url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | have commutes: pi ∘ g' ∘ incl = g := by
ext y
rw [comp_apply, comp_apply] | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | obtain ⟨K''⟩ : Nonempty (PositiveCompacts (E × (Fin (n-m) → ℝ))) := PositiveCompacts.nonempty' | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | let μ' : Measure (E × (Fin (n-m) → ℝ)) := addHaarMeasure K'' | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : Smoot... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | have aux : μ' (incl '' U) = 0 := by sorry | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : Smoot... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | have : ν ((pi ∘ g' ∘ incl) '' U) = 0 := by
sorry | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : Smoot... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | rw [← commutes] | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : Smoot... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | exact this | case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : Smoot... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | ext y | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | case h
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | rw [comp_apply, comp_apply] | case h
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothMan... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | sorry | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_measure_zero_of_C1_dimension_increase | [59, 1] | [90, 13] | sorry | E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWithCorners I M
inst✝¹⁶ : FiniteDimensional ℝ E
inst✝¹⁵ : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace ℝ E
H : Type u_2
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
inst✝¹⁸ : I.Boundaryless
inst✝¹⁷ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | rw [← image_univ] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | suffices hyp : ∀ x : M, MeasureZero J (f '' ((chartAt H x).source ∩ univ)) from
measure_zero_image_iff_chart_domains (J := J) hyp | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | intro x μ hμ y | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | let e := chartAt H x | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | let e' := chartAt G y | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | have aux : J ∘ e' '' (e'.source ∩ f '' e.source) = (J ∘ e' ∘ f) '' e.source := by sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | rw [inter_univ, aux] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | let g : E → F := J ∘ e' ∘ f ∘ e.invFun ∘ I.invFun | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | have : (J ∘ ↑e' ∘ f '' e.source) = g '' (I '' e.target) := by sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | have hopen : IsOpen (I '' e.target) := by sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | have gdiff : ContDiffOn ℝ 1 g (I '' e.target) := by sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | rw [this] | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | apply image_measure_zero_of_C1_dimension_increase hopen μ gdiff hdim | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/fpvandoorn/sard.git | f4a1bb550136e8591dcd20dca5bdcce766a958c7 | Sard/Stuff.lean | image_null_of_C1_of_dimension_increase | [92, 1] | [114, 71] | sorry | E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWithCorners I M
inst✝¹³ : FiniteDimensional ℝ E
inst✝¹² : Secon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type u_1
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹⁷ : TopologicalSpace M
inst✝¹⁶ : ChartedSpace H M
inst✝¹⁵ : I.Boundaryless
inst✝¹⁴ : SmoothManifoldWi... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_west | [201, 1] | [212, 44] | have hmm : GameState.mk s ⟨x,y⟩ w = make_move ⟨s,⟨x+1, y⟩,w⟩ Move.west :=
by have h' : x + 1 - 1 = x := rfl
simp [h', hclear'] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ Escapable { size := s, position := { x := x + 1, y := y }, walls := w } | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x + 1, y := y }, walls := w } Move.west
⊢ Esca... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
TACTIC:
|
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_west | [201, 1] | [212, 44] | rw [hmm] at W | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x + 1, y := y }, walls := w } Move.west
⊢ Esca... | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable (make_move { size := s, position := { x := x + 1, y := y }, walls := w } Move.west)
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x + 1, y := y }, wal... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, posi... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_west | [201, 1] | [212, 44] | exact .Step ⟨s,⟨x+1,y⟩,w⟩ Move.west W | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable (make_move { size := s, position := { x := x + 1, y := y }, walls := w } Move.west)
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x + 1, y := y }, wal... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable (make_move { size := s, position := { x := x + 1, y := y }, walls := w } Move.west)
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
m... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_west | [201, 1] | [212, 44] | have h' : x + 1 - 1 = x := rfl | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x + 1, y := y }, walls := w } Move.west | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
h' : x + 1 - 1 = x
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x + 1, y := y }, walls := w } Mov... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position :... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_west | [201, 1] | [212, 44] | simp [h', hclear'] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
h' : x + 1 - 1 = x
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x + 1, y := y }, walls := w } Mov... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
W : Escapable { size := s, position := { x := x, y := y }, walls := w }
h' : x + 1 - 1 = x
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { si... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_east | [214, 1] | [225, 45] | have hmm : GameState.mk s ⟨x+1,y⟩ w = make_move ⟨s, ⟨x,y⟩,w⟩ Move.east :=
by simp [hclear', hinbounds] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
