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Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.nonUnitalAlgHom_ext'
|
[] |
[
681,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.coe_toOrderHom
|
[] |
[
149,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
9
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
AffineSubspace.isClosed_direction_iff
|
[
{
"state_after": "case inl\nα : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_3\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\n⊢ IsClosed ↑(direction ⊥) ↔ IsClosed ↑⊥\n\ncase inr.intro\nα : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_3\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\ns : AffineSubspace 𝕜 Q\nx : Q\nhx : x ∈ ↑s\n⊢ IsClosed ↑(direction s) ↔ IsClosed ↑s",
"state_before": "α : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_3\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\ns : AffineSubspace 𝕜 Q\n⊢ IsClosed ↑(direction s) ↔ IsClosed ↑s",
"tactic": "rcases s.eq_bot_or_nonempty with (rfl | ⟨x, hx⟩)"
},
{
"state_after": "case inr.intro\nα : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_3\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\ns : AffineSubspace 𝕜 Q\nx : Q\nhx : x ∈ ↑s\n⊢ IsClosed ((fun x_1 => x_1 -ᵥ x) '' ↑s) ↔\n IsClosed (↑(Homeomorph.symm (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst x))) '' ↑s)",
"state_before": "case inr.intro\nα : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_3\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\ns : AffineSubspace 𝕜 Q\nx : Q\nhx : x ∈ ↑s\n⊢ IsClosed ↑(direction s) ↔ IsClosed ↑s",
"tactic": "rw [← (IsometryEquiv.vaddConst x).toHomeomorph.symm.isClosed_image,\n AffineSubspace.coe_direction_eq_vsub_set_right hx]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nα : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_3\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\ns : AffineSubspace 𝕜 Q\nx : Q\nhx : x ∈ ↑s\n⊢ IsClosed ((fun x_1 => x_1 -ᵥ x) '' ↑s) ↔\n IsClosed (↑(Homeomorph.symm (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst x))) '' ↑s)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_3\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\n⊢ IsClosed ↑(direction ⊥) ↔ IsClosed ↑⊥",
"tactic": "simp [isClosed_singleton]"
}
] |
[
44,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_empty
|
[] |
[
51,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.canonicalEquiv_spanSingleton
|
[
{
"state_after": "R : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\n⊢ ∀ (x_1 : P'),\n x_1 ∈ ↑(canonicalEquiv S P P') (spanSingleton S x) ↔\n x_1 ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)",
"state_before": "R : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\n⊢ ↑(canonicalEquiv S P P') (spanSingleton S x) =\n spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)",
"tactic": "apply SetLike.ext_iff.mpr"
},
{
"state_after": "R : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\n⊢ y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x) ↔\n y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)",
"state_before": "R : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\n⊢ ∀ (x_1 : P'),\n x_1 ∈ ↑(canonicalEquiv S P P') (spanSingleton S x) ↔\n x_1 ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)",
"tactic": "intro y"
},
{
"state_after": "case mp\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n\ncase mpr\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x)",
"state_before": "R : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\n⊢ y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x) ↔\n y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ ∃ z, z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x = y",
"state_before": "case mp\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)",
"tactic": "rw [mem_spanSingleton]"
},
{
"state_after": "case mp.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx x' : P\nhx' : x' ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x' ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ ∃ z,\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x'",
"state_before": "case mp\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ ∃ z, z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x = y",
"tactic": "obtain ⟨x', hx', rfl⟩ := (mem_canonicalEquiv_apply _ _ _).mp h"
},
{
"state_after": "case mp.intro.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nhx' : z • x ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ ∃ z_1,\n z_1 • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x)",
"state_before": "case mp.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx x' : P\nhx' : x' ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x' ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ ∃ z,\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x'",
"tactic": "obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp hx'"
},
{
"state_after": "case mp.intro.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nhx' : z • x ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x)",
"state_before": "case mp.intro.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nhx' : z • x ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ ∃ z_1,\n z_1 • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x)",
"tactic": "use z"
},
{
"state_after": "case mp.intro.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nhx' : z • x ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(RingHom.id R) z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"state_before": "case mp.intro.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nhx' : z • x ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x)",
"tactic": "rw [IsLocalization.map_smul]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nhx' : z • x ∈ spanSingleton S x\nh :\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) ∈\n ↑(canonicalEquiv S P P') (spanSingleton S x)\n⊢ z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x =\n ↑(RingHom.id R) z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"tactic": "rfl"
},
{
"state_after": "case mpr\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ ∃ y_1,\n y_1 ∈ spanSingleton S x ∧\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) y_1 = y",
"state_before": "case mpr\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ y ∈ ↑(canonicalEquiv S P P') (spanSingleton S x)",
"tactic": "rw [mem_canonicalEquiv_apply]"
},
{
"state_after": "case mpr.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nh :\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x ∈\n spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ ∃ y,\n y ∈ spanSingleton S x ∧\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) y =\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"state_before": "case mpr\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\ny : P'\nh : y ∈ spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ ∃ y_1,\n y_1 ∈ spanSingleton S x ∧\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) y_1 = y",
"tactic": "obtain ⟨z, rfl⟩ := (mem_spanSingleton _).mp h"
},
{
"state_after": "case mpr.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nh :\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x ∈\n spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ z • x ∈ spanSingleton S x ∧\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) =\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"state_before": "case mpr.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nh :\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x ∈\n spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ ∃ y,\n y ∈ spanSingleton S x ∧\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) y =\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"tactic": "use z • x"
},
{
"state_after": "case mpr.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nh :\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x ∈\n spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) =\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"state_before": "case mpr.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nh :\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x ∈\n spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ z • x ∈ spanSingleton S x ∧\n ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) =\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"tactic": "use (mem_spanSingleton _).mpr ⟨z, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u_2\ninst✝⁹ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁸ : CommRing P\ninst✝⁷ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type ?u.1366656\ninst✝⁶ : CommRing R₁\nK : Type ?u.1366662\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nP' : Type u_1\ninst✝² : CommRing P'\ninst✝¹ : Algebra R P'\ninst✝ : IsLocalization S P'\nx : P\nz : R\nh :\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x ∈\n spanSingleton S (↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x)\n⊢ ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) (z • x) =\n z • ↑(IsLocalization.map P' (RingHom.id R) (_ : ∀ (y : R), y ∈ S → ↑(RingHom.id R) y ∈ S)) x",
"tactic": "simp [IsLocalization.map_smul]"
}
] |
[
1415,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1396,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
apply_abs_le_mul_of_one_le
|
[] |
[
264,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/LinearAlgebra/Contraction.lean
|
homTensorHomEquiv_toLinearMap
|
[
{
"state_after": "case H.h.h.H.h.h\nι : Type w\nR : Type u\nM : Type v₁\nN : Type v₂\nP : Type v₃\nQ : Type v₄\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : AddCommGroup P\ninst✝⁹ : AddCommGroup Q\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Free R N\ninst✝¹ : Module.Finite R N\ninst✝ : Nontrivial R\nm : M →ₗ[R] P\nn : N →ₗ[R] Q\nx✝¹ : M\nx✝ : N\n⊢ ↑(↑(compr₂ (TensorProduct.mk R M N)\n (↑(↑(compr₂ (TensorProduct.mk R (M →ₗ[R] P) (N →ₗ[R] Q)) ↑(homTensorHomEquiv R M N P Q)) m) n))\n x✝¹)\n x✝ =\n ↑(↑(compr₂ (TensorProduct.mk R M N)\n (↑(↑(compr₂ (TensorProduct.mk R (M →ₗ[R] P) (N →ₗ[R] Q)) (homTensorHomMap R M N P Q)) m) n))\n x✝¹)\n x✝",
