Datasets:
pkuAI4M
/
lean_github

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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
refine eventually_nhdsSet_iff_exists.mpr ⟨(t0 ∩ t1) Γ—Λ’ ball 0 q, (ot0.inter ot1).prod isOpen_ball, ?_, ?_⟩
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ (𝓝˒ ({c} Γ—Λ’ closedBall 0 p)).EventuallyEq (uncurry ray) (uncurry (r h))
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ {c} Γ—Λ’ closedBall 0 p βŠ† (t0 ∩ t1) Γ—Λ’ ball 0 q case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ βˆ€ x ∈ (t0 ∩ t1) Γ—Λ’ ball 0 q, uncurry ray x = uncurry (r h) x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ (𝓝˒ ({c} Γ—Λ’ closedBall 0 p)).EventuallyEq (uncurry ray) (uncurry (r h)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
exact prod_mono (singleton_subset_iff.mpr ⟨ct0, ct1⟩) (Metric.closedBall_subset_ball pq)
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ {c} Γ—Λ’ closedBall 0 p βŠ† (t0 ∩ t1) Γ—Λ’ ball 0 q
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ {c} Γ—Λ’ closedBall 0 p βŠ† (t0 ∩ t1) Γ—Λ’ ball 0 q TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
intro ⟨e, x⟩ ⟨⟨et0, et1⟩, xq⟩
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ βˆ€ x ∈ (t0 ∩ t1) Γ—Λ’ ball 0 q, uncurry ray x = uncurry (r h) x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : (e, x).1 ∈ t0 et1 : (e, x).1 ∈ t1 xq : (e, x).2 ∈ ball 0 q ⊒ uncurry ray (e, x) = uncurry (r h) (e, x)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 ⊒ βˆ€ x ∈ (t0 ∩ t1) Γ—Λ’ ball 0 q, uncurry ray x = uncurry (r h) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
simp only [uncurry] at et0 et1 xq ⊒
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : (e, x).1 ∈ t0 et1 : (e, x).1 ∈ t1 xq : (e, x).2 ∈ ball 0 q ⊒ uncurry ray (e, x) = uncurry (r h) (e, x)
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : x ∈ ball 0 q ⊒ ray e x = r h e x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : (e, x).1 ∈ t0 et1 : (e, x).1 ∈ t1 xq : (e, x).2 ∈ ball 0 q ⊒ uncurry ray (e, x) = uncurry (r h) (e, x) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
simp only [mem_ball, Complex.dist_eq, sub_zero] at xq
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : x ∈ ball 0 q ⊒ ray e x = r h e x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q ⊒ ray e x = r h e x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : x ∈ ball 0 q ⊒ ray e x = r h e x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
have hx : 0 ≀ abs x ∧ abs x < s.p e := ⟨Complex.abs.nonneg _, _root_.trans xq (lo _ et1)⟩
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q ⊒ ray e x = r h e x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ ray e x = r h e x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q ⊒ ray e x = r h e x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
simp only [← hray, dif_pos hx.2]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ ray e x = r h e x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ r β‹― e x = r h e x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ ray e x = r h e x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
refine ((g hx).unique (gh _ et0) xq.le).self_of_nhdsSet (x := ⟨e, x⟩) ⟨rfl, ?_⟩
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ r β‹― e x = r h e x
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ (e, x).2 ∈ closedBall 0 (Complex.abs x)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ r β‹― e x = r h e x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Grow.lean
Super.has_ray
[625, 1]
[654, 45]
simp only [mem_closedBall, Complex.dist_eq, sub_zero, le_refl]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ (e, x).2 ∈ closedBall 0 (Complex.abs x)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2 S : Type inst✝⁡ : TopologicalSpace S inst✝⁴ : CompactSpace S inst✝³ : T3Space S inst✝² : ChartedSpace β„‚ S inst✝¹ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c✝ : β„‚ a z : S d n : β„• p✝ : ℝ s✝ : Super f d a r✝ : β„‚ β†’ β„‚ β†’ S s : Super f d a inst✝ : OnePreimage s r : {c : β„‚} β†’ {p : ℝ} β†’ 0 ≀ p ∧ p < s.p c β†’ β„‚ β†’ β„‚ β†’ S g : βˆ€ {c : β„‚} {p : ℝ} (h : 0 ≀ p ∧ p < s.p c), Grow s c p (s.np c p) (r h) ray : β„‚ β†’ β„‚ β†’ S hray : (fun c x => if h : Complex.abs x < s.p c then r β‹― c x else a) = ray c : β„‚ p : ℝ h : 0 ≀ p ∧ p < s.p c q : ℝ pq : p < q qs : q < s.p c q0 : 0 ≀ q gh✝ : βˆ€αΆ  (x : β„‚) in 𝓝 c, Grow s x q (s.np c p) (r h) t0 : Set β„‚ gh : βˆ€ x ∈ t0, Grow s x q (s.np c p) (r h) ot0 : IsOpen t0 ct0 : c ∈ t0 t1 : Set β„‚ lo : βˆ€ x ∈ t1, q < s.p x ot1 : IsOpen t1 ct1 : c ∈ t1 e x : β„‚ et0 : e ∈ t0 et1 : e ∈ t1 xq : Complex.abs x < q hx : 0 ≀ Complex.abs x ∧ Complex.abs x < s.p e ⊒ (e, x).2 ∈ closedBall 0 (Complex.abs x) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
