file stringlengths 21 79 | dependencies sequencelengths 1 16 | definitions listlengths 1 625 | theorem_idx int64 0 574 | theorem stringlengths 3 121 | theorem_loc sequencelengths 2 2 | tactic_idx int64 0 155 | tactic_len int64 1 156 | tactic stringlengths 3 5.76k | state_before stringlengths 7 13.6k | state_after stringlengths 7 13.6k |
|---|---|---|---|---|---|---|---|---|---|---|
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n ∀ y ∈ x, y ⊆ x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive ∅",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 0 | 6 | rw [← sUnion_pair] | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
⊢ (x ∪ y).IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
⊢ (⋃₀ {x, y}).IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n ∀ y ∈ x, y ⊆ x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive ∅",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 1 | 6 | apply IsTransitive.sUnion' fun z => _ | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
⊢ (⋃₀ {x, y}).IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
⊢ ∀ z ∈ {x, y}, z.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n ∀ y ∈ x, y ⊆ x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive ∅",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 2 | 6 | intro | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
⊢ ∀ z ∈ {x, y}, z.IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
z✝ : ZFSet
⊢ z✝ ∈ {x, y} → z✝.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n ∀ y ∈ x, y ⊆ x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive ∅",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 3 | 6 | rw [mem_pair] | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
z✝ : ZFSet
⊢ z✝ ∈ {x, y} → z✝.IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
z✝ : ZFSet
⊢ z✝ = x ∨ z✝ = y → z✝.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n ∀ y ∈ x, y ⊆ x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive ∅",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 4 | 6 | rintro (rfl | rfl) | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
z✝ : ZFSet
⊢ z✝ = x ∨ z✝ = y → z✝.IsTransitive | case inl
y z : ZFSet
hy : y.IsTransitive
z✝ : ZFSet
hx : z✝.IsTransitive
⊢ z✝.IsTransitive
case inr
x z : ZFSet
hx : x.IsTransitive
z✝ : ZFSet
hy : z✝.IsTransitive
⊢ z✝.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n ∀ y ∈ x, y ⊆ x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive ∅",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 5 | 6 | assumption' | case inl
y z : ZFSet
hy : y.IsTransitive
z✝ : ZFSet
hx : z✝.IsTransitive
⊢ z✝.IsTransitive
case inr
x z : ZFSet
hx : x.IsTransitive
z✝ : ZFSet
hy : z✝.IsTransitive
⊢ z✝.IsTransitive | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 0 | Surreal.Multiplication.P3_comm | [
[
96,
52
],
[
98,
34
]
] | 0 | 2 | rw [P3, P3, add_comm] | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ P3 x₁ x₂ y₁ y₂ ↔ P3 y₁ y₂ x₁ x₂ | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ ⟦x₂ * y₁⟧ + ⟦x₁ * y₂⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ ↔ ⟦y₁ * x₂⟧ + ⟦y₂ * x₁⟧ < ⟦y₁ * x₁⟧ + ⟦y₂ * x₂⟧ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 0 | Surreal.Multiplication.P3_comm | [
[
96,
52
],
[
98,
34
]
] | 1 | 2 | congr! 2 <;> rw [quot_mul_comm] | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ ⟦x₂ * y₁⟧ + ⟦x₁ * y₂⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ ↔ ⟦y₁ * x₂⟧ + ⟦y₂ * x₁⟧ < ⟦y₁ * x₁⟧ + ⟦y₂ * x₂⟧ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 1 | Surreal.Multiplication.P3.trans | [
[
100,
80
],
[
103,
44
]
] | 0 | 3 | rw [P3] at h₁ h₂ | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
h₁ : P3 x₁ x₂ y₁ y₂
h₂ : P3 x₂ x₃ y₁ y₂
⊢ P3 x₁ x₃ y₁ y₂ | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
h₁ : ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧
h₂ : ⟦x₂ * y₂⟧ + ⟦x₃ * y₁⟧ < ⟦x₂ * y₁⟧ + ⟦x₃ * y₂⟧
⊢ P3 x₁ x₃ y₁ y₂ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 1 | Surreal.Multiplication.P3.trans | [
[
100,
80
],
[
103,
44
]
] | 1 | 3 | rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)] | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
h₁ : ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧
h₂ : ⟦x₂ * y₂⟧ + ⟦x₃ * y₁⟧ < ⟦x₂ * y₁⟧ + ⟦x₃ * y₂⟧
⊢ P3 x₁ x₃ y₁ y₂ | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
h₁ : ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧
h₂ : ⟦x₂ * y₂⟧ + ⟦x₃ * y₁⟧ < ⟦x₂ * y₁⟧ + ⟦x₃ * y₂⟧
⊢ ⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧ + (⟦x₁ * y₂⟧ + ⟦x₃ * y₁⟧) < ⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧ + (⟦x₁ * y₁⟧ + ⟦x₃ * y₂⟧) |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 1 | Surreal.Multiplication.P3.trans | [
[
100,
80
],
[
103,
44
]
] | 2 | 3 | convert add_lt_add h₁ h₂ using 1 <;> abel | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
h₁ : ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧
h₂ : ⟦x₂ * y₂⟧ + ⟦x₃ * y₁⟧ < ⟦x₂ * y₁⟧ + ⟦x₃ * y₂⟧
⊢ ⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧ + (⟦x₁ * y₂⟧ + ⟦x₃ * y₁⟧) < ⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧ + (⟦x₁ * y₁⟧ + ⟦x₃ * y₂⟧) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 2 | Surreal.Multiplication.P3_neg | [
[
105,
57
],
[
108,
10
]
] | 0 | 3 | simp_rw [P3, quot_neg_mul] | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ P3 x₁ x₂ y₁ y₂ ↔ P3 (-x₂) (-x₁) y₁ y₂ | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ ↔ -⟦x₂ * y₂⟧ + -⟦x₁ * y₁⟧ < -⟦x₂ * y₁⟧ + -⟦x₁ * y₂⟧ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 2 | Surreal.Multiplication.P3_neg | [
[
105,
57
],
[
108,
10
]
] | 1 | 3 | rw [← _root_.neg_lt_neg_iff] | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ ↔ -⟦x₂ * y₂⟧ + -⟦x₁ * y₁⟧ < -⟦x₂ * y₁⟧ + -⟦x₁ * y₂⟧ | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ -(⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧) < -(⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧) ↔ -⟦x₂ * y₂⟧ + -⟦x₁ * y₁⟧ < -⟦x₂ * y₁⟧ + -⟦x₁ * y₂⟧ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 2 | Surreal.Multiplication.P3_neg | [
[
105,
57
],
[
108,
10
]
] | 2 | 3 | abel_nf | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ -(⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧) < -(⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧) ↔ -⟦x₂ * y₂⟧ + -⟦x₁ * y₁⟧ < -⟦x₂ * y₁⟧ + -⟦x₁ * y₂⟧ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def P1 (x₁ x₂ x₃ y₁ y₂ y₃ : PGame) :=\n ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧ - ⟦x₁ * y₂⟧ < ⟦x₃ * y₁⟧ + ⟦x₂ * y₃⟧ - (⟦x₃ * y₃⟧ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
] | 0 | 8 | rw [P2, P2] | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ P2 x₁ x₂ y ↔ P2 (-x₂) (-x₁) y | x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame
⊢ x₁ ≈ x₂ → ⟦x₁ * y⟧ = ⟦x₂ * y⟧ ↔ -x₂ ≈ -x₁ → ⟦-x₂ * y⟧ = ⟦-x₁ * y⟧ |
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