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Mathlib/LinearAlgebra/LinearIndependent.lean | LinearIndependent.group_smul | [
{
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"tactic": "rw [linearIndependent_iff''] at hv⊢"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni : ι\n⊢ g i = 0",
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"tactic": "intro s g hgs hsum i"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni : ι\n⊢ w i • g i = 0",
"state_before": "ι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni : ι\n⊢ g i = 0",
"tactic": "refine' (smul_eq_zero_iff_eq (w i)).1 _"
},
{
"state_after": "case refine'_1\nι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni✝ i : ι\nhi : ¬i ∈ s\n⊢ (fun i => w i • g i) i = 0\n\ncase refine'_2\nι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni : ι\n⊢ ∑ i in s, (fun i => w i • g i) i • v i = 0",
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"tactic": "refine' hv s (fun i => w i • g i) (fun i hi => _) _ i"
},
{
"state_after": "case refine'_1\nι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni✝ i : ι\nhi : ¬i ∈ s\n⊢ w i • g i = 0",
"state_before": "case refine'_1\nι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni✝ i : ι\nhi : ¬i ∈ s\n⊢ (fun i => w i • g i) i = 0",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni✝ i : ι\nhi : ¬i ∈ s\n⊢ w i • g i = 0",
"tactic": "exact (hgs i hi).symm ▸ smul_zero _"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni : ι\n⊢ ∀ (x : ι), x ∈ s → (fun i => w i • g i) x • v x = g x • (w • v) x",
"state_before": "case refine'_2\nι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni : ι\n⊢ ∑ i in s, (fun i => w i • g i) i • v i = 0",
"tactic": "rw [← hsum, Finset.sum_congr rfl _]"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni x✝ : ι\na✝ : x✝ ∈ s\n⊢ (fun i => w i • g i) x✝ • v x✝ = g x✝ • (w • v) x✝",
"state_before": "ι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni : ι\n⊢ ∀ (x : ι), x ∈ s → (fun i => w i • g i) x • v x = g x • (w • v) x",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.270528\nR : Type u_2\nK : Type ?u.270534\nM : Type u_3\nM' : Type ?u.270540\nM'' : Type ?u.270543\nV : Type u\nV' : Type ?u.270548\nv✝ : ι → M\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup M'\ninst✝⁷ : AddCommGroup M''\ninst✝⁶ : Module R M\ninst✝⁵ : Module R M'\ninst✝⁴ : Module R M''\na b : R\nx y : M\nG : Type u_1\nhG : Group G\ninst✝³ : DistribMulAction G R\ninst✝² : DistribMulAction G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\nw : ι → G\ns : Finset ι\ng : ι → R\nhgs : ∀ (i : ι), ¬i ∈ s → g i = 0\nhsum : ∑ i in s, g i • (w • v) i = 0\ni x✝ : ι\na✝ : x✝ ∈ s\n⊢ (fun i => w i • g i) x✝ • v x✝ = g x✝ • (w • v) x✝",
"tactic": "erw [Pi.smul_apply, smul_assoc, smul_comm]"
}
] | [
532,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/Algebra/CharP/Two.lean | CharTwo.neg_eq' | [] | [
72,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean | FiberBundle.totalSpaceMk_closedEmbedding | [
{
"state_after": "ι : Type ?u.6444\nB : Type u_1\nF : Type ?u.6450\nX : Type ?u.6453\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace F\nE : B → Type u_2\ninst✝³ : TopologicalSpace (TotalSpace E)\ninst✝² : (b : B) → TopologicalSpace (E b)\ninst✝¹ : FiberBundle F E\ninst✝ : T1Space B\nx : B\n⊢ IsClosed (Sigma.fst ⁻¹' {x})",
"state_before": "ι : Type ?u.6444\nB : Type u_1\nF : Type ?u.6450\nX : Type ?u.6453\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace F\nE : B → Type u_2\ninst✝³ : TopologicalSpace (TotalSpace E)\ninst✝² : (b : B) → TopologicalSpace (E b)\ninst✝¹ : FiberBundle F E\ninst✝ : T1Space B\nx : B\n⊢ IsClosed (range (totalSpaceMk x))",
"tactic": "rw [range_sigmaMk]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6444\nB : Type u_1\nF : Type ?u.6450\nX : Type ?u.6453\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace F\nE : B → Type u_2\ninst✝³ : TopologicalSpace (TotalSpace E)\ninst✝² : (b : B) → TopologicalSpace (E b)\ninst✝¹ : FiberBundle F E\ninst✝ : T1Space B\nx : B\n⊢ IsClosed (Sigma.fst ⁻¹' {x})",
"tactic": "exact isClosed_singleton.preimage <| continuous_proj F E"
}
] | [
283,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
280,
1
] |
Mathlib/Topology/Homotopy/HomotopyGroup.lean | Cube.zero_mem_boundary | [] | [
75,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean | Nat.eq_one_of_dvd_coprimes | [
{
"state_after": "a b k : ℕ\nh_ab_coprime : gcd a b = 1\nhka : k ∣ a\nhkb : k ∣ b\n⊢ k = 1",
"state_before": "a b k : ℕ\nh_ab_coprime : coprime a b\nhka : k ∣ a\nhkb : k ∣ b\n⊢ k = 1",
"tactic": "rw [coprime_iff_gcd_eq_one] at h_ab_coprime"
},
{
"state_after": "a b k : ℕ\nh_ab_coprime : gcd a b = 1\nhka : k ∣ a\nhkb : k ∣ b\nh1 : k ∣ gcd a b\n⊢ k = 1",
"state_before": "a b k : ℕ\nh_ab_coprime : gcd a b = 1\nhka : k ∣ a\nhkb : k ∣ b\n⊢ k = 1",
"tactic": "have h1 := dvd_gcd hka hkb"
},
{
"state_after": "a b k : ℕ\nh_ab_coprime : gcd a b = 1\nhka : k ∣ a\nhkb : k ∣ b\nh1 : k ∣ 1\n⊢ k = 1",
"state_before": "a b k : ℕ\nh_ab_coprime : gcd a b = 1\nhka : k ∣ a\nhkb : k ∣ b\nh1 : k ∣ gcd a b\n⊢ k = 1",
"tactic": "rw [h_ab_coprime] at h1"
},
{
"state_after": "no goals",
"state_before": "a b k : ℕ\nh_ab_coprime : gcd a b = 1\nhka : k ∣ a\nhkb : k ∣ b\nh1 : k ∣ 1\n⊢ k = 1",
"tactic": "exact Nat.dvd_one.mp h1"
}
] | [
277,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
272,
1
] |
Mathlib/Algebra/Group/Ext.lean | LeftCancelMonoid.ext | [] | [
72,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.map_compose | [] | [
1867,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1866,
1
] |
Mathlib/Algebra/Homology/ShortExact/Preadditive.lean | CategoryTheory.exact_inr_fst | [] | [
167,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean | RingHom.quotientKerEquivOfRightInverse.apply | [] | [
73,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.mem_iUnion_of_mem | [] | [
160,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
] |
Mathlib/Topology/Sets/Opens.lean | TopologicalSpace.Opens.nonempty_coeSort | [] | [
103,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
11
] |
Mathlib/Analysis/MeanInequalities.lean | NNReal.geom_mean_le_arith_mean_weighted | [
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ≥0\nhw' : ∑ i in s, w i = 1\n⊢ ∏ i in s, z i ^ ↑(w i) ≤ ∑ i in s, w i * z i",
"tactic": "exact_mod_cast\n Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg)\n (by assumption_mod_cast) fun i _ => (z i).coe_nonneg"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ≥0\nhw' : ∑ i in s, w i = 1\n⊢ ∑ i in s, ↑(w i) = 1",
"tactic": "assumption_mod_cast"
}
] | [
187,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
183,
1
] |
Mathlib/Data/Vector3.lean | vectorAllP_nil | [] | [
267,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
] |
Mathlib/FieldTheory/Fixed.lean | FixedPoints.isIntegral | [
{
"state_after": "case intro\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Finite G\nx : F\nval✝ : Fintype G\n⊢ IsIntegral { x // x ∈ subfield G F } x",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Finite G\nx : F\n⊢ IsIntegral { x // x ∈ subfield G F } x",
"tactic": "cases nonempty_fintype G"
},
{
"state_after": "no goals",
"state_before": "case intro\nM : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Finite G\nx : F\nval✝ : Fintype G\n⊢ IsIntegral { x // x ∈ subfield G F } x",
"tactic": "exact ⟨minpoly G F x, minpoly.monic G F x, minpoly.eval₂ G F x⟩"
}
] | [
261,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
260,
1
] |
Mathlib/Data/Semiquot.lean | Semiquot.liftOn_ofMem | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nq : Semiquot α\nf : α → β\na : α\naq : a ∈ q\n⊢ ∀ (h : ∀ (a : α), a ∈ q → ∀ (b : α), b ∈ q → f a = f b), liftOn q f h = f a",
"state_before": "α : Type u_1\nβ : Type u_2\nq : Semiquot α\nf : α → β\nh : ∀ (a : α), a ∈ q → ∀ (b : α), b ∈ q → f a = f b\na : α\naq : a ∈ q\n⊢ liftOn q f h = f a",
"tactic": "revert h"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nq : Semiquot α\nf : α → β\na : α\naq : a ∈ q\n⊢ ∀ (h : ∀ (a_1 : α), a_1 ∈ mk aq → ∀ (b : α), b ∈ mk aq → f a_1 = f b), liftOn (mk aq) f h = f a",
"state_before": "α : Type u_1\nβ : Type u_2\nq : Semiquot α\nf : α → β\na : α\naq : a ∈ q\n⊢ ∀ (h : ∀ (a : α), a ∈ q → ∀ (b : α), b ∈ q → f a = f b), liftOn q f h = f a",
"tactic": "rw [eq_mk_of_mem aq]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nq : Semiquot α\nf : α → β\na : α\naq : a ∈ q\nh✝ : ∀ (a_1 : α), a_1 ∈ mk aq → ∀ (b : α), b ∈ mk aq → f a_1 = f b\n⊢ liftOn (mk aq) f h✝ = f a",
"state_before": "α : Type u_1\nβ : Type u_2\nq : Semiquot α\nf : α → β\na : α\naq : a ∈ q\n⊢ ∀ (h : ∀ (a_1 : α), a_1 ∈ mk aq → ∀ (b : α), b ∈ mk aq → f a_1 = f b), liftOn (mk aq) f h = f a",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nq : Semiquot α\nf : α → β\na : α\naq : a ∈ q\nh✝ : ∀ (a_1 : α), a_1 ∈ mk aq → ∀ (b : α), b ∈ mk aq → f a_1 = f b\n⊢ liftOn (mk aq) f h✝ = f a",
"tactic": "rfl"
}
] | [
124,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
1
] |
Mathlib/Data/Set/Basic.lean | Set.disjoint_compl_left_iff_subset | [] | [
1747,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1746,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.two_le_of_mem_cycleType | [
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nn : ℕ\nh : ∃ a, (IsCycle a ∧ ∀ (a_1 : α), a_1 ∈ support a → ↑a a_1 = ↑σ a_1) ∧ Finset.card (support a) = n\n⊢ 2 ≤ n",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nn : ℕ\nh : n ∈ cycleType σ\n⊢ 2 ≤ n",
