theorem_id,theorem_name,lean_proposition,proof_node,graph_edge,lean_proof,kernel_verified,benchmark_route,benchmark_family,machine_route,machine_family,human_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V1,root,"by
have hxy : 0 ≤ x * y := mul_nonneg hx hy
have hs : 0 ≤ Real.sqrt (x * y) := Real.sqrt_nonneg _
have hsq : (Real.sqrt (x * y)) ^ 2 = x * y := Real.sq_sqrt hxy
nlinarith [sq_nonneg (x - y)]",True,reference,baseline,reference,baseline,reference
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V2,T001_V1 -> T001_V2,"by
rw [Real.sqrt_le_iff]
constructor
· nlinarith
· nlinarith [sq_nonneg (x - y)]",True,new_route,route_substitution,new_route,logical_characterization,new_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V3,T001_V1 -> T001_V3,"by
apply (Real.sqrt_le_left (by linarith)).2
nlinarith [sq_nonneg (x - y)]",True,new_route,algebraic_normal_form,new_route,logical_characterization,new_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V4,T001_V2 -> T001_V4,"by
have hmean : 0 ≤ (x + y) / 2 := by linarith
rw [← Real.sqrt_sq hmean]
apply Real.sqrt_le_sqrt
nlinarith [sq_nonneg (x - y)]",True,new_route,operator_functional,new_route,logical_characterization,new_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V5,T001_V1 -> T001_V5,"by
exact id (by
have hxy : 0 ≤ x * y := mul_nonneg hx hy
have hs : 0 ≤ Real.sqrt (x * y) := Real.sqrt_nonneg _
have hsq : (Real.sqrt (x * y)) ^ 2 = x * y := Real.sq_sqrt hxy
nlinarith [sq_nonneg (x - y)]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V6,T001_V2 -> T001_V6,"by
exact (show _ from by
rw [Real.sqrt_le_iff]
constructor
· nlinarith
· nlinarith [sq_nonneg (x - y)]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V1,root,"by
exact Real.tendsto_tsum_powerSeries_nhdsWithin_lt hs",True,reference,baseline,reference,baseline,reference
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V2,T004_V1 -> T004_V2,"by
apply Real.tendsto_tsum_powerSeries_nhdsWithin_lt
exact hs",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V3,T004_V1 -> T004_V3,"by
exact (by
exact Real.tendsto_tsum_powerSeries_nhdsWithin_lt hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V4,T004_V2 -> T004_V4,"by
exact (by
apply Real.tendsto_tsum_powerSeries_nhdsWithin_lt
exact hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V5,T004_V1 -> T004_V5,"by
exact id (by
exact Real.tendsto_tsum_powerSeries_nhdsWithin_lt hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V6,T004_V2 -> T004_V6,"by
exact (show _ from by
apply Real.tendsto_tsum_powerSeries_nhdsWithin_lt
exact hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T005,Abelian and Tauberian theorems,"(c : ℕ → ℝ) (C : ℝ)
(hc : Tendsto c atTop (𝓝 C)) :
Tendsto
(fun N : ℕ => (1 / (N : ℝ)) * ∑ n ∈ Finset.Icc 1 N, c n)
atTop (𝓝 C)",T005_V1,root,"by
have hshift : Tendsto (fun n : ℕ => c (n + 1)) atTop (𝓝 C) :=
hc.comp (tendsto_add_atTop_nat 1)
have hsum : ∀ N : ℕ,
(∑ n ∈ Finset.Icc 1 N, c n) = ∑ n ∈ Finset.range N, c (n + 1) := by
intro N
induction N with
| zero => simp
| succ N ih =>
rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ N + 1), Finset.sum_range_succ, ih]
simpa [one_div, hsum] using hshift.cesaro",True,reference,baseline,reference,baseline,reference
T005,Abelian and Tauberian theorems,"(c : ℕ → ℝ) (C : ℝ)
(hc : Tendsto c atTop (𝓝 C)) :
Tendsto
(fun N : ℕ => (1 / (N : ℝ)) * ∑ n ∈ Finset.Icc 1 N, c n)
atTop (𝓝 C)",T005_V2,T005_V1 -> T005_V2,"by
have hshift : Tendsto (fun n : ℕ => c (n + 1)) atTop (𝓝 C) :=
hc.comp (tendsto_add_atTop_nat 1)
have hsum : ∀ N : ℕ,
(∑ n ∈ Finset.Icc 1 N, c n) = ∑ n ∈ Finset.range N, c (n + 1) := by
intro N
induction N with
| zero => simp
| succ N ih =>
rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ N + 1),
Finset.sum_range_succ, ih]
simpa only [one_div, hsum] using hshift.cesaro",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T005,Abelian and Tauberian theorems,"(c : ℕ → ℝ) (C : ℝ)
(hc : Tendsto c atTop (𝓝 C)) :
Tendsto
(fun N : ℕ => (1 / (N : ℝ)) * ∑ n ∈ Finset.Icc 1 N, c n)
atTop (𝓝 C)",T005_V3,T005_V1 -> T005_V3,"by
exact (by
have hshift : Tendsto (fun n : ℕ => c (n + 1)) atTop (𝓝 C) :=
hc.comp (tendsto_add_atTop_nat 1)
have hsum : ∀ N : ℕ,
(∑ n ∈ Finset.Icc 1 N, c n) = ∑ n ∈ Finset.range N, c (n + 1) := by
intro N
induction N with
| zero => simp
| succ N ih =>
rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ N + 1), Finset.sum_range_succ, ih]
simpa [one_div, hsum] using hshift.cesaro
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T005,Abelian and Tauberian theorems,"(c : ℕ → ℝ) (C : ℝ)
(hc : Tendsto c atTop (𝓝 C)) :
Tendsto
(fun N : ℕ => (1 / (N : ℝ)) * ∑ n ∈ Finset.Icc 1 N, c n)
atTop (𝓝 C)",T005_V4,T005_V2 -> T005_V4,"by
exact (by
have hshift : Tendsto (fun n : ℕ => c (n + 1)) atTop (𝓝 C) :=
hc.comp (tendsto_add_atTop_nat 1)
have hsum : ∀ N : ℕ,
(∑ n ∈ Finset.Icc 1 N, c n) = ∑ n ∈ Finset.range N, c (n + 1) := by
intro N
induction N with
| zero => simp
| succ N ih =>
rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ N + 1),
Finset.sum_range_succ, ih]
simpa only [one_div, hsum] using hshift.cesaro
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T005,Abelian and Tauberian theorems,"(c : ℕ → ℝ) (C : ℝ)
(hc : Tendsto c atTop (𝓝 C)) :
Tendsto
(fun N : ℕ => (1 / (N : ℝ)) * ∑ n ∈ Finset.Icc 1 N, c n)
atTop (𝓝 C)",T005_V5,T005_V1 -> T005_V5,"by
exact id (by
have hshift : Tendsto (fun n : ℕ => c (n + 1)) atTop (𝓝 C) :=
hc.comp (tendsto_add_atTop_nat 1)
have hsum : ∀ N : ℕ,
(∑ n ∈ Finset.Icc 1 N, c n) = ∑ n ∈ Finset.range N, c (n + 1) := by
intro N
induction N with
| zero => simp
| succ N ih =>
rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ N + 1), Finset.sum_range_succ, ih]
simpa [one_div, hsum] using hshift.cesaro
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T005,Abelian and Tauberian theorems,"(c : ℕ → ℝ) (C : ℝ)
(hc : Tendsto c atTop (𝓝 C)) :
Tendsto
(fun N : ℕ => (1 / (N : ℝ)) * ∑ n ∈ Finset.Icc 1 N, c n)
atTop (𝓝 C)",T005_V6,T005_V2 -> T005_V6,"by
exact (show _ from by
have hshift : Tendsto (fun n : ℕ => c (n + 1)) atTop (𝓝 C) :=
hc.comp (tendsto_add_atTop_nat 1)
have hsum : ∀ N : ℕ,
(∑ n ∈ Finset.Icc 1 N, c n) = ∑ n ∈ Finset.range N, c (n + 1) := by
intro N
induction N with
| zero => simp
| succ N ih =>
rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ N + 1),
Finset.sum_range_succ, ih]
simpa only [one_div, hsum] using hshift.cesaro
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T011,Analyticity of holomorphic functions,"(U : Set ℂ) (f : ℂ → ℂ) (a : ℂ)
(hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (ha : a ∈ U) :
∃ r > 0, Metric.ball a r ⊆ U ∧
∀ z ∈ Metric.ball a r,
HasSum
(fun n => (iteratedDeriv n f a / (n.factorial : ℂ)) * (z - a) ^ n)
(f z)",T011_V1,root,"by
let p := FormalMultilinearSeries.ofScalars ℂ
(fun n => iteratedDeriv n f a / (n.factorial : ℂ))
have hp : HasFPowerSeriesAt f p a :=
(hf.analyticAt (hU.mem_nhds ha)).hasFPowerSeriesAt
have hevent :
{z : ℂ | z ∈ U ∧ HasSum (fun n => p n (fun _ => z - a)) (f z)} ∈ 𝓝 a :=
Filter.inter_mem (hU.mem_nhds ha) hp.eventually_hasSum_sub
obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp hevent
refine ⟨r, hr, fun z hz => (hball hz).1, fun z hz => ?_⟩
have hzsum := (hball hz).2
simpa only [p, FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul] using hzsum",True,reference,baseline,reference,baseline,reference
T011,Analyticity of holomorphic functions,"(U : Set ℂ) (f : ℂ → ℂ) (a : ℂ)
(hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (ha : a ∈ U) :
∃ r > 0, Metric.ball a r ⊆ U ∧
∀ z ∈ Metric.ball a r,
HasSum
(fun n => (iteratedDeriv n f a / (n.factorial : ℂ)) * (z - a) ^ n)
(f z)",T011_V2,T011_V1 -> T011_V2,"by
let p := FormalMultilinearSeries.ofScalars ℂ
(fun n => iteratedDeriv n f a / (n.factorial : ℂ))
have hp : HasFPowerSeriesAt f p a :=
(hf.analyticAt (hU.mem_nhds ha)).hasFPowerSeriesAt
have hboth :
U ∩ {z : ℂ | HasSum (fun n => p n (fun _ => z - a)) (f z)} ∈ 𝓝 a :=
Filter.inter_mem (hU.mem_nhds ha) hp.eventually_hasSum_sub
rcases Metric.mem_nhds_iff.mp hboth with ⟨r, hr, hball⟩
refine ⟨r, hr, ?_, ?_⟩
· intro z hz
exact (hball hz).1
· intro z hz
have hzsum := (hball hz).2
simpa only [p, FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul] using hzsum",True,same_route,proof_intermediate,new_route,logical_characterization,same_route
T011,Analyticity of holomorphic functions,"(U : Set ℂ) (f : ℂ → ℂ) (a : ℂ)
(hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (ha : a ∈ U) :
∃ r > 0, Metric.ball a r ⊆ U ∧
∀ z ∈ Metric.ball a r,
HasSum
(fun n => (iteratedDeriv n f a / (n.factorial : ℂ)) * (z - a) ^ n)
(f z)",T011_V3,T011_V1 -> T011_V3,"by
exact (by
let p := FormalMultilinearSeries.ofScalars ℂ
(fun n => iteratedDeriv n f a / (n.factorial : ℂ))
have hp : HasFPowerSeriesAt f p a :=
(hf.analyticAt (hU.mem_nhds ha)).hasFPowerSeriesAt
have hevent :
{z : ℂ | z ∈ U ∧ HasSum (fun n => p n (fun _ => z - a)) (f z)} ∈ 𝓝 a :=
Filter.inter_mem (hU.mem_nhds ha) hp.eventually_hasSum_sub
obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp hevent
refine ⟨r, hr, fun z hz => (hball hz).1, fun z hz => ?_⟩
have hzsum := (hball hz).2
simpa only [p, FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul] using hzsum
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T011,Analyticity of holomorphic functions,"(U : Set ℂ) (f : ℂ → ℂ) (a : ℂ)
(hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (ha : a ∈ U) :
∃ r > 0, Metric.ball a r ⊆ U ∧
∀ z ∈ Metric.ball a r,
HasSum
(fun n => (iteratedDeriv n f a / (n.factorial : ℂ)) * (z - a) ^ n)
(f z)",T011_V4,T011_V2 -> T011_V4,"by
exact (by
let p := FormalMultilinearSeries.ofScalars ℂ
(fun n => iteratedDeriv n f a / (n.factorial : ℂ))
have hp : HasFPowerSeriesAt f p a :=
(hf.analyticAt (hU.mem_nhds ha)).hasFPowerSeriesAt
have hboth :
U ∩ {z : ℂ | HasSum (fun n => p n (fun _ => z - a)) (f z)} ∈ 𝓝 a :=
Filter.inter_mem (hU.mem_nhds ha) hp.eventually_hasSum_sub
rcases Metric.mem_nhds_iff.mp hboth with ⟨r, hr, hball⟩
refine ⟨r, hr, ?_, ?_⟩
· intro z hz
exact (hball hz).1
· intro z hz
have hzsum := (hball hz).2
simpa only [p, FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul] using hzsum
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T011,Analyticity of holomorphic functions,"(U : Set ℂ) (f : ℂ → ℂ) (a : ℂ)
(hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (ha : a ∈ U) :
∃ r > 0, Metric.ball a r ⊆ U ∧
∀ z ∈ Metric.ball a r,
HasSum
(fun n => (iteratedDeriv n f a / (n.factorial : ℂ)) * (z - a) ^ n)
(f z)",T011_V5,T011_V1 -> T011_V5,"by
exact id (by
let p := FormalMultilinearSeries.ofScalars ℂ
(fun n => iteratedDeriv n f a / (n.factorial : ℂ))
have hp : HasFPowerSeriesAt f p a :=
(hf.analyticAt (hU.mem_nhds ha)).hasFPowerSeriesAt
have hevent :
{z : ℂ | z ∈ U ∧ HasSum (fun n => p n (fun _ => z - a)) (f z)} ∈ 𝓝 a :=
Filter.inter_mem (hU.mem_nhds ha) hp.eventually_hasSum_sub
obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp hevent
refine ⟨r, hr, fun z hz => (hball hz).1, fun z hz => ?_⟩
have hzsum := (hball hz).2
simpa only [p, FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul] using hzsum
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T011,Analyticity of holomorphic functions,"(U : Set ℂ) (f : ℂ → ℂ) (a : ℂ)
(hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (ha : a ∈ U) :
∃ r > 0, Metric.ball a r ⊆ U ∧
∀ z ∈ Metric.ball a r,
HasSum
(fun n => (iteratedDeriv n f a / (n.factorial : ℂ)) * (z - a) ^ n)
(f z)",T011_V6,T011_V2 -> T011_V6,"by
exact (show _ from by
let p := FormalMultilinearSeries.ofScalars ℂ
(fun n => iteratedDeriv n f a / (n.factorial : ℂ))
have hp : HasFPowerSeriesAt f p a :=
(hf.analyticAt (hU.mem_nhds ha)).hasFPowerSeriesAt
have hboth :
U ∩ {z : ℂ | HasSum (fun n => p n (fun _ => z - a)) (f z)} ∈ 𝓝 a :=
Filter.inter_mem (hU.mem_nhds ha) hp.eventually_hasSum_sub
rcases Metric.mem_nhds_iff.mp hboth with ⟨r, hr, hball⟩
refine ⟨r, hr, ?_, ?_⟩
· intro z hz
exact (hball hz).1
· intro z hz
have hzsum := (hball hz).2
simpa only [p, FormalMultilinearSeries.ofScalars_apply_eq, smul_eq_mul] using hzsum
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T014,Artin–Rees lemma,"{A : Type*} [CommRing A] [IsNoetherianRing A]
(I : Ideal A) (M : Type*) [AddCommGroup M] [Module A M]
[Module.Finite A M] (N : Submodule A M) :
∃ k : ℕ, ∀ n ≥ k,
(I ^ n • ⊤ : Submodule A M) ⊓ N =
I ^ (n - k) • ((I ^ k • ⊤ : Submodule A M) ⊓ N)",T014_V1,root,"by
exact Ideal.exists_pow_inf_eq_pow_smul I N",True,reference,baseline,reference,baseline,reference
T014,Artin–Rees lemma,"{A : Type*} [CommRing A] [IsNoetherianRing A]
(I : Ideal A) (M : Type*) [AddCommGroup M] [Module A M]
[Module.Finite A M] (N : Submodule A M) :
∃ k : ℕ, ∀ n ≥ k,
(I ^ n • ⊤ : Submodule A M) ⊓ N =
I ^ (n - k) • ((I ^ k • ⊤ : Submodule A M) ⊓ N)",T014_V2,T014_V1 -> T014_V2,"by
simpa only using (Ideal.exists_pow_inf_eq_pow_smul I N)",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T014,Artin–Rees lemma,"{A : Type*} [CommRing A] [IsNoetherianRing A]
(I : Ideal A) (M : Type*) [AddCommGroup M] [Module A M]
[Module.Finite A M] (N : Submodule A M) :
∃ k : ℕ, ∀ n ≥ k,
(I ^ n • ⊤ : Submodule A M) ⊓ N =
I ^ (n - k) • ((I ^ k • ⊤ : Submodule A M) ⊓ N)",T014_V3,T014_V1 -> T014_V3,"by
exact (by
exact Ideal.exists_pow_inf_eq_pow_smul I N
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T014,Artin–Rees lemma,"{A : Type*} [CommRing A] [IsNoetherianRing A]
