full_name
stringlengths 3
121
| state
stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
| target_state
stringlengths 7
19k
| url
stringclasses 1
value | commit
stringclasses 1
value | file_path
stringlengths 21
79
|
|---|---|---|---|---|---|---|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.h.e_6.h.h.refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
y : ℝ
h : ∀ (a : ℚ), x < ↑a → y ≤ ↑a
q : ℚ
hq : x < ↑q
⊢ x < ↑q
|
exact mod_cast hq
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ∀ (i : { r' // x < ↑r' }), MeasurableSet (Iic ↑↑i)
|
exact fun _ ↦ <a>measurableSet_Iic</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ Directed (fun x x_1 => x ⊇ x_1) fun r => Iic ↑↑r
|
refine <a>Monotone.directed_ge</a> fun r r' hrr' ↦ Iic_subset_Iic.mpr ?_
|
case e_f.h.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
r r' : { r' // x < ↑r' }
hrr' : r ≤ r'
⊢ ↑↑r ≤ ↑↑r'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
r r' : { r' // x < ↑r' }
hrr' : r ≤ r'
⊢ ↑↑r ≤ ↑↑r'
|
exact mod_cast hrr'
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.hfin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ⊤
|
obtain ⟨q, hq⟩ := <a>exists_rat_gt</a> x
|
case e_f.h.hfin.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
q : ℚ
hq : x < ↑q
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ⊤
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case e_f.h.hfin.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
q : ℚ
hq : x < ↑q
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ⊤
|
exact ⟨⟨q, hq⟩, <a>MeasureTheory.measure_ne_top</a> _ _⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
⊢ Nonempty { r' // x < ↑r' }
|
obtain ⟨r, hrx⟩ := <a>exists_rat_gt</a> x
|
case intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
r : ℚ
hrx : x < ↑r
⊢ Nonempty { r' // x < ↑r' }
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
r : ℚ
hrx : x < ↑r
⊢ Nonempty { r' // x < ↑r' }
|
exact ⟨⟨r, hrx⟩⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∀ (b : { r' // x < ↑r' }), ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ⊤
|
intro b
|
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
b : { r' // x < ↑r' }
⊢ ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ⊤
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
b : { r' // x < ↑r' }
⊢ ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ⊤
|
rw [<a>ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat_rat</a> hf a _ hs]
|
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
b : { r' // x < ↑r' }
⊢ (κ a) (s ×ˢ Iic ↑↑b) ≠ ⊤
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.hf_int
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
b : { r' // x < ↑r' }
⊢ (κ a) (s ×ˢ Iic ↑↑b) ≠ ⊤
|
exact <a>MeasureTheory.measure_ne_top</a> _ _
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ Directed (fun x x_1 => x ≥ x_1) fun r b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
|
refine <a>Monotone.directed_ge</a> fun i j hij b ↦ ?_
|
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑i) ≤
ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑j)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑i) ≤
ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑j)
|
simp_rw [← <a>ProbabilityTheory.measure_stieltjesOfMeasurableRat_Iic</a>]
|
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≤ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑j)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≤ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑j)
|
refine <a>MeasureTheory.measure_mono</a> (Iic_subset_Iic.mpr ?_)
|
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ ↑↑i ≤ ↑↑j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ ↑↑i ≤ ↑↑j
|
exact mod_cast hij
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
q : { r' // x < ↑r' }
⊢ Measurable fun b => ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)
|
refine <a>Measurable.ennreal_ofReal</a> ?_
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
q : { r' // x < ↑r' }
⊢ Measurable fun b => ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
q : { r' // x < ↑r' }
⊢ Measurable fun b => ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q
|
exact (<a>ProbabilityTheory.measurable_stieltjesOfMeasurableRat</a> hf.measurable _).<a>Measurable.comp</a> <a>measurable_prod_mk_left</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ (κ a) (⋂ i, s ×ˢ Iic ↑↑i) = (κ a) (s ×ˢ Iic x)
|
rw [← <a>Set.prod_iInter</a>]
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ (κ a) (s ×ˢ ⋂ i, Iic ↑↑i) = (κ a) (s ×ˢ Iic x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ (κ a) (s ×ˢ ⋂ i, Iic ↑↑i) = (κ a) (s ×ˢ Iic x)
|
congr with y
|
case neg.h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
y : ℝ
⊢ y ∈ ⋂ i, Iic ↑↑i ↔ y ∈ Iic x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
y : ℝ
⊢ y ∈ ⋂ i, Iic ↑↑i ↔ y ∈ Iic x
|