"tactic": "simp only [cycleType_def, ← Finset.mem_def, Function.comp_apply, Multiset.mem_map,\n mem_cycleFactorsFinset_iff] at h"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ w✝ : Perm α\nhc : IsCycle w✝\n⊢ 2 ≤ Finset.card (support w✝)",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nn : ℕ\nh : ∃ a, (IsCycle a ∧ ∀ (a_1 : α), a_1 ∈ support a → ↑a a_1 = ↑σ a_1) ∧ Finset.card (support a) = n\n⊢ 2 ≤ n",
"tactic": "obtain ⟨_, ⟨hc, -⟩, rfl⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ w✝ : Perm α\nhc : IsCycle w✝\n⊢ 2 ≤ Finset.card (support w✝)",
"tactic": "exact hc.two_le_card_support"
}
] | [
100,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Algebra/Algebra/Operations.lean | Submodule.pow_induction_on_left | [] | [
487,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
481,
11
] |
Mathlib/MeasureTheory/PiSystem.lean | MeasurableSpace.DynkinSystem.generate_le | [] | [
695,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
694,
1
] |
Mathlib/Algebra/Quandle.lean | ShelfHom.map_act | [] | [
375,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
374,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean | LinearIndependent.fin_cons' | [
{
"state_after": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\n⊢ ∀ (g : Fin (m + 1) → R), ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0 → ∀ (i : Fin (m + 1)), g i = 0",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli : LinearIndependent R v\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\n⊢ LinearIndependent R (Fin.cons x v)",
"tactic": "rw [Fintype.linearIndependent_iff] at hli⊢"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\ntotal_eq : ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0\nj : Fin (m + 1)\n⊢ g j = 0",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\n⊢ ∀ (g : Fin (m + 1) → R), ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0 → ∀ (i : Fin (m + 1)), g i = 0",
"tactic": "rintro g total_eq j"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g (Fin.succ x) • v x = 0\n⊢ g j = 0",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\ntotal_eq : ∑ i : Fin (m + 1), g i • Fin.cons x v i = 0\nj : Fin (m + 1)\n⊢ g j = 0",
"tactic": "simp_rw [Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ] at total_eq"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g (Fin.succ x) • v x = 0\nthis : g 0 = 0\n⊢ g j = 0",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g (Fin.succ x) • v x = 0\n⊢ g j = 0",
"tactic": "have : g 0 = 0 := by\n refine' x_ortho (g 0) ⟨∑ i : Fin m, g i.succ • v i, _⟩ total_eq\n exact sum_mem fun i _ => smul_mem _ _ (subset_span ⟨i, rfl⟩)"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : ∑ x : Fin m, g (Fin.succ x) • v x = 0\nthis : g 0 = 0\n⊢ g j = 0",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g (Fin.succ x) • v x = 0\nthis : g 0 = 0\n⊢ g j = 0",
"tactic": "rw [this, zero_smul, zero_add] at total_eq"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : ∑ x : Fin m, g (Fin.succ x) • v x = 0\nthis : g 0 = 0\n⊢ g j = 0",
"tactic": "exact Fin.cases this (hli _ total_eq) j"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g (Fin.succ x) • v x = 0\n⊢ ∑ i : Fin m, g (Fin.succ i) • v i ∈ span R (range v)",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g (Fin.succ x) • v x = 0\n⊢ g 0 = 0",
"tactic": "refine' x_ortho (g 0) ⟨∑ i : Fin m, g i.succ • v i, _⟩ total_eq"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.115828\nR : Type u_1\nK : Type ?u.115834\nM : Type u_2\nM' : Type ?u.115840\nM'' : Type ?u.115843\nV : Type u\nV' : Type ?u.115848\nv✝ : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx✝ y : M\nm : ℕ\nx : M\nv : Fin m → M\nhli✝ : LinearIndependent R v\nhli : ∀ (g : Fin m → R), ∑ i : Fin m, g i • v i = 0 → ∀ (i : Fin m), g i = 0\nx_ortho : ∀ (c : R) (y : { x // x ∈ span R (range v) }), c • x + ↑y = 0 → c = 0\ng : Fin (m + 1) → R\nj : Fin (m + 1)\ntotal_eq : g 0 • x + ∑ x : Fin m, g (Fin.succ x) • v x = 0\n⊢ ∑ i : Fin m, g (Fin.succ i) • v i ∈ span R (range v)",
"tactic": "exact sum_mem fun i _ => smul_mem _ _ (subset_span ⟨i, rfl⟩)"
}
] | [
295,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
] |
Mathlib/Data/Finite/Card.lean | Nat.card_eq | [
{
"state_after": "case inl\nα✝ : Type ?u.215\nβ : Type ?u.218\nγ : Type ?u.221\nα : Type u_1\nh✝ : Finite α\n⊢ Nat.card α = if h : Finite α then Fintype.card α else 0\n\ncase inr\nα✝ : Type ?u.215\nβ : Type ?u.218\nγ : Type ?u.221\nα : Type u_1\nh✝ : Infinite α\n⊢ Nat.card α = if h : Finite α then Fintype.card α else 0",
"state_before": "α✝ : Type ?u.215\nβ : Type ?u.218\nγ : Type ?u.221\nα : Type u_1\n⊢ Nat.card α = if h : Finite α then Fintype.card α else 0",
"tactic": "cases finite_or_infinite α"
},
{
"state_after": "case inl\nα✝ : Type ?u.215\nβ : Type ?u.218\nγ : Type ?u.221\nα : Type u_1\nh✝ : Finite α\nthis : Fintype α := Fintype.ofFinite α\n⊢ Nat.card α = if h : Finite α then Fintype.card α else 0",
"state_before": "case inl\nα✝ : Type ?u.215\nβ : Type ?u.218\nγ : Type ?u.221\nα : Type u_1\nh✝ : Finite α\n⊢ Nat.card α = if h : Finite α then Fintype.card α else 0",
"tactic": "letI := Fintype.ofFinite α"
},
{
"state_after": "no goals",
"state_before": "case inl\nα✝ : Type ?u.215\nβ : Type ?u.218\nγ : Type ?u.221\nα : Type u_1\nh✝ : Finite α\nthis : Fintype α := Fintype.ofFinite α\n⊢ Nat.card α = if h : Finite α then Fintype.card α else 0",
"tactic": "simp only [*, Nat.card_eq_fintype_card, dif_pos]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα✝ : Type ?u.215\nβ : Type ?u.218\nγ : Type ?u.221\nα : Type u_1\nh✝ : Infinite α\n⊢ Nat.card α = if h : Finite α then Fintype.card α else 0",
"tactic": "simp only [*, card_eq_zero_of_infinite, not_finite_iff_infinite.mpr, dite_false]"
}
] | [
57,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/Algebra/CharZero/Defs.lean | Nat.cast_ne_one | [] | [
97,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean | approximatesLinearOn_empty | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.5815\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.5918\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nε : ℝ\nf : E → F\nf' : E →L[𝕜] F\nc : ℝ≥0\n⊢ ApproximatesLinearOn f f' ∅ c",
"tactic": "simp [ApproximatesLinearOn]"
}
] | [
125,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/Algebra/Polynomial/GroupRingAction.lean | Polynomial.smul_eval_smul | [
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝⁴ : Monoid M\nR : Type ?u.35641\ninst✝³ : Semiring R\ninst✝² : MulSemiringAction M R\nS : Type u_1\ninst✝¹ : CommSemiring S\ninst✝ : MulSemiringAction M S\nm : M\nf : S[X]\nx r : S\n⊢ eval (m • x) (m • ↑C r) = m • eval x (↑C r)",
"tactic": "rw [smul_C, eval_C, eval_C]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝⁴ : Monoid M\nR : Type ?u.35641\ninst✝³ : Semiring R\ninst✝² : MulSemiringAction M R\nS : Type u_1\ninst✝¹ : CommSemiring S\ninst✝ : MulSemiringAction M S\nm : M\nf✝ : S[X]\nx : S\nf g : S[X]\nihf : eval (m • x) (m • f) = m • eval x f\nihg : eval (m • x) (m • g) = m • eval x g\n⊢ eval (m • x) (m • (f + g)) = m • eval x (f + g)",
"tactic": "rw [smul_add, eval_add, ihf, ihg, eval_add, smul_add]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝⁴ : Monoid M\nR : Type ?u.35641\ninst✝³ : Semiring R\ninst✝² : MulSemiringAction M R\nS : Type u_1\ninst✝¹ : CommSemiring S\ninst✝ : MulSemiringAction M S\nm : M\nf : S[X]\nx : S\nn : ℕ\nr : S\nx✝ : eval (m • x) (m • (↑C r * X ^ n)) = m • eval x (↑C r * X ^ n)\n⊢ eval (m • x) (m • (↑C r * X ^ (n + 1))) = m • eval x (↑C r * X ^ (n + 1))",
"tactic": "rw [smul_mul', smul_pow', smul_C, smul_X, eval_mul, eval_C, eval_pow, eval_X, eval_mul, eval_C,\n eval_pow, eval_X, smul_mul', smul_pow']"
}
] | [
72,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | Complex.sin_eq_zero_iff | [
{
"state_after": "θ : ℂ\n⊢ (∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2) ↔ ∃ k, θ = ↑k * ↑π",
"state_before": "θ : ℂ\n⊢ sin θ = 0 ↔ ∃ k, θ = ↑k * ↑π",
"tactic": "rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]"
},
{
"state_after": "case mp\nθ : ℂ\n⊢ (∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2) → ∃ k, θ = ↑k * ↑π\n\ncase mpr\nθ : ℂ\n⊢ (∃ k, θ = ↑k * ↑π) → ∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2",
"state_before": "θ : ℂ\n⊢ (∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2) ↔ ∃ k, θ = ↑k * ↑π",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nθ : ℂ\nk : ℤ\nhk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2\n⊢ ∃ k, θ = ↑k * ↑π",
"state_before": "case mp\nθ : ℂ\n⊢ (∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2) → ∃ k, θ = ↑k * ↑π",
"tactic": "rintro ⟨k, hk⟩"
},
{
"state_after": "case mp.intro\nθ : ℂ\nk : ℤ\nhk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2\n⊢ θ = ↑(k + 1) * ↑π",
"state_before": "case mp.intro\nθ : ℂ\nk : ℤ\nhk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2\n⊢ ∃ k, θ = ↑k * ↑π",
"tactic": "use k + 1"
},
{
"state_after": "case mp.intro\nθ : ℂ\nk : ℤ\nhk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2\n⊢ (2 * ↑k + 1) * ↑π + ↑π = (↑k + 1) * ↑π * 2",
"state_before": "case mp.intro\nθ : ℂ\nk : ℤ\nhk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2\n⊢ θ = ↑(k + 1) * ↑π",
"tactic": "field_simp [eq_add_of_sub_eq hk]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nθ : ℂ\nk : ℤ\nhk : θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2\n⊢ (2 * ↑k + 1) * ↑π + ↑π = (↑k + 1) * ↑π * 2",
"tactic": "ring"
},
{
"state_after": "case mpr.intro\nk : ℤ\n⊢ ∃ k_1, ↑k * ↑π - ↑π / 2 = (2 * ↑k_1 + 1) * ↑π / 2",
"state_before": "case mpr\nθ : ℂ\n⊢ (∃ k, θ = ↑k * ↑π) → ∃ k, θ - ↑π / 2 = (2 * ↑k + 1) * ↑π / 2",
"tactic": "rintro ⟨k, rfl⟩"
},
{
"state_after": "case mpr.intro\nk : ℤ\n⊢ ↑k * ↑π - ↑π / 2 = (2 * ↑(k - 1) + 1) * ↑π / 2",
"state_before": "case mpr.intro\nk : ℤ\n⊢ ∃ k_1, ↑k * ↑π - ↑π / 2 = (2 * ↑k_1 + 1) * ↑π / 2",
"tactic": "use k - 1"
},
{
