file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Algebra/IndicatorFunction.lean
|
Set.piecewise_eq_mulIndicator
|
[] |
[
65,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.comap_top
|
[] |
[
1579,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1578,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.zero_mem
|
[] |
[
348,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
11
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lift_id
|
[] |
[
713,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
712,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
Function.Injective.wOppSide_map_iff
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nh : WOppSide (map f s) (↑f x) (↑f y)\n⊢ WOppSide s x y",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\n⊢ WOppSide (map f s) (↑f x) (↑f y) ↔ WOppSide s x y",
"tactic": "refine' ⟨fun h => _, fun h => h.map _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nfp₁ : P'\nhfp₁ : fp₁ ∈ map f s\nfp₂ : P'\nhfp₂ : fp₂ ∈ map f s\nh : SameRay R (↑f x -ᵥ fp₁) (fp₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nh : WOppSide (map f s) (↑f x) (↑f y)\n⊢ WOppSide s x y",
"tactic": "rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nfp₁ : P'\nhfp₁ : ∃ y, y ∈ s ∧ ↑f y = fp₁\nfp₂ : P'\nhfp₂ : ∃ y, y ∈ s ∧ ↑f y = fp₂\nh : SameRay R (↑f x -ᵥ fp₁) (fp₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nfp₁ : P'\nhfp₁ : fp₁ ∈ map f s\nfp₂ : P'\nhfp₂ : fp₂ ∈ map f s\nh : SameRay R (↑f x -ᵥ fp₁) (fp₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"tactic": "rw [mem_map] at hfp₁ hfp₂"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nfp₂ : P'\nhfp₂ : ∃ y, y ∈ s ∧ ↑f y = fp₂\np₁ : P\nhp₁ : p₁ ∈ s\nh : SameRay R (↑f x -ᵥ ↑f p₁) (fp₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nfp₁ : P'\nhfp₁ : ∃ y, y ∈ s ∧ ↑f y = fp₁\nfp₂ : P'\nhfp₂ : ∃ y, y ∈ s ∧ ↑f y = fp₂\nh : SameRay R (↑f x -ᵥ fp₁) (fp₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"tactic": "rcases hfp₁ with ⟨p₁, hp₁, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (↑f x -ᵥ ↑f p₁) (↑f p₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\nfp₂ : P'\nhfp₂ : ∃ y, y ∈ s ∧ ↑f y = fp₂\np₁ : P\nhp₁ : p₁ ∈ s\nh : SameRay R (↑f x -ᵥ ↑f p₁) (fp₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"tactic": "rcases hfp₂ with ⟨p₂, hp₂, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (↑f x -ᵥ ↑f p₁) (↑f p₂ -ᵥ ↑f y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (↑f x -ᵥ ↑f p₁) (↑f p₂ -ᵥ ↑f y)\n⊢ WOppSide s x y",
"tactic": "refine' ⟨p₁, hp₁, p₂, hp₂, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (↑f x -ᵥ ↑f p₁) (↑f p₂ -ᵥ ↑f y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_4\nP : Type u_3\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ↑f\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)",
"tactic": "exact h"
}
] |
[
123,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.totalDegree_rename_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\n⊢ b ∈ support (↑(rename f) p) → (sum b fun x e => e) ≤ totalDegree p",
"tactic": "classical\nintro h\nrw [rename_eq] at h\nhave h' := Finsupp.mapDomain_support h\nrw [Finset.mem_image] at h'\nrcases h' with ⟨s, hs, rfl⟩\nrw [Finsupp.sum_mapDomain_index]\nexact le_trans le_rfl (le_totalDegree hs)\nexact fun _ => rfl\nexact fun _ _ _ => rfl"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (↑(rename f) p)\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\n⊢ b ∈ support (↑(rename f) p) → (sum b fun x e => e) ≤ totalDegree p",
"tactic": "intro h"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (↑(rename f) p)\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"tactic": "rw [rename_eq] at h"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\nh' : b ∈ Finset.image (Finsupp.mapDomain f) p.support\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"tactic": "have h' := Finsupp.mapDomain_support h"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\nh' : ∃ a, a ∈ p.support ∧ Finsupp.mapDomain f a = b\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\nh' : b ∈ Finset.image (Finsupp.mapDomain f) p.support\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"tactic": "rw [Finset.mem_image] at h'"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ (sum (Finsupp.mapDomain f s) fun x e => e) ≤ totalDegree p",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\nb : τ →₀ ℕ\nh : b ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\nh' : ∃ a, a ∈ p.support ∧ Finsupp.mapDomain f a = b\n⊢ (sum b fun x e => e) ≤ totalDegree p",
"tactic": "rcases h' with ⟨s, hs, rfl⟩"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ (sum s fun a m => m) ≤ totalDegree p\n\ncase intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → 0 = 0\n\ncase intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → ∀ (m₁ m₂ : ℕ), m₁ + m₂ = m₁ + m₂",
"state_before": "case intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ (sum (Finsupp.mapDomain f s) fun x e => e) ≤ totalDegree p",
"tactic": "rw [Finsupp.sum_mapDomain_index]"
},
{
"state_after": "case intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → 0 = 0\n\ncase intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → ∀ (m₁ m₂ : ℕ), m₁ + m₂ = m₁ + m₂",
"state_before": "case intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ (sum s fun a m => m) ≤ totalDegree p\n\ncase intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → 0 = 0\n\ncase intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → ∀ (m₁ m₂ : ℕ), m₁ + m₂ = m₁ + m₂",
"tactic": "exact le_trans le_rfl (le_totalDegree hs)"
},
{
"state_after": "case intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → ∀ (m₁ m₂ : ℕ), m₁ + m₂ = m₁ + m₂",
"state_before": "case intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → 0 = 0\n\ncase intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → ∀ (m₁ m₂ : ℕ), m₁ + m₂ = m₁ + m₂",
"tactic": "exact fun _ => rfl"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q : MvPolynomial σ R\nf : σ → τ\np : MvPolynomial σ R\ns : σ →₀ ℕ\nhs : s ∈ p.support\nh : Finsupp.mapDomain f s ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) p)\n⊢ τ → ∀ (m₁ m₂ : ℕ), m₁ + m₂ = m₁ + m₂",
"tactic": "exact fun _ _ _ => rfl"
}
] |
[
795,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
783,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.zero_le_lf
|
[
{
"state_after": "x : PGame\n⊢ ((∀ (i : LeftMoves 0), moveLeft 0 i ⧏ x) ∧ ∀ (j : RightMoves x), 0 ⧏ moveRight x j) ↔\n ∀ (j : RightMoves x), 0 ⧏ moveRight x j",
"state_before": "x : PGame\n⊢ 0 ≤ x ↔ ∀ (j : RightMoves x), 0 ⧏ moveRight x j",
"tactic": "rw [le_iff_forall_lf]"
},
{
"state_after": "no goals",
"state_before": "x : PGame\n⊢ ((∀ (i : LeftMoves 0), moveLeft 0 i ⧏ x) ∧ ∀ (j : RightMoves x), 0 ⧏ moveRight x j) ↔\n ∀ (j : RightMoves x), 0 ⧏ moveRight x j",
"tactic": "simp"
}
] |
[
645,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
643,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.map_matrix_volume_pi_eq_smul_volume_pi
|
[
{
"state_after": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\n⊢ ∀ (D : ι → ℝ),\n det (Matrix.diagonal D) ≠ 0 →\n Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume = ofReal (abs (det (Matrix.diagonal D))⁻¹) • volume\n\ncase htransvec\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\n⊢ ∀ (t : TransvectionStruct ι ℝ),\n Measure.map (↑(↑toLin' (TransvectionStruct.toMatrix t))) volume =\n ofReal (abs (det (TransvectionStruct.toMatrix t))⁻¹) • volume\n\ncase hmul\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\n⊢ ∀ (A B : Matrix ι ι ℝ),\n det A ≠ 0 →\n det B ≠ 0 →\n Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume →\n Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume →\n Measure.map (↑(↑toLin' (A ⬝ B))) volume = ofReal (abs (det (A ⬝ B))⁻¹) • volume",
"state_before": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\n⊢ Measure.map (↑(↑toLin' M)) volume = ofReal (abs (det M)⁻¹) • volume",
"tactic": "apply diagonal_transvection_induction_of_det_ne_zero _ M hM"
},
{
"state_after": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nD : ι → ℝ\nhD : det (Matrix.diagonal D) ≠ 0\n⊢ Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume = ofReal (abs (det (Matrix.diagonal D))⁻¹) • volume",
"state_before": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\n⊢ ∀ (D : ι → ℝ),\n det (Matrix.diagonal D) ≠ 0 →\n Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume = ofReal (abs (det (Matrix.diagonal D))⁻¹) • volume",
"tactic": "intro D hD"
},
{
"state_after": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nD : ι → ℝ\nhD : det (Matrix.diagonal D) ≠ 0\n⊢ Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume =\n ofReal (abs (det (Matrix.diagonal D))⁻¹) •\n ofReal (abs (det (Matrix.diagonal D))) • Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume",
"state_before": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nD : ι → ℝ\nhD : det (Matrix.diagonal D) ≠ 0\n⊢ Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume = ofReal (abs (det (Matrix.diagonal D))⁻¹) • volume",
"tactic": "conv_rhs => rw [← smul_map_diagonal_volume_pi hD]"
},
{
"state_after": "no goals",
"state_before": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nD : ι → ℝ\nhD : det (Matrix.diagonal D) ≠ 0\n⊢ Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume =\n ofReal (abs (det (Matrix.diagonal D))⁻¹) •\n ofReal (abs (det (Matrix.diagonal D))) • Measure.map (↑(↑toLin' (Matrix.diagonal D))) volume",
"tactic": "rw [smul_smul, ← ENNReal.ofReal_mul (abs_nonneg _), ← abs_mul, inv_mul_cancel hD, abs_one,\n ENNReal.ofReal_one, one_smul]"
},
{
"state_after": "case htransvec\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nt : TransvectionStruct ι ℝ\n⊢ Measure.map (↑(↑toLin' (TransvectionStruct.toMatrix t))) volume =\n ofReal (abs (det (TransvectionStruct.toMatrix t))⁻¹) • volume",
"state_before": "case htransvec\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\n⊢ ∀ (t : TransvectionStruct ι ℝ),\n Measure.map (↑(↑toLin' (TransvectionStruct.toMatrix t))) volume =\n ofReal (abs (det (TransvectionStruct.toMatrix t))⁻¹) • volume",
"tactic": "intro t"
},
{
"state_after": "no goals",
"state_before": "case htransvec\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nt : TransvectionStruct ι ℝ\n⊢ Measure.map (↑(↑toLin' (TransvectionStruct.toMatrix t))) volume =\n ofReal (abs (det (TransvectionStruct.toMatrix t))⁻¹) • volume",
"tactic": "simp only [Matrix.TransvectionStruct.det, ENNReal.ofReal_one,\n (volume_preserving_transvectionStruct _).map_eq, one_smul, _root_.inv_one, abs_one]"
},
{
"state_after": "case hmul\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Measure.map (↑(↑toLin' (A ⬝ B))) volume = ofReal (abs (det (A ⬝ B))⁻¹) • volume",
"state_before": "case hmul\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\n⊢ ∀ (A B : Matrix ι ι ℝ),\n det A ≠ 0 →\n det B ≠ 0 →\n Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume →\n Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume →\n Measure.map (↑(↑toLin' (A ⬝ B))) volume = ofReal (abs (det (A ⬝ B))⁻¹) • volume",
"tactic": "intro A B _ _ IHA IHB"
},
{
"state_after": "case hmul.hg\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Measurable ↑(↑toLin' A)\n\ncase hmul.hf\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Measurable ↑(↑toLin' B)",
"state_before": "case hmul\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Measure.map (↑(↑toLin' (A ⬝ B))) volume = ofReal (abs (det (A ⬝ B))⁻¹) • volume",
"tactic": "rw [toLin'_mul, det_mul, LinearMap.coe_comp, ← Measure.map_map, IHB, Measure.map_smul, IHA,\n smul_smul, ← ENNReal.ofReal_mul (abs_nonneg _), ← abs_mul, mul_comm, mul_inv]"
},
{
"state_after": "case hmul.hg.hf\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Continuous ↑(↑toLin' A)",
"state_before": "case hmul.hg\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Measurable ↑(↑toLin' A)",
"tactic": "apply Continuous.measurable"
},
{
"state_after": "no goals",
"state_before": "case hmul.hg.hf\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Continuous ↑(↑toLin' A)",
"tactic": "apply LinearMap.continuous_on_pi"
},
{
"state_after": "case hmul.hf.hf\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Continuous ↑(↑toLin' B)",
"state_before": "case hmul.hf\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Measurable ↑(↑toLin' B)",
"tactic": "apply Continuous.measurable"
},
{
"state_after": "no goals",
"state_before": "case hmul.hf.hf\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : det M ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : det A ≠ 0\na✝ : det B ≠ 0\nIHA : Measure.map (↑(↑toLin' A)) volume = ofReal (abs (det A)⁻¹) • volume\nIHB : Measure.map (↑(↑toLin' B)) volume = ofReal (abs (det B)⁻¹) • volume\n⊢ Continuous ↑(↑toLin' B)",