simp only [<a>Set.mem_iInter</a>, <a>Set.mem_Iic</a>, <a>Subtype.forall</a>, <a>Subtype.coe_mk</a>]
|
case neg.h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
y : ℝ
⊢ (∀ (a : ℚ), x < ↑a → y ≤ ↑a) ↔ y ≤ x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.h.e_6.h.e_a.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
y : ℝ
⊢ (∀ (a : ℚ), x < ↑a → y ≤ ↑a) ↔ y ≤ x
|
exact ⟨<a>le_of_forall_lt_rat_imp_le</a>, fun hyx q hq ↦ hyx.trans hq.le⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∀ (i : { r' // x < ↑r' }), MeasurableSet (s ×ˢ Iic ↑↑i)
|
exact fun i ↦ hs.prod <a>measurableSet_Iic</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ Directed (fun x x_1 => x ⊇ x_1) fun b => s ×ˢ Iic ↑↑b
|
refine <a>Monotone.directed_ge</a> fun i j hij ↦ ?_
|
case neg.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
⊢ s ×ˢ Iic ↑↑i ≤ s ×ˢ Iic ↑↑j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
⊢ s ×ˢ Iic ↑↑i ≤ s ×ˢ Iic ↑↑j
|
refine prod_subset_prod_iff.mpr (<a>Or.inl</a> ⟨<a>subset_rfl</a>, Iic_subset_Iic.mpr ?_⟩)
|
case neg.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
⊢ ↑↑i ≤ ↑↑j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.hd
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
⊢ ↑↑i ≤ ↑↑j
|
exact mod_cast hij
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
|
case neg.hfin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ↥(kernel α (β × ℝ))
ν : ↥(kernel α β)
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ∃ i, (κ a) (s ×ˢ Iic ↑↑i) ≠ ⊤
|
exact ⟨h_nonempty.some, <a>MeasureTheory.measure_ne_top</a> _ _⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
|
BoxIntegral.Prepartition.card_filter_mem_Icc_le
|
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
inst✝ : Fintype ι
x : ι → ℝ
⊢ (filter (fun J => x ∈ Box.Icc J) π.boxes).card ≤ 2 ^ Fintype.card ι
|
rw [← <a>Fintype.card_set</a>]
|
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
inst✝ : Fintype ι
x : ι → ℝ
⊢ (filter (fun J => x ∈ Box.Icc J) π.boxes).card ≤ Fintype.card (Set ι)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.card_filter_mem_Icc_le
|
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
inst✝ : Fintype ι
x : ι → ℝ
⊢ (filter (fun J => x ∈ Box.Icc J) π.boxes).card ≤ Fintype.card (Set ι)
|
refine <a>Finset.card_le_card_of_injOn</a> (fun J : <a>BoxIntegral.Box</a> ι => { i | J.lower i = x i }) (fun _ _ => <a>Finset.mem_univ</a> _) ?_
|
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
inst✝ : Fintype ι
x : ι → ℝ
⊢ InjOn (fun J => {i | J.lower i = x i}) ↑(filter (fun J => x ∈ Box.Icc J) π.boxes)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.card_filter_mem_Icc_le
|
ι : Type u_1
I J J₁ J₂ : Box ι
π π₁ π₂ : Prepartition I
x✝ : ι → ℝ
inst✝ : Fintype ι
x : ι → ℝ
⊢ InjOn (fun J => {i | J.lower i = x i}) ↑(filter (fun J => x ∈ Box.Icc J) π.boxes)
|
simpa using π.injOn_setOf_mem_Icc_setOf_lower_eq x
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
|
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
⊢ map (fun x_1 => x * x_1) μ = μ
|
ext A hA
|
case h
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
⊢ (map (fun x_1 => x * x_1) μ) A = μ A
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
|
case h
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
⊢ (map (fun x_1 => x * x_1) μ) A = μ A
|
obtain ⟨x₁, h⟩ := @<a>Quotient.exists_rep</a> _ (<a>QuotientGroup.leftRel</a> Γ) x
|
case h.intro
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x_1 => x * x_1) μ) A = μ A
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
|
case h.intro
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x_1 => x * x_1) μ) A = μ A
|
convert <a>MeasureTheory.measure_preimage_smul</a> x₁ μ A using 1
|
case h.e'_2
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x_1 => x * x_1) μ) A = μ ((fun x => x₁ • x) ⁻¹' A)
case h.intro
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ SMulInvariantMeasure G (G ⧸ Γ) μ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
|
case h.intro
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ SMulInvariantMeasure G (G ⧸ Γ) μ
|
exact <a>MeasureTheory.QuotientMeasureEqMeasurePreimage.smulInvariantMeasure_quotient</a> ν
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
|
case h.e'_2
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ (map (fun x_1 => x * x_1) μ) A = μ ((fun x => x₁ • x) ⁻¹' A)
|
rw [← h, <a>MeasureTheory.Measure.map_apply</a> (<a>MeasurableMul.measurable_const_mul</a> _) hA]
|
case h.e'_2
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ μ ((fun x => ⟦x₁⟧ * x) ⁻¹' A) = μ ((fun x => x₁ • x) ⁻¹' A)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient
|
case h.e'_2
G : Type u_1
inst✝¹³ : Group G
inst✝¹² : MeasurableSpace G
inst✝¹¹ : TopologicalSpace G
inst✝¹⁰ : TopologicalGroup G
inst✝⁹ : BorelSpace G
inst✝⁸ : PolishSpace G
Γ : Subgroup G
inst✝⁷ : Countable ↥Γ
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝³ : ν.IsMulLeftInvariant