"state_after": "case mpr.intro\nk : ℤ\n⊢ ↑k * ↑π * 2 - ↑π = (2 * (↑k - 1) + 1) * ↑π",
"state_before": "case mpr.intro\nk : ℤ\n⊢ ↑k * ↑π - ↑π / 2 = (2 * ↑(k - 1) + 1) * ↑π / 2",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nk : ℤ\n⊢ ↑k * ↑π * 2 - ↑π = (2 * (↑k - 1) + 1) * ↑π",
"tactic": "ring"
}
] | [
59,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
49,
1
] |
Mathlib/Data/Set/Image.lean | Function.Injective.compl_image_eq | [
{
"state_after": "case h\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\ny : β\n⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ range fᶜ",
"state_before": "ι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\n⊢ (f '' s)ᶜ = f '' sᶜ ∪ range fᶜ",
"tactic": "ext y"
},
{
"state_after": "case h.inl.intro\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\nx : α\n⊢ f x ∈ (f '' s)ᶜ ↔ f x ∈ f '' sᶜ ∪ range fᶜ\n\ncase h.inr\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\ny : β\nhx : ¬y ∈ range f\n⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ range fᶜ",
"state_before": "case h\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\ny : β\n⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ range fᶜ",
"tactic": "rcases em (y ∈ range f) with (⟨x, rfl⟩ | hx)"
},
{
"state_after": "no goals",
"state_before": "case h.inl.intro\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\nx : α\n⊢ f x ∈ (f '' s)ᶜ ↔ f x ∈ f '' sᶜ ∪ range fᶜ",
"tactic": "simp [hf.eq_iff]"
},
{
"state_after": "case h.inr\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\ny : β\nhx : ∀ (x : α), ¬f x = y\n⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ range fᶜ",
"state_before": "case h.inr\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\ny : β\nhx : ¬y ∈ range f\n⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ range fᶜ",
"tactic": "rw [mem_range, not_exists] at hx"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nι : Sort ?u.105472\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set α\ny : β\nhx : ∀ (x : α), ¬f x = y\n⊢ y ∈ (f '' s)ᶜ ↔ y ∈ f '' sᶜ ∪ range fᶜ",
"tactic": "simp [hx]"
}
] | [
1348,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1342,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.eval_zero' | [] | [
1530,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1529,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.equiv_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt₁ t₂ : Ordnode α\nh₁ : Sized t₁\nh₂ : Sized t₂\nh : toList t₁ = toList t₂\n⊢ size t₁ = size t₂",
"tactic": "rw [← length_toList h₁, h, length_toList h₂]"
}
] | [
567,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
565,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | hasFDerivAt_ring_inverse | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type ?u.1084130\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type ?u.1084230\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type ?u.1084325\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type ?u.1084420\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nthis : (fun t => Ring.inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =o[𝓝 0] _root_.id\n⊢ HasFDerivAt Ring.inverse (-↑(↑(mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹) ↑x",
"tactic": "simpa [hasFDerivAt_iff_isLittleO_nhds_zero] using this"
}
] | [
525,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/Algebra/Opposites.lean | MulOpposite.unop_inv | [] | [
303,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
302,
1
] |
Mathlib/Order/Monotone/Monovary.lean | antivaryOn_toDual_right | [] | [
284,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
283,
1
] |
Mathlib/Data/Multiset/Nodup.lean | Multiset.coe_nodup | [] | [
35,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
34,
1
] |
Mathlib/RingTheory/Ideal/Prod.lean | Ideal.ideal_prod_prime_aux | [
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\n⊢ I ≠ ⊤ ∧ J ≠ ⊤ → ¬IsPrime (prod I J)",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\n⊢ IsPrime (prod I J) → I = ⊤ ∨ J = ⊤",
"tactic": "contrapose!"
},
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\n⊢ ¬1 ∈ I ∧ ¬1 ∈ J → ¬1 ∈ prod I J → ∃ x x_1 h, ¬x ∈ prod I J ∧ ¬x_1 ∈ prod I J",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\n⊢ I ≠ ⊤ ∧ J ≠ ⊤ → ¬IsPrime (prod I J)",
"tactic": "simp only [ne_top_iff_one, isPrime_iff, not_and, not_forall, not_or]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\n⊢ ¬1 ∈ I ∧ ¬1 ∈ J → ¬1 ∈ prod I J → ∃ x x_1 h, ¬x ∈ prod I J ∧ ¬x_1 ∈ prod I J",
"tactic": "exact fun ⟨hI, hJ⟩ _ => ⟨⟨0, 1⟩, ⟨1, 0⟩, by simp, by simp [hJ], by simp [hI]⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx✝¹ : ¬1 ∈ I ∧ ¬1 ∈ J\nx✝ : ¬1 ∈ prod I J\nhI : ¬1 ∈ I\nhJ : ¬1 ∈ J\n⊢ (0, 1) * (1, 0) ∈ prod I J",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx✝¹ : ¬1 ∈ I ∧ ¬1 ∈ J\nx✝ : ¬1 ∈ prod I J\nhI : ¬1 ∈ I\nhJ : ¬1 ∈ J\n⊢ ¬(0, 1) ∈ prod I J",
"tactic": "simp [hJ]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI✝ I' : Ideal R\nJ✝ J' : Ideal S\nI : Ideal R\nJ : Ideal S\nx✝¹ : ¬1 ∈ I ∧ ¬1 ∈ J\nx✝ : ¬1 ∈ prod I J\nhI : ¬1 ∈ I\nhJ : ¬1 ∈ J\n⊢ ¬(1, 0) ∈ prod I J",
"tactic": "simp [hI]"
}
] | [
154,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/CategoryTheory/Sites/Plus.lean | CategoryTheory.GrothendieckTopology.plusMap_comp | [
{
"state_after": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nx✝ : Cᵒᵖ\n⊢ (plusMap J (η ≫ γ)).app x✝ = (plusMap J η ≫ plusMap J γ).app x✝",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\n⊢ plusMap J (η ≫ γ) = plusMap J η ≫ plusMap J γ",
"tactic": "ext : 2"
},
{
"state_after": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nx✝ : Cᵒᵖ\nS : (Cover J x✝.unop)ᵒᵖ\n⊢ colimit.ι (diagram J P x✝.unop) S ≫ (plusMap J (η ≫ γ)).app x✝ =\n colimit.ι (diagram J P x✝.unop) S ≫ (plusMap J η ≫ plusMap J γ).app x✝",
"state_before": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nx✝ : Cᵒᵖ\n⊢ (plusMap J (η ≫ γ)).app x✝ = (plusMap J η ≫ plusMap J γ).app x✝",
"tactic": "refine' colimit.hom_ext (fun S => _)"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP✝ : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q R : Cᵒᵖ ⥤ D\nη : P ⟶ Q\nγ : Q ⟶ R\nx✝ : Cᵒᵖ\nS : (Cover J x✝.unop)ᵒᵖ\n⊢ colimit.ι (diagram J P x✝.unop) S ≫ (plusMap J (η ≫ γ)).app x✝ =\n colimit.ι (diagram J P x✝.unop) S ≫ (plusMap J η ≫ plusMap J γ).app x✝",
"tactic": "simp [plusMap, J.diagramNatTrans_comp]"
}
] | [
192,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Logic/Equiv/Basic.lean | Equiv.Perm.subtypeCongr.right_apply | [
{
"state_after": "no goals",
"state_before": "ε : Type u_1\np : ε → Prop\ninst✝ : DecidablePred p\nep ep' : Perm { a // p a }\nen en' : Perm { a // ¬p a }\na : ε\nh : ¬p a\n⊢ ↑(subtypeCongr ep en) a = ↑(↑en { val := a, property := h })",
"tactic": "simp [Perm.subtypeCongr.apply, h]"
}
] | [
576,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
575,
1
] |
Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable | [
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\nε : ℝ\nhε : 0 < ε\n⊢ Tendsto (fun i => ↑↑μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\n⊢ TendstoInMeasure μ f l g",
"tactic": "intro ε hε"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n Tendsto (fun b => ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f b - g) p μ ^ ENNReal.toReal p) l\n (𝓝 (ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * 0 ^ ENNReal.toReal p))\n⊢ Tendsto (fun i => ↑↑μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nhfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)\nε : ℝ\nhε : 0 < ε\n⊢ Tendsto (fun i => ↑↑μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"tactic": "replace hfg := ENNReal.Tendsto.const_mul\n (Tendsto.ennrpow_const p.toReal hfg) (Or.inr <| @ENNReal.ofReal_ne_top (1 / ε ^ p.toReal))"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg : Tendsto (fun b => ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f b - g) p μ ^ ENNReal.toReal p) l (𝓝 0)\n⊢ Tendsto (fun i => ↑↑μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n Tendsto (fun b => ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f b - g) p μ ^ ENNReal.toReal p) l\n (𝓝 (ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * 0 ^ ENNReal.toReal p))\n⊢ Tendsto (fun i => ↑↑μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"tactic": "simp only [MulZeroClass.mul_zero,\n ENNReal.zero_rpow_of_pos (ENNReal.toReal_pos hp_ne_zero hp_ne_top)] at hfg"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\n⊢ ∀ (ε_1 : ℝ≥0∞), ε_1 > 0 → ∀ᶠ (x : ι) in l, ↑↑μ {x_1 | ε ≤ dist (f x x_1) (g x_1)} ≤ ε_1",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg : Tendsto (fun b => ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f b - g) p μ ^ ENNReal.toReal p) l (𝓝 0)\n⊢ Tendsto (fun i => ↑↑μ {x | ε ≤ dist (f i x) (g x)}) l (𝓝 0)",
"tactic": "rw [ENNReal.tendsto_nhds_zero] at hfg⊢"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\n⊢ ∀ᶠ (x : ι) in l, ↑↑μ {x_1 | ε ≤ dist (f x x_1) (g x_1)} ≤ δ",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\n⊢ ∀ (ε_1 : ℝ≥0∞), ε_1 > 0 → ∀ᶠ (x : ι) in l, ↑↑μ {x_1 | ε ≤ dist (f x x_1) (g x_1)} ≤ ε_1",
"tactic": "intro δ hδ"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\n⊢ ∀ᶠ (x : ι) in l, ↑↑μ {x_1 | ε ≤ dist (f x x_1) (g x_1)} ≤ δ",
"tactic": "refine' (hfg δ hδ).mono fun n hn => _"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ",
"tactic": "refine' le_trans _ hn"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ENNReal.ofReal (ε ^ ENNReal.toReal p) * ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ snorm (f n - g) p μ ^ ENNReal.toReal p\n\nα : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ENNReal.ofReal (ε ^ ENNReal.toReal p) ≠ 0 ∨ snorm (f n - g) p μ ^ ENNReal.toReal p ≠ 0",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p",
