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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | exact β¨t, hF ht, hxtβ© | case intro.intro.intro.intro.intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspace : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
x : E
s : Finset E
hs : s β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hsint : combiInterior β s = β sβ β F, combiInterior β sβ
t : Finset E
ht : t β F
hxt : x β combiInterior β t
β’ β i, β (_ : i β Kβ), x β combiInterior β i | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | obtain β¨t, htβ© := hempty β¨_, hsβ© | case mpr.intro.intro.inl
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
s t : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
hs : β
β Kβ
β’ β sβ β Kβ, (convexHull β) ββ
β (convexHull β) βsβ | case mpr.intro.intro.inl.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
s tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
hs : β
β Kβ
t : Finset E
ht : t β Kβ.faces
β’ β sβ β Kβ, (convexHull β) ββ
β (convexHull β) βsβ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | exact β¨t, ht, by simpβ© | case mpr.intro.intro.inl.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
s tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
hs : β
β Kβ
t : Finset E
ht : t β Kβ.faces
β’ β sβ β Kβ, (convexHull β) ββ
β (convexHull β) βsβ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | simp | π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
s tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
hs : β
β Kβ
t : Finset E
ht : t β Kβ.faces
β’ (convexHull β) ββ
β (convexHull β) βt | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | rw [hinterior, mem_iUnionβ] at hxt β’ | π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
hxt' : x' β combiInterior β t'
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
β’ x β combiInterior β t | π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
hxt' : x' β combiInterior β t'
F : Set (Finset E)
hxt : β i, β (_ : i β F), x β combiInterior β i
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
β’ β i, β (_ : i β F), x β combiInterior β i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | obtain β¨u, hu, hxuβ© := hxt | π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
hxt' : x' β combiInterior β t'
F : Set (Finset E)
hxt : β i, β (_ : i β F), x β combiInterior β i
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
β’ β i, β (_ : i β F), x β combiInterior β i | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
hxt' : x' β combiInterior β t'
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F
hxu : x β combiInterior β u
β’ β i, β (_ : i β F), x β combiInterior β i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | exact β¨u, hu, hxuβ© | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
hxt' : x' β combiInterior β t'
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F
hxu : x β combiInterior β u
β’ β i, β (_ : i β F), x β combiInterior β i | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | rw [hinterior', mem_iUnionβ] at hxt' β’ | π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
hxt' : x' β combiInterior β t'
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
β’ x β combiInterior β t' | π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hxt' : β i, β (_ : i β F'), x' β combiInterior β i
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
β’ β i, β (_ : i β F'), x β combiInterior β i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | obtain β¨u, hu, hxuβ© := hxt' | π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hxt' : β i, β (_ : i β F'), x' β combiInterior β i
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
β’ β i, β (_ : i β F'), x β combiInterior β i | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F'
hxu : x' β combiInterior β u
β’ β i, β (_ : i β F'), x β combiInterior β i |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | refine' β¨u, hu, _β© | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F'
hxu : x' β combiInterior β u
β’ β i, β (_ : i β F'), x β combiInterior β i | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F'
hxu : x' β combiInterior β u
β’ x β combiInterior β u |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | rw [β disjoint_interiors hs (hF' hu) hx' hxu] | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F'
hxu : x' β combiInterior β u
β’ x β combiInterior β u | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F'
hxu : x' β combiInterior β u