realCircleMap_eq_circleMap
[35, 1]
[39, 66]
simp only [realCircleMap, circleMap, Complex.equivRealProd_apply, Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.exp_ofReal_mul_I_re, Complex.ofReal_im, Complex.exp_ofReal_mul_I_im, zero_mul, sub_zero, Complex.add_im, Complex.mul_im, add_zero]
c : β„‚ x : ℝ Γ— ℝ ⊒ realCircleMap c x = Complex.equivRealProd (circleMap c x.1 x.2)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ x : ℝ Γ— ℝ ⊒ realCircleMap c x = Complex.equivRealProd (circleMap c x.1 x.2) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
realCircleMap.fderiv
[50, 1]
[54, 73]
simp_rw [realCircleMap]
c : β„‚ x : ℝ Γ— ℝ ⊒ HasFDerivAt (fun x => realCircleMap c x) (rcmDeriv x) x
c : β„‚ x : ℝ Γ— ℝ ⊒ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ x : ℝ Γ— ℝ ⊒ HasFDerivAt (fun x => realCircleMap c x) (rcmDeriv x) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
realCircleMap.fderiv
[50, 1]
[54, 73]
apply_rules [hasFDerivAt_const, hasFDerivAt_fst, hasFDerivAt_snd, HasFDerivAt.cos, HasFDerivAt.sin, HasFDerivAt.add, HasFDerivAt.mul, HasFDerivAt.prod]
c : β„‚ x : ℝ Γ— ℝ ⊒ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ x : ℝ Γ— ℝ ⊒ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcm00
[59, 1]
[60, 53]
simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2]
x : ℝ Γ— ℝ ⊒ rcmMatrix x 0 0 = x.2.cos
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ rcmMatrix x 0 0 = x.2.cos TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcm01
[61, 1]
[62, 53]
simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2]
x : ℝ Γ— ℝ ⊒ rcmMatrix x 0 1 = -x.1 * x.2.sin
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ rcmMatrix x 0 1 = -x.1 * x.2.sin TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcm10
[63, 1]
[64, 53]
simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2]
x : ℝ Γ— ℝ ⊒ rcmMatrix x 1 0 = x.2.sin
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ rcmMatrix x 1 0 = x.2.sin TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcm11
[65, 1]
[66, 53]
simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2]
x : ℝ Γ— ℝ ⊒ rcmMatrix x 1 1 = x.1 * x.2.cos
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ rcmMatrix x 1 1 = x.1 * x.2.cos TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcmDeriv.det
[68, 1]
[74, 62]
rw [ContinuousLinearMap.det, ← LinearMap.det_toMatrix (Basis.finTwoProd ℝ), ←rcmMatrix]
x : ℝ Γ— ℝ ⊒ (rcmDeriv x).det = x.1
x : ℝ Γ— ℝ ⊒ (rcmMatrix x).det = x.1
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ (rcmDeriv x).det = x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcmDeriv.det
[68, 1]
[74, 62]
rw [Matrix.det_fin_two, rcm00, rcm01, rcm10, rcm11]
x : ℝ Γ— ℝ ⊒ (rcmMatrix x).det = x.1
x : ℝ Γ— ℝ ⊒ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ (rcmMatrix x).det = x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcmDeriv.det
[68, 1]
[74, 62]
ring_nf
x : ℝ Γ— ℝ ⊒ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1
x : ℝ Γ— ℝ ⊒ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcmDeriv.det
[68, 1]
[74, 62]
calc cos x.2 ^ 2 * x.1 + x.1 * sin x.2 ^ 2 _ = x.1 * (cos x.2 ^ 2 + sin x.2 ^ 2) := by ring _ = x.1 := by simp only [Real.cos_sq_add_sin_sq, mul_one]
x : ℝ Γ— ℝ ⊒ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcmDeriv.det
[68, 1]
[74, 62]
ring
x : ℝ Γ— ℝ ⊒ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2)
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
rcmDeriv.det
[68, 1]
[74, 62]
simp only [Real.cos_sq_add_sin_sq, mul_one]
x : ℝ Γ— ℝ ⊒ x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) = x.1
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ Γ— ℝ ⊒ x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) = x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
constructor
S : Type inst✝ : RCLike S c : S r : ℝ ⊒ sphere c r = βˆ… ↔ r < 0
case mp S : Type inst✝ : RCLike S c : S r : ℝ ⊒ sphere c r = βˆ… β†’ r < 0 case mpr S : Type inst✝ : RCLike S c : S r : ℝ ⊒ r < 0 β†’ sphere c r = βˆ…
Please generate a tactic in lean4 to solve the state. STATE: S : Type inst✝ : RCLike S c : S r : ℝ ⊒ sphere c r = βˆ… ↔ r < 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
intro rp
case mp S : Type inst✝ : RCLike S c : S r : ℝ ⊒ sphere c r = βˆ… β†’ r < 0
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : sphere c r = βˆ… ⊒ r < 0
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Type inst✝ : RCLike S c : S r : ℝ ⊒ sphere c r = βˆ… β†’ r < 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
contrapose rp
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : sphere c r = βˆ… ⊒ r < 0