"state_before": "ι : Type w\nR : Type u\nM : Type v₁\nN : Type v₂\nP : Type v₃\nQ : Type v₄\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : AddCommGroup P\ninst✝⁹ : AddCommGroup Q\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Free R N\ninst✝¹ : Module.Finite R N\ninst✝ : Nontrivial R\n⊢ ↑(homTensorHomEquiv R M N P Q) = homTensorHomMap R M N P Q",
"tactic": "ext (m n)"
},
{
"state_after": "no goals",
"state_before": "case H.h.h.H.h.h\nι : Type w\nR : Type u\nM : Type v₁\nN : Type v₂\nP : Type v₃\nQ : Type v₄\ninst✝¹³ : CommRing R\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : AddCommGroup N\ninst✝¹⁰ : AddCommGroup P\ninst✝⁹ : AddCommGroup Q\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Free R N\ninst✝¹ : Module.Finite R N\ninst✝ : Nontrivial R\nm : M →ₗ[R] P\nn : N →ₗ[R] Q\nx✝¹ : M\nx✝ : N\n⊢ ↑(↑(compr₂ (TensorProduct.mk R M N)\n (↑(↑(compr₂ (TensorProduct.mk R (M →ₗ[R] P) (N →ₗ[R] Q)) ↑(homTensorHomEquiv R M N P Q)) m) n))\n x✝¹)\n x✝ =\n ↑(↑(compr₂ (TensorProduct.mk R M N)\n (↑(↑(compr₂ (TensorProduct.mk R (M →ₗ[R] P) (N →ₗ[R] Q)) (homTensorHomMap R M N P Q)) m) n))\n x✝¹)\n x✝",
"tactic": "simp only [homTensorHomEquiv, compr₂_apply, mk_apply, LinearEquiv.coe_toLinearMap,\n LinearEquiv.trans_apply, lift.equiv_apply, LinearEquiv.arrowCongr_apply, LinearEquiv.refl_symm,\n LinearEquiv.refl_apply, rTensorHomEquivHomRTensor_apply, lTensorHomEquivHomLTensor_apply,\n lTensorHomToHomLTensor_apply, rTensorHomToHomRTensor_apply, homTensorHomMap_apply,\n map_tmul]"
}
] |
[
322,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
src/lean/Init/Data/String/Basic.lean
|
String.pos_lt_eq
|
[] |
[
133,
75
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
133,
9
] |
Mathlib/Algebra/BigOperators/NatAntidiagonal.lean
|
Finset.Nat.prod_antidiagonal_eq_prod_range_succ_mk
|
[] |
[
66,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Std/Data/String/Lemmas.lean
|
String.Pos.le_iff
|
[] |
[
139,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
139,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.ext
|
[] |
[
181,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
BiheytingHom.cancel_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.147678\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.147690\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf f₁ f₂ : BiheytingHom α β\ng g₁ g₂ : BiheytingHom β γ\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)",
"tactic": "rw [← comp_apply, h, comp_apply]"
}
] |
[
610,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
609,
1
] |
Mathlib/Algebra/Bounds.lean
|
BddAbove.inv
|
[] |
[
48,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.localTrivAsLocalEquiv_trans
|
[
{
"state_after": "case left\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\n⊢ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source =\n (trivChange Z i j).toLocalEquiv.source\n\ncase right\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\n⊢ EqOn (↑(LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)))\n (↑(trivChange Z i j).toLocalEquiv)\n (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source",
"state_before": "ι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\n⊢ LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j) ≈\n (trivChange Z i j).toLocalEquiv",
"tactic": "constructor"
},
{
"state_after": "case left.h\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\nx : B × F\n⊢ x ∈ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source ↔\n x ∈ (trivChange Z i j).toLocalEquiv.source",
"state_before": "case left\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\n⊢ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source =\n (trivChange Z i j).toLocalEquiv.source",
"tactic": "ext x"
},
{
"state_after": "case left.h\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\nx : B × F\n⊢ x.fst ∈ baseSet Z i ∧ ↑(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) x ∈ (localTrivAsLocalEquiv Z j).source ↔\n x.fst ∈ baseSet Z i ∧ x.fst ∈ baseSet Z j",
"state_before": "case left.h\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\nx : B × F\n⊢ x ∈ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source ↔\n x ∈ (trivChange Z i j).toLocalEquiv.source",
"tactic": "simp only [mem_localTrivAsLocalEquiv_target, mfld_simps]"
},
{
"state_after": "no goals",
"state_before": "case left.h\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\nx : B × F\n⊢ x.fst ∈ baseSet Z i ∧ ↑(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) x ∈ (localTrivAsLocalEquiv Z j).source ↔\n x.fst ∈ baseSet Z i ∧ x.fst ∈ baseSet Z j",
"tactic": "rfl"
},
{
"state_after": "case right.mk\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\nx : B\nv : F\nhx : (x, v) ∈ (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source\n⊢ ↑(LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)) (x, v) =\n ↑(trivChange Z i j).toLocalEquiv (x, v)",
"state_before": "case right\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\n⊢ EqOn (↑(LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)))\n (↑(trivChange Z i j).toLocalEquiv)\n (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source",
"tactic": "rintro ⟨x, v⟩ hx"
},
{
"state_after": "no goals",
"state_before": "case right.mk\nι : Type u_3\nB : Type u_1\nF : Type u_2\nX : Type ?u.32818\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni✝ i j : ι\nx : B\nv : F\nhx : x ∈ baseSet Z i ∧ x ∈ baseSet Z j\n⊢ coordChange Z (indexAt Z x) j x (coordChange Z i (indexAt Z x) x v) = coordChange Z i j x v",
"tactic": "simp only [Z.coordChange_comp, hx, mem_inter_iff, and_self_iff, mem_baseSet_at]"
}
] |
[
563,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Algebra/Regular/SMul.lean
|
IsSMulRegular.mul_and_mul_iff
|
[
{
"state_after": "case refine'_1\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) → IsSMulRegular M a ∧ IsSMulRegular M b\n\ncase refine'_2\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M a ∧ IsSMulRegular M b → IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a)",
"state_before": "R : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b",
"tactic": "refine' ⟨_, _⟩"
},
{
"state_after": "case refine'_1.intro\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\nab : IsSMulRegular M (a * b)\nba : IsSMulRegular M (b * a)\n⊢ IsSMulRegular M a ∧ IsSMulRegular M b",
"state_before": "case refine'_1\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) → IsSMulRegular M a ∧ IsSMulRegular M b",
"tactic": "rintro ⟨ab, ba⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\nab : IsSMulRegular M (a * b)\nba : IsSMulRegular M (b * a)\n⊢ IsSMulRegular M a ∧ IsSMulRegular M b",
"tactic": "refine' ⟨ba.of_mul, ab.of_mul⟩"
},
{
"state_after": "case refine'_2.intro\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\nha : IsSMulRegular M a\nhb : IsSMulRegular M b\n⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a)",
"state_before": "case refine'_2\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\n⊢ IsSMulRegular M a ∧ IsSMulRegular M b → IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a)",
"tactic": "rintro ⟨ha, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nR : Type u_1\nS : Type ?u.7631\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\nha : IsSMulRegular M a\nhb : IsSMulRegular M b\n⊢ IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a)",
"tactic": "exact ⟨ha.mul hb, hb.mul ha⟩"
}
] |
[
125,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.semiconj_conj
|
[
{
"state_after": "no goals",
"state_before": "α₁ : Type u_1\nβ₁ : Type u_2\ne : α₁ ≃ β₁\nf✝ : α₁ → α₁ → α₁\nf : α₁ → α₁\nx : α₁\n⊢ ↑e (f x) = ↑(conj e) f (↑e x)",
"tactic": "simp"
}
] |
[
1846,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1846,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_div_assoc
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.18617\nβ : Type ?u.18620\nG : Type u_1\ninst✝ : DivInvMonoid G\na✝ b✝ c✝ a b c : G\n⊢ a * b / c = a * (b / c)",
"tactic": "rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _]"
}
] |
[
308,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.QuasiMeasurePreserving.mono_left
|
[] |
[
2460,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2458,
1
] |
Std/Data/List/Lemmas.lean
|
List.erase_eq_eraseP
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na b : α\nl : List α\n⊢ List.erase (b :: l) a = eraseP (fun b => decide (a = b)) (b :: l)",
"tactic": "if h : a = b then simp [h] else simp [h, Ne.symm h, erase_eq_eraseP a l]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na b : α\nl : List α\nh : a = b\n⊢ List.erase (b :: l) a = eraseP (fun b => decide (a = b)) (b :: l)",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na b : α\nl : List α\nh : ¬a = b\n⊢ List.erase (b :: l) a = eraseP (fun b => decide (a = b)) (b :: l)",
"tactic": "simp [h, Ne.symm h, erase_eq_eraseP a l]"
}
] |
[
1045,
77
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1042,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Real.lean
|
cauchySeq_of_dist_le_of_summable
|
[
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\nε : ℝ\nεpos : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f n) (f N) < ε",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\n⊢ CauchySeq f",
"tactic": "refine' Metric.cauchySeq_iff'.2 fun ε εpos => _"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f n) (f N) < ε",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nhd : Summable d\nε : ℝ\nεpos : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f n) (f N) < ε",
"tactic": "replace hd : CauchySeq fun n : ℕ => ∑ x in range n, d x :=\n let ⟨_, H⟩ := hd\n H.tendsto_sum_nat.cauchySeq"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ dist (f n) (f N) < ε",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f n) (f N) < ε",
"tactic": "refine' (Metric.cauchySeq_iff'.1 hd ε εpos).imp fun N hN n hn => _"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\nn : ℕ\nhn : n ≥ N\nhsum : dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\n⊢ dist (f n) (f N) < ε",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ dist (f n) (f N) < ε",
"tactic": "have hsum := hN n hn"
},
{
"state_after": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\nn : ℕ\nhn : n ≥ N\nhsum : abs (∑ k in Ico N n, d k) < ε\n⊢ dist (f n) (f N) < ε",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\nn : ℕ\nhn : n ≥ N\nhsum : dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\n⊢ dist (f n) (f N) < ε",