"tactic": "apply LinearMap.continuous_on_pi"
}
] |
[
444,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
IsLocalization.integerNormalization_aeval_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁸ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Type ?u.52206\ninst✝⁵ : CommRing P\ninst✝⁴ : IsLocalization M S\nR' : Type u_2\ninst✝³ : CommRing R'\ninst✝² : Algebra R R'\ninst✝¹ : Algebra S R'\ninst✝ : IsScalarTower R S R'\np : S[X]\nx : R'\nhx : ↑(aeval x) p = 0\n⊢ ↑(aeval x) (integerNormalization M p) = 0",
"tactic": "rw [aeval_def, IsScalarTower.algebraMap_eq R S R', integerNormalization_eval₂_eq_zero _ _ _ hx]"
}
] |
[
120,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/GroupTheory/GroupAction/Sum.lean
|
Sum.smul_def
|
[] |
[
42,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.of_bot
|
[] |
[
528,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_map
|
[] |
[
602,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Data/Int/Units.lean
|
Int.isUnit_iff
|
[
{
"state_after": "a : ℤ\nh : a = 1 ∨ a = -1\n⊢ IsUnit a",
"state_before": "a : ℤ\n⊢ IsUnit a ↔ a = 1 ∨ a = -1",
"tactic": "refine' ⟨fun h => isUnit_eq_one_or h, fun h => _⟩"
},
{
"state_after": "case inl\n\n⊢ IsUnit 1\n\ncase inr\n\n⊢ IsUnit (-1)",
"state_before": "a : ℤ\nh : a = 1 ∨ a = -1\n⊢ IsUnit a",
"tactic": "rcases h with (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\n\n⊢ IsUnit 1",
"tactic": "exact isUnit_one"
},
{
"state_after": "no goals",
"state_before": "case inr\n\n⊢ IsUnit (-1)",
"tactic": "exact isUnit_one.neg"
}
] |
[
44,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.top_toSubmonoid
|
[] |
[
892,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_eq_dif
|
[
{
"state_after": "case inl\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : p\n⊢ (∏ᶠ (i : p), f i) = f h\n\ncase inr\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : ¬p\n⊢ (∏ᶠ (i : p), f i) = 1",
"state_before": "G : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\n⊢ (∏ᶠ (i : p), f i) = if h : p then f h else 1",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : p\nthis : Unique p\n⊢ (∏ᶠ (i : p), f i) = f h",
"state_before": "case inl\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : p\n⊢ (∏ᶠ (i : p), f i) = f h",
"tactic": "haveI : Unique p := ⟨⟨h⟩, fun _ => rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : p\nthis : Unique p\n⊢ (∏ᶠ (i : p), f i) = f h",
"tactic": "exact finprod_unique f"
},
{
"state_after": "case inr\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : ¬p\nthis : IsEmpty p\n⊢ (∏ᶠ (i : p), f i) = 1",
"state_before": "case inr\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : ¬p\n⊢ (∏ᶠ (i : p), f i) = 1",
"tactic": "haveI : IsEmpty p := ⟨h⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nG : Type ?u.38544\nM : Type u_1\nN : Type ?u.38550\nα : Sort ?u.38553\nβ : Sort ?u.38556\nι : Sort ?u.38559\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\np : Prop\ninst✝ : Decidable p\nf : p → M\nh : ¬p\nthis : IsEmpty p\n⊢ (∏ᶠ (i : p), f i) = 1",
"tactic": "exact finprod_of_isEmpty f"
}
] |
[
247,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
of_real_exp_ℝ_ℝ
|
[] |
[
674,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
673,
1
] |
Mathlib/Data/Nat/Cast/Defs.lean
|
Nat.cast_bit1
|
[
{
"state_after": "R : Type u_1\ninst✝ : AddMonoidWithOne R\nn : ℕ\n⊢ bit0 ↑n + 1 = bit1 ↑n",
"state_before": "R : Type u_1\ninst✝ : AddMonoidWithOne R\nn : ℕ\n⊢ ↑(bit1 n) = bit1 ↑n",
"tactic": "rw [bit1, cast_add_one, cast_bit0]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : AddMonoidWithOne R\nn : ℕ\n⊢ bit0 ↑n + 1 = bit1 ↑n",
"tactic": "rfl"
}
] |
[
181,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean
|
dist_inv_inv
|
[] |
[
401,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
ContinuousAt.div'
|
[] |
[
1128,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1126,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.map₂_coe
|
[] |
[
216,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
NNReal.sqrt_le_iff
|
[] |
[
68,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean
|
SimplexCategory.δ_comp_δ
|
[
{
"state_after": "case a.h.h.h\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\nk : Fin (len [n] + 1)\n⊢ ↑(↑(Hom.toOrderHom (δ i ≫ δ (Fin.succ j))) k) = ↑(↑(Hom.toOrderHom (δ j ≫ δ (↑Fin.castSucc i))) k)",
"state_before": "n : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\n⊢ δ i ≫ δ (Fin.succ j) = δ j ≫ δ (↑Fin.castSucc i)",
"tactic": "ext k"
},
{
"state_after": "case a.h.h.h\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\nk : Fin (len [n] + 1)\n⊢ ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k) < Fin.succ j then\n ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k) < ↑Fin.castSucc i then\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k))",
"state_before": "case a.h.h.h\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\nk : Fin (len [n] + 1)\n⊢ ↑(↑(Hom.toOrderHom (δ i ≫ δ (Fin.succ j))) k) = ↑(↑(Hom.toOrderHom (δ j ≫ δ (↑Fin.castSucc i))) k)",
"tactic": "dsimp [δ, Fin.succAbove]"
},
{
"state_after": "case a.h.h.h\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\nk : Fin (len [n] + 1)\n⊢ ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k) < Fin.succ j then\n ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k) < ↑Fin.castSucc i then\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k))",
"state_before": "case a.h.h.h\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\nk : Fin (len [n] + 1)\n⊢ ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k) < Fin.succ j then\n ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k) < ↑Fin.castSucc i then\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k))",
"tactic": "simp only [OrderEmbedding.toOrderHom_coe, OrderEmbedding.coe_ofStrictMono, Function.comp_apply,\n SimplexCategory.Hom.toOrderHom_mk, OrderHom.comp_coe]"
},
{
"state_after": "case a.h.h.h.mk\nn : ℕ\nj : Fin (n + 2)\nk : Fin (len [n] + 1)\ni : ℕ\nisLt✝ : i < n + 2\nH : { val := i, isLt := isLt✝ } ≤ j\n⊢ ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k) <\n Fin.succ j then\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < { val := i, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k) <\n ↑Fin.castSucc { val := i, isLt := isLt✝ } then\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k))",
"state_before": "case a.h.h.h\nn : ℕ\ni j : Fin (n + 2)\nH : i ≤ j\nk : Fin (len [n] + 1)\n⊢ ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k) < Fin.succ j then\n ↑Fin.castSucc (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < i then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k) < ↑Fin.castSucc i then\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k))",
"tactic": "rcases i with ⟨i, _⟩"
},
{
"state_after": "case a.h.h.h.mk.mk\nn : ℕ\nk : Fin (len [n] + 1)\ni : ℕ\nisLt✝¹ : i < n + 2\nj : ℕ\nisLt✝ : j < n + 2\nH : { val := i, isLt := isLt✝¹ } ≤ { val := j, isLt := isLt✝ }\n⊢ ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝¹ } then ↑Fin.castSucc k else Fin.succ k) <\n Fin.succ { val := j, isLt := isLt✝ } then\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝¹ } then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < { val := i, isLt := isLt✝¹ } then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := j, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k) <\n ↑Fin.castSucc { val := i, isLt := isLt✝¹ } then\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := j, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < { val := j, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k))",
"state_before": "case a.h.h.h.mk\nn : ℕ\nj : Fin (n + 2)\nk : Fin (len [n] + 1)\ni : ℕ\nisLt✝ : i < n + 2\nH : { val := i, isLt := isLt✝ } ≤ j\n⊢ ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k) <\n Fin.succ j then\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < { val := i, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k) <\n ↑Fin.castSucc { val := i, isLt := isLt✝ } then\n ↑Fin.castSucc (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < j then ↑Fin.castSucc k else Fin.succ k))",
"tactic": "rcases j with ⟨j, _⟩"
},
{
"state_after": "case a.h.h.h.mk.mk.mk\nn i : ℕ\nisLt✝² : i < n + 2\nj : ℕ\nisLt✝¹ : j < n + 2\nH : { val := i, isLt := isLt✝² } ≤ { val := j, isLt := isLt✝¹ }\nk : ℕ\nisLt✝ : k < len [n] + 1\n⊢ ↑(if\n ↑Fin.castSucc\n (if ↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := i, isLt := isLt✝² } then\n ↑Fin.castSucc { val := k, isLt := isLt✝ }\n else Fin.succ { val := k, isLt := isLt✝ }) <\n Fin.succ { val := j, isLt := isLt✝¹ } then\n ↑Fin.castSucc\n (if ↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := i, isLt := isLt✝² } then\n ↑Fin.castSucc { val := k, isLt := isLt✝ }\n else Fin.succ { val := k, isLt := isLt✝ })\n else\n Fin.succ\n (if ↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := i, isLt := isLt✝² } then\n ↑Fin.castSucc { val := k, isLt := isLt✝ }\n else Fin.succ { val := k, isLt := isLt✝ })) =\n ↑(if\n ↑Fin.castSucc\n (if ↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := j, isLt := isLt✝¹ } then\n ↑Fin.castSucc { val := k, isLt := isLt✝ }\n else Fin.succ { val := k, isLt := isLt✝ }) <\n ↑Fin.castSucc { val := i, isLt := isLt✝² } then\n ↑Fin.castSucc\n (if ↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := j, isLt := isLt✝¹ } then\n ↑Fin.castSucc { val := k, isLt := isLt✝ }\n else Fin.succ { val := k, isLt := isLt✝ })\n else\n Fin.succ\n (if ↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := j, isLt := isLt✝¹ } then\n ↑Fin.castSucc { val := k, isLt := isLt✝ }\n else Fin.succ { val := k, isLt := isLt✝ }))",
"state_before": "case a.h.h.h.mk.mk\nn : ℕ\nk : Fin (len [n] + 1)\ni : ℕ\nisLt✝¹ : i < n + 2\nj : ℕ\nisLt✝ : j < n + 2\nH : { val := i, isLt := isLt✝¹ } ≤ { val := j, isLt := isLt✝ }\n⊢ ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝¹ } then ↑Fin.castSucc k else Fin.succ k) <\n Fin.succ { val := j, isLt := isLt✝ } then\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := i, isLt := isLt✝¹ } then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < { val := i, isLt := isLt✝¹ } then ↑Fin.castSucc k else Fin.succ k)) =\n ↑(if\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := j, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k) <\n ↑Fin.castSucc { val := i, isLt := isLt✝¹ } then\n ↑Fin.castSucc (if ↑Fin.castSucc k < { val := j, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k)\n else Fin.succ (if ↑Fin.castSucc k < { val := j, isLt := isLt✝ } then ↑Fin.castSucc k else Fin.succ k))",
"tactic": "rcases k with ⟨k, _⟩"
},
{
"state_after": "no goals",
"state_before": "case a.h.h.h.mk.mk.mk.inr.inr.inr.inr\nn i : ℕ\nisLt✝² : i < n + 2\nj : ℕ\nisLt✝¹ : j < n + 2\nH : { val := i, isLt := isLt✝² } ≤ { val := j, isLt := isLt✝¹ }\nk : ℕ\nisLt✝ : k < len [n] + 1\nh✝³ : ¬↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := i, isLt := isLt✝² }\nh✝² : ¬↑Fin.castSucc (Fin.succ { val := k, isLt := isLt✝ }) < Fin.succ { val := j, isLt := isLt✝¹ }\nh✝¹ : ¬↑Fin.castSucc { val := k, isLt := isLt✝ } < { val := j, isLt := isLt✝¹ }\nh✝ : ¬↑Fin.castSucc (Fin.succ { val := k, isLt := isLt✝ }) < ↑Fin.castSucc { val := i, isLt := isLt✝² }\n⊢ ↑(Fin.succ (Fin.succ { val := k, isLt := isLt✝ })) = ↑(Fin.succ (Fin.succ { val := k, isLt := isLt✝ }))",
"tactic": "simp at * <;> linarith"
}
] |
[
226,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.zero_top
|
[] |
[
710,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Order/Cover.lean
|
Covby.le_of_lt
|
[] |
[
439,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
438,
1
] |
Mathlib/LinearAlgebra/SymplecticGroup.lean
|
Matrix.J_squared
|
[
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : CommRing R\ninst✝ : Fintype l\n⊢ fromBlocks (0 ⬝ 0 + (-1) ⬝ 1) (0 ⬝ (-1) + (-1) ⬝ 0) (1 ⬝ 0 + 0 ⬝ 1) (1 ⬝ (-1) + 0 ⬝ 0) = -1",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : CommRing R\ninst✝ : Fintype l\n⊢ J l R ⬝ J l R = -1",
"tactic": "rw [J, fromBlocks_multiply]"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : CommRing R\ninst✝ : Fintype l\n⊢ fromBlocks (-1) 0 0 (-1) = -1",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : CommRing R\ninst✝ : Fintype l\n⊢ fromBlocks (0 ⬝ 0 + (-1) ⬝ 1) (0 ⬝ (-1) + (-1) ⬝ 0) (1 ⬝ 0 + 0 ⬝ 1) (1 ⬝ (-1) + 0 ⬝ 0) = -1",
"tactic": "simp only [Matrix.zero_mul, Matrix.neg_mul, zero_add, neg_zero, Matrix.one_mul, add_zero]"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : CommRing R\ninst✝ : Fintype l\n⊢ fromBlocks (-1) 0 0 (-1) = -1",
"tactic": "rw [← neg_zero, ← Matrix.fromBlocks_neg, ← fromBlocks_one]"
}
] |
[
58,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Tactic/NormNum/Core.lean
|
