ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G

⊢ borel G ≤ μ.outerMeasure.caratheodory
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G

⊢ ∀ t ∈ {s | IsOpen s}, MeasurableSet t
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}

⊢ MeasurableSet U
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}

⊢ ∀ (U_1 : Opens G), μ.outerMeasure (↑U_1 ∩ U) + μ.outerMeasure (↑U_1 \ U) ≤ μ.outerMeasure ↑U_1
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G

⊢ μ.outerMeasure (↑U' ∩ U) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G

⊢ μ.innerContent { carrier := ↑U' ∩ U, is_open' := ⋯ } + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G

⊢ (⨆ x, ↑(μ.toFun ↑x)) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }

⊢ (⨆ x, ↑(μ.toFun ↑x)) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }

⊢ ⨆ b, ↑(μ.toFun ↑b) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }

⊢ ∀ (i : { i // ↑i ⊆ ↑U' ∩ U }), ↑(μ.toFun ↑i) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL : ↑L ⊆ ↑U' ∩ U

⊢ ↑(μ.toFun ↑⟨L, hL⟩) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL : ↑L ⊆ ↑U' ∩ U
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }

⊢ ↑(μ.toFun ↑⟨L, hL⟩) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
hL : ↑L ⊆ ↑U' ∩ U
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }

⊢ (fun s => ↑(μ.toFun s)) L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U

⊢ (fun s => ↑(μ.toFun s)) L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U

⊢ (fun s => ↑(μ.toFun s)) L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'

⊢ (fun s => ↑(μ.toFun s)) L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'

⊢ (fun s => ↑(μ.toFun s)) L' + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'

⊢ (fun s => ↑(μ.toFun s)) L' + μ.outerMeasure (↑U' \ ↑L') ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'

⊢ (fun s => ↑(μ.toFun s)) L' + μ.innerContent { carrier := ↑U' \ ↑L', is_open' := ⋯ } ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this : ↑U' \ U ⊆ ↑U' \ ↑L'

⊢ ↑(μ.toFun L') + ⨆ x, ↑(μ.toFun ↑x) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }

⊢ ↑(μ.toFun L') + ⨆ x, ↑(μ.toFun ↑x) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }

⊢ ⨆ b, ↑(μ.toFun L') + ↑(μ.toFun ↑b) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }

⊢ ∀ (i : { i // ↑i ⊆ ↑U' \ ↑L' }), ↑(μ.toFun L') + ↑(μ.toFun ↑i) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'

⊢ ↑(μ.toFun L') + ↑(μ.toFun ↑⟨M, hM⟩) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }

⊢ ↑(μ.toFun L') + ↑(μ.toFun ↑⟨M, hM⟩) ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }

⊢ (fun s => ↑(μ.toFun s)) L' + (fun s => ↑(μ.toFun s)) M' ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝ : ↑U' \ U ⊆ ↑U' \ ↑L'
this : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'

⊢ (fun s => ↑(μ.toFun s)) L' + (fun s => ↑(μ.toFun s)) M' ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'

⊢ (fun s => ↑(μ.toFun s)) L' + (fun s => ↑(μ.toFun s)) M' ≤ μ.outerMeasure ↑U'
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'

⊢ (fun s => ↑(μ.toFun s)) L' + (fun s => ↑(μ.toFun s)) M' ≤ μ.innerContent { carrier := ↑U', is_open' := ⋯ }
-- Mathlib/MeasureTheory/Measure/Content.lean
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this✝² : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
L : Compacts G
L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }
hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U
hL'U : ↑L' ⊆ U
hL'U' : ↑L' ⊆ ↑U'
this✝¹ : ↑U' \ U ⊆ ↑U' \ ↑L'
this✝ : Nonempty { M // ↑M ⊆ ↑U' \ closure ↑L }
M : Compacts G
hM : ↑M ⊆ ↑U' \ ↑L'
M' : Compacts G := { carrier := closure ↑M, isCompact' := ⋯ }
hM' : ↑M' ⊆ ↑U' \ ↑L'
this : ↑(L' ⊔ M') ⊆ ↑U'

⊢ (fun s => ↑(μ.toFun s)) (L' ⊔ M') = (fun s => ↑(μ.toFun s)) L' + (fun s => ↑(μ.toFun s)) M'
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : AffineTargetMorphismProperty
inst✝² : P.toProperty.RespectsIso
H : ∀ ⦃X Y S : Scheme⦄ [inst : IsAffine S] [inst_1 : IsAffine X] (f : X ⟶ S) (g : Y ⟶ S), P g → P (pullback.fst f g)
Z X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
f' : Z ⟶ Y
g' : Z ⟶ X
h : IsPullback g' f' f g
hg : P g