"tactic": "rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f (Nat.succ n)) ≤ d n\nε : ℝ\nεpos : ε > 0\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ε\nn : ℕ\nhn : n ≥ N\nhsum : abs (∑ k in Ico N n, d k) < ε\n⊢ dist (f n) (f N) < ε",
"tactic": "calc\n dist (f n) (f N) = dist (f N) (f n) := dist_comm _ _\n _ ≤ ∑ x in Ico N n, d x := dist_le_Ico_sum_of_dist_le hn fun _ _ => hf _\n _ ≤ |∑ x in Ico N n, d x| := le_abs_self _\n _ < ε := hsum"
}
] |
[
68,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
IsOpen.smul_left
|
[
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (⋃ (a : α) (_ : a ∈ s), a • t)",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (s • t)",
"tactic": "rw [← iUnion_smul_set]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (⋃ (a : α) (_ : a ∈ s), a • t)",
"tactic": "exact isOpen_biUnion fun a _ => ht.smul _"
}
] |
[
1228,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1226,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Measure.lean
|
BoxIntegral.Box.measure_coe_lt_top
|
[] |
[
53,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.isOpen_extend_source
|
[
{
"state_after": "𝕜 : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.129280\nM' : Type ?u.129283\nH' : Type ?u.129286\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\n⊢ IsOpen f.source",
"state_before": "𝕜 : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.129280\nM' : Type ?u.129283\nH' : Type ?u.129286\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\n⊢ IsOpen (extend f I).source",
"tactic": "rw [extend_source]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.129280\nM' : Type ?u.129283\nH' : Type ?u.129286\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\n⊢ IsOpen f.source",
"tactic": "exact f.open_source"
}
] |
[
800,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
798,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Prod.card_uIcc
|
[] |
[
1016,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1013,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mul_lt_of_lt_div
|
[] |
[
929,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
928,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.support_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.903481\nδ : Type ?u.903484\ninst✝¹ : MeasurableSpace α\ninst✝ : Zero β\nf : α →ₛ β\nx : α\n⊢ x ∈ support ↑f ↔ x ∈ ⋃ (y : β) (_ : y ∈ filter (fun y => y ≠ 0) (SimpleFunc.range f)), ↑f ⁻¹' {y}",
"tactic": "simp only [mem_support, Set.mem_preimage, mem_filter, mem_range_self, true_and_iff, exists_prop,\n mem_iUnion, Set.mem_range, mem_singleton_iff, exists_eq_right']"
}
] |
[
1172,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1168,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
extChartAt_source_mem_nhdsWithin
|
[] |
[
1082,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1081,
1
] |
Mathlib/Data/QPF/Multivariate/Basic.lean
|
MvQPF.mem_supp
|
[
{
"state_after": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ u ∈ {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i y} ↔\n ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ u ∈ supp x i ↔ ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ",
"tactic": "rw [supp]"
},
{
"state_after": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ (∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u) ↔\n ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ u ∈ {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i y} ↔\n ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ",
"tactic": "dsimp"
},
{
"state_after": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ (∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u) →\n ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ\n\ncase mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ (∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ) →\n ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ (∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u) ↔\n ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ",
"tactic": "constructor"
},
{
"state_after": "case mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ\np : (i : Fin2 n) → α i → Prop\n⊢ LiftP p x → p i u",
"state_before": "case mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ (∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ) →\n ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u",
"tactic": "intro h p"
},
{
"state_after": "case mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ\np : (i : Fin2 n) → α i → Prop\n⊢ (∃ a f, x = abs { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)) → p i u",
"state_before": "case mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ\np : (i : Fin2 n) → α i → Prop\n⊢ LiftP p x → p i u",
"tactic": "rw [liftP_iff]"
},
{
"state_after": "case mpr.intro.intro.intro\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ\np : (i : Fin2 n) → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ fun i => α i\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)\n⊢ p i u",
"state_before": "case mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ\np : (i : Fin2 n) → α i → Prop\n⊢ (∃ a f, x = abs { fst := a, snd := f } ∧ ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)) → p i u",
"tactic": "rintro ⟨a, f, xeq, h'⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro.intro\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni✝ : Fin2 n\nu : α i✝\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i✝ '' univ\np : (i : Fin2 n) → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ fun i => α i\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)\ni : MvPFunctor.B (P F) a i✝\nleft✝ : i ∈ univ\nhi : f i✝ i = u\n⊢ p i✝ u",
"state_before": "case mpr.intro.intro.intro\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ\np : (i : Fin2 n) → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ fun i => α i\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)\n⊢ p i u",
"tactic": "rcases h a f xeq.symm with ⟨i, _, hi⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro.intro\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni✝ : Fin2 n\nu : α i✝\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i✝ '' univ\np : (i : Fin2 n) → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ fun i => α i\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)\ni : MvPFunctor.B (P F) a i✝\nleft✝ : i ∈ univ\nhi : f i✝ i = u\n⊢ p i✝ (f i✝ i)",
"state_before": "case mpr.intro.intro.intro.intro.intro\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni✝ : Fin2 n\nu : α i✝\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i✝ '' univ\np : (i : Fin2 n) → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ fun i => α i\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)\ni : MvPFunctor.B (P F) a i✝\nleft✝ : i ∈ univ\nhi : f i✝ i = u\n⊢ p i✝ u",
"tactic": "rw [← hi]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro.intro\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni✝ : Fin2 n\nu : α i✝\nh : ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i✝ '' univ\np : (i : Fin2 n) → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ fun i => α i\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), p i (f i j)\ni : MvPFunctor.B (P F) a i✝\nleft✝ : i ∈ univ\nhi : f i✝ i = u\n⊢ p i✝ (f i✝ i)",
"tactic": "apply h'"
},
{
"state_after": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\n⊢ u ∈ f i '' univ",
"state_before": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\n⊢ (∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u) →\n ∀ (a : (P F).A) (f : MvPFunctor.B (P F) a ⟹ α), abs { fst := a, snd := f } = x → u ∈ f i '' univ",
"tactic": "intro h a f haf"
},
{
"state_after": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\nthis : LiftP (fun i u => u ∈ f i '' univ) x\n⊢ u ∈ f i '' univ",
"state_before": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\n⊢ u ∈ f i '' univ",
"tactic": "have : LiftP (fun i u => u ∈ f i '' univ) x := by\n rw [liftP_iff]\n refine' ⟨a, f, haf.symm, _⟩\n intro i u\n exact mem_image_of_mem _ (mem_univ _)"
},
{
"state_after": "no goals",
"state_before": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\nthis : LiftP (fun i u => u ∈ f i '' univ) x\n⊢ u ∈ f i '' univ",
"tactic": "exact h this"
},
{
"state_after": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\n⊢ ∃ a_1 f_1, x = abs { fst := a_1, snd := f_1 } ∧ ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a_1 i), f_1 i j ∈ f i '' univ",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\n⊢ LiftP (fun i u => u ∈ f i '' univ) x",
"tactic": "rw [liftP_iff]"
},
{
"state_after": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\n⊢ ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), f i j ∈ f i '' univ",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\n⊢ ∃ a_1 f_1, x = abs { fst := a_1, snd := f_1 } ∧ ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a_1 i), f_1 i j ∈ f i '' univ",
"tactic": "refine' ⟨a, f, haf.symm, _⟩"
},
{
"state_after": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni✝ : Fin2 n\nu✝ : α i✝\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i✝ u✝\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\ni : Fin2 n\nu : MvPFunctor.B (P F) a i\n⊢ f i u ∈ f i '' univ",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni : Fin2 n\nu : α i\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i u\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\n⊢ ∀ (i : Fin2 n) (j : MvPFunctor.B (P F) a i), f i j ∈ f i '' univ",
"tactic": "intro i u"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nx : F α\ni✝ : Fin2 n\nu✝ : α i✝\nh : ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i✝ u✝\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nhaf : abs { fst := a, snd := f } = x\ni : Fin2 n\nu : MvPFunctor.B (P F) a i\n⊢ f i u ∈ f i '' univ",
"tactic": "exact mem_image_of_mem _ (mem_univ _)"
}
] |
[
180,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/MeasureTheory/Measure/AEDisjoint.lean
|
MeasureTheory.AEDisjoint.measure_diff_right
|
[] |
[
138,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
