
inst✝ : Zermelo V
x : V
⊢ Inductive x ↔ ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x | case mp
V : Type u_1
inst✝ : Zermelo V
x : V
⊢ Inductive x → ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x
case mpr
V : Type u_1
inst✝ : Zermelo V
x : V
⊢ (∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x) → Inductive x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
x : V
⊢ Inductive x ↔ ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.inductive_iff | [71, 1] | [76, 21] | intro h | case mp
V : Type u_1
inst✝ : Zermelo V
x : V
⊢ Inductive x → ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x | case mp
V : Type u_1
inst✝ : Zermelo V
x : V
h : Inductive x
⊢ ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
x : V
⊢ Inductive x → ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.inductive_iff | [71, 1] | [76, 21] | exact ⟨h.empty_mem, h.succ_mem⟩ | case mp
V : Type u_1
inst✝ : Zermelo V
x : V
h : Inductive x
⊢ ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
V : Type u_1
inst✝ : Zermelo V
x : V
h : Inductive x
⊢ ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.inductive_iff | [71, 1] | [76, 21] | intro h | case mpr
V : Type u_1
inst✝ : Zermelo V
x : V
⊢ (∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x) → Inductive x | case mpr
V : Type u_1
inst✝ : Zermelo V
x : V
h : ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x
⊢ Inductive x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type u_1
inst✝ : Zermelo V
x : V
⊢ (∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x) → Inductive x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.inductive_iff | [71, 1] | [76, 21] | exact ⟨h.1, h.2⟩ | case mpr
V : Type u_1
inst✝ : Zermelo V
x : V
h : ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x
⊢ Inductive x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
V : Type u_1
inst✝ : Zermelo V
x : V
h : ∅ ∈ x ∧ ∀ (y : V), y ∈ x → succ y ∈ x
⊢ Inductive x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_isInductive | [87, 1] | [91, 7] | unfold BoundedFormula.isInductive | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isInductive x) v l ↔ Inductive (interpretTerm V v l x) | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.and
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqEmpty (Sum.inr (Fin.last n)))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))))
(BoundedFormula.succClosed x))
v l ↔
Inductive (interpretTerm V v l x) | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isInductive x) v l ↔ Inductive (interpretTerm V v l x)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_isInductive | [87, 1] | [91, 7] | rw [inductive_iff] | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.and
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqEmpty (Sum.inr (Fin.last n)))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))))
(BoundedFormula.succClosed x))
v l ↔
Inductive (interpretTerm V v l x) | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.and
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqEmpty (Sum.inr (Fin.last n)))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))))
(BoundedFormula.succClosed x))
v l ↔
∅ ∈ interpretTerm V v l x ∧ ∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.and
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqEmpty (Sum.inr (Fin.last n)))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))))
(BoundedFormula.succClosed x))
v l ↔
Inductive (interpretTerm V v l x)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_isInductive | [87, 1] | [91, 7] | simp | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.and
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqEmpty (Sum.inr (Fin.last n)))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))))
(BoundedFormula.succClosed x))
v l ↔
∅ ∈ interpretTerm V v l x ∧ ∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.and
(BoundedFormula.exists
(BoundedFormula.and (BoundedFormula.eqEmpty (Sum.inr (Fin.last n)))
(BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc x))))