inst✝² : ν.IsMulRightInvariant
inst✝¹ : SigmaFinite ν
hasFun : HasFundamentalDomain (↥Γ.op) G ν
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
x : G ⧸ Γ
A : Set (G ⧸ Γ)
hA : MeasurableSet A
x₁ : G
h : ⟦x₁⟧ = x
⊢ μ ((fun x => ⟦x₁⟧ * x) ⁻¹' A) = μ ((fun x => x₁ • x) ⁻¹' A)
|
simp [← <a>MulAction.Quotient.coe_smul_out'</a>, ← <a>Quotient.mk''_eq_mk</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
|
Set.pairwise_sUnion
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
κ : Sort u_6
r✝ p q : α → α → Prop
f g : ι → α
s✝ t u : Set α
a b : α
r : α → α → Prop
s : Set (Set α)
h : DirectedOn (fun x x_1 => x ⊆ x_1) s
⊢ (⋃₀ s).Pairwise r ↔ ∀ a ∈ s, a.Pairwise r
|
rw [<a>Set.sUnion_eq_iUnion</a>, <a>Set.pairwise_iUnion</a> h.directed_val, <a>SetCoe.forall</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Pairwise/Lattice.lean
|
List.prod_eq_one
|
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a : M
hl : ∀ x ∈ l, x = 1
⊢ l.prod = 1
|
induction' l with i l hil
|
case nil
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a : M
hl : ∀ x ∈ [], x = 1
⊢ [].prod = 1
case cons
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l✝ l₁ l₂ : List M
a i : M
l : List M
hil : (∀ x ∈ l, x = 1) → l.prod = 1
hl : ∀ x ∈ i :: l, x = 1
⊢ (i :: l).prod = 1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/BigOperators/Group/List.lean
|
List.prod_eq_one
|
case cons
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l✝ l₁ l₂ : List M
a i : M
l : List M
hil : (∀ x ∈ l, x = 1) → l.prod = 1
hl : ∀ x ∈ i :: l, x = 1
⊢ (i :: l).prod = 1
|
rw [<a>List.prod_cons</a>, hil fun x hx ↦ hl _ (<a>List.mem_cons_of_mem</a> i hx), hl _ (<a>List.mem_cons_self</a> i l), <a>one_mul</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/BigOperators/Group/List.lean
|
List.prod_eq_one
|
case nil
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a : M
hl : ∀ x ∈ [], x = 1
⊢ [].prod = 1
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/BigOperators/Group/List.lean
|
CategoryTheory.Functor.shiftIso_add
|
C : Type u_1
A : Type u_2
inst✝⁶ : Category.{?u.41942, u_1} C
inst✝⁵ : Category.{?u.41946, u_2} A
F : C ⥤ A
M : Type u_3
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
G : Type u_4
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ m + n + a = a''
|
rw [<a>add_assoc</a>, ha', ha'']
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Shift/ShiftSequence.lean
|
Finset.disjSups_singleton
|
F : Type u_1
α : Type u_2
β : Type u_3
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : SemilatticeSup α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s s₁ s₂ t t₁ t₂ u : Finset α
a b c : α
⊢ {a} ○ {b} = if Disjoint a b then {a ⊔ b} else ∅
|
split_ifs with h <;> simp [<a>Finset.disjSups</a>, <a>Finset.filter_singleton</a>, h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Finset/Sups.lean
|
Stream'.WSeq.tail_cons
|
α : Type u
β : Type v
γ : Type w
a : α
s : WSeq α
⊢ (cons a s).tail = s
|
simp [<a>Stream'.WSeq.tail</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
PythagoreanTriple.mul_isClassified
|
x y z : ℤ
h : PythagoreanTriple x y z
k : ℤ
hc : h.IsClassified
⊢ ⋯.IsClassified
|
obtain ⟨l, m, n, ⟨⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, co⟩⟩ := hc
|
case intro.intro.intro.intro.inl.intro
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ ⋯.IsClassified
case intro.intro.intro.intro.inr.intro
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ ⋯.IsClassified
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
case intro.intro.intro.intro.inl.intro
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ ⋯.IsClassified
|
use k * l, m, n
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ (k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (2 * m * n)) = k * l * (2 * m * n) ∨
k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∧ k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2)) ∧
m.gcd n = 1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ (k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (2 * m * n)) = k * l * (2 * m * n) ∨
k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∧ k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2)) ∧
m.gcd n = 1
|
apply <a>And.intro</a> _ co
|
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (2 * m * n)) = k * l * (2 * m * n) ∨
k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∧ k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (2 * m * n)) = k * l * (2 * m * n) ∨
k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∧ k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2)
|
left
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (2 * m * n)) = k * l * (2 * m * n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (m ^ 2 - n ^ 2)) (l * (2 * m * n)) z
⊢ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (2 * m * n)) = k * l * (2 * m * n)
|
constructor <;> ring
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