EuclideanDomain.dvd_mod_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : EuclideanDomain R\na b c : R\nh : c ∣ b\n⊢ c ∣ a % b ↔ c ∣ a",
"tactic": "rw [← dvd_add_right (h.mul_right _), div_add_mod]"
}
] |
[
70,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
EuclideanSpace.single_apply
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.584231\n𝕜 : Type u_2\ninst✝¹⁰ : IsROrC 𝕜\nE : Type ?u.584240\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : InnerProductSpace 𝕜 E\nE' : Type ?u.584260\ninst✝⁷ : NormedAddCommGroup E'\ninst✝⁶ : InnerProductSpace 𝕜 E'\nF : Type ?u.584278\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ F\nF' : Type ?u.584298\ninst✝³ : NormedAddCommGroup F'\ninst✝² : InnerProductSpace ℝ F'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni : ι\na : 𝕜\nj : ι\n⊢ single i a j = if j = i then a else 0",
"tactic": "rw [EuclideanSpace.single, PiLp.equiv_symm_apply, ← Pi.single_apply i a j]"
}
] |
[
272,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.mem_comap
|
[] |
[
1339,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1338,
1
] |
Mathlib/Topology/Separation.lean
|
PreconnectedSpace.trivial_of_discrete
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ninst✝ : DiscreteTopology α\n⊢ ¬Nontrivial α",
"state_before": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ninst✝ : DiscreteTopology α\n⊢ Subsingleton α",
"tactic": "rw [← not_nontrivial_iff_subsingleton]"
},
{
"state_after": "case mk.intro.intro\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ninst✝ : DiscreteTopology α\nx y : α\nhxy : x ≠ y\n⊢ False",
"state_before": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ninst✝ : DiscreteTopology α\n⊢ ¬Nontrivial α",
"tactic": "rintro ⟨x, y, hxy⟩"
},
{
"state_after": "case mk.intro.intro\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ninst✝ : DiscreteTopology α\nx y : α\nhxy : ¬x ∈ univ\n⊢ False",
"state_before": "case mk.intro.intro\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ninst✝ : DiscreteTopology α\nx y : α\nhxy : x ≠ y\n⊢ False",
"tactic": "rw [Ne.def, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ninst✝ : DiscreteTopology α\nx y : α\nhxy : ¬x ∈ univ\n⊢ False",
"tactic": "exact hxy (mem_univ x)"
}
] |
[
798,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
LocalRing.ResidueField.mapEquiv_refl
|
[] |
[
473,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.option_bind
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type ?u.113880\nδ : Type ?u.113883\nσ : Type u_2\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\nf : α → Option β\ng : α → β → Option σ\nhf : Primrec f\nhg : Primrec₂ g\na : α\n⊢ Option.casesOn (f a) none (g a) = Option.bind (f a) (g a)",
"tactic": "cases f a <;> rfl"
}
] |
[
626,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.comap_id
|
[] |
[
298,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.pair_comm
|
[] |
[
1133,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1132,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.div_pos
|
[
{
"state_after": "no goals",
"state_before": "m n k a b : ℕ\nhba : b ≤ a\nhb : 0 < b\nh : a / b = 0\n⊢ a = a % b",
"tactic": "simpa [h] using (mod_add_div a b).symm"
}
] |
[
639,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
11
] |
Std/Data/Array/Lemmas.lean
|
mkArray_data
|
[] |
[
29,
96
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
29,
9
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
coplanar_insert_iff_of_mem_affineSpan
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\nι : Type ?u.356602\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np : P\nh : p ∈ affineSpan k s\n⊢ Coplanar k (insert p s) ↔ Coplanar k s",
"tactic": "rw [Coplanar, Coplanar, vectorSpan_insert_eq_vectorSpan h]"
}
] |
[
669,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
QuasilinearOn.monotoneOn_or_antitoneOn
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.63373\nF : Type ?u.63376\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : LinearOrderedAddCommMonoid β\ns : Set 𝕜\nf : 𝕜 → β\nhf : QuasilinearOn 𝕜 s f\n⊢ ∀ (a : 𝕜), a ∈ s → ∀ (b : 𝕜), b ∈ s → ∀ (c : 𝕜), c ∈ s → c ∈ segment 𝕜 a b → f c ∈ uIcc (f a) (f b)",
"state_before": "𝕜 : Type u_1\nE : Type ?u.63373\nF : Type ?u.63376\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : LinearOrderedAddCommMonoid β\ns : Set 𝕜\nf : 𝕜 → β\nhf : QuasilinearOn 𝕜 s f\n⊢ MonotoneOn f s ∨ AntitoneOn f s",
"tactic": "simp_rw [monotoneOn_or_antitoneOn_iff_uIcc, ← segment_eq_uIcc]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.63373\nF : Type ?u.63376\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : LinearOrderedAddCommMonoid β\ns : Set 𝕜\nf : 𝕜 → β\nhf : QuasilinearOn 𝕜 s f\na : 𝕜\nha : a ∈ s\nb : 𝕜\nhb : b ∈ s\nc : 𝕜\nx✝ : c ∈ s\nh : c ∈ segment 𝕜 a b\n⊢ f c ∈ uIcc (f a) (f b)",
"state_before": "𝕜 : Type u_1\nE : Type ?u.63373\nF : Type ?u.63376\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : LinearOrderedAddCommMonoid β\ns : Set 𝕜\nf : 𝕜 → β\nhf : QuasilinearOn 𝕜 s f\n⊢ ∀ (a : 𝕜), a ∈ s → ∀ (b : 𝕜), b ∈ s → ∀ (c : 𝕜), c ∈ s → c ∈ segment 𝕜 a b → f c ∈ uIcc (f a) (f b)",
"tactic": "rintro a ha b hb c _ h"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.63373\nF : Type ?u.63376\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : LinearOrderedAddCommMonoid β\ns : Set 𝕜\nf : 𝕜 → β\nhf : QuasilinearOn 𝕜 s f\na : 𝕜\nha : a ∈ s\nb : 𝕜\nhb : b ∈ s\nc : 𝕜\nx✝ : c ∈ s\nh : c ∈ segment 𝕜 a b\n⊢ f c ∈ uIcc (f a) (f b)",
"tactic": "refine' ⟨((hf.2 _).segment_subset _ _ h).2, ((hf.1 _).segment_subset _ _ h).2⟩ <;> simp [*]"
}
] |
[
244,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.Measure.toENNRealVectorMeasure_apply_measurable
|
[] |
[
502,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Ring.lean
|
Summable.tsum_mul_left
|
[] |
[
58,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MulHom.coe_copy_eq
|
[] |
[
850,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.eq_zero_of_bot_submodule
|
[] |
[
1025,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.exists_nat_eq_of_le_nat
|
[] |
[
1734,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1732,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.ofNat
|
[] |
[
285,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
11
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_injOn_pos_of_base_lt_one
|
[] |
[
330,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.ball_smul_ball
|
[
{
"state_after": "R : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\n⊢ ∀ (x : E), x ∈ Metric.ball 0 r₁ • ball p 0 r₂ → x ∈ ball p 0 (r₁ * r₂)",
"state_before": "R : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\n⊢ Metric.ball 0 r₁ • ball p 0 r₂ ⊆ ball p 0 (r₁ * r₂)",
"tactic": "rw [Set.subset_def]"
},
{
"state_after": "R : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\nhx : x ∈ Metric.ball 0 r₁ • ball p 0 r₂\n⊢ x ∈ ball p 0 (r₁ * r₂)",
"state_before": "R : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\n⊢ ∀ (x : E), x ∈ Metric.ball 0 r₁ • ball p 0 r₂ → x ∈ ball p 0 (r₁ * r₂)",
"tactic": "intro x hx"
},
{
"state_after": "R : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\nhx : ∃ x_1 y, x_1 ∈ Metric.ball 0 r₁ ∧ y ∈ ball p 0 r₂ ∧ x_1 • y = x\n⊢ x ∈ ball p 0 (r₁ * r₂)",
"state_before": "R : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\nhx : x ∈ Metric.ball 0 r₁ • ball p 0 r₂\n⊢ x ∈ ball p 0 (r₁ * r₂)",
"tactic": "rw [Set.mem_smul] at hx"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ x ∈ ball p 0 (r₁ * r₂)",
"state_before": "R : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\nhx : ∃ x_1 y, x_1 ∈ Metric.ball 0 r₁ ∧ y ∈ ball p 0 r₂ ∧ x_1 • y = x\n⊢ x ∈ ball p 0 (r₁ * r₂)",
"tactic": "rcases hx with ⟨a, y, ha, hy, hx⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ ‖a‖ * ↑p y < r₁ * r₂",
"state_before": "case intro.intro.intro.intro\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ x ∈ ball p 0 (r₁ * r₂)",
"tactic": "rw [← hx, mem_ball_zero, map_smul_eq_mul]"
},
{
"state_after": "case intro.intro.intro.intro.a\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ ‖a‖ < r₁\n\ncase intro.intro.intro.intro.a\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ ↑p y < r₂",
"state_before": "case intro.intro.intro.intro\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ ‖a‖ * ↑p y < r₁ * r₂",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.a\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ ‖a‖ < r₁",
"tactic": "exact mem_ball_zero_iff.mp ha"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.a\nR : Type ?u.1107787\nR' : Type ?u.1107790\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1107796\n𝕜₃ : Type ?u.1107799\n𝕝 : Type ?u.1107802\nE : Type u_2\nE₂ : Type ?u.1107808\nE₃ : Type ?u.1107811\nF : Type ?u.1107814\nG : Type ?u.1107817\nι : Type ?u.1107820\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np✝ p : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nx : E\na : 𝕜\ny : E\nha : a ∈ Metric.ball 0 r₁\nhy : y ∈ ball p 0 r₂\nhx : a • y = x\n⊢ ↑p y < r₂",
"tactic": "exact p.mem_ball_zero.mp hy"
}
] |
[
891,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.dvd_lcm_right
|
[] |
[
95,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.coe_mk
|
[] |
[
304,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_subsingleton
|
[
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.913497\nE : Type u_1\nEₗ : Type ?u.913503\nF : Type u_2\nFₗ : Type ?u.913509\nG : Type ?u.913512\nGₗ : Type ?u.913515\n𝓕 : Type ?u.913518\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\ninst✝ : Subsingleton E\n⊢ ‖f‖ ≤ 0",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.913497\nE : Type u_1\nEₗ : Type ?u.913503\nF : Type u_2\nFₗ : Type ?u.913509\nG : Type ?u.913512\nGₗ : Type ?u.913515\n𝓕 : Type ?u.913518\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\ninst✝ : Subsingleton E\n⊢ ‖f‖ = 0",
"tactic": "refine' le_antisymm _ (norm_nonneg _)"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.913497\nE : Type u_1\nEₗ : Type ?u.913503\nF : Type u_2\nFₗ : Type ?u.913509\nG : Type ?u.913512\nGₗ : Type ?u.913515\n𝓕 : Type ?u.913518\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\ninst✝ : Subsingleton E\n⊢ ∀ (x : E), ‖↑f x‖ ≤ 0 * ‖x‖",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.913497\nE : Type u_1\nEₗ : Type ?u.913503\nF : Type u_2\nFₗ : Type ?u.913509\nG : Type ?u.913512\nGₗ : Type ?u.913515\n𝓕 : Type ?u.913518\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\ninst✝ : Subsingleton E\n⊢ ‖f‖ ≤ 0",