⊢ Escapable { size := s, position := { x := x, y := y }, walls := w } | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
hmm :
{ size := s, position := { x := x + 1, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y ... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
⊢ Escapable { size := s, position := { x := x, y := y }, walls := w... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_east | [214, 1] | [225, 45] | rw [hmm] at E | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
hmm :
{ size := s, position := { x := x + 1, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y ... | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable (make_move { size := s, position := { x := x, y := y }, walls := w } Move.east)
hmm :
{ size := s, position := { x := x + 1, y := y }, walls := w } =
make_move { size := s, position :=... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
hmm :
{ size := s, position := { x := x + 1, y := y }, walls := w... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_east | [214, 1] | [225, 45] | exact .Step ⟨s, ⟨x,y⟩, w⟩ Move.east E | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable (make_move { size := s, position := { x := x, y := y }, walls := w } Move.east)
hmm :
{ size := s, position := { x := x + 1, y := y }, walls := w } =
make_move { size := s, position :=... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable (make_move { size := s, position := { x := x, y := y }, walls := w } Move.east)
hmm :
{ size := s, position := { x := x + 1, y ... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_east | [214, 1] | [225, 45] | simp [hclear', hinbounds] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
⊢ { size := s, position := { x := x + 1, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y }, wal... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x + 1, y := y } w = true
hinbounds : x + 1 ≤ s.x
E : Escapable { size := s, position := { x := x + 1, y := y }, walls := w }
⊢ { size := s, position := { x := x + 1, y := y }, walls := w } =
... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_north | [227, 1] | [238, 46] | have hmm : GameState.mk s ⟨x,y⟩ w = make_move ⟨s,⟨x, y+1⟩,w⟩ Move.north :=
by have h' : y + 1 - 1 = y := rfl
simp [h', hclear'] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ Escapable { size := s, position := { x := x, y := y + 1 }, walls := w } | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Move.north
⊢ Esc... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
TACTIC:
|
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_north | [227, 1] | [238, 46] | rw [hmm] at N | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Move.north
⊢ Esc... | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable (make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Move.north)
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y + 1 }, wa... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, posi... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_north | [227, 1] | [238, 46] | exact .Step ⟨s,⟨x,y+1⟩,w⟩ Move.north N | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable (make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Move.north)
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y + 1 }, wa... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable (make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Move.north)
hmm :
{ size := s, position := { x := x, y := y }, walls := w } =
... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_north | [227, 1] | [238, 46] | have h' : y + 1 - 1 = y := rfl | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Move.north | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
h' : y + 1 - 1 = y
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Mov... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position :... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_north | [227, 1] | [238, 46] | simp [h', hclear'] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
h' : y + 1 - 1 = y
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { size := s, position := { x := x, y := y + 1 }, walls := w } Mov... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y } w = true
N : Escapable { size := s, position := { x := x, y := y }, walls := w }
h' : y + 1 - 1 = y
⊢ { size := s, position := { x := x, y := y }, walls := w } =
make_move { si... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_south | [240, 1] | [251, 44] | have hmm : GameState.mk s ⟨x,y+1⟩ w = make_move ⟨s,⟨x, y⟩,w⟩ Move.south :=
by simp [hclear', hinbounds] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
⊢ Escapable { size := s, position := { x := x, y := y }, walls := w } | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
hmm :
{ size := s, position := { x := x, y := y + 1 }, walls := w } =
make_move { size := s, position := { x := x, y := y ... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
⊢ Escapable { size := s, position := { x := x, y := y }, walls := w... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_south | [240, 1] | [251, 44] | rw [hmm] at S | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
hmm :
{ size := s, position := { x := x, y := y + 1 }, walls := w } =
make_move { size := s, position := { x := x, y := y ... | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable (make_move { size := s, position := { x := x, y := y }, walls := w } Move.south)
hmm :
{ size := s, position := { x := x, y := y + 1 }, walls := w } =
make_move { size := s, position :... | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
hmm :
{ size := s, position := { x := x, y := y + 1 }, walls := w... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_south | [240, 1] | [251, 44] | exact .Step ⟨s,⟨x,y⟩,w⟩ Move.south S | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable (make_move { size := s, position := { x := x, y := y }, walls := w } Move.south)
hmm :
{ size := s, position := { x := x, y := y + 1 }, walls := w } =
make_move { size := s, position :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable (make_move { size := s, position := { x := x, y := y }, walls := w } Move.south)
hmm :
{ size := s, position := { x := x, y := ... |
