url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | exact ⟨t, hF ht, hxt⟩ | case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiI... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace : K₁.sp... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | obtain ⟨t, ht⟩ := hempty ⟨_, hs⟩ | case mpr.intro.intro.inl
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s t : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ ... | case mpr.intro.intro.inl.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.inl
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s t : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hp... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | exact ⟨t, ht, by simp⟩ | case mpr.intro.intro.inl.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.inl.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.s... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | simp | 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | rw [hinterior, mem_iUnion₂] at hxt ⊢ | 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ... | 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | obtain ⟨u, hu, hxu⟩ := hxt | 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ... | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | exact ⟨u, hu, hxu⟩ | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartiti... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | rw [hinterior', mem_iUnion₂] at hxt' ⊢ | 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ... | 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | obtain ⟨u, hu, hxu⟩ := hxt' | 𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F, combiInterior ℝ... | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | refine' ⟨u, hu, _⟩ | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartiti... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Subdivision.lean | Geometry.SimplicialComplex.subdivides_iff_partition | [68, 1] | [128, 15] | rw [← disjoint_interiors hs (hF' hu) hx' hxu] | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartiti... |
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
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartition : ∀ s₂ ∈ K₂, ∃ F ⊆ K₁.faces, combiInterior ℝ s₂ = ⋃ s₁ ∈ F... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : T2Space E
inst✝ : NormedSpace ℝ E
s✝ t✝ : Finset E
m : ℕ
K₁ K₂ : SimplicialComplex ℝ E
hempty : K₁.faces.Nonempty → K₂.faces.Nonempty
hspace✝ : K₁.space ⊆ K₂.space
hpartiti... |
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, ← image_smul, Submonoid.smul_def,
smul_eq_mul, mul_comm] | α : Type u_1
inst✝¹ : CommGroup α
N : Subgroup α
inst✝ : N.Normal
s : Set α
⊢ mk ⁻¹' (mk '' s) = ⋃ x, x • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : CommGroup α
N : Subgroup α
inst✝ : N.Normal
s : Set α
⊢ mk ⁻¹' (mk '' s) = ⋃ x, x • s
TACTIC:
|
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 l₀ | α : Type u_1
s t : Multiset α
n : ℕ
hst : s ≤ t
hs : card s ≤ n
ht : n ≤ card t
⊢ ∃ u, s ≤ u ∧ u ≤ t ∧ card u = n | case h
α : Type u_1
s t : Multiset α
n : ℕ
ht : n ≤ card t
l₀ : List α
hst : ⟦l₀⟧ ≤ t
hs : card ⟦l₀⟧ ≤ n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ t ∧ card u = n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s t : Multiset α
n : ℕ
hst : s ≤ t
hs : card s ≤ n
ht : n ≤ card t
⊢ ∃ u, s ≤ u ∧ u ≤ t ∧ card u = n
TACTIC:
|
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 l₂ | case h
α : Type u_1
s t : Multiset α
n : ℕ
ht : n ≤ card t
l₀ : List α
hst : ⟦l₀⟧ ≤ t
hs : card ⟦l₀⟧ ≤ n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ t ∧ card u = n | case h.h
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
s t : Multiset α
n : ℕ
ht : n ≤ card t
l₀ : List α
hst : ⟦l₀⟧ ≤ t
hs : card ⟦l₀⟧ ≤ n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ t ∧ card u = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | obtain ⟨l₁, h⟩ := hst.exists_intermediate hs ht | case h.h
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | case h.h.intro
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
l₁ : List α
h : l₀.Subperm l₁ ∧ l₁.Subperm l₂ ∧ l₁.length = n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Data/Multiset/Basic.lean | Multiset.exists_intermediate | [14, 1] | [19, 16] | exact ⟨l₁, h⟩ | case h.h.intro
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
l₁ : List α
h : l₀.Subperm l₁ ∧ l₁.Subperm l₂ ∧ l₁.length = n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.intro
α : Type u_1
s t : Multiset α
n : ℕ
l₀ : List α
hs : card ⟦l₀⟧ ≤ n
l₂ : List α
ht : n ≤ card ⟦l₂⟧
hst : ⟦l₀⟧ ≤ ⟦l₂⟧
l₁ : List α
h : l₀.Subperm l₁ ∧ l₁.Subperm l₂ ∧ l₁.length = n
⊢ ∃ u, ⟦l₀⟧ ≤ u ∧ u ≤ ⟦l₂⟧ ∧ card u = n
TACTIC:
|
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 | α : Type u_1
s t : Multiset α
n : ℕ
hn : n ≤ card s
⊢ ∃ t ≤ s, card t = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s t : Multiset α
n : ℕ
hn : n ≤ card s