⊢ P g'
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : AffineTargetMorphismProperty
inst✝² : P.toProperty.RespectsIso
H : ∀ ⦃X Y S : Scheme⦄ [inst : IsAffine S] [inst_1 : IsAffine X] (f : X ⟶ S) (g : Y ⟶ S), P g → P (pullback.fst f g)
Z X Y S : Scheme
inst✝¹ : IsAffine S
inst✝ : IsAffine X
f : X ⟶ S
g : Y ⟶ S
f' : Z ⟶ Y
g' : Z ⟶ X
h : IsPullback g' f' f g
hg : P g

⊢ P (pullback.fst f g)
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
x y : E

⊢ inner_ 𝕜 (r • x) y = (starRingEnd 𝕜) r * inner_ 𝕜 x y
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : 𝕜
x y : E

⊢ ↑(re r) * inner_ 𝕜 x y + ↑(im r) * (-I * inner_ 𝕜 x y) = (↑(re r) + ↑(im r) * -I) * inner_ 𝕜 x y
-- Mathlib/SetTheory/Game/PGame.lean
xl xr : Type u
xL : xl → PGame
i : xr
xR : xr → PGame
j : (xR i).LeftMoves

⊢ ((xR i).moveLeft j).Subsequent (mk xl xr xL xR)
-- Mathlib/RingTheory/NonUnitalSubring/Basic.lean
F : Type w
R : Type u
S : Type v
T : Type u_1
inst✝⁴ : NonUnitalNonAssocRing R
inst✝³ : NonUnitalNonAssocRing S
inst✝² : NonUnitalNonAssocRing T
inst✝¹ : FunLike F R S
inst✝ : NonUnitalRingHomClass F R S
g : S →ₙ+* T
f : R →ₙ+* S
s : NonUnitalSubring R
x : R × S

⊢ x ∈ s.prod ⊤ ↔ x ∈ comap (NonUnitalRingHom.fst R S) s
-- Mathlib/RingTheory/Polynomial/Basic.lean
R : Type u
S : Type u_1
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n : ℕ

⊢ degreeLT R n = Submodule.span R ↑(image (fun n => X ^ n) (range n))
-- Mathlib/RingTheory/Polynomial/Basic.lean
R : Type u
S : Type u_1
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n : ℕ

⊢ degreeLT R n ≤ Submodule.span R ↑(image (fun n => X ^ n) (range n))
-- Mathlib/RingTheory/Polynomial/Basic.lean
R : Type u
S : Type u_1
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n : ℕ

⊢ Submodule.span R ↑(image (fun n => X ^ n) (range n)) ≤ degreeLT R n
-- Mathlib/RingTheory/Polynomial/Basic.lean
R : Type u
S : Type u_1
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n : ℕ

⊢ ↑(range n) ⊆ (fun n => X ^ n) ⁻¹' ↑(degreeLT R n)
-- Mathlib/RingTheory/Polynomial/Basic.lean
R : Type u
S : Type u_1
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n k : ℕ
hk : k ∈ ↑(range n)

⊢ k ∈ (fun n => X ^ n) ⁻¹' ↑(degreeLT R n)
-- Mathlib/RingTheory/Polynomial/Basic.lean
R : Type u
S : Type u_1
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n k : ℕ
hk : k ∈ ↑(range n)

⊢ ((fun n => X ^ n) k).degree < ↑n
-- Mathlib/Analysis/Normed/Group/Pointwise.lean
E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x y : E

⊢ s * ball 1 δ = thickening δ s
-- Mathlib/Analysis/Normed/Group/Pointwise.lean
E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x y : E

⊢ s * ball 1 δ = ⋃ x ∈ s, ball x δ
-- Mathlib/Analysis/Normed/Group/Pointwise.lean
E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x y : E

⊢ s = ⋃ i ∈ s, {i}
-- Mathlib/Analysis/Normed/Group/Pointwise.lean
E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x y x✝¹ : E
x✝ : x✝¹ ∈ s

⊢ ball x✝¹ δ = {x✝¹} * ball 1 δ
-- Mathlib/Analysis/Normed/Group/Pointwise.lean
E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x✝² y x✝¹ : E
x✝ : x✝¹ ∈ s
x : E

⊢ x ∈ ball x✝¹ δ ↔ x ∈ {x✝¹} * ball 1 δ
-- Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
k n : ℕ

⊢ Pairwise (Pi.Lex (fun x x_1 => x < x_1) fun x x x_1 => x < x_1) (antidiagonalTuple (k + 1) n)
-- Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
k n : ℕ