https://github.com/dwrensha/lean4-maze.git | 79497250a226b05f0c9a9515f5ebe0da341a3121 | Maze.lean | step_south | [240, 1] | [251, 44] | simp [hclear', hinbounds] | s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
⊢ { size := s, position := { x := x, y := y + 1 }, walls := w } =
make_move { size := s, position := { x := x, y := y }, wal... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : Coords
x y : Nat
w : List Coords
hclear' : List.notElem { x := x, y := y + 1 } w = true
hinbounds : y + 1 ≤ s.y
S : Escapable { size := s, position := { x := x, y := y + 1 }, walls := w }
⊢ { size := s, position := { x := x, y := y + 1 }, walls := w } =
... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | have : x ∈ g '' univ := by
contrapose! hx
rw [sbSet, mem_iUnion]
use 0
rw [sbAux, mem_diff]
sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ g (invFun g x) = x | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | have : ∃ y, g y = x := by
sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this✝ : x ∈ g '' univ
this : ∃ y, g y = x
⊢ g (invFun g x) = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | contrapose! hx | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ x ∈ g '' univ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
⊢ x ∈ g '' univ
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | rw [sbSet, mem_iUnion] | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbSet f g
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | use 0 | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ ∃ i, x ∈ sbAux f g i
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | rw [sbAux, mem_diff] | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0 | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ sbAux f g 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | sorry | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ g '' univ
⊢ x ∈ univ ∧ x ∉ g '' univ
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_right_inv | [25, 1] | [34, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ ∃ y, g y = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
x : α
hx : x ∉ sbSet f g
this : x ∈ g '' univ
⊢ ∃ y, g y = x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | set A := sbSet f g with A_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
⊢ Injective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
⊢ Injective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | set h := sbFun f g with h_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Injective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | intro x₁ x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Injective h
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | intro (hxeq : h x₁ = h x₂) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
⊢ h x₁ = h x₂ → x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | simp only [h_def, sbFun, ← A_def] at hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : h x₁ = h x₂
⊢ x₁ = x₂
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | by_cases xA : x₁ ∈ A ∨ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
⊢ x₁ = x₂ | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
⊢ x₁ = x₂
case neg
α :... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | push_neg at xA | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : ¬(x₁ ∈ A ∨ x₂ ∈ A)
⊢ x₁ = x₂ | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | sorry | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∉ A ∧ x₂ ∉ A
⊢ x₁ = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
⊢ x₁ = x₂ | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | have x₂A : x₂ ∈ A := by
apply not_imp_self.mp
intro (x₂nA : x₂ ∉ A)
rw [if_pos x₁A, if_neg x₂nA] at hxeq
rw [A_def, sbSet, mem_iUnion] at x₁A
have x₂eq : x₂ = g (f x₁) := by
sorry
rcases x₁A with ⟨n, hn⟩
rw [A_def, sbSet, mem_iUnion]
use n + 1
simp [sbAux]
exact ⟨x₁, hn, x₂eq.symm⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₁ = x₂ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ =... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂A : x₂ ∈ A
⊢ x₁ =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | symm | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | apply this hxeq.symm xA.symm (xA.resolve_left x₁A) | case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
this :
∀ ⦃x₁ x₂ :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | apply not_imp_self.mp | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∉ A → x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | intro (x₂nA : x₂ ∉ A) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
⊢ x₂ ∉ A → x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rw [if_pos x₁A, if_neg x₂nA] at hxeq | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ... | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : (if x₁ ∈ A then f x₁ else invFun g x₁) = if x₂ ∈ A then f x₂ else invF... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rw [A_def, sbSet, mem_iUnion] at x₁A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : x₁ ∈ A
x₂nA : x₂ ∉ A
⊢ x... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | have x₂eq : x₂ = g (f x₁) := by
sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rcases x₁A with ⟨n, hn⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
⊢ x₂ ∈ A | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | rw [A_def, sbSet, mem_iUnion] | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ A | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ i, x₂ ∈ sbAux... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | use n + 1 | case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ i, x₂ ∈ sbAux... | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ sbAux f g (n +... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | simp [sbAux] | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ x₂ ∈ sbAux f g (n +... | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ a ∈ sbAux f g n, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | exact ⟨x₁, hn, x₂eq.symm⟩ | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂ = g (f x₁)
n : ℕ
hn : x₁ ∈ sbAux f g n
⊢ ∃ a ∈ sbAux f g n, ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₂nA : x₂ ∉ A
x₂eq : x₂... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_injective | [36, 1] | [61, 8] | sorry | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂nA : x₂ ∉ A
⊢ x₂ = g (f x₁) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
