file stringlengths 21 79 | dependencies listlengths 1 16 | definitions listlengths 1 625 | theorem_idx int64 0 574 | theorem stringlengths 3 121 | theorem_loc listlengths 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 β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 0 | 6 | rw [β sUnion_pair] | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ (x βͺ y).IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ (ββ {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 β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"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
β’ (ββ {x, y}).IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ β z β {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 β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 2 | 6 | intro | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ β z β {x, y}, z.IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ β {x, y} β zβ.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 β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 3 | 6 | rw [mem_pair] | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ β {x, y} β zβ.IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ = x β¨ zβ = y β zβ.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 β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 4 | 6 | rintro (rfl | rfl) | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ = x β¨ zβ = y β zβ.IsTransitive | case inl
y z : ZFSet
hy : y.IsTransitive
zβ : ZFSet
hx : zβ.IsTransitive
β’ zβ.IsTransitive
case inr
x z : ZFSet
hx : x.IsTransitive
zβ : ZFSet
hy : zβ.IsTransitive
β’ zβ.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 β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
... | 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
] | 5 | 6 | assumption' | case inl
y z : ZFSet
hy : y.IsTransitive
zβ : ZFSet
hx : zβ.IsTransitive
β’ zβ.IsTransitive
case inr
x z : ZFSet
hx : x.IsTransitive
zβ : ZFSet
hy : zβ.IsTransitive
β’ zβ.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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β β¦yβ * xββ§ + β¦yβ * xββ§ < β¦yβ * xββ§ + β¦yβ * xββ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β β¦yβ * xββ§ + β¦yβ * xββ§ < β¦yβ * xββ§ + β¦yβ * xββ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 hβ hβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ P3 xβ xβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ P3 xβ xβ yβ yβ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ P3 xβ xβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 hβ hβ using 1 <;> abel | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 [β _root_.neg_lt_neg_iff] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ -(β¦xβ * yββ§ + β¦xβ * yββ§) < -(β¦xβ * yββ§ + β¦xβ * yββ§) β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ -(β¦xβ * yββ§ + β¦xβ * yββ§) < -(β¦xβ * yββ§ + β¦xβ * yββ§) β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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 xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P2 xβ xβ y β P2 (-xβ) (-xβ) y | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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
]
] | 1 | 8 | constructor | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§ | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§
case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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
]
] | 2 | 8 | Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]
exact (Β· Β·) | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§
case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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
]
] | 3 | 8 | Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]
exact (Β· Β·) | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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
]
] | 4 | 8 | rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm] | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§ | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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
]
] | 5 | 8 | exact (Β· Β·) | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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
]
] | 6 | 8 | rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm] | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : 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
]
] | 7 | 8 | exact (Β· Β·) | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 4 | Surreal.Multiplication.P2_neg_right | [
[
118,
52
],
[
119,
51
]
] | 0 | 1 | rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P2 xβ xβ y β P2 xβ xβ (-y) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 5 | Surreal.Multiplication.P4_neg_left | [
[
121,
54
],
[
122,
62
]
] | 0 | 1 | simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P4 xβ xβ y β P4 (-xβ) (-xβ) y | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 6 | Surreal.Multiplication.P4_neg_right | [
[
124,
52
],
[
125,
33
]
] | 0 | 1 | rw [P4, P4, neg_neg, and_comm] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P4 xβ xβ y β P4 xβ xβ (-y) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 7 | Surreal.Multiplication.P24_neg_left | [
[
127,
57
],
[
127,
99
]
] | 0 | 1 | rw [P24, P24, P2_neg_left, P4_neg_left] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P24 xβ xβ y β P24 (-xβ) (-xβ) y | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 8 | Surreal.Multiplication.P24_neg_right | [
[
128,
55
],
[
128,
99
]
] | 0 | 1 | rw [P24, P24, P2_neg_right, P4_neg_right] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P24 xβ xβ y β P24 xβ xβ (-y) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 9 | Surreal.Multiplication.mulOption_lt_iff_P1 | [
[
134,
79
],
[
136,
53
]
] | 0 | 2 | dsimp only [P1, mulOption, quot_sub, quot_add] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ β P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
-(β¦x.moveLeft j * -yβ§ + β¦x * (-y).moveLeft lβ§ - β¦x.moveLeft j * (-y).moveLeft lβ§) β
β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
β¦x.moveLeft j * yβ§ + β¦x * -(-y).moveLeft lβ§ - β¦x.moveLeft j * -(-y).moveLeft lβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 9 | Surreal.Multiplication.mulOption_lt_iff_P1 | [