case intro.intro.intro.intro.inr.intro
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ ⋯.IsClassified
|
use k * l, m, n
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ (k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∨
k * (l * (2 * m * n)) = k * l * (2 * m * n) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2)) ∧
m.gcd n = 1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ (k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∨
k * (l * (2 * m * n)) = k * l * (2 * m * n) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2)) ∧
m.gcd n = 1
|
apply <a>And.intro</a> _ co
|
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∨
k * (l * (2 * m * n)) = k * l * (2 * m * n) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ k * (l * (2 * m * n)) = k * l * (m ^ 2 - n ^ 2) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (2 * m * n) ∨
k * (l * (2 * m * n)) = k * l * (2 * m * n) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2)
|
right
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ k * (l * (2 * m * n)) = k * l * (2 * m * n) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.mul_isClassified
|
case h
z k l m n : ℤ
co : m.gcd n = 1
h : PythagoreanTriple (l * (2 * m * n)) (l * (m ^ 2 - n ^ 2)) z
⊢ k * (l * (2 * m * n)) = k * l * (2 * m * n) ∧ k * (l * (m ^ 2 - n ^ 2)) = k * l * (m ^ 2 - n ^ 2)
|
constructor <;> ring
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
toIocDiv_neg'
|
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a✝ b✝ c : α
n : ℤ
a b : α
⊢ toIocDiv hp (-a) b = -(toIcoDiv hp a (-b) + 1)
|
simpa only [<a>neg_neg</a>] using <a>toIocDiv_neg</a> hp (-a) (-b)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/ToIntervalMod.lean
|
Bornology.isVonNBounded_pi_iff
|
𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
E' : Type u_4
F : Type u_5
ι✝ : Type u_6
𝕜 : Type u_7
ι : Type u_8
E : ι → Type u_9
inst✝³ : NormedDivisionRing 𝕜
inst✝² : (i : ι) → AddCommGroup (E i)
inst✝¹ : (i : ι) → Module 𝕜 (E i)
inst✝ : (i : ι) → TopologicalSpace (E i)
S : Set ((i : ι) → E i)
⊢ IsVonNBounded 𝕜 S ↔ ∀ (i : ι), IsVonNBounded 𝕜 (eval i '' S)
|
simp_rw [<a>Bornology.isVonNBounded_iff_tendsto_smallSets_nhds</a>, <a>nhds_pi</a>, <a>Filter.pi</a>, <a>Filter.smallSets_iInf</a>, <a>Filter.smallSets_comap_eq_comap_image</a>, <a>Filter.tendsto_iInf</a>, <a>Filter.tendsto_comap_iff</a>, <a>Function.comp</a>, ← <a>Set.image_smul</a>, <a>Set.image_image</a>, <a>Function.eval</a>, <a>Pi.smul_apply</a>, <a>Pi.zero_apply</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
WittVector.ghostFun_intCast
|
p : ℕ
R : Type u_1
S : Type u_2
T : Type u_3
hp : Fact (Nat.Prime p)
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
α : Type u_4
β : Type u_5
x y : 𝕎 R
i : ℤ
⊢ WittVector.ghostFun i.castDef = ↑i
|
cases i <;> simp [*, <a>Int.castDef</a>, <a>_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_natCast</a>, <a>_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_neg</a>, -<a>Pi.natCast_def</a>, -<a>Pi.intCast_def</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/WittVector/Basic.lean
|
LinearIndependent.repr_eq
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
⊢ hv.repr x = l
|
have : ↑((<a>LinearIndependent.totalEquiv</a> hv : (ι →₀ R) →ₗ[R] <a>Submodule.span</a> R (<a>Set.range</a> v)) l) = <a>Finsupp.total</a> ι M R v l := <a>rfl</a>
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
⊢ hv.repr x = l
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.repr_eq
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
⊢ hv.repr x = l
|
have : (<a>LinearIndependent.totalEquiv</a> hv : (ι →₀ R) →ₗ[R] <a>Submodule.span</a> R (<a>Set.range</a> v)) l = x := by rw [eq] at this exact <a>Subtype.ext_iff</a>.2 this
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this✝ : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
this : ↑hv.totalEquiv l = x
⊢ hv.repr x = l
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.repr_eq
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this✝ : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
this : ↑hv.totalEquiv l = x
⊢ hv.repr x = l
|
rw [← <a>LinearEquiv.symm_apply_apply</a> hv.totalEquiv l]
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this✝ : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
this : ↑hv.totalEquiv l = x
⊢ hv.repr x = hv.totalEquiv.symm (hv.totalEquiv l)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.repr_eq
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this✝ : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
this : ↑hv.totalEquiv l = x
⊢ hv.repr x = hv.totalEquiv.symm (hv.totalEquiv l)
|
rw [← this]
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this✝ : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
this : ↑hv.totalEquiv l = x
⊢ hv.repr (↑hv.totalEquiv l) = hv.totalEquiv.symm (hv.totalEquiv l)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.repr_eq