β’ x β combiInterior β s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | exact hx | case intro.intro
π : Type u_1
E : Type u_2
instβΒ² : SeminormedAddCommGroup E
instβΒΉ : T2Space E
instβ : NormedSpace β E
sβ tβ : Finset E
m : β
Kβ Kβ : SimplicialComplex β E
hempty : Kβ.faces.Nonempty β Kβ.faces.Nonempty
hspaceβ : Kβ.space β Kβ.space
hpartition : β sβ β Kβ, β F β Kβ.faces, combiInterior β sβ = β sβ β F, combiInterior β sβ
hspace : Kβ.space = Kβ.space
s : Finset E
hs : s β Kβ
hsnonempty : s.Nonempty
x : E
hx : x β combiInterior β s
t : Finset E
ht : t β Kβ
hxt : x β combiInterior β t
x' : E
hx' : x' β combiInterior β s
t' : Finset E
ht' : t' β Kβ
F : Set (Finset E)
hF : F β Kβ.faces
hinterior : combiInterior β t = β sβ β F, combiInterior β sβ
F' : Set (Finset E)
hF' : F' β Kβ.faces
hinterior' : combiInterior β t' = β sβ β F', combiInterior β sβ
u : Finset E
hu : u β F'
hxu : x' β combiInterior β u
β’ x β combiInterior β s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/GroupTheory/QuotientGroup.lean | QuotientGroup.preimage_image_mk_eq_iUnion_smul | [11, 1] | [15, 27] | simp_rw [QuotientGroup.preimage_image_mk_eq_iUnion_image N s, β image_smul, Submonoid.smul_def,
smul_eq_mul, mul_comm] | Ξ± : Type u_1
instβΒΉ : CommGroup Ξ±
N : Subgroup Ξ±
instβ : N.Normal
s : Set Ξ±
β’ mk β»ΒΉ' (mk '' s) = β x, x β’ s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | induction' s using Quotient.inductionOn with lβ | Ξ± : Type u_1
s t : Multiset Ξ±
n : β
hst : s β€ t
hs : card s β€ n
ht : n β€ card t
β’ β u, s β€ u β§ u β€ t β§ card u = n | case h
Ξ± : Type u_1
s t : Multiset Ξ±
n : β
ht : n β€ card t
lβ : List Ξ±
hst : β¦lββ§ β€ t
hs : card β¦lββ§ β€ n
β’ β u, β¦lββ§ β€ u β§ u β€ t β§ card u = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | induction' t using Quotient.inductionOn with lβ | case h
Ξ± : Type u_1
s t : Multiset Ξ±
n : β
ht : n β€ card t
lβ : List Ξ±
hst : β¦lββ§ β€ t
hs : card β¦lββ§ β€ n
β’ β u, β¦lββ§ β€ u β§ u β€ t β§ card u = n | case h.h
Ξ± : Type u_1
s t : Multiset Ξ±
n : β
lβ : List Ξ±
hs : card β¦lββ§ β€ n
lβ : List Ξ±
ht : n β€ card β¦lββ§
hst : β¦lββ§ β€ β¦lββ§
β’ β u, β¦lββ§ β€ u β§ u β€ β¦lββ§ β§ card u = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | obtain β¨lβ, hβ© := hst.exists_intermediate hs ht | case h.h
Ξ± : Type u_1
s t : Multiset Ξ±
n : β
lβ : List Ξ±
hs : card β¦lββ§ β€ n
lβ : List Ξ±
ht : n β€ card β¦lββ§
hst : β¦lββ§ β€ β¦lββ§
β’ β u, β¦lββ§ β€ u β§ u β€ β¦lββ§ β§ card u = n | case h.h.intro
Ξ± : Type u_1
s t : Multiset Ξ±
n : β
lβ : List Ξ±
hs : card β¦lββ§ β€ n
lβ : List Ξ±
ht : n β€ card β¦lββ§
hst : β¦lββ§ β€ β¦lββ§
lβ : List Ξ±
h : lβ.Subperm lβ β§ lβ.Subperm lβ β§ lβ.length = n
β’ β u, β¦lββ§ β€ u β§ u β€ β¦lββ§ β§ card u = n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | exact β¨lβ, hβ© | case h.h.intro
Ξ± : Type u_1
s t : Multiset Ξ±
n : β
lβ : List Ξ±
hs : card β¦lββ§ β€ n
lβ : List Ξ±
ht : n β€ card β¦lββ§
hst : β¦lββ§ β€ β¦lββ§
lβ : List Ξ±
h : lβ.Subperm lβ β§ lβ.Subperm lβ β§ lβ.length = n
β’ β u, β¦lββ§ β€ u β§ u β€ β¦lββ§ β§ card u = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_le_card_eq | [21, 1] | [22, 65] | simpa using exists_intermediate (zero_le _) (Nat.zero_le _) hn | Ξ± : Type u_1
s t : Multiset Ξ±
n : β
hn : n β€ card s
β’ β t β€ s, card t = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.disjoint_edgeFinset | [8, 1] | [10, 68] | simp_rw [β Finset.disjoint_coe, coe_edgeFinset, disjoint_edgeSet] | Ξ± : Type u_1
G H : SimpleGraph Ξ±
instβΒΉ : Fintype βG.edgeSet
instβ : Fintype βH.edgeSet
β’ Disjoint G.edgeFinset H.edgeFinset β Disjoint G H | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.edgeFinset_eq_empty | [12, 1] | [13, 40] | rw [β edgeFinset_bot, edgeFinset_inj] | Ξ± : Type u_1
G H : SimpleGraph Ξ±
instβ : Fintype βG.edgeSet
β’ G.edgeFinset = β
β G = β₯ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.edgeFinset_nonempty | [15, 1] | [16, 60] | rw [Finset.nonempty_iff_ne_empty, edgeFinset_eq_empty.ne] | Ξ± : Type u_1
G H : SimpleGraph Ξ±
instβ : Fintype βG.edgeSet
β’ G.edgeFinset.Nonempty β G β β₯ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_zero | [18, 1] | [19, 47] | rw [hasSliceRankLE_iff] | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE 0 f β f = 0 | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ (0 = 0 β§ f = 0 β¨ β n f_1 i g h, HasSliceRankLE n f_1 β§ 0 = n + 1 β§ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) β