"tactic": "apply op_norm_le_bound _ rfl.ge"
},
{
"state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.913497\nE : Type u_1\nEₗ : Type ?u.913503\nF : Type u_2\nFₗ : Type ?u.913509\nG : Type ?u.913512\nGₗ : Type ?u.913515\n𝓕 : Type ?u.913518\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\ninst✝ : Subsingleton E\nx : E\n⊢ ‖↑f x‖ ≤ 0 * ‖x‖",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.913497\nE : Type u_1\nEₗ : Type ?u.913503\nF : Type u_2\nFₗ : Type ?u.913509\nG : Type ?u.913512\nGₗ : Type ?u.913515\n𝓕 : Type ?u.913518\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\ninst✝ : Subsingleton E\n⊢ ∀ (x : E), ‖↑f x‖ ≤ 0 * ‖x‖",
"tactic": "intro x"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.913497\nE : Type u_1\nEₗ : Type ?u.913503\nF : Type u_2\nFₗ : Type ?u.913509\nG : Type ?u.913512\nGₗ : Type ?u.913515\n𝓕 : Type ?u.913518\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\ninst✝ : Subsingleton E\nx : E\n⊢ ‖↑f x‖ ≤ 0 * ‖x‖",
"tactic": "simp [Subsingleton.elim x 0]"
}
] |
[
648,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
star_injective
|
[] |
[
91,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.natDegree_smul_of_smul_regular
|
[
{
"state_after": "case pos\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : p = 0\n⊢ natDegree (k • p) = natDegree p\n\ncase neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : ¬p = 0\n⊢ natDegree (k • p) = natDegree p",
"state_before": "R : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\n⊢ natDegree (k • p) = natDegree p",
"tactic": "by_cases hp : p = 0"
},
{
"state_after": "case neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : ¬p = 0\n⊢ k • p ≠ 0",
"state_before": "case neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : ¬p = 0\n⊢ natDegree (k • p) = natDegree p",
"tactic": "rw [← WithBot.coe_eq_coe, ← Nat.cast_withBot, ←Nat.cast_withBot,\n ← degree_eq_natDegree hp, ← degree_eq_natDegree,\n degree_smul_of_smul_regular p h]"
},
{
"state_after": "case neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : k • p = 0\n⊢ p = 0",
"state_before": "case neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : ¬p = 0\n⊢ k • p ≠ 0",
"tactic": "contrapose! hp"
},
{
"state_after": "case neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp✝ : k • p = 0\nhp : k • p = k • 0\n⊢ p = 0",
"state_before": "case neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : k • p = 0\n⊢ p = 0",
"tactic": "rw [← smul_zero k] at hp"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp✝ : k • p = 0\nhp : k • p = k • 0\n⊢ p = 0",
"tactic": "exact h.polynomial hp"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS✝ : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝² : Semiring R\np✝ : R[X]\nS : Type u_1\ninst✝¹ : Monoid S\ninst✝ : DistribMulAction S R\nk : S\np : R[X]\nh : IsSMulRegular R k\nhp : p = 0\n⊢ natDegree (k • p) = natDegree p",
"tactic": "simp [hp]"
}
] |
[
507,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.enum_inj
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\n⊢ o₁ = o₂",
"tactic": "by_contra hne"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\n⊢ False",
"tactic": "haveI : IsIrrefl _ r := inferInstance"
},
{
"state_after": "case inl.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt : o₁ < o₂\n⊢ r ?inl.a ?inl.a\n\ncase inl.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt : o₁ < o₂\n⊢ α\n\ncase inr.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt : o₁ > o₂\n⊢ r ?inr.a ?inr.a\n\ncase inr.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt : o₁ > o₂\n⊢ α",
"state_before": "α : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\n⊢ False",
"tactic": "cases' lt_or_gt_of_ne hne with hlt hlt <;> apply (this).1"
},
{
"state_after": "no goals",
"state_before": "case inl.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt : o₁ < o₂\n⊢ r ?inl.a ?inl.a",
"tactic": "rwa [← @enum_lt_enum α r _ o₁ o₂ h₁ h₂, h] at hlt"
},
{
"state_after": "case inr.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt✝ : o₁ > o₂\nhlt : o₂ < o₁\n⊢ r ?inr.a ?inr.a",
"state_before": "case inr.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt : o₁ > o₂\n⊢ r ?inr.a ?inr.a",
"tactic": "rw [GT.gt] at hlt"
},
{
"state_after": "no goals",
"state_before": "case inr.a\nα : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : enum r o₁ h₁ = enum r o₂ h₂\nhne : ¬o₁ = o₂\nthis : IsIrrefl α r\nhlt✝ : o₁ > o₂\nhlt : o₂ < o₁\n⊢ r ?inr.a ?inr.a",
"tactic": "rwa [← @enum_lt_enum α r _ o₂ o₁ h₂ h₁, h] at hlt"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.198351\nγ : Type ?u.198354\nr✝ : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nr : α → α → Prop\ninst✝ : IsWellOrder α r\no₁ o₂ : Ordinal\nh₁ : o₁ < type r\nh₂ : o₂ < type r\nh : o₁ = o₂\n⊢ enum r o₁ h₁ = enum r o₂ h₂",
"tactic": "simp_rw [h]"
}
] |
[
1196,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1188,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
not_sbtw_self_left
|
[] |
[
312,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.I_pow_bit0
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ I ^ bit0 n = (-1) ^ n",
"tactic": "rw [pow_bit0', Complex.I_mul_I]"
}
] |
[
471,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Std/Logic.lean
|
imp_self
|
[] |
[
122,
63
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
122,
9
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
Absorbs.inter
|
[
{
"state_after": "case intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b : 𝕜\nht : Absorbs 𝕜 t u\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\n⊢ Absorbs 𝕜 (s ∩ t) u",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na b : 𝕜\nhs : Absorbs 𝕜 s u\nht : Absorbs 𝕜 t u\n⊢ Absorbs 𝕜 (s ∩ t) u",
"tactic": "obtain ⟨a, ha, hs⟩ := hs"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\n⊢ Absorbs 𝕜 (s ∩ t) u",
"state_before": "case intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b : 𝕜\nht : Absorbs 𝕜 t u\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\n⊢ Absorbs 𝕜 (s ∩ t) u",
"tactic": "obtain ⟨b, _hb, ht⟩ := ht"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\nh : 0 < max a b\n⊢ Absorbs 𝕜 (s ∩ t) u",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\n⊢ Absorbs 𝕜 (s ∩ t) u",
"tactic": "have h : 0 < max a b := lt_max_of_lt_left ha"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\nh : 0 < max a b\nc : 𝕜\nhc : max a b ≤ ‖c‖\n⊢ u ⊆ c • (s ∩ t)",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\nh : 0 < max a b\n⊢ Absorbs 𝕜 (s ∩ t) u",
"tactic": "refine' ⟨max a b, lt_max_of_lt_left ha, fun c hc => _⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\nh : 0 < max a b\nc : 𝕜\nhc : max a b ≤ ‖c‖\n⊢ u ⊆ c • s ∩ c • t",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\nh : 0 < max a b\nc : 𝕜\nhc : max a b ≤ ‖c‖\n⊢ u ⊆ c • (s ∩ t)",
"tactic": "rw [smul_set_inter₀ (norm_pos_iff.1 <| h.trans_le hc)]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕝 : Type ?u.125647\nE : Type u_2\nι : Sort ?u.125653\nκ : ι → Sort ?u.125658\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : NormedSpace 𝕜 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\ns t u v A B : Set E\nx : E\na✝ b✝ : 𝕜\na : ℝ\nha : 0 < a\nhs : ∀ (a_1 : 𝕜), a ≤ ‖a_1‖ → u ⊆ a_1 • s\nb : ℝ\n_hb : 0 < b\nht : ∀ (a : 𝕜), b ≤ ‖a‖ → u ⊆ a • t\nh : 0 < max a b\nc : 𝕜\nhc : max a b ≤ ‖c‖\n⊢ u ⊆ c • s ∩ c • t",
"tactic": "exact subset_inter (hs _ <| le_of_max_le_left hc) (ht _ <| le_of_max_le_right hc)"
}
] |
[
324,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
src/lean/Init/Data/Nat/Power2.lean
|
Nat.one_isPowerOfTwo
|
[
{
"state_after": "no goals",
"state_before": "⊢ 1 = 2 ^ 0",
"tactic": "decide"
}
] |
[
30,
17
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
29,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.dvd_prod_of_mem
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.786416\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : α → β\na : α\ns : Finset α\nha : a ∈ s\n⊢ f a ∣ ∏ i in s, f i",
"tactic": "classical\n rw [Finset.prod_eq_mul_prod_diff_singleton ha]\n exact dvd_mul_right _ _"
},
{
"state_after": "ι : Type ?u.786416\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : α → β\na : α\ns : Finset α\nha : a ∈ s\n⊢ f a ∣ f a * ∏ x in s \\ {a}, f x",
"state_before": "ι : Type ?u.786416\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : α → β\na : α\ns : Finset α\nha : a ∈ s\n⊢ f a ∣ ∏ i in s, f i",
"tactic": "rw [Finset.prod_eq_mul_prod_diff_singleton ha]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.786416\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : α → β\na : α\ns : Finset α\nha : a ∈ s\n⊢ f a ∣ f a * ∏ x in s \\ {a}, f x",
"tactic": "exact dvd_mul_right _ _"
}
] |
[
1583,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1580,
1
] |
Mathlib/Algebra/Algebra/Spectrum.lean
|
spectrum.zero_mem_resolventSet_of_unit
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : Aˣ\n⊢ 0 ∈ resolventSet R ↑a",
"tactic": "simpa only [mem_resolventSet_iff, ← not_mem_iff, zero_not_mem_iff] using a.isUnit"
}
] |
[
197,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Topology/Order.lean
|
le_nhdsAdjoint_iff'
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ (∀ (x : α), 𝓝 x ≤ 𝓝 x) ↔ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"state_before": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ t ≤ nhdsAdjoint a f ↔ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"tactic": "rw [le_iff_nhds]"
},
{
"state_after": "case mp\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ (∀ (x : α), 𝓝 x ≤ 𝓝 x) → 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b\n\ncase mpr\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ (𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b) → ∀ (x : α), 𝓝 x ≤ 𝓝 x",
"state_before": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ (∀ (x : α), 𝓝 x ≤ 𝓝 x) ↔ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"tactic": "constructor"
},
{
"state_after": "case mp\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\n⊢ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"state_before": "case mp\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ (∀ (x : α), 𝓝 x ≤ 𝓝 x) → 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"tactic": "intro h"