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this✝ : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
this : ↑hv.totalEquiv l = x
⊢ hv.repr (↑hv.totalEquiv l) = hv.totalEquiv.symm (hv.totalEquiv l)
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.repr_eq
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this : ↑(↑hv.totalEquiv l) = (Finsupp.total ι M R v) l
⊢ ↑hv.totalEquiv l = x
|
rw [eq] at this
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this : ↑(↑hv.totalEquiv l) = ↑x
⊢ ↑hv.totalEquiv l = x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.repr_eq
|
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x✝ y : M
hv : LinearIndependent R v
l : ι →₀ R
x : ↥(span R (range v))
eq : (Finsupp.total ι M R v) l = ↑x
this : ↑(↑hv.totalEquiv l) = ↑x
⊢ ↑hv.totalEquiv l = x
|
exact <a>Subtype.ext_iff</a>.2 this
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/LinearIndependent.lean
|
MulHom.cancel_left
|
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
H : Type u_8
F : Type u_9
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
g : N →ₙ* P
f₁ f₂ : M →ₙ* N
hg : Function.Injective ⇑g
h : g.comp f₁ = g.comp f₂
x : M
⊢ g (f₁ x) = g (f₂ x)
|
rw [← <a>MulHom.comp_apply</a>, h, <a>MulHom.comp_apply</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Group/Hom/Defs.lean
|
card_le_of_injective'
|
R : Type u
inst✝³ : Semiring R
inst✝² : StrongRankCondition R
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : (α →₀ R) →ₗ[R] β →₀ R
i : Injective ⇑f
⊢ Fintype.card α ≤ Fintype.card β
|
let P := <a>Finsupp.linearEquivFunOnFinite</a> R R β
|
R : Type u
inst✝³ : Semiring R
inst✝² : StrongRankCondition R
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : (α →₀ R) →ₗ[R] β →₀ R
i : Injective ⇑f
P : (β →₀ R) ≃ₗ[R] β → R := Finsupp.linearEquivFunOnFinite R R β
⊢ Fintype.card α ≤ Fintype.card β
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/InvariantBasisNumber.lean
|
card_le_of_injective'
|
R : Type u
inst✝³ : Semiring R
inst✝² : StrongRankCondition R
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : (α →₀ R) →ₗ[R] β →₀ R
i : Injective ⇑f
P : (β →₀ R) ≃ₗ[R] β → R := Finsupp.linearEquivFunOnFinite R R β
⊢ Fintype.card α ≤ Fintype.card β
|
let Q := (<a>Finsupp.linearEquivFunOnFinite</a> R R α).<a>LinearEquiv.symm</a>
|
R : Type u
inst✝³ : Semiring R
inst✝² : StrongRankCondition R
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : (α →₀ R) →ₗ[R] β →₀ R
i : Injective ⇑f
P : (β →₀ R) ≃ₗ[R] β → R := Finsupp.linearEquivFunOnFinite R R β
Q : (α → R) ≃ₗ[R] α →₀ R := (Finsupp.linearEquivFunOnFinite R R α).symm
⊢ Fintype.card α ≤ Fintype.card β
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/InvariantBasisNumber.lean
|
card_le_of_injective'
|
R : Type u
inst✝³ : Semiring R
inst✝² : StrongRankCondition R
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : (α →₀ R) →ₗ[R] β →₀ R
i : Injective ⇑f
P : (β →₀ R) ≃ₗ[R] β → R := Finsupp.linearEquivFunOnFinite R R β
Q : (α → R) ≃ₗ[R] α →₀ R := (Finsupp.linearEquivFunOnFinite R R α).symm
⊢ Fintype.card α ≤ Fintype.card β
|
exact <a>card_le_of_injective</a> R ((P.toLinearMap.comp f).<a>LinearMap.comp</a> Q.toLinearMap) ((P.injective.comp i).<a>Function.Injective.comp</a> Q.injective)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/LinearAlgebra/InvariantBasisNumber.lean
|
Stream'.WSeq.join_append
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
⊢ (S.append T).join ~ʷ S.join.append T.join
|
refine ⟨fun s1 s2 => ∃ s S T, s1 = <a>Stream'.WSeq.append</a> s (<a>Stream'.WSeq.join</a> (<a>Stream'.WSeq.append</a> S T)) ∧ s2 = <a>Stream'.WSeq.append</a> s (<a>Stream'.WSeq.append</a> (<a>Stream'.WSeq.join</a> S) (<a>Stream'.WSeq.join</a> T)), ⟨<a>Stream'.WSeq.nil</a>, S, T, by simp, by simp⟩, ?_⟩
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
⊢ ∀ {s t : WSeq α},
(fun s1 s2 => ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)) s t →
Computation.LiftRel
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
s.destruct t.destruct
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
⊢ ∀ {s t : WSeq α},
(fun s1 s2 => ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)) s t →
Computation.LiftRel
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
s.destruct t.destruct
|
intro s1 s2 h
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
⊢ Computation.LiftRel
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
s1.destruct s2.destruct
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
⊢ Computation.LiftRel
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
s1.destruct s2.destruct
|
apply <a>Computation.liftRel_rec</a> (fun c1 c2 => ∃ (s : <a>Stream'.WSeq</a> α) (S T : _), c1 = <a>Stream'.WSeq.destruct</a> (<a>Stream'.WSeq.append</a> s (<a>Stream'.WSeq.join</a> (<a>Stream'.WSeq.append</a> S T))) ∧ c2 = <a>Stream'.WSeq.destruct</a> (<a>Stream'.WSeq.append</a> s (<a>Stream'.WSeq.append</a> (<a>Stream'.WSeq.join</a> S) (<a>Stream'.WSeq.join</a> T)))) _ _ _ (let ⟨s, S, T, h1, h2⟩ := h ⟨s, S, T, <a>congr_arg</a> <a>Stream'.WSeq.destruct</a> h1, <a>congr_arg</a> <a>Stream'.WSeq.destruct</a> h2⟩)
