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https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | apply Filter.tendsto_add_atTop_nat | case a.refine'_3.convert_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ Filter.Tendsto (fun e => e + 2) Filter.atTop Filter.atTop | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | ext n | case h.e'_3
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
⊢ (fun e => ‖(↑e + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑e + 2)⁻¹) • x - x‖) = fun e =>
(↑e + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖ | case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ ‖(↑n + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑n + 2)⁻¹) • x - x‖ = (↑n + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rw [add_sub_assoc, sub_smul, sub_right_comm, one_smul, sub_self, zero_sub, ← smul_neg, ←
smul_add, norm_smul_of_nonneg, ← sub_eq_add_neg] | case h.e'_3.h
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ ‖(↑n + 2)⁻¹ • Finset.centroid ℝ s id + (1 - (↑n + 2)⁻¹) • x - x‖ = (↑n + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖ | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ (↑n + 2)⁻¹ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | rw [inv_nonneg] | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ (↑n + 2)⁻¹ | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ ↑n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | norm_cast | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ ↑n + 2 | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ n + 2 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | apply Nat.zero_le | case h.e'_3.h.ht
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
n : ℕ
⊢ 0 ≤ n + 2 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | subset_closure_combiInterior | [234, 1] | [288, 39] | simpa using this.mul_const _ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s t : Finset E
hs : AffineIndependent ℝ Subtype.val
x : E
hx : x ∈ s
hsnonempty : s.Nonempty
centroid_weights : ∑ i ∈ s, Finset.centroidWeights ℝ s i = 1
this : Filter.Tendsto (fun e => (↑(e + 2))⁻¹) Filter.atTop (nhds 0)
⊢ Filter.Tendsto (fun e => (↑e + 2)⁻¹ * ‖Finset.centroid ℝ s id - x‖) Filter.atTop (nhds 0) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | closure_combiInterior_eq_convexHull | [293, 1] | [299, 40] | refine'
Set.Subset.antisymm _
(convexHull_min (subset_closure_combiInterior hs) (convex_combiInterior hs).closure) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ closure (combiInterior ℝ s) = (convexHull ℝ) ↑s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | closure_combiInterior_eq_convexHull | [293, 1] | [299, 40] | rw [s.isClosed_convexHull.closure_subset_iff] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ (convexHull ℝ) ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | closure_combiInterior_eq_convexHull | [293, 1] | [299, 40] | exact combiInterior_subset_convexHull | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ (convexHull ℝ) ↑s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | rw [← closure_combiInterior_eq_convexHull hs] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → (convexHull ℝ) ↑s ⊆ (convexHull ℝ) ↑t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | rw [← closure_combiInterior_eq_convexHull ht] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ (convexHull ℝ) ↑t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | intro h | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
⊢ combiInterior ℝ s ⊆ combiInterior ℝ t → closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
h : combiInterior ℝ s ⊆ combiInterior ℝ t
⊢ closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Simplex.lean | convexHull_subset_convexHull_of_combiInterior_subset_combiInterior | [301, 1] | [308, 23] | apply closure_mono h | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Finset E
inst✝ : T2Space E
hs : AffineIndependent ℝ Subtype.val
ht : AffineIndependent ℝ Subtype.val
h : combiInterior ℝ s ⊆ combiInterior ℝ t
⊢ closure (combiInterior ℝ s) ⊆ closure (combiInterior ℝ t) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_single | [25, 1] | [26, 81] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P : R[X]
r : R
s : S
⊢ discConv (fun₀ | r => s) P = s • P.comp (X + C r) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_zero | [28, 1] | [28, 82] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P : R[X]
w : R →₀ S