⊢ ∃ t ≤ s, card t = n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.disjoint_edgeFinset | [8, 1] | [10, 68] | simp_rw [← Finset.disjoint_coe, coe_edgeFinset, disjoint_edgeSet] | α : Type u_1
G H : SimpleGraph α
inst✝¹ : Fintype ↑G.edgeSet
inst✝ : Fintype ↑H.edgeSet
⊢ Disjoint G.edgeFinset H.edgeFinset ↔ Disjoint G H | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G H : SimpleGraph α
inst✝¹ : Fintype ↑G.edgeSet
inst✝ : Fintype ↑H.edgeSet
⊢ Disjoint G.edgeFinset H.edgeFinset ↔ Disjoint G H
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Finite.lean | SimpleGraph.edgeFinset_eq_empty | [12, 1] | [13, 40] | rw [← edgeFinset_bot, edgeFinset_inj] | α : Type u_1
G H : SimpleGraph α
inst✝ : Fintype ↑G.edgeSet
⊢ G.edgeFinset = ∅ ↔ G = ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G H : SimpleGraph α
inst✝ : Fintype ↑G.edgeSet
⊢ G.edgeFinset = ∅ ↔ G = ⊥
TACTIC:
|
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] | α : Type u_1
G H : SimpleGraph α
inst✝ : Fintype ↑G.edgeSet
⊢ G.edgeFinset.Nonempty ↔ G ≠ ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
G H : SimpleGraph α
inst✝ : Fintype ↑G.edgeSet
⊢ G.edgeFinset.Nonempty ↔ G ≠ ⊥
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_zero | [18, 1] | [19, 47] | rw [hasSliceRankLE_iff] | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE 0 f ↔ f = 0 | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ (0 = 0 ∧ f = 0 ∨ ∃ n f_1 i g h, HasSliceRankLE n f_1 ∧ 0 = n + 1 ∧ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) ↔
f = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE 0 f ↔ f = 0
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_zero | [18, 1] | [19, 47] | simp [@eq_comm _ 0] | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ (0 = 0 ∧ f = 0 ∨ ∃ n f_1 i g h, HasSliceRankLE n f_1 ∧ 0 = n + 1 ∧ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) ↔
f = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ (0 = 0 ∧ f = 0 ∨ ∃ n f_1 i g h, HasSliceRankLE n f_1 ∧ 0 = n + 1 ∧ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) ↔
f = 0
TACTIC... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_succ | [21, 1] | [25, 8] | rw [hasSliceRankLE_iff] | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE (n + 1) f ↔ ∃ f' i g h, HasSliceRankLE n f' ∧ f = f' + fun x => g (x i) * h fun j x_1 => x j | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ (n + 1 = 0 ∧ f = 0 ∨
∃ n_1 f_1 i g h, HasSliceRankLE n_1 f_1 ∧ n + 1 = n_1 + 1 ∧ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) ↔
∃ f' i g h, HasSliceRankLE n f' ∧ f = f' + fun x => g... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE (n + 1) f ↔ ∃ f' i g h, HasSliceRankLE n f' ∧ f = f' + fun x => g (x i) * h fun j x_1 => x j
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_succ | [21, 1] | [25, 8] | sorry | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ (n + 1 = 0 ∧ f = 0 ∨
∃ n_1 f_1 i g h, HasSliceRankLE n_1 f_1 ∧ n + 1 = n_1 + 1 ∧ f = f_1 + fun x => g (x i) * h fun j x_1 => x j) ↔
∃ f' i g h, HasSliceRankLE n f' ∧ f = f' + fun x => g... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ (n + 1 = 0 ∧ f = 0 ∨
∃ n_1 f_1 i g h, HasSliceRankLE n_1 f_1 ∧ n + 1 = n_1 + 1 ∧ f = f_1 + fun x => g (x i) * h fun j x_1 => x j... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_one | [27, 1] | [29, 77] | simp [hasSliceRankLE_succ] | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE 1 f ↔ ∃ i g h, f = fun x => g (x i) * h fun j x_1 => x j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE 1 f ↔ ∃ i g h, f = fun x => g (x i) * h fun j x_1 => x j
TACTIC:
|
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 | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j | case zero
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ f : ((i : ι) → α i) → R
⊢ HasSliceRankLE 0 f ↔ ∃ i g h, f = ∑ k : Fin 0, fun x => g k (x (i k)) * h k fun j x_1 => x j
case succ
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring ... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
⊢ HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
TACTIC:
|
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
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ι) → α i) → R
⊢ HasSliceRankLE (n + 1) f ↔... | case succ
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ι) → α i) → R
⊢ (∃ f' i g h,
(∃ i g ... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | constructor | case succ
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ι) → α i) → R
⊢ (∃ f' i g h,
(∃ i g ... | case succ.mp
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ι) → α i) → R
⊢ (∃ f' i g h,
(∃ i... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | simp | case zero
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ f : ((i : ι) → α i) → R
⊢ HasSliceRankLE 0 f ↔ ∃ i g h, f = ∑ k : Fin 0, fun x => g k (x (i k)) * h k fun j x_1 => x j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ f : ((i : ι) → α i) → R