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : Β¬r < 0 ⊒ Β¬sphere c r = βˆ…
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : sphere c r = βˆ… ⊒ r < 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
simp at rp
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : Β¬r < 0 ⊒ Β¬sphere c r = βˆ…
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : 0 ≀ r ⊒ Β¬sphere c r = βˆ…
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : Β¬r < 0 ⊒ Β¬sphere c r = βˆ… TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
refine Nonempty.ne_empty ⟨c + r, ?_⟩
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : 0 ≀ r ⊒ Β¬sphere c r = βˆ…
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : 0 ≀ r ⊒ c + ↑r ∈ sphere c r
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : 0 ≀ r ⊒ Β¬sphere c r = βˆ… TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
simpa only [mem_sphere_iff_norm, add_sub_cancel_left, RCLike.norm_ofReal, abs_eq_self]
case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : 0 ≀ r ⊒ c + ↑r ∈ sphere c r
no goals
Please generate a tactic in lean4 to solve the state. STATE: case mp S : Type inst✝ : RCLike S c : S r : ℝ rp : 0 ≀ r ⊒ c + ↑r ∈ sphere c r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
intro n
case mpr S : Type inst✝ : RCLike S c : S r : ℝ ⊒ r < 0 β†’ sphere c r = βˆ…
case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : r < 0 ⊒ sphere c r = βˆ…
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Type inst✝ : RCLike S c : S r : ℝ ⊒ r < 0 β†’ sphere c r = βˆ… TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
contrapose n
case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : r < 0 ⊒ sphere c r = βˆ…
case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : Β¬sphere c r = βˆ… ⊒ Β¬r < 0
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : r < 0 ⊒ sphere c r = βˆ… TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
rw [← not_nonempty_iff_eq_empty] at n
case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : Β¬sphere c r = βˆ… ⊒ Β¬r < 0
case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : ¬¬(sphere c r).Nonempty ⊒ Β¬r < 0
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : Β¬sphere c r = βˆ… ⊒ Β¬r < 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Metric.sphere_eq_empty
[80, 1]
[87, 69]
simpa only [not_lt, NormedSpace.sphere_nonempty, not_le] using n
case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : ¬¬(sphere c r).Nonempty ⊒ Β¬r < 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case mpr S : Type inst✝ : RCLike S c : S r : ℝ n : ¬¬(sphere c r).Nonempty ⊒ Β¬r < 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
by_cases rp : r < 0
c z : β„‚ r : ℝ zs : z ∈ sphere c r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
case pos c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : r < 0 ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t case neg c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : Β¬r < 0 ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ zs : z ∈ sphere c r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
simp only [not_lt] at rp
case neg c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : Β¬r < 0 ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
case neg c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : 0 ≀ r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
Please generate a tactic in lean4 to solve the state. STATE: case neg c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : Β¬r < 0 ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
rw [←abs_of_nonneg rp, ← range_circleMap, mem_range] at zs
case neg c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : 0 ≀ r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
case neg c z : β„‚ r : ℝ zs : βˆƒ y, circleMap c r y = z rp : 0 ≀ r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
Please generate a tactic in lean4 to solve the state. STATE: case neg c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : 0 ≀ r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
rcases zs with ⟨t, ht⟩
case neg c z : β„‚ r : ℝ zs : βˆƒ y, circleMap c r y = z rp : 0 ≀ r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
Please generate a tactic in lean4 to solve the state. STATE: case neg c z : β„‚ r : ℝ zs : βˆƒ y, circleMap c r y = z rp : 0 ≀ r ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
generalize ha : 2 * Ο€ = a
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
have ap : a > 0 := by rw [←ha]; bound
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
generalize hs : t + a - a * ⌈t / aβŒ‰ = s
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
use s