},
{
"state_after": "case mp.left\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\n⊢ 𝓝 a ≤ pure a ⊔ f\n\ncase mp.right\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\n⊢ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"state_before": "case mp\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\n⊢ 𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"tactic": "constructor"
},
{
"state_after": "case mp.left\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ 𝓝 a\n⊢ 𝓝 a ≤ pure a ⊔ f",
"state_before": "case mp.left\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\n⊢ 𝓝 a ≤ pure a ⊔ f",
"tactic": "specialize h a"
},
{
"state_after": "no goals",
"state_before": "case mp.left\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ 𝓝 a\n⊢ 𝓝 a ≤ pure a ⊔ f",
"tactic": "rwa [nhdsAdjoint_nhds] at h"
},
{
"state_after": "case mp.right\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\nb : α\nhb : b ≠ a\n⊢ 𝓝 b = pure b",
"state_before": "case mp.right\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\n⊢ ∀ (b : α), b ≠ a → 𝓝 b = pure b",
"tactic": "intro b hb"
},
{
"state_after": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\nb : α\nhb : b ≠ a\n⊢ 𝓝 b ≤ pure b",
"state_before": "case mp.right\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\nb : α\nhb : b ≠ a\n⊢ 𝓝 b = pure b",
"tactic": "apply le_antisymm _ (pure_le_nhds b)"
},
{
"state_after": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nb : α\nhb : b ≠ a\nh : 𝓝 b ≤ 𝓝 b\n⊢ 𝓝 b ≤ pure b",
"state_before": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : ∀ (x : α), 𝓝 x ≤ 𝓝 x\nb : α\nhb : b ≠ a\n⊢ 𝓝 b ≤ pure b",
"tactic": "specialize h b"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nb : α\nhb : b ≠ a\nh : 𝓝 b ≤ 𝓝 b\n⊢ 𝓝 b ≤ pure b",
"tactic": "rwa [nhdsAdjoint_nhds_of_ne a f hb] at h"
},
{
"state_after": "case mpr.intro\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ pure a ⊔ f\nh' : ∀ (b : α), b ≠ a → 𝓝 b = pure b\nb : α\n⊢ 𝓝 b ≤ 𝓝 b",
"state_before": "case mpr\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\n⊢ (𝓝 a ≤ pure a ⊔ f ∧ ∀ (b : α), b ≠ a → 𝓝 b = pure b) → ∀ (x : α), 𝓝 x ≤ 𝓝 x",
"tactic": "rintro ⟨h, h'⟩ b"
},
{
"state_after": "case pos\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ pure a ⊔ f\nh' : ∀ (b : α), b ≠ a → 𝓝 b = pure b\nb : α\nhb : b = a\n⊢ 𝓝 b ≤ 𝓝 b\n\ncase neg\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ pure a ⊔ f\nh' : ∀ (b : α), b ≠ a → 𝓝 b = pure b\nb : α\nhb : ¬b = a\n⊢ 𝓝 b ≤ 𝓝 b",
"state_before": "case mpr.intro\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ pure a ⊔ f\nh' : ∀ (b : α), b ≠ a → 𝓝 b = pure b\nb : α\n⊢ 𝓝 b ≤ 𝓝 b",
"tactic": "by_cases hb : b = a"
},
{
"state_after": "no goals",
"state_before": "case pos\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ pure a ⊔ f\nh' : ∀ (b : α), b ≠ a → 𝓝 b = pure b\nb : α\nhb : b = a\n⊢ 𝓝 b ≤ 𝓝 b",
"tactic": "rwa [hb, nhdsAdjoint_nhds]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα✝ : Type u\nβ : Type v\nα : Type u_1\na : α\nf : Filter α\nt : TopologicalSpace α\nh : 𝓝 a ≤ pure a ⊔ f\nh' : ∀ (b : α), b ≠ a → 𝓝 b = pure b\nb : α\nhb : ¬b = a\n⊢ 𝓝 b ≤ 𝓝 b",
"tactic": "simp [nhdsAdjoint_nhds_of_ne a f hb, h' b hb]"
}
] |
[
648,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
AEMeasurable.pow_const
|
[] |
[
238,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
RingHom.comap_ker
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type ?u.1721293\nG : Type ?u.1721296\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nrcf : RingHomClass F R S\nrcg : RingHomClass G T S\nf✝ : F\ng✝ : G\nf : S →+* R\ng : T →+* S\n⊢ Ideal.comap g (ker f) = ker (comp f g)",
"tactic": "rw [RingHom.ker_eq_comap_bot, Ideal.comap_comap, RingHom.ker_eq_comap_bot]"
}
] |
[
2013,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2012,
1
] |
Mathlib/GroupTheory/Submonoid/Centralizer.lean
|
Submonoid.centralizer_le
|
[] |
[
84,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.coe_smul
|
[] |
[
423,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.exists_of_LiftRel_left
|
[] |
[
1137,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1135,
1
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
TopologicalSpace.Opens.toTopCat_map
|
[] |
[
125,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Combinatorics/HalesJewett.lean
|
Combinatorics.Line.horizontal_apply
|
[
{
"state_after": "case h\nα : Type u_1\nι : Type u_2\nι' : Type u_3\nl : Line α ι\nv : ι' → α\nx : α\ni : ι ⊕ ι'\n⊢ (fun x i => Option.getD (idxFun (horizontal l v) i) x) x i = Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) v i",
"state_before": "α : Type u_1\nι : Type u_2\nι' : Type u_3\nl : Line α ι\nv : ι' → α\nx : α\n⊢ (fun x i => Option.getD (idxFun (horizontal l v) i) x) x = Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) v",
"tactic": "funext i"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nι : Type u_2\nι' : Type u_3\nl : Line α ι\nv : ι' → α\nx : α\ni : ι ⊕ ι'\n⊢ (fun x i => Option.getD (idxFun (horizontal l v) i) x) x i = Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) v i",
"tactic": "cases i <;> rfl"
}
] |
[
210,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Algebra/Parity.lean
|
Odd.pos
|
[
{
"state_after": "case intro\nF : Type ?u.100603\nα : Type u_1\nβ : Type ?u.100609\nR : Type ?u.100612\ninst✝¹ : CanonicallyOrderedCommSemiring α\ninst✝ : Nontrivial α\nk : α\n⊢ 0 < 2 * k + 1",
"state_before": "F : Type ?u.100603\nα : Type u_1\nβ : Type ?u.100609\nR : Type ?u.100612\ninst✝¹ : CanonicallyOrderedCommSemiring α\ninst✝ : Nontrivial α\nn : α\nhn : Odd n\n⊢ 0 < n",
"tactic": "obtain ⟨k, rfl⟩ := hn"
},
{
"state_after": "case intro\nF : Type ?u.100603\nα : Type u_1\nβ : Type ?u.100609\nR : Type ?u.100612\ninst✝¹ : CanonicallyOrderedCommSemiring α\ninst✝ : Nontrivial α\nk : α\n⊢ 1 = 0 → ¬2 * k = 0",
"state_before": "case intro\nF : Type ?u.100603\nα : Type u_1\nβ : Type ?u.100609\nR : Type ?u.100612\ninst✝¹ : CanonicallyOrderedCommSemiring α\ninst✝ : Nontrivial α\nk : α\n⊢ 0 < 2 * k + 1",
"tactic": "rw [pos_iff_ne_zero, Ne.def, add_eq_zero_iff, not_and']"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type ?u.100603\nα : Type u_1\nβ : Type ?u.100609\nR : Type ?u.100612\ninst✝¹ : CanonicallyOrderedCommSemiring α\ninst✝ : Nontrivial α\nk : α\n⊢ 1 = 0 → ¬2 * k = 0",
"tactic": "exact fun h => (one_ne_zero h).elim"
}
] |
[
406,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.19057\nγ : Type ?u.19060\nδ : Type ?u.19063\ninst✝ : TopologicalSpace α\na : α\ns : Set α\n⊢ 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a",
"tactic": "rw [← singleton_union, nhdsWithin_union, nhdsWithin_singleton]"
}
] |
[
287,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.top_toAddSubmonoid
|
[] |
[
150,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_linear
|
[] |
[
1164,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1163,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.testBit_two_pow_of_ne
|
[
{
"state_after": "n m : ℕ\nhm : n ≠ m\n⊢ bodd (2 ^ n / 2 ^ m) = false",
"state_before": "n m : ℕ\nhm : n ≠ m\n⊢ testBit (2 ^ n) m = false",
"tactic": "rw [testBit, shiftr_eq_div_pow]"
},
{
"state_after": "case inl\nn m : ℕ\nhm✝ : n ≠ m\nhm : n < m\n⊢ bodd (2 ^ n / 2 ^ m) = false\n\ncase inr\nn m : ℕ\nhm✝ : n ≠ m\nhm : m < n\n⊢ bodd (2 ^ n / 2 ^ m) = false",
"state_before": "n m : ℕ\nhm : n ≠ m\n⊢ bodd (2 ^ n / 2 ^ m) = false",
"tactic": "cases' hm.lt_or_lt with hm hm"
},
{
"state_after": "case inl\nn m : ℕ\nhm✝ : n ≠ m\nhm : n < m\n⊢ 2 ^ n < 2 ^ m",
"state_before": "case inl\nn m : ℕ\nhm✝ : n ≠ m\nhm : n < m\n⊢ bodd (2 ^ n / 2 ^ m) = false",
"tactic": "rw [Nat.div_eq_zero, bodd_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl\nn m : ℕ\nhm✝ : n ≠ m\nhm : n < m\n⊢ 2 ^ n < 2 ^ m",
"tactic": "exact Nat.pow_lt_pow_of_lt_right one_lt_two hm"
},
{
"state_after": "case inr\nn m : ℕ\nhm✝ : n ≠ m\nhm : m < n\n⊢ bodd (2 ^ (n - m - succ 0 + succ 0)) = false",
"state_before": "case inr\nn m : ℕ\nhm✝ : n ≠ m\nhm : m < n\n⊢ bodd (2 ^ n / 2 ^ m) = false",
"tactic": "rw [pow_div hm.le zero_lt_two, ← tsub_add_cancel_of_le (succ_le_of_lt <| tsub_pos_of_lt hm)]"
},
{
"state_after": "case inr\nn m : ℕ\nhm✝ : n ≠ m\nhm : m < n\n⊢ bodd (2 ^ (n - m - succ 0 + succ 0)) = false",
"state_before": "case inr\nn m : ℕ\nhm✝ : n ≠ m\nhm : m < n\n⊢ bodd (2 ^ (n - m - succ 0 + succ 0)) = false",
"tactic": "rw [(rfl : succ 0 = 1)]"
},
{
"state_after": "no goals",
"state_before": "case inr\nn m : ℕ\nhm✝ : n ≠ m\nhm : m < n\n⊢ bodd (2 ^ (n - m - succ 0 + succ 0)) = false",
"tactic": "simp only [ge_iff_le, tsub_le_iff_right, pow_succ, bodd_mul,\n Bool.and_eq_false_eq_eq_false_or_eq_false, or_true]"
}
] |
[
155,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Data/Part.lean
|
Part.map_eq_map
|
[] |
[
613,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
612,
1
] |
Mathlib/Topology/Separation.lean
|
t2_separation_compact_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : LocallyCompactSpace α\ninst✝ : T2Space α\nx y : α\nh : x ≠ y\n⊢ ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v",
"tactic": "simpa only [exists_prop, ← exists_and_left, and_comm, and_assoc, and_left_comm] using\n ((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h)"
}
] |
[
943,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
940,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.toFinsupp_inj
|
[] |
[
261,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
antivary_toDual_left
|
[] |
[
259,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.angle_left_midpoint_eq_pi_div_two_of_dist_eq
|
[