(I : Ideal A) (M : Type*) [AddCommGroup M] [Module A M]
[Module.Finite A M] (N : Submodule A M) :
∃ k : ℕ, ∀ n ≥ k,
(I ^ n • ⊤ : Submodule A M) ⊓ N =
I ^ (n - k) • ((I ^ k • ⊤ : Submodule A M) ⊓ N)",T014_V4,T014_V2 -> T014_V4,"by
exact (by
simpa only using (Ideal.exists_pow_inf_eq_pow_smul I N)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T014,Artin–Rees lemma,"{A : Type*} [CommRing A] [IsNoetherianRing A]
(I : Ideal A) (M : Type*) [AddCommGroup M] [Module A M]
[Module.Finite A M] (N : Submodule A M) :
∃ k : ℕ, ∀ n ≥ k,
(I ^ n • ⊤ : Submodule A M) ⊓ N =
I ^ (n - k) • ((I ^ k • ⊤ : Submodule A M) ⊓ N)",T014_V5,T014_V1 -> T014_V5,"by
exact id (by
exact Ideal.exists_pow_inf_eq_pow_smul I N
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T014,Artin–Rees lemma,"{A : Type*} [CommRing A] [IsNoetherianRing A]
(I : Ideal A) (M : Type*) [AddCommGroup M] [Module A M]
[Module.Finite A M] (N : Submodule A M) :
∃ k : ℕ, ∀ n ≥ k,
(I ^ n • ⊤ : Submodule A M) ⊓ N =
I ^ (n - k) • ((I ^ k • ⊤ : Submodule A M) ⊓ N)",T014_V6,T014_V2 -> T014_V6,"by
exact (show _ from by
simpa only using (Ideal.exists_pow_inf_eq_pow_smul I N)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T020,Bayes' theorem,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) (hB : 0 < μ.real B) :
μ.real (A ∩ B) / μ.real B =
(μ.real (B ∩ A) / μ.real A) * μ.real A / μ.real B",T020_V1,root,"by
rw [Set.inter_comm B A]
field_simp",True,reference,baseline,reference,baseline,reference
T020,Bayes' theorem,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) (hB : 0 < μ.real B) :
μ.real (A ∩ B) / μ.real B =
(μ.real (B ∩ A) / μ.real A) * μ.real A / μ.real B",T020_V2,T020_V1 -> T020_V2,"by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
have hB0 : μ.real B ≠ 0 := ne_of_gt hB
have hinter : μ.real (A ∩ B) = μ.real (B ∩ A) := by
rw [Set.inter_comm]
rw [hinter]
field_simp",True,new_route,algebraic_normal_form,same_route,proof_intermediate,same_route
T020,Bayes' theorem,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) (hB : 0 < μ.real B) :
μ.real (A ∩ B) / μ.real B =
(μ.real (B ∩ A) / μ.real A) * μ.real A / μ.real B",T020_V3,T020_V1 -> T020_V3,"by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
have hswap : μ.real (B ∩ A) = μ.real (A ∩ B) :=
congrArg (fun S : Set Ω => μ.real S) (Set.inter_comm B A)
rw [hswap, div_mul_cancel₀ _ hA0]",True,new_route,algebraic_normal_form,new_route,algebraic_normal_form,same_route
T020,Bayes' theorem,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) (hB : 0 < μ.real B) :
μ.real (A ∩ B) / μ.real B =
(μ.real (B ∩ A) / μ.real A) * μ.real A / μ.real B",T020_V4,T020_V2 -> T020_V4,"by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
have hB0 : μ.real B ≠ 0 := ne_of_gt hB
apply (div_eq_iff hB0).2
rw [Set.inter_comm B A]
field_simp",True,new_route,logical_characterization,new_route,logical_characterization,same_route
T020,Bayes' theorem,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) (hB : 0 < μ.real B) :
μ.real (A ∩ B) / μ.real B =
(μ.real (B ∩ A) / μ.real A) * μ.real A / μ.real B",T020_V5,T020_V1 -> T020_V5,"by
exact id (by
rw [Set.inter_comm B A]
field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T020,Bayes' theorem,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) (hB : 0 < μ.real B) :
μ.real (A ∩ B) / μ.real B =
(μ.real (B ∩ A) / μ.real A) * μ.real A / μ.real B",T020_V6,T020_V2 -> T020_V6,"by
exact (show _ from by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
have hB0 : μ.real B ≠ 0 := ne_of_gt hB
have hinter : μ.real (A ∩ B) = μ.real (B ∩ A) := by
rw [Set.inter_comm]
rw [hinter]
field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T021,Bernoulli's inequality,"(x : ℝ) (n : ℕ) (hx : -1 ≤ x) :
1 + (n : ℝ) * x ≤ (1 + x) ^ n",T021_V1,root,"by
exact one_add_mul_le_pow (by linarith) n",True,reference,baseline,reference,baseline,reference
T021,Bernoulli's inequality,"(x : ℝ) (n : ℕ) (hx : -1 ≤ x) :
1 + (n : ℝ) * x ≤ (1 + x) ^ n",T021_V2,T021_V1 -> T021_V2,"by
have hbase : 0 ≤ 1 + x := by linarith
induction n with
| zero => simp
| succ n ih =>
rw [pow_succ]
have hmul := mul_le_mul_of_nonneg_right ih hbase
have hsq : 0 ≤ (n : ℝ) * x ^ 2 := mul_nonneg (Nat.cast_nonneg n) (sq_nonneg x)
push_cast
nlinarith",True,new_route,route_substitution,new_route,route_substitution,new_route
T021,Bernoulli's inequality,"(x : ℝ) (n : ℕ) (hx : -1 ≤ x) :
1 + (n : ℝ) * x ≤ (1 + x) ^ n",T021_V3,T021_V1 -> T021_V3,"by
exact (by
exact one_add_mul_le_pow (by linarith) n
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T021,Bernoulli's inequality,"(x : ℝ) (n : ℕ) (hx : -1 ≤ x) :
1 + (n : ℝ) * x ≤ (1 + x) ^ n",T021_V4,T021_V2 -> T021_V4,"by
exact (by
have hbase : 0 ≤ 1 + x := by linarith
induction n with
| zero => simp
| succ n ih =>
rw [pow_succ]
have hmul := mul_le_mul_of_nonneg_right ih hbase
have hsq : 0 ≤ (n : ℝ) * x ^ 2 := mul_nonneg (Nat.cast_nonneg n) (sq_nonneg x)
push_cast
nlinarith
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T021,Bernoulli's inequality,"(x : ℝ) (n : ℕ) (hx : -1 ≤ x) :
1 + (n : ℝ) * x ≤ (1 + x) ^ n",T021_V5,T021_V1 -> T021_V5,"by
exact id (by
exact one_add_mul_le_pow (by linarith) n
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T021,Bernoulli's inequality,"(x : ℝ) (n : ℕ) (hx : -1 ≤ x) :
1 + (n : ℝ) * x ≤ (1 + x) ^ n",T021_V6,T021_V2 -> T021_V6,"by
exact (show _ from by
have hbase : 0 ≤ 1 + x := by linarith
induction n with
| zero => simp
| succ n ih =>
rw [pow_succ]
have hmul := mul_le_mul_of_nonneg_right ih hbase
have hsq : 0 ≤ (n : ℝ) * x ^ 2 := mul_nonneg (Nat.cast_nonneg n) (sq_nonneg x)
push_cast
nlinarith
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T025,Bertrand's postulate,"(n : ℕ) (hn : 1 < n) :
∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p < 2 * n",T025_V1,root,"by
obtain ⟨p, hp, hnp, hp2n⟩ := Nat.exists_prime_lt_and_le_two_mul n (by omega)
refine ⟨p, hp, hnp, lt_of_le_of_ne hp2n ?_⟩
intro hEq
have heven : Even p := by
rw [hEq]
exact even_two_mul n
have hp2 : p = 2 := hp.even_iff.mp heven
omega",True,reference,baseline,reference,baseline,reference
T025,Bertrand's postulate,"(n : ℕ) (hn : 1 < n) :
∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p < 2 * n",T025_V2,T025_V1 -> T025_V2,"by
obtain ⟨p, hp, hnp, hp2n⟩ := Nat.bertrand n (by omega)
refine ⟨p, hp, hnp, ?_⟩
rcases hp2n.lt_or_eq with hlt | heq
· exact hlt
· have hpEven : Even p := by
rw [heq]
exact even_two_mul n
have : p = 2 := hp.even_iff.mp hpEven
omega",True,same_route,proof_intermediate,new_route,route_substitution,new_route
T025,Bertrand's postulate,"(n : ℕ) (hn : 1 < n) :
∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p < 2 * n",T025_V3,T025_V1 -> T025_V3,"by
exact (by
obtain ⟨p, hp, hnp, hp2n⟩ := Nat.exists_prime_lt_and_le_two_mul n (by omega)
refine ⟨p, hp, hnp, lt_of_le_of_ne hp2n ?_⟩
intro hEq
have heven : Even p := by
rw [hEq]
exact even_two_mul n
have hp2 : p = 2 := hp.even_iff.mp heven
omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T025,Bertrand's postulate,"(n : ℕ) (hn : 1 < n) :
∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p < 2 * n",T025_V4,T025_V2 -> T025_V4,"by
exact (by
obtain ⟨p, hp, hnp, hp2n⟩ := Nat.bertrand n (by omega)
refine ⟨p, hp, hnp, ?_⟩
rcases hp2n.lt_or_eq with hlt | heq
· exact hlt
· have hpEven : Even p := by
rw [heq]
exact even_two_mul n
have : p = 2 := hp.even_iff.mp hpEven
omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T025,Bertrand's postulate,"(n : ℕ) (hn : 1 < n) :
∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p < 2 * n",T025_V5,T025_V1 -> T025_V5,"by
exact id (by
obtain ⟨p, hp, hnp, hp2n⟩ := Nat.exists_prime_lt_and_le_two_mul n (by omega)
refine ⟨p, hp, hnp, lt_of_le_of_ne hp2n ?_⟩
intro hEq
have heven : Even p := by
rw [hEq]
exact even_two_mul n
have hp2 : p = 2 := hp.even_iff.mp heven
omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T025,Bertrand's postulate,"(n : ℕ) (hn : 1 < n) :
∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p < 2 * n",T025_V6,T025_V2 -> T025_V6,"by
exact (show _ from by
obtain ⟨p, hp, hnp, hp2n⟩ := Nat.bertrand n (by omega)
refine ⟨p, hp, hnp, ?_⟩
rcases hp2n.lt_or_eq with hlt | heq
· exact hlt
· have hpEven : Even p := by
rw [heq]
exact even_two_mul n
have : p = 2 := hp.even_iff.mp hpEven
omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T026,Besicovitch covering theorem,"{α : Type*} [MetricSpace α] [HasBesicovitchCovering α]
{β : Type*} (q : Besicovitch.BallPackage β α) :
∃ N : ℕ, ∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧
Set.range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)",T026_V1,root,"by
obtain ⟨N, τ, hτ, hN⟩ :=
HasBesicovitchCovering.no_satelliteConfig (α := α)
obtain ⟨s, hs⟩ := Besicovitch.exist_disjoint_covering_families hτ hN q
exact ⟨N, s, hs⟩",True,reference,baseline,reference,baseline,reference
T026,Besicovitch covering theorem,"{α : Type*} [MetricSpace α] [HasBesicovitchCovering α]
{β : Type*} (q : Besicovitch.BallPackage β α) :
∃ N : ℕ, ∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧
Set.range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)",T026_V2,T026_V1 -> T026_V2,"by
rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
rcases Besicovitch.exist_disjoint_covering_families hτ hN q with ⟨s, hs⟩
refine ⟨N, s, ?_⟩
exact hs",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T026,Besicovitch covering theorem,"{α : Type*} [MetricSpace α] [HasBesicovitchCovering α]
{β : Type*} (q : Besicovitch.BallPackage β α) :
∃ N : ℕ, ∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧
Set.range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)",T026_V3,T026_V1 -> T026_V3,"by
exact (by
obtain ⟨N, τ, hτ, hN⟩ :=
HasBesicovitchCovering.no_satelliteConfig (α := α)
obtain ⟨s, hs⟩ := Besicovitch.exist_disjoint_covering_families hτ hN q
exact ⟨N, s, hs⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T026,Besicovitch covering theorem,"{α : Type*} [MetricSpace α] [HasBesicovitchCovering α]
{β : Type*} (q : Besicovitch.BallPackage β α) :
∃ N : ℕ, ∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧
Set.range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)",T026_V4,T026_V2 -> T026_V4,"by
exact (by
rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
rcases Besicovitch.exist_disjoint_covering_families hτ hN q with ⟨s, hs⟩
refine ⟨N, s, ?_⟩
exact hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T026,Besicovitch covering theorem,"{α : Type*} [MetricSpace α] [HasBesicovitchCovering α]
{β : Type*} (q : Besicovitch.BallPackage β α) :
∃ N : ℕ, ∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧
Set.range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)",T026_V5,T026_V1 -> T026_V5,"by
exact id (by
obtain ⟨N, τ, hτ, hN⟩ :=
HasBesicovitchCovering.no_satelliteConfig (α := α)
obtain ⟨s, hs⟩ := Besicovitch.exist_disjoint_covering_families hτ hN q
exact ⟨N, s, hs⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T026,Besicovitch covering theorem,"{α : Type*} [MetricSpace α] [HasBesicovitchCovering α]
{β : Type*} (q : Besicovitch.BallPackage β α) :
∃ N : ℕ, ∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧
Set.range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)",T026_V6,T026_V2 -> T026_V6,"by
exact (show _ from by
rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
rcases Besicovitch.exist_disjoint_covering_families hτ hN q with ⟨s, hs⟩
refine ⟨N, s, ?_⟩
exact hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T027,Bhatia–Davis inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (m M : ℝ)
(hbounds : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc m M)
(hX : AEMeasurable X μ) :
ProbabilityTheory.variance X μ ≤
(M - μ[X]) * (μ[X] - m)",T027_V1,root,"by
exact ProbabilityTheory.variance_le_sub_mul_sub hbounds hX",True,reference,baseline,reference,baseline,reference
T027,Bhatia–Davis inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (m M : ℝ)
(hbounds : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc m M)
(hX : AEMeasurable X μ) :
ProbabilityTheory.variance X μ ≤
(M - μ[X]) * (μ[X] - m)",T027_V2,T027_V1 -> T027_V2,"by
apply ProbabilityTheory.variance_le_sub_mul_sub
· exact hbounds
· exact hX",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T027,Bhatia–Davis inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (m M : ℝ)
(hbounds : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc m M)
(hX : AEMeasurable X μ) :
ProbabilityTheory.variance X μ ≤
(M - μ[X]) * (μ[X] - m)",T027_V3,T027_V1 -> T027_V3,"by
exact (by
exact ProbabilityTheory.variance_le_sub_mul_sub hbounds hX
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T027,Bhatia–Davis inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (m M : ℝ)
(hbounds : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc m M)
(hX : AEMeasurable X μ) :
ProbabilityTheory.variance X μ ≤
(M - μ[X]) * (μ[X] - m)",T027_V4,T027_V2 -> T027_V4,"by
exact (by
apply ProbabilityTheory.variance_le_sub_mul_sub
· exact hbounds
· exact hX
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T027,Bhatia–Davis inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (m M : ℝ)
(hbounds : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc m M)
(hX : AEMeasurable X μ) :
ProbabilityTheory.variance X μ ≤
(M - μ[X]) * (μ[X] - m)",T027_V5,T027_V1 -> T027_V5,"by
exact id (by
exact ProbabilityTheory.variance_le_sub_mul_sub hbounds hX