"tactic": "rw [ENNReal.ofReal_div_of_pos (Real.rpow_pos_of_pos hε _), ENNReal.ofReal_one, mul_comm,\n mul_one_div, ENNReal.le_div_iff_mul_le _ (Or.inl ENNReal.ofReal_ne_top), mul_comm]"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ENNReal.ofReal ε ^ ENNReal.toReal p * ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ snorm (f n - g) p μ ^ ENNReal.toReal p",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ENNReal.ofReal (ε ^ ENNReal.toReal p) * ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ snorm (f n - g) p μ ^ ENNReal.toReal p",
"tactic": "rw [← ENNReal.ofReal_rpow_of_pos hε]"
},
{
"state_after": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\nα : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\nx✝ : α\n⊢ ε ≤ dist (f n x✝) (g x✝) ↔ ENNReal.ofReal ε ≤ ↑‖(f n - g) x✝‖₊",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ENNReal.ofReal ε ^ ENNReal.toReal p * ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ snorm (f n - g) p μ ^ ENNReal.toReal p",
"tactic": "convert mul_meas_ge_le_pow_snorm' μ hp_ne_zero hp_ne_top ((hf n).sub hg).aestronglyMeasurable\n (ENNReal.ofReal ε)"
},
{
"state_after": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\nα : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\nx✝ : α\n⊢ ENNReal.ofReal ε ≤ ↑‖f n x✝ - g x✝‖₊ ↔ ENNReal.ofReal ε ≤ ↑‖(f n - g) x✝‖₊",
"state_before": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\nα : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\nx✝ : α\n⊢ ε ≤ dist (f n x✝) (g x✝) ↔ ENNReal.ofReal ε ≤ ↑‖(f n - g) x✝‖₊",
"tactic": "rw [dist_eq_norm, ← ENNReal.ofReal_le_ofReal_iff (norm_nonneg _), ofReal_norm_eq_coe_nnnorm]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6.h.e'_3.h.e'_2.h.a\nα : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\nx✝ : α\n⊢ ENNReal.ofReal ε ≤ ↑‖f n x✝ - g x✝‖₊ ↔ ENNReal.ofReal ε ≤ ↑‖(f n - g) x✝‖₊",
"tactic": "exact Iff.rfl"
},
{
"state_after": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ 0 < ε ^ ENNReal.toReal p ∨ snorm (f n - g) p μ ^ ENNReal.toReal p ≠ 0",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ ENNReal.ofReal (ε ^ ENNReal.toReal p) ≠ 0 ∨ snorm (f n - g) p μ ^ ENNReal.toReal p ≠ 0",
"tactic": "rw [Ne, ENNReal.ofReal_eq_zero, not_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type u_3\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ\nhε : 0 < ε\nhfg :\n ∀ (ε_1 : ℝ≥0∞),\n ε_1 > 0 → ∀ᶠ (x : ι) in l, ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f x - g) p μ ^ ENNReal.toReal p ≤ ε_1\nδ : ℝ≥0∞\nhδ : δ > 0\nn : ι\nhn : ENNReal.ofReal (1 / ε ^ ENNReal.toReal p) * snorm (f n - g) p μ ^ ENNReal.toReal p ≤ δ\n⊢ 0 < ε ^ ENNReal.toReal p ∨ snorm (f n - g) p μ ^ ENNReal.toReal p ≠ 0",
"tactic": "exact Or.inl (Real.rpow_pos_of_pos hε _)"
}
] | [
307,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
286,
1
] |
Mathlib/Algebra/Star/Basic.lean | star_eq_iff_star_eq | [] | [
113,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Data/Vector/Basic.lean | Vector.toList_empty | [] | [
202,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.comapDomain_zero | [
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.304416\nι : Type ?u.304419\nM : Type u_3\nM' : Type ?u.304425\nN : Type ?u.304428\nP : Type ?u.304431\nG : Type ?u.304434\nH : Type ?u.304437\nR : Type ?u.304440\nS : Type ?u.304443\ninst✝ : Zero M\nf : α → β\nhif : optParam (Set.InjOn f (f ⁻¹' ↑0.support)) (_ : Set.InjOn f (f ⁻¹' ↑∅))\na✝ : α\n⊢ ↑(comapDomain f 0 hif) a✝ = ↑0 a✝",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.304416\nι : Type ?u.304419\nM : Type u_3\nM' : Type ?u.304425\nN : Type ?u.304428\nP : Type ?u.304431\nG : Type ?u.304434\nH : Type ?u.304437\nR : Type ?u.304440\nS : Type ?u.304443\ninst✝ : Zero M\nf : α → β\nhif : optParam (Set.InjOn f (f ⁻¹' ↑0.support)) (_ : Set.InjOn f (f ⁻¹' ↑∅))\n⊢ comapDomain f 0 hif = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.304416\nι : Type ?u.304419\nM : Type u_3\nM' : Type ?u.304425\nN : Type ?u.304428\nP : Type ?u.304431\nG : Type ?u.304434\nH : Type ?u.304437\nR : Type ?u.304440\nS : Type ?u.304443\ninst✝ : Zero M\nf : α → β\nhif : optParam (Set.InjOn f (f ⁻¹' ↑0.support)) (_ : Set.InjOn f (f ⁻¹' ↑∅))\na✝ : α\n⊢ ↑(comapDomain f 0 hif) a✝ = ↑0 a✝",
"tactic": "rfl"
}
] | [
738,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
734,
1
] |
Mathlib/Order/Bounded.lean | Set.bounded_lt_Ioc | [] | [
222,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
221,
1
] |
Mathlib/Order/Circular.lean | sbtw_iff_not_btw | [
{
"state_after": "α : Type u_1\ninst✝ : CircularOrder α\na b c : α\n⊢ btw a b c ∧ ¬btw c b a ↔ ¬btw c b a",
"state_before": "α : Type u_1\ninst✝ : CircularOrder α\na b c : α\n⊢ sbtw a b c ↔ ¬btw c b a",
"tactic": "rw [sbtw_iff_btw_not_btw]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CircularOrder α\na b c : α\n⊢ btw a b c ∧ ¬btw c b a ↔ ¬btw c b a",
"tactic": "exact and_iff_right_of_imp (btw_total _ _ _).resolve_left"
}
] | [
320,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inverse.lean | HasStrictDerivAt.of_local_left_inverse | [] | [
73,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Std/Data/List/Lemmas.lean | List.get_range | [
{
"state_after": "no goals",
"state_before": "n i : Nat\nH : i < length (range n)\n⊢ some (get (range n) { val := i, isLt := H }) = some i",
"tactic": "rw [← get?_eq_get _, get?_range (by simpa using H)]"
},
{
"state_after": "no goals",
"state_before": "n i : Nat\nH : i < length (range n)\n⊢ i < n",
"tactic": "simpa using H"
}
] | [
1949,
76
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1948,
9
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.VectorMeasure.map_add | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : Measurable f\n⊢ map (v + w) f = map v f + map w f\n\ncase neg\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : ¬Measurable f\n⊢ map (v + w) f = map v f + map w f",
"state_before": "α : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\n⊢ map (v + w) f = map v f + map w f",
"tactic": "by_cases hf : Measurable f"
},
{
"state_after": "case pos.h\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : Measurable f\ni : Set β\n⊢ MeasurableSet i → ↑(map (v + w) f) i = ↑(map v f + map w f) i",
"state_before": "case pos\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : Measurable f\n⊢ map (v + w) f = map v f + map w f",
"tactic": "ext i"
},
{
"state_after": "case pos.h\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : Measurable f\ni : Set β\nhi : MeasurableSet i\n⊢ ↑(map (v + w) f) i = ↑(map v f + map w f) i",
"state_before": "case pos.h\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : Measurable f\ni : Set β\n⊢ MeasurableSet i → ↑(map (v + w) f) i = ↑(map v f + map w f) i",
"tactic": "intro hi"
},
{
"state_after": "no goals",
"state_before": "case pos.h\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : Measurable f\ni : Set β\nhi : MeasurableSet i\n⊢ ↑(map (v + w) f) i = ↑(map v f + map w f) i",
"tactic": "simp [map_apply _ hf hi]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type u_3\nm inst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv✝ : VectorMeasure α M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nf : α → β\nhf : ¬Measurable f\n⊢ map (v + w) f = map v f + map w f",
"tactic": "simp [map, dif_neg hf]"
}
] | [
740,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
735,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Summable.compl_add | [] | [
382,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
380,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.pred_castSucc_succ | [
{
"state_after": "no goals",
"state_before": "n m : ℕ\ni : Fin n\n⊢ pred (↑castSucc (succ i)) (_ : ↑castSucc (succ i) ≠ 0) = ↑castSucc i",
"tactic": "simp [eq_iff_veq]"
}
] | [
1592,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1590,
1
] |
Mathlib/Order/Filter/Extr.lean | IsMaxOn.min | [] | [
605,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
603,
1
] |
Mathlib/Data/Nat/Totient.lean | Nat.totient_div_of_dvd | [
{
"state_after": "case inl\nn : ℕ\nhnd : 0 ∣ n\n⊢ φ (n / 0) = Finset.card (filter (fun k => gcd n k = 0) (range n))\n\ncase inr\nn d : ℕ\nhnd : d ∣ n\nhd0 : d > 0\n⊢ φ (n / d) = Finset.card (filter (fun k => gcd n k = d) (range n))",
"state_before": "n d : ℕ\nhnd : d ∣ n\n⊢ φ (n / d) = Finset.card (filter (fun k => gcd n k = d) (range n))",
"tactic": "rcases d.eq_zero_or_pos with (rfl | hd0)"
},
{
"state_after": "case inr.intro\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ φ (d * x / d) = Finset.card (filter (fun k => gcd (d * x) k = d) (range (d * x)))",
"state_before": "case inr\nn d : ℕ\nhnd : d ∣ n\nhd0 : d > 0\n⊢ φ (n / d) = Finset.card (filter (fun k => gcd n k = d) (range n))",
"tactic": "rcases hnd with ⟨x, rfl⟩"
},
{
"state_after": "case inr.intro\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ φ x = Finset.card (filter (fun k => gcd (d * x) k = d) (range (d * x)))",
"state_before": "case inr.intro\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ φ (d * x / d) = Finset.card (filter (fun k => gcd (d * x) k = d) (range (d * x)))",
"tactic": "rw [Nat.mul_div_cancel_left x hd0]"
},
{
"state_after": "case inr.intro.h₁\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (a : ℕ), a ∈ filter (coprime x) (range x) → d * a ∈ filter (fun k => gcd (d * x) k = d) (range (d * x))\n\ncase inr.intro.h₂\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (a b : ℕ), a ∈ filter (coprime x) (range x) → b ∈ filter (coprime x) (range x) → d * a = d * b → a = b\n\ncase inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (b : ℕ), b ∈ filter (fun k => gcd (d * x) k = d) (range (d * x)) → ∃ a ha, d * a = b",