⊢ HasSliceRankLE 0 f ↔ ∃ i g h, f = ∑ k : Fin 0, fun x => g k (x (i k)) * h k fun j x_1 => x j
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | rintro ⟨f', iₙ, gₙ, hₙ, ⟨i, g, h, rfl⟩, rfl⟩ | case succ.mp
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ι) → α i) → R
⊢ (∃ f' i g h,
(∃ i... | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mp
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | refine ⟨Fin.cons iₙ i, Fin.cons gₙ g, Fin.cons hₙ h, ?_⟩ | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i... | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f =... |
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
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i... | case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f =... |
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
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, 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.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f... |
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
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mp.intro.intro.intro.intro.intro.intro.intro.intro.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | rintro ⟨i, g, h, rfl⟩ | case succ.mpr
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
f : ((i : ι) → α i) → R
⊢ (∃ i g h, f = ∑ k : Fi... | case succ.mpr.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k : F... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mpr
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_iff_exists_sum | [31, 1] | [49, 10] | refine ⟨_, i 0, g 0, h 0, ⟨Fin.tail i, Fin.tail g, Fin.tail h, rfl⟩, ?_⟩ | case succ.mpr.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k : F... | case succ.mpr.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k : F... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mpr.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x... |
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
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k : F... | case succ.mpr.intro.intro.intro.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k :... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mpr.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (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
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k :... | case succ.mpr.intro.intro.intro.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k :... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mpr.intro.intro.intro.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k ... |
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
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k (x (i k)) * h k fun j x_1 => x j
i : Fin (n + 1) → ι
g : (k :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mpr.intro.intro.intro.h
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m : ℕ
f₁ f₂ : ((i : ι) → α i) → R
n : ℕ
ih :
∀ {f : ((i : ι) → α i) → R},
HasSliceRankLE n f ↔ ∃ i g h, f = ∑ k : Fin n, fun x => g k ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | HasSliceRankLE.add | [51, 1] | [54, 74] | simpa | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
h₁ : HasSliceRankLE m f₁
⊢ HasSliceRankLE (m + 0) (f₁ + 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
h₁ : HasSliceRankLE m f₁
⊢ HasSliceRankLE (m + 0) (f₁ + 0)
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | HasSliceRankLE.add | [51, 1] | [54, 74] | simpa [add_assoc] using (h₁.add h₂).succ g h | ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
h₁ : HasSliceRankLE m f₁
n✝ : ℕ
f✝ : ((i : ι) → (fun i => α i) i) → R
i✝ : ι
g : α i✝ → R
h : ((j : ι) → j ≠ i✝ → α j) → R
h₂ : HasSliceRankLE n✝ f✝
⊢ HasSliceRankLE (m + (n✝ + 1)) (f₁ + (f✝ + fun ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝¹ : DecidableEq ι
α : ι → Type u_3
inst✝ : Semiring R
m n : ℕ
f f₁ f₂ : ((i : ι) → α i) → R
h₁ : HasSliceRankLE m f₁
n✝ : ℕ
f✝ : ((i : ι) → (fun i => α i) i) → R
i✝ : ι
g : α i✝ → R
h : ((j : ι) → j ≠ i✝ → α j) → R
h₂ : HasSlice... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_card | [56, 1] | [60, 8] | rw [hasSliceRankLE_iff_exists_sum] | ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Fintype ι
inst✝ : (i : ι) → Fintype (α i)
f : ((i : ι) → α i) → R
⊢ HasSliceRankLE (Fintype.card ((i : ι) → α i)) f | ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Fintype ι
inst✝ : (i : ι) → Fintype (α i)
f : ((i : ι) → α i) → R
⊢ ∃ i g h, f = ∑ k : Fin (Fintype.card ((i : ι) → α i)), fun x => g k (x (i k)) * h k fun j x_1 => x j | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Fintype ι
inst✝ : (i : ι) → Fintype (α i)
f : ((i : ι) → α i) → R
⊢ HasSliceRankLE (Fintype.card ((i : ι) → α i)) f
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | hasSliceRankLE_card | [56, 1] | [60, 8] | sorry | ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Fintype ι
inst✝ : (i : ι) → Fintype (α i)
f : ((i : ι) → α i) → R