(BoundedFormula.succClosed x))
v l ↔
∅ ∈ interpretTerm V v l x ∧ ∀ (y : V), y ∈ interpretTerm V v l x → succ y ∈ interpretTerm V v l x
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_isNat' | [99, 1] | [102, 7] | unfold BoundedFormula.isNat | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isNat x) v l ↔ ∀ (y : V), Inductive y → interpretTerm V v l x ∈ y | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.isInductive (Sum.inr (Fin.last n)))
(BoundedFormula.mem (termSucc x) (Sum.inr (Fin.last n)))))
v l ↔
∀ (y : V), Inductive y → interpretTerm V v l x ∈ y | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isNat x) v l ↔ ∀ (y : V), Inductive y → interpretTerm V v l x ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_isNat' | [99, 1] | [102, 7] | simp | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.isInductive (Sum.inr (Fin.last n)))
(BoundedFormula.mem (termSucc x) (Sum.inr (Fin.last n)))))
v l ↔
∀ (y : V), Inductive y → interpretTerm V v l x ∈ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.isInductive (Sum.inr (Fin.last n)))
(BoundedFormula.mem (termSucc x) (Sum.inr (Fin.last n)))))
v l ↔
∀ (y : V), Inductive y → interpretTerm V v l x ∈ y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.mem_ω_iff | [111, 1] | [115, 45] | unfold ω | V : Type u_1
inst✝ : Zermelo V
n : V
⊢ n ∈ ω ↔ ∀ (s : V), Inductive s → n ∈ s | V : Type u_1
inst✝ : Zermelo V
n : V
⊢ n ∈ sep (BoundedFormula.isNat (Sum.inr 0)) (fun i => Empty.elim i) Infinity.inductiveSet ↔
∀ (s : V), Inductive s → n ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
n : V
⊢ n ∈ ω ↔ ∀ (s : V), Inductive s → n ∈ s
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.mem_ω_iff | [111, 1] | [115, 45] | simp only [mem_sep_iff, interpret_isNat', interpret_inr, and_iff_right_iff_imp] | V : Type u_1
inst✝ : Zermelo V
n : V
⊢ n ∈ sep (BoundedFormula.isNat (Sum.inr 0)) (fun i => Empty.elim i) Infinity.inductiveSet ↔
∀ (s : V), Inductive s → n ∈ s | V : Type u_1
inst✝ : Zermelo V
n : V
⊢ (∀ (y : V), Inductive y → n ∈ y) → n ∈ Infinity.inductiveSet | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
n : V
⊢ n ∈ sep (BoundedFormula.isNat (Sum.inr 0)) (fun i => Empty.elim i) Infinity.inductiveSet ↔
∀ (s : V), Inductive s → n ∈ s
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.mem_ω_iff | [111, 1] | [115, 45] | intro hn | V : Type u_1
inst✝ : Zermelo V
n : V
⊢ (∀ (y : V), Inductive y → n ∈ y) → n ∈ Infinity.inductiveSet | V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (y : V), Inductive y → n ∈ y
⊢ n ∈ Infinity.inductiveSet | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
n : V
⊢ (∀ (y : V), Inductive y → n ∈ y) → n ∈ Infinity.inductiveSet
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.mem_ω_iff | [111, 1] | [115, 45] | exact hn _ Infinity.inductiveSet_inductive | V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (y : V), Inductive y → n ∈ y
⊢ n ∈ Infinity.inductiveSet | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (y : V), Inductive y → n ∈ y
⊢ n ∈ Infinity.inductiveSet
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.interpret_isNat | [118, 1] | [120, 35] | rw [mem_ω_iff, interpret_isNat'] | V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isNat x) v l ↔ interpretTerm V v l x ∈ ω | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
α : Type
n : Nat
v : α → V
l : Fin n → V
x : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isNat x) v l ↔ interpretTerm V v l x ∈ ω
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.empty_mem_ω | [122, 1] | [125, 21] | rw [mem_ω_iff] | V : Type u_1
inst✝ : Zermelo V
⊢ ∅ ∈ ω | V : Type u_1
inst✝ : Zermelo V
⊢ ∀ (s : V), Inductive s → ∅ ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
⊢ ∅ ∈ ω
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.empty_mem_ω | [122, 1] | [125, 21] | intro s hs | V : Type u_1
inst✝ : Zermelo V
⊢ ∀ (s : V), Inductive s → ∅ ∈ s | V : Type u_1
inst✝ : Zermelo V
s : V
hs : Inductive s
⊢ ∅ ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