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint ℝ p1 p2\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"tactic": "let m : P := midpoint ℝ p1 p2"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint ℝ p1 p2\nh1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint ℝ p1 p2\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"tactic": "have h1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m) := (vsub_sub_vsub_cancel_right p3 p1 m).symm"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint ℝ p1 p2\nh1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)\nh2 : p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m)\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint ℝ p1 p2\nh1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"tactic": "have h2 : p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m) := by\n rw [left_vsub_midpoint, ← midpoint_vsub_right, vsub_add_vsub_cancel]"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nm : P := midpoint ℝ p1 p2\nh : ‖p3 -ᵥ m - (p1 -ᵥ m)‖ = ‖p3 -ᵥ m + (p1 -ᵥ m)‖\nh1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)\nh2 : p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m)\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint ℝ p1 p2\nh1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)\nh2 : p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m)\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"tactic": "rw [dist_eq_norm_vsub V p3 p1, dist_eq_norm_vsub V p3 p2, h1, h2] at h"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nm : P := midpoint ℝ p1 p2\nh : ‖p3 -ᵥ m - (p1 -ᵥ m)‖ = ‖p3 -ᵥ m + (p1 -ᵥ m)‖\nh1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)\nh2 : p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m)\n⊢ ∠ p3 (midpoint ℝ p1 p2) p1 = π / 2",
"tactic": "exact (norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (p3 -ᵥ m) (p1 -ᵥ m)).mp h.symm"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint ℝ p1 p2\nh1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)\n⊢ p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m)",
"tactic": "rw [left_vsub_midpoint, ← midpoint_vsub_right, vsub_add_vsub_cancel]"
}
] |
[
269,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integral_comp_mul_right_Ioi
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\nf : ℝ → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ng : ℝ → E\na b : ℝ\nhb : 0 < b\n⊢ (∫ (x : ℝ) in Ioi a, g (x * b)) = abs b⁻¹ • ∫ (x : ℝ) in Ioi (a * b), g x",
"tactic": "simpa only [mul_comm] using integral_comp_mul_left_Ioi g a hb"
}
] |
[
891,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
889,
1
] |
Mathlib/Deprecated/Subring.lean
|
Ring.mem_closure
|
[] |
[
188,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.splitFun_inj
|
[
{
"state_after": "n : ℕ\nα : TypeVec (n + 1)\nα' : TypeVec (n + 1)\nf f' : drop α ⟹ drop α'\ng g' : last α → last α'\nH : splitFun f g = splitFun f' g'\n⊢ dropFun (splitFun f' g') = f' ∧ lastFun (splitFun f' g') = g'",
"state_before": "n : ℕ\nα : TypeVec (n + 1)\nα' : TypeVec (n + 1)\nf f' : drop α ⟹ drop α'\ng g' : last α → last α'\nH : splitFun f g = splitFun f' g'\n⊢ f = f' ∧ g = g'",
"tactic": "rw [← dropFun_splitFun f g, H, ← lastFun_splitFun f g, H]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : TypeVec (n + 1)\nα' : TypeVec (n + 1)\nf f' : drop α ⟹ drop α'\ng g' : last α → last α'\nH : splitFun f g = splitFun f' g'\n⊢ dropFun (splitFun f' g') = f' ∧ lastFun (splitFun f' g') = g'",
"tactic": "simp"
}
] |
[
234,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.ofList_toList
|
[
{
"state_after": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\n⊢ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length = s.length\n\ncase refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\n⊢ ∀ (i : Fin ((ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length + 1)),\n series (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))) i =\n series s\n (↑(Fin.cast\n (_ :\n Nat.succ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length =\n Nat.succ s.length))\n i)",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\n⊢ ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s)) = s",
"tactic": "refine' ext_fun _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\n⊢ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length = s.length",
"tactic": "rw [length_ofList, length_toList, Nat.succ_sub_one]"
},
{
"state_after": "case refine'_2.mk\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi : i < (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length + 1\n⊢ series (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))) { val := i, isLt := hi } =\n series s\n (↑(Fin.cast\n (_ :\n Nat.succ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length =\n Nat.succ s.length))\n { val := i, isLt := hi })",
"state_before": "case refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\n⊢ ∀ (i : Fin ((ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length + 1)),\n series (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))) i =\n series s\n (↑(Fin.cast\n (_ :\n Nat.succ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length =\n Nat.succ s.length))\n i)",
"tactic": "rintro ⟨i, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : ℕ\nhi : i < (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length + 1\n⊢ series (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))) { val := i, isLt := hi } =\n series s\n (↑(Fin.cast\n (_ :\n Nat.succ (ofList (toList s) (_ : toList s ≠ []) (_ : List.Chain' IsMaximal (toList s))).length =\n Nat.succ s.length))\n { val := i, isLt := hi })",
"tactic": "simp [ofList, toList, -List.ofFn_succ]"
}
] |
[
296,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_ack_lt_ack_max_add_two
|
[] |
[
296,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Analysis/Subadditive.lean
|
Subadditive.tendsto_lim
|
[
{
"state_after": "case refine'_1\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nl : ℝ\nhl : l < Subadditive.lim h\n⊢ ∀ᶠ (b : ℕ) in atTop, l < u b / ↑b\n\ncase refine'_2\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > Subadditive.lim h\n⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < L",
"state_before": "u : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\n⊢ Tendsto (fun n => u n / ↑n) atTop (𝓝 (Subadditive.lim h))",
"tactic": "refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nl : ℝ\nhl : l < Subadditive.lim h\n⊢ ∀ᶠ (b : ℕ) in atTop, l < u b / ↑b",
"tactic": "refine' eventually_atTop.2\n ⟨1, fun n hn => hl.trans_le (h.lim_le_div hbdd (zero_lt_one.trans_le hn).ne')⟩"
},
{
"state_after": "case refine'_2.intro.intro\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > Subadditive.lim h\nn : ℕ\nnpos : 0 < n\nhn : u n / ↑n < L\n⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < L",
"state_before": "case refine'_2\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > Subadditive.lim h\n⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < L",
"tactic": "obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ u n / n < L := by\n rw [Subadditive.lim] at hL\n rcases exists_lt_of_csInf_lt (by simp) hL with ⟨x, hx, xL⟩\n rcases (mem_image _ _ _).1 hx with ⟨n, hn, rfl⟩\n exact ⟨n, zero_lt_one.trans_le hn, xL⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > Subadditive.lim h\nn : ℕ\nnpos : 0 < n\nhn : u n / ↑n < L\n⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < L",
"tactic": "exact h.eventually_div_lt_of_div_lt npos.ne' hn"
},
{
"state_after": "u : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > sInf ((fun n => u n / ↑n) '' Ici 1)\n⊢ ∃ n, 0 < n ∧ u n / ↑n < L",
"state_before": "u : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > Subadditive.lim h\n⊢ ∃ n, 0 < n ∧ u n / ↑n < L",
"tactic": "rw [Subadditive.lim] at hL"
},
{
"state_after": "case intro.intro\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > sInf ((fun n => u n / ↑n) '' Ici 1)\nx : ℝ\nhx : x ∈ (fun n => u n / ↑n) '' Ici 1\nxL : x < L\n⊢ ∃ n, 0 < n ∧ u n / ↑n < L",
"state_before": "u : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > sInf ((fun n => u n / ↑n) '' Ici 1)\n⊢ ∃ n, 0 < n ∧ u n / ↑n < L",
"tactic": "rcases exists_lt_of_csInf_lt (by simp) hL with ⟨x, hx, xL⟩"
},
{
"state_after": "case intro.intro.intro.intro\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > sInf ((fun n => u n / ↑n) '' Ici 1)\nn : ℕ\nhn : n ∈ Ici 1\nhx : u n / ↑n ∈ (fun n => u n / ↑n) '' Ici 1\nxL : u n / ↑n < L\n⊢ ∃ n, 0 < n ∧ u n / ↑n < L",
"state_before": "case intro.intro\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > sInf ((fun n => u n / ↑n) '' Ici 1)\nx : ℝ\nhx : x ∈ (fun n => u n / ↑n) '' Ici 1\nxL : x < L\n⊢ ∃ n, 0 < n ∧ u n / ↑n < L",
"tactic": "rcases (mem_image _ _ _).1 hx with ⟨n, hn, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nu : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > sInf ((fun n => u n / ↑n) '' Ici 1)\nn : ℕ\nhn : n ∈ Ici 1\nhx : u n / ↑n ∈ (fun n => u n / ↑n) '' Ici 1\nxL : u n / ↑n < L\n⊢ ∃ n, 0 < n ∧ u n / ↑n < L",
"tactic": "exact ⟨n, zero_lt_one.trans_le hn, xL⟩"
},
{
"state_after": "no goals",
"state_before": "u : ℕ → ℝ\nh : Subadditive u\nhbdd : BddBelow (range fun n => u n / ↑n)\nL : ℝ\nhL : L > sInf ((fun n => u n / ↑n) '' Ici 1)\n⊢ Set.Nonempty ((fun n => u n / ↑n) '' Ici 1)",
"tactic": "simp"
}
] |
[
98,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
RingHom.map_iterate_pthRoot
|
[] |
[
120,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.singleton_repr
|
[
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.612908\nι' : Type ?u.612911\nR✝ : Type ?u.612914\nR₂ : Type ?u.612917\nK : Type ?u.612920\nM : Type ?u.612923\nM' : Type ?u.612926\nM'' : Type ?u.612929\nV : Type u\nV' : Type ?u.612934\ninst✝⁶ : Semiring R✝\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R✝ M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R✝ M'\nb b₁ : Basis ι✝ R✝ M\ni✝ : ι✝\nc : R✝\nx✝ : M\nι : Type u_1\nR : Type u_2\ninst✝¹ : Unique ι\ninst✝ : Semiring R\nx : R\ni : ι\n⊢ ↑(↑(Basis.singleton ι R).repr x) i = x",
"tactic": "simp [Basis.singleton, Unique.eq_default i]"
}
] |
[
829,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Std/Data/String/Lemmas.lean
|
String.drop_eq
|
[] |
[
1090,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1089,
1
] |
Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
|
TensorAlgebra.ringQuot_mkAlgHom_freeAlgebra_ι_eq_ι