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
⊢ ∀ {ca cb : Computation (Option (α × WSeq α))},
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
ca cb →
LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 =>
∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
ca.destruct cb.destruct
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
⊢ ∀ {ca cb : Computation (Option (α × WSeq α))},
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
ca cb →
LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 =>
∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
ca.destruct cb.destruct
|
rintro c1 c2 ⟨s, S, T, rfl, rfl⟩
|
case intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
s : WSeq α
S T : WSeq (WSeq α)
⊢ LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
(s.append (S.append T).join).destruct.destruct (s.append (S.join.append T.join)).destruct.destruct
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
s : WSeq α
S T : WSeq (WSeq α)
⊢ LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
(s.append (S.append T).join).destruct.destruct (s.append (S.join.append T.join)).destruct.destruct
|
induction' s using <a>Stream'.WSeq.recOn</a> with a s s <;> simp
|
case intro.intro.intro.intro.h1
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
S T : WSeq (WSeq α)
⊢ LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
(S.append T).join.destruct.destruct (S.join.append T.join).destruct.destruct
case intro.intro.intro.intro.h2
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
S T : WSeq (WSeq α)
a : α
s : WSeq α
⊢ ∃ s_1 S_1 T_1,
s.append (S.append T).join = s_1.append (S_1.append T_1).join ∧
s.append (S.join.append T.join) = s_1.append (S_1.join.append T_1.join)
case intro.intro.intro.intro.h3
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
S T : WSeq (WSeq α)
s : WSeq α
⊢ ∃ s_1 S_1 T_1,
(s.append (S.append T).join).destruct = (s_1.append (S_1.append T_1).join).destruct ∧
(s.append (S.join.append T.join)).destruct = (s_1.append (S_1.join.append T_1.join)).destruct
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
⊢ (S.append T).join = nil.append (S.append T).join
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
α : Type u
β : Type v
γ : Type w
S T : WSeq (WSeq α)
⊢ S.join.append T.join = nil.append (S.join.append T.join)
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h1
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
S T : WSeq (WSeq α)
⊢ LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
(S.append T).join.destruct.destruct (S.join.append T.join).destruct.destruct
|
induction' S using <a>Stream'.WSeq.recOn</a> with s S S <;> simp
|
case intro.intro.intro.intro.h1.h1
α : Type u
β : Type v
γ : Type w
S T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T : WSeq (WSeq α)
⊢ LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
T.join.destruct.destruct T.join.destruct.destruct
case intro.intro.intro.intro.h1.h2
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T : WSeq (WSeq α)
s : WSeq α
S : WSeq (WSeq α)
⊢ ∃ s_1 S_1 T_1,
(s.append (S.append T).join).destruct = (s_1.append (S_1.append T_1).join).destruct ∧
(s.append (S.join.append T.join)).destruct = (s_1.append (S_1.join.append T_1.join)).destruct
case intro.intro.intro.intro.h1.h3
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T S : WSeq (WSeq α)
⊢ ∃ s S_1 T_1,
(S.append T).join.destruct = (s.append (S_1.append T_1).join).destruct ∧
(S.join.append T.join).destruct = (s.append (S_1.join.append T_1.join)).destruct
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h1.h1
α : Type u
β : Type v
γ : Type w
S T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T : WSeq (WSeq α)
⊢ LiftRelAux
(LiftRelO (fun x x_1 => x = x_1) fun s1 s2 =>
∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join))
(fun c1 c2 => ∃ s S T, c1 = (s.append (S.append T).join).destruct ∧ c2 = (s.append (S.join.append T.join)).destruct)
T.join.destruct.destruct T.join.destruct.destruct
|
induction' T using <a>Stream'.WSeq.recOn</a> with s T T <;> simp
|
case intro.intro.intro.intro.h1.h1.h2
α : Type u
β : Type v
γ : Type w
S T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
s : WSeq α
T : WSeq (WSeq α)
⊢ ∃ s_1 S T_1,
(s.append T.join).destruct = (s_1.append (S.append T_1).join).destruct ∧
(s.append T.join).destruct = (s_1.append (S.join.append T_1.join)).destruct
case intro.intro.intro.intro.h1.h1.h3
α : Type u
β : Type v
γ : Type w
S T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T : WSeq (WSeq α)
⊢ ∃ s S T_1,
T.join.destruct = (s.append (S.append T_1).join).destruct ∧
T.join.destruct = (s.append (S.join.append T_1.join)).destruct
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h1.h1.h2