⊢ discConv w 0 = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_add | [30, 1] | [31, 75] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P Q : R[X]
⊢ discConv w (P + Q) = discConv w P + discConv w Q | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sub | [33, 1] | [34, 85] | simp [discConv, smul_sub] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P Q : R[X]
⊢ discConv w (P - Q) = discConv w P - discConv w Q | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_neg | [36, 1] | [37, 18] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ discConv w (-P) = -discConv w P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum | [39, 1] | [41, 64] | simp [discConv, smul_sum, Finsupp.sum] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
s : Finset ι
P : ι → R[X]
⊢ discConv w (∑ i ∈ s, P i) = ∑ i ∈ s, discConv w (P i) | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
s : Finset ι
P : ι → R[X]
⊢ ∑ x ∈ w.support, ∑ x_1 ∈ s, w x • (P x_1).comp (X + C x) = ∑ x ∈ s, ∑ x_1 ∈ w.support, w x_1 • (P x).comp (X + C x_1) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum | [39, 1] | [41, 64] | exact Finset.sum_comm | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
s : Finset ι
P : ι → R[X]
⊢ ∑ x ∈ w.support, ∑ x_1 ∈ s, w x • (P x_1).comp (X + C x) = ∑ x ∈ s, ∑ x_1 ∈ w.support, w x_1 • (P x).comp (X + C x_1) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_zero_weight | [43, 1] | [43, 98] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
⊢ discConv 0 P = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_add_weight | [45, 1] | [47, 56] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : R →₀ S
P : R[X]
⊢ ∀ (a : R), 0 • P.comp (X + C a) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_add_weight | [45, 1] | [47, 56] | simp [add_smul] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : R →₀ S
P : R[X]
⊢ ∀ (a : R) (b₁ b₂ : S), (b₁ + b₂) • P.comp (X + C a) = b₁ • P.comp (X + C a) + b₂ • P.comp (X + C a) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sub_weight | [49, 1] | [50, 99] | simp [sub_smul] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : R →₀ S
P : R[X]
⊢ ∀ (a : R) (b₁ b₂ : S), (b₁ - b₂) • P.comp (X + C a) = b₁ • P.comp (X + C a) - b₂ • P.comp (X + C a) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_neg_weight | [52, 1] | [53, 63] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ ∀ (a : R), 0 • P.comp (X + C a) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_neg_weight | [52, 1] | [53, 63] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ (w.sum fun a b => -b • P.comp (X + C a)) = -discConv w P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum_weight | [55, 1] | [57, 56] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
s : Finset ι
w : ι → R →₀ S
P : R[X]
⊢ ∀ (i : R), 0 • P.comp (X + C i) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_sum_weight | [55, 1] | [57, 56] | simp [add_smul] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
s : Finset ι
w : ι → R →₀ S
P : R[X]
⊢ ∀ (i : R) (x y : S), (x + y) • P.comp (X + C i) = x • P.comp (X + C i) + y • P.comp (X + C i) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discConv | [59, 1] | [61, 47] | simp [discConv, Finsupp.sum,eval_finset_sum] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
x : R
⊢ eval x (discConv w P) = w.sum fun r s => s • eval (x + r) P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discForwardDiff | [63, 1] | [65, 72] | simp_rw [discForwardDiff, discConv_sub_weight, discConv_single] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x P.discForwardDiff = eval (x + 1) P - eval x P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 1) - 1 • P.comp (X + C 0)) = eval (x + 1) P - eval x P |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discForwardDiff | [63, 1] | [65, 72] | simp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 1) - 1 • P.comp (X + C 0)) = eval (x + 1) P - eval x P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discBackwardDiff | [67, 1] | [69, 90] | simp_rw [discBackwardDiff, discConv_sub_weight, discConv_single] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x P.discBackwardDiff = eval x P - eval (x - 1) P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 0) - 1 • P.comp (X + C (-1))) = eval x P - eval (x - 1) P |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.eval_discBackwardDiff | [67, 1] | [69, 90] | simp [sub_eq_add_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w : R →₀ S