case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t
case h c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ s ∈ Ioc 0 a ∧ z = circleMap c r s
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ βˆƒ t ∈ Ioc 0 a, z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
constructor
case h c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ s ∈ Ioc 0 a ∧ z = circleMap c r s
case h.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ s ∈ Ioc 0 a case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ z = circleMap c r s
Please generate a tactic in lean4 to solve the state. STATE: case h c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ s ∈ Ioc 0 a ∧ z = circleMap c r s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
simp only [Metric.sphere_eq_empty.mpr rp, mem_empty_iff_false] at zs
case pos c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : r < 0 ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos c z : β„‚ r : ℝ zs : z ∈ sphere c r rp : r < 0 ⊒ βˆƒ t ∈ Ioc 0 (2 * Ο€), z = circleMap c r t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
rw [←ha]
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ a > 0
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ 2 * Ο€ > 0
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ a > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
bound
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ 2 * Ο€ > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ⊒ 2 * Ο€ > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
simp only [mem_Ioc, sub_pos, tsub_le_iff_right, ← hs]
case h.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ s ∈ Ioc 0 a
case h.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < t + a ∧ t + a ≀ a + a * β†‘βŒˆt / aβŒ‰
Please generate a tactic in lean4 to solve the state. STATE: case h.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ s ∈ Ioc 0 a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
constructor
case h.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < t + a ∧ t + a ≀ a + a * β†‘βŒˆt / aβŒ‰
case h.left.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < t + a case h.left.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ t + a ≀ a + a * β†‘βŒˆt / aβŒ‰
Please generate a tactic in lean4 to solve the state. STATE: case h.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < t + a ∧ t + a ≀ a + a * β†‘βŒˆt / aβŒ‰ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
calc a * ⌈t / aβŒ‰ _ < a * (t / a + 1) := by bound _ = a / a * t + a := by ring _ = t + a := by field_simp [ap.ne']
case h.left.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < t + a
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left.left c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < t + a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
bound
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < a * (t / a + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * β†‘βŒˆt / aβŒ‰ < a * (t / a + 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
ring
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * (t / a + 1) = a / a * t + a
no goals
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a * (t / a + 1) = a / a * t + a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
field_simp [ap.ne']
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a / a * t + a = t + a
no goals
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a / a * t + a = t + a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
calc a + a * ⌈t / aβŒ‰ _ β‰₯ a + a * (t / a) := by bound _ = a / a * t + a := by ring _ = t + a := by field_simp [ap.ne']
case h.left.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ t + a ≀ a + a * β†‘βŒˆt / aβŒ‰
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ t + a ≀ a + a * β†‘βŒˆt / aβŒ‰ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
bound
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a + a * β†‘βŒˆt / aβŒ‰ β‰₯ a + a * (t / a)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a + a * β†‘βŒˆt / aβŒ‰ β‰₯ a + a * (t / a) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
ring
c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a + a * (t / a) = a / a * t + a
no goals
Please generate a tactic in lean4 to solve the state. STATE: c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ a + a * (t / a) = a / a * t + a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
simp only [←ht, circleMap, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_mul, Complex.ofReal_intCast, add_right_inj, mul_eq_mul_left_iff, Complex.ofReal_eq_zero, ← hs]
case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ z = circleMap c r s