⊢ ∀ (s : V), Inductive s → ∅ ∈ s
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.empty_mem_ω | [122, 1] | [125, 21] | exact hs.empty_mem | V : Type u_1
inst✝ : Zermelo V
s : V
hs : Inductive s
⊢ ∅ ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
s : V
hs : Inductive s
⊢ ∅ ∈ s
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.succ_mem_ω | [127, 1] | [130, 32] | rw [mem_ω_iff] at hn ⊢ | V : Type u_1
inst✝ : Zermelo V
n : V
hn : n ∈ ω
⊢ succ n ∈ ω | V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (s : V), Inductive s → n ∈ s
⊢ ∀ (s : V), Inductive s → succ n ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
n : V
hn : n ∈ ω
⊢ succ n ∈ ω
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.succ_mem_ω | [127, 1] | [130, 32] | intro s hs | V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (s : V), Inductive s → n ∈ s
⊢ ∀ (s : V), Inductive s → succ n ∈ s | V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (s : V), Inductive s → n ∈ s
s : V
hs : Inductive s
⊢ succ n ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (s : V), Inductive s → n ∈ s
⊢ ∀ (s : V), Inductive s → succ n ∈ s
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.succ_mem_ω | [127, 1] | [130, 32] | exact hs.succ_mem n (hn s hs) | V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (s : V), Inductive s → n ∈ s
s : V
hs : Inductive s
⊢ succ n ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
n : V
hn : ∀ (s : V), Inductive s → n ∈ s
s : V
hs : Inductive s
⊢ succ n ∈ s
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.one_def | [145, 1] | [147, 7] | show ∅ ∪ {∅} = {∅} | V : Type u_1
inst✝ : Zermelo V
⊢ 1 = {∅} | V : Type u_1
inst✝ : Zermelo V
⊢ ∅ ∪ {∅} = {∅} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
⊢ 1 = {∅}
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Nat.lean | SetTheory.one_def | [145, 1] | [147, 7] | simp | V : Type u_1
inst✝ : Zermelo V
⊢ ∅ ∪ {∅} = {∅} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
⊢ ∅ ∪ {∅} = {∅}
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.interpret_isRelation | [17, 1] | [20, 7] | unfold BoundedFormula.isRelation IsRelation | V : Type u_1
inst✝ : Zermelo V
r✝ x y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isRelation r) v l ↔ IsRelation (interpretTerm V v l r) | V : Type u_1
inst✝ : Zermelo V
r✝ x y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc r))
(BoundedFormula.isOPair (Sum.inr (Fin.last n)))))
v l ↔
∀ (p : V), p ∈ interpretTerm V v l r → IsOPair p | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r✝ x y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.isRelation r) v l ↔ IsRelation (interpretTerm V v l r)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.interpret_isRelation | [17, 1] | [20, 7] | simp | V : Type u_1
inst✝ : Zermelo V
r✝ x y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc r))
(BoundedFormula.isOPair (Sum.inr (Fin.last n)))))
v l ↔
∀ (p : V), p ∈ interpretTerm V v l r → IsOPair p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r✝ x y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.all
(BoundedFormula.imp (BoundedFormula.mem (Sum.inr (Fin.last n)) (termSucc r))
(BoundedFormula.isOPair (Sum.inr (Fin.last n)))))
v l ↔
∀ (p : V), p ∈ interpretTerm V v l r → IsOPair p
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_left_mem_sUnion_sUnion | [22, 1] | [27, 27] | simp only [mem_sUnion_iff] | V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ x ∈ ⋃ ⋃ r | V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ ∃ t, (∃ t_1, t_1 ∈ r ∧ t ∈ t_1) ∧ x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ x ∈ ⋃ ⋃ r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_left_mem_sUnion_sUnion | [22, 1] | [27, 27] | refine ⟨{x}, ⟨_, hxyr, ?_⟩, ?_⟩ | V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ ∃ t, (∃ t_1, t_1 ∈ r ∧ t ∈ t_1) ∧ x ∈ t | case refine_1
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ {x} ∈ opair x y
case refine_2
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ x ∈ {x} | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ ∃ t, (∃ t_1, t_1 ∈ r ∧ t ∈ t_1) ∧ x ∈ t