⊢ (∀ a ∈ antidiagonal n, Pairwise (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a.1 a_2) (Fin.cons a.1 b)) (antidiagonalTuple k a.2)) ∧ Pairwise (fun a₁ a₂ => ∀ a ∈ antidiagonalTuple k a₁.2, ∀ a_2 ∈ antidiagonalTuple k a₂.2, Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a₁.1 a) (Fin.cons a₂.1 a_2)) (antidiagonal n)
-- Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
k n : ℕ

⊢ (∀ (a b : ℕ), a + b = n → Pairwise (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a a_2) (Fin.cons a b)) (antidiagonalTuple k b)) ∧ Pairwise (fun a₁ a₂ => ∀ a ∈ antidiagonalTuple k a₁.2, ∀ a_2 ∈ antidiagonalTuple k a₂.2, Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a₁.1 a) (Fin.cons a₂.1 a_2)) (antidiagonal n)
-- Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
k n : ℕ

⊢ (∀ (a b : ℕ), a + b = n → Pairwise (fun a b => Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a b) (antidiagonalTuple k b)) ∧ Pairwise (fun a₁ a₂ => ∀ a ∈ antidiagonalTuple k a₁.2, ∀ a_2 ∈ antidiagonalTuple k a₂.2, a₁.1 < a₂.1 ∨ a₁.1 = a₂.1 ∧ Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2) (antidiagonal n)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

⊢ Tendsto cos (𝓝[>] (-(π / 2))) (𝓝[>] 0)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

⊢ Tendsto cos (𝓝[>] (-(π / 2))) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

⊢ ∀ᶠ (x : ℝ) in 𝓝[>] (-(π / 2)), cos x ∈ Ioi 0
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b

⊢ closure (Ioc a b) = Icc a b
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b

⊢ closure (Ioc a b) ⊆ Icc a b
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b✝ : α
s : Set α
a b : α
hab : a ≠ b

⊢ Icc a b ⊆ closure (Ioc a b)
-- Mathlib/Combinatorics/SimpleGraph/Walk.lean
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
u✝ v✝ w x y : V
inst✝ : DecidableEq V
u v : V
c : G.Walk v v
h : u ∈ c.support

⊢ (c.rotate h).darts ~r c.darts
-- Mathlib/Combinatorics/SimpleGraph/Walk.lean
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
u✝ v✝ w x y : V
inst✝ : DecidableEq V
u v : V
c : G.Walk v v
h : u ∈ c.support

⊢ ((c.dropUntil u h).darts ++ (c.takeUntil u h).darts) ~r c.darts
-- Mathlib/Combinatorics/SimpleGraph/Walk.lean
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
u✝ v✝ w x y : V
inst✝ : DecidableEq V
u v : V
c : G.Walk v v
h : u ∈ c.support

⊢ ((c.takeUntil u h).darts ++ (c.dropUntil u h).darts) ~r c.darts
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N

⊢ dualDistribInvOfBasis b c ∘ₗ dualDistrib R M N = LinearMap.id
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N

⊢ ∀ (i : ι × κ), (dualDistribInvOfBasis b c ∘ₗ dualDistrib R M N) ((b.dualBasis.tensorProduct c.dualBasis) i) = LinearMap.id ((b.dualBasis.tensorProduct c.dualBasis) i)
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
i : ι
j : κ

⊢ (dualDistribInvOfBasis b c ∘ₗ dualDistrib R M N) ((b.dualBasis.tensorProduct c.dualBasis) (i, j)) = LinearMap.id ((b.dualBasis.tensorProduct c.dualBasis) (i, j))
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
i : ι
j : κ

⊢ ∑ x : ι, ∑ x_1 : κ, ((Finsupp.single x 1) i * (Finsupp.single x_1 1) j) • b.coord x ⊗ₜ[R] c.coord x_1 = b.coord i ⊗ₜ[R] c.coord j
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
i : ι
j : κ

⊢ ((Finsupp.single i 1) i * (Finsupp.single j 1) j) • b.coord i ⊗ₜ[R] c.coord j = b.coord i ⊗ₜ[R] c.coord j
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
i : ι
j : κ

⊢ ∀ b_1 ∈ Finset.univ, b_1 ≠ j → ((Finsupp.single i 1) i * (Finsupp.single b_1 1) j) • b.coord i ⊗ₜ[R] c.coord b_1 = 0
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
i : ι
j : κ

⊢ j ∉ Finset.univ → ((Finsupp.single i 1) i * (Finsupp.single j 1) j) • b.coord i ⊗ₜ[R] c.coord j = 0
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u_1
A : Type u_2
M : Type u_3
N : Type u_4
ι : Type u_5
κ : Type u_6
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
i : ι
j : κ

⊢ ∀ b_1 ∈ Finset.univ, b_1 ≠ i → ∑ x : κ, ((Finsupp.single b_1 1) i * (Finsupp.single x 1) j) • b.coord b_1 ⊗ₜ[R] c.coord x = 0