module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.EReal.Basic | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 13
} | {
"line": 368,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : 0 < ↑x\nh'x : ↑x ≠ ⊤\n⊢ 0 < (↑x).toReal",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instLT",
"id",
"LT.lt",
"EReal.toReal",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.toEReal"
],
... | [
"x : ℝ\nhx : 0 < ↑x\nh'x : ↑x ≠ ⊤\n⊢ 0 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 13
} | {
"line": 372,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : ↑x < 0\nh'x : ↑x ≠ ⊥\n⊢ (↑x).toReal < 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instLT",
"id",
"LT.lt",
"EReal.toReal",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.toEReal"
],
... | [
"x : ℝ\nhx : ↑x < 0\nh'x : ↑x ≠ ⊥\n⊢ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 15
} | {
"line": 379,
"column": 16
} | [
{
"pp": "case mp\ny : ℝ\nhy0 : 0 < ↑y\nright✝ : ↑y < ⊤\n⊢ (↑y).toReal ∈ Ioi 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Preorder.toLT",
"Real.instZero",
"Membership.mem",
"Set.mem_Ioi._simp_1",
"id",
"L... | [
"case mp\ny : ℝ\nhy0 : 0 < ↑y\nright✝ : ↑y < ⊤\n⊢ 0 < y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 15
} | {
"line": 382,
"column": 16
} | [
{
"pp": "case h\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ ↑x ∈ Ioo 0 ⊤ ∧ (↑x).toReal = x",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"and_true",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"Real.... | [
"case h\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ 0 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 15
} | {
"line": 389,
"column": 16
} | [
{
"pp": "case mp\ny : ℝ\nleft✝ : ⊥ < ↑y\nhy0 : ↑y < 0\n⊢ (↑y).toReal ∈ Iio 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instZero",
"Membership.mem",
"id",
"LT.lt",
"EReal.toReal",
"Zero.toOfNa... | [
"case mp\ny : ℝ\nleft✝ : ⊥ < ↑y\nhy0 : ↑y < 0\n⊢ y < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 392,
"column": 4
} | {
"line": 392,
"column": 15
} | {
"line": 392,
"column": 16
} | [
{
"pp": "case h\nx : ℝ\nhx : x ∈ Iio 0\n⊢ ↑x ∈ Ioo ⊥ 0 ∧ (↑x).toReal = x",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"and_true",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"Real.... | [
"case h\nx : ℝ\nhx : x ∈ Iio 0\n⊢ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 13
} | {
"line": 398,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : ↑x ≠ ⊥\ny : ℝ\nhy : ↑y ≠ ⊤\nh : ↑x ≤ ↑y\n⊢ (↑x).toReal ≤ (↑y).toReal",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"id",
"LE.le",
"EReal.toReal",
"Real.toEReal"
],
"usedFVars": [
"x",
"y"
... | [
"x : ℝ\nhx : ↑x ≠ ⊥\ny : ℝ\nhy : ↑y ≠ ⊤\nh : ↑x ≤ ↑y\n⊢ x ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 818,
"column": 11
} | {
"line": 818,
"column": 22
} | {
"line": 818,
"column": 23
} | [
{
"pp": "a : ℝ\nx✝ : ↑a < ⊤\nb : ℚ\nhab : a < ↑b\n⊢ ↑a < ↑↑b",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instRatCast",
"PartialOrder.toPreorder",
"EReal",
"Real.instLT",
"EReal.coe_lt_coe_iff._simp... | [
"a : ℝ\nx✝ : ↑a < ⊤\nb : ℚ\nhab : a < ↑b\n⊢ a < ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 822,
"column": 25
} | {
"line": 822,
"column": 36
} | {
"line": 822,
"column": 37
} | [
{
"pp": "a : ℝ\nx✝ : ⊥ < ↑a\nb : ℚ\nhab : ↑b < a\n⊢ ↑↑b < ↑a",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instRatCast",
"PartialOrder.toPreorder",
"EReal",
"Real.instLT",
"EReal.coe_lt_coe_iff._simp... | [
"a : ℝ\nx✝ : ⊥ < ↑a\nb : ℚ\nhab : ↑b < a\n⊢ ↑b < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 841,
"column": 6
} | {
"line": 841,
"column": 36
} | {
"line": 841,
"column": 37
} | [
{
"pp": "x✝ : ↑{⊥, ⊤}ᶜ\nx : EReal\nhx : x ∈ {⊥, ⊤}ᶜ\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal",
"_private.Mathlib.Data.EReal.Basic.0.EReal.neTopBotEquivReal._simp_5",
"instTopEReal",
"id",
"Ne",
"Bot.bot",
... | [
"x✝ : ↑{⊥, ⊤}ᶜ\nx : EReal\nhx : x ∈ {⊥, ⊤}ᶜ\n⊢ ¬x = ⊥ ∧ ¬x = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 44
} | {
"line": 84,
"column": 45
} | [
{
"pp": "r p : ℝ≥0\n⊢ ↑(p / r) ≤ ↑p / ↑r",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"Monoid.toMulOneClass",
"congrArg",
"C... | [
"r p : ℝ≥0\n⊢ ↑p * ↑r⁻¹ ≤ ↑p * (↑r)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 90,
"column": 18
} | {
"line": 90,
"column": 65
} | {
"line": 90,
"column": 66
} | [
{