α : Type u
β : Type v
γ : Type w
S T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
s : WSeq α
T : WSeq (WSeq α)
⊢ ∃ s_1 S T_1,
(s.append T.join).destruct = (s_1.append (S.append T_1).join).destruct ∧
(s.append T.join).destruct = (s_1.append (S.join.append T_1.join)).destruct
|
refine ⟨s, <a>Stream'.WSeq.nil</a>, T, ?_, ?_⟩ <;> simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h1.h1.h3
α : Type u
β : Type v
γ : Type w
S T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T : WSeq (WSeq α)
⊢ ∃ s S T_1,
T.join.destruct = (s.append (S.append T_1).join).destruct ∧
T.join.destruct = (s.append (S.join.append T_1.join)).destruct
|
refine ⟨<a>Stream'.WSeq.nil</a>, <a>Stream'.WSeq.nil</a>, T, ?_, ?_⟩ <;> simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h1.h2
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T : WSeq (WSeq α)
s : WSeq α
S : WSeq (WSeq α)
⊢ ∃ s_1 S_1 T_1,
(s.append (S.append T).join).destruct = (s_1.append (S_1.append T_1).join).destruct ∧
(s.append (S.join.append T.join)).destruct = (s_1.append (S_1.join.append T_1.join)).destruct
|
exact ⟨s, S, T, <a>rfl</a>, <a>rfl</a>⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h1.h3
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
T S : WSeq (WSeq α)
⊢ ∃ s S_1 T_1,
(S.append T).join.destruct = (s.append (S_1.append T_1).join).destruct ∧
(S.join.append T.join).destruct = (s.append (S_1.join.append T_1.join)).destruct
|
refine ⟨<a>Stream'.WSeq.nil</a>, S, T, ?_, ?_⟩ <;> simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h2
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
S T : WSeq (WSeq α)
a : α
s : WSeq α
⊢ ∃ s_1 S_1 T_1,
s.append (S.append T).join = s_1.append (S_1.append T_1).join ∧
s.append (S.join.append T.join) = s_1.append (S_1.join.append T_1.join)
|
exact ⟨s, S, T, <a>rfl</a>, <a>rfl</a>⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.join_append
|
case intro.intro.intro.intro.h3
α : Type u
β : Type v
γ : Type w
S✝ T✝ : WSeq (WSeq α)
s1 s2 : WSeq α
h : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)
S T : WSeq (WSeq α)
s : WSeq α
⊢ ∃ s_1 S_1 T_1,
(s.append (S.append T).join).destruct = (s_1.append (S_1.append T_1).join).destruct ∧
(s.append (S.join.append T.join)).destruct = (s_1.append (S_1.join.append T_1.join)).destruct
|
exact ⟨s, S, T, <a>rfl</a>, <a>rfl</a>⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Seq/WSeq.lean
|
Matroid.indep_or_dep
|
α : Type u_1
M : Matroid α
X : Set α
hX : autoParam (X ⊆ M.E) _auto✝
⊢ M.Indep X ∨ M.Dep X
|
rw [<a>Matroid.Dep</a>, <a>and_iff_left</a> hX]
|
α : Type u_1
M : Matroid α
X : Set α
hX : autoParam (X ⊆ M.E) _auto✝
⊢ M.Indep X ∨ ¬M.Indep X
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Matroid/Basic.lean
|
Matroid.indep_or_dep
|
α : Type u_1
M : Matroid α
X : Set α
hX : autoParam (X ⊆ M.E) _auto✝
⊢ M.Indep X ∨ ¬M.Indep X
|
apply <a>em</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Matroid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_whiskerRight_comp_evaluation
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ (tensorRightHomEquiv Y X Xᘁ (𝟙_ C)) (f ▷ X ≫ ε_ X Xᘁ) = 𝟙 Y ⊗≫ (Y ◁ η_ X Xᘁ ≫ f ▷ (X ⊗ Xᘁ)) ⊗≫ ε_ X Xᘁ ▷ Xᘁ
|
dsimp [<a>CategoryTheory.tensorRightHomEquiv</a>]
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ (ρ_ Y).inv ≫ Y ◁ η_ X Xᘁ ≫ (α_ Y X Xᘁ).inv ≫ (f ▷ X ≫ ε_ X Xᘁ) ▷ Xᘁ =
𝟙 Y ⊗≫ (Y ◁ η_ X Xᘁ ≫ f ▷ (X ⊗ Xᘁ)) ⊗≫ ε_ X Xᘁ ▷ Xᘁ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_whiskerRight_comp_evaluation
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ (ρ_ Y).inv ≫ Y ◁ η_ X Xᘁ ≫ (α_ Y X Xᘁ).inv ≫ (f ▷ X ≫ ε_ X Xᘁ) ▷ Xᘁ =
𝟙 Y ⊗≫ (Y ◁ η_ X Xᘁ ≫ f ▷ (X ⊗ Xᘁ)) ⊗≫ ε_ X Xᘁ ▷ Xᘁ
|
coherence
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_whiskerRight_comp_evaluation
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ 𝟙 Y ⊗≫ (Y ◁ η_ X Xᘁ ≫ f ▷ (X ⊗ Xᘁ)) ⊗≫ ε_ X Xᘁ ▷ Xᘁ = f ⊗≫ Xᘁ ◁ η_ X Xᘁ ⊗≫ ε_ X Xᘁ ▷ Xᘁ
|
rw [<a>CategoryTheory.MonoidalCategory.whisker_exchange</a>]
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ 𝟙 Y ⊗≫ (f ▷ 𝟙_ C ≫ Xᘁ ◁ η_ X Xᘁ) ⊗≫ ε_ X Xᘁ ▷ Xᘁ = f ⊗≫ Xᘁ ◁ η_ X Xᘁ ⊗≫ ε_ X Xᘁ ▷ Xᘁ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_whiskerRight_comp_evaluation
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ 𝟙 Y ⊗≫ (f ▷ 𝟙_ C ≫ Xᘁ ◁ η_ X Xᘁ) ⊗≫ ε_ X Xᘁ ▷ Xᘁ = f ⊗≫ Xᘁ ◁ η_ X Xᘁ ⊗≫ ε_ X Xᘁ ▷ Xᘁ
|
coherence
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_whiskerRight_comp_evaluation
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ f ⊗≫ Xᘁ ◁ η_ X Xᘁ ⊗≫ ε_ X Xᘁ ▷ Xᘁ = f ≫ (λ_ Xᘁ).inv
|
rw [<a>CategoryTheory.ExactPairing.coevaluation_evaluation''</a>]
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ f ⊗≫ ⊗𝟙 = f ≫ (λ_ Xᘁ).inv
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
CategoryTheory.tensorRightHomEquiv_whiskerRight_comp_evaluation
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y : C
inst✝ : HasRightDual X
f : Y ⟶ Xᘁ