Mathlib.Meta.NormNum.IsInt.to_isNat
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Ring α\na✝ : ℕ\n⊢ ↑(Int.ofNat a✝) = ↑a✝",
"tactic": "simp"
}
] |
[
78,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.comap_inf_map_of_injective
|
[] |
[
406,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sigmaToiUnion_bijective
|
[] |
[
2188,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2186,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.coe_trans
|
[] |
[
901,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
900,
1
] |
Mathlib/LinearAlgebra/InvariantBasisNumber.lean
|
nontrivial_of_invariantBasisNumber
|
[
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\n⊢ False",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\n⊢ Nontrivial R",
"tactic": "by_contra h"
},
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\n⊢ (Fin 0 → R) ≃ₗ[R] Fin 1 → R",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\n⊢ False",
"tactic": "refine' zero_ne_one (eq_of_fin_equiv R _)"
},
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis : Subsingleton R\n⊢ (Fin 0 → R) ≃ₗ[R] Fin 1 → R",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\n⊢ (Fin 0 → R) ≃ₗ[R] Fin 1 → R",
"tactic": "haveI := not_nontrivial_iff_subsingleton.1 h"
},
{
"state_after": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis✝ : Subsingleton R\nthis : Subsingleton (Fin 1 → R)\n⊢ (Fin 0 → R) ≃ₗ[R] Fin 1 → R",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis : Subsingleton R\n⊢ (Fin 0 → R) ≃ₗ[R] Fin 1 → R",
"tactic": "haveI : Subsingleton (Fin 1 → R) :=\n Subsingleton.intro <| fun a b => funext fun x => Subsingleton.elim _ _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis✝ : Subsingleton R\nthis : Subsingleton (Fin 1 → R)\n⊢ (Fin 0 → R) ≃ₗ[R] Fin 1 → R",
"tactic": "exact\n { toFun := 0\n invFun := 0\n map_add' := by aesop\n map_smul' := by aesop\n left_inv := fun _ => by simp\n right_inv := fun _ => by simp }"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis✝ : Subsingleton R\nthis : Subsingleton (Fin 1 → R)\n⊢ ∀ (x y : Fin 0 → R), OfNat.ofNat 0 (x + y) = OfNat.ofNat 0 x + OfNat.ofNat 0 y",
"tactic": "aesop"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis✝ : Subsingleton R\nthis : Subsingleton (Fin 1 → R)\n⊢ ∀ (r : R) (x : Fin 0 → R),\n AddHom.toFun { toFun := 0, map_add' := (_ : ∀ (x y : Fin 0 → R), OfNat.ofNat 0 (x + y) = 0 + 0) } (r • x) =\n ↑(RingHom.id R) r •\n AddHom.toFun { toFun := 0, map_add' := (_ : ∀ (x y : Fin 0 → R), OfNat.ofNat 0 (x + y) = 0 + 0) } x",
"tactic": "aesop"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis✝ : Subsingleton R\nthis : Subsingleton (Fin 1 → R)\nx✝ : Fin 0 → R\n⊢ OfNat.ofNat 0\n (AddHom.toFun\n { toAddHom := { toFun := 0, map_add' := (_ : ∀ (x y : Fin 0 → R), OfNat.ofNat 0 (x + y) = 0 + 0) },\n map_smul' := (_ : ∀ (r : R) (x : Fin 0 → R), OfNat.ofNat 0 (r • x) = r • 0) }.toAddHom\n x✝) =\n x✝",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : InvariantBasisNumber R\nh : ¬Nontrivial R\nthis✝ : Subsingleton R\nthis : Subsingleton (Fin 1 → R)\nx✝ : Fin 1 → R\n⊢ AddHom.toFun\n { toAddHom := { toFun := 0, map_add' := (_ : ∀ (x y : Fin 0 → R), OfNat.ofNat 0 (x + y) = 0 + 0) },\n map_smul' := (_ : ∀ (r : R) (x : Fin 0 → R), OfNat.ofNat 0 (r • x) = r • 0) }.toAddHom\n (OfNat.ofNat 0 x✝) =\n x✝",
"tactic": "simp"
}
] |
[
212,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.naturalityProperty.stableUnderComposition
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : F₁.map f ≫ app Y = app X ≫ F₂.map f\nhg : F₁.map g ≫ app Z = app Y ≫ F₂.map g\n⊢ F₁.map (f ≫ g) ≫ app Z = app X ≫ F₂.map (f ≫ g)",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : naturalityProperty app f\nhg : naturalityProperty app g\n⊢ naturalityProperty app (f ≫ g)",
"tactic": "simp only [naturalityProperty] at hf hg⊢"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : F₁.map f ≫ app Y = app X ≫ F₂.map f\nhg : F₁.map g ≫ app Z = app Y ≫ F₂.map g\n⊢ F₁.map f ≫ app Y ≫ F₂.map g = app X ≫ F₂.map f ≫ F₂.map g",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : F₁.map f ≫ app Y = app X ≫ F₂.map f\nhg : F₁.map g ≫ app Z = app Y ≫ F₂.map g\n⊢ F₁.map (f ≫ g) ≫ app Z = app X ≫ F₂.map (f ≫ g)",
"tactic": "simp only [Functor.map_comp, Category.assoc, hg]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : F₁.map f ≫ app Y = app X ≫ F₂.map f\nhg : F₁.map g ≫ app Z = app Y ≫ F₂.map g\n⊢ (app X ≫ F₂.map f) ≫ F₂.map g = app X ≫ F₂.map f ≫ F₂.map g",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : F₁.map f ≫ app Y = app X ≫ F₂.map f\nhg : F₁.map g ≫ app Z = app Y ≫ F₂.map g\n⊢ F₁.map f ≫ app Y ≫ F₂.map g = app X ≫ F₂.map f ≫ F₂.map g",
"tactic": "slice_lhs 1 2 => rw [hf]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : F₁.map f ≫ app Y = app X ≫ F₂.map f\nhg : F₁.map g ≫ app Z = app Y ≫ F₂.map g\n⊢ (app X ≫ F₂.map f) ≫ F₂.map g = app X ≫ F₂.map f ≫ F₂.map g",
"tactic": "rw [Category.assoc]"
}
] |
[
355,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.sub_mem_sup
|
[
{
"state_after": "R : Type ?u.157648\nS✝ : Type ?u.157651\nM : Type ?u.157654\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring S✝\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : Module S✝ M\ninst✝⁴ : SMul S✝ R\ninst✝³ : IsScalarTower S✝ R M\np q : Submodule R M\nR' : Type u_1\nM' : Type u_2\ninst✝² : Ring R'\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R' M'\nS T : Submodule R' M'\ns t : M'\nhs : s ∈ S\nht : t ∈ T\n⊢ s + -t ∈ S ⊔ T",
"state_before": "R : Type ?u.157648\nS✝ : Type ?u.157651\nM : Type ?u.157654\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring S✝\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : Module S✝ M\ninst✝⁴ : SMul S✝ R\ninst✝³ : IsScalarTower S✝ R M\np q : Submodule R M\nR' : Type u_1\nM' : Type u_2\ninst✝² : Ring R'\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R' M'\nS T : Submodule R' M'\ns t : M'\nhs : s ∈ S\nht : t ∈ T\n⊢ s - t ∈ S ⊔ T",
"tactic": "rw [sub_eq_add_neg]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.157648\nS✝ : Type ?u.157651\nM : Type ?u.157654\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring S✝\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : Module S✝ M\ninst✝⁴ : SMul S✝ R\ninst✝³ : IsScalarTower S✝ R M\np q : Submodule R M\nR' : Type u_1\nM' : Type u_2\ninst✝² : Ring R'\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R' M'\nS T : Submodule R' M'\ns t : M'\nhs : s ∈ S\nht : t ∈ T\n⊢ s + -t ∈ S ⊔ T",
"tactic": "exact add_mem_sup hs (neg_mem ht)"
}
] |
[
297,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/NumberTheory/Padics/PadicNorm.lean
|
padicNorm.neg
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nq : ℚ\nhq : q = 0\n⊢ padicNorm p (-q) = padicNorm p q",
"tactic": "simp [hq]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nq : ℚ\nhq : ¬q = 0\n⊢ padicNorm p (-q) = padicNorm p q",
"tactic": "simp [padicNorm, hq]"
}
] |
[
131,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
11
] |
Mathlib/Algebra/DirectSum/Internal.lean
|
Submodule.setLike.coe_galgebra_toFun
|
[] |
[
305,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
inf_compl_self
|
[] |
[
873,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
872,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty
|
[] |
[
401,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioo_inter_Ioc_of_right_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.196902\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\nh : b₂ < b₁\n⊢ Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioc (max a₁ a₂) b₂",
"tactic": "rw [inter_comm, Ioc_inter_Ioo_of_left_lt h, max_comm]"
}
] |
[
1811,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1810,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.total_zero_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_3\nN : Type ?u.294176\nP : Type ?u.294179\nR : Type u_2\nS : Type ?u.294185\ninst✝¹² : Semiring R\ninst✝¹¹ : Semiring S\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommMonoid N\ninst✝⁷ : Module R N\ninst✝⁶ : AddCommMonoid P\ninst✝⁵ : Module R P\nα' : Type ?u.294277\nM' : Type ?u.294280\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M'\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M'\ninst✝ : Module R M\nv : α → M\nv' : α' → M'\nx : α →₀ R\n⊢ ↑(Finsupp.total α M R 0) x = 0",
"tactic": "simp [Finsupp.total_apply]"
}
] |
[
563,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
562,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.image_emetric_closedBall
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.750293\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nh : α ≃ᵢ β\nx : α\nr : ℝ≥0∞\n⊢ ↑h '' EMetric.closedBall x r = EMetric.closedBall (↑h x) r",
"tactic": "rw [← h.preimage_symm, h.symm.preimage_emetric_closedBall, symm_symm]"
}
] |
[
499,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Data/List/Destutter.lean
|
List.destutter_nil
|
[] |
[
126,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.symm_apply_apply
|
[] |
[
355,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Equiv.realize_term
|
[] |
[
242,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_lt_tsub_iff_right
|
[] |
[
474,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
Rel.interedges_mono
|
[
{
"state_after": "𝕜 : Type ?u.2805\nι : Type ?u.2808\nκ : Type ?u.2811\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nx : α × β\n⊢ x.fst ∈ s₂ ∧ x.snd ∈ t₂ ∧ r x.fst x.snd → x.fst ∈ s₁ ∧ x.snd ∈ t₁ ∧ r x.fst x.snd",
"state_before": "𝕜 : Type ?u.2805\nι : Type ?u.2808\nκ : Type ?u.2811\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nx : α × β\n⊢ x ∈ interedges r s₂ t₂ → x ∈ interedges r s₁ t₁",
"tactic": "simp_rw [mem_interedges_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.2805\nι : Type ?u.2808\nκ : Type ?u.2811\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nx : α × β\n⊢ x.fst ∈ s₂ ∧ x.snd ∈ t₂ ∧ r x.fst x.snd → x.fst ∈ s₁ ∧ x.snd ∈ t₁ ∧ r x.fst x.snd",
"tactic": "exact fun h ↦ ⟨hs h.1, ht h.2.1, h.2.2⟩"
}
] |
[
76,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.IndepSets.iInter
|
[
{
"state_after": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\ns : ι → Set (Set Ω)\ns' : Set (Set Ω)\nμ : MeasureTheory.Measure Ω\nh : ∃ n, IndepSets (s n) s'\nt1 t2 : Set Ω\nht1 : t1 ∈ ⋂ (n : ι), s n\nht2 : t2 ∈ s'\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\ns : ι → Set (Set Ω)\ns' : Set (Set Ω)\nμ : MeasureTheory.Measure Ω\nh : ∃ n, IndepSets (s n) s'\n⊢ IndepSets (⋂ (n : ι), s n) s'",
"tactic": "intro t1 t2 ht1 ht2"
},
{
"state_after": "case intro\nΩ : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\ns : ι → Set (Set Ω)\ns' : Set (Set Ω)\nμ : MeasureTheory.Measure Ω\nt1 t2 : Set Ω\nht1 : t1 ∈ ⋂ (n : ι), s n\nht2 : t2 ∈ s'\nn : ι\nh : IndepSets (s n) s'\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\ns : ι → Set (Set Ω)\ns' : Set (Set Ω)\nμ : MeasureTheory.Measure Ω\nh : ∃ n, IndepSets (s n) s'\nt1 t2 : Set Ω\nht1 : t1 ∈ ⋂ (n : ι), s n\nht2 : t2 ∈ s'\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "cases' h with n h"
},
{
"state_after": "no goals",
"state_before": "case intro\nΩ : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\ns : ι → Set (Set Ω)\ns' : Set (Set Ω)\nμ : MeasureTheory.Measure Ω\nt1 t2 : Set Ω\nht1 : t1 ∈ ⋂ (n : ι), s n\nht2 : t2 ∈ s'\nn : ι\nh : IndepSets (s n) s'\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "exact h t1 t2 (Set.mem_iInter.mp ht1 n) ht2"
}
] |
[
243,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.one_gcd
|
[
{
"state_after": "n : ℕ+\n⊢ 1 ∣ n",
"state_before": "n : ℕ+\n⊢ gcd 1 n = 1",
"tactic": "rw [← gcd_eq_left_iff_dvd]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ+\n⊢ 1 ∣ n",
"tactic": "apply one_dvd"
}
] |
[
232,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.IsEquivalence.fun_inv_map
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\ninst✝ : IsEquivalence F\nX Y : D\nf : X ⟶ Y\n⊢ F.map ((Functor.inv F).map f) = ((asEquivalence F).inverse ⋙ (asEquivalence F).functor).map f",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\ninst✝ : IsEquivalence F\nX Y : D\nf : X ⟶ Y\n⊢ F.map ((Functor.inv F).map f) =\n (Equivalence.counit (asEquivalence F)).app X ≫ f ≫ (Equivalence.counitInv (asEquivalence F)).app Y",
"tactic": "erw [NatIso.naturality_2]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\ninst✝ : IsEquivalence F\nX Y : D\nf : X ⟶ Y\n⊢ F.map ((Functor.inv F).map f) = ((asEquivalence F).inverse ⋙ (asEquivalence F).functor).map f",
"tactic": "rfl"
}
] |
[
605,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/Data/FinEnum.lean
|
FinEnum.nodup_toList
|
[
{
"state_after": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\n⊢ List.Nodup (List.map (↑Equiv.symm) (List.finRange (card α)))",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\n⊢ List.Nodup (toList α)",