f = 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_zero | [18, 1] | [19, 47] | simp [@eq_comm _ 0] | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ (0 = 0 β§ f = 0 β¨ β n f_1 i g h, HasSliceRankLE n f_1 β§ 0 = n + 1 β§ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) β
f = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_succ | [21, 1] | [25, 8] | rw [hasSliceRankLE_iff] | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE (n + 1) f β β f' i g h, HasSliceRankLE n f' β§ f = f' + fun x => g (x i) * h fun j x_1 => x j | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ (n + 1 = 0 β§ f = 0 β¨
β n_1 f_1 i g h, HasSliceRankLE n_1 f_1 β§ n + 1 = n_1 + 1 β§ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) β
β f' i g h, HasSliceRankLE n f' β§ f = f' + fun x => g (x i) * h fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_succ | [21, 1] | [25, 8] | sorry | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ (n + 1 = 0 β§ f = 0 β¨
β n_1 f_1 i g h, HasSliceRankLE n_1 f_1 β§ n + 1 = n_1 + 1 β§ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) β
β f' i g h, HasSliceRankLE n f' β§ f = f' + fun x => g (x i) * h fun j x_1 => x j | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_one | [27, 1] | [29, 77] | simp [hasSliceRankLE_succ] | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE 1 f β β i g h, f = fun x => g (x i) * h fun j x_1 => x j | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | induction' n with n ih generalizing f | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j | case zero
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ f : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE 0 f β β i g h, f = β k : Fin 0, fun x => g k (x (i k)) * h k fun j x_1 => x j
case succ
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE (n + 1) f β β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | simp_rw [hasSliceRankLE_succ, ih] | case succ
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE (n + 1) f β β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j | case succ
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ (β f' i g h,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
f = f' + fun x => g (x i) * h fun j x_1 => x j) β
β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | constructor | case succ
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ (β f' i g h,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
f = f' + fun x => g (x i) * h fun j x_1 => x j) β
β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j | case succ.mp
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ (β f' i g h,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
f = f' + fun x => g (x i) * h fun j x_1 => x j) β
β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j
case succ.mpr
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ (β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) β
β f' i g h,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
f = f' + fun x => g (x i) * h fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | simp | case zero
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ f : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE 0 f β β i g h, f = β k : Fin 0, fun x => g k (x (i k)) * h k fun j x_1 => x j | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | rintro β¨f', iβ, gβ, hβ, β¨i, g, h, rflβ©, rflβ© | case succ.mp
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ (β f' i g h,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
f = f' + fun x => g (x i) * h fun j x_1 => x j) β
β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ β i_1 g_1 h_1,
((β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) + fun x => gβ (x iβ) * hβ fun j x_1 => x j) =
β k : Fin (n + 1), fun x => g_1 k (x (i_1 k)) * h_1 k fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | refine β¨Fin.cons iβ i, Fin.cons gβ g, Fin.cons hβ h, ?_β© | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ β i_1 g_1 h_1,
((β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) + fun x => gβ (x iβ) * hβ fun j x_1 => x j) =
β k : Fin (n + 1), fun x => g_1 k (x (i_1 k)) * h_1 k fun j x_1 => x j | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ ((β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) + fun x => gβ (x iβ) * hβ fun j x_1 => x j) =
β k : Fin (n + 1), fun x => Fin.cons gβ g k (x (Fin.cons iβ i k)) * Fin.cons hβ h k fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | ext x | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ ((β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) + fun x => gβ (x iβ) * hβ fun j x_1 => x j) =