P✝ P : R[X]
x : R
⊢ eval x (1 • P.comp (X + C 0) - 1 • P.comp (X + C (-1))) = eval x P - eval (x - 1) P | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_discConv | [71, 1] | [76, 8] | simp [discConv] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : AddMonoidAlgebra S R
P : R[X]
⊢ discConv v (discConv w P) = discConv (v * w) P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : AddMonoidAlgebra S R
P : R[X]
⊢ (Finsupp.sum v fun r s => s • (Finsupp.sum w fun r s => s • P.comp (X + C r)).comp (X + C r)) =
Finsupp.sum (v * w) fun r s => s • P.comp (X + C r) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discConv_discConv | [71, 1] | [76, 8] | sorry | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
v w : AddMonoidAlgebra S R
P : R[X]
⊢ (Finsupp.sum v fun r s => s • (Finsupp.sum w fun r s => s • P.comp (X + C r)).comp (X + C r)) =
Finsupp.sum (v * w) fun r s => s • P.comp (X + C r) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.coeff_discConv_natDegree | [78, 1] | [80, 8] | sorry | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra S R
w✝ : R →₀ S
P✝ : R[X]
w : R →₀ S
P : R[X]
⊢ (discConv w P).coeff P.natDegree = (w.sum fun x => id) • P.leadingCoeff | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discForwardDiff_aux | [84, 1] | [86, 62] | classical simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 1 => 1) - fun₀ | 0 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discForwardDiff_aux | [84, 1] | [86, 62] | simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 1 => 1) - fun₀ | 0 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discBackwardDiff_aux | [88, 1] | [90, 62] | classical simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 0 => 1) - fun₀ | -1 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.discBackwardDiff_aux | [88, 1] | [90, 62] | simp [Finsupp.sum_sub_index, -Finsupp.single_neg] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra S R
w : R →₀ S
P : R[X]
inst✝ : Nontrivial S
⊢ (((fun₀ | 0 => 1) - fun₀ | -1 => 1).sum fun x => id) = 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | refine' (natDegree_sum_le _ _).trans $ Finset.sup_le fun r _ ↦ _ | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w P).natDegree ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (natDegree ∘ fun a => (fun r s => s • P.comp (X + C r)) a (w a)) r ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | dsimp | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (natDegree ∘ fun a => (fun r s => s • P.comp (X + C r)) a (w a)) r ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (w r • P.comp (X + C r)).natDegree ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | simp_rw [algebra_compatible_smul R (w r)] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (w r • P.comp (X + C r)).natDegree ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ ((algebraMap S R) (w r) • P.comp (X + C r)).natDegree ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | refine (natDegree_smul_le _ _).trans ?_ | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ ((algebraMap S R) (w r) • P.comp (X + C r)).natDegree ≤ P.natDegree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (P.comp (X + C r)).natDegree ≤ P.natDegree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.natDegree_discConv_le | [94, 1] | [99, 24] | simp [natDegree_comp] | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
r : R
x✝ : r ∈ w.support
⊢ (P.comp (X + C r)).natDegree ≤ P.natDegree | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | obtain rfl | hP := eq_or_ne P 0 | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w P).degree ≤ P.degree | case inl
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w 0).degree ≤ degree 0
case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ (discConv w P).degree ≤ P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | refine degree_le_natDegree.trans $ (Nat.cast_le.2 natDegree_discConv_le).trans ?_ | case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ (discConv w P).degree ≤ P.degree | case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ ↑P.natDegree ≤ P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | rw [degree_eq_natDegree hP] | case inr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hP : P ≠ 0
⊢ ↑P.natDegree ≤ P.degree | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_le | [101, 1] | [105, 30] | simp | case inl