"state_before": "case inr.intro\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ φ x = Finset.card (filter (fun k => gcd (d * x) k = d) (range (d * x)))",
"tactic": "apply Finset.card_congr fun k _ => d * k"
},
{
"state_after": "no goals",
"state_before": "case inl\nn : ℕ\nhnd : 0 ∣ n\n⊢ φ (n / 0) = Finset.card (filter (fun k => gcd n k = 0) (range n))",
"tactic": "simp [eq_zero_of_zero_dvd hnd]"
},
{
"state_after": "case inr.intro.h₁\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (a : ℕ), a < x → gcd x a = 1 → d * a < d * x ∧ gcd (d * x) (d * a) = d",
"state_before": "case inr.intro.h₁\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (a : ℕ), a ∈ filter (coprime x) (range x) → d * a ∈ filter (fun k => gcd (d * x) k = d) (range (d * x))",
"tactic": "simp only [mem_filter, mem_range, and_imp, coprime]"
},
{
"state_after": "case inr.intro.h₁\nd : ℕ\nhd0 : d > 0\nx a : ℕ\nha1 : a < x\nha2 : gcd x a = 1\n⊢ gcd (d * x) (d * a) = d",
"state_before": "case inr.intro.h₁\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (a : ℕ), a < x → gcd x a = 1 → d * a < d * x ∧ gcd (d * x) (d * a) = d",
"tactic": "refine' fun a ha1 ha2 => ⟨(mul_lt_mul_left hd0).2 ha1, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.h₁\nd : ℕ\nhd0 : d > 0\nx a : ℕ\nha1 : a < x\nha2 : gcd x a = 1\n⊢ gcd (d * x) (d * a) = d",
"tactic": "rw [gcd_mul_left, ha2, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.h₂\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (a b : ℕ), a ∈ filter (coprime x) (range x) → b ∈ filter (coprime x) (range x) → d * a = d * b → a = b",
"tactic": "simp [hd0.ne']"
},
{
"state_after": "case inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (b : ℕ), b < d * x → gcd (d * x) b = d → ∃ a, (a < x ∧ coprime x a) ∧ d * a = b",
"state_before": "case inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (b : ℕ), b ∈ filter (fun k => gcd (d * x) k = d) (range (d * x)) → ∃ a ha, d * a = b",
"tactic": "simp only [mem_filter, mem_range, exists_prop, and_imp]"
},
{
"state_after": "case inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx b : ℕ\nhb1 : b < d * x\nhb2 : gcd (d * x) b = d\n⊢ ∃ a, (a < x ∧ coprime x a) ∧ d * a = b",
"state_before": "case inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx : ℕ\n⊢ ∀ (b : ℕ), b < d * x → gcd (d * x) b = d → ∃ a, (a < x ∧ coprime x a) ∧ d * a = b",
"tactic": "refine' fun b hb1 hb2 => _"
},
{
"state_after": "case inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx b : ℕ\nhb1 : b < d * x\nhb2 : gcd (d * x) b = d\nthis : d ∣ b\n⊢ ∃ a, (a < x ∧ coprime x a) ∧ d * a = b",
"state_before": "case inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx b : ℕ\nhb1 : b < d * x\nhb2 : gcd (d * x) b = d\n⊢ ∃ a, (a < x ∧ coprime x a) ∧ d * a = b",
"tactic": "have : d ∣ b := by\n rw [← hb2]\n apply gcd_dvd_right"
},
{
"state_after": "case inr.intro.h₃.intro\nd : ℕ\nhd0 : d > 0\nx q : ℕ\nhb1 : d * q < d * x\nhb2 : gcd (d * x) (d * q) = d\n⊢ ∃ a, (a < x ∧ coprime x a) ∧ d * a = d * q",
"state_before": "case inr.intro.h₃\nd : ℕ\nhd0 : d > 0\nx b : ℕ\nhb1 : b < d * x\nhb2 : gcd (d * x) b = d\nthis : d ∣ b\n⊢ ∃ a, (a < x ∧ coprime x a) ∧ d * a = b",
"tactic": "rcases this with ⟨q, rfl⟩"
},
{
"state_after": "case inr.intro.h₃.intro\nd : ℕ\nhd0 : d > 0\nx q : ℕ\nhb1 : d * q < d * x\nhb2 : gcd (d * x) (d * q) = d\n⊢ coprime x q",
"state_before": "case inr.intro.h₃.intro\nd : ℕ\nhd0 : d > 0\nx q : ℕ\nhb1 : d * q < d * x\nhb2 : gcd (d * x) (d * q) = d\n⊢ ∃ a, (a < x ∧ coprime x a) ∧ d * a = d * q",
"tactic": "refine' ⟨q, ⟨⟨(mul_lt_mul_left hd0).1 hb1, _⟩, rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.h₃.intro\nd : ℕ\nhd0 : d > 0\nx q : ℕ\nhb1 : d * q < d * x\nhb2 : gcd (d * x) (d * q) = d\n⊢ coprime x q",
"tactic": "rwa [gcd_mul_left, mul_right_eq_self_iff hd0] at hb2"
},
{
"state_after": "d : ℕ\nhd0 : d > 0\nx b : ℕ\nhb1 : b < d * x\nhb2 : gcd (d * x) b = d\n⊢ gcd (d * x) b ∣ b",
"state_before": "d : ℕ\nhd0 : d > 0\nx b : ℕ\nhb1 : b < d * x\nhb2 : gcd (d * x) b = d\n⊢ d ∣ b",
"tactic": "rw [← hb2]"
},
{
"state_after": "no goals",
"state_before": "d : ℕ\nhd0 : d > 0\nx b : ℕ\nhb1 : b < d * x\nhb2 : gcd (d * x) b = d\n⊢ gcd (d * x) b ∣ b",
"tactic": "apply gcd_dvd_right"
}
] | [
170,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
] |
Mathlib/Order/Disjoint.lean | codisjoint_bot | [] | [
294,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
] |
Mathlib/Data/List/Basic.lean | List.last_eq_of_concat_eq | [
{
"state_after": "ι : Type ?u.24637\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na b : α\nl : List α\nh : concat l a = concat l b\n⊢ a = b",
"state_before": "ι : Type ?u.24637\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na b : α\nl : List α\n⊢ concat l a = concat l b → a = b",
"tactic": "intro h"
},
{
"state_after": "ι : Type ?u.24637\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na b : α\nl : List α\nh : l ++ [a] = l ++ [b]\n⊢ a = b",
"state_before": "ι : Type ?u.24637\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na b : α\nl : List α\nh : concat l a = concat l b\n⊢ a = b",
"tactic": "rw [concat_eq_append, concat_eq_append] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.24637\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na b : α\nl : List α\nh : l ++ [a] = l ++ [b]\n⊢ a = b",
"tactic": "exact head_eq_of_cons_eq (append_left_cancel h)"
}
] | [
565,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
562,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean | CategoryTheory.MorphismProperty.StableUnderBaseChange.unop | [] | [
290,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_add_const_Ioo | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x + a) ⁻¹' Ioo b c = Ioo (b - a) (c - a)",
"tactic": "simp [← Ioi_inter_Iio]"
}
] | [
124,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Algebra/AddTorsor.lean | vsub_ne_zero | [] | [
144,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.EventuallyLE.diff | [] | [
1733,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1731,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | MonoidHom.eq_of_eqOn_dense | [] | [
2918,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2917,
1
] |
Mathlib/Analysis/Convex/Side.lean | AffineSubspace.SOppSide.right_not_mem | [] | [
187,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
1
] |
Mathlib/Topology/Order/Basic.lean | not_tendsto_nhds_of_tendsto_atBot | [] | [
1564,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1562,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.cospanCompIso_app_one | [] | [
299,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
299,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean | Polynomial.aeval_monomial | [] | [
212,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
] |
Mathlib/Algebra/Lie/Quotient.lean | LieSubmodule.Quotient.mk'_ker | [
{
"state_after": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nm✝ : M\n⊢ m✝ ∈ LieModuleHom.ker (mk' N) ↔ m✝ ∈ N",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\n⊢ LieModuleHom.ker (mk' N) = N",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nm✝ : M\n⊢ m✝ ∈ LieModuleHom.ker (mk' N) ↔ m✝ ∈ N",
"tactic": "simp"
}
] | [
202,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.le_div_iff_mul_le | [
{
"state_after": "no goals",
"state_before": "k x y : Nat\nk0 : 0 < k\n⊢ x ≤ y / k ↔ x * k ≤ y",
"tactic": "induction y, k using mod.inductionOn generalizing x with\n (rw [div_eq]; simp [h]; cases x with simp [zero_le] | succ x => ?_)\n| base y k h =>\n simp [not_succ_le_zero x, succ_mul, Nat.add_comm]\n refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_right ..)\n exact Nat.not_le.1 fun h' => h ⟨k0, h'⟩\n| ind y k h IH =>\n rw [← add_one, Nat.add_le_add_iff_le_right, IH k0, succ_mul,\n ← Nat.add_sub_cancel (x*k) k, Nat.sub_le_sub_right_iff h.2, Nat.add_sub_cancel]"
},
{
"state_after": "case ind\ny k : Nat\nh : 0 < k ∧ k ≤ y\nIH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)\nx : Nat\nk0 : 0 < k\n⊢ (x ≤ if 0 < k ∧ k ≤ y then (y - k) / k + 1 else 0) ↔ x * k ≤ y",
"state_before": "case ind\ny k : Nat\nh : 0 < k ∧ k ≤ y\nIH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)\nx : Nat\nk0 : 0 < k\n⊢ x ≤ y / k ↔ x * k ≤ y",
"tactic": "rw [div_eq]"
},
{
"state_after": "case ind\ny k : Nat\nh : 0 < k ∧ k ≤ y\nIH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)\nx : Nat\nk0 : 0 < k\n⊢ x ≤ (y - k) / k + 1 ↔ x * k ≤ y",
"state_before": "case ind\ny k : Nat\nh : 0 < k ∧ k ≤ y\nIH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)\nx : Nat\nk0 : 0 < k\n⊢ (x ≤ if 0 < k ∧ k ≤ y then (y - k) / k + 1 else 0) ↔ x * k ≤ y",
"tactic": "simp [h]"
},
{
"state_after": "case ind.succ\ny k : Nat\nh : 0 < k ∧ k ≤ y\nIH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)\nk0 : 0 < k\nx : Nat\n⊢ succ x ≤ (y - k) / k + 1 ↔ succ x * k ≤ y",
"state_before": "case ind\ny k : Nat\nh : 0 < k ∧ k ≤ y\nIH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)\nx : Nat\nk0 : 0 < k\n⊢ x ≤ (y - k) / k + 1 ↔ x * k ≤ y",
"tactic": "cases x with simp [zero_le] | succ x => ?_"
},
{
"state_after": "case base.succ\ny k : Nat\nh : ¬(0 < k ∧ k ≤ y)\nk0 : 0 < k\nx : Nat\n⊢ y < k + x * k",
"state_before": "case base.succ\ny k : Nat\nh : ¬(0 < k ∧ k ≤ y)\nk0 : 0 < k\nx : Nat\n⊢ y < succ x * k",
"tactic": "simp [not_succ_le_zero x, succ_mul, Nat.add_comm]"
},
{
"state_after": "case base.succ\ny k : Nat\nh : ¬(0 < k ∧ k ≤ y)\nk0 : 0 < k\nx : Nat\n⊢ y < k",
"state_before": "case base.succ\ny k : Nat\nh : ¬(0 < k ∧ k ≤ y)\nk0 : 0 < k\nx : Nat\n⊢ y < k + x * k",
"tactic": "refine Nat.lt_of_lt_of_le ?_ (Nat.le_add_right ..)"