case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = ((↑t + ↑a - ↑a * β†‘βŒˆt / aβŒ‰) * I).exp ∨ r = 0
Please generate a tactic in lean4 to solve the state. STATE: case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ z = circleMap c r s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
rw [mul_sub_right_distrib, right_distrib, Complex.exp_sub, Complex.exp_add]
case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = ((↑t + ↑a - ↑a * β†‘βŒˆt / aβŒ‰) * I).exp ∨ r = 0
case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = (↑t * I).exp * (↑a * I).exp / (↑a * β†‘βŒˆt / aβŒ‰ * I).exp ∨ r = 0
Please generate a tactic in lean4 to solve the state. STATE: case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = ((↑t + ↑a - ↑a * β†‘βŒˆt / aβŒ‰) * I).exp ∨ r = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
rw [mul_comm _ (⌈_βŒ‰:β„‚), mul_assoc, Complex.exp_int_mul, ← ha]
case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = (↑t * I).exp * (↑a * I).exp / (↑a * β†‘βŒˆt / aβŒ‰ * I).exp ∨ r = 0
case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = (↑t * I).exp * (↑(2 * Ο€) * I).exp / (↑(2 * Ο€) * I).exp ^ ⌈t / (2 * Ο€)βŒ‰ ∨ r = 0
Please generate a tactic in lean4 to solve the state. STATE: case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = (↑t * I).exp * (↑a * I).exp / (↑a * β†‘βŒˆt / aβŒ‰ * I).exp ∨ r = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
circleMap_Ioc
[90, 1]
[116, 34]
simp only [Complex.ofReal_mul, Complex.ofReal_ofNat, Complex.exp_two_pi_mul_I, mul_one, one_zpow, div_one, true_or]
case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = (↑t * I).exp * (↑(2 * Ο€) * I).exp / (↑(2 * Ο€) * I).exp ^ ⌈t / (2 * Ο€)βŒ‰ ∨ r = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right c z : β„‚ r : ℝ rp : 0 ≀ r t : ℝ ht : circleMap c r t = z a : ℝ ha : 2 * Ο€ = a ap : a > 0 s : ℝ hs : t + a - a * β†‘βŒˆt / aβŒ‰ = s ⊒ (↑t * I).exp = (↑t * I).exp * (↑(2 * Ο€) * I).exp / (↑(2 * Ο€) * I).exp ^ ⌈t / (2 * Ο€)βŒ‰ ∨ r = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square.rp
[124, 1]
[127, 26]
simp only [square, gt_iff_lt, not_lt, ge_iff_le, zero_lt_two, mul_pos_iff_of_pos_left, mem_prod, mem_Ioc, and_imp]
r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 ⊒ x ∈ square r0 r1 β†’ 0 < x.1
r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 ⊒ r0 < x.1 β†’ x.1 ≀ r1 β†’ 0 < x.2 β†’ x.2 ≀ 2 * Ο€ β†’ 0 < x.1
Please generate a tactic in lean4 to solve the state. STATE: r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 ⊒ x ∈ square r0 r1 β†’ 0 < x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square.rp
[124, 1]
[127, 26]
intro h _ _ _
r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 ⊒ r0 < x.1 β†’ x.1 ≀ r1 β†’ 0 < x.2 β†’ x.2 ≀ 2 * Ο€ β†’ 0 < x.1
r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 h : r0 < x.1 a✝² : x.1 ≀ r1 a✝¹ : 0 < x.2 a✝ : x.2 ≀ 2 * Ο€ ⊒ 0 < x.1
Please generate a tactic in lean4 to solve the state. STATE: r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 ⊒ r0 < x.1 β†’ x.1 ≀ r1 β†’ 0 < x.2 β†’ x.2 ≀ 2 * Ο€ β†’ 0 < x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square.rp
[124, 1]
[127, 26]
linarith
r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 h : r0 < x.1 a✝² : x.1 ≀ r1 a✝¹ : 0 < x.2 a✝ : x.2 ≀ 2 * Ο€ ⊒ 0 < x.1
no goals
Please generate a tactic in lean4 to solve the state. STATE: r0 r1 : ℝ x : ℝ Γ— ℝ r0p : 0 ≀ r0 h : r0 < x.1 a✝² : x.1 ≀ r1 a✝¹ : 0 < x.2 a✝ : x.2 ≀ 2 * Ο€ ⊒ 0 < x.1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Measurable.square
[129, 1]
[130, 54]
apply_rules [MeasurableSet.prod, measurableSet_Ioc]
r0 r1 : ℝ ⊒ MeasurableSet (_root_.square r0 r1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: r0 r1 : ℝ ⊒ MeasurableSet (_root_.square r0 r1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
rw [← MeasurableEquiv.image_eq_preimage]
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 ⊒ ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 ⊒ ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
have e : realCircleMap c = fun x : ℝ Γ— ℝ ↦ Complex.measurableEquivRealProd (circleMap c x.1 x.2) := by funext simp only [realCircleMap_eq_circleMap, Complex.measurableEquivRealProd, Complex.equivRealProd_apply, Homeomorph.toMeasurableEquiv_coe, ContinuousLinearEquiv.coe_toHomeomorph, Complex.equivRealProdCLM_apply]
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
have im := image_comp Complex.measurableEquivRealProd (fun x : ℝ Γ— ℝ ↦ circleMap c x.1 x.2) (square r0 r1)
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 im : (⇑Complex.measurableEquivRealProd ∘ fun x => circleMap c x.1 x.2) '' square r0 r1 = ⇑Complex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1) ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only [Function.comp] at im