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ (discConv w 0).degree ≤ degree 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | have := coeff_discConv_natDegree w P | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
⊢ (discConv w P).degree < P.degree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = (w.sum fun x => id) • P.leadingCoeff
⊢ (discConv w P).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | rw [hw, zero_smul] at this | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = (w.sum fun x => id) • P.leadingCoeff
⊢ (discConv w P).degree < P.degree | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ (discConv w P).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | refine' (degree_sum_le _ _).trans_lt $ (Finset.sup_lt_iff _).2 _ | ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ (discConv w P).degree < P.degree | case refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ⊥ < P.degree
case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | sorry | case refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ⊥ < P.degree
case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree | case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/DiscreteDeriv.lean | Polynomial.degree_discConv_lt | [108, 1] | [115, 8] | sorry | case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra S R
w : R →₀ S
P : R[X]
inst✝² : Nontrivial S
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hw : (w.sum fun x => id) = 0
hP : P ≠ 0
this : (discConv w P).coeff P.natDegree = 0
⊢ ∀ b ∈ w.support, ((fun r s => s • P.comp (X + C r)) b (w b)).degree < P.degree | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | have :
((univ : Finset β) : Set β).PairwiseDisjoint fun b ↦
univ.filter fun a : α ↦ g a = b := by
refine' fun b₁ _ b₂ _ hb ↦ disjoint_left.2 fun a ha₁ ha₂ ↦ _
simp only [mem_filter, mem_univ, true_and_iff] at ha₁ ha₂
exact hb (ha₁.symm.trans ha₂) | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
⊢ ∑ a : β, (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) a = 1 | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ a : β, (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) a = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | rw [← sum_disjiUnion _ _ this] | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ a : β, (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) a = 1 | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ x ∈ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this, f x = 1 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | convert h | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ ∑ x ∈ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this, f x = 1 | case h.e'_2.h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this = univ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | exact eq_univ_of_forall fun a ↦
mem_disjiUnion.2 ⟨_, mem_univ _, mem_filter.2 ⟨mem_univ _, rfl⟩⟩ | case h.e'_2.h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
this : (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ
⊢ univ.disjiUnion (fun b => Finset.filter (fun a => g a = b) univ) this = univ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | refine' fun b₁ _ b₂ _ hb ↦ disjoint_left.2 fun a ha₁ ha₂ ↦ _ | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
⊢ (↑univ).PairwiseDisjoint fun b => Finset.filter (fun a => g a = b) univ | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₁
ha₂ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₂
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | simp only [mem_filter, mem_univ, true_and_iff] at ha₁ ha₂ | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₁
ha₂ : a ∈ (fun b => Finset.filter (fun a => g a = b) univ) b₂
⊢ False | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : g a = b₁
ha₂ : g a = b₂
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | exact hb (ha₁.symm.trans ha₂) | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b₁ : β
x✝¹ : b₁ ∈ ↑univ
b₂ : β
x✝ : b₂ ∈ ↑univ
hb : b₁ ≠ b₂
a : α
ha₁ : g a = b₁
ha₂ : g a = b₂
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | ext b : 1 | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