[
134,
79
],
[
136,
53
]
] | 1 | 2 | simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
-(β¦x.moveLeft j * -yβ§ + β¦x * (-y).moveLeft lβ§ - β¦x.moveLeft j * (-y).moveLeft lβ§) β
β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
β¦x.moveLeft j * yβ§ + β¦x * -(-y).moveLeft lβ§ - β¦x.moveLeft j * -(-y).moveLeft lβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 10 | Surreal.Multiplication.mulOption_lt_mul_iff_P3 | [
[
139,
86
],
[
141,
27
]
] | 0 | 2 | dsimp only [mulOption, quot_sub, quot_add] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i : x.LeftMoves
j : y.LeftMoves
β’ β¦x.mulOption y i jβ§ < β¦x * yβ§ β P3 (x.moveLeft i) x (y.moveLeft j) y | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i : x.LeftMoves
j : y.LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft jβ§ - β¦x.moveLeft i * y.moveLeft jβ§ < β¦x * yβ§ β
P3 (x.moveLeft i) x (y.moveLeft j) y |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 10 | Surreal.Multiplication.mulOption_lt_mul_iff_P3 | [
[
139,
86
],
[
141,
27
]
] | 1 | 2 | exact sub_lt_iff_lt_add' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i : x.LeftMoves
j : y.LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft jβ§ - β¦x.moveLeft i * y.moveLeft jβ§ < β¦x * yβ§ β
P3 (x.moveLeft i) x (y.moveLeft j) y | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 11 | Surreal.Multiplication.P1_of_eq | [
[
144,
29
],
[
146,
56
]
] | 0 | 2 | rw [P1, β hβ he, β hβ he, sub_lt_sub_iff] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
he : xβ β xβ
hβ : P2 xβ xβ yβ
hβ : P2 xβ xβ yβ
h3 : P3 xβ xβ yβ yβ
β’ P1 xβ xβ xβ yβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
he : xβ β xβ
hβ : P2 xβ xβ yβ
hβ : P2 xβ xβ yβ
h3 : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 11 | Surreal.Multiplication.P1_of_eq | [
[
144,
29
],
[
146,
56
]
] | 1 | 2 | convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
he : xβ β xβ
hβ : P2 xβ xβ yβ
hβ : P2 xβ xβ yβ
h3 : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 12 | Surreal.Multiplication.P1_of_lt | [
[
148,
86
],
[
150,
44
]
] | 0 | 2 | rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ P1 xβ xβ xβ yβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 12 | Surreal.Multiplication.P1_of_lt | [
[
148,
86
],
[
150,
44
]
] | 1 | 2 | convert add_lt_add hβ hβ using 1 <;> abel | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 13 | Surreal.Multiplication.Args.numeric_P1 | [
[
165,
80
],
[
166,
39
]
] | 0 | 1 | simp [Args.Numeric, Args.toMultiset] | xβ xβ xβ xβ x' yβ yβ yβ yβ y' : PGame
x y : PGame
β’ (P1 x y).Numeric β x.Numeric β§ y.Numeric | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 14 | Surreal.Multiplication.Args.numeric_P24 | [
[
169,
73
],
[
170,
39
]
] | 0 | 1 | simp [Args.Numeric, Args.toMultiset] | x xββ xββ xβ x' yβ yβ yβ yβ y' : PGame
xβ xβ y : PGame
β’ (P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
] | 0 | 5 | rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
β’ IH1 x y | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y) |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
] | 1 | 5 | on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy) | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y) | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
] | 2 | 5 | all_goals exact TransGen.single (cutExpand_double_left hβ hβ) | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
] | 3 | 5 | refine TransGen.tail ?_ (cutExpand_pair_right hy) | case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y) | case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
] | 4 | 5 | exact TransGen.single (cutExpand_double_left hβ hβ) | case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 0 | 9 | obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i < x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 1 | 9 | Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h | case inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i < x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 2 | 9 | Β· have ml := @IsOption.moveLeft
exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1
(ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l) | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 3 | 9 | Β· rw [mulOption_neg_neg, lt_neg]
exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 4 | 9 | exact mulOption_lt_of_lt hy ihxy ihyx i j k l h | case inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i < x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 5 | 9 | have ml := @IsOption.moveLeft | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
ml : β {x : PGame} (i : x.LeftMoves), (x.moveLeft i).IsOption x
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 6 | 9 | exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1
(ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l) | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
ml : β {x : PGame} (i : x.LeftMoves), (x.moveLeft i).IsOption x
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 7 | 9 | rw [mulOption_neg_neg, lt_neg] | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption (-y) j lβ§ < -β¦x.mulOption (- -y) i (toLeftMovesNeg (toRightMovesNeg k))β§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
] | 8 | 9 | exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption (-y) j lβ§ < -β¦x.mulOption (- -y) i (toLeftMovesNeg (toRightMovesNeg k))β§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 0 | 29 | have ihxy := ih1 ih | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
β’ (x * y).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 1 | 29 | have ihyx := ih1_swap ih | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
β’ (x * y).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 2 | 29 | have ihxyn := ih1_neg_left (ih1_neg_right ihxy) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
β’ (x * y).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 3 | 29 | have ihyxn := ih1_neg_left (ih1_neg_right ihyx) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ (x * y).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 4 | 29 | refine numeric_def.mpr β¨?_, ?_, ?_β© | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ (x * y).Numeric | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 5 | 29 | Β· simp_rw [lt_iff_game_lt]