"tactic": "simp [toList]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\n⊢ List.Nodup (List.map (↑Equiv.symm) (List.finRange (card α)))",
"tactic": "apply List.Nodup.map <;> [apply Equiv.injective; apply List.nodup_finRange]"
}
] |
[
82,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Algebra/Hom/NonUnitalAlg.lean
|
NonUnitalAlgHom.coe_comp
|
[] |
[
285,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_smul_measure_iff
|
[] |
[
293,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.roots_list_prod
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nL : List R[X]\nhd : R[X]\ntl : List R[X]\nih : ¬0 ∈ tl → roots (List.prod tl) = Multiset.bind (↑tl) roots\nH : ¬0 = hd ∧ ¬0 ∈ tl\n⊢ roots (List.prod (hd :: tl)) = Multiset.bind (↑(hd :: tl)) roots",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nL : List R[X]\nhd : R[X]\ntl : List R[X]\nih : ¬0 ∈ tl → roots (List.prod tl) = Multiset.bind (↑tl) roots\nH : ¬0 ∈ hd :: tl\n⊢ roots (List.prod (hd :: tl)) = Multiset.bind (↑(hd :: tl)) roots",
"tactic": "rw [List.mem_cons, not_or] at H"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nL : List R[X]\nhd : R[X]\ntl : List R[X]\nih : ¬0 ∈ tl → roots (List.prod tl) = Multiset.bind (↑tl) roots\nH : ¬0 = hd ∧ ¬0 ∈ tl\n⊢ roots (List.prod (hd :: tl)) = Multiset.bind (↑(hd :: tl)) roots",
"tactic": "rw [List.prod_cons, roots_mul (mul_ne_zero (Ne.symm H.1) <| List.prod_ne_zero H.2), ←\n Multiset.cons_coe, Multiset.cons_bind, ih H.2]"
}
] |
[
691,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos
|
[
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\n⊢ trailingDegree p = ↑n → natTrailingDegree p = n\n\ncase mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\n⊢ natTrailingDegree p = n → trailingDegree p = ↑n",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\n⊢ trailingDegree p = ↑n ↔ natTrailingDegree p = n",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : trailingDegree p = ↑n\n⊢ natTrailingDegree p = n",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\n⊢ trailingDegree p = ↑n → natTrailingDegree p = n",
"tactic": "intro H"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : trailingDegree p = ↑n\n⊢ p ≠ 0",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : trailingDegree p = ↑n\n⊢ natTrailingDegree p = n",
"tactic": "rwa [← trailingDegree_eq_iff_natTrailingDegree_eq]"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : trailingDegree 0 = ↑n\n⊢ False",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : trailingDegree p = ↑n\n⊢ p ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : ⊤ = ↑n\n⊢ False",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : trailingDegree 0 = ↑n\n⊢ False",
"tactic": "rw [trailingDegree_zero] at H"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : ⊤ = ↑n\n⊢ False",
"tactic": "exact Option.noConfusion H"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : natTrailingDegree p = n\n⊢ trailingDegree p = ↑n",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\n⊢ natTrailingDegree p = n → trailingDegree p = ↑n",
"tactic": "intro H"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : natTrailingDegree p = n\n⊢ p ≠ 0",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : natTrailingDegree p = n\n⊢ trailingDegree p = ↑n",
"tactic": "rwa [trailingDegree_eq_iff_natTrailingDegree_eq]"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : natTrailingDegree 0 = n\n⊢ False",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nhn : 0 < n\nH : natTrailingDegree p = n\n⊢ p ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : 0 = n\n⊢ False",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : natTrailingDegree 0 = n\n⊢ False",
"tactic": "rw [natTrailingDegree_zero] at H"
},
{
"state_after": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : n < n\nH : 0 = n\n⊢ False",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : 0 < n\nH : 0 = n\n⊢ False",
"tactic": "rw [H] at hn"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nhn : n < n\nH : 0 = n\n⊢ False",
"tactic": "exact lt_irrefl _ hn"
}
] |
[
133,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.div_zero_subset
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.652865\nα : Type u_1\nβ : Type ?u.652871\nγ : Type ?u.652874\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : GroupWithZero α\ns✝ t s : Finset α\n⊢ s / 0 ⊆ 0",
"tactic": "simp [subset_iff, mem_div]"
}
] |
[
1189,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1189,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.coe_sub
|
[] |
[
486,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean
|
Finpartition.equitabilise_aux
|
[
{
"state_after": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = 0 ∨ card x = 0 + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ 0) ∧\n card (filter (fun i => card i = 0 + 1) Q.parts) = b\n\ncase inr\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nm_pos : m > 0\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "obtain rfl | m_pos := m.eq_zero_or_pos"
},
{
"state_after": "case inr.H\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case inr\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nm_pos : m > 0\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "induction' s using Finset.strongInduction with s ih generalizing a b"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : a = 0 ∧ b = 0\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : ¬(a = 0 ∧ b = 0)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case inr.H\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "by_cases hab : a = 0 ∧ b = 0"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : ¬(a = 0 ∧ b = 0)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "simp_rw [not_and_or, ← Ne.def, ← pos_iff_ne_zero] at hab"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "set n := if 0 < a then m else m + 1 with hn"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ¬∀ (u : Finset α), u ∈ P.parts → card u < m + 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "by_cases h : ∀ u ∈ P.parts, card u < m + 1"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∃ u, u ∈ P.parts ∧ m + 1 ≤ card u\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ¬∀ (u : Finset α), u ∈ P.parts → card u < m + 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "push_neg at h"
},
{
"state_after": "case neg.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∃ u, u ∈ P.parts ∧ m + 1 ≤ card u\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "obtain ⟨u, hu₁, hu₂⟩ := h"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "obtain ⟨t, htu, htn⟩ := exists_smaller_set _ _ (hn₁.trans hu₂)"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "have ht : t.Nonempty := by rwa [← card_pos, htn]"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "have hcard : ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = (s \\ t).card := by\n rw [card_sdiff (htu.trans <| P.le hu₁), htn, hn₃]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "obtain ⟨R, hR₁, hR₂, hR₃⟩ :=\n @ih (s \\ t) (sdiff_ssubset (htu.trans <| P.le hu₁) ht) (if 0 < a then a - 1 else a)\n (if 0 < a then b else b - 1) (P.avoid t) hcard"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts → card x = m ∨ card x = m + 1\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ P.parts →\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y ⊆ x) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) id) ≤\n m\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (filter (fun i => card i = m + 1) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) = b",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "refine' ⟨R.extend ht.ne_empty sdiff_disjoint (sdiff_sup_cancel <| htu.trans <| P.le hu₁), _, _, _⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card\n (if ¬0 < a then insert t (filter (fun i => card i = m + 1) R.parts)\n else filter (fun i => card i = m + 1) R.parts) =\n b",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (filter (fun i => card i = m + 1) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) = b",
"tactic": "simp only [extend_parts, filter_insert, htn, hn, m.succ_ne_self.symm.ite_eq_right_iff]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : 0 < a\n⊢ card (filter (fun i => card i = m + 1) R.parts) = b\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : ¬0 < a\n⊢ card (insert t (filter (fun i => card i = m + 1) R.parts)) = b",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card\n (if ¬0 < a then insert t (filter (fun i => card i = m + 1) R.parts)\n else filter (fun i => card i = m + 1) R.parts) =\n b",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ ∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) ⊥.parts) id) ≤ 0",
"state_before": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = 0 ∨ card x = 0 + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ 0) ∧\n card (filter (fun i => card i = 0 + 1) Q.parts) = b",
"tactic": "refine' ⟨⊥, by simp, _, by simpa using hs.symm⟩"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ ∀ (x : Finset α), x ∈ P.parts → ∀ ⦃x_1 : α⦄, x_1 ∈ x → ∃ a, a ∈ ⊥.parts ∧ (∀ ⦃x_2 : α⦄, x_2 ∈ a → x_2 ∈ x) ∧ x_1 ∈ a",
"state_before": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ ∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) ⊥.parts) id) ≤ 0",
"tactic": "simp only [le_zero_iff, card_eq_zero, mem_biUnion, exists_prop, mem_filter, id.def, and_assoc,\n sdiff_eq_empty_iff_subset, subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ ∀ (x : Finset α), x ∈ P.parts → ∀ ⦃x_1 : α⦄, x_1 ∈ x → ∃ a, a ∈ ⊥.parts ∧ (∀ ⦃x_2 : α⦄, x_2 ∈ a → x_2 ∈ x) ∧ x_1 ∈ a",
"tactic": "exact fun x hx a ha =>\n ⟨{a}, mem_map_of_mem _ (P.le hx ha), singleton_subset_iff.2 ha, mem_singleton_self _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ ∀ (x : Finset α), x ∈ ⊥.parts → card x = 0 ∨ card x = 0 + 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nn a b : ℕ\nP : Finpartition s\nhs : a * 0 + b * (0 + 1) = card s\n⊢ card (filter (fun i => card i = 0 + 1) ⊥.parts) = b",
"tactic": "simpa using hs.symm"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhab : a = 0 ∧ b = 0\nhs : s = ∅\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : a = 0 ∧ b = 0\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "simp only [hab.1, hab.2, add_zero, zero_mul, eq_comm, card_eq_zero, Finset.bot_eq_empty] at hs"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s\nhs : a✝ * m + b✝ * (m + 1) = card s\nm_pos : m > 0\na b : ℕ\nhab : a = 0 ∧ b = 0\nih :\n ∀ (t : Finset α),\n t ⊂ ∅ →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition ∅\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhab : a = 0 ∧ b = 0\nhs : s = ∅\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "subst hs"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s\nhs : a✝ * m + b✝ * (m + 1) = card s\nm_pos : m > 0\na b : ℕ\nhab : a = 0 ∧ b = 0\nih :\n ∀ (t : Finset α),\n t ⊂ ∅ →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition ∅\nthis : P = Finpartition.empty (Finset α)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s\nhs : a✝ * m + b✝ * (m + 1) = card s\nm_pos : m > 0\na b : ℕ\nhab : a = 0 ∧ b = 0\nih :\n ∀ (t : Finset α),\n t ⊂ ∅ →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition ∅\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "have : P = Finpartition.empty _ := Unique.eq_default (α := Finpartition ⊥) P"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s\nhs : a✝ * m + b✝ * (m + 1) = card s\nm_pos : m > 0\na b : ℕ\nhab : a = 0 ∧ b = 0\nih :\n ∀ (t : Finset α),\n t ⊂ ∅ →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition ∅\nthis : P = Finpartition.empty (Finset α)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "exact ⟨Finpartition.empty _, by simp, by simp [this], by simp [hab.2]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s\nhs : a✝ * m + b✝ * (m + 1) = card s\nm_pos : m > 0\na b : ℕ\nhab : a = 0 ∧ b = 0\nih :\n ∀ (t : Finset α),\n t ⊂ ∅ →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition ∅\nthis : P = Finpartition.empty (Finset α)\n⊢ ∀ (x : Finset α), x ∈ (Finpartition.empty (Finset α)).parts → card x = m ∨ card x = m + 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s\nhs : a✝ * m + b✝ * (m + 1) = card s\nm_pos : m > 0\na b : ℕ\nhab : a = 0 ∧ b = 0\nih :\n ∀ (t : Finset α),\n t ⊂ ∅ →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition ∅\nthis : P = Finpartition.empty (Finset α)\n⊢ ∀ (x : Finset α),\n x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) (Finpartition.empty (Finset α)).parts) id) ≤ m",
