file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
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|---|---|---|---|---|---|---|
Mathlib/Data/Real/EReal.lean
|
EReal.toReal_neg
|
[
{
"state_after": "no goals",
"state_before": "⊢ toReal (-⊤) = -toReal ⊤",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "⊢ toReal (-⊥) = -toReal ⊥",
"tactic": "simp"
}
] |
[
773,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/Analysis/Calculus/Series.lean
|
contDiff_tsum_of_eventually
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ContDiff 𝕜 N fun x => ∑' (i : α), f i x",
"tactic": "classical\n refine contDiff_iff_forall_nat_le.2 fun m hm => ?_\n let t : Set α :=\n { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }\n have ht : Set.Finite t :=\n haveI A :\n ∀ᶠ i in (Filter.cofinite : Filter α),\n ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by\n rw [eventually_all_finset]\n intro i hi\n apply h'f\n simp only [Finset.mem_range_succ_iff] at hi \n exact (WithTop.coe_le_coe.2 hi).trans hm\n eventually_cofinite.2 A\n let T : Finset α := ht.toFinset\n have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +\n fun x => ∑' i : { i // i ∉ T }, f i x := by\n ext1 x\n refine' (sum_add_tsum_subtype_compl _ T).symm\n refine' summable_of_norm_bounded_eventually _ (hv 0 (zero_le _)) _\n filter_upwards [h'f 0 (zero_le _)]with i hi\n simpa only [norm_iteratedFDeriv_zero] using hi x\n rw [this]\n apply (ContDiff.sum fun i _ => (hf i).of_le hm).add\n have h'u : ∀ k : ℕ, (k : ℕ∞) ≤ m → Summable (v k ∘ ((↑) : { i // i ∉ T } → α)) := fun k hk =>\n (hv k (hk.trans hm)).subtype _\n refine' contDiff_tsum (fun i => (hf i).of_le hm) h'u _\n rintro k ⟨i, hi⟩ x hk\n dsimp\n simp only [Finite.mem_toFinset, mem_setOf_eq, Finset.mem_range, not_forall, not_le,\n exists_prop, not_exists, not_and, not_lt] at hi \n exact hi k (Nat.lt_succ_iff.2 (WithTop.coe_le_coe.1 hk)) x"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ContDiff 𝕜 N fun x => ∑' (i : α), f i x",
"tactic": "refine contDiff_iff_forall_nat_le.2 fun m hm => ?_"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"tactic": "let t : Set α :=\n { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"tactic": "have ht : Set.Finite t :=\n haveI A :\n ∀ᶠ i in (Filter.cofinite : Filter α),\n ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by\n rw [eventually_all_finset]\n intro i hi\n apply h'f\n simp only [Finset.mem_range_succ_iff] at hi \n exact (WithTop.coe_le_coe.2 hi).trans hm\n eventually_cofinite.2 A"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"tactic": "let T : Finset α := ht.toFinset"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"tactic": "have : (fun x => ∑' i, f i x) = (fun x => ∑ i in T, f i x) +\n fun x => ∑' i : { i // i ∉ T }, f i x := by\n ext1 x\n refine' (sum_add_tsum_subtype_compl _ T).symm\n refine' summable_of_norm_bounded_eventually _ (hv 0 (zero_le _)) _\n filter_upwards [h'f 0 (zero_le _)]with i hi\n simpa only [norm_iteratedFDeriv_zero] using hi x"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\n⊢ ContDiff 𝕜 (↑m) ((fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x)",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\n⊢ ContDiff 𝕜 ↑m fun x => ∑' (i : α), f i x",
"tactic": "rw [this]"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\n⊢ ContDiff 𝕜 ↑m fun x => (fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x) x",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\n⊢ ContDiff 𝕜 (↑m) ((fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x)",
"tactic": "apply (ContDiff.sum fun i _ => (hf i).of_le hm).add"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\n⊢ ContDiff 𝕜 ↑m fun x => (fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x) x",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\n⊢ ContDiff 𝕜 ↑m fun x => (fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x) x",
"tactic": "have h'u : ∀ k : ℕ, (k : ℕ∞) ≤ m → Summable (v k ∘ ((↑) : { i // i ∉ T } → α)) := fun k hk =>\n (hv k (hk.trans hm)).subtype _"
},
{
"state_after": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\n⊢ ∀ (k : ℕ) (i : { i // ¬i ∈ T }) (x : E), ↑k ≤ ↑m → ‖iteratedFDeriv 𝕜 k (fun x => f (↑i) x) x‖ ≤ (v k ∘ Subtype.val) i",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\n⊢ ContDiff 𝕜 ↑m fun x => (fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x) x",
"tactic": "refine' contDiff_tsum (fun i => (hf i).of_le hm) h'u _"
},
{
"state_after": "case mk\nα : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\nk : ℕ\ni : α\nhi : ¬i ∈ T\nx : E\nhk : ↑k ≤ ↑m\n⊢ ‖iteratedFDeriv 𝕜 k (fun x => f (↑{ val := i, property := hi }) x) x‖ ≤\n (v k ∘ Subtype.val) { val := i, property := hi }",
"state_before": "α : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\n⊢ ∀ (k : ℕ) (i : { i // ¬i ∈ T }) (x : E), ↑k ≤ ↑m → ‖iteratedFDeriv 𝕜 k (fun x => f (↑i) x) x‖ ≤ (v k ∘ Subtype.val) i",
"tactic": "rintro k ⟨i, hi⟩ x hk"
},
{
"state_after": "case mk\nα : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\nk : ℕ\ni : α\nhi : ¬i ∈ T\nx : E\nhk : ↑k ≤ ↑m\n⊢ ‖iteratedFDeriv 𝕜 k (fun x => f i x) x‖ ≤ v k i",
"state_before": "case mk\nα : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\nk : ℕ\ni : α\nhi : ¬i ∈ T\nx : E\nhk : ↑k ≤ ↑m\n⊢ ‖iteratedFDeriv 𝕜 k (fun x => f (↑{ val := i, property := hi }) x) x‖ ≤\n (v k ∘ Subtype.val) { val := i, property := hi }",
"tactic": "dsimp"
},
{
"state_after": "case mk\nα : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\nk : ℕ\ni : α\nx : E\nhk : ↑k ≤ ↑m\nhi : ∀ (x : ℕ), x < m + 1 → ∀ (x_1 : E), ‖iteratedFDeriv 𝕜 x (f i) x_1‖ ≤ v x i\n⊢ ‖iteratedFDeriv 𝕜 k (fun x => f i x) x‖ ≤ v k i",
"state_before": "case mk\nα : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\nk : ℕ\ni : α\nhi : ¬i ∈ T\nx : E\nhk : ↑k ≤ ↑m\n⊢ ‖iteratedFDeriv 𝕜 k (fun x => f i x) x‖ ≤ v k i",
"tactic": "simp only [Finite.mem_toFinset, mem_setOf_eq, Finset.mem_range, not_forall, not_le,\n exists_prop, not_exists, not_and, not_lt] at hi"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nthis : (fun x => ∑' (i : α), f i x) = (fun x => ∑ i in T, f i x) + fun x => ∑' (i : { i // ¬i ∈ T }), f (↑i) x\nh'u : ∀ (k : ℕ), ↑k ≤ ↑m → Summable (v k ∘ Subtype.val)\nk : ℕ\ni : α\nx : E\nhk : ↑k ≤ ↑m\nhi : ∀ (x : ℕ), x < m + 1 → ∀ (x_1 : E), ‖iteratedFDeriv 𝕜 x (f i) x_1‖ ≤ v x i\n⊢ ‖iteratedFDeriv 𝕜 k (fun x => f i x) x‖ ≤ v k i",
"tactic": "exact hi k (Nat.lt_succ_iff.2 (WithTop.coe_le_coe.1 hk)) x"
},
{
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"tactic": "refine' summable_of_norm_bounded_eventually _ (hv 0 (zero_le _)) _"
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{
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{
"state_after": "no goals",
"state_before": "case h\nα : Type u_4\nβ : Type ?u.153529\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nt : Set α := {i | ¬∀ (k : ℕ), k ∈ Finset.range (m + 1) → ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i}\nht : Set.Finite t\nT : Finset α := Finite.toFinset ht\nx : E\ni : α\nhi : ∀ (x : E), ‖iteratedFDeriv 𝕜 0 (f i) x‖ ≤ v 0 i\n⊢ ‖f i x‖ ≤ v 0 i",
"tactic": "simpa only [norm_iteratedFDeriv_zero] using hi x"
}
] |
[
296,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Data/Multiset/Pi.lean
|
Multiset.Nodup.pi
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\n⊢ Nodup (pi (a ::ₘ s) t)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns : Multiset α\nt : (a : α) → Multiset (β a)\n⊢ ∀ ⦃a : α⦄ {s : Multiset α},\n (Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)) →\n Nodup (a ::ₘ s) → (∀ (a_3 : α), a_3 ∈ a ::ₘ s → Nodup (t a_3)) → Nodup (pi (a ::ₘ s) t)",
"tactic": "intro a s ih hs ht"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\n⊢ Nodup (pi (a ::ₘ s) t)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\n⊢ Nodup (pi (a ::ₘ s) t)",
"tactic": "have has : a ∉ s := by simp at hs; exact hs.1"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ Nodup (pi (a ::ₘ s) t)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\n⊢ Nodup (pi (a ::ₘ s) t)",
"tactic": "have hs : Nodup s := by simp at hs; exact hs.2"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ (∀ (a_1 : β a), a_1 ∈ t a → Nodup (Multiset.map (Pi.cons s a a_1) (pi s t))) ∧\n Pairwise (fun a_1 b => Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t)))\n (t a)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ Nodup (pi (a ::ₘ s) t)",
"tactic": "simp"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ Pairwise (fun a_1 b => Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t)))\n (t a)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ (∀ (a_1 : β a), a_1 ∈ t a → Nodup (Multiset.map (Pi.cons s a a_1) (pi s t))) ∧\n Pairwise (fun a_1 b => Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t)))\n (t a)",
"tactic": "refine'\n ⟨fun b _ => ((ih hs) fun a' h' => ht a' <| mem_cons_of_mem h').map (Pi.cons_injective has),\n _⟩"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ ∀ (a_1 : β a),\n a_1 ∈ t a →\n ∀ (b : β a),\n b ∈ t a → a_1 ≠ b → Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t))",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ Pairwise (fun a_1 b => Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t)))\n (t a)",
"tactic": "refine' (ht a <| mem_cons_self _ _).pairwise _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\n⊢ ∀ (a_1 : β a),\n a_1 ∈ t a →\n ∀ (b : β a),\n b ∈ t a → a_1 ≠ b → Disjoint (Multiset.map (Pi.cons s a a_1) (pi s t)) (Multiset.map (Pi.cons s a b) (pi s t))",
"tactic": "exact fun b₁ _ b₂ _ neb =>\n disjoint_map_map.2 fun f _ g _ eq =>\n have : Pi.cons s a b₁ f a (mem_cons_self _ _) = Pi.cons s a b₂ g a (mem_cons_self _ _) :=\n by rw [eq]\n neb <| show b₁ = b₂ by rwa [Pi.cons_same, Pi.cons_same] at this"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhs : ¬a ∈ s ∧ Nodup s\n⊢ ¬a ∈ s",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\n⊢ ¬a ∈ s",
"tactic": "simp at hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhs : ¬a ∈ s ∧ Nodup s\n⊢ ¬a ∈ s",
"tactic": "exact hs.1"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : ¬a ∈ s ∧ Nodup s\n⊢ Nodup s",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\n⊢ Nodup s",
"tactic": "simp at hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : ¬a ∈ s ∧ Nodup s\n⊢ Nodup s",
"tactic": "exact hs.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\nb₁ : β a\nx✝³ : b₁ ∈ t a\nb₂ : β a\nx✝² : b₂ ∈ t a\nneb : b₁ ≠ b₂\nf : (a : α) → a ∈ s → β a\nx✝¹ : f ∈ pi s t\ng : (a : α) → a ∈ s → β a\nx✝ : g ∈ pi s t\neq : Pi.cons s a b₁ f = Pi.cons s a b₂ g\n⊢ Pi.cons s a b₁ f a (_ : a ∈ a ::ₘ s) = Pi.cons s a b₂ g a (_ : a ∈ a ::ₘ s)",
"tactic": "rw [eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nβ : α → Type u\nδ : α → Sort v\ns✝ : Multiset α\nt : (a : α) → Multiset (β a)\na : α\ns : Multiset α\nih : Nodup s → (∀ (a : α), a ∈ s → Nodup (t a)) → Nodup (pi s t)\nhs✝ : Nodup (a ::ₘ s)\nht : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → Nodup (t a_1)\nhas : ¬a ∈ s\nhs : Nodup s\nb₁ : β a\nx✝³ : b₁ ∈ t a\nb₂ : β a\nx✝² : b₂ ∈ t a\nneb : b₁ ≠ b₂\nf : (a : α) → a ∈ s → β a\nx✝¹ : f ∈ pi s t\ng : (a : α) → a ∈ s → β a\nx✝ : g ∈ pi s t\neq : Pi.cons s a b₁ f = Pi.cons s a b₂ g\nthis : Pi.cons s a b₁ f a (_ : a ∈ a ::ₘ s) = Pi.cons s a b₂ g a (_ : a ∈ a ::ₘ s)\n⊢ b₁ = b₂",
"tactic": "rwa [Pi.cons_same, Pi.cons_same] at this"
}
] |
[
139,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
11
] |
Mathlib/Algebra/Homology/Single.lean
|
CochainComplex.single₀_obj_X_d
|
[] |
[
358,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
exists_seq_antitone_tendsto_atTop_atBot
|
[] |
[
109,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean
|
Real.deriv_arcsin_aux
|
[
{