},
{
"state_after": "no goals",
"state_before": "case base.succ\ny k : Nat\nh : ¬(0 < k ∧ k ≤ y)\nk0 : 0 < k\nx : Nat\n⊢ y < k",
"tactic": "exact Nat.not_le.1 fun h' => h ⟨k0, h'⟩"
},
{
"state_after": "no goals",
"state_before": "case ind.succ\ny k : Nat\nh : 0 < k ∧ k ≤ y\nIH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)\nk0 : 0 < k\nx : Nat\n⊢ succ x ≤ (y - k) / k + 1 ↔ succ x * k ≤ y",
"tactic": "rw [← add_one, Nat.add_le_add_iff_le_right, IH k0, succ_mul,\n ← Nat.add_sub_cancel (x*k) k, Nat.sub_le_sub_right_iff h.2, Nat.add_sub_cancel]"
}
] | [
276,
88
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
267,
1
] |
Mathlib/Topology/SubsetProperties.lean | IsCompact.insert | [] | [
457,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
456,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.liftRel_mk_mk | [] | [
430,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
428,
1
] |
Mathlib/Data/QPF/Univariate/Basic.lean | Qpf.Wequiv.refl | [
{
"state_after": "case mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a → WType (P F).B\n⊢ Wequiv (WType.mk a f) (WType.mk a f)",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx : PFunctor.W (P F)\n⊢ Wequiv x x",
"tactic": "cases' x with a f"
},
{
"state_after": "no goals",
"state_before": "case mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a → WType (P F).B\n⊢ Wequiv (WType.mk a f) (WType.mk a f)",
"tactic": "exact Wequiv.abs a f a f rfl"
}
] | [
215,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Data/Fin/Interval.lean | Fin.card_Ioi | [
{
"state_after": "case b\nn : ℕ\na b : Fin n\n⊢ Fin n",
"state_before": "n : ℕ\na b : Fin n\n⊢ card (Ioi a) = n - 1 - ↑a",
"tactic": "rw [← card_map, map_valEmbedding_Ioi, Nat.card_Ioc]"
},
{
"state_after": "no goals",
"state_before": "case b\nn : ℕ\na b : Fin n\n⊢ Fin n",
"tactic": "assumption"
}
] | [
186,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
184,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean | Ideal.map_quotient_self | [] | [
99,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean | OrthogonalFamily.summable_of_lp | [
{
"state_after": "ι : Type u_1\n𝕜 : Type u_4\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nf : { x // x ∈ lp G 2 }\n⊢ Summable fun i => ‖↑f i‖ ^ 2",
"state_before": "ι : Type u_1\n𝕜 : Type u_4\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nf : { x // x ∈ lp G 2 }\n⊢ Summable fun i => ↑(V i) (↑f i)",
"tactic": "rw [hV.summable_iff_norm_sq_summable]"
},
{
"state_after": "case h.e'_5.h\nι : Type u_1\n𝕜 : Type u_4\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nf : { x // x ∈ lp G 2 }\nx✝ : ι\n⊢ ‖↑f x✝‖ ^ 2 = ‖↑f x✝‖ ^ ENNReal.toReal 2\n\nι : Type u_1\n𝕜 : Type u_4\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nf : { x // x ∈ lp G 2 }\n⊢ 0 < ENNReal.toReal 2",
"state_before": "ι : Type u_1\n𝕜 : Type u_4\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nf : { x // x ∈ lp G 2 }\n⊢ Summable fun i => ‖↑f i‖ ^ 2",
"tactic": "convert (lp.memℓp f).summable _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nι : Type u_1\n𝕜 : Type u_4\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nf : { x // x ∈ lp G 2 }\nx✝ : ι\n⊢ ‖↑f x✝‖ ^ 2 = ‖↑f x✝‖ ^ ENNReal.toReal 2",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\n𝕜 : Type u_4\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nf : { x // x ∈ lp G 2 }\n⊢ 0 < ENNReal.toReal 2",
"tactic": "norm_num"
}
] | [
201,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
11
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean | IsHilbertSum.linearIsometryEquiv_apply_dfinsupp_sum_single | [
{
"state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\nW₀ : Π₀ (i : ι), G i\n⊢ ↑(↑(linearIsometryEquiv hV) (↑(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) (Dfinsupp.sum W₀ (lp.single 2)))) =\n ↑W₀",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\nW₀ : Π₀ (i : ι), G i\n⊢ ↑(↑(linearIsometryEquiv hV) (Dfinsupp.sum W₀ fun i => ↑(V i))) = ↑W₀",
"tactic": "rw [← hV.linearIsometryEquiv_symm_apply_dfinsupp_sum_single]"
},
{
"state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\nW₀ : Π₀ (i : ι), G i\n⊢ ↑(Dfinsupp.sum W₀ (lp.single 2)) = ↑W₀",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\nW₀ : Π₀ (i : ι), G i\n⊢ ↑(↑(linearIsometryEquiv hV) (↑(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) (Dfinsupp.sum W₀ (lp.single 2)))) =\n ↑W₀",
"tactic": "rw [LinearIsometryEquiv.apply_symm_apply]"
},
{
"state_after": "case h\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\nW₀ : Π₀ (i : ι), G i\ni : ι\n⊢ ↑(Dfinsupp.sum W₀ (lp.single 2)) i = ↑W₀ i",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\nW₀ : Π₀ (i : ι), G i\n⊢ ↑(Dfinsupp.sum W₀ (lp.single 2)) = ↑W₀",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\nW₀ : Π₀ (i : ι), G i\ni : ι\n⊢ ↑(Dfinsupp.sum W₀ (lp.single 2)) i = ↑W₀ i",
"tactic": "simp (config := { contextual := true }) [Dfinsupp.sum, lp.single_apply]"
}
] | [
377,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
371,
11
] |
Mathlib/Data/Real/Irrational.lean | Irrational.add_int | [] | [
243,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
1
] |
Mathlib/Topology/ContinuousFunction/Weierstrass.lean | continuousMap_mem_polynomialFunctions_closure | [
{
"state_after": "a b : ℝ\nf : C(↑(Set.Icc a b), ℝ)\n⊢ f ∈ ⊤",
"state_before": "a b : ℝ\nf : C(↑(Set.Icc a b), ℝ)\n⊢ f ∈ Subalgebra.topologicalClosure (polynomialFunctions (Set.Icc a b))",
"tactic": "rw [polynomialFunctions_closure_eq_top _ _]"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nf : C(↑(Set.Icc a b), ℝ)\n⊢ f ∈ ⊤",
"tactic": "simp"
}
] | [
96,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Order/Hom/Basic.lean | OrderHomClass.mono | [] | [
160,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
11
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | Complex.hasStrictDerivAt_exp | [] | [
76,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Order/GaloisConnection.lean | GaloisInsertion.l_inf_u | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.39919\na✝ a₁ a₂ : α\nb✝ b₁ b₂ : β\nl : α → β\nu : β → α\ninst✝¹ : SemilatticeInf α\ninst✝ : SemilatticeInf β\ngi : GaloisInsertion l u\na b : β\n⊢ l (u (a ⊓ b)) = a ⊓ b",
"tactic": "simp only [gi.l_u_eq]"
}
] | [
560,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
556,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean | AffineIndependent.comp_embedding | [
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\n⊢ AffineIndependent k (p ∘ ↑f)",
"tactic": "classical\n intro fs w hw hs i0 hi0\n let fs' := fs.map f\n let w' i := if h : ∃ i2, f i2 = i then w h.choose else 0\n have hw' : ∀ i2 : ι2, w' (f i2) = w i2 := by\n intro i2\n have h : ∃ i : ι2, f i = f i2 := ⟨i2, rfl⟩\n have hs : h.choose = i2 := f.injective h.choose_spec\n simp_rw [dif_pos h, hs]\n have hw's : (∑ i in fs', w' i) = 0 := by\n rw [← hw, Finset.sum_map]\n simp [hw']\n have hs' : fs'.weightedVSub p w' = (0 : V) := by\n rw [← hs, Finset.weightedVSub_map]\n congr with i\n simp_all only [comp_apply, EmbeddingLike.apply_eq_iff_eq, exists_eq, dite_true]\n rw [← ha fs' w' hw's hs' (f i0) ((Finset.mem_map' _).2 hi0), hw']"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\n⊢ w i0 = 0",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\n⊢ AffineIndependent k (p ∘ ↑f)",
"tactic": "intro fs w hw hs i0 hi0"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\n⊢ w i0 = 0",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\n⊢ w i0 = 0",
"tactic": "let fs' := fs.map f"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\n⊢ w i0 = 0",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\n⊢ w i0 = 0",
"tactic": "let w' i := if h : ∃ i2, f i2 = i then w h.choose else 0"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\n⊢ w i0 = 0",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\n⊢ w i0 = 0",
"tactic": "have hw' : ∀ i2 : ι2, w' (f i2) = w i2 := by\n intro i2\n have h : ∃ i : ι2, f i = f i2 := ⟨i2, rfl⟩\n have hs : h.choose = i2 := f.injective h.choose_spec\n simp_rw [dif_pos h, hs]"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\n⊢ w i0 = 0",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\n⊢ w i0 = 0",
"tactic": "have hw's : (∑ i in fs', w' i) = 0 := by\n rw [← hw, Finset.sum_map]\n simp [hw']"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\nhs' : ↑(Finset.weightedVSub fs' p) w' = 0\n⊢ w i0 = 0",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\n⊢ w i0 = 0",
"tactic": "have hs' : fs'.weightedVSub p w' = (0 : V) := by\n rw [← hs, Finset.weightedVSub_map]\n congr with i\n simp_all only [comp_apply, EmbeddingLike.apply_eq_iff_eq, exists_eq, dite_true]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\nhs' : ↑(Finset.weightedVSub fs' p) w' = 0\n⊢ w i0 = 0",
"tactic": "rw [← ha fs' w' hw's hs' (f i0) ((Finset.mem_map' _).2 hi0), hw']"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\ni2 : ι2\n⊢ w' (↑f i2) = w i2",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\n⊢ ∀ (i2 : ι2), w' (↑f i2) = w i2",
"tactic": "intro i2"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\ni2 : ι2\nh : ∃ i, ↑f i = ↑f i2\n⊢ w' (↑f i2) = w i2",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\ni2 : ι2\n⊢ w' (↑f i2) = w i2",
"tactic": "have h : ∃ i : ι2, f i = f i2 := ⟨i2, rfl⟩"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs✝ : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\ni2 : ι2\nh : ∃ i, ↑f i = ↑f i2\nhs : Exists.choose h = i2\n⊢ w' (↑f i2) = w i2",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\ni2 : ι2\nh : ∃ i, ↑f i = ↑f i2\n⊢ w' (↑f i2) = w i2",
"tactic": "have hs : h.choose = i2 := f.injective h.choose_spec"
},
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs✝ : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\ni2 : ι2\nh : ∃ i, ↑f i = ↑f i2\nhs : Exists.choose h = i2\n⊢ w' (↑f i2) = w i2",
"tactic": "simp_rw [dif_pos h, hs]"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\n⊢ ∑ x in fs, w' (↑f x) = ∑ i in fs, w i",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\n⊢ ∑ i in fs', w' i = 0",
"tactic": "rw [← hw, Finset.sum_map]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\n⊢ ∑ x in fs, w' (↑f x) = ∑ i in fs, w i",
"tactic": "simp [hw']"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\n⊢ ↑(Finset.weightedVSub fs (p ∘ ↑f)) (w' ∘ ↑f) = ↑(Finset.weightedVSub fs (p ∘ ↑f)) w",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\n⊢ ↑(Finset.weightedVSub fs' p) w' = 0",
"tactic": "rw [← hs, Finset.weightedVSub_map]"
},
{
"state_after": "case h.e_6.h.h\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\ni : ι2\n⊢ (w' ∘ ↑f) i = w i",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\n⊢ ↑(Finset.weightedVSub fs (p ∘ ↑f)) (w' ∘ ↑f) = ↑(Finset.weightedVSub fs (p ∘ ↑f)) w",
"tactic": "congr with i"