β k : Fin (n + 1), fun x => Fin.cons gβ g k (x (Fin.cons iβ i k)) * Fin.cons hβ h k fun j x_1 => x j | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ ((β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) + fun x => gβ (x iβ) * hβ fun j x_1 => x j) x =
(β k : Fin (n + 1), fun x => Fin.cons gβ g k (x (Fin.cons iβ i k)) * Fin.cons hβ h k fun j x_1 => x j) x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | simp only [ne_eq, Pi.add_apply, Finset.sum_apply, add_comm (_ * _), Fin.sum_univ_succ,
Fin.cons_zero, Fin.cons_succ] | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ ((β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) + fun x => gβ (x iβ) * hβ fun j x_1 => x j) x =
(β k : Fin (n + 1), fun x => Fin.cons gβ g k (x (Fin.cons iβ i k)) * Fin.cons hβ h k fun j x_1 => x j) x | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ ((β c : Fin n, g c (x (i c)) * h c fun j x_1 => x j) + gβ (x iβ) * hβ fun j x_1 => x j) =
(β x_1 : Fin n, g x_1 (x (Fin.cons iβ i x_1.succ)) * h x_1 fun j x_2 => x j) +
gβ (x (Fin.cons iβ i 0)) * hβ fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | congr | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
iβ : ΞΉ
gβ : Ξ± iβ β R
hβ : ((j : ΞΉ) β j β iβ β Ξ± j) β R
i : Fin n β ΞΉ
g : (k : Fin n) β Ξ± (i k) β R
h : (k : Fin n) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ ((β c : Fin n, g c (x (i c)) * h c fun j x_1 => x j) + gβ (x iβ) * hβ fun j x_1 => x j) =
(β x_1 : Fin n, g x_1 (x (Fin.cons iβ i x_1.succ)) * h x_1 fun j x_2 => x j) +
gβ (x (Fin.cons iβ i 0)) * hβ fun j x_1 => x j | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | rintro β¨i, g, h, rflβ© | case succ.mpr
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ΞΉ) β Ξ± i) β R
β’ (β i g h, f = β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) β
β f' i g h,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
f = f' + fun x => g (x i) * h fun j x_1 => x j | case succ.mpr.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ β f' i_1 g_1 h_1,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
(β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) =
f' + fun x => g_1 (x i_1) * h_1 fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | refine β¨_, i 0, g 0, h 0, β¨Fin.tail i, Fin.tail g, Fin.tail h, rflβ©, ?_β© | case succ.mpr.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ β f' i_1 g_1 h_1,
(β i g h, f' = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j) β§
(β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) =
f' + fun x => g_1 (x i_1) * h_1 fun j x_1 => x j | case succ.mpr.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ (β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) =
(β k : Fin n, fun x => Fin.tail g k (x (Fin.tail i k)) * Fin.tail h k fun j x_1 => x j) + fun x =>
g 0 (x (i 0)) * h 0 fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | ext x | case succ.mpr.intro.intro.intro
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
β’ (β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) =
(β k : Fin n, fun x => Fin.tail g k (x (Fin.tail i k)) * Fin.tail h k fun j x_1 => x j) + fun x =>
g 0 (x (i 0)) * h 0 fun j x_1 => x j | case succ.mpr.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ (β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) x =
((β k : Fin n, fun x => Fin.tail g k (x (Fin.tail i k)) * Fin.tail h k fun j x_1 => x j) + fun x =>
g 0 (x (i 0)) * h 0 fun j x_1 => x j)
x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | simp only [ne_eq, Pi.add_apply, Finset.sum_apply, add_comm (_ * _), Fin.sum_univ_succ,
Fin.cons_zero, Fin.cons_succ] | case succ.mpr.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ (β k : Fin (n + 1), fun x => g k (x (i k)) * h k fun j x_1 => x j) x =
((β k : Fin n, fun x => Fin.tail g k (x (Fin.tail i k)) * Fin.tail h k fun j x_1 => x j) + fun x =>
g 0 (x (i 0)) * h 0 fun j x_1 => x j)
x | case succ.mpr.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ ((β i_1 : Fin n, g i_1.succ (x (i i_1.succ)) * h i_1.succ fun j x_1 => x j) + g 0 (x (i 0)) * h 0 fun j x_1 => x j) =
(β c : Fin n, Fin.tail g c (x (Fin.tail i c)) * Fin.tail h c fun j x_1 => x j) +
g 0 (x (i 0)) * h 0 fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | congr | case succ.mpr.intro.intro.intro.h
ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m : β
fβ fβ : ((i : ΞΉ) β Ξ± i) β R
n : β
ih :
β {f : ((i : ΞΉ) β Ξ± i) β R},
HasSliceRankLE n f β β i g h, f = β k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) β ΞΉ
g : (k : Fin (n + 1)) β Ξ± (i k) β R