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T027,Bhatia–Davis inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (m M : ℝ)
(hbounds : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc m M)
(hX : AEMeasurable X μ) :
ProbabilityTheory.variance X μ ≤
(M - μ[X]) * (μ[X] - m)",T027_V6,T027_V2 -> T027_V6,"by
exact (show _ from by
apply ProbabilityTheory.variance_le_sub_mul_sub
· exact hbounds
· exact hX
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T029,Binomial sum variance inequality,"(m : ℕ) (p : Fin m → ℝ)
(hp : ∀ i, 0 ≤ p i ∧ p i ≤ 1) :
(∑ i, p i * (1 - p i)) ≤
(∑ i, p i) * (1 - (∑ i, p i) / m)",T029_V1,root,"by
by_cases hm : m = 0
· subst m
simp
have hmpos : (0 : ℝ) < m := by positivity
have hsq : (∑ i, p i) ^ 2 ≤ (m : ℝ) * ∑ i, p i ^ 2 := by
simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := p))
have hdiv : (∑ i, p i) ^ 2 / (m : ℝ) ≤ ∑ i, p i ^ 2 := by
rw [div_le_iff₀ hmpos]
simpa [mul_comm] using hsq
calc
(∑ i, p i * (1 - p i)) = (∑ i, p i) - ∑ i, p i ^ 2 := by
simp_rw [mul_sub, mul_one, sq]
rw [Finset.sum_sub_distrib]
_ ≤ (∑ i, p i) - (∑ i, p i) ^ 2 / (m : ℝ) := sub_le_sub_left hdiv _
_ = (∑ i, p i) * (1 - (∑ i, p i) / m) := by ring",True,reference,baseline,reference,baseline,reference
T029,Binomial sum variance inequality,"(m : ℕ) (p : Fin m → ℝ)
(hp : ∀ i, 0 ≤ p i ∧ p i ≤ 1) :
(∑ i, p i * (1 - p i)) ≤
(∑ i, p i) * (1 - (∑ i, p i) / m)",T029_V2,T029_V1 -> T029_V2,"by
by_cases hm : m = 0
· subst m
simp
have hmpos : (0 : ℝ) < m := by positivity
have hcs : (∑ i, p i) ^ 2 ≤ (m : ℝ) * ∑ i, p i ^ 2 := by
simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := p))
have hdiv : (∑ i, p i) ^ 2 / (m : ℝ) ≤ ∑ i, p i ^ 2 := by
apply (div_le_iff₀ hmpos).2
simpa [mul_comm] using hcs
calc
(∑ i, p i * (1 - p i)) = (∑ i, p i) - ∑ i, p i ^ 2 := by
simp_rw [mul_sub, mul_one, sq]
rw [Finset.sum_sub_distrib]
_ ≤
(∑ i, p i) - (∑ i, p i) ^ 2 / (m : ℝ) := by
exact sub_le_sub_left hdiv _
_ = (∑ i, p i) * (1 - (∑ i, p i) / m) := by ring",True,same_route,residual_nonnegativity,same_route,proof_term_structure,same_route
T029,Binomial sum variance inequality,"(m : ℕ) (p : Fin m → ℝ)
(hp : ∀ i, 0 ≤ p i ∧ p i ≤ 1) :
(∑ i, p i * (1 - p i)) ≤
(∑ i, p i) * (1 - (∑ i, p i) / m)",T029_V3,T029_V1 -> T029_V3,"by
by_cases hm : m = 0
· subst m
simp
have hmpos : (0 : ℝ) < m := by positivity
have hcs : (∑ i, p i) ^ 2 ≤ (m : ℝ) * ∑ i, p i ^ 2 := by
simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := p))
have hres : (∑ i, p i) ^ 2 / (m : ℝ) ≤ ∑ i, p i ^ 2 :=
(div_le_iff₀ hmpos).2 (by simpa [mul_comm] using hcs)
calc
(∑ i, p i * (1 - p i)) = (∑ i, p i) - ∑ i, p i ^ 2 := by
simp_rw [mul_sub, mul_one, sq]
rw [Finset.sum_sub_distrib]
_ ≤ (∑ i, p i) - (∑ i, p i) ^ 2 / (m : ℝ) := sub_le_sub_left hres _
_ = (∑ i, p i) * (1 - (∑ i, p i) / m) := by ring",True,new_route,residual_nonnegativity,same_route,proof_term_structure,same_route
T029,Binomial sum variance inequality,"(m : ℕ) (p : Fin m → ℝ)
(hp : ∀ i, 0 ≤ p i ∧ p i ≤ 1) :
(∑ i, p i * (1 - p i)) ≤
(∑ i, p i) * (1 - (∑ i, p i) / m)",T029_V4,T029_V2 -> T029_V4,"by
exact (by
by_cases hm : m = 0
· subst m
simp
have hmpos : (0 : ℝ) < m := by positivity
have hcs : (∑ i, p i) ^ 2 ≤ (m : ℝ) * ∑ i, p i ^ 2 := by
simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := p))
have hdiv : (∑ i, p i) ^ 2 / (m : ℝ) ≤ ∑ i, p i ^ 2 := by
apply (div_le_iff₀ hmpos).2
simpa [mul_comm] using hcs
calc
(∑ i, p i * (1 - p i)) = (∑ i, p i) - ∑ i, p i ^ 2 := by
simp_rw [mul_sub, mul_one, sq]
rw [Finset.sum_sub_distrib]
_ ≤
(∑ i, p i) - (∑ i, p i) ^ 2 / (m : ℝ) := by
exact sub_le_sub_left hdiv _
_ = (∑ i, p i) * (1 - (∑ i, p i) / m) := by ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T029,Binomial sum variance inequality,"(m : ℕ) (p : Fin m → ℝ)
(hp : ∀ i, 0 ≤ p i ∧ p i ≤ 1) :
(∑ i, p i * (1 - p i)) ≤
(∑ i, p i) * (1 - (∑ i, p i) / m)",T029_V5,T029_V1 -> T029_V5,"by
exact id (by
by_cases hm : m = 0
· subst m
simp
have hmpos : (0 : ℝ) < m := by positivity
have hsq : (∑ i, p i) ^ 2 ≤ (m : ℝ) * ∑ i, p i ^ 2 := by
simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := p))
have hdiv : (∑ i, p i) ^ 2 / (m : ℝ) ≤ ∑ i, p i ^ 2 := by
rw [div_le_iff₀ hmpos]
simpa [mul_comm] using hsq
calc
(∑ i, p i * (1 - p i)) = (∑ i, p i) - ∑ i, p i ^ 2 := by
simp_rw [mul_sub, mul_one, sq]
rw [Finset.sum_sub_distrib]
_ ≤ (∑ i, p i) - (∑ i, p i) ^ 2 / (m : ℝ) := sub_le_sub_left hdiv _
_ = (∑ i, p i) * (1 - (∑ i, p i) / m) := by ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T029,Binomial sum variance inequality,"(m : ℕ) (p : Fin m → ℝ)
(hp : ∀ i, 0 ≤ p i ∧ p i ≤ 1) :
(∑ i, p i * (1 - p i)) ≤
(∑ i, p i) * (1 - (∑ i, p i) / m)",T029_V6,T029_V2 -> T029_V6,"by
exact (show _ from by
by_cases hm : m = 0
· subst m
simp
have hmpos : (0 : ℝ) < m := by positivity
have hcs : (∑ i, p i) ^ 2 ≤ (m : ℝ) * ∑ i, p i ^ 2 := by
simpa using (sq_sum_le_card_mul_sum_sq (s := Finset.univ) (f := p))
have hdiv : (∑ i, p i) ^ 2 / (m : ℝ) ≤ ∑ i, p i ^ 2 := by
apply (div_le_iff₀ hmpos).2
simpa [mul_comm] using hcs
calc
(∑ i, p i * (1 - p i)) = (∑ i, p i) - ∑ i, p i ^ 2 := by
simp_rw [mul_sub, mul_one, sq]
rw [Finset.sum_sub_distrib]
_ ≤
(∑ i, p i) - (∑ i, p i) ^ 2 / (m : ℝ) := by
exact sub_le_sub_left hdiv _
_ = (∑ i, p i) * (1 - (∑ i, p i) / m) := by ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T030,Binomial theorem,"{R : Type*} [CommSemiring R] (x y : R) (n : ℕ) :
(x + y) ^ n =
∑ k ∈ Finset.range (n + 1), (Nat.choose n k : R) * x ^ k * y ^ (n - k)",T030_V1,root,"by
simpa [mul_assoc, mul_comm, mul_left_comm] using (add_pow x y n)",True,reference,baseline,reference,baseline,reference
T030,Binomial theorem,"{R : Type*} [CommSemiring R] (x y : R) (n : ℕ) :
(x + y) ^ n =
∑ k ∈ Finset.range (n + 1), (Nat.choose n k : R) * x ^ k * y ^ (n - k)",T030_V2,T030_V1 -> T030_V2,"by
rw [add_pow]
apply Finset.sum_congr rfl
intro k hk
ring",True,same_route,proof_intermediate,same_route,proof_intermediate,same_route
T030,Binomial theorem,"{R : Type*} [CommSemiring R] (x y : R) (n : ℕ) :
(x + y) ^ n =
∑ k ∈ Finset.range (n + 1), (Nat.choose n k : R) * x ^ k * y ^ (n - k)",T030_V3,T030_V1 -> T030_V3,"by
simpa [mul_assoc, mul_comm, mul_left_comm] using
(Commute.all x y).add_pow n",True,new_route,operator_functional,new_route,route_substitution,new_route
T030,Binomial theorem,"{R : Type*} [CommSemiring R] (x y : R) (n : ℕ) :
(x + y) ^ n =
∑ k ∈ Finset.range (n + 1), (Nat.choose n k : R) * x ^ k * y ^ (n - k)",T030_V4,T030_V2 -> T030_V4,"by
exact (by
rw [add_pow]
apply Finset.sum_congr rfl
intro k hk
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T030,Binomial theorem,"{R : Type*} [CommSemiring R] (x y : R) (n : ℕ) :
(x + y) ^ n =
∑ k ∈ Finset.range (n + 1), (Nat.choose n k : R) * x ^ k * y ^ (n - k)",T030_V5,T030_V1 -> T030_V5,"by
exact id (by
simpa [mul_assoc, mul_comm, mul_left_comm] using (add_pow x y n)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T030,Binomial theorem,"{R : Type*} [CommSemiring R] (x y : R) (n : ℕ) :
(x + y) ^ n =
∑ k ∈ Finset.range (n + 1), (Nat.choose n k : R) * x ^ k * y ^ (n - k)",T030_V6,T030_V2 -> T030_V6,"by
exact (show _ from by
rw [add_pow]
apply Finset.sum_congr rfl
intro k hk
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T038,Bohr–Mollerup theorem,"(f : ℝ → ℝ)
(hpos : ∀ x > 0, 0 < f x)
(h1 : f 1 = 1)
(hrec : ∀ x > 0, f (x + 1) = x * f x)
(hlog : ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ f)) :
∀ x > 0, f x = Real.Gamma x",T038_V1,root,"by
intro x hx
exact Real.eq_Gamma_of_log_convex hlog (fun hy => hrec _ hy) (fun hy => hpos _ hy) h1 hx",True,reference,baseline,reference,baseline,reference
T038,Bohr–Mollerup theorem,"(f : ℝ → ℝ)
(hpos : ∀ x > 0, 0 < f x)
(h1 : f 1 = 1)
(hrec : ∀ x > 0, f (x + 1) = x * f x)
(hlog : ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ f)) :
∀ x > 0, f x = Real.Gamma x",T038_V2,T038_V1 -> T038_V2,"by
intro x hx
apply Real.eq_Gamma_of_log_convex hlog
· intro y hy
exact hrec y hy
· intro y hy
exact hpos y hy
· exact h1
· exact hx",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T038,Bohr–Mollerup theorem,"(f : ℝ → ℝ)
(hpos : ∀ x > 0, 0 < f x)
(h1 : f 1 = 1)
(hrec : ∀ x > 0, f (x + 1) = x * f x)
(hlog : ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ f)) :
∀ x > 0, f x = Real.Gamma x",T038_V3,T038_V1 -> T038_V3,"by
exact (by
intro x hx
exact Real.eq_Gamma_of_log_convex hlog (fun hy => hrec _ hy) (fun hy => hpos _ hy) h1 hx
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T038,Bohr–Mollerup theorem,"(f : ℝ → ℝ)
(hpos : ∀ x > 0, 0 < f x)
(h1 : f 1 = 1)
(hrec : ∀ x > 0, f (x + 1) = x * f x)
(hlog : ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ f)) :
∀ x > 0, f x = Real.Gamma x",T038_V4,T038_V2 -> T038_V4,"by
exact (by
intro x hx
apply Real.eq_Gamma_of_log_convex hlog
· intro y hy
exact hrec y hy
· intro y hy
exact hpos y hy
· exact h1
· exact hx
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T038,Bohr–Mollerup theorem,"(f : ℝ → ℝ)
(hpos : ∀ x > 0, 0 < f x)
(h1 : f 1 = 1)
(hrec : ∀ x > 0, f (x + 1) = x * f x)
(hlog : ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ f)) :
∀ x > 0, f x = Real.Gamma x",T038_V5,T038_V1 -> T038_V5,"by
exact id (by
intro x hx
exact Real.eq_Gamma_of_log_convex hlog (fun hy => hrec _ hy) (fun hy => hpos _ hy) h1 hx
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T038,Bohr–Mollerup theorem,"(f : ℝ → ℝ)
(hpos : ∀ x > 0, 0 < f x)
(h1 : f 1 = 1)
(hrec : ∀ x > 0, f (x + 1) = x * f x)
(hlog : ConvexOn ℝ (Set.Ioi 0) (Real.log ∘ f)) :
∀ x > 0, f x = Real.Gamma x",T038_V6,T038_V2 -> T038_V6,"by
exact (show _ from by
intro x hx
apply Real.eq_Gamma_of_log_convex hlog
· intro y hy
exact hrec y hy
· intro y hy
exact hpos y hy
· exact h1
· exact hx
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T046,Bézout's identity,"(a b : ℤ) :
∃ x y : ℤ, Int.gcd a b = a * x + b * y",T046_V1,root,"by
exact ⟨Int.gcdA a b, Int.gcdB a b, Int.gcd_eq_gcd_ab a b⟩",True,reference,baseline,reference,baseline,reference
T046,Bézout's identity,"(a b : ℤ) :
∃ x y : ℤ, Int.gcd a b = a * x + b * y",T046_V2,T046_V1 -> T046_V2,"by
refine ⟨Int.gcdA a b, Int.gcdB a b, ?_⟩
simpa [mul_comm] using (Int.gcd_eq_gcd_ab a b)",True,same_route,witness_construction,same_route,proof_term_structure,same_route
T046,Bézout's identity,"(a b : ℤ) :
∃ x y : ℤ, Int.gcd a b = a * x + b * y",T046_V3,T046_V1 -> T046_V3,"by
exact (by
exact ⟨Int.gcdA a b, Int.gcdB a b, Int.gcd_eq_gcd_ab a b⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T046,Bézout's identity,"(a b : ℤ) :
∃ x y : ℤ, Int.gcd a b = a * x + b * y",T046_V4,T046_V2 -> T046_V4,"by
exact (by
refine ⟨Int.gcdA a b, Int.gcdB a b, ?_⟩
simpa [mul_comm] using (Int.gcd_eq_gcd_ab a b)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T046,Bézout's identity,"(a b : ℤ) :
∃ x y : ℤ, Int.gcd a b = a * x + b * y",T046_V5,T046_V1 -> T046_V5,"by
exact id (by
exact ⟨Int.gcdA a b, Int.gcdB a b, Int.gcd_eq_gcd_ab a b⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T046,Bézout's identity,"(a b : ℤ) :
∃ x y : ℤ, Int.gcd a b = a * x + b * y",T046_V6,T046_V2 -> T046_V6,"by
exact (show _ from by
refine ⟨Int.gcdA a b, Int.gcdB a b, ?_⟩
simpa [mul_comm] using (Int.gcd_eq_gcd_ab a b)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T048,Candido's identity,"(x y : ℝ) :
(x ^ 2 + y ^ 2) ^ 2 + (2 * x * y) ^ 2 =
(x ^ 2 - y ^ 2) ^ 2 + (2 * x * y) ^ 2 + 4 * x ^ 2 * y ^ 2",T048_V1,root,"by
ring",True,reference,baseline,reference,baseline,reference
T048,Candido's identity,"(x y : ℝ) :
(x ^ 2 + y ^ 2) ^ 2 + (2 * x * y) ^ 2 =
(x ^ 2 - y ^ 2) ^ 2 + (2 * x * y) ^ 2 + 4 * x ^ 2 * y ^ 2",T048_V2,T048_V1 -> T048_V2,"by
ring_nf",True,new_route,algebraic_normal_form,new_route,algebraic_normal_form,same_route
T048,Candido's identity,"(x y : ℝ) :
(x ^ 2 + y ^ 2) ^ 2 + (2 * x * y) ^ 2 =
(x ^ 2 - y ^ 2) ^ 2 + (2 * x * y) ^ 2 + 4 * x ^ 2 * y ^ 2",T048_V3,T048_V1 -> T048_V3,"by
nlinarith",True,new_route,algebraic_normal_form,new_route,operator_functional,new_route
T048,Candido's identity,"(x y : ℝ) :
(x ^ 2 + y ^ 2) ^ 2 + (2 * x * y) ^ 2 =
(x ^ 2 - y ^ 2) ^ 2 + (2 * x * y) ^ 2 + 4 * x ^ 2 * y ^ 2",T048_V4,T048_V2 -> T048_V4,"by
exact (by
ring_nf
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T048,Candido's identity,"(x y : ℝ) :
(x ^ 2 + y ^ 2) ^ 2 + (2 * x * y) ^ 2 =
(x ^ 2 - y ^ 2) ^ 2 + (2 * x * y) ^ 2 + 4 * x ^ 2 * y ^ 2",T048_V5,T048_V1 -> T048_V5,"by
exact id (by
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T048,Candido's identity,"(x y : ℝ) :
(x ^ 2 + y ^ 2) ^ 2 + (2 * x * y) ^ 2 =
(x ^ 2 - y ^ 2) ^ 2 + (2 * x * y) ^ 2 + 4 * x ^ 2 * y ^ 2",T048_V6,T048_V2 -> T048_V6,"by
exact (show _ from by
ring_nf
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T049,Cantor's intersection theorem,"{X : Type*} [TopologicalSpace X] [T2Space X] (K : ℕ → Set X)
(hne : ∀ n, (K n).Nonempty)
(hcompact : ∀ n, IsCompact (K n))
(hdec : ∀ n, K (n + 1) ⊆ K n) :
(⋂ n, K n).Nonempty",T049_V1,root,"by
exact IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed K hdec hne
(hcompact 0) (fun n => (hcompact n).isClosed)",True,reference,baseline,reference,baseline,reference
T049,Cantor's intersection theorem,"{X : Type*} [TopologicalSpace X] [T2Space X] (K : ℕ → Set X)
(hne : ∀ n, (K n).Nonempty)
(hcompact : ∀ n, IsCompact (K n))
(hdec : ∀ n, K (n + 1) ⊆ K n) :
(⋂ n, K n).Nonempty",T049_V2,T049_V1 -> T049_V2,"by
apply IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed K hdec hne
· exact hcompact 0
· intro n