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_left_mem_sUnion_sUnion | [22, 1] | [27, 27] | simp only [opair, mem_pair_iff, true_or] | case refine_1
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ {x} ∈ opair x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ {x} ∈ opair x y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_left_mem_sUnion_sUnion | [22, 1] | [27, 27] | rw [mem_singleton_iff] | case refine_2
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ x ∈ {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ x ∈ {x}
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_right_mem_sUnion_sUnion | [29, 1] | [34, 25] | simp only [mem_sUnion_iff] | V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ y ∈ ⋃ ⋃ r | V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ ∃ t, (∃ t_1, t_1 ∈ r ∧ t ∈ t_1) ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ y ∈ ⋃ ⋃ r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_right_mem_sUnion_sUnion | [29, 1] | [34, 25] | refine ⟨pair x y, ⟨_, hxyr, ?_⟩, ?_⟩ | V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ ∃ t, (∃ t_1, t_1 ∈ r ∧ t ∈ t_1) ∧ y ∈ t | case refine_1
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ pair x y ∈ opair x y
case refine_2
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ y ∈ pair x y | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ ∃ t, (∃ t_1, t_1 ∈ r ∧ t ∈ t_1) ∧ y ∈ t
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_right_mem_sUnion_sUnion | [29, 1] | [34, 25] | simp only [opair, mem_pair_iff, or_true] | case refine_1
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ pair x y ∈ opair x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ pair x y ∈ opair x y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.opair_right_mem_sUnion_sUnion | [29, 1] | [34, 25] | exact right_mem_pair | case refine_2
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ y ∈ pair x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
V : Type u_1
inst✝ : Zermelo V
r x y z : V
hxyr : opair x y ∈ r
⊢ y ∈ pair x y
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.interpret_memDom' | [43, 1] | [47, 7] | unfold BoundedFormula.memDom | V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memDom x r) v l ↔ ∃ z, opair (interpretTerm V v l x) z ∈ interpretTerm V v l r | V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqOPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc r))))))
v l ↔
∃ z, opair (interpretTerm V v l x) z ∈ interpretTerm V v l r | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memDom x r) v l ↔ ∃ z, opair (interpretTerm V v l x) z ∈ interpretTerm V v l r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.interpret_memDom' | [43, 1] | [47, 7] | simp | V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqOPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc r))))))
v l ↔
∃ z, opair (interpretTerm V v l x) z ∈ interpretTerm V v l r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqOPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc x)) (Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc r))))))
v l ↔
∃ z, opair (interpretTerm V v l x) z ∈ interpretTerm V v l r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.mem_dom_iff | [52, 1] | [57, 37] | unfold dom | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ dom r ↔ ∃ z, opair x z ∈ r | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ sep (BoundedFormula.memDom (Sum.inr 0) (Sum.inl ())) (fun x => r) (⋃ ⋃ r) ↔ ∃ z, opair x z ∈ r | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ dom r ↔ ∃ z, opair x z ∈ r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.mem_dom_iff | [52, 1] | [57, 37] | simp only [mem_sep_iff, interpret_memDom', interpret_inr, interpret_inl,