intro i
rw [rightMoves_mul_iff]
constructor <;> (intro j l; revert i; rw [leftMoves_mul_iff (_ > Β·)]; constructor <;> intro i k)
Β· apply mulOption_lt hx hy ihxy ihyx
Β· simp_rw [β mulOption_symm (-y), mulOption_neg_neg x]
apply mulOption_lt hy.neg hx.neg ihyxn ihxyn
Β· simp only [β mulOption_symm y]
apply mulOption_lt hy hx ihyx ihxy
Β· rw [mulOption_neg_neg y]
apply mulOption_lt hx.neg hy.neg ihxyn ihyxn | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 6 | 29 | all_goals
cases x; cases y
rintro (β¨i,jβ©|β¨i,jβ©) <;>
refine ((numeric_option_mul ih ?_).add <| numeric_mul_option ih ?_).sub
(numeric_option_mul_option ih ?_ ?_) <;>
solve_by_elim [IsOption.mk_left, IsOption.mk_right] | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 7 | 29 | simp_rw [lt_iff_game_lt] | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 8 | 29 | intro i | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ β (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 9 | 29 | rw [rightMoves_mul_iff] | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ β (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ (β (i_1 : x.LeftMoves) (j : (-y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦x.mulOption (-y) i_1 jβ§) β§
β (i_1 : (-x).LeftMoves) (j : y.LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y i_1 jβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 10 | 29 | constructor <;> (intro j l; revert i; rw [leftMoves_mul_iff (_ > Β·)]; constructor <;> intro i k) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ (β (i_1 : x.LeftMoves) (j : (-y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦x.mulOption (-y) i_1 jβ§) β§
β (i_1 : (-x).LeftMoves) (j : y.LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y i_1 jβ§ | case refine_1.left.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦x.mulOption y i kβ§
case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 11 | 29 | Β· apply mulOption_lt hx hy ihxy ihyx | case refine_1.left.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦x.mulOption y i kβ§
case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 12 | 29 | Β· simp_rw [β mulOption_symm (-y), mulOption_neg_neg x]
apply mulOption_lt hy.neg hx.neg ihyxn ihxyn | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 13 | 29 | Β· simp only [β mulOption_symm y]
apply mulOption_lt hy hx ihyx ihxy | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 14 | 29 | Β· rw [mulOption_neg_neg y]
apply mulOption_lt hx.neg hy.neg ihxyn ihyxn | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 15 | 29 | intro j l | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ β (i_1 : (-x).LeftMoves) (j : y.LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y i_1 jβ§ | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
j : (-x).LeftMoves
l : y.LeftMoves
β’ β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 16 | 29 | revert i | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
j : (-x).LeftMoves
l : y.LeftMoves
β’ β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ β (i : (x * y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 17 | 29 | rw [leftMoves_mul_iff (_ > Β·)] | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ β (i : (x * y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ (β (i : x.LeftMoves) (j_1 : y.LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i j_1β§) β§
β (i : (-x).LeftMoves) (j_1 : (-y).LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i j_1β§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 18 | 29 | constructor <;> intro i k | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ (β (i : x.LeftMoves) (j_1 : y.LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i j_1β§) β§
β (i : (-x).LeftMoves) (j_1 : (-y).LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i j_1β§ | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 19 | 29 | apply mulOption_lt hx hy ihxy ihyx | case refine_1.left.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦x.mulOption y i kβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 20 | 29 | simp_rw [β mulOption_symm (-y), mulOption_neg_neg x] | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-y).mulOption (- -x) l (toLeftMovesNeg (toRightMovesNeg j))β§ > β¦(-y).mulOption (-x) k iβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 21 | 29 | apply mulOption_lt hy.neg hx.neg ihyxn ihxyn | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-y).mulOption (- -x) l (toLeftMovesNeg (toRightMovesNeg j))β§ > β¦(-y).mulOption (-x) k iβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 22 | 29 | simp only [β mulOption_symm y] | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§ | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦y.mulOption (-x) l jβ§ > β¦y.mulOption x k iβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 23 | 29 | apply mulOption_lt hy hx ihyx ihxy | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦y.mulOption (-x) l jβ§ > β¦y.mulOption x k iβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 24 | 29 | rw [mulOption_neg_neg y] | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption (- -y) j (toLeftMovesNeg (toRightMovesNeg l))β§ > β¦(-x).mulOption (-y) i kβ§ |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 25 | 29 | apply mulOption_lt hx.neg hy.neg ihxyn ihyxn | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption (- -y) j (toLeftMovesNeg (toRightMovesNeg l))β§ > β¦(-x).mulOption (-y) i kβ§ | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 26 | 29 | cases x | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric | case refine_3.mk
xβ xβ xβ x' y yβ yβ yβ y' : PGame
hy : y.Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y) β P124 a
hx : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ihxy : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y
ihyx : IH1 y (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihxyn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-y)
ihyxn : IH1 (-y) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
β’ β (j : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).RightMoves), ((PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).moveRight j).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 27 | 29 | cases y | case refine_3.mk