h : (k : Fin (n + 1)) β ((j : ΞΉ) β j β i k β Ξ± j) β R
x : (i : ΞΉ) β Ξ± i
β’ ((β i_1 : Fin n, g i_1.succ (x (i i_1.succ)) * h i_1.succ fun j x_1 => x j) + g 0 (x (i 0)) * h 0 fun j x_1 => x j) =
(β c : Fin n, Fin.tail g c (x (Fin.tail i c)) * Fin.tail h c fun j x_1 => x j) +
g 0 (x (i 0)) * h 0 fun j x_1 => x j | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | HasSliceRankLE.add | [51, 1] | [54, 74] | simpa | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
hβ : HasSliceRankLE m fβ
β’ HasSliceRankLE (m + 0) (fβ + 0) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | HasSliceRankLE.add | [51, 1] | [54, 74] | simpa [add_assoc] using (hβ.add hβ).succ g h | ΞΉ : Type u_1
R : Type u_2
instβΒΉ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβ : Semiring R
m n : β
f fβ fβ : ((i : ΞΉ) β Ξ± i) β R
hβ : HasSliceRankLE m fβ
nβ : β
fβ : ((i : ΞΉ) β (fun i => Ξ± i) i) β R
iβ : ΞΉ
g : Ξ± iβ β R
h : ((j : ΞΉ) β j β iβ β Ξ± j) β R
hβ : HasSliceRankLE nβ fβ
β’ HasSliceRankLE (m + (nβ + 1)) (fβ + (fβ + fun x => g (x iβ) * h fun j x_1 => x j)) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_card | [56, 1] | [60, 8] | rw [hasSliceRankLE_iff_exists_sum] | ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Fintype ΞΉ
instβ : (i : ΞΉ) β Fintype (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
β’ HasSliceRankLE (Fintype.card ((i : ΞΉ) β Ξ± i)) f | ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Fintype ΞΉ
instβ : (i : ΞΉ) β Fintype (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
β’ β i g h, f = β k : Fin (Fintype.card ((i : ΞΉ) β Ξ± i)), fun x => g k (x (i k)) * h k fun j x_1 => x j |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_card | [56, 1] | [60, 8] | sorry | ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Fintype ΞΉ
instβ : (i : ΞΉ) β Fintype (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
β’ β i g h, f = β k : Fin (Fintype.card ((i : ΞΉ) β Ξ± i)), fun x => g k (x (i k)) * h k fun j x_1 => x j | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | exists_hasSliceRankLE | [62, 1] | [67, 35] | cases nonempty_fintype ΞΉ | ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Finite ΞΉ
instβ : β (i : ΞΉ), Finite (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
β’ β n, HasSliceRankLE n f | case intro
ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Finite ΞΉ
instβ : β (i : ΞΉ), Finite (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
valβ : Fintype ΞΉ
β’ β n, HasSliceRankLE n f |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | exists_hasSliceRankLE | [62, 1] | [67, 35] | have (i) := Fintype.ofFinite (Ξ± i) | case intro
ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Finite ΞΉ
instβ : β (i : ΞΉ), Finite (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
valβ : Fintype ΞΉ
β’ β n, HasSliceRankLE n f | case intro
ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Finite ΞΉ
instβ : β (i : ΞΉ), Finite (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
valβ : Fintype ΞΉ
this : (i : ΞΉ) β Fintype (Ξ± i)
β’ β n, HasSliceRankLE n f |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | exists_hasSliceRankLE | [62, 1] | [67, 35] | exact β¨_, hasSliceRankLE_card _β© | case intro
ΞΉ : Type u_1
R : Type u_2
instβΒ³ : DecidableEq ΞΉ
Ξ± : ΞΉ β Type u_3
instβΒ² : Semiring R
m n : β
fβ fβ fβ : ((i : ΞΉ) β Ξ± i) β R
instβΒΉ : Finite ΞΉ
instβ : β (i : ΞΉ), Finite (Ξ± i)
f : ((i : ΞΉ) β Ξ± i) β R
valβ : Fintype ΞΉ
this : (i : ΞΉ) β Fintype (Ξ± i)
β’ β n, HasSliceRankLE n f | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | one_le_schnirelmannDensity_iff | [10, 1] | [12, 76] | rw [schnirelmannDensity_le_one.ge_iff_eq, schnirelmannDensity_eq_one_iff] | A B : Set β
n : β
β’ 1 β€ schnirelmannDensity A β {0}αΆ β A | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | one_le_schnirelmannDensity_iff_of_zero_mem | [14, 1] | [17, 91] | rw [schnirelmannDensity_le_one.ge_iff_eq, schnirelmannDensity_eq_one_iff_of_zero_mem hA] | A B : Set β
n : β
hA : 0 β A
β’ 1 β€ schnirelmannDensity A β A = Set.univ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_nonneg | [22, 1] | [22, 90] | positivity | Aβ B : Set β
nβ : β
A : Set β
n : β
β’ 0 β€ countelements A n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | card_Icc_one_n_n | [24, 1] | [25, 47] | rw [Nat.card_Icc 1 n, add_tsub_cancel_right] | A B : Set β
nβ n : β
β’ (Icc 1 n).card = n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_le_n | [27, 1] | [28, 57] | simpa [countelements] using card_filter_le (Icc 1 n) _ | Aβ B : Set β
nβ : β