⊢ ∃ i g h, f = ∑ k : Fin (Fintype.card ((i : ι) → α i)), fun x => g k (x (i k)) * h k fun j x_1 => x j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Fintype ι
inst✝ : (i : ι) → Fintype (α i)
f : ((i : ι) → α i) → R
⊢ ∃ i g h, f = ∑ k : Fin (Fintype.card ((i : ι) → α i)), fu... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | exists_hasSliceRankLE | [62, 1] | [67, 35] | cases nonempty_fintype ι | ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
⊢ ∃ n, HasSliceRankLE n f | case intro
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
val✝ : Fintype ι
⊢ ∃ n, HasSliceRankLE n f | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
⊢ ∃ n, HasSliceRankLE n f
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | exists_hasSliceRankLE | [62, 1] | [67, 35] | have (i) := Fintype.ofFinite (α i) | case intro
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
val✝ : Fintype ι
⊢ ∃ n, HasSliceRankLE n f | case intro
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
val✝ : Fintype ι
this : (i : ι) → Fintype (α i)
⊢ ∃ n, HasSliceRankLE n f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
val✝ : Fintype ι
⊢ ∃ n, HasSliceRankLE n f
TACTIC... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SliceRank.lean | exists_hasSliceRankLE | [62, 1] | [67, 35] | exact ⟨_, hasSliceRankLE_card _⟩ | case intro
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
val✝ : Fintype ι
this : (i : ι) → Fintype (α i)
⊢ ∃ n, HasSliceRankLE n f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type u_1
R : Type u_2
inst✝³ : DecidableEq ι
α : ι → Type u_3
inst✝² : Semiring R
m n : ℕ
f✝ f₁ f₂ : ((i : ι) → α i) → R
inst✝¹ : Finite ι
inst✝ : ∀ (i : ι), Finite (α i)
f : ((i : ι) → α i) → R
val✝ : Fintype ι
this : (i : ι) → Fintype (α i)
⊢... |
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 ℕ
n : ℕ
⊢ 1 ≤ schnirelmannDensity A ↔ {0}ᶜ ⊆ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
⊢ 1 ≤ schnirelmannDensity A ↔ {0}ᶜ ⊆ A
TACTIC:
|
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 ℕ
n : ℕ
hA : 0 ∈ A
⊢ 1 ≤ schnirelmannDensity A ↔ A = Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
⊢ 1 ≤ schnirelmannDensity A ↔ A = Set.univ
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_nonneg | [22, 1] | [22, 90] | positivity | A✝ B : Set ℕ
n✝ : ℕ
A : Set ℕ
n : ℕ
⊢ 0 ≤ countelements A n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ B : Set ℕ
n✝ : ℕ
A : Set ℕ
n : ℕ
⊢ 0 ≤ countelements A n
TACTIC:
|
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 ℕ
n✝ n : ℕ
⊢ (Icc 1 n).card = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n✝ n : ℕ
⊢ (Icc 1 n).card = n
TACTIC:
|
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) _ | A✝ B : Set ℕ
n✝ : ℕ
A : Set ℕ
n : ℕ
⊢ countelements A n ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A✝ B : Set ℕ
n✝ : ℕ
A : Set ℕ
n : ℕ
⊢ countelements A n ≤ n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | repeat rw [countelements] | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ countelements A (n - 1) = countelements A n | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card = (filter (fun x => x ∈ A) (Icc 1 n)).card | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ countelements A (n - 1) = countelements A n
TACTIC:
|
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 ↦ ?_ | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 n)).card ≤ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ filter (fun x => x ∈ A) (Icc 1 (n - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 n)).card ≤ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card
TACTIC:
|
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 ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ filter (fun x => x ∈ A) (Icc 1 (n - 1)) | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ filter (fun x => x ∈ A) (Icc 1 (n - 1))
TACTIC:
|
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 ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : x ∈ filter (fun x => x ∈ A) (Icc 1 n)
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | refine ⟨mem_Icc.2 ⟨hx.1.1, Nat.le_pred_of_lt $ hx.1.2.lt_of_ne ?_⟩, hx.2⟩ | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A | case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ≠ n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ∈ Icc 1 (n - 1) ∧ x ∈ A
TACTIC:
|
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 ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ≠ n | case refine_2
A B : Set ℕ
x : ℕ
hn : x ∉ A
hx : (1 ≤ x ∧ x ≤ x) ∧ x ∈ A
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
A B : Set ℕ
n : ℕ
hn : n ∉ A
x : ℕ
hx : (1 ≤ x ∧ x ≤ n) ∧ x ∈ A
⊢ x ≠ n
TACTIC:
|
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 ℕ
x : ℕ
hn : x ∉ A
hx : (1 ≤ x ∧ x ≤ x) ∧ x ∈ A
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
A B : Set ℕ
x : ℕ
hn : x ∉ A
hx : (1 ≤ x ∧ x ≤ x) ∧ x ∈ A