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m) =\n ↑{\n toAddHom :=\n { toFun := fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m),\n map_add' :=\n (_ :\n ∀ (x y : M),\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x +\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y) },\n map_smul' :=\n (_ :\n ∀ (r : R) (x : M),\n AddHom.toFun\n { toFun := fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m),\n map_add' :=\n (_ :\n ∀ (x y : M),\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x +\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y) }\n (r • x) =\n ↑(RingHom.id R) r •\n AddHom.toFun\n { toFun := fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m),\n map_add' :=\n (_ :\n ∀ (x y : M),\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x +\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y) }\n x) }\n m",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m) = ↑(ι R) m",
"tactic": "rw [ι]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m) =\n ↑{\n toAddHom :=\n { toFun := fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m),\n map_add' :=\n (_ :\n ∀ (x y : M),\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x +\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y) },\n map_smul' :=\n (_ :\n ∀ (r : R) (x : M),\n AddHom.toFun\n { toFun := fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m),\n map_add' :=\n (_ :\n ∀ (x y : M),\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x +\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y) }\n (r • x) =\n ↑(RingHom.id R) r •\n AddHom.toFun\n { toFun := fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m),\n map_add' :=\n (_ :\n ∀ (x y : M),\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x +\n (fun m => ↑(RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y) }\n x) }\n m",
"tactic": "rfl"
}
] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformOnFun.postcomp_uniformInducing
|
[
{
"state_after": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformInducing f\n⊢ comap\n (fun x =>\n ((↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) x.fst,\n (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) x.snd))\n (𝓤 (α →ᵤ[𝔖] β)) =\n 𝓤 (α →ᵤ[𝔖] γ)",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformInducing f\n⊢ UniformInducing (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖))",
"tactic": "constructor"
},
{
"state_after": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : comap (Prod.map f f) (𝓤 β) = 𝓤 γ\n⊢ comap\n (fun x =>\n ((↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) x.fst,\n (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) x.snd))\n (𝓤 (α →ᵤ[𝔖] β)) =\n 𝓤 (α →ᵤ[𝔖] γ)",
"state_before": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : UniformInducing f\n⊢ comap\n (fun x =>\n ((↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) x.fst,\n (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) x.snd))\n (𝓤 (α →ᵤ[𝔖] β)) =\n 𝓤 (α →ᵤ[𝔖] γ)",
"tactic": "replace hf : (𝓤 β).comap (Prod.map f f) = _ := hf.comap_uniformity"
},
{
"state_after": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ 𝓤 (α →ᵤ[𝔖] γ) = 𝓤 (α →ᵤ[𝔖] γ)",
"state_before": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : comap (Prod.map f f) (𝓤 β) = 𝓤 γ\n⊢ comap\n (Prod.map (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖))\n (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)))\n (𝓤 (α →ᵤ[𝔖] β)) =\n 𝓤 (α →ᵤ[𝔖] γ)",
"tactic": "rw [← uniformity_comap] at hf⊢"
},
{
"state_after": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) (uniformSpace α β 𝔖) = uniformSpace α γ 𝔖",
"state_before": "case comap_uniformity\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ 𝓤 (α →ᵤ[𝔖] γ) = 𝓤 (α →ᵤ[𝔖] γ)",
"tactic": "congr"
},
{
"state_after": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) (uniformSpace α β 𝔖) =\n UniformSpace.comap ((fun x x_1 => x ∘ x_1) f) (uniformSpace α β 𝔖)",
"state_before": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) (uniformSpace α β 𝔖) = uniformSpace α γ 𝔖",
"tactic": "rw [← uniformSpace_eq hf, UniformOnFun.comap_eq]"
},
{
"state_after": "no goals",
"state_before": "case comap_uniformity.h.e_2.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.78533\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : UniformSpace γ\nf : γ → β\nhf : 𝓤 γ = 𝓤 γ\n⊢ UniformSpace.comap (↑(ofFun 𝔖) ∘ (fun x x_1 => x ∘ x_1) f ∘ ↑(toFun 𝔖)) (uniformSpace α β 𝔖) =\n UniformSpace.comap ((fun x x_1 => x ∘ x_1) f) (uniformSpace α β 𝔖)",
"tactic": "rfl"
}
] |
[
783,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
11
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.le_principal_iff
|
[] |
[
646,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/GroupTheory/SpecificGroups/Alternating.lean
|
Equiv.Perm.finRotate_bit1_mem_alternatingGroup
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.27282\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\n⊢ finRotate (bit1 n) ∈ alternatingGroup (Fin (bit1 n))",
"tactic": "rw [mem_alternatingGroup, bit1, sign_finRotate, pow_bit0', Int.units_mul_self, one_pow]"
}
] |
[
92,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
AlgEquiv.ofInjective_apply
|
[] |
[
729,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
1
] |
Mathlib/FieldTheory/Tower.lean
|
FiniteDimensional.finrank_mul_finrank'
|
[
{
"state_after": "F : Type u\nK : Type v\nA : Type w\ninst✝¹³ : CommRing F\ninst✝¹² : Ring K\ninst✝¹¹ : AddCommGroup A\ninst✝¹⁰ : Algebra F K\ninst✝⁹ : Module K A\ninst✝⁸ : Module F A\ninst✝⁷ : IsScalarTower F K A\ninst✝⁶ : StrongRankCondition F\ninst✝⁵ : StrongRankCondition K\ninst✝⁴ : Module.Free F K\ninst✝³ : Module.Free K A\ninst✝² : Nontrivial K\ninst✝¹ : Module.Finite F K\ninst✝ : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\n⊢ finrank F K * finrank K A = finrank F A",
"state_before": "F : Type u\nK : Type v\nA : Type w\ninst✝¹³ : CommRing F\ninst✝¹² : Ring K\ninst✝¹¹ : AddCommGroup A\ninst✝¹⁰ : Algebra F K\ninst✝⁹ : Module K A\ninst✝⁸ : Module F A\ninst✝⁷ : IsScalarTower F K A\ninst✝⁶ : StrongRankCondition F\ninst✝⁵ : StrongRankCondition K\ninst✝⁴ : Module.Free F K\ninst✝³ : Module.Free K A\ninst✝² : Nontrivial K\ninst✝¹ : Module.Finite F K\ninst✝ : Module.Finite K A\n⊢ finrank F K * finrank K A = finrank F A",
"tactic": "letI := nontrivial_of_invariantBasisNumber F"
},
{
"state_after": "F : Type u\nK : Type v\nA : Type w\ninst✝¹³ : CommRing F\ninst✝¹² : Ring K\ninst✝¹¹ : AddCommGroup A\ninst✝¹⁰ : Algebra F K\ninst✝⁹ : Module K A\ninst✝⁸ : Module F A\ninst✝⁷ : IsScalarTower F K A\ninst✝⁶ : StrongRankCondition F\ninst✝⁵ : StrongRankCondition K\ninst✝⁴ : Module.Free F K\ninst✝³ : Module.Free K A\ninst✝² : Nontrivial K\ninst✝¹ : Module.Finite F K\ninst✝ : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\nb : Basis (Module.Free.ChooseBasisIndex F K) F K := Module.Free.chooseBasis F K\n⊢ finrank F K * finrank K A = finrank F A",
"state_before": "F : Type u\nK : Type v\nA : Type w\ninst✝¹³ : CommRing F\ninst✝¹² : Ring K\ninst✝¹¹ : AddCommGroup A\ninst✝¹⁰ : Algebra F K\ninst✝⁹ : Module K A\ninst✝⁸ : Module F A\ninst✝⁷ : IsScalarTower F K A\ninst✝⁶ : StrongRankCondition F\ninst✝⁵ : StrongRankCondition K\ninst✝⁴ : Module.Free F K\ninst✝³ : Module.Free K A\ninst✝² : Nontrivial K\ninst✝¹ : Module.Finite F K\ninst✝ : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\n⊢ finrank F K * finrank K A = finrank F A",
"tactic": "let b := Module.Free.chooseBasis F K"
},
{
"state_after": "F : Type u\nK : Type v\nA : Type w\ninst✝¹³ : CommRing F\ninst✝¹² : Ring K\ninst✝¹¹ : AddCommGroup A\ninst✝¹⁰ : Algebra F K\ninst✝⁹ : Module K A\ninst✝⁸ : Module F A\ninst✝⁷ : IsScalarTower F K A\ninst✝⁶ : StrongRankCondition F\ninst✝⁵ : StrongRankCondition K\ninst✝⁴ : Module.Free F K\ninst✝³ : Module.Free K A\ninst✝² : Nontrivial K\ninst✝¹ : Module.Finite F K\ninst✝ : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\nb : Basis (Module.Free.ChooseBasisIndex F K) F K := Module.Free.chooseBasis F K\nc : Basis (Module.Free.ChooseBasisIndex K A) K A := Module.Free.chooseBasis K A\n⊢ finrank F K * finrank K A = finrank F A",
"state_before": "F : Type u\nK : Type v\nA : Type w\ninst✝¹³ : CommRing F\ninst✝¹² : Ring K\ninst✝¹¹ : AddCommGroup A\ninst✝¹⁰ : Algebra F K\ninst✝⁹ : Module K A\ninst✝⁸ : Module F A\ninst✝⁷ : IsScalarTower F K A\ninst✝⁶ : StrongRankCondition F\ninst✝⁵ : StrongRankCondition K\ninst✝⁴ : Module.Free F K\ninst✝³ : Module.Free K A\ninst✝² : Nontrivial K\ninst✝¹ : Module.Finite F K\ninst✝ : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\nb : Basis (Module.Free.ChooseBasisIndex F K) F K := Module.Free.chooseBasis F K\n⊢ finrank F K * finrank K A = finrank F A",
"tactic": "let c := Module.Free.chooseBasis K A"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nA : Type w\ninst✝¹³ : CommRing F\ninst✝¹² : Ring K\ninst✝¹¹ : AddCommGroup A\ninst✝¹⁰ : Algebra F K\ninst✝⁹ : Module K A\ninst✝⁸ : Module F A\ninst✝⁷ : IsScalarTower F K A\ninst✝⁶ : StrongRankCondition F\ninst✝⁵ : StrongRankCondition K\ninst✝⁴ : Module.Free F K\ninst✝³ : Module.Free K A\ninst✝² : Nontrivial K\ninst✝¹ : Module.Finite F K\ninst✝ : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\nb : Basis (Module.Free.ChooseBasisIndex F K) F K := Module.Free.chooseBasis F K\nc : Basis (Module.Free.ChooseBasisIndex K A) K A := Module.Free.chooseBasis K A\n⊢ finrank F K * finrank K A = finrank F A",
"tactic": "rw [finrank_eq_card_basis b, finrank_eq_card_basis c, finrank_eq_card_basis (b.smul c),\n Fintype.card_prod]"
}
] |
[
87,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.min_zero
|
[] |
[
194,
83
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
194,
11
] |
Mathlib/Algebra/Group/Pi.lean
|
Function.const_ne_one
|
[] |
[
652,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sub_lt_of_sub_lt
|
[] |
[
1185,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1184,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.map_comp_includeRight
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁸ : CommSemiring R\nA : Type v₁\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nB : Type v₂\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\nC : Type v₃\ninst✝³ : Semiring C\ninst✝² : Algebra R C\nD : Type v₄\ninst✝¹ : Semiring D\ninst✝ : Algebra R D\nf : A →ₐ[R] B\ng : C →ₐ[R] D\n⊢ ∀ (x : C), ↑(AlgHom.comp (map f g) includeRight) x = ↑(AlgHom.comp includeRight g) x",
"tactic": "simp"
}
] |
[
881,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
1
] |
Mathlib/Computability/Language.lean
|
Language.mem_kstar
|
[] |
[
125,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
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