"state_after": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x\n\ncase inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : -1 < x\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "x : ℝ\nh₁ : x ≠ -1\nh₂ : x ≠ 1\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "cases' h₁.lt_or_lt with h₁ h₁"
},
{
"state_after": "case inr.inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x\n\ncase inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : -1 < x\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "cases' h₂.lt_or_lt with h₂ h₂"
},
{
"state_after": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "have : 1 - x ^ 2 < 0 := by nlinarith [h₁]"
},
{
"state_after": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "rw [sqrt_eq_zero'.2 this.le, div_zero]"
},
{
"state_after": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\nthis✝ : 1 - x ^ 2 < 0\nthis : arcsin =ᶠ[𝓝 x] fun x => -(π / 2)\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "have : arcsin =ᶠ[𝓝 x] fun _ => -(π / 2) :=\n (gt_mem_nhds h₁).mono fun y hy => arcsin_of_le_neg_one hy.le"
},
{
"state_after": "no goals",
"state_before": "case inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\nthis✝ : 1 - x ^ 2 < 0\nthis : arcsin =ᶠ[𝓝 x] fun x => -(π / 2)\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "exact ⟨(hasStrictDerivAt_const _ _).congr_of_eventuallyEq this.symm,\n contDiffAt_const.congr_of_eventuallyEq this⟩"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh₁✝ : x ≠ -1\nh₂ : x ≠ 1\nh₁ : x < -1\n⊢ 1 - x ^ 2 < 0",
"tactic": "nlinarith [h₁]"
},
{
"state_after": "case inr.inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\nthis : 0 < sqrt (1 - x ^ 2)\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inr.inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "have : 0 < sqrt (1 - x ^ 2) := sqrt_pos.2 (by nlinarith [h₁, h₂])"
},
{
"state_after": "case inr.inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\nthis : 0 < cos (arcsin x)\n⊢ HasStrictDerivAt arcsin (cos (arcsin x))⁻¹ x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inr.inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\nthis : 0 < sqrt (1 - x ^ 2)\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "simp only [← cos_arcsin, one_div] at this ⊢"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\nthis : 0 < cos (arcsin x)\n⊢ HasStrictDerivAt arcsin (cos (arcsin x))⁻¹ x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "exact ⟨sinLocalHomeomorph.hasStrictDerivAt_symm ⟨h₁, h₂⟩ this.ne' (hasStrictDerivAt_sin _),\n sinLocalHomeomorph.contDiffAt_symm_deriv this.ne' ⟨h₁, h₂⟩ (hasDerivAt_sin _)\n contDiff_sin.contDiffAt⟩"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : x < 1\n⊢ 0 < 1 - x ^ 2",
"tactic": "nlinarith [h₁, h₂]"
},
{
"state_after": "case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "have : 1 - x ^ 2 < 0 := by nlinarith [h₂]"
},
{
"state_after": "case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "rw [sqrt_eq_zero'.2 this.le, div_zero]"
},
{
"state_after": "case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\nthis✝ : 1 - x ^ 2 < 0\nthis : arcsin =ᶠ[𝓝 x] fun x => π / 2\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"state_before": "case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\nthis : 1 - x ^ 2 < 0\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "have : arcsin =ᶠ[𝓝 x] fun _ => π / 2 := (lt_mem_nhds h₂).mono fun y hy => arcsin_of_one_le hy.le"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nx : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\nthis✝ : 1 - x ^ 2 < 0\nthis : arcsin =ᶠ[𝓝 x] fun x => π / 2\n⊢ HasStrictDerivAt arcsin 0 x ∧ ContDiffAt ℝ ⊤ arcsin x",
"tactic": "exact ⟨(hasStrictDerivAt_const _ _).congr_of_eventuallyEq this.symm,\n contDiffAt_const.congr_of_eventuallyEq this⟩"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh₁✝ : x ≠ -1\nh₂✝ : x ≠ 1\nh₁ : -1 < x\nh₂ : 1 < x\n⊢ 1 - x ^ 2 < 0",
"tactic": "nlinarith [h₂]"
}
] |
[
52,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
dist_one_left
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.50544\n𝕜 : Type ?u.50547\nα : Type ?u.50550\nι : Type ?u.50553\nκ : Type ?u.50556\nE : Type u_1\nF : Type ?u.50562\nG : Type ?u.50565\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\na : E\n⊢ dist 1 a = ‖a‖",
"tactic": "rw [dist_comm, dist_one_right]"
}
] |
[
406,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.add_lt_add_of_lt_of_le
|
[] |
[
718,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
716,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.edist_le_mul
|
[] |
[
134,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Analysis/NormedSpace/Ray.lean
|
sameRay_iff_inv_norm_smul_eq
|
[
{
"state_after": "case inl\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ny : F\n⊢ SameRay ℝ 0 y ↔ 0 = 0 ∨ y = 0 ∨ ‖0‖⁻¹ • 0 = ‖y‖⁻¹ • y\n\ncase inr\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : F\nhx : x ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y",
"state_before": "E : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : F\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y",
"tactic": "rcases eq_or_ne x 0 with (rfl | hx)"
},
{
"state_after": "case inr.inl\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\nhx : x ≠ 0\n⊢ SameRay ℝ x 0 ↔ x = 0 ∨ 0 = 0 ∨ ‖x‖⁻¹ • x = ‖0‖⁻¹ • 0\n\ncase inr.inr\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y",
"state_before": "case inr\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : F\nhx : x ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y",
"tactic": "rcases eq_or_ne y 0 with (rfl | hy)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y",
"tactic": "simp only [sameRay_iff_inv_norm_smul_eq_of_ne hx hy, *, false_or_iff]"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ny : F\n⊢ SameRay ℝ 0 y ↔ 0 = 0 ∨ y = 0 ∨ ‖0‖⁻¹ • 0 = ‖y‖⁻¹ • y",
"tactic": "simp [SameRay.zero_left]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nE : Type ?u.236691\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\nhx : x ≠ 0\n⊢ SameRay ℝ x 0 ↔ x = 0 ∨ 0 = 0 ∨ ‖x‖⁻¹ • x = ‖0‖⁻¹ • 0",
"tactic": "simp [SameRay.zero_right]"
}
] |
[
98,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.continuous_deriv
|
[] |
[
2168,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2167,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurableSet_prod_of_nonempty
|
[
{
"state_after": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nh : Set.Nonempty (s ×ˢ t)\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t",
"tactic": "rcases h with ⟨⟨x, y⟩, hx, hy⟩"
},
{
"state_after": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\nhst : MeasurableSet (s ×ˢ t)\n⊢ MeasurableSet s ∧ MeasurableSet t",
"state_before": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t",
"tactic": "refine' ⟨fun hst => _, fun h => h.1.prod h.2⟩"
},
{
"state_after": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\nhst : MeasurableSet (s ×ˢ t)\nthis : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t)\n⊢ MeasurableSet s ∧ MeasurableSet t",
"state_before": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\nhst : MeasurableSet (s ×ˢ t)\n⊢ MeasurableSet s ∧ MeasurableSet t",
"tactic": "have : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t) := measurable_prod_mk_right hst"
},
{
"state_after": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\nhst : MeasurableSet (s ×ˢ t)\nthis✝ : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t)\nthis : MeasurableSet (Prod.mk x ⁻¹' s ×ˢ t)\n⊢ MeasurableSet s ∧ MeasurableSet t",
"state_before": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\nhst : MeasurableSet (s ×ˢ t)\nthis : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t)\n⊢ MeasurableSet s ∧ MeasurableSet t",
"tactic": "have : MeasurableSet (Prod.mk x ⁻¹' s ×ˢ t) := measurable_prod_mk_left hst"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.83086\nδ : Type ?u.83089\nδ' : Type ?u.83092\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nx : α\ny : β\nhx : (x, y).fst ∈ s\nhy : (x, y).snd ∈ t\nhst : MeasurableSet (s ×ˢ t)\nthis✝ : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t)\nthis : MeasurableSet (Prod.mk x ⁻¹' s ×ˢ t)\n⊢ MeasurableSet s ∧ MeasurableSet t",
"tactic": "simp_all"
}
] |
[
731,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
725,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.sub_mem
|
[] |
[
141,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
11
] |
Mathlib/Data/Set/Function.lean
|
Set.BijOn.injOn
|
[] |
[
912,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
911,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.argmax_singleton
|
[] |
[
122,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Analysis/Calculus/Deriv/Pow.lean
|
HasDerivWithinAt.pow
|
[] |
[
106,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.ofNat_add_ofNat
|
[] |
[
51,
80
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
51,
9
] |
Mathlib/Data/MvPolynomial/Funext.lean
|
MvPolynomial.funext_fin
|
[
{
"state_after": "case zero\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn : ℕ\np✝ : MvPolynomial (Fin n) R\nh✝ : ∀ (x : Fin n → R), ↑(eval x) p✝ = 0\np : MvPolynomial (Fin Nat.zero) R\nh : ∀ (x : Fin Nat.zero → R), ↑(eval x) p = 0\n⊢ p = 0\n\ncase succ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\n⊢ p = 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn : ℕ\np : MvPolynomial (Fin n) R\nh : ∀ (x : Fin n → R), ↑(eval x) p = 0\n⊢ p = 0",
"tactic": "induction' n with n ih"
},
{
"state_after": "case zero.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn : ℕ\np✝ : MvPolynomial (Fin n) R\nh✝ : ∀ (x : Fin n → R), ↑(eval x) p✝ = 0\np : MvPolynomial (Fin Nat.zero) R\nh : ∀ (x : Fin Nat.zero → R), ↑(eval x) p = 0\n⊢ ↑(isEmptyRingEquiv R (Fin 0)) p = ↑(isEmptyRingEquiv R (Fin 0)) 0",
"state_before": "case zero\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn : ℕ\np✝ : MvPolynomial (Fin n) R\nh✝ : ∀ (x : Fin n → R), ↑(eval x) p✝ = 0\np : MvPolynomial (Fin Nat.zero) R\nh : ∀ (x : Fin Nat.zero → R), ↑(eval x) p = 0\n⊢ p = 0",
"tactic": "apply (MvPolynomial.isEmptyRingEquiv R (Fin 0)).injective"
},
{
"state_after": "case zero.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn : ℕ\np✝ : MvPolynomial (Fin n) R\nh✝ : ∀ (x : Fin n → R), ↑(eval x) p✝ = 0\np : MvPolynomial (Fin Nat.zero) R\nh : ∀ (x : Fin Nat.zero → R), ↑(eval x) p = 0\n⊢ ↑(isEmptyRingEquiv R (Fin 0)) p = 0",
"state_before": "case zero.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn : ℕ\np✝ : MvPolynomial (Fin n) R\nh✝ : ∀ (x : Fin n → R), ↑(eval x) p✝ = 0\np : MvPolynomial (Fin Nat.zero) R\nh : ∀ (x : Fin Nat.zero → R), ↑(eval x) p = 0\n⊢ ↑(isEmptyRingEquiv R (Fin 0)) p = ↑(isEmptyRingEquiv R (Fin 0)) 0",
"tactic": "rw [RingEquiv.map_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn : ℕ\np✝ : MvPolynomial (Fin n) R\nh✝ : ∀ (x : Fin n → R), ↑(eval x) p✝ = 0\np : MvPolynomial (Fin Nat.zero) R\nh : ∀ (x : Fin Nat.zero → R), ↑(eval x) p = 0\n⊢ ↑(isEmptyRingEquiv R (Fin 0)) p = 0",
"tactic": "convert h finZeroElim"
},
{
"state_after": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\n⊢ ↑(finSuccEquiv R n) p = ↑(finSuccEquiv R n) 0",
"state_before": "case succ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\n⊢ p = 0",
"tactic": "apply (finSuccEquiv R n).injective"
},
{
"state_after": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\n⊢ ↑(finSuccEquiv R n) p = 0",
"state_before": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\n⊢ ↑(finSuccEquiv R n) p = ↑(finSuccEquiv R n) 0",
"tactic": "simp only [AlgEquiv.map_zero]"
},
{
"state_after": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\nq : MvPolynomial (Fin n) R\n⊢ Polynomial.eval q (↑(finSuccEquiv R n) p) = Polynomial.eval q 0",
"state_before": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\n⊢ ↑(finSuccEquiv R n) p = 0",
"tactic": "refine Polynomial.funext fun q => ?_"
},
{