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | countelements_pred | [30, 1] | [39, 16] | rw [countelements] | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card = countelements A n | A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card = (filter (fun x => x ∈ A) (Icc 1 n)).card | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ (filter (fun x => x ∈ A) (Icc 1 (n - 1))).card = countelements A n
TACTIC:
|
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 ℕ
n : ℕ
hn : n ∉ A
⊢ n - 1 ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
A B : Set ℕ
n : ℕ
hn : n ∉ A
⊢ n - 1 ≤ n
TACTIC:
|
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
⊢ n ∈ A + B | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
⊢ n ∈ A + B
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnA : n ∉ A := Set.not_mem_subset (Set.subset_add_left _ hB) h | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnB : n ∉ B := Set.not_mem_subset (Set.subset_add_right _ hA) h | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
⊢ False
TACTIC:
|
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
⊢ False
TACTIC:
|
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
⊢ False | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
⊢ False
TACTIC:
|
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
⊢ False | case inl
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : 0 ≤ countelements A 0 + countelements B 0
h : 0 ∉ A + B
hnA : 0 ∉ A
hnB : 0 ∉ B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
⊢ False
case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + count... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
⊢ False
TACTIC:
|
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ False | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ n ∈ A + B | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ False
TACTI... |
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n... | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filte... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have lem31 : (A ∩ ({n} - B) ∩ Set.Ioo 0 n).Nonempty := by
rw [← filter_and, ← 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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n... | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filte... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | obtain ⟨_, ⟨hxA, n, rfl, x, hxB, rfl⟩, hx⟩ := lem31 | case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n... | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n}... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filte... |
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 ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n}... | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n}... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = c... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact ⟨_, hxA, x, hxB, tsub_add_cancel_of_le hx.1.le⟩ | case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n}... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro.intro.intro.intro.intro.intro
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
n : ℕ
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = c... |
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 ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : 0 ≤ countelements A 0 + countelements B 0
h : 0 ∉ A + B
hnA : 0 ∉ A
hnB : 0 ∉ B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : 0 ≤ countelements A 0 + countelements B 0
h : 0 ∉ A + B
hnA : 0 ∉ A
hnB : 0 ∉ B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
⊢ False
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [countelements] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1) | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ (filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card = (filter (fun x => x ∈ B) (Icc 1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ (filter (fun x => x ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [← hfim, card_image_of_injOn] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ B) (Ioo 0 n)) = filter (fun x =... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ B) (Ioo 0 n)) = filter (fun x =... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x =>... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | congr | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ B) (Ioo 0 n)) = filter (fun x =... | case e_s.e_s.e_b
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ B) (Ioo 0 n)) ... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x =>... |
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ B) (Ioo 0 n)) ... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ B) (Ioo 0 n)) = filter (fun x =... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_s.e_s.e_b
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | ext | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) = filter (fun x => x ∈... | case a
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
a✝ : ℕ