"tactic": "simp [this]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a✝ b✝ : ℕ\nP✝ : Finpartition s\nhs : a✝ * m + b✝ * (m + 1) = card s\nm_pos : m > 0\na b : ℕ\nhab : a = 0 ∧ b = 0\nih :\n ∀ (t : Finset α),\n t ⊂ ∅ →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\nP : Finpartition ∅\nthis : P = Finpartition.empty (Finset α)\n⊢ card (filter (fun i => card i = m + 1) (Finpartition.empty (Finset α)).parts) = b",
"tactic": "simp [hab.2]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\n⊢ (0 < if 0 < a then m else m + 1) ∧\n (if 0 < a then m else m + 1) ≤ m + 1 ∧\n (if 0 < a then m else m + 1) ≤ a * m + b * (m + 1) ∧\n (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) =\n a * m + b * (m + 1) - if 0 < a then m else m + 1",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\n⊢ 0 < n ∧\n n ≤ m + 1 ∧\n n ≤ a * m + b * (m + 1) ∧ (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n",
"tactic": "rw [hn, ← hs]"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : 0 < a\n⊢ 0 < m ∧ m ≤ m + 1 ∧ m ≤ a * m + b * (m + 1) ∧ a * m - m + b * (m + 1) = a * m + b * (m + 1) - m\n\ncase inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : ¬0 < a\n⊢ 0 < m + 1 ∧\n m + 1 ≤ m + 1 ∧ m + 1 ≤ a * m + b * (m + 1) ∧ a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\n⊢ (0 < if 0 < a then m else m + 1) ∧\n (if 0 < a then m else m + 1) ≤ m + 1 ∧\n (if 0 < a then m else m + 1) ≤ a * m + b * (m + 1) ∧\n (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) =\n a * m + b * (m + 1) - if 0 < a then m else m + 1",
"tactic": "split_ifs with h <;> rw [tsub_mul, one_mul]"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : 0 < a\n⊢ a * m - m + b * (m + 1) = a * m + b * (m + 1) - m",
"state_before": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : 0 < a\n⊢ 0 < m ∧ m ≤ m + 1 ∧ m ≤ a * m + b * (m + 1) ∧ a * m - m + b * (m + 1) = a * m + b * (m + 1) - m",
"tactic": "refine' ⟨m_pos, le_succ _, le_add_right (le_mul_of_pos_left ‹0 < a›), _⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : 0 < a\n⊢ a * m - m + b * (m + 1) = a * m + b * (m + 1) - m",
"tactic": "rw [tsub_add_eq_add_tsub (le_mul_of_pos_left h)]"
},
{
"state_after": "case inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : ¬0 < a\n⊢ a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)",
"state_before": "case inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : ¬0 < a\n⊢ 0 < m + 1 ∧\n m + 1 ≤ m + 1 ∧ m + 1 ≤ a * m + b * (m + 1) ∧ a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)",
"tactic": "refine' ⟨succ_pos', le_rfl, le_add_left (le_mul_of_pos_left <| hab.resolve_left ‹¬0 < a›), _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nh : ¬0 < a\n⊢ a * m + (b * (m + 1) - (m + 1)) = a * m + b * (m + 1) - (m + 1)",
"tactic": "rw [← add_tsub_assoc_of_le (le_mul_of_pos_left <| hab.resolve_left ‹¬0 < a›)]"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "obtain ⟨t, hts, htn⟩ := exists_smaller_set s n (hn₂.trans_eq hs)"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "have ht : t.Nonempty := by rwa [← card_pos, htn]"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "have hcard : ite (0 < a) (a - 1) a * m + ite (0 < a) b (b - 1) * (m + 1) = (s \\ t).card := by\n rw [card_sdiff ‹t ⊆ s›, htn, hn₃]"
},
{
"state_after": "case pos.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "obtain ⟨R, hR₁, _, hR₃⟩ :=\n @ih (s \\ t) (sdiff_ssubset hts ‹t.Nonempty›) (if 0 < a then a - 1 else a)\n (if 0 < a then b else b - 1) (P.avoid t) hcard"
},
{
"state_after": "case pos.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts → card x = m ∨ card x = m + 1\n\ncase pos.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ P.parts →\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y ⊆ x) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) id) ≤\n m\n\ncase pos.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (filter (fun i => card i = m + 1) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) = b",
"state_before": "case pos.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b",
"tactic": "refine' ⟨R.extend ht.ne_empty sdiff_disjoint (sdiff_sup_cancel hts), _, _, _⟩"
},
{
"state_after": "case pos.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card\n (if ¬0 < a then insert t (filter (fun i => card i = m + 1) R.parts)\n else filter (fun i => card i = m + 1) R.parts) =\n b",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (filter (fun i => card i = m + 1) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) = b",
"tactic": "simp_rw [extend_parts, filter_insert, htn, m.succ_ne_self.symm.ite_eq_right_iff]"
},
{
"state_after": "case pos.intro.intro.intro.intro.intro.refine'_3.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : 0 < a\n⊢ card (filter (fun i => card i = m + 1) R.parts) = b\n\ncase pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\n⊢ card (insert t (filter (fun i => card i = m + 1) R.parts)) = b",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card\n (if ¬0 < a then insert t (filter (fun i => card i = m + 1) R.parts)\n else filter (fun i => card i = m + 1) R.parts) =\n b",
"tactic": "split_ifs with ha"
},
{
"state_after": "case pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\n⊢ 1 ≤ b\n\ncase pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\n⊢ ¬t ∈ filter (fun i => card i = m + 1) R.parts",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\n⊢ card (insert t (filter (fun i => card i = m + 1) R.parts)) = b",
"tactic": "rw [card_insert_of_not_mem, hR₃, if_neg ha, tsub_add_cancel_of_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\n⊢ Finset.Nonempty t",
"tactic": "rwa [← card_pos, htn]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\n⊢ (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)",
"tactic": "rw [card_sdiff ‹t ⊆ s›, htn, hn₃]"
},
{
"state_after": "case pos.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ (if 0 < a then m else m + 1) = m ∨ (if 0 < a then m else m + 1) = m + 1",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts → card x = m ∨ card x = m + 1",
"tactic": "simp only [extend_parts, mem_insert, forall_eq_or_imp, and_iff_left hR₁, htn, hn]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ (if 0 < a then m else m + 1) = m ∨ (if 0 < a then m else m + 1) = m + 1",
"tactic": "exact ite_eq_or_eq _ _ _"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ P.parts →\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y ⊆ x) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) id) ≤\n m",
"tactic": "exact fun x hx => (card_le_of_subset <| sdiff_subset _ _).trans (lt_succ_iff.1 <| h _ hx)"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_3.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : 0 < a\n⊢ card (filter (fun i => card i = m + 1) R.parts) = b",
"tactic": "rw [hR₃, if_pos ha]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\n⊢ 1 ≤ b",
"tactic": "exact hab.resolve_left ha"
},
{
"state_after": "case pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\nH : t ∈ filter (fun i => card i = m + 1) R.parts\n⊢ False",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\n⊢ ¬t ∈ filter (fun i => card i = m + 1) R.parts",
"tactic": "intro H"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nh : ∀ (u : Finset α), u ∈ P.parts → card u < m + 1\nt : Finset α\nhts : t ⊆ s\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nleft✝ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nha : ¬0 < a\nH : t ∈ filter (fun i => card i = m + 1) R.parts\n⊢ False",
"tactic": "exact ht.ne_empty (le_sdiff_iff.1 <| R.le <| filter_subset _ _ H)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\n⊢ Finset.Nonempty t",
"tactic": "rwa [← card_pos, htn]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\n⊢ (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)",
"tactic": "rw [card_sdiff (htu.trans <| P.le hu₁), htn, hn₃]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ (if 0 < a then m else m + 1) = m ∨ (if 0 < a then m else m + 1) = m + 1",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts → card x = m ∨ card x = m + 1",
"tactic": "simp only [mem_insert, forall_eq_or_imp, extend_parts, and_iff_left hR₁, htn, hn]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ (if 0 < a then m else m + 1) = m ∨ (if 0 < a then m else m + 1) = m + 1",
"tactic": "exact ite_eq_or_eq _ _ _"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ insert u (erase P.parts u) →\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y ⊆ x) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) id) ≤\n m",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ P.parts →\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y ⊆ x) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) id) ≤\n m",
"tactic": "conv in _ ∈ _ => rw [← insert_erase hu₁]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ Finset.biUnion (filter (fun y => y ⊆ u) (insert t R.parts)) id) ≤ m ∧\n ∀ (a : Finset α),\n a ∈ erase P.parts u → card (a \\ Finset.biUnion (filter (fun y => y ⊆ a) (insert t R.parts)) id) ≤ m",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ ∀ (x : Finset α),\n x ∈ insert u (erase P.parts u) →\n card\n (x \\\n Finset.biUnion\n (filter (fun y => y ⊆ x) (extend R (_ : t ≠ ∅) (_ : Disjoint (s \\ t) t) (_ : s \\ t ⊔ t = s)).parts) id) ≤\n m",
"tactic": "simp only [and_imp, mem_insert, forall_eq_or_imp, Ne.def, extend_parts]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ Finset.biUnion (filter (fun y => y ⊆ u) (insert t R.parts)) id) ≤ m\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ x \\ Finset.biUnion (filter (fun y => y ⊆ x) (insert t R.parts)) id ⊆\n x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ x ∈ (avoid P t).parts",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ Finset.biUnion (filter (fun y => y ⊆ u) (insert t R.parts)) id) ≤ m ∧\n ∀ (a : Finset α),\n a ∈ erase P.parts u → card (a \\ Finset.biUnion (filter (fun y => y ⊆ a) (insert t R.parts)) id) ≤ m",
"tactic": "refine' ⟨_, fun x hx => (card_le_of_subset _).trans <| hR₂ x _⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ x ≠ ∅ ∧ ∃ a, a ∈ P.parts ∧ a \\ t = x",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ x ∈ (avoid P t).parts",
"tactic": "simp only [avoid, ofErase, mem_erase, mem_image, bot_eq_empty]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_3\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ x ≠ ∅ ∧ ∃ a, a ∈ P.parts ∧ a \\ t = x",
"tactic": "exact\n ⟨(nonempty_of_mem_parts _ <| mem_of_mem_erase hx).ne_empty, _, mem_of_mem_erase hx,\n (disjoint_of_subset_right htu <|\n P.disjoint (mem_of_mem_erase hx) hu₁ <| ne_of_mem_erase hx).sdiff_eq_left⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ (t ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id)) ≤ m",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ Finset.biUnion (filter (fun y => y ⊆ u) (insert t R.parts)) id) ≤ m",
"tactic": "simp only [filter_insert, if_pos htu, biUnion_insert, mem_erase, id.def]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nhtu : u ⊆ u\nhtn : card u = n\nht : Finset.Nonempty u\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ u)\nR : Finpartition (s \\ u)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P u).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ (u ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id)) ≤ m\n\ncase neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u ≠ t\n⊢ card (u \\ (t ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id)) ≤ m",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ (t ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id)) ≤ m",
"tactic": "obtain rfl | hut := eq_or_ne u t"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u ≠ t\ni : α\n⊢ i ∈ u \\ (t ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id) →\n i ∈ (u \\ t) \\ Finset.biUnion (filter (fun y => y ⊆ u \\ t) R.parts) id",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u ≠ t\n⊢ card (u \\ (t ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id)) ≤ m",
"tactic": "refine'\n (card_le_of_subset fun i => _).trans\n (hR₂ (u \\ t) <| P.mem_avoid.2 ⟨u, hu₁, fun i => hut <| i.antisymm htu, rfl⟩)"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u ≠ t\ni : α\n⊢ i ∈ u →\n ¬i ∈ t →\n (∀ (x : Finset α), x ∈ R.parts → x ⊆ u → ¬i ∈ x) →\n (i ∈ u ∧ ¬i ∈ t) ∧ ∀ (x : Finset α), x ∈ R.parts → x ⊆ u \\ t → ¬i ∈ x",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u ≠ t\ni : α\n⊢ i ∈ u \\ (t ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id) →\n i ∈ (u \\ t) \\ Finset.biUnion (filter (fun y => y ⊆ u \\ t) R.parts) id",
"tactic": "simp only [not_exists, not_and, mem_biUnion, and_imp, mem_union, mem_filter, mem_sdiff,\n id.def, not_or]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nhut : u ≠ t\ni : α\n⊢ i ∈ u →\n ¬i ∈ t →\n (∀ (x : Finset α), x ∈ R.parts → x ⊆ u → ¬i ∈ x) →\n (i ∈ u ∧ ¬i ∈ t) ∧ ∀ (x : Finset α), x ∈ R.parts → x ⊆ u \\ t → ¬i ∈ x",