"state_after": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\nq : MvPolynomial (Fin n) R\n⊢ Polynomial.eval q (↑(finSuccEquiv R n) p) = 0",
"state_before": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\nq : MvPolynomial (Fin n) R\n⊢ Polynomial.eval q (↑(finSuccEquiv R n) p) = Polynomial.eval q 0",
"tactic": "rw [Polynomial.eval_zero]"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\nq : MvPolynomial (Fin n) R\nx : Fin n → R\n⊢ ↑(eval x) (Polynomial.eval q (↑(finSuccEquiv R n) p)) = 0",
"state_before": "case succ.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\nq : MvPolynomial (Fin n) R\n⊢ Polynomial.eval q (↑(finSuccEquiv R n) p) = 0",
"tactic": "apply ih fun x => ?_"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Infinite R\nn✝ : ℕ\np✝ : MvPolynomial (Fin n✝) R\nh✝ : ∀ (x : Fin n✝ → R), ↑(eval x) p✝ = 0\nn : ℕ\nih : ∀ {p : MvPolynomial (Fin n) R}, (∀ (x : Fin n → R), ↑(eval x) p = 0) → p = 0\np : MvPolynomial (Fin (Nat.succ n)) R\nh : ∀ (x : Fin (Nat.succ n) → R), ↑(eval x) p = 0\nq : MvPolynomial (Fin n) R\nx : Fin n → R\n⊢ ↑(eval x) (Polynomial.eval q (↑(finSuccEquiv R n) p)) = 0",
"tactic": "calc _ = _ := eval_polynomial_eval_finSuccEquiv p _\n _ = 0 := h _"
}
] |
[
45,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
9
] |
Mathlib/Order/Disjoint.lean
|
codisjoint_inf_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : OrderTop α\na b c : α\n⊢ Codisjoint a (b ⊓ c) ↔ Codisjoint a b ∧ Codisjoint a c",
"tactic": "simp only [codisjoint_iff, sup_inf_left, inf_eq_top_iff]"
}
] |
[
379,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Embedding.comp_toHom
|
[
{
"state_after": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type u_1\ninst✝¹ : Structure L P\nQ : Type ?u.151174\ninst✝ : Structure L Q\nhnp : N ↪[L] P\nhmn : M ↪[L] N\nx✝ : M\n⊢ ↑(toHom (comp hnp hmn)) x✝ = ↑(Hom.comp (toHom hnp) (toHom hmn)) x✝",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type u_1\ninst✝¹ : Structure L P\nQ : Type ?u.151174\ninst✝ : Structure L Q\nhnp : N ↪[L] P\nhmn : M ↪[L] N\n⊢ toHom (comp hnp hmn) = Hom.comp (toHom hnp) (toHom hmn)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type u_1\ninst✝¹ : Structure L P\nQ : Type ?u.151174\ninst✝ : Structure L Q\nhnp : N ↪[L] P\nhmn : M ↪[L] N\nx✝ : M\n⊢ ↑(toHom (comp hnp hmn)) x✝ = ↑(Hom.comp (toHom hnp) (toHom hmn)) x✝",
"tactic": "simp only [coe_toHom, comp_apply, Hom.comp_apply]"
}
] |
[
731,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/Deprecated/Group.lean
|
IsGroupHom.mk'
|
[] |
[
271,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
src/lean/Init/Data/Nat/Power2.lean
|
Nat.isPowerOfTwo_nextPowerOfTwo
|
[
{
"state_after": "case h₂\nn : Nat\n⊢ isPowerOfTwo 1",
"state_before": "n : Nat\n⊢ isPowerOfTwo (nextPowerOfTwo n)",
"tactic": "apply isPowerOfTwo_go"
},
{
"state_after": "no goals",
"state_before": "case h₂\nn : Nat\n⊢ isPowerOfTwo 1",
"tactic": "apply one_isPowerOfTwo"
},
{
"state_after": "n power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\n⊢ isPowerOfTwo (if power < n then nextPowerOfTwo.go n (power * 2) (_ : power * 2 > 0) else power)",
"state_before": "n power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\n⊢ isPowerOfTwo (nextPowerOfTwo.go n power h₁)",
"tactic": "unfold nextPowerOfTwo.go"
},
{
"state_after": "case inl\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : power < n\n⊢ isPowerOfTwo (nextPowerOfTwo.go n (power * 2) (_ : power * 2 > 0))\n\ncase inr\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : ¬power < n\n⊢ isPowerOfTwo power",
"state_before": "n power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\n⊢ isPowerOfTwo (if power < n then nextPowerOfTwo.go n (power * 2) (_ : power * 2 > 0) else power)",
"tactic": "split"
},
{
"state_after": "case inr\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : ¬power < n\n⊢ isPowerOfTwo power",
"state_before": "case inl\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : power < n\n⊢ isPowerOfTwo (nextPowerOfTwo.go n (power * 2) (_ : power * 2 > 0))\n\ncase inr\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : ¬power < n\n⊢ isPowerOfTwo power",
"tactic": ". exact isPowerOfTwo_go (power*2) (Nat.mul_pos h₁ (by decide)) (Nat.mul2_isPowerOfTwo_of_isPowerOfTwo h₂)"
},
{
"state_after": "no goals",
"state_before": "case inr\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : ¬power < n\n⊢ isPowerOfTwo power",
"tactic": ". assumption"
},
{
"state_after": "no goals",
"state_before": "case inl\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : power < n\n⊢ isPowerOfTwo (nextPowerOfTwo.go n (power * 2) (_ : power * 2 > 0))",
"tactic": "exact isPowerOfTwo_go (power*2) (Nat.mul_pos h₁ (by decide)) (Nat.mul2_isPowerOfTwo_of_isPowerOfTwo h₂)"
},
{
"state_after": "no goals",
"state_before": "n power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : power < n\n⊢ 2 > 0",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "case inr\nn power : Nat\nh₁ : power > 0\nh₂ : isPowerOfTwo power\nh✝ : ¬power < n\n⊢ isPowerOfTwo power",
"tactic": "assumption"
},
{
"state_after": "n : Nat\n_x✝ : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power\na✝² :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y _x✝ →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\npower : Nat\nh₁✝ : (_ : power > 0) ×' isPowerOfTwo power\na✝¹ :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y { fst := power, snd := h₁✝ } →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\nh₁ : power > 0\nh₂ : isPowerOfTwo power\na✝ :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y { fst := power, snd := { fst := h₁, snd := h₂ } } →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\nh✝ : power < n\n⊢ n - power * 2 < n - power",
"state_before": "n : Nat\n_x✝ : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power\na✝² :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y _x✝ →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\npower : Nat\nh₁✝ : (_ : power > 0) ×' isPowerOfTwo power\na✝¹ :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y { fst := power, snd := h₁✝ } →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\nh₁ : power > 0\nh₂ : isPowerOfTwo power\na✝ :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y { fst := power, snd := { fst := h₁, snd := h₂ } } →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\nh✝ : power < n\n⊢ (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n { fst := power * 2, snd := { fst := (_ : power * 2 > 0), snd := (_ : isPowerOfTwo (power * 2)) } }\n { fst := power, snd := { fst := h₁, snd := h₂ } }",
"tactic": "simp_wf"
},
{
"state_after": "no goals",
"state_before": "n : Nat\n_x✝ : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power\na✝² :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y _x✝ →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\npower : Nat\nh₁✝ : (_ : power > 0) ×' isPowerOfTwo power\na✝¹ :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y { fst := power, snd := h₁✝ } →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\nh₁ : power > 0\nh₂ : isPowerOfTwo power\na✝ :\n ∀ (y : (power : Nat) ×' (_ : power > 0) ×' isPowerOfTwo power),\n (invImage (fun a => PSigma.casesOn a fun p snd => PSigma.casesOn snd fun h₁ snd => n - p) instWellFoundedRelation).1\n y { fst := power, snd := { fst := h₁, snd := h₂ } } →\n isPowerOfTwo (nextPowerOfTwo.go n y.1 (_ : y.1 > 0))\nh✝ : power < n\n⊢ n - power * 2 < n - power",
"tactic": "apply nextPowerOfTwo_dec <;> assumption"
}
] |
[
52,
63
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
42,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isLUB_prod
|
[
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : α\nha : a ∈ upperBounds (Prod.fst '' s)\n⊢ p.fst ≤ a\n\ncase refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : β\nha : a ∈ upperBounds (Prod.snd '' s)\n⊢ p.snd ≤ a\n\ncase refine'_3\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB (Prod.fst '' s) p.fst ∧ IsLUB (Prod.snd '' s) p.snd\n⊢ p ∈ upperBounds s\n\ncase refine'_4\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB (Prod.fst '' s) p.fst ∧ IsLUB (Prod.snd '' s) p.snd\n⊢ p ∈ lowerBounds (upperBounds s)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\n⊢ IsLUB s p ↔ IsLUB (Prod.fst '' s) p.fst ∧ IsLUB (Prod.snd '' s) p.snd",
"tactic": "refine'\n ⟨fun H =>\n ⟨⟨monotone_fst.mem_upperBounds_image H.1, fun a ha => _⟩,\n ⟨monotone_snd.mem_upperBounds_image H.1, fun a ha => _⟩⟩,\n fun H => ⟨_, _⟩⟩"
},
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : α\nha : a ∈ upperBounds (Prod.fst '' s)\n⊢ (a, p.snd) ∈ upperBounds s",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : α\nha : a ∈ upperBounds (Prod.fst '' s)\n⊢ p.fst ≤ a",
"tactic": "suffices h : (a, p.2) ∈ upperBounds s from (H.2 h).1"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : α\nha : a ∈ upperBounds (Prod.fst '' s)\n⊢ (a, p.snd) ∈ upperBounds s",
"tactic": "exact fun q hq => ⟨ha <| mem_image_of_mem _ hq, (H.1 hq).2⟩"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : β\nha : a ∈ upperBounds (Prod.snd '' s)\n⊢ (p.fst, a) ∈ upperBounds s",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : β\nha : a ∈ upperBounds (Prod.snd '' s)\n⊢ p.snd ≤ a",
"tactic": "suffices h : (p.1, a) ∈ upperBounds s from (H.2 h).2"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB s p\na : β\nha : a ∈ upperBounds (Prod.snd '' s)\n⊢ (p.fst, a) ∈ upperBounds s",
"tactic": "exact fun q hq => ⟨(H.1 hq).1, ha <| mem_image_of_mem _ hq⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB (Prod.fst '' s) p.fst ∧ IsLUB (Prod.snd '' s) p.snd\n⊢ p ∈ upperBounds s",
"tactic": "exact fun q hq => ⟨H.1.1 <| mem_image_of_mem _ hq, H.2.1 <| mem_image_of_mem _ hq⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_4\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set (α × β)\np : α × β\nH : IsLUB (Prod.fst '' s) p.fst ∧ IsLUB (Prod.snd '' s) p.snd\n⊢ p ∈ lowerBounds (upperBounds s)",
"tactic": "exact fun q hq =>\n ⟨H.1.2 <| monotone_fst.mem_upperBounds_image hq,\n H.2.2 <| monotone_snd.mem_upperBounds_image hq⟩"
}
] |
[
1599,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1584,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
map_extChartAt_symm_nhdsWithin
|
[] |
[
1195,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1192,
1
] |
Mathlib/CategoryTheory/Monoidal/Category.lean
|
CategoryTheory.MonoidalCategory.tensorLeftTensor_inv_app
|
[
{
"state_after": "no goals",
"state_before": "C✝¹ : Type u\n𝒞 : Category C✝¹\ninst✝⁴ : MonoidalCategory C✝¹\nC✝ : Type u\ninst✝³ : Category C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nX Y Z : C\n⊢ (tensorLeftTensor X Y).inv.app Z = (α_ X Y Z).inv",
"tactic": "simp [tensorLeftTensor]"
}
] |
[
531,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/CategoryTheory/Monoidal/Types/Basic.lean
|
CategoryTheory.rightUnitor_inv_apply
|
[] |
[
60,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean
|
DirectSum.decompose_of_mem_ne
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nR : Type ?u.124931\nM : Type u_1\nσ : Type u_2\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddCommMonoid M\ninst✝² : SetLike σ M\ninst✝¹ : AddSubmonoidClass σ M\nℳ : ι → σ\ninst✝ : Decomposition ℳ\nx : M\ni j : ι\nhx : x ∈ ℳ i\nhij : i ≠ j\n⊢ ↑(↑(↑(decompose ℳ) x) j) = 0",
"tactic": "rw [decompose_of_mem _ hx, DirectSum.of_eq_of_ne _ _ _ _ hij, ZeroMemClass.coe_zero]"
}
] |
[
127,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.IsEquiv.ne_top
|
[] |
[
865,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_iff_lt_or_fuzzy
|
[
{
"state_after": "x y : PGame\n⊢ ¬y ≤ x ↔ x ≤ y ∧ ¬y ≤ x ∨ ¬y ≤ x ∧ ¬x ≤ y",
"state_before": "x y : PGame\n⊢ x ⧏ y ↔ x < y ∨ x ‖ y",
"tactic": "simp only [lt_iff_le_and_lf, Fuzzy, ← PGame.not_le]"
},
{
"state_after": "no goals",
"state_before": "x y : PGame\n⊢ ¬y ≤ x ↔ x ≤ y ∧ ¬y ≤ x ∨ ¬y ≤ x ∧ ¬x ≤ y",
"tactic": "tauto"
}
] |
[
924,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isIntegral_algHom_iff
|
[
{
"state_after": "R : Type u_3\nA✝ : Type ?u.158436\nB✝ : Type ?u.158439\nS : Type ?u.158442\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A✝\ninst✝⁷ : CommRing B✝\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B✝\nf✝ : R →+* S\nA : Type u_1\nB : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nx : A\n⊢ IsIntegral R (↑f x) → IsIntegral R x",
"state_before": "R : Type u_3\nA✝ : Type ?u.158436\nB✝ : Type ?u.158439\nS : Type ?u.158442\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A✝\ninst✝⁷ : CommRing B✝\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B✝\nf✝ : R →+* S\nA : Type u_1\nB : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nx : A\n⊢ IsIntegral R (↑f x) ↔ IsIntegral R x",
"tactic": "refine' ⟨_, map_isIntegral f⟩"
},
{
"state_after": "case intro.intro\nR : Type u_3\nA✝ : Type ?u.158436\nB✝ : Type ?u.158439\nS : Type ?u.158442\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A✝\ninst✝⁷ : CommRing B✝\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B✝\nf✝ : R →+* S\nA : Type u_1\nB : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nx : A\np : R[X]\nhp : Monic p\nhx : eval₂ (algebraMap R ((fun x => B) x)) (↑f x) p = 0\n⊢ IsIntegral R x",
"state_before": "R : Type u_3\nA✝ : Type ?u.158436\nB✝ : Type ?u.158439\nS : Type ?u.158442\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A✝\ninst✝⁷ : CommRing B✝\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B✝\nf✝ : R →+* S\nA : Type u_1\nB : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nx : A\n⊢ IsIntegral R (↑f x) → IsIntegral R x",
"tactic": "rintro ⟨p, hp, hx⟩"
},
{
"state_after": "case intro.intro\nR : Type u_3\nA✝ : Type ?u.158436\nB✝ : Type ?u.158439\nS : Type ?u.158442\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A✝\ninst✝⁷ : CommRing B✝\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B✝\nf✝ : R →+* S\nA : Type u_1\nB : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nx : A\np : R[X]\nhp : Monic p\nhx : eval₂ (algebraMap R ((fun x => B) x)) (↑f x) p = 0\n⊢ eval₂ (algebraMap R A) x p = 0",
"state_before": "case intro.intro\nR : Type u_3\nA✝ : Type ?u.158436\nB✝ : Type ?u.158439\nS : Type ?u.158442\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A✝\ninst✝⁷ : CommRing B✝\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B✝\nf✝ : R →+* S\nA : Type u_1\nB : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nx : A\np : R[X]\nhp : Monic p\nhx : eval₂ (algebraMap R ((fun x => B) x)) (↑f x) p = 0\n⊢ IsIntegral R x",
"tactic": "use p, hp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_3\nA✝ : Type ?u.158436\nB✝ : Type ?u.158439\nS : Type ?u.158442\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A✝\ninst✝⁷ : CommRing B✝\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R A✝\ninst✝⁴ : Algebra R B✝\nf✝ : R →+* S\nA : Type u_1\nB : Type u_2\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nhf : Function.Injective ↑f\nx : A\np : R[X]\nhp : Monic p\nhx : eval₂ (algebraMap R ((fun x => B) x)) (↑f x) p = 0\n⊢ eval₂ (algebraMap R A) x p = 0",
"tactic": "rwa [← f.comp_algebraMap, ← AlgHom.coe_toRingHom, ← Polynomial.hom_eval₂, AlgHom.coe_toRingHom,\n map_eq_zero_iff f hf] at hx"
}
] |
[
153,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Control/Bifunctor.lean
|
Bifunctor.fst_snd
|
[
{
"state_after": "no goals",
"state_before": "F : Type u₀ → Type u₁ → Type u₂\ninst✝¹ : Bifunctor F\ninst✝ : LawfulBifunctor F\nα₀ α₁ : Type u₀\nβ₀ β₁ : Type u₁\nf : α₀ → α₁\nf' : β₀ → β₁\nx : F α₀ β₀\n⊢ fst f (snd f' x) = bimap f f' x",
"tactic": "simp [fst, bimap_bimap]"
}
] |
[
99,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/CategoryTheory/Limits/Final.lean
|
CategoryTheory.Functor.initial_of_final_op
|
[] |
[
122,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/RingTheory/Derivation/Basic.lean
|
Derivation.coe_zero_linearMap
|
[] |
[
193,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Order/Antichain.lean
|
IsAntichain.insert_of_symmetric
|
[] |
[
120,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integral_integral_sub
|
[] |
[
410,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieSubmodule.ucs_mono
|
[
{
"state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\n⊢ ucs Nat.zero N₁ ≤ ucs Nat.zero N₂\n\ncase succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\nk : ℕ\nih : ucs k N₁ ≤ ucs k N₂\n⊢ ucs (Nat.succ k) N₁ ≤ ucs (Nat.succ k) N₂",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nk : ℕ\nh : N₁ ≤ N₂\n⊢ ucs k N₁ ≤ ucs k N₂",
"tactic": "induction' k with k ih"
},
{
"state_after": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\nk : ℕ\nih : ucs k N₁ ≤ ucs k N₂\n⊢ normalizer (ucs k N₁) ≤ normalizer (ucs k N₂)",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\nk : ℕ\nih : ucs k N₁ ≤ ucs k N₂\n⊢ ucs (Nat.succ k) N₁ ≤ ucs (Nat.succ k) N₂",
"tactic": "simp only [ucs_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk✝ : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\nk : ℕ\nih : ucs k N₁ ≤ ucs k N₂\n⊢ normalizer (ucs k N₁) ≤ normalizer (ucs k N₂)",
"tactic": "apply monotone_normalizer ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nN N₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\n⊢ ucs Nat.zero N₁ ≤ ucs Nat.zero N₂",
"tactic": "simpa"
}
] |
[
382,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Topology/MetricSpace/PartitionOfUnity.lean
|
EMetric.exists_forall_closedBall_subset_aux₂
|
[] |
[
82,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.norm_def
|
[] |
[
145,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
comap_nhdsWithin_range
|
[] |
[
688,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
687,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
StrictConvexOn.translate_right
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.700544\nα : Type ?u.700547\nβ : Type u_3\nι : Type ?u.700553\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCancelCommMonoid E\ninst✝² : OrderedAddCommMonoid β\ninst✝¹ : Module 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nhf : StrictConvexOn 𝕜 s f\nc x : E\nhx : x ∈ (fun z => c + z) ⁻¹' s\ny : E\nhy : y ∈ (fun z => c + z) ⁻¹' s\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ f (c + (a • x + b • y)) = f (a • (c + x) + b • (c + y))",
"tactic": "rw [smul_add, smul_add, add_add_add_comm, Convex.combo_self hab]"
}
] |
[
941,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
935,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.list_foldl
|
[] |
[
1027,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/Order/Max.lean
|
IsMax.not_lt
|
[] |
[
316,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.const_mul_right
|
[] |
[
1523,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1521,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
TopCat.coequalizer_isOpen_iff
|
[
{
"state_after": "J : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\n⊢ (∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)) ↔\n IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\n⊢ IsOpen U ↔ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"tactic": "rw [colimit_isOpen_iff]"
},
{
"state_after": "case mp\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\n⊢ (∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)) →\n IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\n\ncase mpr\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U) →\n ∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\n⊢ (∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)) ↔\n IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"tactic": "constructor"
},
{
"state_after": "case mp\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : ∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"state_before": "case mp\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\n⊢ (∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)) →\n IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"tactic": "intro H"
},
{
"state_after": "no goals",
"state_before": "case mp\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : ∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"tactic": "exact H _"
},
{
"state_after": "case mpr\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\nj : WalkingParallelPair\n⊢ IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"state_before": "case mpr\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U) →\n ∀ (j : WalkingParallelPair), IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"tactic": "intro H j"
},
{
"state_after": "case mpr.zero\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.zero) ⁻¹' U)\n\ncase mpr.one\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"state_before": "case mpr\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\nj : WalkingParallelPair\n⊢ IsOpen ((forget TopCat).map (colimit.ι F j) ⁻¹' U)",
"tactic": "cases j"
},
{
"state_after": "case mpr.zero\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (F.map WalkingParallelPairHom.left ≫ colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"state_before": "case mpr.zero\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.zero) ⁻¹' U)",
"tactic": "rw [← colimit.w F WalkingParallelPairHom.left]"
},
{
"state_after": "no goals",
"state_before": "case mpr.zero\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (F.map WalkingParallelPairHom.left ≫ colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"tactic": "exact (F.map WalkingParallelPairHom.left).continuous_toFun.isOpen_preimage _ H"
},
{
"state_after": "no goals",
"state_before": "case mpr.one\nJ : Type v\ninst✝ : SmallCategory J\nF : WalkingParallelPair ⥤ TopCat\nU : Set ↑(colimit F)\nH : IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)\n⊢ IsOpen ((forget TopCat).map (colimit.ι F WalkingParallelPair.one) ⁻¹' U)",
"tactic": "exact H"
}
] |
[
432,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.ValidFor.takeWhile
|
[
{
"state_after": "l m r : List Char\np : Char → Bool\n⊢ ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) } }",
"state_before": "l m r : List Char\np : Char → Bool\n⊢ ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)\n (Substring.takeWhile\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } }\n p)",
"tactic": "simp only [Substring.takeWhile, takeWhileAux_of_valid]"
},
{
"state_after": "case refine'_1\nl m r : List Char\np : Char → Bool\n⊢ m ++ r = List.takeWhile p m ++ (List.dropWhile p m ++ r)",
"state_before": "l m r : List Char\np : Char → Bool\n⊢ ValidFor l (List.takeWhile p m) (List.dropWhile p m ++ r)\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len (List.takeWhile p m) } }",
"tactic": "refine' .of_eq .. <;> simp"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nl m r : List Char\np : Char → Bool\n⊢ m ++ r = List.takeWhile p m ++ (List.dropWhile p m ++ r)",
"tactic": "rw [← List.append_assoc, List.takeWhile_append_dropWhile]"
}
] |
[
942,
62
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
937,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.ValidFor.any
|
[
{
"state_after": "no goals",
"state_before": "l m r : List Char\nf : Char → Bool\n⊢ Substring.any\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } }\n f =\n List.any m f",
"tactic": "simp [-List.append_assoc, Substring.any, anyAux_of_valid]"
}
] |
[
929,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
928,
1
] |
Mathlib/Data/Real/Sign.lean
|
Real.sign_int_cast
|
[
{
"state_after": "case inl\nz : ℤ\nhn : z < 0\n⊢ sign ↑z = ↑(Int.sign z)\n\ncase inr.inl\n\n⊢ sign ↑0 = ↑(Int.sign 0)\n\ncase inr.inr\nz : ℤ\nhp : 0 < z\n⊢ sign ↑z = ↑(Int.sign z)",
"state_before": "z : ℤ\n⊢ sign ↑z = ↑(Int.sign z)",
"tactic": "obtain hn | rfl | hp := lt_trichotomy z (0 : ℤ)"
},
{
"state_after": "no goals",
"state_before": "case inl\nz : ℤ\nhn : z < 0\n⊢ sign ↑z = ↑(Int.sign z)",
"tactic": "rw [sign_of_neg (Int.cast_lt_zero.mpr hn), Int.sign_eq_neg_one_of_neg hn, Int.cast_neg,\n Int.cast_one]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\n\n⊢ sign ↑0 = ↑(Int.sign 0)",
"tactic": "rw [Int.cast_zero, sign_zero, Int.sign_zero, Int.cast_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nz : ℤ\nhp : 0 < z\n⊢ sign ↑z = ↑(Int.sign z)",
"tactic": "rw [sign_of_pos (Int.cast_pos.mpr hp), Int.sign_eq_one_of_pos hp, Int.cast_one]"
}
] |
[
82,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
|
ContinuousAt.clog
|
[] |
[
259,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.sub_eq_fold_erase
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.160973\nγ : Type ?u.160976\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ ↑(List.diff l₁ l₂) = foldl erase (_ : ∀ (s : Multiset α) (a b : α), erase (erase s a) b = erase (erase s b) a) ↑l₁ ↑l₂",
"state_before": "α : Type u_1\nβ : Type ?u.160973\nγ : Type ?u.160976\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ Quotient.mk (isSetoid α) l₁ - Quotient.mk (isSetoid α) l₂ =\n foldl erase (_ : ∀ (s : Multiset α) (a b : α), erase (erase s a) b = erase (erase s b) a)\n (Quotient.mk (isSetoid α) l₁) (Quotient.mk (isSetoid α) l₂)",
"tactic": "show ofList (l₁.diff l₂) = foldl erase erase_comm l₁ l₂"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.160973\nγ : Type ?u.160976\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ ↑(List.foldl List.erase l₁ l₂) =\n foldl erase (_ : ∀ (s : Multiset α) (a b : α), erase (erase s a) b = erase (erase s b) a) ↑l₁ ↑l₂",
"state_before": "α : Type u_1\nβ : Type ?u.160973\nγ : Type ?u.160976\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ ↑(List.diff l₁ l₂) = foldl erase (_ : ∀ (s : Multiset α) (a b : α), erase (erase s a) b = erase (erase s b) a) ↑l₁ ↑l₂",
"tactic": "rw [diff_eq_foldl l₁ l₂]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.160973\nγ : Type ?u.160976\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ foldl erase (_ : ∀ (s : Multiset α) (a b : α), erase (erase s a) b = erase (erase s b) a) ↑l₁ ↑l₂ =\n ↑(List.foldl List.erase l₁ l₂)",
"state_before": "α : Type u_1\nβ : Type ?u.160973\nγ : Type ?u.160976\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ ↑(List.foldl List.erase l₁ l₂) =\n foldl erase (_ : ∀ (s : Multiset α) (a b : α), erase (erase s a) b = erase (erase s b) a) ↑l₁ ↑l₂",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.160973\nγ : Type ?u.160976\ninst✝ : DecidableEq α\ns✝ t✝ u : Multiset α\na b : α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ foldl erase (_ : ∀ (s : Multiset α) (a b : α), erase (erase s a) b = erase (erase s b) a) ↑l₁ ↑l₂ =\n ↑(List.foldl List.erase l₁ l₂)",
"tactic": "exact foldl_hom _ _ _ _ _ fun x y => rfl"
}
] |
[
1673,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1668,
1
] |
Mathlib/Order/SuccPred/LinearLocallyFinite.lean
|
toZ_mono
|
[
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : IsMax i\n⊢ toZ i0 i ≤ toZ i0 j\n\ncase neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "by_cases hi_max : IsMax i"
},
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : IsMin j\n⊢ toZ i0 i ≤ toZ i0 j\n\ncase neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "by_cases hj_min : IsMin j"
},
{
"state_after": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\n⊢ toZ i0 i ≤ toZ i0 j\n\ncase neg.inl.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : j < i0\n⊢ toZ i0 i ≤ toZ i0 j\n\ncase neg.inr.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : i0 ≤ j\n⊢ toZ i0 i ≤ toZ i0 j\n\ncase neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "cases' le_or_lt i0 i with hi hi <;> cases' le_or_lt i0 j with hj hj"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : IsMax i\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "rw [le_antisymm h_le (hi_max h_le)]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : IsMin j\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "rw [le_antisymm h_le (hj_min h_le)]"
},
{
"state_after": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "let m := Nat.find (exists_succ_iterate_of_le h_le)"
},
{
"state_after": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "have hm : (succ^[m]) i = j := Nat.find_spec (exists_succ_iterate_of_le h_le)"
},
{
"state_after": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "have hj_eq : j = (succ^[(toZ i0 i).toNat + m]) i0 := by\n rw [← hm, add_comm]\n nth_rw 1 [← iterate_succ_toZ i hi]\n rw [Function.iterate_add]\n rfl"
},
{
"state_after": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\n⊢ False",
"state_before": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "by_contra h"
},
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False\n\ncase neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ False",
"state_before": "case neg.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\n⊢ False",
"tactic": "by_cases hm0 : m = 0"
},
{
"state_after": "case neg.refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ succ i ≤ j\n\ncase neg.refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ toZ i0 j ≤ toZ i0 i",
"state_before": "case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ False",
"tactic": "refine' hi_max (max_of_succ_le (le_trans _ (@le_of_toZ_le _ _ _ _ _ i0 j i _)))"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ (succ^[m]) i = (succ^[m + Int.toNat (toZ i0 i)]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ j = (succ^[Int.toNat (toZ i0 i) + m]) i0",
"tactic": "rw [← hm, add_comm]"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ (succ^[m]) ((succ^[Int.toNat (toZ i0 i)]) i0) = (succ^[m + Int.toNat (toZ i0 i)]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ (succ^[m]) i = (succ^[m + Int.toNat (toZ i0 i)]) i0",
"tactic": "nth_rw 1 [← iterate_succ_toZ i hi]"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ (succ^[m]) ((succ^[Int.toNat (toZ i0 i)]) i0) = (succ^[m] ∘ succ^[Int.toNat (toZ i0 i)]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ (succ^[m]) ((succ^[Int.toNat (toZ i0 i)]) i0) = (succ^[m + Int.toNat (toZ i0 i)]) i0",
"tactic": "rw [Function.iterate_add]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\n⊢ (succ^[m]) ((succ^[Int.toNat (toZ i0 i)]) i0) = (succ^[m] ∘ succ^[Int.toNat (toZ i0 i)]) i0",
"tactic": "rfl"
},
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"tactic": "rw [hm0, Function.iterate_zero, id.def] at hm"
},
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 j ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"tactic": "rw [hm] at h"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 j ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"tactic": "exact h (le_of_eq rfl)"
},
{
"state_after": "case neg.refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\nh_succ_le : (succ^[Int.toNat (toZ i0 i) + 1]) i0 ≤ j\n⊢ succ i ≤ j",
"state_before": "case neg.refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ succ i ≤ j",
"tactic": "have h_succ_le : (succ^[(toZ i0 i).toNat + 1]) i0 ≤ j := by\n rw [hj_eq]\n refine' Monotone.monotone_iterate_of_le_map succ_mono (le_succ i0) (add_le_add_left _ _)\n exact Nat.one_le_iff_ne_zero.mpr hm0"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\nh_succ_le : (succ^[Int.toNat (toZ i0 i) + 1]) i0 ≤ j\n⊢ succ i ≤ j",
"tactic": "rwa [Function.iterate_succ', Function.comp_apply, iterate_succ_toZ i hi] at h_succ_le"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ (succ^[Int.toNat (toZ i0 i) + 1]) i0 ≤ (succ^[Int.toNat (toZ i0 i) + m]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ (succ^[Int.toNat (toZ i0 i) + 1]) i0 ≤ j",
"tactic": "rw [hj_eq]"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ 1 ≤ m",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ (succ^[Int.toNat (toZ i0 i) + 1]) i0 ≤ (succ^[Int.toNat (toZ i0 i) + m]) i0",
"tactic": "refine' Monotone.monotone_iterate_of_le_map succ_mono (le_succ i0) (add_le_add_left _ _)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ 1 ≤ m",
"tactic": "exact Nat.one_le_iff_ne_zero.mpr hm0"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find (_ : ∃ n, (succ^[n]) i = j)\nhm : (succ^[m]) i = j\nhj_eq : j = (succ^[Int.toNat (toZ i0 i) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ toZ i0 j ≤ toZ i0 i",
"tactic": "exact le_of_not_le h"
},
{
"state_after": "no goals",
"state_before": "case neg.inl.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : j < i0\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "exact absurd h_le (not_le.mpr (hj.trans_le hi))"
},
{
"state_after": "no goals",
"state_before": "case neg.inr.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : i0 ≤ j\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "exact (toZ_neg hi).le.trans (toZ_nonneg hj)"
},
{
"state_after": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "let m := Nat.find (exists_pred_iterate_of_le h_le)"
},
{
"state_after": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "have hm : (pred^[m]) j = i := Nat.find_spec (exists_pred_iterate_of_le h_le)"
},
{
"state_after": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\n⊢ toZ i0 i ≤ toZ i0 j",
"state_before": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "have hj_eq : i = (pred^[(-toZ i0 j).toNat + m]) i0 := by\n rw [← hm, add_comm]\n nth_rw 1 [← iterate_pred_toZ j hj]\n rw [Function.iterate_add]\n rfl"
},
{
"state_after": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\n⊢ False",
"state_before": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\n⊢ toZ i0 i ≤ toZ i0 j",
"tactic": "by_contra h"
},
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False\n\ncase neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ False",
"state_before": "case neg.inr.inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\n⊢ False",
"tactic": "by_cases hm0 : m = 0"
},
{
"state_after": "case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ j ≤ pred j",
"state_before": "case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ False",
"tactic": "refine' hj_min (min_of_le_pred _)"
},
{
"state_after": "case neg.refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ toZ i0 j ≤ toZ i0 i\n\ncase neg.refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ i ≤ pred j",
"state_before": "case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ j ≤ pred j",
"tactic": "refine' (@le_of_toZ_le _ _ _ _ _ i0 j i _).trans _"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ (pred^[m]) j = (pred^[m + Int.toNat (-toZ i0 j)]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ i = (pred^[Int.toNat (-toZ i0 j) + m]) i0",
"tactic": "rw [← hm, add_comm]"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ (pred^[m]) ((pred^[Int.toNat (-toZ i0 j)]) i0) = (pred^[m + Int.toNat (-toZ i0 j)]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ (pred^[m]) j = (pred^[m + Int.toNat (-toZ i0 j)]) i0",
"tactic": "nth_rw 1 [← iterate_pred_toZ j hj]"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ (pred^[m]) ((pred^[Int.toNat (-toZ i0 j)]) i0) = (pred^[m] ∘ pred^[Int.toNat (-toZ i0 j)]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ (pred^[m]) ((pred^[Int.toNat (-toZ i0 j)]) i0) = (pred^[m + Int.toNat (-toZ i0 j)]) i0",
"tactic": "rw [Function.iterate_add]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\n⊢ (pred^[m]) ((pred^[Int.toNat (-toZ i0 j)]) i0) = (pred^[m] ∘ pred^[Int.toNat (-toZ i0 j)]) i0",
"tactic": "rfl"
},
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"tactic": "rw [hm0, Function.iterate_zero, id.def] at hm"
},
{
"state_after": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 i\nhm0 : m = 0\n⊢ False",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : m = 0\n⊢ False",
"tactic": "rw [hm] at h"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 i\nhm0 : m = 0\n⊢ False",
"tactic": "exact h (le_of_eq rfl)"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ toZ i0 j ≤ toZ i0 i",
"tactic": "exact le_of_not_le h"
},
{
"state_after": "case neg.refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\nh_le_pred : i ≤ (pred^[Int.toNat (-toZ i0 j) + 1]) i0\n⊢ i ≤ pred j",
"state_before": "case neg.refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ i ≤ pred j",
"tactic": "have h_le_pred : i ≤ (pred^[(-toZ i0 j).toNat + 1]) i0 := by\n rw [hj_eq]\n refine' Monotone.antitone_iterate_of_map_le pred_mono (pred_le i0) (add_le_add_left _ _)\n exact Nat.one_le_iff_ne_zero.mpr hm0"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_2\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\nh_le_pred : i ≤ (pred^[Int.toNat (-toZ i0 j) + 1]) i0\n⊢ i ≤ pred j",
"tactic": "rwa [Function.iterate_succ', Function.comp_apply, iterate_pred_toZ j hj] at h_le_pred"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ (pred^[Int.toNat (-toZ i0 j) + m]) i0 ≤ (pred^[Int.toNat (-toZ i0 j) + 1]) i0",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ i ≤ (pred^[Int.toNat (-toZ i0 j) + 1]) i0",
"tactic": "rw [hj_eq]"
},
{
"state_after": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ 1 ≤ m",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ (pred^[Int.toNat (-toZ i0 j) + m]) i0 ≤ (pred^[Int.toNat (-toZ i0 j) + 1]) i0",
"tactic": "refine' Monotone.antitone_iterate_of_map_le pred_mono (pred_le i0) (add_le_add_left _ _)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i✝ i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find (_ : ∃ n, (pred^[n]) j = i)\nhm : (pred^[m]) j = i\nhj_eq : i = (pred^[Int.toNat (-toZ i0 j) + m]) i0\nh : ¬toZ i0 i ≤ toZ i0 j\nhm0 : ¬m = 0\n⊢ 1 ≤ m",
"tactic": "exact Nat.one_le_iff_ne_zero.mpr hm0"
}
] |
[
347,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
ContinuousLinearMap.compLeftContinuousBounded_apply
|
[] |
[
1207,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1206,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
LinearMap.rank_finset_sum_le
|
[] |
[
1386,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1383,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
subset_interior_mul
|
[] |
[
1273,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1272,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.map_mul
|
[] |
[
677,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.nodup_toList
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.479712\nγ : Type ?u.479715\ns : Finset α\n⊢ Nodup s.val",
"state_before": "α : Type u_1\nβ : Type ?u.479712\nγ : Type ?u.479715\ns : Finset α\n⊢ List.Nodup (toList s)",
"tactic": "rw [toList, ← Multiset.coe_nodup, Multiset.coe_toList]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.479712\nγ : Type ?u.479715\ns : Finset α\n⊢ Nodup s.val",
"tactic": "exact s.nodup"
}
] |
[
3350,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3348,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
pow_bit1_nonneg_iff
|
[] |
[
721,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
720,
1
] |
Mathlib/RingTheory/Finiteness.lean
|
Submodule.FG.mul
|
[] |
[
452,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Algebra/Hom/Freiman.lean
|
MonoidHom.toFreimanHom_coe
|
[] |
[
476,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.zeroLocus_singleton_one
|
[] |
[
300,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.indep_bot_left
|
[] |
[
169,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.inclusion_injective
|
[] |
[
1046,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1045,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.minimalPeriod_apply_iterate
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.23216\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhx : x ∈ periodicPts f\nn : ℕ\n⊢ IsPeriodicPt f (minimalPeriod f x) ((f^[n]) x)\n\nα : Type u_1\nβ : Type ?u.23216\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhx : x ∈ periodicPts f\nn : ℕ\n⊢ (f^[n]) x ∈ periodicPts f",
"state_before": "α : Type u_1\nβ : Type ?u.23216\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhx : x ∈ periodicPts f\nn : ℕ\n⊢ minimalPeriod f ((f^[n]) x) = minimalPeriod f x",
"tactic": "apply\n (IsPeriodicPt.minimalPeriod_le (minimalPeriod_pos_of_mem_periodicPts hx) _).antisymm\n ((isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate hx\n (isPeriodicPt_minimalPeriod f _)).minimalPeriod_le\n (minimalPeriod_pos_of_mem_periodicPts _))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.23216\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhx : x ∈ periodicPts f\nn : ℕ\n⊢ IsPeriodicPt f (minimalPeriod f x) ((f^[n]) x)",
"tactic": "exact (isPeriodicPt_minimalPeriod f x).apply_iterate n"
},
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type ?u.23216\nf fa : α → α\nfb : β → β\nx y : α\nm✝ n✝ n m : ℕ\nhm : m > 0\nhx : IsPeriodicPt f m x\n⊢ (f^[n]) x ∈ periodicPts f",
"state_before": "α : Type u_1\nβ : Type ?u.23216\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhx : x ∈ periodicPts f\nn : ℕ\n⊢ (f^[n]) x ∈ periodicPts f",
"tactic": "rcases hx with ⟨m, hm, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type ?u.23216\nf fa : α → α\nfb : β → β\nx y : α\nm✝ n✝ n m : ℕ\nhm : m > 0\nhx : IsPeriodicPt f m x\n⊢ (f^[n]) x ∈ periodicPts f",
"tactic": "exact ⟨m, hm, hx.apply_iterate n⟩"
}
] |
[
338,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.frobenius_pow
|
[
{
"state_after": "case a\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nhcard : q = p ^ n\nx : K\n⊢ ↑(frobenius K p ^ n) x = ↑1 x",
"state_before": "K : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nhcard : q = p ^ n\n⊢ frobenius K p ^ n = 1",
"tactic": "ext x"
},
{
"state_after": "case a\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nhcard : q = p ^ n\nx : K\n⊢ ↑(frobenius K p ^ n) x = x ^ p ^ n",
"state_before": "case a\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nhcard : q = p ^ n\nx : K\n⊢ ↑(frobenius K p ^ n) x = ↑1 x",
"tactic": "conv_rhs => rw [RingHom.one_def, RingHom.id_apply, ← pow_card x, hcard]"
},
{
"state_after": "case a\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ ↑(frobenius K p ^ n) x = x ^ p ^ n",
"state_before": "case a\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nhcard : q = p ^ n\nx : K\n⊢ ↑(frobenius K p ^ n) x = x ^ p ^ n",
"tactic": "clear hcard"
},
{
"state_after": "case a.zero\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : K\n⊢ ↑(frobenius K p ^ Nat.zero) x = x ^ p ^ Nat.zero\n\ncase a.succ\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : K\nn : ℕ\nhn : ↑(frobenius K p ^ n) x = x ^ p ^ n\n⊢ ↑(frobenius K p ^ Nat.succ n) x = x ^ p ^ Nat.succ n",
"state_before": "case a\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\n⊢ ↑(frobenius K p ^ n) x = x ^ p ^ n",
"tactic": "induction' n with n hn"
},
{
"state_after": "no goals",
"state_before": "case a.zero\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : K\n⊢ ↑(frobenius K p ^ Nat.zero) x = x ^ p ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case a.succ\nK : Type u_1\nR : Type ?u.742366\ninst✝⁵ : Field K\ninst✝⁴ : Fintype K\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : Algebra (ZMod p✝) K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : K\nn : ℕ\nhn : ↑(frobenius K p ^ n) x = x ^ p ^ n\n⊢ ↑(frobenius K p ^ Nat.succ n) x = x ^ p ^ Nat.succ n",
"tactic": "rw [pow_succ, pow_succ', pow_mul, RingHom.mul_def, RingHom.comp_apply, frobenius_def, hn]"
}
] |
[
290,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Topology/Constructions.lean
|
closure_prod_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.59142\nδ : Type ?u.59145\nε : Type ?u.59148\nζ : Type ?u.59151\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\ns : Set α\nt : Set β\nx✝ : α × β\na : α\nb : β\n⊢ (a, b) ∈ closure (s ×ˢ t) ↔ (a, b) ∈ closure s ×ˢ closure t",
"tactic": "simp_rw [mem_prod, mem_closure_iff_nhdsWithin_neBot, nhdsWithin_prod_eq, prod_neBot]"
}
] |
[
756,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
ContinuousLinearMap.comp_memℒp'
|
[] |
[
963,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.card_Ioo_eq_card_Ico_sub_one
|
[] |
[
235,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.pbind_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.6432\np : α → Prop\nf✝ : (a : α) → p a → β\nx✝ : Option α\nf : α → β\nx : Option α\ng : (b : β) → b ∈ Option.map f x → Option γ\n⊢ pbind (Option.map f x) g = pbind x fun a h => g (f a) (_ : f a ∈ Option.map f x)",
"tactic": "cases x <;> rfl"
}
] |
[
163,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.NumDenSameDeg.deg_pow
|
[] |
[
244,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_inter_mulSupport
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.15893\nι : Type ?u.15896\nM : Type u_2\nN : Type ?u.15902\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g : α → M\na : α\ns : Set α\nf : α → M\n⊢ mulIndicator (s ∩ mulSupport f) f = mulIndicator s f",
"tactic": "rw [← mulIndicator_mulIndicator, mulIndicator_mulSupport]"
}
] |
[
247,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/RingTheory/Algebraic.lean
|
Subalgebra.inv_mem_of_algebraic
|
[
{
"state_after": "case intro.intro\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\nne_zero : p ≠ 0\naeval_eq : ↑(aeval ↑x) p = 0\n⊢ (↑x)⁻¹ ∈ A",
"state_before": "R : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\nhx : _root_.IsAlgebraic K ↑x\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "obtain ⟨p, ne_zero, aeval_eq⟩ := hx"
},
{
"state_after": "case intro.intro\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\nne_zero : p ≠ 0\naeval_eq : ↑(aeval x) p = 0\n⊢ (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\nne_zero : p ≠ 0\naeval_eq : ↑(aeval ↑x) p = 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "rw [Subalgebra.aeval_coe, Subalgebra.coe_eq_zero] at aeval_eq"
},
{
"state_after": "case intro.intro\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\nne_zero : p ≠ 0\naeval_eq : ↑(aeval x) p = 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "revert ne_zero aeval_eq"
},
{
"state_after": "case intro.intro.refine'_1\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ 0 ≠ 0 → ↑(aeval x) 0 = 0 → (↑x)⁻¹ ∈ A\n\ncase intro.intro.refine'_2\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ ∀ (p : K[X]) (a : K),\n coeff p 0 = 0 →\n a ≠ 0 → (p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A) → p + ↑C a ≠ 0 → ↑(aeval x) (p + ↑C a) = 0 → (↑x)⁻¹ ∈ A\n\ncase intro.intro.refine'_3\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ ∀ (p : K[X]), p ≠ 0 → (p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A) → p * X ≠ 0 → ↑(aeval x) (p * X) = 0 → (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A",
"tactic": "refine' p.recOnHorner _ _ _"
},
{
"state_after": "case intro.intro.refine'_1\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\nh : 0 ≠ 0\n⊢ ↑(aeval x) 0 = 0 → (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro.refine'_1\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ 0 ≠ 0 → ↑(aeval x) 0 = 0 → (↑x)⁻¹ ∈ A",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\nh : 0 ≠ 0\n⊢ ↑(aeval x) 0 = 0 → (↑x)⁻¹ ∈ A",
"tactic": "contradiction"
},
{
"state_after": "case intro.intro.refine'_2\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\na : K\nhp : coeff p 0 = 0\nha : a ≠ 0\n_ih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p + ↑C a ≠ 0\naeval_eq : ↑(aeval x) (p + ↑C a) = 0\n⊢ (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro.refine'_2\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ ∀ (p : K[X]) (a : K),\n coeff p 0 = 0 →\n a ≠ 0 → (p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A) → p + ↑C a ≠ 0 → ↑(aeval x) (p + ↑C a) = 0 → (↑x)⁻¹ ∈ A",
"tactic": "intro p a hp ha _ih _ne_zero aeval_eq"
},
{
"state_after": "case intro.intro.refine'_2\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\na : K\nhp : coeff p 0 = 0\nha : a ≠ 0\n_ih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p + ↑C a ≠ 0\naeval_eq : ↑(aeval x) (p + ↑C a) = 0\n⊢ coeff (p + ↑C a) 0 ≠ 0",
"state_before": "case intro.intro.refine'_2\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\na : K\nhp : coeff p 0 = 0\nha : a ≠ 0\n_ih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p + ↑C a ≠ 0\naeval_eq : ↑(aeval x) (p + ↑C a) = 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "refine' A.inv_mem_of_root_of_coeff_zero_ne_zero aeval_eq _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\na : K\nhp : coeff p 0 = 0\nha : a ≠ 0\n_ih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p + ↑C a ≠ 0\naeval_eq : ↑(aeval x) (p + ↑C a) = 0\n⊢ coeff (p + ↑C a) 0 ≠ 0",
"tactic": "rwa [coeff_add, hp, zero_add, coeff_C, if_pos rfl]"
},
{
"state_after": "case intro.intro.refine'_3\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\naeval_eq : ↑(aeval x) (p * X) = 0\n⊢ (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro.refine'_3\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\n⊢ ∀ (p : K[X]), p ≠ 0 → (p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A) → p * X ≠ 0 → ↑(aeval x) (p * X) = 0 → (↑x)⁻¹ ∈ A",
"tactic": "intro p hp ih _ne_zero aeval_eq"
},
{
"state_after": "case intro.intro.refine'_3\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\naeval_eq : ↑(aeval x) p = 0 ∨ x = 0\n⊢ (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro.refine'_3\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\naeval_eq : ↑(aeval x) (p * X) = 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "rw [AlgHom.map_mul, aeval_X, mul_eq_zero] at aeval_eq"
},
{
"state_after": "case intro.intro.refine'_3.inl\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\naeval_eq : ↑(aeval x) p = 0\n⊢ (↑x)⁻¹ ∈ A\n\ncase intro.intro.refine'_3.inr\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\nx_eq : x = 0\n⊢ (↑x)⁻¹ ∈ A",
"state_before": "case intro.intro.refine'_3\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\naeval_eq : ↑(aeval x) p = 0 ∨ x = 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "cases' aeval_eq with aeval_eq x_eq"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_3.inl\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\naeval_eq : ↑(aeval x) p = 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "exact ih hp aeval_eq"
},
{
"state_after": "case intro.intro.refine'_3.inr\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\nx_eq : x = 0\n⊢ 0 ∈ A",
"state_before": "case intro.intro.refine'_3.inr\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\nx_eq : x = 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "rw [x_eq, Subalgebra.coe_zero, inv_zero]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_3.inr\nR : Type ?u.864898\nS : Type ?u.864901\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np✝ p : K[X]\nhp : p ≠ 0\nih : p ≠ 0 → ↑(aeval x) p = 0 → (↑x)⁻¹ ∈ A\n_ne_zero : p * X ≠ 0\nx_eq : x = 0\n⊢ 0 ∈ A",
"tactic": "exact A.zero_mem"
}
] |
[
391,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.one_le_ceil_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.209524\nα : Type u_1\nβ : Type ?u.209530\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ 1 ≤ ⌈a⌉ ↔ 0 < a",
"tactic": "rw [← zero_add (1 : ℤ), add_one_le_ceil_iff, cast_zero]"
}
] |
[
1117,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Combinatorics/SetFamily/Shadow.lean
|
Finset.erase_mem_shadow
|
[] |
[
97,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.right_unique_forall₂'
|
[] |
[
151,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.log_one_right
|
[] |
[
87,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
StrictMono.mul_monotone
|
[] |
[
607,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
605,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.mul_mem_right
|
[] |
[
566,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/NumberTheory/PythagoreanTriples.lean
|
PythagoreanTriple.isPrimitiveClassified_of_coprime
|
[
{
"state_after": "case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : 0 < z\n⊢ IsPrimitiveClassified h\n\ncase neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\n⊢ IsPrimitiveClassified h",
"state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n⊢ IsPrimitiveClassified h",
"tactic": "by_cases hz : 0 < z"
},
{
"state_after": "case neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\nh' : PythagoreanTriple x y (-z)\n⊢ IsPrimitiveClassified h",
"state_before": "case neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\n⊢ IsPrimitiveClassified h",
"tactic": "have h' : PythagoreanTriple x y (-z) := by simpa [PythagoreanTriple, neg_mul_neg] using h.eq"
},
{
"state_after": "case neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\nh' : PythagoreanTriple x y (-z)\n⊢ 0 < -z",
"state_before": "case neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\nh' : PythagoreanTriple x y (-z)\n⊢ IsPrimitiveClassified h",
"tactic": "apply h'.isPrimitiveClassified_of_coprime_of_pos hc"
},
{
"state_after": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\nh' : PythagoreanTriple x y (-z)\n⊢ 0 ≤ -z",
"state_before": "case neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\nh' : PythagoreanTriple x y (-z)\n⊢ 0 < -z",
"tactic": "apply lt_of_le_of_ne _ (h'.ne_zero_of_coprime hc).symm"
},
{
"state_after": "no goals",
"state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\nh' : PythagoreanTriple x y (-z)\n⊢ 0 ≤ -z",
"tactic": "exact le_neg.mp (not_lt.mp hz)"
},
{
"state_after": "no goals",
"state_before": "case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : 0 < z\n⊢ IsPrimitiveClassified h",
"tactic": "exact h.isPrimitiveClassified_of_coprime_of_pos hc hz"
},
{
"state_after": "no goals",
"state_before": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhz : ¬0 < z\n⊢ PythagoreanTriple x y (-z)",
"tactic": "simpa [PythagoreanTriple, neg_mul_neg] using h.eq"
}
] |
[
578,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mem_carrier
|
[] |
[
212,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.le_piecewise_of_le_of_le
|
[] |
[
2564,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2562,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
right_vsub_midpoint
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nV : Type u_1\nV' : Type ?u.52855\nP : Type u_2\nP' : Type ?u.52861\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z p₁ p₂ : P\n⊢ p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ p₁)",
"tactic": "rw [midpoint_comm, left_vsub_midpoint]"
}
] |
[
124,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_le_logb
|
[
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nhb : 1 < b\nh : 0 < x\nh₁ : 0 < y\n⊢ logb b x ≤ logb b y ↔ x ≤ y",
"tactic": "rw [logb, logb, div_le_div_right (log_pos hb), log_le_log h h₁]"
}
] |
[
149,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
sub_div
|
[] |
[
173,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Analysis/Complex/RemovableSingularity.lean
|
Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable
|
[
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀",
"tactic": "have hf' : DifferentiableOn ℂ (dslope f w₀) U :=\n (differentiableOn_dslope (hU.mem_nhds ((ball_subset_closedBall.trans hc) hw₀))).mpr hf"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀",
"tactic": "have h0 := (hf'.diffContOnCl_ball hc).two_pi_i_inv_smul_circleIntegral_sub_inv_smul hw₀"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) =\n (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀",
"tactic": "rw [← dslope_same, ← h0]"
},
{
"state_after": "case e_a\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) =\n (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z",
"tactic": "congr 1"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)\n\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)) = ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z",
"state_before": "case e_a\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z",
"tactic": "trans ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"tactic": "have h1 : ContinuousOn (fun z : ℂ => ((z - w₀) ^ 2)⁻¹) (sphere c R) := by\n refine' ((continuous_id'.sub continuous_const).pow 2).continuousOn.inv₀ fun w hw h => _\n exact sphere_disjoint_ball.ne_of_mem hw hw₀ (sub_eq_zero.mp (sq_eq_zero_iff.mp h))"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"tactic": "have h2 : CircleIntegrable (fun z : ℂ => ((z - w₀) ^ 2)⁻¹ • f z) c R := by\n refine' ContinuousOn.circleIntegrable (pos_of_mem_ball hw₀).le _\n exact h1.smul (hf.continuousOn.mono (sphere_subset_closedBall.trans hc))"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\nh3 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f w₀) c R\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"tactic": "have h3 : CircleIntegrable (fun z : ℂ => ((z - w₀) ^ 2)⁻¹ • f w₀) c R :=\n ContinuousOn.circleIntegrable (pos_of_mem_ball hw₀).le (h1.smul continuousOn_const)"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\nh3 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f w₀) c R\nh4 : (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹) = 0\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\nh3 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f w₀) c R\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"tactic": "have h4 : (∮ z : ℂ in C(c, R), ((z - w₀) ^ 2)⁻¹) = 0 := by\n simpa using circleIntegral.integral_sub_zpow_of_ne (by decide : (-2 : ℤ) ≠ -1) c w₀ R"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\nh3 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f w₀) c R\nh4 : (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹) = 0\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = ∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)",
"tactic": "simp only [smul_sub, circleIntegral.integral_sub h2 h3, h4, circleIntegral.integral_smul_const,\n zero_smul, sub_zero]"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nw : ℂ\nhw : w ∈ sphere c R\nh : (w - w₀) ^ 2 = 0\n⊢ False",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)",
"tactic": "refine' ((continuous_id'.sub continuous_const).pow 2).continuousOn.inv₀ fun w hw h => _"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nw : ℂ\nhw : w ∈ sphere c R\nh : (w - w₀) ^ 2 = 0\n⊢ False",
"tactic": "exact sphere_disjoint_ball.ne_of_mem hw hw₀ (sub_eq_zero.mp (sq_eq_zero_iff.mp h))"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\n⊢ ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹ • f z) (sphere c R)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\n⊢ CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R",
"tactic": "refine' ContinuousOn.circleIntegrable (pos_of_mem_ball hw₀).le _"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\n⊢ ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹ • f z) (sphere c R)",
"tactic": "exact h1.smul (hf.continuousOn.mono (sphere_subset_closedBall.trans hc))"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\nh3 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f w₀) c R\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹) = 0",
"tactic": "simpa using circleIntegral.integral_sub_zpow_of_ne (by decide : (-2 : ℤ) ≠ -1) c w₀ R"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nh1 : ContinuousOn (fun z => ((z - w₀) ^ 2)⁻¹) (sphere c R)\nh2 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f z) c R\nh3 : CircleIntegrable (fun z => ((z - w₀) ^ 2)⁻¹ • f w₀) c R\n⊢ -2 ≠ -1",
"tactic": "decide"
},
{
"state_after": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nz : ℂ\nhz : z ∈ sphere c R\n⊢ ((z - w₀) ^ 2)⁻¹ • (f z - f w₀) = (z - w₀)⁻¹ • dslope f w₀ z",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\n⊢ (∮ (z : ℂ) in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)) = ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z",
"tactic": "refine' circleIntegral.integral_congr (pos_of_mem_ball hw₀).le fun z hz => _"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w₀ z) = dslope f w₀ w₀\nz : ℂ\nhz : z ∈ sphere c R\n⊢ ((z - w₀) ^ 2)⁻¹ • (f z - f w₀) = (z - w₀)⁻¹ • dslope f w₀ z",
"tactic": "simp only [dslope_of_ne, Metric.sphere_disjoint_ball.ne_of_mem hz hw₀, slope, ← smul_assoc, sq,\n mul_inv, Ne.def, not_false_iff, vsub_eq_sub, Algebra.id.smul_eq_mul]"
}
] |
[
166,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/LinearAlgebra/StdBasis.lean
|
LinearMap.coe_stdBasis
|
[] |
[
66,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.rel_of_LiftRel
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nl : ∀ {a : α}, a ∈ ca → ∃ b, b ∈ cb ∧ R a b\nright✝ : ∀ {b : β}, b ∈ cb → ∃ a, a ∈ ca ∧ R a b\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\nb' : β\nmb' : b' ∈ cb\nab' : R a b'\n⊢ R a b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nl : ∀ {a : α}, a ∈ ca → ∃ b, b ∈ cb ∧ R a b\nright✝ : ∀ {b : β}, b ∈ cb → ∃ a, a ∈ ca ∧ R a b\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\n⊢ R a b",
"tactic": "let ⟨b', mb', ab'⟩ := l ma"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nl : ∀ {a : α}, a ∈ ca → ∃ b, b ∈ cb ∧ R a b\nright✝ : ∀ {b : β}, b ∈ cb → ∃ a, a ∈ ca ∧ R a b\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\nb' : β\nmb' : b' ∈ cb\nab' : R a b'\n⊢ R a b'",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nl : ∀ {a : α}, a ∈ ca → ∃ b, b ∈ cb ∧ R a b\nright✝ : ∀ {b : β}, b ∈ cb → ∃ a, a ∈ ca ∧ R a b\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\nb' : β\nmb' : b' ∈ cb\nab' : R a b'\n⊢ R a b",
"tactic": "rw [mem_unique mb mb']"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\nl : ∀ {a : α}, a ∈ ca → ∃ b, b ∈ cb ∧ R a b\nright✝ : ∀ {b : β}, b ∈ cb → ∃ a, a ∈ ca ∧ R a b\na : α\nb : β\nma : a ∈ ca\nmb : b ∈ cb\nb' : β\nmb' : b' ∈ cb\nab' : R a b'\n⊢ R a b'",
"tactic": "exact ab'"
}
] |
[
1126,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1122,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.ndinsert_of_mem
|
[] |
[
49,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.eq_top_iff'
|
[] |
[
709,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean
|
CauSeq.Completion.mul_inv_cancel
|
[
{
"state_after": "α : Type u_1\ninst✝² : LinearOrderedField α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf : CauSeq β abv\nhf : ¬LimZero f\n⊢ Quotient.mk equiv f * (Quotient.mk equiv f)⁻¹ = 1",
"state_before": "α : Type u_1\ninst✝² : LinearOrderedField α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf : CauSeq β abv\nhf : Quotient.mk equiv f ≠ 0\n⊢ Quotient.mk equiv f * (Quotient.mk equiv f)⁻¹ = 1",
"tactic": "simp only [mk_eq_mk, ne_eq, mk_eq_zero] at hf"
},
{
"state_after": "α : Type u_1\ninst✝² : LinearOrderedField α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf : CauSeq β abv\nhf : ¬LimZero f\n⊢ mk (f * inv f (_ : ¬LimZero f)) = 1",
"state_before": "α : Type u_1\ninst✝² : LinearOrderedField α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf : CauSeq β abv\nhf : ¬LimZero f\n⊢ Quotient.mk equiv f * (Quotient.mk equiv f)⁻¹ = 1",
"tactic": "simp [hf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : LinearOrderedField α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf : CauSeq β abv\nhf : ¬LimZero f\n⊢ mk (f * inv f (_ : ¬LimZero f)) = 1",
"tactic": "exact Quotient.sound (CauSeq.mul_inv_cancel hf)"
}
] |
[
265,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
11
] |
Mathlib/Algebra/Hom/Freiman.lean
|
FreimanHom.cancel_right_on
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.51726\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.51738\nG : Type ?u.51741\ninst✝⁵ : FunLike F α fun x => β\ninst✝⁴ : CommMonoid α\ninst✝³ : CommMonoid β\ninst✝² : CommMonoid γ\ninst✝¹ : CommMonoid δ\ninst✝ : CommGroup G\nA : Set α\nB : Set β\nC : Set γ\nn : ℕ\na b c d : α\ng₁ g₂ : B →*[n] γ\nf : A →*[n] β\nhf : Set.SurjOn (↑f) A B\nhf' : Set.MapsTo (↑f) A B\n⊢ Set.EqOn (↑(FreimanHom.comp g₁ f hf')) (↑(FreimanHom.comp g₂ f hf')) A ↔ Set.EqOn (↑g₁) (↑g₂) B",
"tactic": "simp [hf.cancel_right hf']"
}
] |
[
267,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.countp_True
|
[] |
[
2282,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2281,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.orthogonalProjectionFn_mem_orthogonal
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑s ∩ ↑(mk' p (direction s)ᗮ) ⊆ ↑(mk' p (direction s)ᗮ)",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ orthogonalProjectionFn s p ∈ mk' p (direction s)ᗮ",
"tactic": "rw [← mem_coe, ← Set.singleton_subset_iff, ← inter_eq_singleton_orthogonalProjectionFn]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑s ∩ ↑(mk' p (direction s)ᗮ) ⊆ ↑(mk' p (direction s)ᗮ)",
"tactic": "exact Set.inter_subset_right _ _"
}
] |
[
276,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Topology/Support.lean
|
hasCompactMulSupport_def
|
[
{
"state_after": "no goals",
"state_before": "X : Type ?u.8941\nα : Type u_1\nα' : Type ?u.8947\nβ : Type u_2\nγ : Type ?u.8953\nδ : Type ?u.8956\nM : Type ?u.8959\nE : Type ?u.8962\nR : Type ?u.8965\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : One β\ninst✝¹ : One γ\ninst✝ : One δ\ng : β → γ\nf : α → β\nf₂ : α → γ\nm : β → γ → δ\nx : α\n⊢ HasCompactMulSupport f ↔ IsCompact (closure (mulSupport f))",
"tactic": "rfl"
}
] |
[
143,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Ioc.mk_one
|
[] |
[
263,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.isFractional_of_le_one
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\n⊢ ∀ (b : P), b ∈ I → IsInteger R (1 • b)",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\n⊢ IsFractional S I",
"tactic": "use 1, S.one_mem"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\nb : P\nhb : b ∈ I\n⊢ IsInteger R (1 • b)",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\n⊢ ∀ (b : P), b ∈ I → IsInteger R (1 • b)",
"tactic": "intro b hb"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\nb : P\nhb : b ∈ I\n⊢ IsInteger R b",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\nb : P\nhb : b ∈ I\n⊢ IsInteger R (1 • b)",
"tactic": "rw [one_smul]"
},
{
"state_after": "case intro.refl\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\nb' : R\nhb : ↑(Algebra.linearMap R P) b' ∈ I\n⊢ IsInteger R (↑(Algebra.linearMap R P) b')",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\nb : P\nhb : b ∈ I\n⊢ IsInteger R b",
"tactic": "obtain ⟨b', b'_mem, rfl⟩ := h hb"
},
{
"state_after": "no goals",
"state_before": "case intro.refl\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : Submodule R P\nh : I ≤ 1\nb' : R\nhb : ↑(Algebra.linearMap R P) b' ∈ I\n⊢ IsInteger R (↑(Algebra.linearMap R P) b')",
"tactic": "exact Set.mem_range_self b'"
}
] |
[
217,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.normSq_one
|
[] |
[
617,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.map_C
|
[] |
[
698,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
697,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuous_if_const
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.383429\nδ : Type ?u.383432\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : Prop\nf g : α → β\ninst✝ : Decidable p\nhf : p → Continuous f\nhg : ¬p → Continuous g\nh : p\n⊢ Continuous fun a => f a\n\ncase inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.383429\nδ : Type ?u.383432\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : Prop\nf g : α → β\ninst✝ : Decidable p\nhf : p → Continuous f\nhg : ¬p → Continuous g\nh : ¬p\n⊢ Continuous fun a => g a",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.383429\nδ : Type ?u.383432\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : Prop\nf g : α → β\ninst✝ : Decidable p\nhf : p → Continuous f\nhg : ¬p → Continuous g\n⊢ Continuous fun a => if p then f a else g a",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.383429\nδ : Type ?u.383432\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : Prop\nf g : α → β\ninst✝ : Decidable p\nhf : p → Continuous f\nhg : ¬p → Continuous g\nh : p\n⊢ Continuous fun a => f a\n\ncase inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.383429\nδ : Type ?u.383432\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\np : Prop\nf g : α → β\ninst✝ : Decidable p\nhf : p → Continuous f\nhg : ¬p → Continuous g\nh : ¬p\n⊢ Continuous fun a => g a",
"tactic": "exacts [hf h, hg h]"
}
] |
[
1194,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1191,
1
] |
Mathlib/Analysis/Convex/Strict.lean
|
StrictConvex.linear_image
|
[
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_4\n𝕝 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.14448\ninst✝¹⁰ : OrderedSemiring 𝕜\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜 F\ns : Set E\ninst✝³ : Semiring 𝕝\ninst✝² : Module 𝕝 E\ninst✝¹ : Module 𝕝 F\ninst✝ : LinearMap.CompatibleSMul E F 𝕜 𝕝\nhs : StrictConvex 𝕜 s\nf : E →ₗ[𝕝] F\nhf : IsOpenMap ↑f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : ↑f x ≠ ↑f y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • ↑f x + b • ↑f y ∈ interior (↑f '' s)",
"state_before": "𝕜 : Type u_4\n𝕝 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.14448\ninst✝¹⁰ : OrderedSemiring 𝕜\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜 F\ns : Set E\ninst✝³ : Semiring 𝕝\ninst✝² : Module 𝕝 E\ninst✝¹ : Module 𝕝 F\ninst✝ : LinearMap.CompatibleSMul E F 𝕜 𝕝\nhs : StrictConvex 𝕜 s\nf : E →ₗ[𝕝] F\nhf : IsOpenMap ↑f\n⊢ StrictConvex 𝕜 (↑f '' s)",
"tactic": "rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ hxy a b ha hb hab"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_4\n𝕝 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.14448\ninst✝¹⁰ : OrderedSemiring 𝕜\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜 F\ns : Set E\ninst✝³ : Semiring 𝕝\ninst✝² : Module 𝕝 E\ninst✝¹ : Module 𝕝 F\ninst✝ : LinearMap.CompatibleSMul E F 𝕜 𝕝\nhs : StrictConvex 𝕜 s\nf : E →ₗ[𝕝] F\nhf : IsOpenMap ↑f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : ↑f x ≠ ↑f y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ ↑f (a • x + b • y) = a • ↑f x + b • ↑f y",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_4\n𝕝 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.14448\ninst✝¹⁰ : OrderedSemiring 𝕜\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜 F\ns : Set E\ninst✝³ : Semiring 𝕝\ninst✝² : Module 𝕝 E\ninst✝¹ : Module 𝕝 F\ninst✝ : LinearMap.CompatibleSMul E F 𝕜 𝕝\nhs : StrictConvex 𝕜 s\nf : E →ₗ[𝕝] F\nhf : IsOpenMap ↑f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : ↑f x ≠ ↑f y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a • ↑f x + b • ↑f y ∈ interior (↑f '' s)",
"tactic": "refine' hf.image_interior_subset _ ⟨a • x + b • y, hs hx hy (ne_of_apply_ne _ hxy) ha hb hab, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_4\n𝕝 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.14448\ninst✝¹⁰ : OrderedSemiring 𝕜\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜 F\ns : Set E\ninst✝³ : Semiring 𝕝\ninst✝² : Module 𝕝 E\ninst✝¹ : Module 𝕝 F\ninst✝ : LinearMap.CompatibleSMul E F 𝕜 𝕝\nhs : StrictConvex 𝕜 s\nf : E →ₗ[𝕝] F\nhf : IsOpenMap ↑f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\nhxy : ↑f x ≠ ↑f y\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ ↑f (a • x + b • y) = a • ↑f x + b • ↑f y",
"tactic": "rw [map_add, f.map_smul_of_tower a, f.map_smul_of_tower b]"
}
] |
[
136,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.sdiff_subset_sdiff
|
[] |
[
2111,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2110,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
Balanced.smul_eq
|
[] |
[
298,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
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