⊢ a✝ ∈ image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) ↔ a... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ image (fun x => 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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
a✝ : ℕ
⊢ a✝ ∈ image (fun x => n - x) (filter (fun x => x ∈ B) (Ioo 0 n)) ↔ a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
a✝ : ℕ
⊢ a✝ ∈ i... |
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 ↦ (mem_Ioo.1 (mem_filter.1 hx).1).2.le) $
fun x hx y hy ↦ tsub_inj_right hx hy | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ B) (Ioo 0 n)) = filter (fun x =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x =>... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [← hca, ← hcb] at hc | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
⊢ (... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rwa [← Finset.card_pos] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (fi... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [← filter_or, ← tsub_zero n, ← Nat.card_Ioo] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (fi... |
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (fi... |
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, ← lem1, countelements] | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (fi... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | congr | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | case e_a.e_s.e_s.e_b
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).c... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (fi... |
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_s.e_s.e_b
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
h... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [← hui] at hc | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A (n - 1) + countelements B (n - 1)
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (fi... |
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnn : n ≤ (n - 1) := le_trans hip0 hip1 | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [← not_lt] at hnn | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | apply hnn | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB ... |
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc :
n ≤
(filter (fun x => x ∈ A) (Ioo 0 n) ∪ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card +
(filter (fun x => x ∈ A) (Ioo 0 n) ∩ filter (fun x => x ∈ {n} - B) (Ioo 0 n)).card
h : n ∉ A + B
hnA : n ∉ A
hnB ... |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [← filter_and, ← coe_nonempty, coe_filter, Set.setOf_and, Set.setOf_and, Set.setOf_mem_eq,
Set.inter_comm] at lem3 | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem... | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem... | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x ... |
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 ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ 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 ∈ {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x ... |
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 ↦ sumset_contains_n hA hB ?_ | A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
⊢ A + B = Set.univ | A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
⊢ n ≤ countelements A n + countelements B n | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
⊢ A + B = Set.univ
TACTIC:
|
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 ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
⊢ n ≤ countelements A n + countelements B n | case inl
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
⊢ 0 ≤ countelements A 0 + countelements B 0
case inr
A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
hn : n ≠ 0
⊢ n ≤ countelements A n + countelements B n | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
⊢ n ≤ countelements A n + countelements B n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | rw [← Nat.cast_le (α := ℝ), ← one_le_div (by positivity)] | case inr
A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
hn : n ≠ 0
⊢ n ≤ countelements A n + countelements B n | case inr
A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
hn : n ≠ 0
⊢ 1 ≤ ↑(countelements A n + countelements B n) / ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
hn : n ≠ 0
⊢ n ≤ countelements A n + countelements B n
TACTIC:
|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | calc
_ ≤ _ := hAB
_ ≤ _ := add_le_add (schnirelmannDensity_le_div hn) (schnirelmannDensity_le_div hn)
_ = _ := by push_cast; rw [add_div]; rfl | case inr
A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
hn : n ≠ 0
⊢ 1 ≤ ↑(countelements A n + countelements B n) / ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
A B : Set ℕ
n✝ : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hAB : 1 ≤ schnirelmannDensity A + schnirelmannDensity B
n : ℕ
hn : n ≠ 0
⊢ 1 ≤ ↑(countelements A n + countelements B n) / ↑n
TACTIC:
|
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