A : Set β
n : β
β’ countelements A n β€ n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | repeat rw [countelements] | A B : Set β
n : β
hn : n β A
β’ countelements A (n - 1) = countelements A n | A B : Set β
n : β
hn : n β A
β’ (filter (fun x => x β A) (Icc 1 (n - 1))).card = (filter (fun x => x β A) (Icc 1 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | refine card_le_card fun x hx β¦ ?_ | case refine_2
A B : Set β
n : β
hn : n β A
β’ (filter (fun x => x β A) (Icc 1 n)).card β€ (filter (fun x => x β A) (Icc 1 (n - 1))).card | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : x β filter (fun x => x β A) (Icc 1 n)
β’ x β filter (fun x => x β A) (Icc 1 (n - 1)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rw [mem_filter] | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : x β filter (fun x => x β A) (Icc 1 n)
β’ x β filter (fun x => x β A) (Icc 1 (n - 1)) | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : x β filter (fun x => x β A) (Icc 1 n)
β’ x β Icc 1 (n - 1) β§ x β A |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rw [mem_filter, mem_Icc] at hx | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : x β filter (fun x => x β A) (Icc 1 n)
β’ x β Icc 1 (n - 1) β§ x β A | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : (1 β€ x β§ x β€ n) β§ x β A
β’ x β Icc 1 (n - 1) β§ x β A |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | refine β¨mem_Icc.2 β¨hx.1.1, Nat.le_pred_of_lt $ hx.1.2.lt_of_ne ?_β©, hx.2β© | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : (1 β€ x β§ x β€ n) β§ x β A
β’ x β Icc 1 (n - 1) β§ x β A | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : (1 β€ x β§ x β€ n) β§ x β A
β’ x β n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rintro rfl | case refine_2
A B : Set β
n : β
hn : n β A
x : β
hx : (1 β€ x β§ x β€ n) β§ x β A
β’ x β n | case refine_2
A B : Set β
x : β
hn : x β A
hx : (1 β€ x β§ x β€ x) β§ x β A
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | exact hn hx.2 | case refine_2
A B : Set β
x : β
hn : x β A
hx : (1 β€ x β§ x β€ x) β§ x β A
β’ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rw [countelements] | A B : Set β
n : β
hn : n β A
β’ (filter (fun x => x β A) (Icc 1 (n - 1))).card = countelements A n | A B : Set β
n : β
hn : n β A
β’ (filter (fun x => x β A) (Icc 1 (n - 1))).card = (filter (fun x => x β A) (Icc 1 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | simp only [tsub_le_iff_right, le_add_iff_nonneg_right, zero_le_one] | case refine_1
A B : Set β
n : β
hn : n β A
β’ n - 1 β€ n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | by_contra! h | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
β’ n β A + B | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnA : n β A := Set.not_mem_subset (Set.subset_add_left _ hB) h | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnB : n β B := Set.not_mem_subset (Set.subset_add_right _ hA) h | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hca : countelements A (n - 1) = countelements A n := countelements_pred hnA | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hcb : countelements B (n - 1) = countelements B n := countelements_pred hnB | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | obtain rfl | hn1 := n.eq_zero_or_pos | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
β’ False | case inl
A B : Set β
hA : 0 β A
hB : 0 β B
hc : 0 β€ countelements A 0 + countelements B 0
h : 0 β A + B
hnA : 0 β A
hnB : 0 β B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
β’ False
case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | apply h | case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
β’ False | case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
β’ n β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | simp only [Nat.lt_one_iff, tsub_eq_zero_iff_le, mem_Ioo, and_imp, Set.singleton_sub,
Set.mem_image, ne_eq] at lem3 | case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).Nonempty
β’ n β A + B | case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
β’ n β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have lem31 : (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty := by
rw [β filter_and, β coe_nonempty, coe_filter, Set.setOf_and, Set.setOf_and, Set.setOf_mem_eq,
Set.inter_comm] at lem3
convert lem3 using 3 <;> ext <;> simp | case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
β’ n β A + B | case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
lem31 : (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty
β’ n β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | obtain β¨_, β¨hxA, n, rfl, x, hxB, rflβ©, hxβ© := lem31 | case inr
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
lem31 : (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty
β’ n β A + B | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set β
hA : 0 β A
hB : 0 β B
n : β
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : β
hxB : x β B
hxA : (fun x x_1 => x - x_1) n x β A
hx : (fun x x_1 => x - x_1) n x β Set.Ioo 0 n
β’ n β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | simp only [Set.mem_Ioo, Nat.succ_le_iff, tsub_pos_iff_lt, tsub_le_iff_right] at hx | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set β
hA : 0 β A
hB : 0 β B
n : β
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : β
hxB : x β B
hxA : (fun x x_1 => x - x_1) n x β A
hx : (fun x x_1 => x - x_1) n x β Set.Ioo 0 n
β’ n β A + B | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set β
hA : 0 β A
hB : 0 β B
n : β
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : β
hxB : x β B
hxA : (fun x x_1 => x - x_1) n x β A
hx : x < n β§ n - x < n
β’ n β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact β¨_, hxA, x, hxB, tsub_add_cancel_of_le hx.1.leβ© | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set β
hA : 0 β A
hB : 0 β B
n : β
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
x : β
hxB : x β B
hxA : (fun x x_1 => x - x_1) n x β A
hx : x < n β§ n - x < n
β’ n β A + B | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | contradiction | case inl
A B : Set β
hA : 0 β A
hB : 0 β B
hc : 0 β€ countelements A 0 + countelements B 0
h : 0 β A + B
hnA : 0 β A
hnB : 0 β B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
β’ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [countelements] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
β’ (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1) | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
β’ (filter (fun x => x β {n} - B) (Ioo 0 n)).card = (filter (fun x => x β B) (Icc 1 (n - 1))).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β hfim, card_image_of_injOn] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ (filter (fun x => x β {n} - B) (Ioo 0 n)).card = (filter (fun x => x β B) (Icc 1 (n - 1))).card | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ (filter (fun x => x β B) (Ioo 0 n)).card = (filter (fun x => x β B) (Icc 1 (n - 1))).card
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ Set.InjOn (fun x => n - x) β(filter (fun x => x β B) (Ioo 0 n)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | congr | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ (filter (fun x => x β B) (Ioo 0 n)).card = (filter (fun x => x β B) (Icc 1 (n - 1))).card
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ Set.InjOn (fun x => n - x) β(filter (fun x => x β B) (Ioo 0 n)) | case e_s.e_s.e_b
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ n = ((n - 1).add 0).succ
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ Set.InjOn (fun x => n - x) β(filter (fun x => x β B) (Ioo 0 n)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact (tsub_add_cancel_of_le $ Nat.succ_le_iff.2 hn1).symm | case e_s.e_s.e_b
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ n = ((n - 1).add 0).succ
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ Set.InjOn (fun x => n - x) β(filter (fun x => x β B) (Ioo 0 n)) | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ Set.InjOn (fun x => n - x) β(filter (fun x => x β B) (Ioo 0 n)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | ext | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
β’ image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n) | case a
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
aβ : β
β’ aβ β image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) β aβ β filter (fun x => x β {n} - B) (Ioo 0 n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | aesop | case a
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
aβ : β
β’ aβ β image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) β aβ β filter (fun x => x β {n} - B) (Ioo 0 n) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact Set.InjOn.mono (fun x hx β¦ (mem_Ioo.1 (mem_filter.1 hx).1).2.le) $
fun x hx y hy β¦ tsub_inj_right hx hy | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ Set.InjOn (fun x => n - x) β(filter (fun x => x β B) (Ioo 0 n)) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β hca, β hcb] at hc | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).Nonempty | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rwa [β Finset.card_pos] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hin : 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