exact (hcompact n).isClosed",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T049,Cantor's intersection theorem,"{X : Type*} [TopologicalSpace X] [T2Space X] (K : ℕ → Set X)
(hne : ∀ n, (K n).Nonempty)
(hcompact : ∀ n, IsCompact (K n))
(hdec : ∀ n, K (n + 1) ⊆ K n) :
(⋂ n, K n).Nonempty",T049_V3,T049_V1 -> T049_V3,"by
have hant : Antitone K := antitone_nat_of_succ_le hdec
apply IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed K
· exact hant.directed_ge
· exact hne
· exact hcompact
· intro n
exact (hcompact n).isClosed",True,new_route,logical_characterization,new_route,route_substitution,new_route
T049,Cantor's intersection theorem,"{X : Type*} [TopologicalSpace X] [T2Space X] (K : ℕ → Set X)
(hne : ∀ n, (K n).Nonempty)
(hcompact : ∀ n, IsCompact (K n))
(hdec : ∀ n, K (n + 1) ⊆ K n) :
(⋂ n, K n).Nonempty",T049_V4,T049_V2 -> T049_V4,"by
exact (by
apply IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed K hdec hne
· exact hcompact 0
· intro n
exact (hcompact n).isClosed
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T049,Cantor's intersection theorem,"{X : Type*} [TopologicalSpace X] [T2Space X] (K : ℕ → Set X)
(hne : ∀ n, (K n).Nonempty)
(hcompact : ∀ n, IsCompact (K n))
(hdec : ∀ n, K (n + 1) ⊆ K n) :
(⋂ n, K n).Nonempty",T049_V5,T049_V1 -> T049_V5,"by
exact id (by
exact IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed K hdec hne
(hcompact 0) (fun n => (hcompact n).isClosed)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T049,Cantor's intersection theorem,"{X : Type*} [TopologicalSpace X] [T2Space X] (K : ℕ → Set X)
(hne : ∀ n, (K n).Nonempty)
(hcompact : ∀ n, IsCompact (K n))
(hdec : ∀ n, K (n + 1) ⊆ K n) :
(⋂ n, K n).Nonempty",T049_V6,T049_V2 -> T049_V6,"by
exact (show _ from by
apply IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed K hdec hne
· exact hcompact 0
· intro n
exact (hcompact n).isClosed
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T052,Cassini and Catalan identities,"(n : ℤ) :
Int.fib (n + 1) * Int.fib (n - 1) - Int.fib n ^ 2 = (-1) ^ n.natAbs",T052_V1,root,"by
exact Int.fib_succ_mul_fib_pred_sub_fib_sq n",True,reference,baseline,reference,baseline,reference
T052,Cassini and Catalan identities,"(n : ℤ) :
Int.fib (n + 1) * Int.fib (n - 1) - Int.fib n ^ 2 = (-1) ^ n.natAbs",T052_V2,T052_V1 -> T052_V2,"by
simpa only using (Int.fib_succ_mul_fib_pred_sub_fib_sq n)",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T052,Cassini and Catalan identities,"(n : ℤ) :
Int.fib (n + 1) * Int.fib (n - 1) - Int.fib n ^ 2 = (-1) ^ n.natAbs",T052_V3,T052_V1 -> T052_V3,"by
exact (by
exact Int.fib_succ_mul_fib_pred_sub_fib_sq n
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T052,Cassini and Catalan identities,"(n : ℤ) :
Int.fib (n + 1) * Int.fib (n - 1) - Int.fib n ^ 2 = (-1) ^ n.natAbs",T052_V4,T052_V2 -> T052_V4,"by
exact (by
simpa only using (Int.fib_succ_mul_fib_pred_sub_fib_sq n)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T052,Cassini and Catalan identities,"(n : ℤ) :
Int.fib (n + 1) * Int.fib (n - 1) - Int.fib n ^ 2 = (-1) ^ n.natAbs",T052_V5,T052_V1 -> T052_V5,"by
exact id (by
exact Int.fib_succ_mul_fib_pred_sub_fib_sq n
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T052,Cassini and Catalan identities,"(n : ℤ) :
Int.fib (n + 1) * Int.fib (n - 1) - Int.fib n ^ 2 = (-1) ^ n.natAbs",T052_V6,T052_V2 -> T052_V6,"by
exact (show _ from by
simpa only using (Int.fib_succ_mul_fib_pred_sub_fib_sq n)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T056,Cauchy–Hadamard theorem,"(c : ℕ → ℂ) :
let p := FormalMultilinearSeries.ofScalars ℂ c
p.radius⁻¹ =
limsup (fun n => ((‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop",T056_V1,root,"by
exact FormalMultilinearSeries.radius_inv_eq_limsup
(p := FormalMultilinearSeries.ofScalars ℂ c)",True,reference,baseline,reference,baseline,reference
T056,Cauchy–Hadamard theorem,"(c : ℕ → ℂ) :
let p := FormalMultilinearSeries.ofScalars ℂ c
p.radius⁻¹ =
limsup (fun n => ((‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop",T056_V2,T056_V1 -> T056_V2,"by
apply FormalMultilinearSeries.radius_inv_eq_limsup
(p := FormalMultilinearSeries.ofScalars ℂ c)",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T056,Cauchy–Hadamard theorem,"(c : ℕ → ℂ) :
let p := FormalMultilinearSeries.ofScalars ℂ c
p.radius⁻¹ =
limsup (fun n => ((‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop",T056_V3,T056_V1 -> T056_V3,"by
exact (by
exact FormalMultilinearSeries.radius_inv_eq_limsup
(p := FormalMultilinearSeries.ofScalars ℂ c)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T056,Cauchy–Hadamard theorem,"(c : ℕ → ℂ) :
let p := FormalMultilinearSeries.ofScalars ℂ c
p.radius⁻¹ =
limsup (fun n => ((‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop",T056_V4,T056_V2 -> T056_V4,"by
exact (by
apply FormalMultilinearSeries.radius_inv_eq_limsup
(p := FormalMultilinearSeries.ofScalars ℂ c)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T056,Cauchy–Hadamard theorem,"(c : ℕ → ℂ) :
let p := FormalMultilinearSeries.ofScalars ℂ c
p.radius⁻¹ =
limsup (fun n => ((‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop",T056_V5,T056_V1 -> T056_V5,"by
exact id (by
exact FormalMultilinearSeries.radius_inv_eq_limsup
(p := FormalMultilinearSeries.ofScalars ℂ c)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T056,Cauchy–Hadamard theorem,"(c : ℕ → ℂ) :
let p := FormalMultilinearSeries.ofScalars ℂ c
p.radius⁻¹ =
limsup (fun n => ((‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop",T056_V6,T056_V2 -> T056_V6,"by
exact (show _ from by
apply FormalMultilinearSeries.radius_inv_eq_limsup
(p := FormalMultilinearSeries.ofScalars ℂ c)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T057,Chain rule,"(f : ℝ → ℝ) (g : ℝ → ℝ) (x : ℝ)
(hf : DifferentiableAt ℝ f (g x)) (hg : DifferentiableAt ℝ g x) :
deriv (fun t => f (g t)) x = deriv f (g x) * deriv g x",T057_V1,root,"by
exact HasDerivAt.deriv (hf.hasDerivAt.comp x hg.hasDerivAt)",True,reference,baseline,reference,baseline,reference
T057,Chain rule,"(f : ℝ → ℝ) (g : ℝ → ℝ) (x : ℝ)
(hf : DifferentiableAt ℝ f (g x)) (hg : DifferentiableAt ℝ g x) :
deriv (fun t => f (g t)) x = deriv f (g x) * deriv g x",T057_V2,T057_V1 -> T057_V2,"by
simpa only [Function.comp_def] using (deriv_comp x hf hg)",True,new_route,operator_functional,new_route,operator_functional,new_route
T057,Chain rule,"(f : ℝ → ℝ) (g : ℝ → ℝ) (x : ℝ)
(hf : DifferentiableAt ℝ f (g x)) (hg : DifferentiableAt ℝ g x) :
deriv (fun t => f (g t)) x = deriv f (g x) * deriv g x",T057_V3,T057_V1 -> T057_V3,"by
change deriv (f ∘ g) x = deriv f (g x) * deriv g x
exact deriv_comp x hf hg",True,new_route,logical_characterization,new_route,operator_functional,new_route
T057,Chain rule,"(f : ℝ → ℝ) (g : ℝ → ℝ) (x : ℝ)
(hf : DifferentiableAt ℝ f (g x)) (hg : DifferentiableAt ℝ g x) :
deriv (fun t => f (g t)) x = deriv f (g x) * deriv g x",T057_V4,T057_V2 -> T057_V4,"by
exact (by
simpa only [Function.comp_def] using (deriv_comp x hf hg)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T057,Chain rule,"(f : ℝ → ℝ) (g : ℝ → ℝ) (x : ℝ)
(hf : DifferentiableAt ℝ f (g x)) (hg : DifferentiableAt ℝ g x) :
deriv (fun t => f (g t)) x = deriv f (g x) * deriv g x",T057_V5,T057_V1 -> T057_V5,"by
exact id (by
exact HasDerivAt.deriv (hf.hasDerivAt.comp x hg.hasDerivAt)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T057,Chain rule,"(f : ℝ → ℝ) (g : ℝ → ℝ) (x : ℝ)
(hf : DifferentiableAt ℝ f (g x)) (hg : DifferentiableAt ℝ g x) :
deriv (fun t => f (g t)) x = deriv f (g x) * deriv g x",T057_V6,T057_V2 -> T057_V6,"by
exact (show _ from by
simpa only [Function.comp_def] using (deriv_comp x hf hg)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T058,Chain rule (probability),"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) :
μ.real (A ∩ B) = μ.real A * (μ.real (A ∩ B) / μ.real A)",T058_V1,root,"by
field_simp",True,reference,baseline,reference,baseline,reference
T058,Chain rule (probability),"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) :
μ.real (A ∩ B) = μ.real A * (μ.real (A ∩ B) / μ.real A)",T058_V2,T058_V1 -> T058_V2,"by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
rw [mul_div_cancel₀ _ hA0]",True,new_route,algebraic_normal_form,new_route,algebraic_normal_form,new_route
T058,Chain rule (probability),"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) :
μ.real (A ∩ B) = μ.real A * (μ.real (A ∩ B) / μ.real A)",T058_V3,T058_V1 -> T058_V3,"by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
exact (mul_div_cancel₀ (μ.real (A ∩ B)) hA0).symm",True,new_route,algebraic_normal_form,new_route,algebraic_normal_form,new_route
T058,Chain rule (probability),"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) :
μ.real (A ∩ B) = μ.real A * (μ.real (A ∩ B) / μ.real A)",T058_V4,T058_V2 -> T058_V4,"by
exact (by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
rw [mul_div_cancel₀ _ hA0]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T058,Chain rule (probability),"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) :
μ.real (A ∩ B) = μ.real A * (μ.real (A ∩ B) / μ.real A)",T058_V5,T058_V1 -> T058_V5,"by
exact id (by
field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T058,Chain rule (probability),"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (A B : Set Ω)
(hA : 0 < μ.real A) :
μ.real (A ∩ B) = μ.real A * (μ.real (A ∩ B) / μ.real A)",T058_V6,T058_V2 -> T058_V6,"by
exact (show _ from by
have hA0 : μ.real A ≠ 0 := ne_of_gt hA
rw [mul_div_cancel₀ _ hA0]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T059,Chebyshev's inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hX : MemLp X 2 μ) (σ k : ℝ) (hσ : 0 < σ) (hk : 0 < k)
(hvar : ProbabilityTheory.variance X μ = σ ^ 2) :
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤ ENNReal.ofReal (1 / k ^ 2)",T059_V1,root,"by
have hcheb := ProbabilityTheory.meas_ge_le_variance_div_sq hX (mul_pos hk hσ)
convert hcheb using 1
congr 1
rw [hvar]
field_simp",True,reference,baseline,reference,baseline,reference
T059,Chebyshev's inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hX : MemLp X 2 μ) (σ k : ℝ) (hσ : 0 < σ) (hk : 0 < k)
(hvar : ProbabilityTheory.variance X μ = σ ^ 2) :
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤ ENNReal.ofReal (1 / k ^ 2)",T059_V2,T059_V1 -> T059_V2,"by
calc
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤
ENNReal.ofReal (ProbabilityTheory.variance X μ / (k * σ) ^ 2) :=
ProbabilityTheory.meas_ge_le_variance_div_sq hX (mul_pos hk hσ)
_ = ENNReal.ofReal (1 / k ^ 2) := by
congr 1
rw [hvar]
field_simp",True,same_route,proof_intermediate,same_route,proof_intermediate,same_route
T059,Chebyshev's inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hX : MemLp X 2 μ) (σ k : ℝ) (hσ : 0 < σ) (hk : 0 < k)
(hvar : ProbabilityTheory.variance X μ = σ ^ 2) :
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤ ENNReal.ofReal (1 / k ^ 2)",T059_V3,T059_V1 -> T059_V3,"by
exact (by
have hcheb := ProbabilityTheory.meas_ge_le_variance_div_sq hX (mul_pos hk hσ)
convert hcheb using 1
congr 1
rw [hvar]
field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T059,Chebyshev's inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hX : MemLp X 2 μ) (σ k : ℝ) (hσ : 0 < σ) (hk : 0 < k)
(hvar : ProbabilityTheory.variance X μ = σ ^ 2) :
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤ ENNReal.ofReal (1 / k ^ 2)",T059_V4,T059_V2 -> T059_V4,"by
exact (by
calc
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤
ENNReal.ofReal (ProbabilityTheory.variance X μ / (k * σ) ^ 2) :=
ProbabilityTheory.meas_ge_le_variance_div_sq hX (mul_pos hk hσ)
_ = ENNReal.ofReal (1 / k ^ 2) := by
congr 1
rw [hvar]
field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T059,Chebyshev's inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hX : MemLp X 2 μ) (σ k : ℝ) (hσ : 0 < σ) (hk : 0 < k)
(hvar : ProbabilityTheory.variance X μ = σ ^ 2) :
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤ ENNReal.ofReal (1 / k ^ 2)",T059_V5,T059_V1 -> T059_V5,"by
exact id (by
have hcheb := ProbabilityTheory.meas_ge_le_variance_div_sq hX (mul_pos hk hσ)
convert hcheb using 1
congr 1
rw [hvar]
field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T059,Chebyshev's inequality,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ)
(hX : MemLp X 2 μ) (σ k : ℝ) (hσ : 0 < σ) (hk : 0 < k)
(hvar : ProbabilityTheory.variance X μ = σ ^ 2) :
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤ ENNReal.ofReal (1 / k ^ 2)",T059_V6,T059_V2 -> T059_V6,"by
exact (show _ from by
calc
μ {ω | k * σ ≤ |X ω - μ[X]|} ≤
ENNReal.ofReal (ProbabilityTheory.variance X μ / (k * σ) ^ 2) :=
ProbabilityTheory.meas_ge_le_variance_div_sq hX (mul_pos hk hσ)
_ = ENNReal.ofReal (1 / k ^ 2) := by
congr 1
rw [hvar]
field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T060,Chebyshev's sum inequality,"(n : ℕ) (a b : Fin n → ℝ)
(ha : Monotone a) (hb : Monotone b) :
(1 / (n : ℝ)) * ∑ i, a i * b i ≥
((1 / (n : ℝ)) * ∑ i, a i) *
((1 / (n : ℝ)) * ∑ i, b i)",T060_V1,root,"by
by_cases hn : n = 0
· subst n
simp
have hnpos : (0 : ℝ) < n := by positivity
have hcore : (∑ i, a i) * ∑ i, b i ≤ (n : ℝ) * ∑ i, a i * b i := by
simpa using (ha.monovary hb).sum_mul_sum_le_card_mul_sum
calc
((1 / (n : ℝ)) * ∑ i, a i) * ((1 / (n : ℝ)) * ∑ i, b i) =
((∑ i, a i) * ∑ i, b i) / (n : ℝ) ^ 2 := by field_simp
_ ≤ ((n : ℝ) * ∑ i, a i * b i) / (n : ℝ) ^ 2 := by