},
{
"state_after": "no goals",
"state_before": "case h.e_6.h.h\nk : Type u_3\nV : Type u_4\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\nι2 : Type u_1\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i in fs, w i = 0\nhs : ↑(Finset.weightedVSub fs (p ∘ ↑f)) w = 0\ni0 : ι2\nhi0 : i0 ∈ fs\nfs' : Finset ι := Finset.map f fs\nw' : ι → k := fun i => if h : ∃ i2, ↑f i2 = i then w (Exists.choose h) else 0\nhw' : ∀ (i2 : ι2), w' (↑f i2) = w i2\nhw's : ∑ i in fs', w' i = 0\ni : ι2\n⊢ (w' ∘ ↑f) i = w i",
"tactic": "simp_all only [comp_apply, EmbeddingLike.apply_eq_iff_eq, exists_eq, dite_true]"
}
] | [
309,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
291,
1
] |
Mathlib/Data/Rat/Defs.lean | Rat.eq_iff_mul_eq_mul | [
{
"state_after": "a b c p q : ℚ\n⊢ p.num /. ↑p.den = q.num /. ↑q.den ↔ p.num * ↑q.den = q.num * ↑p.den",
"state_before": "a b c p q : ℚ\n⊢ p = q ↔ p.num * ↑q.den = q.num * ↑p.den",
"tactic": "conv =>\n lhs\n rw [← @num_den p, ← @num_den q]"
},
{
"state_after": "case z₂\na b c p q : ℚ\n⊢ ¬q.den = 0",
"state_before": "case z₂\na b c p q : ℚ\n⊢ ↑q.den ≠ 0",
"tactic": "rw [← Nat.cast_zero, Ne, Int.ofNat_inj]"
},
{
"state_after": "no goals",
"state_before": "case z₂\na b c p q : ℚ\n⊢ ¬q.den = 0",
"tactic": "apply den_nz"
}
] | [
393,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
1
] |
Mathlib/Topology/Constructions.lean | nhds_toMul | [] | [
136,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
] |
Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_inv | [] | [
529,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
527,
1
] |
Mathlib/CategoryTheory/Preadditive/Mat.lean | CategoryTheory.Mat_.add_apply | [] | [
175,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.copy_nil | [
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' : V\n⊢ Walk.copy nil (_ : u' = u') (_ : u' = u') = nil",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu u' : V\nhu : u = u'\n⊢ Walk.copy nil hu hu = nil",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu' : V\n⊢ Walk.copy nil (_ : u' = u') (_ : u' = u') = nil",
"tactic": "rfl"
}
] | [
152,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean | Nat.bitwise'_bit | [
{
"state_after": "f : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit a m)\n (bit b n) =\n bit (f a b)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m n)",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\n⊢ bitwise' f (bit a m) (bit b n) = bit (f a b) (bitwise' f m n)",
"tactic": "unfold bitwise'"
},
{
"state_after": "case h\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif f true false then bit a m else 0\n\ncase h\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\n⊢ (binaryRec (bif f true false then bit false 0 else 0) fun b n x => bit (f false b) (bif f false true then n else 0)) =\n fun n => bif f false true then n else 0",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) (bit a m)\n (bit b n) =\n bit (f a b)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m n)",
"tactic": "rw [binaryRec_eq, binaryRec_eq]"
},
{
"state_after": "case h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif false then bit a m else 0\n\ncase h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0",
"state_before": "case h\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif f true false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif f true false then bit a m else 0",
"tactic": "induction' ftf : f true false"
},
{
"state_after": "case h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ bit false\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif false then bit a m else 0\n\ncase h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0",
"state_before": "case h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif false then bit a m else 0\n\ncase h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0",
"tactic": "rw [show f a false = false by cases a <;> assumption]"
},
{
"state_after": "case h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n 0\n\ncase h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0",
"state_before": "case h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ bit false\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif false then bit a m else 0\n\ncase h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0",
"tactic": "apply @congr_arg _ _ _ 0 (bit false)"
},
{
"state_after": "case h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0\n\ncase h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n 0",
"state_before": "case h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n 0\n\ncase h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0",
"tactic": "swap"
},
{
"state_after": "case h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit a\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0\n\ncase h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n 0",
"state_before": "case h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit (f a false)\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0\n\ncase h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n 0",
"tactic": "rw [show f a false = a by cases a <;> assumption]"
},
{
"state_after": "case h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n m\n\ncase h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n 0",
"state_before": "case h.true\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ bit a\n (binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif true then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0) =\n bif true then bit a m else 0\n\ncase h.false\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ binaryRec (fun n => bif f false true then n else 0)\n (fun a m Ia => binaryRec (bif false then bit a m else 0) fun b n x => bit (f a b) (Ia n)) m 0 =\n 0",
"tactic": "apply congr_arg (bit a)"
},
{
"state_after": "no goals",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\n⊢ f a false = false",
"tactic": "cases a <;> assumption"
},
{
"state_after": "no goals",
"state_before": "f : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = true\n⊢ f a false = a",
"tactic": "cases a <;> assumption"
},
{
"state_after": "no goals",
"state_before": "case h.false\nf : Bool → Bool → Bool\nh : f false false = false\na✝ : Bool\nm✝ : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\na : Bool\nm : ℕ\n⊢ (bif false then bit a m else 0) = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.false.h\nf : Bool → Bool → Bool\nh : f false false = false\na✝ : Bool\nm✝ : ℕ\nb : Bool\nn : ℕ\nx✝ : Bool\nftf✝ : f true false = x✝\nftf : f true false = false\na : Bool\nm : ℕ\n⊢ (binaryRec (bif false then bit false 0 else 0) fun b n x => bit (f false b) (bif f false true then n else 0)) =\n fun n => bif f false true then n else 0",
"tactic": "rw [← bitwise'_bit_aux h, ftf]"
},
{
"state_after": "no goals",
"state_before": "case h\nf : Bool → Bool → Bool\nh : f false false = false\na : Bool\nm : ℕ\nb : Bool\nn : ℕ\n⊢ (binaryRec (bif f true false then bit false 0 else 0) fun b n x => bit (f false b) (bif f false true then n else 0)) =\n fun n => bif f false true then n else 0",
"tactic": "exact bitwise'_bit_aux h"
}
] | [
443,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
427,
1
] |
Mathlib/Data/List/FinRange.lean | List.finRange_succ_eq_map | [
{
"state_after": "case a\nα : Type u\nn : ℕ\n⊢ map Fin.val (finRange (Nat.succ n)) = map Fin.val (0 :: map Fin.succ (finRange n))",
"state_before": "α : Type u\nn : ℕ\n⊢ finRange (Nat.succ n) = 0 :: map Fin.succ (finRange n)",
"tactic": "apply map_injective_iff.mpr Fin.val_injective"
},
{
"state_after": "case a\nα : Type u\nn : ℕ\n⊢ 0 :: map (Nat.succ ∘ Fin.val) (finRange n) = 0 :: map (Fin.val ∘ Fin.succ) (finRange n)",
"state_before": "case a\nα : Type u\nn : ℕ\n⊢ map Fin.val (finRange (Nat.succ n)) = map Fin.val (0 :: map Fin.succ (finRange n))",
"tactic": "rw [map_cons, map_coe_finRange, range_succ_eq_map, Fin.val_zero, ← map_coe_finRange, map_map,\n map_map]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nn : ℕ\n⊢ 0 :: map (Nat.succ ∘ Fin.val) (finRange n) = 0 :: map (Fin.val ∘ Fin.succ) (finRange n)",
"tactic": "simp only [Function.comp, Fin.val_succ]"
}
] | [
37,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
33,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean | Matrix.TransvectionStruct.inv_mul | [
{
"state_after": "case mk\nn : Type u_1\np : Type ?u.32998\nR : Type u₂\n𝕜 : Type ?u.33003\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\ni✝ j✝ : n\nt_hij : i✝ ≠ j✝\nc✝ : R\n⊢ toMatrix (TransvectionStruct.inv { i := i✝, j := j✝, hij := t_hij, c := c✝ }) ⬝\n toMatrix { i := i✝, j := j✝, hij := t_hij, c := c✝ } =\n 1",
"state_before": "n : Type u_1\np : Type ?u.32998\nR : Type u₂\n𝕜 : Type ?u.33003\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nt : TransvectionStruct n R\n⊢ toMatrix (TransvectionStruct.inv t) ⬝ toMatrix t = 1",
"tactic": "rcases t with ⟨_, _, t_hij⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nn : Type u_1\np : Type ?u.32998\nR : Type u₂\n𝕜 : Type ?u.33003\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\ni✝ j✝ : n\nt_hij : i✝ ≠ j✝\nc✝ : R\n⊢ toMatrix (TransvectionStruct.inv { i := i✝, j := j✝, hij := t_hij, c := c✝ }) ⬝\n toMatrix { i := i✝, j := j✝, hij := t_hij, c := c✝ } =\n 1",
"tactic": "simp [toMatrix, transvection_mul_transvection_same, t_hij]"
}
] | [
211,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.div_subset_div | [] | [
638,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
637,
1
] |
Mathlib/Init/Function.lean | Function.curry_uncurry | [] | [
144,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
9
] |
Mathlib/Computability/Primrec.lean | Primrec.nat_lt | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.137547\nβ : Type ?u.137550\nγ : Type ?u.137553\nδ : Type ?u.137556\nσ : Type ?u.137559\ninst✝⁴ : Primcodable α\ninst✝³ : Primcodable β\ninst✝² : Primcodable γ\ninst✝¹ : Primcodable δ\ninst✝ : Primcodable σ\np : ℕ × ℕ\n⊢ ¬p.snd ≤ p.fst ↔ (fun x x_1 => x < x_1) p.fst p.snd",
"tactic": "simp"
}
] | [
756,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
755,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | LinearIsometryEquiv.differentiableWithinAt | [] | [
282,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
281,
11
] |
Mathlib/Algebra/BigOperators/Ring.lean | Finset.prod_sub_ordered | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nb : β\nf✝ g✝ : α → β\ninst✝² : CommSemiring β\nι : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrder ι\ns : Finset ι\nf g : ι → R\n⊢ ∏ x in s, (f x + -g x) =\n ∏ i in s, f i +\n -∑ x in s, (g x * ∏ x in filter (fun x_1 => x_1 < x) s, (f x + -g x)) * ∏ i in filter (fun j => x < j) s, f i",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nb : β\nf✝ g✝ : α → β\ninst✝² : CommSemiring β\nι : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrder ι\ns : Finset ι\nf g : ι → R\n⊢ ∏ i in s, (f i - g i) =\n ∏ i in s, f i -\n ∑ i in s, (g i * ∏ j in filter (fun x => x < i) s, (f j - g j)) * ∏ j in filter (fun j => i < j) s, f j",