β’ (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β filter_or, β tsub_zero n, β Nat.card_Ioo] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1 | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun a => a β A β¨ a β {n - 0} - B) (Ioo 0 (n - 0))).card β€ (Ioo 0 n).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact card_filter_le _ _ | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun a => a β A β¨ a β {n - 0} - B) (Ioo 0 (n - 0))).card β€ (Ioo 0 n).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [card_union_add_card_inter, β lem1, countelements] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1) | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ (filter (fun x => x β A) (Ioo 0 n)).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card =
(filter (fun x => x β A) (Icc 1 (n - 1))).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | congr | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ (filter (fun x => x β A) (Ioo 0 n)).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card =
(filter (fun x => x β A) (Icc 1 (n - 1))).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card | case e_a.e_s.e_s.e_b
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ n = ((n - 1).add 0).succ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact (tsub_add_cancel_of_le $ Nat.succ_le_iff.2 hn1).symm | case e_a.e_s.e_s.e_b
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ n = ((n - 1).add 0).succ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β hui] at hc | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
β’ 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
β’ 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | by_contra! hip | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
β’ 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnn : n β€ (n - 1) := le_trans hip0 hip1 | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : n β€ n - 1
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β not_lt] at hnn | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : n β€ n - 1
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | apply hnn | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ n - 1 < n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [propext (Nat.lt_iff_le_pred hn1)] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ n - 1 < n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β filter_and, β coe_nonempty, coe_filter, Set.setOf_and, Set.setOf_and, Set.setOf_mem_eq,
Set.inter_comm] at lem3 | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
β’ (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (A β© {a | β x β B, n - x = a} β© {a | a β Ioo 0 n}).Nonempty
β’ (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | convert lem3 using 3 <;> ext <;> simp | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (A β© {a | β x β B, n - x = a} β© {a | a β Ioo 0 n}).Nonempty
β’ (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | refine Set.eq_univ_of_forall $ fun n β¦ sumset_contains_n hA hB ?_ | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
β’ A + B = Set.univ | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
β’ n β€ countelements A n + countelements B n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | obtain rfl | hn := eq_or_ne n 0 | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
β’ n β€ countelements A n + countelements B n | case inl
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
β’ 0 β€ countelements A 0 + countelements B 0
case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ n β€ countelements A n + countelements B n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | rw [β Nat.cast_le (Ξ± := β), β one_le_div (by positivity)] | case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ n β€ countelements A n + countelements B n | case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ 1 β€ β(countelements A n + countelements B n) / βn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | calc
_ β€ _ := hAB
_ β€ _ := add_le_add (schnirelmannDensity_le_div hn) (schnirelmannDensity_le_div hn)
_ = _ := by push_cast; rw [add_div]; rfl | case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ 1 β€ β(countelements A n + countelements B n) / βn | no goals |
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