⊢ f ⊗≫ ⊗𝟙 = f ≫ (λ_ Xᘁ).inv
|
coherence
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
|
Set.Finite.exists_maximal_wrt
|
case H0
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝ t : Set α
inst✝ : PartialOrder β
f : α → β
s : Set α
hs : ∅.Nonempty
⊢ ∃ a ∈ ∅, ∀ a' ∈ ∅, f a ≤ f a' → f a = f a'
|
exact <a>absurd</a> hs <a>Set.not_nonempty_empty</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case H1
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
rcases s.eq_empty_or_nonempty with h | h
|
case H1.inl
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s = ∅
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
case H1.inr
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s.Nonempty
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case H1.inr
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s.Nonempty
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
rcases ih h with ⟨b, hb, ih⟩
|
case H1.inr.intro.intro
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case H1.inr.intro.intro
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
by_cases h : f b ≤ f a
|
case pos
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : f b ≤ f a
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
case neg
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : ¬f b ≤ f a
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case H1.inl
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s = ∅
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
use a
|
case h
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s = ∅
⊢ a ∈ insert a s ∧ ∀ a' ∈ insert a s, f a ≤ f a' → f a = f a'
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case h
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝ : s.Finite
ih : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h : s = ∅
⊢ a ∈ insert a s ∧ ∀ a' ∈ insert a s, f a ≤ f a' → f a = f a'
|
simp [h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case pos
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : f b ≤ f a
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
refine ⟨a, <a>Set.mem_insert</a> _ _, fun c hc hac => <a>le_antisymm</a> hac ?_⟩
|
case pos
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : f b ≤ f a
c : α
hc : c ∈ insert a s
hac : f a ≤ f c
⊢ f c ≤ f a
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case pos
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : f b ≤ f a
c : α
hc : c ∈ insert a s
hac : f a ≤ f c
⊢ f c ≤ f a
|
rcases <a>Set.mem_insert_iff</a>.1 hc with (rfl | hcs)
|
case pos.inl
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ s : Set α
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
c : α
his : c ∉ s
hs : (insert c s).Nonempty
h : f b ≤ f c
hc : c ∈ insert c s
hac : f c ≤ f c
⊢ f c ≤ f c
case pos.inr
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : f b ≤ f a
c : α
hc : c ∈ insert a s
hac : f a ≤ f c
hcs : c ∈ s
⊢ f c ≤ f a
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case pos.inl
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ s : Set α
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
c : α
his : c ∉ s
hs : (insert c s).Nonempty
h : f b ≤ f c
hc : c ∈ insert c s
hac : f c ≤ f c
⊢ f c ≤ f c
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case pos.inr
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : f b ≤ f a
c : α
hc : c ∈ insert a s
hac : f a ≤ f c
hcs : c ∈ s
⊢ f c ≤ f a
|
rwa [← ih c hcs (<a>le_trans</a> h hac)]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case neg
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : ¬f b ≤ f a
⊢ ∃ a_1 ∈ insert a s, ∀ a' ∈ insert a s, f a_1 ≤ f a' → f a_1 = f a'
|
refine ⟨b, <a>Set.mem_insert_of_mem</a> _ hb, fun c hc hbc => ?_⟩
|
case neg
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : ¬f b ≤ f a
c : α
hc : c ∈ insert a s
hbc : f b ≤ f c
⊢ f b = f c
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Set.Finite.exists_maximal_wrt
|
case neg
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : ¬f b ≤ f a
c : α
hc : c ∈ insert a s
hbc : f b ≤ f c
⊢ f b = f c
|
rcases <a>Set.mem_insert_iff</a>.1 hc with (rfl | hcs)
|
case neg.inl
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ s : Set α
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
c : α
hbc : f b ≤ f c
his : c ∉ s
hs : (insert c s).Nonempty
h : ¬f b ≤ f c
hc : c ∈ insert c s
⊢ f b = f c
case neg.inr
α : Type u
β : Type v
ι : Sort w
γ : Type x
s✝¹ t : Set α
inst✝ : PartialOrder β
f : α → β
s✝ : Set α
a : α
s : Set α
his : a ∉ s
h✝¹ : s.Finite
ih✝ : s.Nonempty → ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a'
hs : (insert a s).Nonempty
h✝ : s.Nonempty
b : α
hb : b ∈ s
ih : ∀ a' ∈ s, f b ≤ f a' → f b = f a'
h : ¬f b ≤ f a
c : α
hc : c ∈ insert a s
hbc : f b ≤ f c
hcs : c ∈ s
⊢ f b = f c
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Set/Finite.lean
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.