"tactic": "exact fun hi₁ hi₂ hi₃ =>\n ⟨⟨hi₁, hi₂⟩, fun x hx hx' => hi₃ _ hx <| hx'.trans <| sdiff_subset _ _⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nhtu : u ⊆ u\nhtn : card u = n\nht : Finset.Nonempty u\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ u)\nR : Finpartition (s \\ u)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P u).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card ∅ ≤ m",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nhtu : u ⊆ u\nhtn : card u = n\nht : Finset.Nonempty u\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ u)\nR : Finpartition (s \\ u)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P u).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card (u \\ (u ∪ Finset.biUnion (filter (fun y => y ⊆ u) R.parts) id)) ≤ m",
"tactic": "rw [sdiff_eq_empty_iff_subset.2 (subset_union_left _ _)]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_1.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nhtu : u ⊆ u\nhtn : card u = n\nht : Finset.Nonempty u\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ u)\nR : Finpartition (s \\ u)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P u).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\n⊢ card ∅ ≤ m",
"tactic": "exact bot_le"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ filter (fun y => y ⊆ x) R.parts ⊆ filter (fun y => y ⊆ x) (insert t R.parts)",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_2.refine'_2\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ x \\ Finset.biUnion (filter (fun y => y ⊆ x) (insert t R.parts)) id ⊆\n x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id",
"tactic": "apply sdiff_subset_sdiff Subset.rfl (biUnion_subset_biUnion_of_subset_left _ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nx : Finset α\nhx : x ∈ erase P.parts u\n⊢ filter (fun y => y ⊆ x) R.parts ⊆ filter (fun y => y ⊆ x) (insert t R.parts)",
"tactic": "exact filter_subset_filter _ (subset_insert _ _)"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : 0 < a\n⊢ card (filter (fun i => card i = m + 1) R.parts) = b",
"tactic": "rw [hR₃, if_pos h]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : ¬0 < a\n⊢ ¬t ∈ filter (fun i => card i = m + 1) R.parts",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : ¬0 < a\n⊢ card (insert t (filter (fun i => card i = m + 1) R.parts)) = b",
"tactic": "rw [card_insert_of_not_mem, hR₃, if_neg h, Nat.sub_add_cancel (hab.resolve_left h)]"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : ¬0 < a\nH : t ∈ filter (fun i => card i = m + 1) R.parts\n⊢ False",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : ¬0 < a\n⊢ ¬t ∈ filter (fun i => card i = m + 1) R.parts",
"tactic": "intro H"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ t✝ : Finset α\nm n✝ a✝ b✝ : ℕ\nP✝ : Finpartition s✝\nhs✝ : a✝ * m + b✝ * (m + 1) = card s✝\nm_pos : m > 0\ns : Finset α\nih :\n ∀ (t : Finset α),\n t ⊂ s →\n ∀ {a b : ℕ} {P : Finpartition t},\n a * m + b * (m + 1) = card t →\n ∃ Q,\n (∀ (x : Finset α), x ∈ Q.parts → card x = m ∨ card x = m + 1) ∧\n (∀ (x : Finset α), x ∈ P.parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) Q.parts) id) ≤ m) ∧\n card (filter (fun i => card i = m + 1) Q.parts) = b\na b : ℕ\nP : Finpartition s\nhs : a * m + b * (m + 1) = card s\nhab : 0 < a ∨ 0 < b\nn : ℕ := if 0 < a then m else m + 1\nhn : n = if 0 < a then m else m + 1\nhn₀ : 0 < n\nhn₁ : n ≤ m + 1\nhn₂ : n ≤ a * m + b * (m + 1)\nhn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card s - n\nu : Finset α\nhu₁ : u ∈ P.parts\nhu₂ : m + 1 ≤ card u\nt : Finset α\nhtu : t ⊆ u\nhtn : card t = n\nht : Finset.Nonempty t\nhcard : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = card (s \\ t)\nR : Finpartition (s \\ t)\nhR₁ : ∀ (x : Finset α), x ∈ R.parts → card x = m ∨ card x = m + 1\nhR₂ : ∀ (x : Finset α), x ∈ (avoid P t).parts → card (x \\ Finset.biUnion (filter (fun y => y ⊆ x) R.parts) id) ≤ m\nhR₃ : card (filter (fun i => card i = m + 1) R.parts) = if 0 < a then b else b - 1\nh : ¬0 < a\nH : t ∈ filter (fun i => card i = m + 1) R.parts\n⊢ False",
"tactic": "exact ht.ne_empty (le_sdiff_iff.1 <| R.le <| filter_subset _ _ H)"
}
] |
[
140,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.add_one_lt_exp_of_neg
|
[
{
"state_after": "x : ℝ\nh : x < 0\nh1 : 0 < -x\n⊢ x + 1 < exp x",
"state_before": "x : ℝ\nh : x < 0\n⊢ x + 1 < exp x",
"tactic": "have h1 : 0 < -x := by linarith"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : x < 0\nh1 : 0 < -x\n⊢ x + 1 < exp x",
"tactic": "simpa [add_comm] using one_sub_lt_exp_minus_of_pos h1"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : x < 0\n⊢ 0 < -x",
"tactic": "linarith"
}
] |
[
1978,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1976,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.image.isoStrongEpiMono_hom_comp_ι
|
[
{
"state_after": "C : Type u\ninst✝³ : Category C\ninst✝² : HasStrongEpiMonoFactorisations C\nX Y : C\nf : X ⟶ Y\nI' : C\ne : X ⟶ I'\nm : I' ⟶ Y\ncomm : e ≫ m = f\ninst✝¹ : StrongEpi e\ninst✝ : Mono m\n⊢ IsImage.lift (StrongEpiMonoFactorisation.toMonoIsImage (StrongEpiMonoFactorisation.mk (MonoFactorisation.mk I' m e)))\n (Image.monoFactorisation f) ≫\n ι f =\n m",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasStrongEpiMonoFactorisations C\nX Y : C\nf : X ⟶ Y\nI' : C\ne : X ⟶ I'\nm : I' ⟶ Y\ncomm : e ≫ m = f\ninst✝¹ : StrongEpi e\ninst✝ : Mono m\n⊢ (isoStrongEpiMono e m comm).hom ≫ ι f = m",
"tactic": "dsimp [isoStrongEpiMono]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\ninst✝² : HasStrongEpiMonoFactorisations C\nX Y : C\nf : X ⟶ Y\nI' : C\ne : X ⟶ I'\nm : I' ⟶ Y\ncomm : e ≫ m = f\ninst✝¹ : StrongEpi e\ninst✝ : Mono m\n⊢ IsImage.lift (StrongEpiMonoFactorisation.toMonoIsImage (StrongEpiMonoFactorisation.mk (MonoFactorisation.mk I' m e)))\n (Image.monoFactorisation f) ≫\n ι f =\n m",
"tactic": "apply IsImage.lift_fac"
}
] |
[
1032,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1029,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupNorm.coe_le_coe
|
[] |
[
796,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.map_map
|
[] |
[
272,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformly.prod_map
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np'✝ : Filter α\ng : ι → α\nι' : Type u_1\nα' : Type u_2\nβ' : Type u_3\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\nf' : α' → β'\np' : Filter ι'\nh : TendstoUniformlyOn F f p univ\nh' : TendstoUniformlyOn F' f' p' univ\n⊢ TendstoUniformlyOn (fun i => Prod.map (F i.fst) (F' i.snd)) (Prod.map f f') (p ×ˢ p') (univ ×ˢ univ)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np'✝ : Filter α\ng : ι → α\nι' : Type u_1\nα' : Type u_2\nβ' : Type u_3\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\nf' : α' → β'\np' : Filter ι'\nh : TendstoUniformly F f p\nh' : TendstoUniformly F' f' p'\n⊢ TendstoUniformly (fun i => Prod.map (F i.fst) (F' i.snd)) (Prod.map f f') (p ×ˢ p')",
"tactic": "rw [← tendstoUniformlyOn_univ, ← univ_prod_univ] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np'✝ : Filter α\ng : ι → α\nι' : Type u_1\nα' : Type u_2\nβ' : Type u_3\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\nf' : α' → β'\np' : Filter ι'\nh : TendstoUniformlyOn F f p univ\nh' : TendstoUniformlyOn F' f' p' univ\n⊢ TendstoUniformlyOn (fun i => Prod.map (F i.fst) (F' i.snd)) (Prod.map f f') (p ×ˢ p') (univ ×ˢ univ)",
"tactic": "exact h.prod_map h'"
}
] |
[
298,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.coe_unbot
|
[
{
"state_after": "case none\nα : Type u_1\nβ : Type ?u.7014\nγ : Type ?u.7017\nδ : Type ?u.7020\na b : α\nh : none ≠ ⊥\n⊢ ↑(unbot none h) = none\n\ncase some\nα : Type u_1\nβ : Type ?u.7014\nγ : Type ?u.7017\nδ : Type ?u.7020\na b val✝ : α\nh : Option.some val✝ ≠ ⊥\n⊢ ↑(unbot (Option.some val✝) h) = Option.some val✝",
"state_before": "α : Type u_1\nβ : Type ?u.7014\nγ : Type ?u.7017\nδ : Type ?u.7020\na b : α\nx : WithBot α\nh : x ≠ ⊥\n⊢ ↑(unbot x h) = x",
"tactic": "cases x"
},
{
"state_after": "case some\nα : Type u_1\nβ : Type ?u.7014\nγ : Type ?u.7017\nδ : Type ?u.7020\na b val✝ : α\nh : Option.some val✝ ≠ ⊥\n⊢ ↑(unbot (Option.some val✝) h) = Option.some val✝",
"state_before": "case none\nα : Type u_1\nβ : Type ?u.7014\nγ : Type ?u.7017\nδ : Type ?u.7020\na b : α\nh : none ≠ ⊥\n⊢ ↑(unbot none h) = none\n\ncase some\nα : Type u_1\nβ : Type ?u.7014\nγ : Type ?u.7017\nδ : Type ?u.7020\na b val✝ : α\nh : Option.some val✝ ≠ ⊥\n⊢ ↑(unbot (Option.some val✝) h) = Option.some val✝",
"tactic": "exact (h rfl).elim"
},
{
"state_after": "no goals",
"state_before": "case some\nα : Type u_1\nβ : Type ?u.7014\nγ : Type ?u.7017\nδ : Type ?u.7020\na b val✝ : α\nh : Option.some val✝ ≠ ⊥\n⊢ ↑(unbot (Option.some val✝) h) = Option.some val✝",
"tactic": "rfl"
}
] |
[
182,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
RingHom.FiniteType.id
|
[] |
[
215,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_const_iff
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.832965\nδ : Type ?u.832968\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nc : β\nthis : AEStronglyMeasurable (fun x => c) μ\n⊢ (Integrable fun x => c) ↔ c = 0 ∨ ↑↑μ univ < ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.832965\nδ : Type ?u.832968\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nc : β\n⊢ (Integrable fun x => c) ↔ c = 0 ∨ ↑↑μ univ < ⊤",
"tactic": "have : AEStronglyMeasurable (fun _ : α => c) μ := aestronglyMeasurable_const"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.832965\nδ : Type ?u.832968\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nc : β\nthis : AEStronglyMeasurable (fun x => c) μ\n⊢ (Integrable fun x => c) ↔ c = 0 ∨ ↑↑μ univ < ⊤",
"tactic": "rw [Integrable, and_iff_right this, hasFiniteIntegral_const_iff]"
}
] |
[
495,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
differentiableAt_zpow
|
[] |
[
79,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.add_to_nat
|
[] |
[
313,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.toFinite
|
[] |
[
86,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
Set.pairwise_iff_exists_forall
|
[
{
"state_after": "case inl\nα : Type u_2\nβ : Type ?u.6322\nγ : Type ?u.6325\nι : Type u_1\nι' : Type ?u.6331\nr✝ p q : α → α → Prop\nf✝ g : ι → α\ns t u : Set α\na b : α\ninst✝¹ : Nonempty ι\nf : α → ι\nr : ι → ι → Prop\ninst✝ : IsEquiv ι r\n⊢ Set.Pairwise ∅ (r on f) ↔ ∃ z, ∀ (x : α), x ∈ ∅ → r (f x) z\n\ncase inr\nα : Type u_2\nβ : Type ?u.6322\nγ : Type ?u.6325\nι : Type u_1\nι' : Type ?u.6331\nr✝ p q : α → α → Prop\nf✝ g : ι → α\ns✝ t u : Set α\na b : α\ninst✝¹ : Nonempty ι\ns : Set α\nf : α → ι\nr : ι → ι → Prop\ninst✝ : IsEquiv ι r\nhne : Set.Nonempty s\n⊢ Set.Pairwise s (r on f) ↔ ∃ z, ∀ (x : α), x ∈ s → r (f x) z",
"state_before": "α : Type u_2\nβ : Type ?u.6322\nγ : Type ?u.6325\nι : Type u_1\nι' : Type ?u.6331\nr✝ p q : α → α → Prop\nf✝ g : ι → α\ns✝ t u : Set α\na b : α\ninst✝¹ : Nonempty ι\ns : Set α\nf : α → ι\nr : ι → ι → Prop\ninst✝ : IsEquiv ι r\n⊢ Set.Pairwise s (r on f) ↔ ∃ z, ∀ (x : α), x ∈ s → r (f x) z",
"tactic": "rcases s.eq_empty_or_nonempty with (rfl | hne)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_2\nβ : Type ?u.6322\nγ : Type ?u.6325\nι : Type u_1\nι' : Type ?u.6331\nr✝ p q : α → α → Prop\nf✝ g : ι → α\ns t u : Set α\na b : α\ninst✝¹ : Nonempty ι\nf : α → ι\nr : ι → ι → Prop\ninst✝ : IsEquiv ι r\n⊢ Set.Pairwise ∅ (r on f) ↔ ∃ z, ∀ (x : α), x ∈ ∅ → r (f x) z",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_2\nβ : Type ?u.6322\nγ : Type ?u.6325\nι : Type u_1\nι' : Type ?u.6331\nr✝ p q : α → α → Prop\nf✝ g : ι → α\ns✝ t u : Set α\na b : α\ninst✝¹ : Nonempty ι\ns : Set α\nf : α → ι\nr : ι → ι → Prop\ninst✝ : IsEquiv ι r\nhne : Set.Nonempty s\n⊢ Set.Pairwise s (r on f) ↔ ∃ z, ∀ (x : α), x ∈ s → r (f x) z",
"tactic": "exact hne.pairwise_iff_exists_forall"
}
] |
[
128,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Topology/Instances/AddCircle.lean
|
AddCircle.coe_period
|
[] |
[
190,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ring.isField_iff_isSimpleOrder_ideal
|
[
{
"state_after": "case inl\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Subsingleton R\n⊢ IsField R ↔ IsSimpleOrder (Ideal R)\n\ncase inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ IsField R ↔ IsSimpleOrder (Ideal R)",
"state_before": "α : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\n⊢ IsField R ↔ IsSimpleOrder (Ideal R)",
"tactic": "cases subsingleton_or_nontrivial R"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ (¬∃ I, ⊥ < I ∧ I < ⊤) ↔ ¬¬IsSimpleOrder (Ideal R)",
"state_before": "case inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ IsField R ↔ IsSimpleOrder (Ideal R)",