xβ xβ xβ x' y yβ yβ yβ y' : PGame
hy : y.Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y) β P124 a
hx : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ihxy : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y
ihyx : IH1 y (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihxyn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-y)
ihyxn : IH1 (-y) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
β’ β (j : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).RightMoves), ((PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).moveRight j).Numeric | case refine_3.mk.mk
xβ xβ xβ x' yβ yβ yβ y' : PGame
Ξ±βΒΉ Ξ²βΒΉ : Type u
aβΒ³ : Ξ±βΒΉ β PGame
aβΒ² : Ξ²βΒΉ β PGame
hx : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²).Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
hy : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)) β P124 a
ihxy : IH1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyx : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
ihxyn : IH1 (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyxn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
β’ β (j : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).RightMoves),
((PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).moveRight j).Numeric |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
] | 28 | 29 | rintro (β¨i,jβ©|β¨i,jβ©) <;>
refine ((numeric_option_mul ih ?_).add <| numeric_mul_option ih ?_).sub
(numeric_option_mul_option ih ?_ ?_) <;>
solve_by_elim [IsOption.mk_left, IsOption.mk_right] | case refine_3.mk.mk
xβ xβ xβ x' yβ yβ yβ y' : PGame
Ξ±βΒΉ Ξ²βΒΉ : Type u
aβΒ³ : Ξ±βΒΉ β PGame
aβΒ² : Ξ²βΒΉ β PGame
hx : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²).Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
hy : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)) β P124 a
ihxy : IH1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyx : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
ihxyn : IH1 (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyxn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
β’ β (j : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).RightMoves),
((PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).moveRight j).Numeric | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 0 | 8 | rw [IH24] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ IH24 xβ xβ y | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ β β¦z : PGameβ¦, (z.IsOption xβ β P24 z xβ y) β§ (z.IsOption xβ β P24 xβ z y) β§ (z.IsOption y β P24 xβ xβ z) |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 1 | 8 | refine fun z β¦ β¨?_, ?_, ?_β© <;>
refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ β β¦z : PGameβ¦, (z.IsOption xβ β P24 z xβ y) β§ (z.IsOption xβ β P24 xβ z y) β§ (z.IsOption y β P24 xβ xβ z) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 2 | 8 | Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 3 | 8 | Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h) | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 4 | 8 | Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h) | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 5 | 8 | exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 6 | 8 | exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h) | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
] | 7 | 8 | exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h) | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 0 | 8 | refine fun z w h β¦ β¨?_, ?_β© | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ IH4 xβ xβ y | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 z xβ w
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 xβ z w |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 1 | 8 | all_goals
intro h'
apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|
(cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1
try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'
try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h' | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 z xβ w
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 xβ z w | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 2 | 8 | intro h' | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 xβ z w | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ P2 xβ z w |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 3 | 8 | apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|
(cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1 | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ P2 xβ z w | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z w).toMultiset ({xβ} + {xβ, w}) |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 4 | 8 | try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ w).toMultiset ({xβ} + {xβ, w}) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 5 | 8 | try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z w).toMultiset ({xβ} + {xβ, w}) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 6 | 8 | exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ w).toMultiset ({xβ} + {xβ, w}) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
] | 7 | 8 | exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z w).toMultiset ({xβ} + {xβ, w}) | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
] | 0 | 5 | constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ (xβ * y).Numeric β§ (xβ * y).Numeric | case left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
] | 1 | 5 | Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero | case left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset |
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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
] | 2 | 5 | Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero | case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | 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 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": ... | 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
] | 3 | 5 | exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero | case left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | no goals |
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