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 im : (⇑Complex.measurableEquivRealProd ∘ fun x => circleMap c x.1 x.2) '' square r0 r1 = ⇑Complex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1) ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 im : (fun a => Complex.measurableEquivRealProd (circleMap c a.1 a.2)) '' square r0 r1 = ⇑Complex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1) ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 im : (⇑Complex.measurableEquivRealProd ∘ fun x => circleMap c x.1 x.2) '' square r0 r1 = ⇑Complex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1) ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only [e, im, i]
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 im : (fun a => Complex.measurableEquivRealProd (circleMap c a.1 a.2)) '' square r0 r1 = ⇑Complex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1) ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) i : (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 im : (fun a => Complex.measurableEquivRealProd (circleMap c a.1 a.2)) '' square r0 r1 = ⇑Complex.measurableEquivRealProd '' ((fun x => circleMap c x.1 x.2) '' square r0 r1) ⊒ ⇑Complex.measurableEquivRealProd '' annulus_oc c r0 r1 = realCircleMap c '' square r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
funext
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 ⊒ realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2)
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 x✝ : ℝ Γ— ℝ ⊒ realCircleMap c x✝ = Complex.measurableEquivRealProd (circleMap c x✝.1 x✝.2)
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 ⊒ realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only [realCircleMap_eq_circleMap, Complex.measurableEquivRealProd, Complex.equivRealProd_apply, Homeomorph.toMeasurableEquiv_coe, ContinuousLinearEquiv.coe_toHomeomorph, Complex.equivRealProdCLM_apply]
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 x✝ : ℝ Γ— ℝ ⊒ realCircleMap c x✝ = Complex.measurableEquivRealProd (circleMap c x✝.1 x✝.2)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 x✝ : ℝ Γ— ℝ ⊒ realCircleMap c x✝ = Complex.measurableEquivRealProd (circleMap c x✝.1 x✝.2) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
ext z
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) ⊒ (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ z ∈ (fun x => circleMap c x.1 x.2) '' square r0 r1 ↔ z ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) ⊒ (fun x => circleMap c x.1 x.2) '' square r0 r1 = annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
rw [mem_image]
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ z ∈ (fun x => circleMap c x.1 x.2) '' square r0 r1 ↔ z ∈ annulus_oc c r0 r1
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ (βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z) ↔ z ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ z ∈ (fun x => circleMap c x.1 x.2) '' square r0 r1 ↔ z ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
constructor
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ (βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z) ↔ z ∈ annulus_oc c r0 r1
case h.mp c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ (βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z) β†’ z ∈ annulus_oc c r0 r1 case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ z ∈ annulus_oc c r0 r1 β†’ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ (βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z) ↔ z ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
intro gp
case h.mp c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ (βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z) β†’ z ∈ annulus_oc c r0 r1
case h.mp c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ gp : βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z ⊒ z ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: case h.mp c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ (βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z) β†’ z ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
rcases gp with ⟨⟨s, t⟩, ss, tz⟩
case h.mp c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ gp : βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z ⊒ z ∈ annulus_oc c r0 r1
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ ss : (s, t) ∈ square r0 r1 tz : circleMap c (s, t).1 (s, t).2 = z ⊒ z ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: case h.mp c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ gp : βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z ⊒ z ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only at tz