⊢ map g (ofFintype f h) = ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) ⋯ | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (map g (ofFintype f h)) b = (ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) ⋯) b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | simp only [sum_filter, eq_comm, map_apply, ofFintype_apply] | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (map g (ofFintype f h)) b = (ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => g a = b) univ, f a) ⋯) b | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (∑' (a : α), if b = g a then f a else 0) = ∑ a : α, if b = g a then f a else 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_ofFintype | [16, 1] | [33, 52] | exact tsum_eq_sum fun _ h ↦ (h $ mem_univ _).elim | case h
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
f : α → ℝ≥0∞
h : ∑ a : α, f a = 1
g : α → β
b : β
⊢ (∑' (a : α), if b = g a then f a else 0) = ∑ a : α, if b = g a then f a else 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_toMeasure | [41, 1] | [43, 82] | ext s hs : 1 | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : PMF α
hf : Measurable f
⊢ Measure.map f p.toMeasure = (map f p).toMeasure | case h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : PMF α
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (Measure.map f p.toMeasure) s = (map f p).toMeasure s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_toMeasure | [41, 1] | [43, 82] | rw [PMF.toMeasure_map_apply _ _ _ hf hs, Measure.map_apply hf hs] | case h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : PMF α
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (Measure.map f p.toMeasure) s = (map f p).toMeasure s | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.support_bernoulli' | [56, 1] | [57, 80] | by_cases b <;> simp [*, hp.lt_iff_ne, tsub_eq_zero_iff_le] | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b✝ b : Prop
⊢ b ∈ (bernoulli' p hp).support ↔ b ∈ {b | if b then p ≠ 0 else p ≠ 1} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.mem_support_bernoulli'_iff | [59, 1] | [60, 7] | simp | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
⊢ b ∈ (bernoulli' p hp).support ↔ if b then p ≠ 0 else p ≠ 1 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | have : ∀ p : Prop, (Finset.univ.filter fun q : Prop ↦ ¬ q ↔ p) = {¬ p} := by
rintro p
ext q
by_cases p <;> by_cases q <;> simp [*] | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
⊢ map Not (bernoulli' p hp) = bernoulli' (1 - p) ⋯ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ map Not (bernoulli' p hp) = bernoulli' (1 - p) ⋯ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | refine' (map_ofFintype _ _ _).trans _ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ map Not (bernoulli' p hp) = bernoulli' (1 - p) ⋯ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => (¬a) = b) Finset.univ, if a then ↑p else 1 - ↑p) ⋯ =
bernoulli' (1 - p) ⋯ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | simp only [this, bernoulli', Finset.mem_filter, Finset.mem_univ, true_and_iff,
Finset.mem_disjiUnion, tsub_le_self, eq_iff_iff, Finset.sum_singleton, WithTop.coe_sub,
ENNReal.coe_one] | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => ∑ a ∈ Finset.filter (fun a => (¬a) = b) Finset.univ, if a then ↑p else 1 - ↑p) ⋯ =
bernoulli' (1 - p) ⋯ | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => if ¬b then ↑p else 1 - ↑p) ⋯ = ofFintype (fun b => if b then ↑(1 - p) else 1 - ↑(1 - p)) ⋯ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | congr 1 with q | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
⊢ ofFintype (fun b => if ¬b then ↑p else 1 - ↑p) ⋯ = ofFintype (fun b => if b then ↑(1 - p) else 1 - ↑(1 - p)) ⋯ | case e_f.h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
q : Prop
⊢ (if ¬q then ↑p else 1 - ↑p) = if q then ↑(1 - p) else 1 - ↑(1 - p) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | split_ifs <;> simp [ENNReal.sub_sub_cancel WithTop.one_ne_top (ENNReal.coe_le_coe.2 hp)] | case e_f.h
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
this : ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p}
q : Prop
⊢ (if ¬q then ↑p else 1 - ↑p) = if q then ↑(1 - p) else 1 - ↑(1 - p) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | rintro p | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p : ℝ≥0
hp : p ≤ 1
b : Prop
⊢ ∀ (p : Prop), Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p} | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p : Prop
⊢ Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | ext q | α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p : Prop
⊢ Finset.filter (fun q => ¬q ↔ p) Finset.univ = {¬p} | case a
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p q : Prop