"tactic": "rw [← not_iff_not, Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top, ← not_iff_not]"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ (∀ (I : Ideal R), ⊥ < I → ¬I < ⊤) ↔ IsSimpleOrder (Ideal R)",
"state_before": "case inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ (¬∃ I, ⊥ < I ∧ I < ⊤) ↔ ¬¬IsSimpleOrder (Ideal R)",
"tactic": "push_neg"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ (∀ (I : Ideal R), I = ⊥ ∨ I = ⊤) ↔ IsSimpleOrder (Ideal R)",
"state_before": "case inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ (∀ (I : Ideal R), ⊥ < I → ¬I < ⊤) ↔ IsSimpleOrder (Ideal R)",
"tactic": "simp_rw [lt_top_iff_ne_top, bot_lt_iff_ne_bot, ← or_iff_not_imp_left, not_ne_iff]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Nontrivial R\n⊢ (∀ (I : Ideal R), I = ⊥ ∨ I = ⊤) ↔ IsSimpleOrder (Ideal R)",
"tactic": "exact ⟨fun h => ⟨h⟩, fun h => h.2⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nR : Type u_1\ninst✝ : CommSemiring R\nh✝ : Subsingleton R\n⊢ IsField R ↔ IsSimpleOrder (Ideal R)",
"tactic": "exact\n ⟨fun h => (not_isField_of_subsingleton _ h).elim, fun h =>\n (false_of_nontrivial_of_subsingleton <| Ideal R).elim⟩"
}
] |
[
790,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isCoseparating_iff_mono
|
[
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : IsCoseparating 𝒢\nA Z✝ : C\nu v : Z✝ ⟶ A\nhuv : u ≫ Pi.lift Sigma.snd = v ≫ Pi.lift Sigma.snd\nG : C\nhG : G ∈ 𝒢\nf : A ⟶ G\n⊢ u ≫ f = v ≫ f\n\ncase refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : ∀ (A : C), Mono (Pi.lift Sigma.snd)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (G : C), G ∈ 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ f = g",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\n⊢ IsCoseparating 𝒢 ↔ ∀ (A : C), Mono (Pi.lift Sigma.snd)",
"tactic": "refine' ⟨fun h A => ⟨fun u v huv => h _ _ fun G hG f => _⟩, fun h X Y f g hh => _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : IsCoseparating 𝒢\nA Z✝ : C\nu v : Z✝ ⟶ A\nhuv : u ≫ Pi.lift Sigma.snd = v ≫ Pi.lift Sigma.snd\nG : C\nhG : G ∈ 𝒢\nf : A ⟶ G\n⊢ u ≫ f = v ≫ f",
"tactic": "simpa using huv =≫ Pi.π (fun f : ΣG : 𝒢, A ⟶ (G : C) => (f.1 : C)) ⟨⟨G, hG⟩, f⟩"
},
{
"state_after": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : ∀ (A : C), Mono (Pi.lift Sigma.snd)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (G : C), G ∈ 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nthis : Mono (Pi.lift Sigma.snd)\n⊢ f = g",
"state_before": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : ∀ (A : C), Mono (Pi.lift Sigma.snd)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (G : C), G ∈ 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ f = g",
"tactic": "haveI := h Y"
},
{
"state_after": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : ∀ (A : C), Mono (Pi.lift Sigma.snd)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (G : C), G ∈ 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nthis : Mono (Pi.lift Sigma.snd)\nj : Discrete ((G : ↑𝒢) × (Y ⟶ ↑G))\n⊢ (f ≫ Pi.lift Sigma.snd) ≫ limit.π (Discrete.functor fun b => ↑b.fst) j =\n (g ≫ Pi.lift Sigma.snd) ≫ limit.π (Discrete.functor fun b => ↑b.fst) j",
"state_before": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : ∀ (A : C), Mono (Pi.lift Sigma.snd)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (G : C), G ∈ 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nthis : Mono (Pi.lift Sigma.snd)\n⊢ f = g",
"tactic": "refine' (cancel_mono (Pi.lift (@Sigma.snd 𝒢 fun G => Y ⟶ (G : C)))).1 (limit.hom_ext fun j => _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\n𝒢 : Set C\ninst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst\nh : ∀ (A : C), Mono (Pi.lift Sigma.snd)\nX Y : C\nf g : X ⟶ Y\nhh : ∀ (G : C), G ∈ 𝒢 → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\nthis : Mono (Pi.lift Sigma.snd)\nj : Discrete ((G : ↑𝒢) × (Y ⟶ ↑G))\n⊢ (f ≫ Pi.lift Sigma.snd) ≫ limit.π (Discrete.functor fun b => ↑b.fst) j =\n (g ≫ Pi.lift Sigma.snd) ≫ limit.π (Discrete.functor fun b => ↑b.fst) j",
"tactic": "simpa using hh j.as.1.1 j.as.1.2 j.as.2"
}
] |
[
277,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
SeminormFamily.finset_sup_comp
|
[
{
"state_after": "case h\n𝕜 : Type u_4\n𝕜₂ : Type u_1\n𝕝 : Type ?u.705364\n𝕝₂ : Type ?u.705367\nE : Type u_5\nF : Type u_2\nG : Type ?u.705376\nι : Type u_3\nι' : Type ?u.705382\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : NormedField 𝕜₂\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nq : SeminormFamily 𝕜₂ F ι\ns : Finset ι\nf : E →ₛₗ[σ₁₂] F\nx : E\n⊢ ↑(Seminorm.comp (Finset.sup s q) f) x = ↑(Finset.sup s (comp q f)) x",
"state_before": "𝕜 : Type u_4\n𝕜₂ : Type u_1\n𝕝 : Type ?u.705364\n𝕝₂ : Type ?u.705367\nE : Type u_5\nF : Type u_2\nG : Type ?u.705376\nι : Type u_3\nι' : Type ?u.705382\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : NormedField 𝕜₂\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nq : SeminormFamily 𝕜₂ F ι\ns : Finset ι\nf : E →ₛₗ[σ₁₂] F\n⊢ Seminorm.comp (Finset.sup s q) f = Finset.sup s (comp q f)",
"tactic": "ext x"
},
{
"state_after": "case h\n𝕜 : Type u_4\n𝕜₂ : Type u_1\n𝕝 : Type ?u.705364\n𝕝₂ : Type ?u.705367\nE : Type u_5\nF : Type u_2\nG : Type ?u.705376\nι : Type u_3\nι' : Type ?u.705382\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : NormedField 𝕜₂\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nq : SeminormFamily 𝕜₂ F ι\ns : Finset ι\nf : E →ₛₗ[σ₁₂] F\nx : E\n⊢ ↑(Finset.sup s fun i => { val := ↑(q i) (↑f x), property := (_ : 0 ≤ ↑(q i) (↑f x)) }) =\n ↑(Finset.sup s fun i => { val := ↑(comp q f i) x, property := (_ : 0 ≤ ↑(comp q f i) x) })",
"state_before": "case h\n𝕜 : Type u_4\n𝕜₂ : Type u_1\n𝕝 : Type ?u.705364\n𝕝₂ : Type ?u.705367\nE : Type u_5\nF : Type u_2\nG : Type ?u.705376\nι : Type u_3\nι' : Type ?u.705382\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : NormedField 𝕜₂\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nq : SeminormFamily 𝕜₂ F ι\ns : Finset ι\nf : E →ₛₗ[σ₁₂] F\nx : E\n⊢ ↑(Seminorm.comp (Finset.sup s q) f) x = ↑(Finset.sup s (comp q f)) x",
"tactic": "rw [Seminorm.comp_apply, Seminorm.finset_sup_apply, Seminorm.finset_sup_apply]"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_4\n𝕜₂ : Type u_1\n𝕝 : Type ?u.705364\n𝕝₂ : Type ?u.705367\nE : Type u_5\nF : Type u_2\nG : Type ?u.705376\nι : Type u_3\nι' : Type ?u.705382\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : NormedField 𝕜₂\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\nq : SeminormFamily 𝕜₂ F ι\ns : Finset ι\nf : E →ₛₗ[σ₁₂] F\nx : E\n⊢ ↑(Finset.sup s fun i => { val := ↑(q i) (↑f x), property := (_ : 0 ≤ ↑(q i) (↑f x)) }) =\n ↑(Finset.sup s fun i => { val := ↑(comp q f i) x, property := (_ : 0 ≤ ↑(comp q f i) x) })",
"tactic": "rfl"
}
] |
[
730,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
726,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.elim_some
|
[] |
[
195,
79
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
195,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.ofFiberEquiv_map
|
[] |
[
782,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp
|
[
{
"state_after": "α : Type u_2\nE : Type u_3\nF : Type ?u.8125054\nG : Type ?u.8125057\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : α → E\nf_lim_ℒp : Memℒp f_lim p\n⊢ Tendsto (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) fi (𝓝 0) ↔\n Tendsto (fun n => snorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8125054\nG : Type ?u.8125057\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : α → E\nf_lim_ℒp : Memℒp f_lim p\n⊢ Tendsto f fi (𝓝 (Memℒp.toLp f_lim f_lim_ℒp)) ↔ Tendsto (fun n => snorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)",
"tactic": "rw [tendsto_Lp_iff_tendsto_ℒp']"
},
{
"state_after": "α : Type u_2\nE : Type u_3\nF : Type ?u.8125054\nG : Type ?u.8125057\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : α → E\nf_lim_ℒp : Memℒp f_lim p\nh_eq : (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) = fun n => snorm (↑↑(f n) - f_lim) p μ\n⊢ Tendsto (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) fi (𝓝 0) ↔\n Tendsto (fun n => snorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)\n\ncase h_eq\nα : Type u_2\nE : Type u_3\nF : Type ?u.8125054\nG : Type ?u.8125057\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : α → E\nf_lim_ℒp : Memℒp f_lim p\n⊢ (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) = fun n => snorm (↑↑(f n) - f_lim) p μ",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8125054\nG : Type ?u.8125057\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : α → E\nf_lim_ℒp : Memℒp f_lim p\n⊢ Tendsto (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) fi (𝓝 0) ↔\n Tendsto (fun n => snorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)",
"tactic": "suffices h_eq :\n (fun n => snorm (⇑(f n) - ⇑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) =\n (fun n => snorm (⇑(f n) - f_lim) p μ)"
},
{
"state_after": "no goals",
"state_before": "case h_eq\nα : Type u_2\nE : Type u_3\nF : Type ?u.8125054\nG : Type ?u.8125057\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : α → E\nf_lim_ℒp : Memℒp f_lim p\n⊢ (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) = fun n => snorm (↑↑(f n) - f_lim) p μ",
"tactic": "exact funext fun n => snorm_congr_ae (EventuallyEq.rfl.sub (Memℒp.coeFn_toLp f_lim_ℒp))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_3\nF : Type ?u.8125054\nG : Type ?u.8125057\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\nι : Type u_1\nfi : Filter ι\ninst✝ : Fact (1 ≤ p)\nf : ι → { x // x ∈ Lp E p }\nf_lim : α → E\nf_lim_ℒp : Memℒp f_lim p\nh_eq : (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) = fun n => snorm (↑↑(f n) - f_lim) p μ\n⊢ Tendsto (fun n => snorm (↑↑(f n) - ↑↑(Memℒp.toLp f_lim f_lim_ℒp)) p μ) fi (𝓝 0) ↔\n Tendsto (fun n => snorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)",
"tactic": "rw [h_eq]"
}
] |
[
1252,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1243,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_iUnion_fintype_le
|
[
{
"state_after": "case h.e'_3.h.e'_3.h.e'_3.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.19449\nδ : Type ?u.19452\nι : Type ?u.19455\ninst✝¹ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\ninst✝ : Fintype β\nf : β → Set α\nx✝ : β\n⊢ f x✝ = ⋃ (_ : x✝ ∈ Finset.univ), f x✝",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.19449\nδ : Type ?u.19452\nι : Type ?u.19455\ninst✝¹ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\ninst✝ : Fintype β\nf : β → Set α\n⊢ ↑↑μ (⋃ (b : β), f b) ≤ ∑ p : β, ↑↑μ (f p)",
"tactic": "convert measure_biUnion_finset_le Finset.univ f"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_3.h.e'_3.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.19449\nδ : Type ?u.19452\nι : Type ?u.19455\ninst✝¹ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\ninst✝ : Fintype β\nf : β → Set α\nx✝ : β\n⊢ f x✝ = ⋃ (_ : x✝ ∈ Finset.univ), f x✝",
"tactic": "simp"
}
] |
[
255,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
sSupHom.dual_id
|
[] |
[
801,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
800,
1
] |
Mathlib/Analysis/Calculus/Dslope.lean
|
ContinuousAt.of_dslope
|
[] |
[
102,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isIntegral_of_pow
|
[
{
"state_after": "case intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.664590\nS : Type ?u.664593\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nn : ℕ\nhn : 0 < n\np : R[X]\nhmonic : Monic p\nheval : eval₂ (algebraMap R A) (x ^ n) p = 0\n⊢ IsIntegral R x",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.664590\nS : Type ?u.664593\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nn : ℕ\nhn : 0 < n\nhx : IsIntegral R (x ^ n)\n⊢ IsIntegral R x",
"tactic": "rcases hx with ⟨p, ⟨hmonic, heval⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.664590\nS : Type ?u.664593\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nn : ℕ\nhn : 0 < n\np : R[X]\nhmonic : Monic p\nheval : eval₂ (algebraMap R A) (x ^ n) p = 0\n⊢ IsIntegral R x",
"tactic": "exact\n ⟨expand R n p, Monic.expand hn hmonic, by\n rwa [eval₂_eq_eval_map, map_expand, expand_eval, ← eval₂_eq_eval_map]⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.664590\nS : Type ?u.664593\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx : A\nn : ℕ\nhn : 0 < n\np : R[X]\nhmonic : Monic p\nheval : eval₂ (algebraMap R A) (x ^ n) p = 0\n⊢ eval₂ (algebraMap R A) x (↑(expand R n) p) = 0",
"tactic": "rwa [eval₂_eq_eval_map, map_expand, expand_eval, ← eval₂_eq_eval_map]"
}
] |
[
543,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.continuous_eval
|
[] |
[
332,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.MutuallySingular.smul_left