and_iff_right_iff_imp, forall_exists_index] | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ sep (BoundedFormula.memDom (Sum.inr 0) (Sum.inl ())) (fun x => r) (⋃ ⋃ r) ↔ ∃ z, opair x z ∈ r | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ ∀ (x_1 : V), opair x x_1 ∈ r → x ∈ ⋃ ⋃ r | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ sep (BoundedFormula.memDom (Sum.inr 0) (Sum.inl ())) (fun x => r) (⋃ ⋃ r) ↔ ∃ z, opair x z ∈ r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.mem_dom_iff | [52, 1] | [57, 37] | intro t | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ ∀ (x_1 : V), opair x x_1 ∈ r → x ∈ ⋃ ⋃ r | V : Type u_1
inst✝ : Zermelo V
r x y z t : V
⊢ opair x t ∈ r → x ∈ ⋃ ⋃ r | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ ∀ (x_1 : V), opair x x_1 ∈ r → x ∈ ⋃ ⋃ r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.mem_dom_iff | [52, 1] | [57, 37] | exact opair_left_mem_sUnion_sUnion | V : Type u_1
inst✝ : Zermelo V
r x y z t : V
⊢ opair x t ∈ r → x ∈ ⋃ ⋃ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z t : V
⊢ opair x t ∈ r → x ∈ ⋃ ⋃ r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.interpret_memDom | [60, 1] | [62, 38] | rw [mem_dom_iff, interpret_memDom'] | V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memDom x r) v l ↔ interpretTerm V v l x ∈ dom (interpretTerm V v l r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memDom x r) v l ↔ interpretTerm V v l x ∈ dom (interpretTerm V v l r)
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.interpret_memRan' | [71, 1] | [75, 7] | unfold BoundedFormula.memRan | V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memRan x r) v l ↔ ∃ z, opair z (interpretTerm V v l x) ∈ interpretTerm V v l r | V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqOPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc x)))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc r))))))
v l ↔
∃ z, opair z (interpretTerm V v l x) ∈ interpretTerm V v l r | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V (BoundedFormula.memRan x r) v l ↔ ∃ z, opair z (interpretTerm V v l x) ∈ interpretTerm V v l r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.interpret_memRan' | [71, 1] | [75, 7] | simp | V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqOPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc x)))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc r))))))
v l ↔
∃ z, opair z (interpretTerm V v l x) ∈ interpretTerm V v l r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r✝ x✝ y z : V
α : Type
n : Nat
v : α → V
l : Fin n → V
x r : α ⊕ Fin n
⊢ Interpret V
(BoundedFormula.exists
(BoundedFormula.exists
(BoundedFormula.and
(BoundedFormula.eqOPair (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(Sum.inr { val := n, isLt := (_ : n < n + (1 + 1)) }) (termSucc (termSucc x)))
(BoundedFormula.mem (Sum.inr { val := n + 1, isLt := (_ : n + 1 < n + 1 + (0 + 1)) })
(termSucc (termSucc r))))))
v l ↔
∃ z, opair z (interpretTerm V v l x) ∈ interpretTerm V v l r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.mem_ran_iff | [80, 1] | [85, 38] | unfold ran | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ ran r ↔ ∃ z, opair z x ∈ r | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ sep (BoundedFormula.memRan (Sum.inr 0) (Sum.inl ())) (fun x => r) (⋃ ⋃ r) ↔ ∃ z, opair z x ∈ r | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ ran r ↔ ∃ z, opair z x ∈ r
TACTIC:
|
https://github.com/zeramorphic/set-theory.git | 37e9d6e920ff687317f50fa33984b273f1637716 | SetTheory/Relation.lean | SetTheory.mem_ran_iff | [80, 1] | [85, 38] | simp only [mem_sep_iff, interpret_memRan', interpret_inr, interpret_inl,
and_iff_right_iff_imp, forall_exists_index] | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ sep (BoundedFormula.memRan (Sum.inr 0) (Sum.inl ())) (fun x => r) (⋃ ⋃ r) ↔ ∃ z, opair z x ∈ r | V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ ∀ (x_1 : V), opair x_1 x ∈ r → x ∈ ⋃ ⋃ r | Please generate a tactic in lean4 to solve the state.
STATE:
V : Type u_1
inst✝ : Zermelo V
r x y z : V
⊢ x ∈ sep (BoundedFormula.memRan (Sum.inr 0) (Sum.inl ())) (fun x => r) (⋃ ⋃ r) ↔ ∃ z, opair z x ∈ r
TACTIC:
|
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