gcongr
_ = (1 / (n : ℝ)) * ∑ i, a i * b i := by field_simp",True,reference,baseline,reference,baseline,reference
T060,Chebyshev's sum inequality,"(n : ℕ) (a b : Fin n → ℝ)
(ha : Monotone a) (hb : Monotone b) :
(1 / (n : ℝ)) * ∑ i, a i * b i ≥
((1 / (n : ℝ)) * ∑ i, a i) *
((1 / (n : ℝ)) * ∑ i, b i)",T060_V2,T060_V1 -> T060_V2,"by
by_cases hn : n = 0
· subst n
simp
have hnpos : (0 : ℝ) < n := by positivity
have hcore : (∑ i, a i) * ∑ i, b i ≤ (n : ℝ) * ∑ i, a i * b i :=
(by simpa using (ha.monovary hb).sum_mul_sum_le_card_mul_sum)
rw [ge_iff_le]
field_simp
nlinarith",True,same_route,residual_nonnegativity,new_route,algebraic_normal_form,same_route
T060,Chebyshev's sum inequality,"(n : ℕ) (a b : Fin n → ℝ)
(ha : Monotone a) (hb : Monotone b) :
(1 / (n : ℝ)) * ∑ i, a i * b i ≥
((1 / (n : ℝ)) * ∑ i, a i) *
((1 / (n : ℝ)) * ∑ i, b i)",T060_V3,T060_V1 -> T060_V3,"by
exact (by
by_cases hn : n = 0
· subst n
simp
have hnpos : (0 : ℝ) < n := by positivity
have hcore : (∑ i, a i) * ∑ i, b i ≤ (n : ℝ) * ∑ i, a i * b i := by
simpa using (ha.monovary hb).sum_mul_sum_le_card_mul_sum
calc
((1 / (n : ℝ)) * ∑ i, a i) * ((1 / (n : ℝ)) * ∑ i, b i) =
((∑ i, a i) * ∑ i, b i) / (n : ℝ) ^ 2 := by field_simp
_ ≤ ((n : ℝ) * ∑ i, a i * b i) / (n : ℝ) ^ 2 := by
gcongr
_ = (1 / (n : ℝ)) * ∑ i, a i * b i := by field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T060,Chebyshev's sum inequality,"(n : ℕ) (a b : Fin n → ℝ)
(ha : Monotone a) (hb : Monotone b) :
(1 / (n : ℝ)) * ∑ i, a i * b i ≥
((1 / (n : ℝ)) * ∑ i, a i) *
((1 / (n : ℝ)) * ∑ i, b i)",T060_V4,T060_V2 -> T060_V4,"by
exact (by
by_cases hn : n = 0
· subst n
simp
have hnpos : (0 : ℝ) < n := by positivity
have hcore : (∑ i, a i) * ∑ i, b i ≤ (n : ℝ) * ∑ i, a i * b i :=
(by simpa using (ha.monovary hb).sum_mul_sum_le_card_mul_sum)
rw [ge_iff_le]
field_simp
nlinarith
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T060,Chebyshev's sum inequality,"(n : ℕ) (a b : Fin n → ℝ)
(ha : Monotone a) (hb : Monotone b) :
(1 / (n : ℝ)) * ∑ i, a i * b i ≥
((1 / (n : ℝ)) * ∑ i, a i) *
((1 / (n : ℝ)) * ∑ i, b i)",T060_V5,T060_V1 -> T060_V5,"by
exact id (by
by_cases hn : n = 0
· subst n
simp
have hnpos : (0 : ℝ) < n := by positivity
have hcore : (∑ i, a i) * ∑ i, b i ≤ (n : ℝ) * ∑ i, a i * b i := by
simpa using (ha.monovary hb).sum_mul_sum_le_card_mul_sum
calc
((1 / (n : ℝ)) * ∑ i, a i) * ((1 / (n : ℝ)) * ∑ i, b i) =
((∑ i, a i) * ∑ i, b i) / (n : ℝ) ^ 2 := by field_simp
_ ≤ ((n : ℝ) * ∑ i, a i * b i) / (n : ℝ) ^ 2 := by
gcongr
_ = (1 / (n : ℝ)) * ∑ i, a i * b i := by field_simp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T060,Chebyshev's sum inequality,"(n : ℕ) (a b : Fin n → ℝ)
(ha : Monotone a) (hb : Monotone b) :
(1 / (n : ℝ)) * ∑ i, a i * b i ≥
((1 / (n : ℝ)) * ∑ i, a i) *
((1 / (n : ℝ)) * ∑ i, b i)",T060_V6,T060_V2 -> T060_V6,"by
exact (show _ from by
by_cases hn : n = 0
· subst n
simp
have hnpos : (0 : ℝ) < n := by positivity
have hcore : (∑ i, a i) * ∑ i, b i ≤ (n : ℝ) * ∑ i, a i * b i :=
(by simpa using (ha.monovary hb).sum_mul_sum_le_card_mul_sum)
rw [ge_iff_le]
field_simp
nlinarith
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T061,Chernoff bound,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (a : ℝ) :
∀ t : ℝ, 0 ≤ t → Integrable (fun ω => Real.exp (t * X ω)) μ →
μ.real {ω | a ≤ X ω} ≤
Real.exp (-t * a) * ProbabilityTheory.mgf X μ t",T061_V1,root,"by
intro t ht hint
exact ProbabilityTheory.measure_ge_le_exp_mul_mgf a ht hint",True,reference,baseline,reference,baseline,reference
T061,Chernoff bound,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (a : ℝ) :
∀ t : ℝ, 0 ≤ t → Integrable (fun ω => Real.exp (t * X ω)) μ →
μ.real {ω | a ≤ X ω} ≤
Real.exp (-t * a) * ProbabilityTheory.mgf X μ t",T061_V2,T061_V1 -> T061_V2,"by
intro t ht hint
apply ProbabilityTheory.measure_ge_le_exp_mul_mgf (X := X) a
· exact ht
· exact hint",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T061,Chernoff bound,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (a : ℝ) :
∀ t : ℝ, 0 ≤ t → Integrable (fun ω => Real.exp (t * X ω)) μ →
μ.real {ω | a ≤ X ω} ≤
Real.exp (-t * a) * ProbabilityTheory.mgf X μ t",T061_V3,T061_V1 -> T061_V3,"by
exact (by
intro t ht hint
exact ProbabilityTheory.measure_ge_le_exp_mul_mgf a ht hint
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T061,Chernoff bound,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (a : ℝ) :
∀ t : ℝ, 0 ≤ t → Integrable (fun ω => Real.exp (t * X ω)) μ →
μ.real {ω | a ≤ X ω} ≤
Real.exp (-t * a) * ProbabilityTheory.mgf X μ t",T061_V4,T061_V2 -> T061_V4,"by
exact (by
intro t ht hint
apply ProbabilityTheory.measure_ge_le_exp_mul_mgf (X := X) a
· exact ht
· exact hint
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T061,Chernoff bound,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (a : ℝ) :
∀ t : ℝ, 0 ≤ t → Integrable (fun ω => Real.exp (t * X ω)) μ →
μ.real {ω | a ≤ X ω} ≤
Real.exp (-t * a) * ProbabilityTheory.mgf X μ t",T061_V5,T061_V1 -> T061_V5,"by
exact id (by
intro t ht hint
exact ProbabilityTheory.measure_ge_le_exp_mul_mgf a ht hint
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T061,Chernoff bound,"{Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → ℝ) (a : ℝ) :
∀ t : ℝ, 0 ≤ t → Integrable (fun ω => Real.exp (t * X ω)) μ →
μ.real {ω | a ≤ X ω} ≤
Real.exp (-t * a) * ProbabilityTheory.mgf X μ t",T061_V6,T061_V2 -> T061_V6,"by
exact (show _ from by
intro t ht hint
apply ProbabilityTheory.measure_ge_le_exp_mul_mgf (X := X) a
· exact ht
· exact hint
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T062,Chinese remainder theorem,"{ι : Type*} [Fintype ι] (m a : ι → ℕ)
(hm : ∀ i, m i ≠ 0)
(hpair : Pairwise fun i j => Nat.Coprime (m i) (m j)) :
∃ x : ℕ, ∀ i, x % m i = a i % m i",T062_V1,root,"by
classical
let x := Nat.chineseRemainderOfFinset a m Finset.univ
(fun i _ => hm i) (fun i _ j _ hij => hpair hij)
refine ⟨x, fun i => ?_⟩
exact (x.property i (Finset.mem_univ i))",True,reference,baseline,reference,baseline,reference
T062,Chinese remainder theorem,"{ι : Type*} [Fintype ι] (m a : ι → ℕ)
(hm : ∀ i, m i ≠ 0)
(hpair : Pairwise fun i j => Nat.Coprime (m i) (m j)) :
∃ x : ℕ, ∀ i, x % m i = a i % m i",T062_V2,T062_V1 -> T062_V2,"by
classical
let hzero : ∀ i ∈ (Finset.univ : Finset ι), m i ≠ 0 := fun i _ => hm i
let hcop : ∀ i ∈ (Finset.univ : Finset ι), ∀ j ∈ Finset.univ, i ≠ j → Nat.Coprime (m i) (m j) :=
fun i _ j _ hij => hpair hij
refine ⟨Nat.chineseRemainderOfFinset a m Finset.univ hzero hcop, ?_⟩
intro i
exact (Nat.chineseRemainderOfFinset a m Finset.univ hzero hcop).property i
(Finset.mem_univ i)",True,same_route,witness_construction,same_route,proof_term_structure,same_route
T062,Chinese remainder theorem,"{ι : Type*} [Fintype ι] (m a : ι → ℕ)
(hm : ∀ i, m i ≠ 0)
(hpair : Pairwise fun i j => Nat.Coprime (m i) (m j)) :
∃ x : ℕ, ∀ i, x % m i = a i % m i",T062_V3,T062_V1 -> T062_V3,"by
exact (by
classical
let x := Nat.chineseRemainderOfFinset a m Finset.univ
(fun i _ => hm i) (fun i _ j _ hij => hpair hij)
refine ⟨x, fun i => ?_⟩
exact (x.property i (Finset.mem_univ i))
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T062,Chinese remainder theorem,"{ι : Type*} [Fintype ι] (m a : ι → ℕ)
(hm : ∀ i, m i ≠ 0)
(hpair : Pairwise fun i j => Nat.Coprime (m i) (m j)) :
∃ x : ℕ, ∀ i, x % m i = a i % m i",T062_V4,T062_V2 -> T062_V4,"by
exact (by
classical
let hzero : ∀ i ∈ (Finset.univ : Finset ι), m i ≠ 0 := fun i _ => hm i
let hcop : ∀ i ∈ (Finset.univ : Finset ι), ∀ j ∈ Finset.univ, i ≠ j → Nat.Coprime (m i) (m j) :=
fun i _ j _ hij => hpair hij
refine ⟨Nat.chineseRemainderOfFinset a m Finset.univ hzero hcop, ?_⟩
intro i
exact (Nat.chineseRemainderOfFinset a m Finset.univ hzero hcop).property i
(Finset.mem_univ i)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T062,Chinese remainder theorem,"{ι : Type*} [Fintype ι] (m a : ι → ℕ)
(hm : ∀ i, m i ≠ 0)
(hpair : Pairwise fun i j => Nat.Coprime (m i) (m j)) :
∃ x : ℕ, ∀ i, x % m i = a i % m i",T062_V5,T062_V1 -> T062_V5,"by
exact id (by
classical
let x := Nat.chineseRemainderOfFinset a m Finset.univ
(fun i _ => hm i) (fun i _ j _ hij => hpair hij)
refine ⟨x, fun i => ?_⟩
exact (x.property i (Finset.mem_univ i))
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T062,Chinese remainder theorem,"{ι : Type*} [Fintype ι] (m a : ι → ℕ)
(hm : ∀ i, m i ≠ 0)
(hpair : Pairwise fun i j => Nat.Coprime (m i) (m j)) :
∃ x : ℕ, ∀ i, x % m i = a i % m i",T062_V6,T062_V2 -> T062_V6,"by
exact (show _ from by
classical
let hzero : ∀ i ∈ (Finset.univ : Finset ι), m i ≠ 0 := fun i _ => hm i
let hcop : ∀ i ∈ (Finset.univ : Finset ι), ∀ j ∈ Finset.univ, i ≠ j → Nat.Coprime (m i) (m j) :=
fun i _ j _ hij => hpair hij
refine ⟨Nat.chineseRemainderOfFinset a m Finset.univ hzero hcop, ?_⟩
intro i
exact (Nat.chineseRemainderOfFinset a m Finset.univ hzero hcop).property i
(Finset.mem_univ i)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T064,Closed graph theorem,"{X Y : Type*}
[NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]
[NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]
(T : X →ₗ[ℝ] Y) (hgraph : IsClosed {p : X × Y | p.2 = T p.1}) :
Continuous T",T064_V1,root,"by
apply LinearMap.continuous_of_isClosed_graph
simpa only [Set.setOf_eq_eq_singleton, Prod.swap_prod_mk] using hgraph",True,reference,baseline,reference,baseline,reference
T064,Closed graph theorem,"{X Y : Type*}
[NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]
[NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]
(T : X →ₗ[ℝ] Y) (hgraph : IsClosed {p : X × Y | p.2 = T p.1}) :
Continuous T",T064_V2,T064_V1 -> T064_V2,"by
refine LinearMap.continuous_of_isClosed_graph T ?_
simpa only [Set.setOf_eq_eq_singleton, Prod.swap_prod_mk] using hgraph",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T064,Closed graph theorem,"{X Y : Type*}
[NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]
[NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]
(T : X →ₗ[ℝ] Y) (hgraph : IsClosed {p : X × Y | p.2 = T p.1}) :
Continuous T",T064_V3,T064_V1 -> T064_V3,"by
exact (by
apply LinearMap.continuous_of_isClosed_graph
simpa only [Set.setOf_eq_eq_singleton, Prod.swap_prod_mk] using hgraph
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T064,Closed graph theorem,"{X Y : Type*}
[NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]
[NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]
(T : X →ₗ[ℝ] Y) (hgraph : IsClosed {p : X × Y | p.2 = T p.1}) :
Continuous T",T064_V4,T064_V2 -> T064_V4,"by
exact (by
refine LinearMap.continuous_of_isClosed_graph T ?_
simpa only [Set.setOf_eq_eq_singleton, Prod.swap_prod_mk] using hgraph
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T064,Closed graph theorem,"{X Y : Type*}
[NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]
[NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]
(T : X →ₗ[ℝ] Y) (hgraph : IsClosed {p : X × Y | p.2 = T p.1}) :
Continuous T",T064_V5,T064_V1 -> T064_V5,"by
exact id (by
apply LinearMap.continuous_of_isClosed_graph
simpa only [Set.setOf_eq_eq_singleton, Prod.swap_prod_mk] using hgraph
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T064,Closed graph theorem,"{X Y : Type*}
[NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]
[NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]
(T : X →ₗ[ℝ] Y) (hgraph : IsClosed {p : X × Y | p.2 = T p.1}) :
Continuous T",T064_V6,T064_V2 -> T064_V6,"by
exact (show _ from by
refine LinearMap.continuous_of_isClosed_graph T ?_
simpa only [Set.setOf_eq_eq_singleton, Prod.swap_prod_mk] using hgraph
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T065,Collage theorem,"{X : Type*} [MetricSpace X]
(L A : Set X) (W : Set X → Set X) (s ε : ℝ)
(hs0 : 0 ≤ s) (hs1 : s < 1)
(hLne : L.Nonempty) (hWLne : (W L).Nonempty)
(hLb : Bornology.IsBounded L) (hWLb : Bornology.IsBounded (W L))
(hcontract : ∀ C D, Metric.hausdorffDist (W C) (W D) ≤
s * Metric.hausdorffDist C D)
(hfix : W A = A)
(hcollage : Metric.hausdorffDist L (W L) ≤ ε) :
Metric.hausdorffDist L A ≤ ε / (1 - s)",T065_V1,root,"by
have hfin : Metric.hausdorffEDist L (W L) ≠ ⊤ :=
Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hLne hWLne hLb hWLb
have htri := Metric.hausdorffDist_triangle (s := L) (t := W L) (u := A) hfin
have hcon := hcontract L A
rw [hfix] at hcon
have hden : 0 < 1 - s := sub_pos.mpr hs1
rw [le_div_iff₀ hden]
nlinarith",True,reference,baseline,reference,baseline,reference
T065,Collage theorem,"{X : Type*} [MetricSpace X]
(L A : Set X) (W : Set X → Set X) (s ε : ℝ)
(hs0 : 0 ≤ s) (hs1 : s < 1)
(hLne : L.Nonempty) (hWLne : (W L).Nonempty)
(hLb : Bornology.IsBounded L) (hWLb : Bornology.IsBounded (W L))
(hcontract : ∀ C D, Metric.hausdorffDist (W C) (W D) ≤
s * Metric.hausdorffDist C D)
(hfix : W A = A)
(hcollage : Metric.hausdorffDist L (W L) ≤ ε) :
Metric.hausdorffDist L A ≤ ε / (1 - s)",T065_V2,T065_V1 -> T065_V2,"by
have hfinite : Metric.hausdorffEDist L (W L) ≠ ⊤ :=
Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hLne hWLne hLb hWLb
have htriangle : Metric.hausdorffDist L A ≤
Metric.hausdorffDist L (W L) + Metric.hausdorffDist (W L) A :=
Metric.hausdorffDist_triangle hfinite
have hWA : Metric.hausdorffDist (W L) A ≤ s * Metric.hausdorffDist L A := by
simpa [hfix] using hcontract L A
have hden : 0 < 1 - s := sub_pos.mpr hs1
apply (le_div_iff₀ hden).2
nlinarith",True,same_route,residual_nonnegativity,same_route,proof_term_structure,same_route