|
[] |
[
1243,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1241,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.piecewise_insert
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ Set.piecewise (↑(insert j s)) f g = Set.piecewise (insert j ↑s) f g",
"state_before": "α : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ piecewise (insert j s) f g = update (piecewise s f g) j (f j)",
"tactic": "classical simp only [← piecewise_coe, coe_insert, ← Set.piecewise_insert]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\nx✝ : α\n⊢ Set.piecewise (↑(insert j s)) f g x✝ = Set.piecewise (insert j ↑s) f g x✝",
"state_before": "α : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ Set.piecewise (↑(insert j s)) f g = Set.piecewise (insert j ↑s) f g",
"tactic": "ext"
},
{
"state_after": "case h.e_s\nα : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\nx✝ : α\n⊢ ↑(insert j s) = insert j ↑s",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\nx✝ : α\n⊢ Set.piecewise (↑(insert j s)) f g x✝ = Set.piecewise (insert j ↑s) f g x✝",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.e_s\nα : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\nx✝ : α\n⊢ ↑(insert j s) = insert j ↑s",
"tactic": "simp"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ Set.piecewise (↑(insert j s)) f g = Set.piecewise (insert j ↑s) f g",
"state_before": "α : Type u_1\nβ : Type ?u.298194\nγ : Type ?u.298197\nδ : α → Sort u_2\ns : Finset α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ piecewise (insert j s) f g = update (piecewise s f g) j (f j)",
"tactic": "simp only [← piecewise_coe, coe_insert, ← Set.piecewise_insert]"
}
] |
[
2489,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2484,
1
] |
Mathlib/RingTheory/Ideal/Over.lean
|
Ideal.IsIntegralClosure.comap_lt_comap
|
[] |
[
316,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Data/Finset/Functor.lean
|
Finset.seqLeft_def
|
[] |
[
85,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean
|
CauSeq.Completion.mk_neg
|
[] |
[
95,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
bot_sdiff
|
[] |
[
606,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
605,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
|
[] |
[
182,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Std/Data/List/Lemmas.lean
|
List.insert_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\n⊢ List.insert a l = l",
"tactic": "simp only [List.insert, if_pos h]"
}
] |
[
893,
36
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
892,
9
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_zero
|
[
{
"state_after": "case a.h\nl : Type ?u.135132\nm : Type u_1\nn : Type u_2\no : Type u_3\np : Type ?u.135144\nq : Type ?u.135147\nm' : o → Type ?u.135152\nn' : o → Type ?u.135157\np' : o → Type ?u.135162\nR : Type ?u.135165\nS : Type ?u.135168\nα : Type u_4\nβ : Type ?u.135174\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\ni✝ : m × o\nx✝ : n × o\n⊢ blockDiagonal 0 i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"state_before": "l : Type ?u.135132\nm : Type u_1\nn : Type u_2\no : Type u_3\np : Type ?u.135144\nq : Type ?u.135147\nm' : o → Type ?u.135152\nn' : o → Type ?u.135157\np' : o → Type ?u.135162\nR : Type ?u.135165\nS : Type ?u.135168\nα : Type u_4\nβ : Type ?u.135174\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\n⊢ blockDiagonal 0 = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.135132\nm : Type u_1\nn : Type u_2\no : Type u_3\np : Type ?u.135144\nq : Type ?u.135147\nm' : o → Type ?u.135152\nn' : o → Type ?u.135157\np' : o → Type ?u.135162\nR : Type ?u.135165\nS : Type ?u.135168\nα : Type u_4\nβ : Type ?u.135174\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\ni✝ : m × o\nx✝ : n × o\n⊢ blockDiagonal 0 i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"tactic": "simp [blockDiagonal_apply]"
}
] |
[
401,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Mathlib/LinearAlgebra/Matrix/Reindex.lean
|
Matrix.reindexAlgEquiv_apply
|
[] |
[
137,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.dist_map_zero_translationNumber_le
|
[] |
[
689,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.iUnion₂_div
|
[] |
[
771,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
Memℓp.star_mem
|
[
{
"state_after": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f 0\n⊢ Memℓp (star f) 0\n\ncase inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ⊤\n⊢ Memℓp (star f) ⊤\n\ncase inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\nhp : 0 < ENNReal.toReal p\n⊢ Memℓp (star f) p",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\n⊢ Memℓp (star f) p",
"tactic": "rcases p.trichotomy with (rfl | rfl | hp)"
},
{
"state_after": "case inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f 0\n⊢ Set.Finite {i | star f i ≠ 0}",
"state_before": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f 0\n⊢ Memℓp (star f) 0",
"tactic": "apply memℓp_zero"
},
{
"state_after": "no goals",
"state_before": "case inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f 0\n⊢ Set.Finite {i | star f i ≠ 0}",
"tactic": "simp [hf.finite_dsupport]"
},
{
"state_after": "case inr.inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ⊤\n⊢ BddAbove (Set.range fun i => ‖star f i‖)",
"state_before": "case inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ⊤\n⊢ Memℓp (star f) ⊤",
"tactic": "apply memℓp_infty"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.hf\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ⊤\n⊢ BddAbove (Set.range fun i => ‖star f i‖)",
"tactic": "simpa using hf.bddAbove"
},
{
"state_after": "case inr.inr.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\nhp : 0 < ENNReal.toReal p\n⊢ Summable fun i => ‖star f i‖ ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\nhp : 0 < ENNReal.toReal p\n⊢ Memℓp (star f) p",
"tactic": "apply memℓp_gen"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.hf\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\nhp : 0 < ENNReal.toReal p\n⊢ Summable fun i => ‖star f i‖ ^ ENNReal.toReal p",
"tactic": "simpa using hf.summable hp"
}
] |
[
733,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
726,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Bounded.of_gt
|
[] |
[
987,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
984,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_zero
|
[] |
[
107,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.sub_le_sub_right
|
[] |
[
111,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
109,
11
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.coeFn_mk
|
[] |
[
187,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Topology/Basic.lean
|
Disjoint.closure_right
|
[] |
[
438,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
linearIndependent_iff_card_eq_finrank_span
|
[
{
"state_after": "case mp\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ LinearIndependent K b → Fintype.card ι = Set.finrank K (Set.range b)\n\ncase mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ Fintype.card ι = Set.finrank K (Set.range b) → LinearIndependent K b",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ LinearIndependent K b ↔ Fintype.card ι = Set.finrank K (Set.range b)",
"tactic": "constructor"
},
{
"state_after": "case mp\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nh : LinearIndependent K b\n⊢ Fintype.card ι = Set.finrank K (Set.range b)",
"state_before": "case mp\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ LinearIndependent K b → Fintype.card ι = Set.finrank K (Set.range b)",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nh : LinearIndependent K b\n⊢ Fintype.card ι = Set.finrank K (Set.range b)",
"tactic": "exact (finrank_span_eq_card h).symm"
},
{
"state_after": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\n⊢ LinearIndependent K b",
"state_before": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\n⊢ Fintype.card ι = Set.finrank K (Set.range b) → LinearIndependent K b",
"tactic": "intro hc"
},
{
"state_after": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\n⊢ LinearIndependent K b",
"state_before": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\n⊢ LinearIndependent K b",
"tactic": "let f := Submodule.subtype (span K (Set.range b))"
},
{
"state_after": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\n⊢ LinearIndependent K b",
"state_before": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\n⊢ LinearIndependent K b",
"tactic": "let b' : ι → span K (Set.range b) := fun i =>\n ⟨b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)⟩"
},
{
"state_after": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nhs : ⊤ ≤ span K (Set.range b')\n⊢ LinearIndependent K b",
"state_before": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\n⊢ LinearIndependent K b",
"tactic": "have hs : ⊤ ≤ span K (Set.range b') := by\n intro x\n have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f\n have hf : f '' Set.range b' = Set.range b := by\n ext x\n simp [Set.mem_image, Set.mem_range]\n rw [hf] at h\n have hx : (x : V) ∈ span K (Set.range b) := x.property\n conv at hx =>\n arg 2\n rw [h]\n simpa [mem_map] using hx"
},
{
"state_after": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nhs : ⊤ ≤ span K (Set.range b')\nhi : LinearMap.ker f = ⊥\n⊢ LinearIndependent K b",
"state_before": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nhs : ⊤ ≤ span K (Set.range b')\n⊢ LinearIndependent K b",
"tactic": "have hi : LinearMap.ker f = ⊥ := ker_subtype _"
},
{
"state_after": "no goals",
"state_before": "case mpr\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nhs : ⊤ ≤ span K (Set.range b')\nhi : LinearMap.ker f = ⊥\n⊢ LinearIndependent K b",
"tactic": "convert (linearIndependent_of_top_le_span_of_card_eq_finrank hs hc).map' _ hi"
},
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\n⊢ ⊤ ≤ span K (Set.range b')",
"tactic": "intro x"
},
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (↑f '' Set.range b') = Submodule.map f (span K (Set.range b'))\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"tactic": "have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f"
},
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (↑f '' Set.range b') = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (↑f '' Set.range b') = Submodule.map f (span K (Set.range b'))\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"tactic": "have hf : f '' Set.range b' = Set.range b := by\n ext x\n simp [Set.mem_image, Set.mem_range]"
},
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (↑f '' Set.range b') = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"tactic": "rw [hf] at h"
},
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\nhx : ↑x ∈ span K (Set.range b)\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"tactic": "have hx : (x : V) ∈ span K (Set.range b) := x.property"
},
{
"state_after": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\nhx : ↑x ∈ Submodule.map f (span K (Set.range b'))\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\nhx : ↑x ∈ span K (Set.range b)\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"tactic": "conv at hx =>\n arg 2\n rw [h]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : ↑f '' Set.range b' = Set.range b\nhx : ↑x ∈ Submodule.map f (span K (Set.range b'))\n⊢ x ∈ ⊤ → x ∈ span K (Set.range b')",
"tactic": "simpa [mem_map] using hx"
},
{
"state_after": "case h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx✝ : { x // x ∈ span K (Set.range b) }\nh : span K (↑f '' Set.range b') = Submodule.map f (span K (Set.range b'))\nx : V\n⊢ x ∈ ↑f '' Set.range b' ↔ x ∈ Set.range b",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx : { x // x ∈ span K (Set.range b) }\nh : span K (↑f '' Set.range b') = Submodule.map f (span K (Set.range b'))\n⊢ ↑f '' Set.range b' = Set.range b",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nK : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → V\nhc : Fintype.card ι = Set.finrank K (Set.range b)\nf : { x // x ∈ span K (Set.range b) } →ₗ[K] V := Submodule.subtype (span K (Set.range b))\nb' : ι → { x // x ∈ span K (Set.range b) } := fun i => { val := b i, property := (_ : b i ∈ span K (Set.range b)) }\nx✝ : { x // x ∈ span K (Set.range b) }\nh : span K (↑f '' Set.range b') = Submodule.map f (span K (Set.range b'))\nx : V\n⊢ x ∈ ↑f '' Set.range b' ↔ x ∈ Set.range b",
"tactic": "simp [Set.mem_image, Set.mem_range]"
}
] |
[
445,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.supported_univ
|
[] |
[
278,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.ker_id
|
[] |
[
1329,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1328,
1
] |
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