"tactic": "simp only [sub_eq_add_neg]"
},
{
"state_after": "case h.e'_3.h.e'_6\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nb : β\nf✝ g✝ : α → β\ninst✝² : CommSemiring β\nι : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrder ι\ns : Finset ι\nf g : ι → R\n⊢ -∑ x in s, (g x * ∏ x in filter (fun x_1 => x_1 < x) s, (f x + -g x)) * ∏ i in filter (fun j => x < j) s, f i =\n ∑ i in s, (-g i * ∏ j in filter (fun x => x < i) s, (f j + -g j)) * ∏ j in filter (fun j => i < j) s, f j",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nb : β\nf✝ g✝ : α → β\ninst✝² : CommSemiring β\nι : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrder ι\ns : Finset ι\nf g : ι → R\n⊢ ∏ x in s, (f x + -g x) =\n ∏ i in s, f i +\n -∑ x in s, (g x * ∏ x in filter (fun x_1 => x_1 < x) s, (f x + -g x)) * ∏ i in filter (fun j => x < j) s, f i",
"tactic": "convert prod_add_ordered s f fun i => -g i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_6\nα : Type u\nβ : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nb : β\nf✝ g✝ : α → β\ninst✝² : CommSemiring β\nι : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrder ι\ns : Finset ι\nf g : ι → R\n⊢ -∑ x in s, (g x * ∏ x in filter (fun x_1 => x_1 < x) s, (f x + -g x)) * ∏ i in filter (fun j => x < j) s, f i =\n ∑ i in s, (-g i * ∏ j in filter (fun x => x < i) s, (f j + -g j)) * ∏ j in filter (fun j => i < j) s, f j",
"tactic": "simp"
}
] | [
194,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/Algebra/CovariantAndContravariant.lean | Group.covariant_iff_contravariant | [
{
"state_after": "case refine_1\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Covariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)\n⊢ r b c\n\ncase refine_2\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Contravariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r b c\n⊢ r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)",
"state_before": "M : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\n⊢ Covariant N N (fun x x_1 => x * x_1) r ↔ Contravariant N N (fun x x_1 => x * x_1) r",
"tactic": "refine ⟨fun h a b c bc ↦ ?_, fun h a b c bc ↦ ?_⟩"
},
{
"state_after": "case refine_1\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Covariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)\n⊢ r (a⁻¹ * (a * b)) (a⁻¹ * (a * c))",
"state_before": "case refine_1\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Covariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)\n⊢ r b c",
"tactic": "rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c]"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Covariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)\n⊢ r (a⁻¹ * (a * b)) (a⁻¹ * (a * c))",
"tactic": "exact h a⁻¹ bc"
},
{
"state_after": "case refine_2\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Contravariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r (a⁻¹ * (a * b)) (a⁻¹ * (a * c))\n⊢ r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)",
"state_before": "case refine_2\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Contravariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r b c\n⊢ r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)",
"tactic": "rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c] at bc"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nM : Type ?u.747\nN : Type u_1\nμ : M → N → N\nr : N → N → Prop\ninst✝¹ : CovariantClass M N μ r\ninst✝ : Group N\nh : Contravariant N N (fun x x_1 => x * x_1) r\na b c : N\nbc : r (a⁻¹ * (a * b)) (a⁻¹ * (a * c))\n⊢ r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)",
"tactic": "exact h a⁻¹ bc"
}
] | [
164,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean | CategoryTheory.Limits.IsColimit.uniq_cocone_morphism | [
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\ns t : Cocone F\nh : IsColimit t\nf f' g : t ⟶ s\n⊢ g = descCoconeMorphism h s",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\ns t : Cocone F\nh : IsColimit t\nf f' : t ⟶ s\n⊢ ∀ {g : t ⟶ s}, g = descCoconeMorphism h s",
"tactic": "intro g"
},
{
"state_after": "case w\nJ : Type u₁\ninst : Category J\nK : Type u₂\ninst_1 : Category K\nC : Type u₃\ninst_2 : Category C\nF : J ⥤ C\ns t : Cocone F\nh : IsColimit t\nf f' g : t ⟶ s\n⊢ g.Hom = desc h s",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\ns t : Cocone F\nh : IsColimit t\nf f' g : t ⟶ s\n⊢ g = descCoconeMorphism h s",
"tactic": "aesop_cat_nonterminal"
},
{
"state_after": "no goals",
"state_before": "case w\nJ : Type u₁\ninst : Category J\nK : Type u₂\ninst_1 : Category K\nC : Type u₃\ninst_2 : Category C\nF : J ⥤ C\ns t : Cocone F\nh : IsColimit t\nf f' g : t ⟶ s\n⊢ g.Hom = desc h s",
"tactic": "exact h.uniq _ _ g.w"
}
] | [
612,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
609,
1
] |
Mathlib/Data/Polynomial/Expand.lean | Polynomial.expand_mul | [] | [
77,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.coe_add | [] | [
88,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/Data/Set/Function.lean | Set.MapsTo.val_restrict_apply | [] | [
353,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.nfpFamily_fp | [
{
"state_after": "ι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\n⊢ f i (sup (List.foldr f a)) = sup (List.foldr f a)",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\n⊢ f i (nfpFamily f a) = nfpFamily f a",
"tactic": "unfold nfpFamily"
},
{
"state_after": "ι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\n⊢ sup (f i ∘ List.foldr f a) = sup (List.foldr f a)",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\n⊢ f i (sup (List.foldr f a)) = sup (List.foldr f a)",
"tactic": "rw [@IsNormal.sup.{u, v, v} _ H _ _ ⟨[]⟩]"
},
{
"state_after": "case a\nι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\nl : List ι\n⊢ (f i ∘ List.foldr f a) l ≤ sup (List.foldr f a)\n\ncase a\nι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\nl : List ι\n⊢ List.foldr f a l ≤ sup (f i ∘ List.foldr f a)",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\n⊢ sup (f i ∘ List.foldr f a) = sup (List.foldr f a)",
"tactic": "apply le_antisymm <;> refine' Ordinal.sup_le fun l => _"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\nl : List ι\n⊢ (f i ∘ List.foldr f a) l ≤ sup (List.foldr f a)",
"tactic": "exact le_sup _ (i::l)"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type u\nf : ι → Ordinal → Ordinal\ni : ι\nH : IsNormal (f i)\na : Ordinal\nl : List ι\n⊢ List.foldr f a l ≤ sup (f i ∘ List.foldr f a)",
"tactic": "exact (H.self_le _).trans (le_sup _ _)"
}
] | [
128,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
1
] |
Mathlib/Data/Vector3.lean | Vector3.append_nil | [] | [
164,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
] |
Mathlib/Data/PFunctor/Univariate/M.lean | PFunctor.M.ext' | [
{
"state_after": "case mk\nF : PFunctor\ny : M F\napprox✝ : (n : ℕ) → CofixA F n\nconsistent✝ : AllAgree approx✝\nH : ∀ (i : ℕ), MIntl.approx { approx := approx✝, consistent := consistent✝ } i = MIntl.approx y i\n⊢ { approx := approx✝, consistent := consistent✝ } = y",
"state_before": "F : PFunctor\nx y : M F\nH : ∀ (i : ℕ), MIntl.approx x i = MIntl.approx y i\n⊢ x = y",
"tactic": "cases x"
},
{
"state_after": "case mk.mk\nF : PFunctor\napprox✝¹ : (n : ℕ) → CofixA F n\nconsistent✝¹ : AllAgree approx✝¹\napprox✝ : (n : ℕ) → CofixA F n\nconsistent✝ : AllAgree approx✝\nH :\n ∀ (i : ℕ),\n MIntl.approx { approx := approx✝¹, consistent := consistent✝¹ } i =\n MIntl.approx { approx := approx✝, consistent := consistent✝ } i\n⊢ { approx := approx✝¹, consistent := consistent✝¹ } = { approx := approx✝, consistent := consistent✝ }",
"state_before": "case mk\nF : PFunctor\ny : M F\napprox✝ : (n : ℕ) → CofixA F n\nconsistent✝ : AllAgree approx✝\nH : ∀ (i : ℕ), MIntl.approx { approx := approx✝, consistent := consistent✝ } i = MIntl.approx y i\n⊢ { approx := approx✝, consistent := consistent✝ } = y",
"tactic": "cases y"
},
{
"state_after": "case mk.mk.e_approx.h\nF : PFunctor\napprox✝¹ : (n : ℕ) → CofixA F n\nconsistent✝¹ : AllAgree approx✝¹\napprox✝ : (n : ℕ) → CofixA F n\nconsistent✝ : AllAgree approx✝\nH :\n ∀ (i : ℕ),\n MIntl.approx { approx := approx✝¹, consistent := consistent✝¹ } i =\n MIntl.approx { approx := approx✝, consistent := consistent✝ } i\nn : ℕ\n⊢ approx✝¹ n = approx✝ n",
"state_before": "case mk.mk\nF : PFunctor\napprox✝¹ : (n : ℕ) → CofixA F n\nconsistent✝¹ : AllAgree approx✝¹\napprox✝ : (n : ℕ) → CofixA F n\nconsistent✝ : AllAgree approx✝\nH :\n ∀ (i : ℕ),\n MIntl.approx { approx := approx✝¹, consistent := consistent✝¹ } i =\n MIntl.approx { approx := approx✝, consistent := consistent✝ } i\n⊢ { approx := approx✝¹, consistent := consistent✝¹ } = { approx := approx✝, consistent := consistent✝ }",
"tactic": "congr with n"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.e_approx.h\nF : PFunctor\napprox✝¹ : (n : ℕ) → CofixA F n\nconsistent✝¹ : AllAgree approx✝¹\napprox✝ : (n : ℕ) → CofixA F n\nconsistent✝ : AllAgree approx✝\nH :\n ∀ (i : ℕ),\n MIntl.approx { approx := approx✝¹, consistent := consistent✝¹ } i =\n MIntl.approx { approx := approx✝, consistent := consistent✝ } i\nn : ℕ\n⊢ approx✝¹ n = approx✝ n",
"tactic": "apply H"
}
] | [
225,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
221,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.lf_of_le_of_lf | [
{
"state_after": "x y z : PGame\nh₁ : x ≤ y\nh₂ : ¬z ≤ y\n⊢ ¬z ≤ x",
"state_before": "x y z : PGame\nh₁ : x ≤ y\nh₂ : y ⧏ z\n⊢ x ⧏ z",
"tactic": "rw [← PGame.not_le] at h₂⊢"
},
{
"state_after": "no goals",
"state_before": "x y z : PGame\nh₁ : x ≤ y\nh₂ : ¬z ≤ y\n⊢ ¬z ≤ x",
"tactic": "exact fun h₃ => h₂ (h₃.trans h₁)"
}
] | [
560,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
558,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.blsub_id | [] | [
1917,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1916,
1
] |
Mathlib/Data/Polynomial/Div.lean | Polynomial.rootMultiplicity_eq_zero_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\nx : R\n⊢ rootMultiplicity x p = 0 ↔ IsRoot p x → p = 0",
"tactic": "simp only [rootMultiplicity_eq_multiplicity, dite_eq_left_iff, PartENat.get_eq_iff_eq_coe,\n Nat.cast_zero, multiplicity.multiplicity_eq_zero, dvd_iff_isRoot, not_imp_not]"
}
] | [
565,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
562,
1
] |
Mathlib/Algebra/Group/Commute.lean | Commute.inv_left_iff | [] | [
374,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
373,
1
] |
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