x₁ x₂ : α
hxeq : f x₁ = invFun g x₂
xA : x₁ ∈ A ∨ x₂ ∈ A
x₁A : ∃ i, x₁ ∈ sbAux f g i
x₂... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | set A := sbSet f g with A_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
⊢ Surjective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Surjective (sbFun f g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
⊢ Surjective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | set h := sbFun f g with h_def | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Surjective (sbFun f g) | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Surjective h | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
⊢ Surjective (sbFun f g)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | intro y | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Surjective h | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
⊢ Surjective h
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | by_cases gyA : g y ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
⊢ ∃ a, h a = y | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∈ A
⊢ ∃ a, h a = y
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injecti... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | sorry | case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∉ A
⊢ ∃ a, h a = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∉ A
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rw [A_def, sbSet, mem_iUnion] at gyA | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∈ A
⊢ ∃ a, h a = y | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : ∃ i, g y ∈ sbAux f g i
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : g y ∈ A
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rcases gyA with ⟨n, hn⟩ | case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : ∃ i, g y ∈ sbAux f g i
⊢ ∃ a, h a = y | case pos.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g n
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
gyA : ∃ i, g y ∈ sbAux f g i
⊢ ∃ a, h a = y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rcases n with _ | n | case pos.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g n
⊢ ∃ a, h a = y | case pos.intro.zero
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
hn : g y ∈ sbAux f g Nat.zero
⊢ ∃ a, h a = y
case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonem... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g n
⊢ ∃ a, h a = y
TACTI... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | simp [sbAux] at hn | case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g (Nat.succ n)
⊢ ∃ a, h a = y | case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : ∃ a ∈ sbAux f g n, g (f a) = g y
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : g y ∈ sbAux f g (Nat.succ n)
⊢ ∃ ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rcases hn with ⟨x, xmem, hx⟩ | case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : ∃ a ∈ sbAux f g n, g (f a) = g y
⊢ ∃ a, h a = y | case pos.intro.succ.intro.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ a, h a = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.succ
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
hn : ∃ a ∈ sbAux f g n, g (f a) = g y
... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | use x | case pos.intro.succ.intro.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ a, h a = y | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ h x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.succ.intro.intro
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | have : x ∈ A := by
rw [A_def, sbSet, mem_iUnion]
exact ⟨n, xmem⟩ | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ h x = y | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ h x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ h... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | simp only [h_def, sbFun, if_pos this] | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ h x = y | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ f x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
thi... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | exact hg hx | case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
this : x ∈ A
⊢ f x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
thi... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | simp [sbAux] at hn | case pos.intro.zero
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
hn : g y ∈ sbAux f g Nat.zero
⊢ ∃ a, h a = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.zero
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
hn : g y ∈ sbAux f g Nat.zero
⊢ ∃ a, h a = y... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | rw [A_def, sbSet, mem_iUnion] | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ x ∈ A | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ i, x ∈ sbAux f g i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ x ∈ A
TA... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S03_The_Schroeder_Bernstein_Theorem.lean | sb_surjective | [63, 1] | [80, 8] | exact ⟨n, xmem⟩ | α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ i, x ∈ sbAux f g i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
f : α → β
g : β → α
hf : Injective f
hg : Injective g
A : Set α := sbSet f g
A_def : A = sbSet f g
h : α → β := sbFun f g
h_def : h = sbFun f g
y : β
n : ℕ
x : α
xmem : x ∈ sbAux f g n
hx : g (f x) = g y
⊢ ∃ i, x ∈... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_zero | [39, 1] | [39, 67] | rw [add_comm, zero_add] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a + 0 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a + 0 = a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_neg | [41, 1] | [41, 77] | rw [add_comm, add_left_neg] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a + -a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a + -a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_add_cancel_left | [51, 1] | [52, 43] | rw [← add_assoc, add_left_neg, zero_add] | R : Type u_1
inst✝ : Ring R
a b : R
⊢ -a + (a + b) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
⊢ -a + (a + b) = b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_neg_cancel_right | [55, 1] | [56, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
⊢ a + b + -b = a
TACTIC:
|
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