T065,Collage theorem,"{X : Type*} [MetricSpace X]
(L A : Set X) (W : Set X → Set X) (s ε : ℝ)
(hs0 : 0 ≤ s) (hs1 : s < 1)
(hLne : L.Nonempty) (hWLne : (W L).Nonempty)
(hLb : Bornology.IsBounded L) (hWLb : Bornology.IsBounded (W L))
(hcontract : ∀ C D, Metric.hausdorffDist (W C) (W D) ≤
s * Metric.hausdorffDist C D)
(hfix : W A = A)
(hcollage : Metric.hausdorffDist L (W L) ≤ ε) :
Metric.hausdorffDist L A ≤ ε / (1 - s)",T065_V3,T065_V1 -> T065_V3,"by
have hfinite : Metric.hausdorffEDist L (W L) ≠ ⊤ :=
Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hLne hWLne hLb hWLb
have htriangle : Metric.hausdorffDist L A ≤
Metric.hausdorffDist L (W L) + Metric.hausdorffDist (W L) A :=
Metric.hausdorffDist_triangle hfinite
have hcontract' : Metric.hausdorffDist (W L) A ≤
s * Metric.hausdorffDist L A := by
simpa [hfix] using hcontract L A
have hself : Metric.hausdorffDist L A ≤
ε + s * Metric.hausdorffDist L A :=
htriangle.trans (add_le_add hcollage hcontract')
have hden : 0 < 1 - s := sub_pos.mpr hs1
apply (le_div_iff₀ hden).2
nlinarith",True,new_route,residual_nonnegativity,new_route,route_substitution,same_route
T065,Collage theorem,"{X : Type*} [MetricSpace X]
(L A : Set X) (W : Set X → Set X) (s ε : ℝ)
(hs0 : 0 ≤ s) (hs1 : s < 1)
(hLne : L.Nonempty) (hWLne : (W L).Nonempty)
(hLb : Bornology.IsBounded L) (hWLb : Bornology.IsBounded (W L))
(hcontract : ∀ C D, Metric.hausdorffDist (W C) (W D) ≤
s * Metric.hausdorffDist C D)
(hfix : W A = A)
(hcollage : Metric.hausdorffDist L (W L) ≤ ε) :
Metric.hausdorffDist L A ≤ ε / (1 - s)",T065_V4,T065_V2 -> T065_V4,"by
exact (by
have hfinite : Metric.hausdorffEDist L (W L) ≠ ⊤ :=
Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hLne hWLne hLb hWLb
have htriangle : Metric.hausdorffDist L A ≤
Metric.hausdorffDist L (W L) + Metric.hausdorffDist (W L) A :=
Metric.hausdorffDist_triangle hfinite
have hWA : Metric.hausdorffDist (W L) A ≤ s * Metric.hausdorffDist L A := by
simpa [hfix] using hcontract L A
have hden : 0 < 1 - s := sub_pos.mpr hs1
apply (le_div_iff₀ hden).2
nlinarith
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T065,Collage theorem,"{X : Type*} [MetricSpace X]
(L A : Set X) (W : Set X → Set X) (s ε : ℝ)
(hs0 : 0 ≤ s) (hs1 : s < 1)
(hLne : L.Nonempty) (hWLne : (W L).Nonempty)
(hLb : Bornology.IsBounded L) (hWLb : Bornology.IsBounded (W L))
(hcontract : ∀ C D, Metric.hausdorffDist (W C) (W D) ≤
s * Metric.hausdorffDist C D)
(hfix : W A = A)
(hcollage : Metric.hausdorffDist L (W L) ≤ ε) :
Metric.hausdorffDist L A ≤ ε / (1 - s)",T065_V5,T065_V1 -> T065_V5,"by
exact id (by
have hfin : Metric.hausdorffEDist L (W L) ≠ ⊤ :=
Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hLne hWLne hLb hWLb
have htri := Metric.hausdorffDist_triangle (s := L) (t := W L) (u := A) hfin
have hcon := hcontract L A
rw [hfix] at hcon
have hden : 0 < 1 - s := sub_pos.mpr hs1
rw [le_div_iff₀ hden]
nlinarith
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T065,Collage theorem,"{X : Type*} [MetricSpace X]
(L A : Set X) (W : Set X → Set X) (s ε : ℝ)
(hs0 : 0 ≤ s) (hs1 : s < 1)
(hLne : L.Nonempty) (hWLne : (W L).Nonempty)
(hLb : Bornology.IsBounded L) (hWLb : Bornology.IsBounded (W L))
(hcontract : ∀ C D, Metric.hausdorffDist (W C) (W D) ≤
s * Metric.hausdorffDist C D)
(hfix : W A = A)
(hcollage : Metric.hausdorffDist L (W L) ≤ ε) :
Metric.hausdorffDist L A ≤ ε / (1 - s)",T065_V6,T065_V2 -> T065_V6,"by
exact (show _ from by
have hfinite : Metric.hausdorffEDist L (W L) ≠ ⊤ :=
Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hLne hWLne hLb hWLb
have htriangle : Metric.hausdorffDist L A ≤
Metric.hausdorffDist L (W L) + Metric.hausdorffDist (W L) A :=
Metric.hausdorffDist_triangle hfinite
have hWA : Metric.hausdorffDist (W L) A ≤ s * Metric.hausdorffDist L A := by
simpa [hfix] using hcontract L A
have hden : 0 < 1 - s := sub_pos.mpr hs1
apply (le_div_iff₀ hden).2
nlinarith
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T069,Cramer's rule,"{K : Type*} [Field K] (n : ℕ)
(A : Matrix (Fin n) (Fin n) K) (b : Fin n → K)
(hA : IsUnit A.det) :
∃! x : Fin n → K,
A.mulVec x = b ∧
∀ i, x i = (A.updateCol i b).det / A.det",T069_V1,root,"by
classical
have hA' : IsUnit A := A.isUnit_iff_isUnit_det.mpr hA
have hinj : Function.Injective A.mulVec := Matrix.mulVec_injective_of_isUnit hA'
have hsurj : Function.Surjective A.mulVec := Matrix.mulVec_surjective_iff_isUnit.mpr hA'
obtain ⟨x, hx⟩ := hsurj b
have hcr : A.cramer b = A.det • x := by
apply hinj
rw [Matrix.mulVec_cramer, Matrix.mulVec_smul, hx]
refine ⟨x, ⟨hx, fun i => ?_⟩, ?_⟩
· have hi := congr_fun hcr i
rw [Matrix.cramer_apply, Pi.smul_apply, smul_eq_mul] at hi
exact (eq_div_iff (hA.ne_zero)).2 (by simpa [mul_comm] using hi.symm)
· intro y hy
exact hinj (hy.1.trans hx.symm)",True,reference,baseline,reference,baseline,reference
T069,Cramer's rule,"{K : Type*} [Field K] (n : ℕ)
(A : Matrix (Fin n) (Fin n) K) (b : Fin n → K)
(hA : IsUnit A.det) :
∃! x : Fin n → K,
A.mulVec x = b ∧
∀ i, x i = (A.updateCol i b).det / A.det",T069_V2,T069_V1 -> T069_V2,"by
classical
have hdet : A.det ≠ 0 := hA.ne_zero
have hAunit : IsUnit A := A.isUnit_iff_isUnit_det.mpr hA
have hinj : Function.Injective A.mulVec := Matrix.mulVec_injective_of_isUnit hAunit
let x : Fin n → K := A.det⁻¹ • A.cramer b
have hx : A.mulVec x = b := by
change A.mulVec (A.det⁻¹ • A.cramer b) = b
rw [Matrix.mulVec_smul, Matrix.mulVec_cramer]
simp [hdet]
refine ⟨x, ⟨hx, ?_⟩, ?_⟩
· intro i
change (A.det⁻¹ • A.cramer b) i = (A.updateCol i b).det / A.det
simp [Matrix.cramer_apply, div_eq_mul_inv, mul_comm]
· intro y hy
exact hinj (hy.1.trans hx.symm)",True,new_route,witness_construction,new_route,witness_construction,new_route
T069,Cramer's rule,"{K : Type*} [Field K] (n : ℕ)
(A : Matrix (Fin n) (Fin n) K) (b : Fin n → K)
(hA : IsUnit A.det) :
∃! x : Fin n → K,
A.mulVec x = b ∧
∀ i, x i = (A.updateCol i b).det / A.det",T069_V3,T069_V1 -> T069_V3,"by
classical
have hinj : Function.Injective A.mulVec :=
Matrix.mulVec_injective_of_isUnit (A.isUnit_iff_isUnit_det.mpr hA)
let x : Fin n → K := (A⁻¹).mulVec b
have hx : A.mulVec x = b := by
dsimp [x]
rw [Matrix.mulVec_mulVec, Matrix.mul_nonsing_inv A hA, Matrix.one_mulVec]
have hcr : A.det • (A⁻¹).mulVec b = A.cramer b :=
Matrix.det_smul_inv_mulVec_eq_cramer A b hA
refine ⟨x, ⟨hx, ?_⟩, ?_⟩
· intro i
have hi := congr_fun hcr i
rw [Pi.smul_apply, smul_eq_mul, Matrix.cramer_apply] at hi
apply (eq_div_iff hA.ne_zero).2
simpa [x, mul_comm] using hi
· intro y hy
exact hinj (hy.1.trans hx.symm)",True,new_route,witness_construction,new_route,witness_construction,new_route
T069,Cramer's rule,"{K : Type*} [Field K] (n : ℕ)
(A : Matrix (Fin n) (Fin n) K) (b : Fin n → K)
(hA : IsUnit A.det) :
∃! x : Fin n → K,
A.mulVec x = b ∧
∀ i, x i = (A.updateCol i b).det / A.det",T069_V4,T069_V2 -> T069_V4,"by
exact (by
classical
have hdet : A.det ≠ 0 := hA.ne_zero
have hAunit : IsUnit A := A.isUnit_iff_isUnit_det.mpr hA
have hinj : Function.Injective A.mulVec := Matrix.mulVec_injective_of_isUnit hAunit
let x : Fin n → K := A.det⁻¹ • A.cramer b
have hx : A.mulVec x = b := by
change A.mulVec (A.det⁻¹ • A.cramer b) = b
rw [Matrix.mulVec_smul, Matrix.mulVec_cramer]
simp [hdet]
refine ⟨x, ⟨hx, ?_⟩, ?_⟩
· intro i
change (A.det⁻¹ • A.cramer b) i = (A.updateCol i b).det / A.det
simp [Matrix.cramer_apply, div_eq_mul_inv, mul_comm]
· intro y hy
exact hinj (hy.1.trans hx.symm)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T069,Cramer's rule,"{K : Type*} [Field K] (n : ℕ)
(A : Matrix (Fin n) (Fin n) K) (b : Fin n → K)
(hA : IsUnit A.det) :
∃! x : Fin n → K,
A.mulVec x = b ∧
∀ i, x i = (A.updateCol i b).det / A.det",T069_V5,T069_V1 -> T069_V5,"by
exact id (by
classical
have hA' : IsUnit A := A.isUnit_iff_isUnit_det.mpr hA
have hinj : Function.Injective A.mulVec := Matrix.mulVec_injective_of_isUnit hA'
have hsurj : Function.Surjective A.mulVec := Matrix.mulVec_surjective_iff_isUnit.mpr hA'
obtain ⟨x, hx⟩ := hsurj b
have hcr : A.cramer b = A.det • x := by
apply hinj
rw [Matrix.mulVec_cramer, Matrix.mulVec_smul, hx]
refine ⟨x, ⟨hx, fun i => ?_⟩, ?_⟩
· have hi := congr_fun hcr i
rw [Matrix.cramer_apply, Pi.smul_apply, smul_eq_mul] at hi
exact (eq_div_iff (hA.ne_zero)).2 (by simpa [mul_comm] using hi.symm)
· intro y hy
exact hinj (hy.1.trans hx.symm)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T069,Cramer's rule,"{K : Type*} [Field K] (n : ℕ)
(A : Matrix (Fin n) (Fin n) K) (b : Fin n → K)
(hA : IsUnit A.det) :
∃! x : Fin n → K,
A.mulVec x = b ∧
∀ i, x i = (A.updateCol i b).det / A.det",T069_V6,T069_V2 -> T069_V6,"by
exact (show _ from by
classical
have hdet : A.det ≠ 0 := hA.ne_zero
have hAunit : IsUnit A := A.isUnit_iff_isUnit_det.mpr hA
have hinj : Function.Injective A.mulVec := Matrix.mulVec_injective_of_isUnit hAunit
let x : Fin n → K := A.det⁻¹ • A.cramer b
have hx : A.mulVec x = b := by
change A.mulVec (A.det⁻¹ • A.cramer b) = b
rw [Matrix.mulVec_smul, Matrix.mulVec_cramer]
simp [hdet]
refine ⟨x, ⟨hx, ?_⟩, ?_⟩
· intro i
change (A.det⁻¹ • A.cramer b) i = (A.updateCol i b).det / A.det
simp [Matrix.cramer_apply, div_eq_mul_inv, mul_comm]
· intro y hy
exact hinj (hy.1.trans hx.symm)
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T076,Darboux's theorem,"(f : ℝ → ℝ) (a b y : ℝ)
(hab : a < b) (hf : ∀ x ∈ Set.Icc a b, DifferentiableAt ℝ f x)
(hy : y ∈ Set.Icc (deriv f a) (deriv f b) ∨
y ∈ Set.Icc (deriv f b) (deriv f a)) :
∃ c ∈ Set.Icc a b, deriv f c = y",T076_V1,root,"by
have himg : Set.OrdConnected (deriv f '' Set.Icc a b) :=
Set.ordConnected_Icc.image_deriv hf
have haimg : deriv f a ∈ deriv f '' Set.Icc a b :=
⟨a, Set.left_mem_Icc.mpr hab.le, rfl⟩
have hbimg : deriv f b ∈ deriv f '' Set.Icc a b :=
⟨b, Set.right_mem_Icc.mpr hab.le, rfl⟩
have hyimg : y ∈ deriv f '' Set.Icc a b := by
rcases hy with hy | hy
· exact himg.out haimg hbimg hy
· exact himg.out hbimg haimg hy
rcases hyimg with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩",True,reference,baseline,reference,baseline,reference
T076,Darboux's theorem,"(f : ℝ → ℝ) (a b y : ℝ)
(hab : a < b) (hf : ∀ x ∈ Set.Icc a b, DifferentiableAt ℝ f x)
(hy : y ∈ Set.Icc (deriv f a) (deriv f b) ∨
y ∈ Set.Icc (deriv f b) (deriv f a)) :
∃ c ∈ Set.Icc a b, deriv f c = y",T076_V2,T076_V1 -> T076_V2,"by
have himg : Set.OrdConnected (deriv f '' Set.Icc a b) :=
Set.ordConnected_Icc.image_deriv hf
have ha : deriv f a ∈ deriv f '' Set.Icc a b :=
⟨a, Set.left_mem_Icc.mpr hab.le, rfl⟩
have hb : deriv f b ∈ deriv f '' Set.Icc a b :=
⟨b, Set.right_mem_Icc.mpr hab.le, rfl⟩
rcases hy with hy | hy
· rcases himg.out ha hb hy with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩
· rcases himg.out hb ha hy with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T076,Darboux's theorem,"(f : ℝ → ℝ) (a b y : ℝ)
(hab : a < b) (hf : ∀ x ∈ Set.Icc a b, DifferentiableAt ℝ f x)
(hy : y ∈ Set.Icc (deriv f a) (deriv f b) ∨
y ∈ Set.Icc (deriv f b) (deriv f a)) :
∃ c ∈ Set.Icc a b, deriv f c = y",T076_V3,T076_V1 -> T076_V3,"by
exact (by
have himg : Set.OrdConnected (deriv f '' Set.Icc a b) :=
Set.ordConnected_Icc.image_deriv hf
have haimg : deriv f a ∈ deriv f '' Set.Icc a b :=
⟨a, Set.left_mem_Icc.mpr hab.le, rfl⟩
have hbimg : deriv f b ∈ deriv f '' Set.Icc a b :=
⟨b, Set.right_mem_Icc.mpr hab.le, rfl⟩
have hyimg : y ∈ deriv f '' Set.Icc a b := by
rcases hy with hy | hy
· exact himg.out haimg hbimg hy
· exact himg.out hbimg haimg hy
rcases hyimg with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T076,Darboux's theorem,"(f : ℝ → ℝ) (a b y : ℝ)
(hab : a < b) (hf : ∀ x ∈ Set.Icc a b, DifferentiableAt ℝ f x)
(hy : y ∈ Set.Icc (deriv f a) (deriv f b) ∨
y ∈ Set.Icc (deriv f b) (deriv f a)) :
∃ c ∈ Set.Icc a b, deriv f c = y",T076_V4,T076_V2 -> T076_V4,"by
exact (by
have himg : Set.OrdConnected (deriv f '' Set.Icc a b) :=
Set.ordConnected_Icc.image_deriv hf
have ha : deriv f a ∈ deriv f '' Set.Icc a b :=
⟨a, Set.left_mem_Icc.mpr hab.le, rfl⟩
have hb : deriv f b ∈ deriv f '' Set.Icc a b :=
⟨b, Set.right_mem_Icc.mpr hab.le, rfl⟩
rcases hy with hy | hy
· rcases himg.out ha hb hy with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩
· rcases himg.out hb ha hy with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T076,Darboux's theorem,"(f : ℝ → ℝ) (a b y : ℝ)
(hab : a < b) (hf : ∀ x ∈ Set.Icc a b, DifferentiableAt ℝ f x)
(hy : y ∈ Set.Icc (deriv f a) (deriv f b) ∨
y ∈ Set.Icc (deriv f b) (deriv f a)) :
∃ c ∈ Set.Icc a b, deriv f c = y",T076_V5,T076_V1 -> T076_V5,"by
exact id (by
have himg : Set.OrdConnected (deriv f '' Set.Icc a b) :=
Set.ordConnected_Icc.image_deriv hf
have haimg : deriv f a ∈ deriv f '' Set.Icc a b :=
⟨a, Set.left_mem_Icc.mpr hab.le, rfl⟩
have hbimg : deriv f b ∈ deriv f '' Set.Icc a b :=
⟨b, Set.right_mem_Icc.mpr hab.le, rfl⟩
have hyimg : y ∈ deriv f '' Set.Icc a b := by
rcases hy with hy | hy
· exact himg.out haimg hbimg hy
· exact himg.out hbimg haimg hy
rcases hyimg with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T076,Darboux's theorem,"(f : ℝ → ℝ) (a b y : ℝ)
(hab : a < b) (hf : ∀ x ∈ Set.Icc a b, DifferentiableAt ℝ f x)
(hy : y ∈ Set.Icc (deriv f a) (deriv f b) ∨
y ∈ Set.Icc (deriv f b) (deriv f a)) :
∃ c ∈ Set.Icc a b, deriv f c = y",T076_V6,T076_V2 -> T076_V6,"by
exact (show _ from by
have himg : Set.OrdConnected (deriv f '' Set.Icc a b) :=
Set.ordConnected_Icc.image_deriv hf
have ha : deriv f a ∈ deriv f '' Set.Icc a b :=
⟨a, Set.left_mem_Icc.mpr hab.le, rfl⟩
have hb : deriv f b ∈ deriv f '' Set.Icc a b :=
⟨b, Set.right_mem_Icc.mpr hab.le, rfl⟩
rcases hy with hy | hy
· rcases himg.out ha hb hy with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩
· rcases himg.out hb ha hy with ⟨c, hc, hcy⟩
exact ⟨c, hc, hcy⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T079,De Moivre's formula,"(x : ℝ) (n : ℤ) :
(Complex.cos x + Complex.I * Complex.sin x) ^ n =
Complex.cos (n * x) + Complex.I * Complex.sin (n * x)",T079_V1,root,"by
rw [mul_comm Complex.I, ← Complex.exp_mul_I]
rw [mul_comm Complex.I, ← Complex.exp_mul_I]
rw [← Complex.exp_int_mul]
congr 1
push_cast
ring",True,reference,baseline,reference,baseline,reference
T079,De Moivre's formula,"(x : ℝ) (n : ℤ) :
(Complex.cos x + Complex.I * Complex.sin x) ^ n =
Complex.cos (n * x) + Complex.I * Complex.sin (n * x)",T079_V2,T079_V1 -> T079_V2,"by
have hleft : Complex.cos x + Complex.I * Complex.sin x =
Complex.exp (Complex.I * x) := by
simpa [mul_comm] using (Complex.exp_mul_I x).symm
have hright : Complex.cos (n * x) + Complex.I * Complex.sin (n * x) =
Complex.exp (Complex.I * (n * x)) := by
simpa [mul_comm] using (Complex.exp_mul_I (n * x)).symm
rw [hleft, hright, ← Complex.exp_int_mul]
congr 1
push_cast
ring",True,same_route,operator_functional,new_route,proof_intermediate,same_route
T079,De Moivre's formula,"(x : ℝ) (n : ℤ) :
(Complex.cos x + Complex.I * Complex.sin x) ^ n =
Complex.cos (n * x) + Complex.I * Complex.sin (n * x)",T079_V3,T079_V1 -> T079_V3,"by
exact (by
rw [mul_comm Complex.I, ← Complex.exp_mul_I]
rw [mul_comm Complex.I, ← Complex.exp_mul_I]
rw [← Complex.exp_int_mul]
congr 1
push_cast
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T079,De Moivre's formula,"(x : ℝ) (n : ℤ) :
(Complex.cos x + Complex.I * Complex.sin x) ^ n =
Complex.cos (n * x) + Complex.I * Complex.sin (n * x)",T079_V4,T079_V2 -> T079_V4,"by
exact (by
have hleft : Complex.cos x + Complex.I * Complex.sin x =
Complex.exp (Complex.I * x) := by
simpa [mul_comm] using (Complex.exp_mul_I x).symm
have hright : Complex.cos (n * x) + Complex.I * Complex.sin (n * x) =
Complex.exp (Complex.I * (n * x)) := by
simpa [mul_comm] using (Complex.exp_mul_I (n * x)).symm
rw [hleft, hright, ← Complex.exp_int_mul]
congr 1
push_cast
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T079,De Moivre's formula,"(x : ℝ) (n : ℤ) :
(Complex.cos x + Complex.I * Complex.sin x) ^ n =
Complex.cos (n * x) + Complex.I * Complex.sin (n * x)",T079_V5,T079_V1 -> T079_V5,"by
exact id (by
rw [mul_comm Complex.I, ← Complex.exp_mul_I]
rw [mul_comm Complex.I, ← Complex.exp_mul_I]
rw [← Complex.exp_int_mul]
congr 1
push_cast
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T079,De Moivre's formula,"(x : ℝ) (n : ℤ) :
(Complex.cos x + Complex.I * Complex.sin x) ^ n =
Complex.cos (n * x) + Complex.I * Complex.sin (n * x)",T079_V6,T079_V2 -> T079_V6,"by
exact (show _ from by
have hleft : Complex.cos x + Complex.I * Complex.sin x =
Complex.exp (Complex.I * x) := by
simpa [mul_comm] using (Complex.exp_mul_I x).symm
have hright : Complex.cos (n * x) + Complex.I * Complex.sin (n * x) =
Complex.exp (Complex.I * (n * x)) := by
simpa [mul_comm] using (Complex.exp_mul_I (n * x)).symm
rw [hleft, hright, ← Complex.exp_int_mul]
congr 1
push_cast
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T080,Degen's eight-square identity,"(x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ : ℝ) :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃ + x₅ * y₆ - x₆ * y₅ - x₇ * y₈ + x₈ * y₇) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂ + x₅ * y₇ + x₆ * y₈ - x₇ * y₅ - x₈ * y₆) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁ + x₅ * y₈ - x₆ * y₇ + x₇ * y₆ - x₈ * y₅) ^ 2 +
(x₁ * y₅ - x₂ * y₆ - x₃ * y₇ - x₄ * y₈ + x₅ * y₁ + x₆ * y₂ + x₇ * y₃ + x₈ * y₄) ^ 2 +
(x₁ * y₆ + x₂ * y₅ - x₃ * y₈ + x₄ * y₇ - x₅ * y₂ + x₆ * y₁ - x₇ * y₄ + x₈ * y₃) ^ 2 +
(x₁ * y₇ + x₂ * y₈ + x₃ * y₅ - x₄ * y₆ - x₅ * y₃ + x₆ * y₄ + x₇ * y₁ - x₈ * y₂) ^ 2 +
(x₁ * y₈ - x₂ * y₇ + x₃ * y₆ + x₄ * y₅ - x₅ * y₄ - x₆ * y₃ + x₇ * y₂ + x₈ * y₁) ^ 2",T080_V1,root,"by
exact sum_eight_sq_mul_sum_eight_sq",True,reference,baseline,reference,baseline,reference
T080,Degen's eight-square identity,"(x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ : ℝ) :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃ + x₅ * y₆ - x₆ * y₅ - x₇ * y₈ + x₈ * y₇) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂ + x₅ * y₇ + x₆ * y₈ - x₇ * y₅ - x₈ * y₆) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁ + x₅ * y₈ - x₆ * y₇ + x₇ * y₆ - x₈ * y₅) ^ 2 +
(x₁ * y₅ - x₂ * y₆ - x₃ * y₇ - x₄ * y₈ + x₅ * y₁ + x₆ * y₂ + x₇ * y₃ + x₈ * y₄) ^ 2 +
(x₁ * y₆ + x₂ * y₅ - x₃ * y₈ + x₄ * y₇ - x₅ * y₂ + x₆ * y₁ - x₇ * y₄ + x₈ * y₃) ^ 2 +
(x₁ * y₇ + x₂ * y₈ + x₃ * y₅ - x₄ * y₆ - x₅ * y₃ + x₆ * y₄ + x₇ * y₁ - x₈ * y₂) ^ 2 +
(x₁ * y₈ - x₂ * y₇ + x₃ * y₆ + x₄ * y₅ - x₅ * y₄ - x₆ * y₃ + x₇ * y₂ + x₈ * y₁) ^ 2",T080_V2,T080_V1 -> T080_V2,"by
ring",True,new_route,algebraic_normal_form,new_route,algebraic_normal_form,new_route
T080,Degen's eight-square identity,"(x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ : ℝ) :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃ + x₅ * y₆ - x₆ * y₅ - x₇ * y₈ + x₈ * y₇) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂ + x₅ * y₇ + x₆ * y₈ - x₇ * y₅ - x₈ * y₆) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁ + x₅ * y₈ - x₆ * y₇ + x₇ * y₆ - x₈ * y₅) ^ 2 +
(x₁ * y₅ - x₂ * y₆ - x₃ * y₇ - x₄ * y₈ + x₅ * y₁ + x₆ * y₂ + x₇ * y₃ + x₈ * y₄) ^ 2 +
(x₁ * y₆ + x₂ * y₅ - x₃ * y₈ + x₄ * y₇ - x₅ * y₂ + x₆ * y₁ - x₇ * y₄ + x₈ * y₃) ^ 2 +
(x₁ * y₇ + x₂ * y₈ + x₃ * y₅ - x₄ * y₆ - x₅ * y₃ + x₆ * y₄ + x₇ * y₁ - x₈ * y₂) ^ 2 +
(x₁ * y₈ - x₂ * y₇ + x₃ * y₆ + x₄ * y₅ - x₅ * y₄ - x₆ * y₃ + x₇ * y₂ + x₈ * y₁) ^ 2",T080_V3,T080_V1 -> T080_V3,"by
ring_nf",True,new_route,algebraic_normal_form,new_route,algebraic_normal_form,new_route
T080,Degen's eight-square identity,"(x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ : ℝ) :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃ + x₅ * y₆ - x₆ * y₅ - x₇ * y₈ + x₈ * y₇) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂ + x₅ * y₇ + x₆ * y₈ - x₇ * y₅ - x₈ * y₆) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁ + x₅ * y₈ - x₆ * y₇ + x₇ * y₆ - x₈ * y₅) ^ 2 +
(x₁ * y₅ - x₂ * y₆ - x₃ * y₇ - x₄ * y₈ + x₅ * y₁ + x₆ * y₂ + x₇ * y₃ + x₈ * y₄) ^ 2 +
(x₁ * y₆ + x₂ * y₅ - x₃ * y₈ + x₄ * y₇ - x₅ * y₂ + x₆ * y₁ - x₇ * y₄ + x₈ * y₃) ^ 2 +
(x₁ * y₇ + x₂ * y₈ + x₃ * y₅ - x₄ * y₆ - x₅ * y₃ + x₆ * y₄ + x₇ * y₁ - x₈ * y₂) ^ 2 +
(x₁ * y₈ - x₂ * y₇ + x₃ * y₆ + x₄ * y₅ - x₅ * y₄ - x₆ * y₃ + x₇ * y₂ + x₈ * y₁) ^ 2",T080_V4,T080_V2 -> T080_V4,"by
exact (by
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T080,Degen's eight-square identity,"(x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ : ℝ) :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃ + x₅ * y₆ - x₆ * y₅ - x₇ * y₈ + x₈ * y₇) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂ + x₅ * y₇ + x₆ * y₈ - x₇ * y₅ - x₈ * y₆) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁ + x₅ * y₈ - x₆ * y₇ + x₇ * y₆ - x₈ * y₅) ^ 2 +
(x₁ * y₅ - x₂ * y₆ - x₃ * y₇ - x₄ * y₈ + x₅ * y₁ + x₆ * y₂ + x₇ * y₃ + x₈ * y₄) ^ 2 +
(x₁ * y₆ + x₂ * y₅ - x₃ * y₈ + x₄ * y₇ - x₅ * y₂ + x₆ * y₁ - x₇ * y₄ + x₈ * y₃) ^ 2 +
(x₁ * y₇ + x₂ * y₈ + x₃ * y₅ - x₄ * y₆ - x₅ * y₃ + x₆ * y₄ + x₇ * y₁ - x₈ * y₂) ^ 2 +
(x₁ * y₈ - x₂ * y₇ + x₃ * y₆ + x₄ * y₅ - x₅ * y₄ - x₆ * y₃ + x₇ * y₂ + x₈ * y₁) ^ 2",T080_V5,T080_V1 -> T080_V5,"by
exact id (by
exact sum_eight_sq_mul_sum_eight_sq
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T080,Degen's eight-square identity,"(x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ : ℝ) :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃ + x₅ * y₆ - x₆ * y₅ - x₇ * y₈ + x₈ * y₇) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂ + x₅ * y₇ + x₆ * y₈ - x₇ * y₅ - x₈ * y₆) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁ + x₅ * y₈ - x₆ * y₇ + x₇ * y₆ - x₈ * y₅) ^ 2 +
(x₁ * y₅ - x₂ * y₆ - x₃ * y₇ - x₄ * y₈ + x₅ * y₁ + x₆ * y₂ + x₇ * y₃ + x₈ * y₄) ^ 2 +
(x₁ * y₆ + x₂ * y₅ - x₃ * y₈ + x₄ * y₇ - x₅ * y₂ + x₆ * y₁ - x₇ * y₄ + x₈ * y₃) ^ 2 +
(x₁ * y₇ + x₂ * y₈ + x₃ * y₅ - x₄ * y₆ - x₅ * y₃ + x₆ * y₄ + x₇ * y₁ - x₈ * y₂) ^ 2 +
(x₁ * y₈ - x₂ * y₇ + x₃ * y₆ + x₄ * y₅ - x₅ * y₄ - x₆ * y₃ + x₇ * y₂ + x₈ * y₁) ^ 2",T080_V6,T080_V2 -> T080_V6,"by
exact (show _ from by
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T081,Difference of two squares,"(a b : ℝ) :
a ^ 2 - b ^ 2 = (a - b) * (a + b)",T081_V1,root,"by
ring",True,reference,baseline,reference,baseline,reference
T081,Difference of two squares,"(a b : ℝ) :
a ^ 2 - b ^ 2 = (a - b) * (a + b)",T081_V2,T081_V1 -> T081_V2,"by
ring_nf",True,new_route,algebraic_normal_form,new_route,algebraic_normal_form,same_route
T081,Difference of two squares,"(a b : ℝ) :
a ^ 2 - b ^ 2 = (a - b) * (a + b)",T081_V3,T081_V1 -> T081_V3,"by
nlinarith",True,new_route,algebraic_normal_form,new_route,route_substitution,new_route
T081,Difference of two squares,"(a b : ℝ) :
a ^ 2 - b ^ 2 = (a - b) * (a + b)",T081_V4,T081_V2 -> T081_V4,"by
exact (by
ring_nf
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T081,Difference of two squares,"(a b : ℝ) :
a ^ 2 - b ^ 2 = (a - b) * (a + b)",T081_V5,T081_V1 -> T081_V5,"by
exact id (by
ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T081,Difference of two squares,"(a b : ℝ) :
a ^ 2 - b ^ 2 = (a - b) * (a + b)",T081_V6,T081_V2 -> T081_V6,"by
exact (show _ from by
ring_nf
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T083,Dirichlet's approximation theorem,"(α : ℝ) (N : ℕ) (hN : 1 ≤ N) :
∃ p : ℤ, ∃ q : ℕ, 1 ≤ q ∧ q ≤ N ∧
|(q : ℝ) * α - p| ≤ 1 / (N + 1 : ℝ)",T083_V1,root,"by
obtain ⟨q, hq0, hqN, hbound⟩ := Real.exists_nat_abs_mul_sub_round_le α (by omega : 0 < N)
exact ⟨round (q * α), q, hq0, hqN, by simpa using hbound⟩",True,reference,baseline,reference,baseline,reference
T083,Dirichlet's approximation theorem,"(α : ℝ) (N : ℕ) (hN : 1 ≤ N) :
∃ p : ℤ, ∃ q : ℕ, 1 ≤ q ∧ q ≤ N ∧
|(q : ℝ) * α - p| ≤ 1 / (N + 1 : ℝ)",T083_V2,T083_V1 -> T083_V2,"by
rcases Real.exists_nat_abs_mul_sub_round_le α (by omega : 0 < N) with
⟨q, hqpos, hqN, hq⟩
refine ⟨round (q * α), q, hqpos, hqN, ?_⟩
simpa only using hq",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T083,Dirichlet's approximation theorem,"(α : ℝ) (N : ℕ) (hN : 1 ≤ N) :
∃ p : ℤ, ∃ q : ℕ, 1 ≤ q ∧ q ≤ N ∧
|(q : ℝ) * α - p| ≤ 1 / (N + 1 : ℝ)",T083_V3,T083_V1 -> T083_V3,"by
obtain ⟨p, q, hq0, hqN, hbound⟩ :=
Real.exists_int_int_abs_mul_sub_le α (by omega : 0 < N)
have hqcast : ((q.toNat : ℕ) : ℤ) = q := Int.toNat_of_nonneg hq0.le
have hqreal : (q.toNat : ℝ) = (q : ℝ) := by exact_mod_cast hqcast
rw [← hqcast] at hq0 hqN
refine ⟨p, q.toNat, ?_, ?_, ?_⟩
· have hq0nat : 0 < q.toNat := by exact_mod_cast hq0
omega
· exact_mod_cast hqN
· simpa [hqreal] using hbound",True,new_route,witness_construction,new_route,route_substitution,new_route
T083,Dirichlet's approximation theorem,"(α : ℝ) (N : ℕ) (hN : 1 ≤ N) :
∃ p : ℤ, ∃ q : ℕ, 1 ≤ q ∧ q ≤ N ∧
|(q : ℝ) * α - p| ≤ 1 / (N + 1 : ℝ)",T083_V4,T083_V2 -> T083_V4,"by
exact (by
rcases Real.exists_nat_abs_mul_sub_round_le α (by omega : 0 < N) with
⟨q, hqpos, hqN, hq⟩
refine ⟨round (q * α), q, hqpos, hqN, ?_⟩
simpa only using hq
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T083,Dirichlet's approximation theorem,"(α : ℝ) (N : ℕ) (hN : 1 ≤ N) :
∃ p : ℤ, ∃ q : ℕ, 1 ≤ q ∧ q ≤ N ∧
|(q : ℝ) * α - p| ≤ 1 / (N + 1 : ℝ)",T083_V5,T083_V1 -> T083_V5,"by
exact id (by
obtain ⟨q, hq0, hqN, hbound⟩ := Real.exists_nat_abs_mul_sub_round_le α (by omega : 0 < N)
exact ⟨round (q * α), q, hq0, hqN, by simpa using hbound⟩
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T083,Dirichlet's approximation theorem,"(α : ℝ) (N : ℕ) (hN : 1 ≤ N) :
∃ p : ℤ, ∃ q : ℕ, 1 ≤ q ∧ q ≤ N ∧
|(q : ℝ) * α - p| ≤ 1 / (N + 1 : ℝ)",T083_V6,T083_V2 -> T083_V6,"by
exact (show _ from by
rcases Real.exists_nat_abs_mul_sub_round_le α (by omega : 0 < N) with
⟨q, hqpos, hqN, hq⟩
refine ⟨round (q * α), q, hqpos, hqN, ?_⟩
simpa only using hq
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T084,Division theorem,"(a b : ℤ) (hb : b ≠ 0) :
∃! qr : ℤ × ℤ,
a = b * qr.1 + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < |b|",T084_V1,root,"by
refine ⟨(a / b, a % b), ?_, ?_⟩
· exact ⟨by rw [add_comm]; exact (Int.emod_add_mul_ediv a b).symm,
Int.emod_nonneg _ hb, Int.emod_lt_abs _ hb⟩
· rintro ⟨q, r⟩ ⟨ha, hr0, hrb⟩
have huniq := (Int.ediv_emod_unique'' hb).mpr ⟨by simpa [add_comm, mul_comm] using ha.symm, hr0, hrb⟩
ext <;> simp_all",True,reference,baseline,reference,baseline,reference
T084,Division theorem,"(a b : ℤ) (hb : b ≠ 0) :
∃! qr : ℤ × ℤ,
a = b * qr.1 + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < |b|",T084_V2,T084_V1 -> T084_V2,"by
refine ⟨(a / b, a % b), ?_, ?_⟩
· exact ⟨by rw [add_comm]; exact (Int.emod_add_mul_ediv a b).symm,
Int.emod_nonneg _ hb, Int.emod_lt_abs _ hb⟩
· rintro ⟨q, r⟩ hqr
have hcanonical : a / b = q ∧ a % b = r :=
(Int.ediv_emod_unique'' hb).mpr
⟨by simpa [add_comm] using hqr.1.symm, hqr.2.1, hqr.2.2⟩
ext <;> simp [hcanonical]",True,same_route,witness_construction,same_route,proof_term_structure,same_route
T084,Division theorem,"(a b : ℤ) (hb : b ≠ 0) :
∃! qr : ℤ × ℤ,
a = b * qr.1 + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < |b|",T084_V3,T084_V1 -> T084_V3,"by
exact (by