⊢ q ∈ Finset.filter (fun q => ¬q ↔ p) Finset.univ ↔ q ∈ {¬p} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | PMF.map_not_bernoulli' | [62, 1] | [72, 91] | by_cases p <;> by_cases q <;> simp [*] | case a
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
p✝ : ℝ≥0
hp : p✝ ≤ 1
b p q : Prop
⊢ q ∈ Finset.filter (fun q => ¬q ↔ p) Finset.univ ↔ q ∈ {¬p} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.map_id | [17, 1] | [17, 99] | ext <;> simp | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H : SimpleGraph V
G : SimpleGraph α
G' : G.Subgraph
⊢ Subgraph.map Hom.id G' = G' | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.map_comp | [19, 1] | [21, 30] | ext <;> simp [Subgraph.map] | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
I : SimpleGraph γ
G' : G.Subgraph
f : G →g H
g : H →g I
⊢ Subgraph.map (g.comp f) G' = Subgraph.map g (Subgraph.map f G') | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_map | [23, 1] | [31, 35] | ext e | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
⊢ (Subgraph.map f G').edgeSet = Sym2.map ⇑f '' G'.edgeSet | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
e : Sym2 β
⊢ e ∈ (Subgraph.map f G').edgeSet ↔ e ∈ Sym2.map ⇑f '' G'.edgeSet |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_map | [23, 1] | [31, 35] | induction' e using Sym2.ind with a b | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
e : Sym2 β
⊢ e ∈ (Subgraph.map f G').edgeSet ↔ e ∈ Sym2.map ⇑f '' G'.edgeSet | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : β
⊢ s(a, b) ∈ (Subgraph.map f G').edgeSet ↔ s(a, b) ∈ Sym2.map ⇑f '' G'.edgeSet |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_map | [23, 1] | [31, 35] | simp only [mem_edgeSet, Sym2.exists, Relation.Map, and_or_left, exists_or, map_adj,
Set.mem_image, Sym2.map_pair_eq, Sym2.eq, Sym2.rel_iff] | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : β
⊢ s(a, b) ∈ (Subgraph.map f G').edgeSet ↔ s(a, b) ∈ Sym2.map ⇑f '' G'.edgeSet | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : β
⊢ (∃ a_1 b_1, G'.Adj a_1 b_1 ∧ f a_1 = a ∧ f b_1 = b) ↔
(∃ a_1 b_1, G'.Adj a_1 b_1 ∧ f a_1 = a ∧ f b_1 = b) ∨ ∃ x x_1, G'.Adj x x_1 ∧ f x = b ∧ f x_1 = a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_map | [23, 1] | [31, 35] | refine' (or_iff_left_of_imp _).symm | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : β
⊢ (∃ a_1 b_1, G'.Adj a_1 b_1 ∧ f a_1 = a ∧ f b_1 = b) ↔
(∃ a_1 b_1, G'.Adj a_1 b_1 ∧ f a_1 = a ∧ f b_1 = b) ∨ ∃ x x_1, G'.Adj x x_1 ∧ f x = b ∧ f x_1 = a | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : β
⊢ (∃ x x_1, G'.Adj x x_1 ∧ f x = b ∧ f x_1 = a) → ∃ a_2 b_1, G'.Adj a_2 b_1 ∧ f a_2 = a ∧ f b_1 = b |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_map | [23, 1] | [31, 35] | rintro ⟨a, b, hab, rfl, rfl⟩ | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : β
⊢ (∃ x x_1, G'.Adj x x_1 ∧ f x = b ∧ f x_1 = a) → ∃ a_2 b_1, G'.Adj a_2 b_1 ∧ f a_2 = a ∧ f b_1 = b | case h.h.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : α
hab : G'.Adj a b
⊢ ∃ a_1 b_1, G'.Adj a_1 b_1 ∧ f a_1 = f b ∧ f b_1 = f a |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_map | [23, 1] | [31, 35] | exact ⟨b, a, hab.symm, rfl, rfl⟩ | case h.h.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G✝ H✝ : SimpleGraph V
G : SimpleGraph α
H : SimpleGraph β
f : G →g H
G' : G.Subgraph
a b : α
hab : G'.Adj a b
⊢ ∃ a_1 b_1, G'.Adj a_1 b_1 ∧ f a_1 = f b ∧ f b_1 = f a | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_coe | [33, 1] | [34, 52] | ext e | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
⊢ G'.coe.edgeSet = Sym2.map Subtype.val ⁻¹' G'.edgeSet | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
e : Sym2 ↑G'.verts
⊢ e ∈ G'.coe.edgeSet ↔ e ∈ Sym2.map Subtype.val ⁻¹' G'.edgeSet |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_coe | [33, 1] | [34, 52] | induction' e using Sym2.ind with a b | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
e : Sym2 ↑G'.verts
⊢ e ∈ G'.coe.edgeSet ↔ e ∈ Sym2.map Subtype.val ⁻¹' G'.edgeSet | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
a b : ↑G'.verts
⊢ s(a, b) ∈ G'.coe.edgeSet ↔ s(a, b) ∈ Sym2.map Subtype.val ⁻¹' G'.edgeSet |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.edgeSet_coe | [33, 1] | [34, 52] | simp | case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
a b : ↑G'.verts