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ ss : (s, t) ∈ square r0 r1 tz : circleMap c (s, t).1 (s, t).2 = z ⊒ z ∈ annulus_oc c r0 r1
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ ss : (s, t) ∈ square r0 r1 tz : circleMap c s t = z ⊒ z ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ ss : (s, t) ∈ square r0 r1 tz : circleMap c (s, t).1 (s, t).2 = z ⊒ z ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only [square, prod_mk_mem_set_prod_eq, mem_Ioc] at ss
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ ss : (s, t) ∈ square r0 r1 tz : circleMap c s t = z ⊒ z ∈ annulus_oc c r0 r1
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ z ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ ss : (s, t) ∈ square r0 r1 tz : circleMap c s t = z ⊒ z ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
rw [← tz]
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ z ∈ annulus_oc c r0 r1
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ circleMap c s t ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ z ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
have s0 : 0 < s := by linarith
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ circleMap c s t ∈ annulus_oc c r0 r1
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ s0 : 0 < s ⊒ circleMap c s t ∈ annulus_oc c r0 r1
Please generate a tactic in lean4 to solve the state. STATE: case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ circleMap c s t ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only [circleMap, add_comm c, annulus_oc, mem_diff, Metric.mem_closedBall, dist_add_self_left, norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, Complex.abs_exp_ofReal_mul_I, mul_one, not_le, abs_of_pos s0, ss.1, true_and]
case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ s0 : 0 < s ⊒ circleMap c s t ∈ annulus_oc c r0 r1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.mp.intro.mk.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ s0 : 0 < s ⊒ circleMap c s t ∈ annulus_oc c r0 r1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
linarith
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ 0 < s
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ s t : ℝ tz : circleMap c s t = z ss : (r0 < s ∧ s ≀ r1) ∧ 0 < t ∧ t ≀ 2 * Ο€ ⊒ 0 < s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
intro zr
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ z ∈ annulus_oc c r0 r1 β†’ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : z ∈ annulus_oc c r0 r1 ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ ⊒ z ∈ annulus_oc c r0 r1 β†’ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only [mem_diff, Metric.mem_closedBall, mem_singleton_iff, annulus_oc, not_le] at zr
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : z ∈ annulus_oc c r0 r1 ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist z c ≀ r1 ∧ r0 < dist z c ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : z ∈ annulus_oc c r0 r1 ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
rw [dist_comm] at zr
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist z c ≀ r1 ∧ r0 < dist z c ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist z c ≀ r1 ∧ r0 < dist z c ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
have zz : z ∈ sphere c (dist c z) := by simp only [Complex.dist_eq, mem_sphere_iff_norm, Complex.norm_eq_abs, Complex.abs.map_sub]
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
rcases circleMap_Ioc zz with ⟨t, ts, tz⟩
case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
case h.mpr.intro.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) t : ℝ ts : t ∈ Ioc 0 (2 * Ο€) tz : z = circleMap c (dist c z) t ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
use (dist c z, t)
case h.mpr.intro.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) t : ℝ ts : t ∈ Ioc 0 (2 * Ο€) tz : z = circleMap c (dist c z) t ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) t : ℝ ts : t ∈ Ioc 0 (2 * Ο€) tz : z = circleMap c (dist c z) t ⊒ (dist c z, t) ∈ square r0 r1 ∧ circleMap c (dist c z, t).1 (dist c z, t).2 = z
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr.intro.intro c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) t : ℝ ts : t ∈ Ioc 0 (2 * Ο€) tz : z = circleMap c (dist c z) t ⊒ βˆƒ x ∈ square r0 r1, circleMap c x.1 x.2 = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simpa only [square, gt_iff_lt, not_lt, ge_iff_le, zero_lt_two, mul_pos_iff_of_pos_left, mem_prod, mem_Ioc, dist_pos, ne_eq, not_false_eq_true, zr, and_self, true_and, tz.symm, and_true] using ts