refine ⟨(a / b, a % b), ?_, ?_⟩
· exact ⟨by rw [add_comm]; exact (Int.emod_add_mul_ediv a b).symm,
Int.emod_nonneg _ hb, Int.emod_lt_abs _ hb⟩
· rintro ⟨q, r⟩ ⟨ha, hr0, hrb⟩
have huniq := (Int.ediv_emod_unique'' hb).mpr ⟨by simpa [add_comm, mul_comm] using ha.symm, hr0, hrb⟩
ext <;> simp_all
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T084,Division theorem,"(a b : ℤ) (hb : b ≠ 0) :
∃! qr : ℤ × ℤ,
a = b * qr.1 + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < |b|",T084_V4,T084_V2 -> T084_V4,"by
exact (by
refine ⟨(a / b, a % b), ?_, ?_⟩
· exact ⟨by rw [add_comm]; exact (Int.emod_add_mul_ediv a b).symm,
Int.emod_nonneg _ hb, Int.emod_lt_abs _ hb⟩
· rintro ⟨q, r⟩ hqr
have hcanonical : a / b = q ∧ a % b = r :=
(Int.ediv_emod_unique'' hb).mpr
⟨by simpa [add_comm] using hqr.1.symm, hqr.2.1, hqr.2.2⟩
ext <;> simp [hcanonical]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T084,Division theorem,"(a b : ℤ) (hb : b ≠ 0) :
∃! qr : ℤ × ℤ,
a = b * qr.1 + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < |b|",T084_V5,T084_V1 -> T084_V5,"by
exact id (by
refine ⟨(a / b, a % b), ?_, ?_⟩
· exact ⟨by rw [add_comm]; exact (Int.emod_add_mul_ediv a b).symm,
Int.emod_nonneg _ hb, Int.emod_lt_abs _ hb⟩
· rintro ⟨q, r⟩ ⟨ha, hr0, hrb⟩
have huniq := (Int.ediv_emod_unique'' hb).mpr ⟨by simpa [add_comm, mul_comm] using ha.symm, hr0, hrb⟩
ext <;> simp_all
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T084,Division theorem,"(a b : ℤ) (hb : b ≠ 0) :
∃! qr : ℤ × ℤ,
a = b * qr.1 + qr.2 ∧ 0 ≤ qr.2 ∧ qr.2 < |b|",T084_V6,T084_V2 -> T084_V6,"by
exact (show _ from by
refine ⟨(a / b, a % b), ?_, ?_⟩
· exact ⟨by rw [add_comm]; exact (Int.emod_add_mul_ediv a b).symm,
Int.emod_nonneg _ hb, Int.emod_lt_abs _ hb⟩
· rintro ⟨q, r⟩ hqr
have hcanonical : a / b = q ∧ a % b = r :=
(Int.ediv_emod_unique'' hb).mpr
⟨by simpa [add_comm] using hqr.1.symm, hqr.2.1, hqr.2.2⟩
ext <;> simp [hcanonical]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T086,Doob–Dynkin lemma,"{Ω S R : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[MeasurableSpace R] [Nonempty R] [StandardBorelSpace R]
(T : Ω → S) (f : Ω → R)
(hsigma : @Measurable Ω R ((inferInstance : MeasurableSpace S).comap T)
(inferInstance : MeasurableSpace R) f) :
∃ g : S → R, Measurable g ∧ f = g ∘ T",T086_V1,root,"by
exact hsigma.exists_eq_measurable_comp",True,reference,baseline,reference,baseline,reference
T086,Doob–Dynkin lemma,"{Ω S R : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[MeasurableSpace R] [Nonempty R] [StandardBorelSpace R]
(T : Ω → S) (f : Ω → R)
(hsigma : @Measurable Ω R ((inferInstance : MeasurableSpace S).comap T)
(inferInstance : MeasurableSpace R) f) :
∃ g : S → R, Measurable g ∧ f = g ∘ T",T086_V2,T086_V1 -> T086_V2,"by
obtain ⟨g, hg, hfg⟩ := hsigma.exists_eq_measurable_comp
refine ⟨g, ?_, ?_⟩
· exact hg
· exact hfg",True,same_route,proof_intermediate,same_route,proof_intermediate,same_route
T086,Doob–Dynkin lemma,"{Ω S R : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[MeasurableSpace R] [Nonempty R] [StandardBorelSpace R]
(T : Ω → S) (f : Ω → R)
(hsigma : @Measurable Ω R ((inferInstance : MeasurableSpace S).comap T)
(inferInstance : MeasurableSpace R) f) :
∃ g : S → R, Measurable g ∧ f = g ∘ T",T086_V3,T086_V1 -> T086_V3,"by
exact (by
exact hsigma.exists_eq_measurable_comp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T086,Doob–Dynkin lemma,"{Ω S R : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[MeasurableSpace R] [Nonempty R] [StandardBorelSpace R]
(T : Ω → S) (f : Ω → R)
(hsigma : @Measurable Ω R ((inferInstance : MeasurableSpace S).comap T)
(inferInstance : MeasurableSpace R) f) :
∃ g : S → R, Measurable g ∧ f = g ∘ T",T086_V4,T086_V2 -> T086_V4,"by
exact (by
obtain ⟨g, hg, hfg⟩ := hsigma.exists_eq_measurable_comp
refine ⟨g, ?_, ?_⟩
· exact hg
· exact hfg
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T086,Doob–Dynkin lemma,"{Ω S R : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[MeasurableSpace R] [Nonempty R] [StandardBorelSpace R]
(T : Ω → S) (f : Ω → R)
(hsigma : @Measurable Ω R ((inferInstance : MeasurableSpace S).comap T)
(inferInstance : MeasurableSpace R) f) :
∃ g : S → R, Measurable g ∧ f = g ∘ T",T086_V5,T086_V1 -> T086_V5,"by
exact id (by
exact hsigma.exists_eq_measurable_comp
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T086,Doob–Dynkin lemma,"{Ω S R : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
[MeasurableSpace R] [Nonempty R] [StandardBorelSpace R]
(T : Ω → S) (f : Ω → R)
(hsigma : @Measurable Ω R ((inferInstance : MeasurableSpace S).comap T)
(inferInstance : MeasurableSpace R) f) :
∃ g : S → R, Measurable g ∧ f = g ∘ T",T086_V6,T086_V2 -> T086_V6,"by
exact (show _ from by
obtain ⟨g, hg, hfg⟩ := hsigma.exists_eq_measurable_comp
refine ⟨g, ?_, ?_⟩
· exact hg
· exact hfg
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T097,Erdős–Ko–Rado theorem,"(n r : ℕ) (F : Finset (Finset (Fin n)))
(hn : 2 * r ≤ n)
(hr : ∀ A ∈ F, A.card = r)
(hint : ∀ A ∈ F, ∀ B ∈ F, (A ∩ B).Nonempty) :
F.card ≤ Nat.choose (n - 1) (r - 1)",T097_V1,root,"by
apply Finset.erdos_ko_rado
· intro A hA B hB hdis
rw [Finset.disjoint_iff_inter_eq_empty] at hdis
exact (hint A hA B hB).ne_empty hdis
· intro A hA
exact hr A hA
· omega",True,reference,baseline,reference,baseline,reference
T097,Erdős–Ko–Rado theorem,"(n r : ℕ) (F : Finset (Finset (Fin n)))
(hn : 2 * r ≤ n)
(hr : ∀ A ∈ F, A.card = r)
(hint : ∀ A ∈ F, ∀ B ∈ F, (A ∩ B).Nonempty) :
F.card ≤ Nat.choose (n - 1) (r - 1)",T097_V2,T097_V1 -> T097_V2,"by
refine Finset.erdos_ko_rado ?_ ?_ ?_
· intro A hA B hB hdis
have hinter := hint A hA B hB
rw [Finset.disjoint_iff_inter_eq_empty] at hdis
exact hinter.ne_empty hdis
· exact hr
· omega",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T097,Erdős–Ko–Rado theorem,"(n r : ℕ) (F : Finset (Finset (Fin n)))
(hn : 2 * r ≤ n)
(hr : ∀ A ∈ F, A.card = r)
(hint : ∀ A ∈ F, ∀ B ∈ F, (A ∩ B).Nonempty) :
F.card ≤ Nat.choose (n - 1) (r - 1)",T097_V3,T097_V1 -> T097_V3,"by
exact (by
apply Finset.erdos_ko_rado
· intro A hA B hB hdis
rw [Finset.disjoint_iff_inter_eq_empty] at hdis
exact (hint A hA B hB).ne_empty hdis
· intro A hA
exact hr A hA
· omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T097,Erdős–Ko–Rado theorem,"(n r : ℕ) (F : Finset (Finset (Fin n)))
(hn : 2 * r ≤ n)
(hr : ∀ A ∈ F, A.card = r)
(hint : ∀ A ∈ F, ∀ B ∈ F, (A ∩ B).Nonempty) :
F.card ≤ Nat.choose (n - 1) (r - 1)",T097_V4,T097_V2 -> T097_V4,"by
exact (by
refine Finset.erdos_ko_rado ?_ ?_ ?_
· intro A hA B hB hdis
have hinter := hint A hA B hB
rw [Finset.disjoint_iff_inter_eq_empty] at hdis
exact hinter.ne_empty hdis
· exact hr
· omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T097,Erdős–Ko–Rado theorem,"(n r : ℕ) (F : Finset (Finset (Fin n)))
(hn : 2 * r ≤ n)
(hr : ∀ A ∈ F, A.card = r)
(hint : ∀ A ∈ F, ∀ B ∈ F, (A ∩ B).Nonempty) :
F.card ≤ Nat.choose (n - 1) (r - 1)",T097_V5,T097_V1 -> T097_V5,"by
exact id (by
apply Finset.erdos_ko_rado
· intro A hA B hB hdis
rw [Finset.disjoint_iff_inter_eq_empty] at hdis
exact (hint A hA B hB).ne_empty hdis
· intro A hA
exact hr A hA
· omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T097,Erdős–Ko–Rado theorem,"(n r : ℕ) (F : Finset (Finset (Fin n)))
(hn : 2 * r ≤ n)
(hr : ∀ A ∈ F, A.card = r)
(hint : ∀ A ∈ F, ∀ B ∈ F, (A ∩ B).Nonempty) :
F.card ≤ Nat.choose (n - 1) (r - 1)",T097_V6,T097_V2 -> T097_V6,"by
exact (show _ from by
refine Finset.erdos_ko_rado ?_ ?_ ?_
· intro A hA B hB hdis
have hinter := hint A hA B hB
rw [Finset.disjoint_iff_inter_eq_empty] at hdis
exact hinter.ne_empty hdis
· exact hr
· omega
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T100,Euler's formula,"(x : ℝ) :
Complex.exp (Complex.I * x) = Complex.cos x + Complex.I * Complex.sin x",T100_V1,root,"by
simpa [mul_comm] using Complex.exp_mul_I x",True,reference,baseline,reference,baseline,reference
T100,Euler's formula,"(x : ℝ) :
Complex.exp (Complex.I * x) = Complex.cos x + Complex.I * Complex.sin x",T100_V2,T100_V1 -> T100_V2,"by
calc
Complex.exp (Complex.I * x) = Complex.exp (x * Complex.I) := by rw [mul_comm]
_ = Complex.cos x + Complex.sin x * Complex.I := Complex.exp_mul_I x
_ = Complex.cos x + Complex.I * Complex.sin x := by ring",True,same_route,proof_intermediate,same_route,proof_intermediate,same_route
T100,Euler's formula,"(x : ℝ) :
Complex.exp (Complex.I * x) = Complex.cos x + Complex.I * Complex.sin x",T100_V3,T100_V1 -> T100_V3,"by
apply Complex.ext
· simp [Complex.exp_re, Complex.mul_re, Complex.cos_ofReal_re,
Complex.sin_ofReal_re, Complex.cos_ofReal_im, Complex.sin_ofReal_im]
· simp [Complex.exp_im, Complex.mul_im, Complex.cos_ofReal_re,
Complex.sin_ofReal_re, Complex.cos_ofReal_im, Complex.sin_ofReal_im]",True,new_route,operator_functional,new_route,logical_characterization,new_route
T100,Euler's formula,"(x : ℝ) :
Complex.exp (Complex.I * x) = Complex.cos x + Complex.I * Complex.sin x",T100_V4,T100_V2 -> T100_V4,"by
exact (by
calc
Complex.exp (Complex.I * x) = Complex.exp (x * Complex.I) := by rw [mul_comm]
_ = Complex.cos x + Complex.sin x * Complex.I := Complex.exp_mul_I x
_ = Complex.cos x + Complex.I * Complex.sin x := by ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T100,Euler's formula,"(x : ℝ) :
Complex.exp (Complex.I * x) = Complex.cos x + Complex.I * Complex.sin x",T100_V5,T100_V1 -> T100_V5,"by
exact id (by
simpa [mul_comm] using Complex.exp_mul_I x
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T100,Euler's formula,"(x : ℝ) :
Complex.exp (Complex.I * x) = Complex.cos x + Complex.I * Complex.sin x",T100_V6,T100_V2 -> T100_V6,"by
exact (show _ from by
calc
Complex.exp (Complex.I * x) = Complex.exp (x * Complex.I) := by rw [mul_comm]
_ = Complex.cos x + Complex.sin x * Complex.I := Complex.exp_mul_I x
_ = Complex.cos x + Complex.I * Complex.sin x := by ring
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
theorem_id,theorem_name,lean_proposition,proof_node,graph_edge,lean_proof,kernel_verified,benchmark_route,benchmark_family,machine_route,machine_family,human_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V1,root,"by
have hxy : 0 ≤ x * y := mul_nonneg hx hy
have hs : 0 ≤ Real.sqrt (x * y) := Real.sqrt_nonneg _
have hsq : (Real.sqrt (x * y)) ^ 2 = x * y := Real.sq_sqrt hxy
nlinarith [sq_nonneg (x - y)]",True,reference,baseline,reference,baseline,reference
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V2,T001_V1 -> T001_V2,"by
rw [Real.sqrt_le_iff]
constructor
· nlinarith
· nlinarith [sq_nonneg (x - y)]",True,new_route,route_substitution,new_route,logical_characterization,new_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V3,T001_V1 -> T001_V3,"by
apply (Real.sqrt_le_left (by linarith)).2
nlinarith [sq_nonneg (x - y)]",True,new_route,algebraic_normal_form,new_route,logical_characterization,new_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V4,T001_V2 -> T001_V4,"by
have hmean : 0 ≤ (x + y) / 2 := by linarith
rw [← Real.sqrt_sq hmean]
apply Real.sqrt_le_sqrt
nlinarith [sq_nonneg (x - y)]",True,new_route,operator_functional,new_route,logical_characterization,new_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V5,T001_V1 -> T001_V5,"by
exact id (by
have hxy : 0 ≤ x * y := mul_nonneg hx hy
have hs : 0 ≤ Real.sqrt (x * y) := Real.sqrt_nonneg _
have hsq : (Real.sqrt (x * y)) ^ 2 = x * y := Real.sq_sqrt hxy
nlinarith [sq_nonneg (x - y)]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T001,AM–GM inequality,"(x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) :
Real.sqrt (x * y) ≤ (x + y) / 2",T001_V6,T001_V2 -> T001_V6,"by
exact (show _ from by
rw [Real.sqrt_le_iff]
constructor
· nlinarith
· nlinarith [sq_nonneg (x - y)]
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V1,root,"by
exact Real.tendsto_tsum_powerSeries_nhdsWithin_lt hs",True,reference,baseline,reference,baseline,reference
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V2,T004_V1 -> T004_V2,"by
apply Real.tendsto_tsum_powerSeries_nhdsWithin_lt
exact hs",True,same_route,proof_intermediate,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V3,T004_V1 -> T004_V3,"by
exact (by
exact Real.tendsto_tsum_powerSeries_nhdsWithin_lt hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V4,T004_V2 -> T004_V4,"by
exact (by
apply Real.tendsto_tsum_powerSeries_nhdsWithin_lt
exact hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V5,T004_V1 -> T004_V5,"by
exact id (by
exact Real.tendsto_tsum_powerSeries_nhdsWithin_lt hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T004,Abel's theorem,"(a : ℕ → ℝ) (s : ℝ)
(hs : Tendsto (fun N => ∑ n ∈ Finset.range N, a n) atTop (𝓝 s)) :
Tendsto
(fun x : ℝ => ∑' n : ℕ, a n * x ^ n)
(𝓝[<] 1) (𝓝 s)",T004_V6,T004_V2 -> T004_V6,"by
exact (show _ from by
apply Real.tendsto_tsum_powerSeries_nhdsWithin_lt
exact hs
)",True,same_route,proof_term_structure,same_route,proof_term_structure,same_route
T005,Abelian and Tauberian theorems,"(c : ℕ → ℝ) (C : ℝ)
(hc : Tendsto c atTop (𝓝 C)) :
Tendsto
(fun N : ℕ => (1 / (N : ℝ)) * ∑ n ∈ Finset.Icc 1 N, c n)
atTop (𝓝 C)",T005_V1,root,"by
have hshift : Tendsto (fun n : ℕ => c (n + 1)) atTop (𝓝 C) :=
hc.comp (tendsto_add_atTop_nat 1)
have hsum : ∀ N : ℕ,
(∑ n ∈ Finset.Icc 1 N, c n) = ∑ n ∈ Finset.range N, c (n + 1) := by
intro N
induction N with
| zero => simp
| succ N ih =>
rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ N + 1), Finset.sum_range_succ, ih]