⊢ s(a, b) ∈ G'.coe.edgeSet ↔ s(a, b) ∈ Sym2.map Subtype.val ⁻¹' G'.edgeSet | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.image_coe_edgeSet_coe | [36, 1] | [41, 83] | rw [edgeSet_coe, Set.image_preimage_eq_iff] | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
⊢ Sym2.map Subtype.val '' G'.coe.edgeSet = G'.edgeSet | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
⊢ G'.edgeSet ⊆ Set.range (Sym2.map Subtype.val) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.image_coe_edgeSet_coe | [36, 1] | [41, 83] | rintro e he | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
⊢ G'.edgeSet ⊆ Set.range (Sym2.map Subtype.val) | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
e : Sym2 V
he : e ∈ G'.edgeSet
⊢ e ∈ Set.range (Sym2.map Subtype.val) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.image_coe_edgeSet_coe | [36, 1] | [41, 83] | induction' e using Sym2.ind with a b | α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
e : Sym2 V
he : e ∈ G'.edgeSet
⊢ e ∈ Set.range (Sym2.map Subtype.val) | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
a b : V
he : s(a, b) ∈ G'.edgeSet
⊢ s(a, b) ∈ Set.range (Sym2.map Subtype.val) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.image_coe_edgeSet_coe | [36, 1] | [41, 83] | rw [Subgraph.mem_edgeSet] at he | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
a b : V
he : s(a, b) ∈ G'.edgeSet
⊢ s(a, b) ∈ Set.range (Sym2.map Subtype.val) | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
a b : V
he : G'.Adj a b
⊢ s(a, b) ∈ Set.range (Sym2.map Subtype.val) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.Subgraph.image_coe_edgeSet_coe | [36, 1] | [41, 83] | exact ⟨s(⟨a, edge_vert _ he⟩, ⟨b, edge_vert _ he.symm⟩), Sym2.map_pair_eq _ _ _⟩ | case h
α : Type u_1
β : Type u_2
γ : Type u_3
V : Type u_4
G H : SimpleGraph V
G' : G.Subgraph
a b : V
he : G'.Adj a b
⊢ s(a, b) ∈ Set.range (Sym2.map Subtype.val) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/RingTheory/Int/Basic.lean | Nat.prime_composite_induction | [5, 1] | [12, 69] | refine induction_on_primes zero one ?_ _ | P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n : ℕ
⊢ P n | P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n : ℕ
⊢ ∀ (p a : ℕ), p.Prime → P a → P (p * a) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/RingTheory/Int/Basic.lean | Nat.prime_composite_induction | [5, 1] | [12, 69] | rintro p (_ | _ | a) hp ha | P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n : ℕ
⊢ ∀ (p a : ℕ), p.Prime → P a → P (p * a) | case zero
P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n p : ℕ
hp : p.Prime
ha : P 0
⊢ P (p * 0)
case succ.zero
P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n p : ℕ
hp : p.Prime
ha : P (0 + 1)
⊢ P (p * (0 + 1))
case succ.succ
P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n p a : ℕ
hp : p.Prime
ha : P (a + 1 + 1)
⊢ P (p * (a + 1 + 1)) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/RingTheory/Int/Basic.lean | Nat.prime_composite_induction | [5, 1] | [12, 69] | simpa | case zero
P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n p : ℕ
hp : p.Prime
ha : P 0
⊢ P (p * 0) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/RingTheory/Int/Basic.lean | Nat.prime_composite_induction | [5, 1] | [12, 69] | simpa using prime _ hp | case succ.zero
P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n p : ℕ
hp : p.Prime
ha : P (0 + 1)
⊢ P (p * (0 + 1)) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/RingTheory/Int/Basic.lean | Nat.prime_composite_induction | [5, 1] | [12, 69] | exact composite _ hp.two_le (prime _ hp) _ a.one_lt_succ_succ ha | case succ.succ
P : ℕ → Prop
zero : P 0
one : P 1
prime : ∀ (p : ℕ), p.Prime → P p
composite : ∀ (a : ℕ), 2 ≤ a → P a → ∀ (b : ℕ), 2 ≤ b → P b → P (a * b)
n p a : ℕ
hp : p.Prime
ha : P (a + 1 + 1)
⊢ P (p * (a + 1 + 1)) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Independence.lean | AffineIndependent.convexIndependent | [11, 1] | [15, 8] | intro s x hx | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
⊢ ConvexIndependent 𝕜 p | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
⊢ x ∈ s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Analysis/Convex/Independence.lean | AffineIndependent.convexIndependent | [11, 1] | [15, 8] | by_contra | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
⊢ x ∈ s | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hp : AffineIndependent 𝕜 p
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
x✝ : x ∉ s
⊢ False |
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