case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) t : ℝ ts : t ∈ Ioc 0 (2 * Ο€) tz : z = circleMap c (dist c z) t ⊒ (dist c z, t) ∈ square r0 r1 ∧ circleMap c (dist c z, t).1 (dist c z, t).2 = z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z zz : z ∈ sphere c (dist c z) t : ℝ ts : t ∈ Ioc 0 (2 * Ο€) tz : z = circleMap c (dist c z) t ⊒ (dist c z, t) ∈ square r0 r1 ∧ circleMap c (dist c z, t).1 (dist c z, t).2 = z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
square_eq
[139, 1]
[175, 23]
simp only [Complex.dist_eq, mem_sphere_iff_norm, Complex.norm_eq_abs, Complex.abs.map_sub]
c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z ⊒ z ∈ sphere c (dist c z)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ r0 r1 : ℝ r0p : 0 ≀ r0 e : realCircleMap c = fun x => Complex.measurableEquivRealProd (circleMap c x.1 x.2) z : β„‚ zr : dist c z ≀ r1 ∧ r0 < dist c z ⊒ z ∈ sphere c (dist c z) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
exp_of_im
[178, 1]
[179, 71]
simp [Complex.ext_iff, Complex.cos_ofReal_re, Complex.sin_ofReal_re]
t : ℝ ⊒ (↑t * I).exp = ↑t.cos + ↑t.sin * I
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ ⊒ (↑t * I).exp = ↑t.cos + ↑t.sin * I TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Complex.cos_eq_cos
[181, 1]
[181, 78]
simp
t : ℝ ⊒ (↑t).cos = ↑t.cos
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ ⊒ (↑t).cos = ↑t.cos TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
Complex.sin_eq_sin
[183, 1]
[183, 78]
simp
t : ℝ ⊒ (↑t).sin = ↑t.sin
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ ⊒ (↑t).sin = ↑t.sin TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
generalize hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n
t : ℝ ⊒ βˆƒ n, (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ βˆƒ n, (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ ⊒ βˆƒ n, (↑t * I).exp.arg = t - 2 * Ο€ * ↑n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
exists n
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ βˆƒ n, (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ βˆƒ n, (↑t * I).exp.arg = t - 2 * Ο€ * ↑n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
have en : exp (2 * Ο€ * n * I) = 1 := by rw [mul_comm _ (n:β„‚), mul_assoc, Complex.exp_int_mul] simp only [Complex.exp_two_pi_mul_I, one_zpow]
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
have e : exp (t * I) = exp (↑(t - 2 * Ο€ * n) * I) := by simp [mul_sub_right_distrib, Complex.exp_sub, en]
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
rw [e, exp_of_im, ← Complex.cos_eq_cos, ← Complex.sin_eq_sin, Complex.arg_cos_add_sin_mul_I ts]
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ts : t - 2 * Ο€ * ↑n ∈ Ioc (-Ο€) Ο€ ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ts : t - 2 * Ο€ * ↑n ∈ Ioc (-Ο€) Ο€ ⊒ (↑t * I).exp.arg = t - 2 * Ο€ * ↑n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
rw [mul_comm _ (n:β„‚), mul_assoc, Complex.exp_int_mul]
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (2 * ↑π * ↑n * I).exp = 1
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (2 * ↑π * I).exp ^ n = 1
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (2 * ↑π * ↑n * I).exp = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
simp only [Complex.exp_two_pi_mul_I, one_zpow]
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (2 * ↑π * I).exp ^ n = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n ⊒ (2 * ↑π * I).exp ^ n = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
simp [mul_sub_right_distrib, Complex.exp_sub, en]
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 ⊒ (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp
no goals
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 ⊒ (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
simp only [mem_Ioc, neg_lt_sub_iff_lt_add, tsub_le_iff_right]
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ t - 2 * Ο€ * ↑n ∈ Ioc (-Ο€) Ο€
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ 2 * Ο€ * ↑n < Ο€ + t ∧ t ≀ Ο€ + 2 * Ο€ * ↑n
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ t - 2 * Ο€ * ↑n ∈ Ioc (-Ο€) Ο€ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/FubiniBall.lean
arg_exp_of_im
[186, 1]
[206, 98]
constructor
t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ 2 * Ο€ * ↑n < Ο€ + t ∧ t ≀ Ο€ + 2 * Ο€ * ↑n
case left t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ 2 * Ο€ * ↑n < Ο€ + t case right t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ t ≀ Ο€ + 2 * Ο€ * ↑n
Please generate a tactic in lean4 to solve the state. STATE: t : ℝ n : β„€ hn : ⌈t / (2 * Ο€) - 1 / 2βŒ‰ = n en : (2 * ↑π * ↑n * I).exp = 1 e : (↑t * I).exp = (↑(t - 2 * Ο€ * ↑n) * I).exp ⊒ 2 * Ο€ * ↑n < Ο€ + t ∧ t ≀ Ο€ + 2 * Ο€ * ↑n TACTIC: