module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Order.Bornology | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 38
} | {
"line": 87,
"column": 39
} | [
{
"pp": "α : Type u_1\ns✝ t : Set α\ninst✝² : Bornology α\ninst✝¹ : Preorder α\ninst✝ : IsOrderBornology α\ns : Set αᵒᵈ\n⊢ IsBounded (⇑toDual ⁻¹' s) ↔ BddAbove (⇑toDual ⁻¹' s) ∧ BddBelow (⇑toDual ⁻¹' s)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr... | [
"α : Type u_1\ns✝ t : Set α\ninst✝² : Bornology α\ninst✝¹ : Preorder α\ninst✝ : IsOrderBornology α\ns : Set αᵒᵈ\n⊢ BddBelow (⇑toDual ⁻¹' s) ∧ BddAbove (⇑toDual ⁻¹' s) ↔ BddAbove (⇑toDual ⁻¹' s) ∧ BddBelow (⇑toDual ⁻¹' s)"
] | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 38
} | {
"line": 93,
"column": 39
} | [
{
"pp": "α : Type u_1\ns✝ t : Set α\ninst✝⁵ : Bornology α\ninst✝⁴ : Preorder α\ninst✝³ : IsOrderBornology α\nβ : Type u_2\ninst✝² : Preorder β\ninst✝¹ : Bornology β\ninst✝ : IsOrderBornology β\ns : Set (α × β)\n⊢ IsBounded (fst '' s) ∧ IsBounded (snd '' s) ↔\n (BddBelow (fst '' s) ∧ BddAbove (fst '' s)) ∧ Bd... | [
"α : Type u_1\ns✝ t : Set α\ninst✝⁵ : Bornology α\ninst✝⁴ : Preorder α\ninst✝³ : IsOrderBornology α\nβ : Type u_2\ninst✝² : Preorder β\ninst✝¹ : Bornology β\ninst✝ : IsOrderBornology β\ns : Set (α × β)\n⊢ (BddBelow (fst '' s) ∧ BddAbove (fst '' s)) ∧ IsBounded (snd '' s) ↔\n (BddBelow (fst '' s) ∧ BddAbove (fst ... | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 109,
"column": 40
} | {
"line": 109,
"column": 72
} | {
"line": 109,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : Bornology α\ninst✝³ : Nonempty α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderBornology α\ninst✝ : NoMaxOrder α\ns : Set α\n⊢ IsBounded sᶜ → sᶜᶜ ∈ Filter.atTop",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"... | [
"α : Type u_1\ninst✝⁴ : Bornology α\ninst✝³ : Nonempty α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderBornology α\ninst✝ : NoMaxOrder α\ns : Set α\n⊢ BddBelow sᶜ ∧ BddAbove sᶜ → sᶜᶜ ∈ Filter.atTop"
] | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 124,
"column": 38
} | {
"line": 124,
"column": 70
} | {
"line": 124,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝³ : Bornology α\ninst✝² : Nonempty α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderBornology α\ns : Set α\n⊢ ((∃ i, True ∧ Iic i ⊆ sᶜᶜ) ∧ ∃ i, True ∧ Ici i ⊆ sᶜᶜ) → IsBounded sᶜ",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"co... | [
"α : Type u_1\ninst✝³ : Bornology α\ninst✝² : Nonempty α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderBornology α\ns : Set α\n⊢ ((∃ i, True ∧ Iic i ⊆ sᶜᶜ) ∧ ∃ i, True ∧ Ici i ⊆ sᶜᶜ) → BddBelow sᶜ ∧ BddAbove sᶜ"
] | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 140,
"column": 38
} | {
"line": 140,
"column": 70
} | {
"line": 140,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Bornology α\ninst✝⁴ : Nonempty α\ninst✝³ : LinearOrder α\ninst✝² : IsOrderBornology α\ninst✝¹ : NoMaxOrder α\ninst✝ : OrderBot α\ns : Set α\n⊢ (∃ i, True ∧ Ici i ⊆ sᶜᶜ) → IsBounded sᶜ",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set... | [
"α : Type u_1\ninst✝⁵ : Bornology α\ninst✝⁴ : Nonempty α\ninst✝³ : LinearOrder α\ninst✝² : IsOrderBornology α\ninst✝¹ : NoMaxOrder α\ninst✝ : OrderBot α\ns : Set α\n⊢ (∃ i, True ∧ Ici i ⊆ sᶜᶜ) → BddBelow sᶜ ∧ BddAbove sᶜ"
] | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Monotone | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | {
"line": 93,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\nf : α → β\ninst✝ : SecondCountableTopology α\nhf : MonotoneOn f univ\n⊢ {c | ∃ x y, x < y ∧ f x = c ∧ f y = c}.Countable",
"ppTerm": "?m.32",
"assigned": false,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\nf : α → β\ninst✝ : SecondCountableTopology α\nhf : MonotoneOn f univ\n⊢ {c | ∃ x y, x < y ∧ f x = c ∧ f y = c}.Countable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 39
} | {
"line": 154,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : SecondCountableTopology β\nhf : Monotone f\n⊢ {x | ¬ContinuousAt f x}.Countable",
"ppTerm": "?... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : SecondCountableTopology β\nhf : Monotone f\n⊢ {x | ¬ContinuousAt f x}.Countable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 435,
"column": 6
} | {
"line": 435,
"column": 73
} | {
"line": 436,
"column": 8
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OrderTopology α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nx y : α\nh_nonempty : (Ioo y x).Nonempty\nMf : MonotoneOn f (Ioo y x)\nh_bdd : BddAbov... | [
"α : Type u_3\nβ : Type u_4\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OrderTopology α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nx y : α\nh_nonempty : (Ioo y x).Nonempty\nMf : MonotoneOn f (Ioo y x)\nh_bdd : BddAbove (f '' Ioo ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 453,
"column": 6
} | {
"line": 453,
"column": 68
} | {
"line": 454,
"column": 8
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OrderTopology α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nx y : α\nh_nonempty : (Ioo x y).Nonempty\nMf : MonotoneOn f (Ioo x y)\nh_bdd : BddBelo... | [
"α : Type u_3\nβ : Type u_4\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OrderTopology α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nx y : α\nh_nonempty : (Ioo x y).Nonempty\nMf : MonotoneOn f (Ioo x y)\nh_bdd : BddBelow (f '' Ioo ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 49,
"column": 19
} | {
"line": 49,
"column": 47
} | {
"line": 49,
"column": 48
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nr : ℝ\nhpos : 0 < r\nhr : ∀ (x : E), x ≠ 1 → r ≤ ‖x‖\nx y : E\nhne : x ≠ y\n⊢ x⁻¹ * y ≠ 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nr : ℝ\nhpos : 0 < r\nhr : ∀ (x : E), x ≠ 1 → r ≤ ‖x‖\nx y : E\nhne : x ≠ y\n⊢ ¬x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 70,
"column": 46
} | {
"line": 70,
"column": 57
} | {
"line": 70,
"column": 58
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ ‖a⁻¹‖ = ‖a‖",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ ‖a⁻¹‖ = ‖a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 36
} | {
"line": 99,
"column": 37
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a * b‖ ≤ ‖a‖ + ‖b‖",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a * b‖ ≤ ‖a‖ + ‖b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 13
} | {
"line": 117,
"column": 14
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b c : E\n⊢ ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b c : E\n⊢ ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 28
} | {
"line": 121,
"column": 29
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a‖ ≤ ‖a / b‖ + ‖b‖",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a‖ ≤ ‖a / b‖ + ‖b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 30
} | {
"line": 153,
"column": 31
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a / b‖ ≤ ‖a‖ + ‖b‖",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClas... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a * b⁻¹‖ ≤ ‖a‖ + ‖b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 36
} | {
"line": 163,
"column": 37
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ dist a b ≤ ‖a‖ + ‖b‖",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a⁻¹ * b‖ ≤ ‖a‖ + ‖b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 36
} | {
"line": 167,
"column": 37
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ |‖a‖ - ‖b‖| ≤ ‖a⁻¹ * b‖",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ |‖a‖ - ‖b‖| ≤ ‖a⁻¹ * b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 13
} | {
"line": 175,
"column": 14
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a‖ - ‖b‖ ≤ ‖a * b‖",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"Real.instSub",
"covariant_swa... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a‖ ≤ ‖a * b‖ + ‖b‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 19
} | {
"line": 218,
"column": 20
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖x‖ = 0\n⊢ ‖x * y‖ ≤ ‖y‖",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖x‖ = 0\n⊢ ‖x * y‖ ≤ ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 19
} | {
"line": 219,
"column": 20
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖x‖ = 0\n⊢ ‖y‖ ≤ ‖x * y‖",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖x‖ = 0\n⊢ ‖y‖ ≤ ‖x * y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 19
} | {
"line": 224,
"column": 20
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x * y‖ ≤ ‖x‖",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x * y‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 19
} | {
"line": 225,
"column": 20
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 44
} | {
"line": 227,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"Monoid.toMulOneClass",
... | [] | simpa [h] using norm_le_mul_norm_add x y | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 44
} | {
"line": 227,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"Monoid.toMulOneClass",
... | [] | simpa [h] using norm_le_mul_norm_add x y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 44
} | {
"line": 227,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"Monoid.toMulOneClass",
... | [] | simpa [h] using norm_le_mul_norm_add x y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 13
} | {
"line": 281,
"column": 14
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nu v w : E\n⊢ ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"Real.instSub",
"covariant_swap_add_of_covariant_ad... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nu v w : E\n⊢ ‖u / w‖ ≤ ‖u / v‖ + ‖v / w‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 286,
"column": 2
} | {
"line": 287,
"column": 9
} | {
"line": 287,
"column": 10
} | [
{
"pp": "E : Type u_8\ninst✝ : SeminormedGroup E\nu v : E\n⊢ ‖u * v‖ - ‖u / v‖ ≤ 2 * ‖v‖",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg"... | [
"E : Type u_8\ninst✝ : SeminormedGroup E\nu v : E\n⊢ ‖u * v‖ ≤ ‖u / v‖ + (‖v‖ + ‖v‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 446,
"column": 16
} | {
"line": 446,
"column": 47
} | {
"line": 446,
"column": 48
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\nn : ℕ\n⊢ ‖a ^ (n + 1)‖ ≤ ↑(n + 1) * ‖a‖",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"add_mul",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na : E\nn : ℕ\n⊢ ‖a ^ n * a‖ ≤ ↑n * ‖a‖ + ‖a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 40
} | {
"line": 487,
"column": 41
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a / b‖ₑ ≤ ‖a‖ₑ + ‖b‖ₑ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"instHDiv",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup",
"PartialOrder.toPreorder",
"Pre... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a / b‖₊ ≤ ‖a‖₊ + ‖b‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 546,
"column": 63
} | {
"line": 546,
"column": 74
} | {
"line": 546,
"column": 75
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx✝ : ∃ x, ‖x‖₊ ≠ 0\nx : E\nhx : ‖x‖₊ ≠ 0\n⊢ ¬‖x⁻¹ * 1‖₊ = 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nx✝ : ∃ x, ‖x‖₊ ≠ 0\nx : E\nhx : ‖x‖₊ ≠ 0\n⊢ ¬‖x‖₊ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 13
} | {
"line": 551,
"column": 14
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\n⊢ IndiscreteTopology E ↔ ∀ (x : E), ‖x‖₊ = 0",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\n⊢ IndiscreteTopology E ↔ ∀ (x : E), ‖x‖₊ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 584,
"column": 2
} | {
"line": 584,
"column": 13
} | {
"line": 584,
"column": 14
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\n⊢ IndiscreteTopology E ↔ ∀ (x : E), ‖x‖ = 0",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedGroup E\n⊢ IndiscreteTopology E ↔ ∀ (x : E), ‖x‖ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 700,
"column": 2
} | {
"line": 700,
"column": 13
} | {
"line": 700,
"column": 14
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nf : Filter E\n⊢ Disjoint (𝓝 1) f ↔ ∃ δ > 0, ∀ᶠ (y : E) in f, δ ≤ ‖y‖",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"InvOneClass.toOne",
"DivInvOneMonoid.to... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\nf : Filter E\n⊢ Disjoint (𝓝 1) f ↔ ∃ δ, 0 < δ ∧ ∀ᶠ (y : E) in f, δ ≤ ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 47
} | {
"line": 773,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\n⊢ ‖a⁻¹ * b‖ = ‖a / b‖",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toI... | [] | rw [← dist_eq_norm_inv_mul, dist_eq_norm_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 47
} | {
"line": 773,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\n⊢ ‖a⁻¹ * b‖ = ‖a / b‖",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toI... | [] | rw [← dist_eq_norm_inv_mul, dist_eq_norm_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 47
} | {
"line": 773,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\n⊢ ‖a⁻¹ * b‖ = ‖a / b‖",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toI... | [] | rw [← dist_eq_norm_inv_mul, dist_eq_norm_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 747,
"column": 18
} | {
"line": 747,
"column": 29
} | {
"line": 747,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J J' : FractionalIdeal R⁰ K\nh : J ≤ I\nhJ' : J' ≠ 0\nhI : I ≠ 0\nH : I * J' = 0 * J\nh' : J' ≤ 0\nthis : (J' ⊓ spanSingleton R⁰ (... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J J' : FractionalIdeal R⁰ K\nh : J ≤ I\nhJ' : J' ≠ 0\nhI : I ≠ 0\nH : I * J' = 0 * J\nh' : J' ≤ 0\nthis : (J' ⊓ spanSingleton R⁰ (divMod 0 I J... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Compact | {
"line": 530,
"column": 2
} | {
"line": 532,
"column": 37
} | {
"line": 534,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : TopologicalSpace β\ninst✝² : DenselyOrdered α\ninst✝¹ : ConditionallyCompleteLinearOrder β\ninst✝ : OrderTopology β\nf : α → β\na b : α\nh : ContinuousOn f [[a, b]]\n... | [] | refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm
refine csInf_le_csSup (nonempty_uIcc.image _) ?_ ?_ <;> rw [h.image_uIcc_eq_Icc]
exacts [bddBelow_Icc, bddAbove_Icc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Compact | {
"line": 530,
"column": 2
} | {
"line": 532,
"column": 37
} | {
"line": 534,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : TopologicalSpace β\ninst✝² : DenselyOrdered α\ninst✝¹ : ConditionallyCompleteLinearOrder β\ninst✝ : OrderTopology β\nf : α → β\na b : α\nh : ContinuousOn f [[a, b]]\n... | [] | refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm
refine csInf_le_csSup (nonempty_uIcc.image _) ?_ ?_ <;> rw [h.image_uIcc_eq_Icc]
exacts [bddBelow_Icc, bddAbove_Icc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 33
} | {
"line": 921,
"column": 34
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\nr : ℝ\nn : ℕ\nh : ‖a⁻¹ * b‖ ≤ r\n⊢ ↑n * ‖a⁻¹ * b‖ ≤ n • r",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSMul",
... | [
"E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\nr : ℝ\nn : ℕ\nh : ‖a⁻¹ * b‖ ≤ r\n⊢ ↑n * ‖a⁻¹ * b‖ ≤ ↑n * r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ProperSpace | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 59
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nh : ∀ (b : β), ProperSpace (X b)\n⊢ ProperSpace ((b : β) → X b)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
... | [] | refine .of_isCompact_closedBall_of_le 0 fun x r hr => ?_
rw [closedBall_pi _ hr]
exact isCompact_univ_pi fun _ => isCompact_closedBall _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.ProperSpace | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 59
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nh : ∀ (b : β), ProperSpace (X b)\n⊢ ProperSpace ((b : β) → X b)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
... | [] | refine .of_isCompact_closedBall_of_le 0 fun x r hr => ?_
rw [closedBall_pi _ hr]
exact isCompact_univ_pi fun _ => isCompact_closedBall _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 28
} | {
"line": 972,
"column": 29
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nf : E → F\nx : E\ny : F\n⊢ Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x' / x‖ < δ → ‖f x' / y‖ < ε",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.... | [
"E : Type u_5\nF : Type u_6\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nf : E → F\nx : E\ny : F\n⊢ Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ (ε : ℝ), 0 < ε → ∃ δ, 0 < δ ∧ ∀ (x' : E), ‖x' / x‖ < δ → ‖f x' / y‖ < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 977,
"column": 2
} | {
"line": 977,
"column": 28
} | {
"line": 977,
"column": 29
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ (𝓝 x).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {y | ‖y / x‖ < ε}",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ (𝓝 x).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {y | ‖y / x‖ < ε}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 982,
"column": 2
} | {
"line": 982,
"column": 28
} | {
"line": 982,
"column": 29
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\n⊢ (𝓤 E).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {p | ‖p.1 / p.2‖ < ε}",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_5\ninst✝ : SeminormedCommGroup E\n⊢ (𝓤 E).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {p | ‖p.1 / p.2‖ < ε}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 1020,
"column": 2
} | {
"line": 1020,
"column": 23
} | {
"line": 1020,
"column": 24
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedGroup E\na : E\n⊢ a = 1 ∨ 0 < ‖a‖",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Real.instZero",
"congrArg",
"Real.ins... | [
"E : Type u_5\ninst✝ : NormedGroup E\na : E\n⊢ a = 1 ∨ ¬a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 13
} | {
"line": 120,
"column": 14
} | [
{
"pp": "X : Type u_2\ns t : Set X\ninst✝ : PseudoEMetricSpace X\nh : (s ∩ t).Nonempty\nx : X\nxs : x ∈ s\nxt : x ∈ t\n⊢ ediam (s ∪ t) ≤ ediam s + ediam t",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Set.instUnion",
"id",
"LE.l... | [
"X : Type u_2\ns t : Set X\ninst✝ : PseudoEMetricSpace X\nh : (s ∩ t).Nonempty\nx : X\nxs : x ∈ s\nxt : x ∈ t\n⊢ ediam (s ∪ t) ≤ ediam s + ediam t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Cauchy | {
"line": 155,
"column": 2
} | {
"line": 161,
"column": 50
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nu : ℕ → α\nhu : CauchySeq u\nb : ℕ → ℝ\nhb : ∀ (n : ℕ), 0 < b n\n⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Metric.cauchySeq_iff",
"Eq.mpr",
"Na... | [] | rw [cauchySeq_iff] at hu
have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
intro k
rw [eventually_atTop]
obtain ⟨N, hN⟩ := hu (b k) (hb k)
exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩
exact Filter.extraction_forall_of_eventually hu' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Cauchy | {
"line": 155,
"column": 2
} | {
"line": 161,
"column": 50
} | {
"line": 163,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nu : ℕ → α\nhu : CauchySeq u\nb : ℕ → ℝ\nhb : ∀ (n : ℕ), 0 < b n\n⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Metric.cauchySeq_iff",
"Eq.mpr",
"Na... | [] | rw [cauchySeq_iff] at hu
have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
intro k
rw [eventually_atTop]
obtain ⟨N, hN⟩ := hu (b k) (hb k)
exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩
exact Filter.extraction_forall_of_eventually hu' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 389,
"column": 21
} | {
"line": 389,
"column": 39
} | {
"line": 389,
"column": 40
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nhs : IsClosed[inst✝³] (s ∩ Icc a b)\nha : a ∈ s\nh : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t\nhab : a ≤ b\nA : Set α := {t | t ∈ Icc a b ∧ Icc a... | [
"α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nhs : IsClosed[inst✝³] (s ∩ Icc a b)\nha : a ∈ s\nh : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t\nhab : a ≤ b\nA : Set α := {t | t ∈ Icc a b ∧ Icc a t ⊆ s}\na_m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 766,
"column": 4
} | {
"line": 766,
"column": 36
} | {
"line": 767,
"column": 4
} | [
{
"pp": "case refine_3\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓... | [
"case pos\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ spanSingleton R⁰... | by_cases H' : I'.divMod I J' = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 767,
"column": 33
} | {
"line": 767,
"column": 49
} | {
"line": 767,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ spanSingleton ... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ spanSingleton R⁰ (I'.divMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 770,
"column": 37
} | {
"line": 770,
"column": 76
} | {
"line": 770,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ spanSingleton ... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ spanSingleton R⁰ (I'.divMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 771,
"column": 44
} | {
"line": 771,
"column": 83
} | {
"line": 771,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis :\n (spanSingleton R... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis :\n (spanSingleton R⁰ (I'.divMod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 140,
"column": 7
} | {
"line": 140,
"column": 18
} | {
"line": 140,
"column": 19
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\ns : Set α\nhs : x ∈ s ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\n⊢ Bornology.IsBounded (s ∩ ball x 1) ∧ s ∩ ball x 1 ⊆ s",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Pseud... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\ns : Set α\nhs : x ∈ s ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\n⊢ Bornology.IsBounded (s ∩ ball x 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 148,
"column": 67
} | {
"line": 148,
"column": 78
} | {
"line": 148,
"column": 79
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\nx✝² x✝¹ : ℝ\nhr : x✝² ≤ x✝¹\nx✝ : α\n⊢ x✝ ∈ (closedBall c x✝¹)ᶜ → x✝ ∈ (closedBall c x✝²)ᶜ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\nx✝² x✝¹ : ℝ\nhr : x✝² ≤ x✝¹\nx✝ : α\n⊢ x✝¹ < dist x✝ c → x✝² < dist x✝ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 156,
"column": 61
} | {
"line": 156,
"column": 72
} | {
"line": 156,
"column": 73
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\nx✝² x✝¹ : ℝ\nhr : x✝² ≤ x✝¹\nx✝ : α\n⊢ x✝ ∈ (ball c x✝¹)ᶜ → x✝ ∈ (ball c x✝²)ᶜ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\nx✝² x✝¹ : ℝ\nhr : x✝² ≤ x✝¹\nx✝ : α\n⊢ x✝¹ ≤ dist x✝ c → x✝² ≤ dist x✝ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 34
} | {
"line": 165,
"column": 35
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ comap (dist c) atTop = cobounded α",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ comap (dist c) atTop = cobounded α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 68,
"column": 12
} | {
"line": 68,
"column": 32
} | {
"line": 68,
"column": 33
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nr : R\nn : ℕ\nhn : -r ≤ ↑n\n⊢ ↑(-↑n) ≤ r",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
... | [
"R : Type u_2\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nr : R\nn : ℕ\nhn : -r ≤ ↑n\n⊢ -r ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 31
} | {
"line": 301,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : TopologicalSpace β\nk : Set β\nf : β → α\nhk : IsCompact k\nhf : ∀ x ∈ k, ContinuousWithinAt f univ x\n⊢ ∃ t, k ⊆ t ∧ IsOpen[inst✝] t ∧ Bornology.IsBounded (f '' t)",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : TopologicalSpace β\nk : Set β\nf : β → α\nhk : IsCompact k\nhf : ∀ x ∈ k, ContinuousWithinAt f univ x\n⊢ ∃ t, k ⊆ t ∧ IsOpen[inst✝] t ∧ Bornology.IsBounded (f '' t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 523,
"column": 2
} | {
"line": 523,
"column": 13
} | {
"line": 523,
"column": 14
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nt : Set α\nx : α\nxs : x ∈ s\nxt : x ∈ t\n⊢ diam (s ∪ t) ≤ diam s + diam t",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nt : Set α\nx : α\nxs : x ∈ s\nxt : x ∈ t\n⊢ diam (s ∪ t) ≤ diam s + diam t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 51
} | {
"line": 220,
"column": 52
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x * r) l atBot",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x * r) l atBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 555,
"column": 4
} | {
"line": 555,
"column": 42
} | {
"line": 556,
"column": 4
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s n)\nh's : ∀ (n : ℕ), Bornology.IsBounded (s n)\nh : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty\nh' : Tendsto (fun n ↦ diam (s n)) atTop (𝓝 ... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s n)\nh's : ∀ (n : ℕ), Bornology.IsBounded (s n)\nh : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty\nh' : Tendsto (fun n ↦ diam (s n)) atTop (𝓝 0)\nu : ℕ → ... | apply cauchySeq_of_le_tendsto_0 _ _ h' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 51
} | {
"line": 233,
"column": 52
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : r < 0\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x * r) l atTop",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : r < 0\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x * r) l atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 254,
"column": 45
} | {
"line": 254,
"column": 56
} | {
"line": 254,
"column": 57
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants":... | [
"α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 264,
"column": 45
} | {
"line": 264,
"column": 56
} | {
"line": 264,
"column": 57
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants":... | [
"α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 272,
"column": 45
} | {
"line": 272,
"column": 56
} | {
"line": 272,
"column": 57
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x • r) l atTop",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants":... | [
"α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x • r) l atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Int | {
"line": 24,
"column": 4
} | {
"line": 24,
"column": 37
} | {
"line": 26,
"column": 0
} | [
{
"pp": "α : Type u_1\nm n : ℤ\n⊢ |↑m - ↑n| = |-↑m + ↑n|",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"AddMonoid.toAddSemigroup",
"AddGroup... | [] | rw [abs_sub_comm, neg_add_eq_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Int | {
"line": 55,
"column": 49
} | {
"line": 55,
"column": 79
} | {
"line": 55,
"column": 80
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : SeminormedCommGroup α\na : α\nn : ℕ\n⊢ ‖a ^ ↑n‖ ≤ ‖↑n‖ * ‖a‖",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"congrArg",
"DivIn... | [
"case inl\nα : Type u_1\ninst✝ : SeminormedCommGroup α\na : α\nn : ℕ\n⊢ ‖a ^ n‖ ≤ ↑n * ‖a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Int | {
"line": 55,
"column": 49
} | {
"line": 55,
"column": 79
} | {
"line": 55,
"column": 80
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : SeminormedCommGroup α\na : α\nn : ℕ\n⊢ ‖a ^ (-↑n)‖ ≤ ‖-↑n‖ * ‖a‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"... | [
"case inr\nα : Type u_1\ninst✝ : SeminormedCommGroup α\na : α\nn : ℕ\n⊢ ‖a ^ n‖ ≤ ↑n * ‖a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Archimedean | {
"line": 60,
"column": 33
} | {
"line": 60,
"column": 44
} | {
"line": 60,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\na : G\na_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}\na_in : a ∈ H\na_pos : 1 < a\ng : G\ng_in : g ∈ H\nk : ℤ\nright✝ : ∀ (y : ℤ), (fun k ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y ... | [
"G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\na : G\na_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}\na_in : a ∈ H\na_pos : 1 < a\ng : G\ng_in : g ∈ H\nk : ℤ\nright✝ : ∀ (y : ℤ), (fun k ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k\nnonneg ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Archimedean | {
"line": 61,
"column": 49
} | {
"line": 61,
"column": 95
} | {
"line": 61,
"column": 96
} | [
{
"pp": "G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\na : G\na_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}\na_in : a ∈ H\na_pos : 1 < a\ng : G\ng_in : g ∈ H\nk : ℤ\nright✝ : ∀ (y : ℤ), (fun k ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y ... | [
"G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\na : G\na_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}\na_in : a ∈ H\na_pos : 1 < a\ng : G\ng_in : g ∈ H\nk : ℤ\nright✝ : ∀ (y : ℤ), (fun k ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y = k\nnonneg ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 825,
"column": 4
} | {
"line": 826,
"column": 63
} | {
"line": 826,
"column": 64
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\nS : Type u_3\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra S R\ninst✝³ : Algebra.IsIntegral S R\ninst✝² : IsDomain S\ninst✝¹ : Module.IsTorsionFree S R\np : Ideal S\ninst✝ : p.IsMaximal\nhp : p ≠ 0\nh : map (algebraMap S R) p ≠ 0\nhF : Fi... | [
"case h\nR : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\nS : Type u_3\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra S R\ninst✝³ : Algebra.IsIntegral S R\ninst✝² : IsDomain S\ninst✝¹ : Module.IsTorsionFree S R\np : Ideal S\ninst✝ : p.IsMaximal\nhp : p ≠ 0\nh : map (algebraMap S R) p ≠ 0\nhF : Fintype ↑{v | ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 897,
"column": 2
} | {
"line": 898,
"column": 9
} | {
"line": 898,
"column": 10
} | [
{
"pp": "R : Type u_3\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ ⊥\n⊢ ∏ᶠ (p : HeightOneSpectrum R), p.asIdeal ^ multiplicity p.asIdeal I = I",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_3\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ ⊥\n⊢ ∏ᶠ (p : HeightOneSpectrum R), p.asIdeal ^ multiplicity p.asIdeal I = I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Ring.Real | {
"line": 49,
"column": 28
} | {
"line": 49,
"column": 86
} | {
"line": 49,
"column": 87
} | [
{
"pp": "ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℝ\nh : dist x✝¹ x✝ < ε\n⊢ dist (-x✝¹) (-x✝) < ε",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.lattice",
"abs",
"congrArg",
"Real.instSub",
"HSub.hSub",
"Real.instLT",
"id",
... | [
"ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℝ\nh : dist x✝¹ x✝ < ε\n⊢ |x✝¹ - x✝| < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 58,
"column": 63
} | {
"line": 58,
"column": 74
} | {
"line": 58,
"column": 75
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"id",
"LE.le",
"Nat.cast",
"Nat.dist_cast_rat",
"Real.instOne",
"funext",
... | [
"⊢ Pairwise fun x y ↦ 1 ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 61,
"column": 58
} | {
"line": 61,
"column": 69
} | {
"line": 61,
"column": 70
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"id",
"LE.le",
"Nat.cast",
"Nat.dist_cast_rat",
"Real.instOne",
"funext",
... | [
"⊢ Pairwise fun x y ↦ 1 ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 68,
"column": 63
} | {
"line": 68,
"column": 74
} | {
"line": 68,
"column": 75
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"Int.dist_cast_rat",
"Rat.instIntCast",
"id",
"Int",
"LE.le",
... | [
"⊢ Pairwise fun x y ↦ 1 ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 71,
"column": 58
} | {
"line": 71,
"column": 69
} | {
"line": 71,
"column": 70
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"Int.dist_cast_rat",
"Rat.instIntCast",
"id",
"Int",
"LE.le",
... | [
"⊢ Pairwise fun x y ↦ 1 ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 69
} | {
"line": 86,
"column": 70
} | [
{
"pp": "ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℚ\nh : dist x✝¹ x✝ < ε\n⊢ dist (-x✝¹) (-x✝) < ε",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"DivisionRing.toRatCast",
"abs",
"congrArg",
"Real.in... | [
"ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℚ\nh : dist x✝¹ x✝ < ε\n⊢ |↑x✝¹ - ↑x✝| < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 98,
"column": 25
} | {
"line": 98,
"column": 50
} | {
"line": 98,
"column": 51
} | [
{
"pp": "ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℚ\nh : dist x✝¹ x✝ < ε\n⊢ dist |x✝¹| |x✝| ≤ dist x✝¹ x✝",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.lattice",
"DivisionRing.toRatCast",
"AddGroupWithOne.toAddGroup",
"abs",
"congrArg"... | [
"ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℚ\nh : dist x✝¹ x✝ < ε\n⊢ ||↑x✝¹| - |↑x✝|| ≤ |↑x✝¹ - ↑x✝|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 103,
"column": 2
} | {
"line": 104,
"column": 9
} | {
"line": 104,
"column": 10
} | [
{
"pp": "a b : ℚ\n⊢ TotallyBounded (Icc a b)",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℚ\n⊢ TotallyBounded (Icc a b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.IsBounded | {
"line": 116,
"column": 28
} | {
"line": 116,
"column": 53
} | {
"line": 116,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\nf : Filter β\nu : β → α\ns : Set β\nhs : ∀ᶠ (x : β) in f, x ∈ s\nb : α\nhb : b ∈ upperBounds (u '' s)\n⊢ ∀ x ∈ s, u x ∈ {x | (fun x ↦ (fun x1 x2 ↦ x1 ≤ x2) x b) x}",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"setOf",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\nf : Filter β\nu : β → α\ns : Set β\nhs : ∀ᶠ (x : β) in f, x ∈ s\nb : α\nhb : b ∈ upperBounds (u '' s)\n⊢ ∀ x ∈ s, u x ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 134,
"column": 18
} | {
"line": 134,
"column": 29
} | {
"line": 134,
"column": 30
} | [
{
"pp": "x : ℝ\nx✝ : EReal\nh : ↑x + x✝ ≤ ↑x + ⊥\n⊢ x✝ ≤ ⊥",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"EReal",
"Preorder.toLE",
"id",
"Bot.bot",
"LE.le",
"instCompleteLinearOrd... | [
"x : ℝ\nx✝ : EReal\nh : ↑x + x✝ ≤ ↑x + ⊥\n⊢ x✝ = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 75
} | {
"line": 136,
"column": 76
} | [
{
"pp": "x y z : ℝ\nh : ↑x + ↑y ≤ ↑x + ↑z\n⊢ ↑y ≤ ↑z",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"PartialOrder.toPreorder",
"EReal",
"Preorder.toLE",
"id",
"LE.le",
"_private.Mathlib.Data.EReal.Opera... | [
"x y z : ℝ\nh : ↑x + ↑y ≤ ↑x + ↑z\n⊢ y ≤ z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 24
} | {
"line": 143,
"column": 25
} | [
{
"pp": "x y : EReal\nh : x < y\nz : ℝ\n⊢ ↑z + x < ↑z + y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"id",
"instAddCommMonoidEReal",
"add_comm",
"instHAdd",... | [
"x y : EReal\nh : x < y\nz : ℝ\n⊢ x + ↑z < y + ↑z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.IsBounded | {
"line": 341,
"column": 4
} | {
"line": 341,
"column": 38
} | {
"line": 341,
"column": 39
} | [
{
"pp": "α : Type u_5\nf : Filter α\nR : Type u_6\nκ : Type u_7\ninst✝ : AddCommMonoid R\nr : R → R → Prop\nhr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)\nhr₀ : r 0 0\nu : κ → α → R\nk₀ : κ\ns : Finset κ\nk₀_notin_s : k₀ ∉ s\nih : (∀ k ∈ s, IsBoundedUnder r... | [
"α : Type u_5\nf : Filter α\nR : Type u_6\nκ : Type u_7\ninst✝ : AddCommMonoid R\nr : R → R → Prop\nhr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)\nhr₀ : r 0 0\nu : κ → α → R\nk₀ : κ\ns : Finset κ\nk₀_notin_s : k₀ ∉ s\nih : (∀ k ∈ s, IsBoundedUnder r f (u k)) → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 312,
"column": 28
} | {
"line": 312,
"column": 39
} | {
"line": 312,
"column": 40
} | [
{
"pp": "motive : EReal → Sort u_1\ncoe : (x : ℝ≥0∞) → motive ↑x\nneg_coe : (x : ℝ≥0∞) → 0 < x → motive (-↑x)\nx : EReal\nhx : ¬0 ≤ x\n⊢ 0 < -x",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal.instNeg",
... | [
"motive : EReal → Sort u_1\ncoe : (x : ℝ≥0∞) → motive ↑x\nneg_coe : (x : ℝ≥0∞) → 0 < x → motive (-↑x)\nx : EReal\nhx : ¬0 ≤ x\n⊢ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.LiminfLimsup | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 33
} | {
"line": 239,
"column": 0
} | [
{
"pp": "ι : Type u_6\nα : Type u_7\nβ : Type u_8\ninst✝ : ConditionallyCompleteLattice β\nv : ι → α\nu : α → β\nf : Filter ι\ng : Filter α\nhv : Tendsto v f g\nhvf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) (map v f) u\nhg : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) g u\n⊢ limsup u (map v f) ≤ limsup u g",
"ppTer... | [] | exact limsup_le_limsup_of_le hv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.LiminfLimsup | {
"line": 360,
"column": 30
} | {
"line": 360,
"column": 62
} | {
"line": 360,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\n⊢ ∀ b ∈ {a | ∀ᶠ (n : α) in ⊤, n ≤ a}, ⊤ ≤ b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.eventually_top._simp_1",
"Filter.Eventually",
"PartialOrder.toPreorder",
"setOf",
... | [
"α : Type u_1\ninst✝ : CompleteLattice α\n⊢ ∀ (b : α), (∀ (x : α), x ≤ b) → b = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.LiminfLimsup | {
"line": 364,
"column": 30
} | {
"line": 364,
"column": 62
} | {
"line": 364,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\n⊢ ∀ b ∈ {a | ∀ᶠ (n : α) in ⊤, a ≤ n}, b ≤ ⊥",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.eventually_top._simp_1",
"Filter.Eventually",
"OrderBot.toBot",
"PartialOrder.toPr... | [
"α : Type u_1\ninst✝ : CompleteLattice α\n⊢ ∀ (b : α), (∀ (x : α), b ≤ x) → b = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.MonotoneConvergence | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 31
} | {
"line": 284,
"column": 2
} | [
{
"pp": "case left\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : Preorder α\ninst✝³ : OrderClosedTopology α\ninst✝² : Preorder β\ninst✝¹ : IsDirectedOrder β\ninst✝ : Nonempty β\nf : β → α\na : α\nhf : Monotone f\nha : Tendsto f atTop (𝓝 a)\nb : β\n⊢ f b ≤ a",
"ppTerm": "?left",
"as... | [] | exact hf.ge_of_tendsto ha b | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.LiminfLimsup | {
"line": 668,
"column": 2
} | {
"line": 668,
"column": 28
} | {
"line": 668,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : CompleteDistribLattice α\nf : Filter β\np : β → Prop\nu : β → α\n⊢ (blimsup u f fun x ↦ ¬p x) ⊔ blimsup u f p = limsup u f",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝ : CompleteDistribLattice α\nf : Filter β\np : β → Prop\nu : β → α\n⊢ (blimsup u f fun x ↦ ¬p x) ⊔ blimsup u f p = limsup u f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.LiminfLimsup | {
"line": 735,
"column": 2
} | {
"line": 736,
"column": 9
} | {
"line": 736,
"column": 10
} | [
{
"pp": "α : Type u_1\nι : Type u_4\ns : ι → Set α\n𝓕 : Filter ι\na : α\n⊢ a ∈ liminf s 𝓕 ↔ ∀ᶠ (i : ι) in 𝓕, a ∈ s i",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"iInf",
"Filter.liminf",
"Iff.of_eq",
"congrArg",
... | [
"α : Type u_1\nι : Type u_4\ns : ι → Set α\n𝓕 : Filter ι\na : α\n⊢ (∃ i, ∃ (_ : i ∈ 𝓕), ∀ i_1 ∈ i, a ∈ s i_1) ↔ ∀ᶠ (i : ι) in 𝓕, a ∈ s i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.LiminfLimsup | {
"line": 748,
"column": 20
} | {
"line": 748,
"column": 31
} | {
"line": 748,
"column": 32
} | [
{
"pp": "α : Type u_1\nι : Type u_4\np : ι → Prop\ns : ι → Set α\nx : α\nh : x ∉ {x | {n | p n ∧ x ∈ s n}.Infinite}\n⊢ {x_1 | p x_1 ∧ ¬s x_1 ⊆ {x}ᶜ}.Finite",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Classical.not_not._simp_1",
"congrArg",
"Compl.comp... | [
"α : Type u_1\nι : Type u_4\np : ι → Prop\ns : ι → Set α\nx : α\nh : x ∉ {x | {n | p n ∧ x ∈ s n}.Infinite}\n⊢ {x_1 | p x_1 ∧ x ∈ s x_1}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.LiminfLimsup | {
"line": 859,
"column": 2
} | {
"line": 859,
"column": 13
} | {
"line": 859,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nu : α → β\nb : β\nhu : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh : ∀ᶠ (x : α) in f, u x ≤ b\n⊢ ∀ᶠ (n : β) in map u f, n ≤ b",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nu : α → β\nb : β\nhu : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh : ∀ᶠ (x : α) in f, u x ≤ b\n⊢ ∀ᶠ (a : α) in f, u a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.LiminfLimsup | {
"line": 880,
"column": 4
} | {
"line": 880,
"column": 31
} | {
"line": 881,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nu : α → β\nx : β\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh : ∀ y > x, ∀ᶠ (a : α) in f, u a < y\nh' : ∃ y > x, ∀ (z : β), z ≤ x ∨ y ≤ z\n⊢ ∀ᶠ (n : α) i... | [
"case neg\nα : Type u_1\nβ : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nu : α → β\nx : β\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh : ∀ y > x, ∀ᶠ (a : α) in f, u a < y\nz : β\nx_z : z > x\nhz : ∀ (z_1 : β), z_1 ≤ x ∨ z ≤ z_1\n⊢ ∀ᶠ (n :... | rcases h' with ⟨z, x_z, hz⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.LiminfLimsup | {
"line": 990,
"column": 6
} | {
"line": 990,
"column": 26
} | {
"line": 991,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : ConditionallyCompleteLinearOrder β\nf✝¹ : Filter α\nu : α → β\ninst✝² : ConditionallyCompleteLinearOrder α\nf✝ : Filter α\nb : α\nf : ι → α\ns : ι' → Set ι\np : ι' → Prop\ninst✝¹ : Countable (Subtype p)\ninst✝ : N... | [] | exact ⟨n, Or.inl hj⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.LiminfLimsup | {
"line": 1008,
"column": 4
} | {
"line": 1008,
"column": 25
} | {
"line": 1009,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nhs : ∀ (j : Subtype p), (s ↑j).Nonempty\nj0 : Subtype p\nhj0 : BddBelow (ran... | [
"case h₁\nα : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nhs : ∀ (j : Subtype p), (s ↑j).Nonempty\nj0 : Subtype p\nhj0 : BddBelow (range ... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 55
} | {
"line": 250,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ edist ((fun x ↦ (f x, g x)) x) ((fun x ↦ (f x, g x)) y) ≤ ↑(max Kf Kg) *... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ max (edist ((fun x ↦ (f x, g x)) x).1 ((fun x ↦ (f x, g x)) y).1)\n (edist ((fu... | rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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