"pp": "a b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ 1⁻¹ = 1",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ 1⁻¹ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 271,
"column": 28
} | {
"line": 271,
"column": 39
} | {
"line": 271,
"column": 40
} | [
{
"pp": "a b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ 0\n⊢ 0 < b → b < a → c⁻¹ ≠ ∞",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"ENNReal.inv_eq_top._simp_1",
"id",
... | [
"a b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ 0\n⊢ 0 < b → b < a → ¬c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 28
} | {
"line": 291,
"column": 29
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a⁻¹ < b ↔ b⁻¹ < a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0∞\n⊢ a⁻¹ < b ↔ b⁻¹ < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 28
} | {
"line": 294,
"column": 29
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a < b⁻¹ ↔ b < a⁻¹",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0∞\n⊢ a < b⁻¹ ↔ b < a⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 28
} | {
"line": 301,
"column": 29
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a⁻¹ ≤ b ↔ b⁻¹ ≤ a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0∞\n⊢ a⁻¹ ≤ b ↔ b⁻¹ ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 28
} | {
"line": 304,
"column": 29
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a ≤ b⁻¹ ↔ b ≤ a⁻¹",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0∞\n⊢ a ≤ b⁻¹ ↔ b ≤ a⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 388,
"column": 23
} | {
"line": 388,
"column": 39
} | {
"line": 388,
"column": 40
} | [
{
"pp": "a b c : ℝ≥0∞\nh : a ≤ b * c\nh0 : c = 0\n⊢ a = 0",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b c : ℝ≥0∞\nh : a ≤ b * c\nh0 : c = 0\n⊢ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 445,
"column": 89
} | {
"line": 450,
"column": 43
} | {
"line": 452,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nha : a ≠ ∞\nhb₀ : b ≠ 0\n⊢ ∃ c, 0 < c ∧ c * a < b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"zero_le",
"Eq.mpr",
"ENNReal.instIsOrderedRing",
"ENNReal.instAdd",
"Preorder.toLT",
"instH... | [] | by
obtain rfl | hb := eq_or_ne b ∞
· exact ⟨1, by simpa [lt_top_iff_ne_top]⟩
refine ⟨b / (a + 1), ENNReal.div_pos hb₀ (by finiteness), ENNReal.mul_lt_of_lt_div ?_⟩
gcongr
exact ENNReal.lt_add_right ha one_ne_zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 453,
"column": 6
} | {
"line": 453,
"column": 16
} | {
"line": 453,
"column": 17
} | [
{
"pp": "a b : ℝ≥0∞\nh₁ : b = ∞ → a ≠ 0\nh₂ : a = ∞ → b ≠ 0\n⊢ a⁻¹ ≤ b ↔ 1 ≤ a * b",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"C... | [
"a b : ℝ≥0∞\nh₁ : b = ∞ → a ≠ 0\nh₂ : a = ∞ → b ≠ 0\n⊢ 1 / a ≤ b ↔ 1 ≤ a * b"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 16
} | {
"line": 458,
"column": 17
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a ≤ b⁻¹ ↔ a * b ≤ 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"id",
... | [
"a b : ℝ≥0∞\n⊢ a ≤ 1 / b ↔ a * b ≤ 1"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 472,
"column": 6
} | {
"line": 472,
"column": 18
} | {
"line": 472,
"column": 19
} | [
{
"pp": "a b : ℝ≥0∞\nh : a * b = 1\n⊢ a = b⁻¹",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"MulOne.toMul",
"ENNReal.instCommSemiring",
"MulZeroOneCla... | [
"a b : ℝ≥0∞\nh : a * b = 1\n⊢ a * 1 = b⁻¹"
] | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 16
} | {
"line": 579,
"column": 17
} | [
{
"pp": "⊢ 1 - 2⁻¹ = 2⁻¹",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"id",
"HDiv.hDiv"... | [
"⊢ 1 - 1 / 2 = 1 / 2"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 94
} | {
"line": 609,
"column": 2
} | [
{
"pp": "case refine_1\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y : ℝ≥0∞\nhxy : x < y\n⊢ (fun x ↦ ⟨(x⁻¹ + 1)⁻¹, ⋯⟩) x < (fun x ↦ ⟨(x⁻¹ + 1)⁻¹, ⋯⟩) y",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Iff.mpr",
"Eq.mpr",
"ENNReal.instAdd... | [
"case refine_2\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ↑(Iic 1)\n⊢ ↑((fun x ↦ ⟨(x⁻¹ + 1)⁻¹, ⋯⟩) ((fun x ↦ ((↑x)⁻¹ - 1)⁻¹) x)) = ↑x"
] | · simpa only [Subtype.mk_lt_mk, ENNReal.inv_lt_inv, ENNReal.add_lt_add_iff_right one_ne_top] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.ENNReal.Inv | {
"line": 687,
"column": 4
} | {
"line": 687,
"column": 15
} | {
"line": 687,
"column": 16
} | [
{
"pp": "case ofNat\na : ℝ≥0∞\nha : a ≠ 0\nh'a : a ≠ ∞\na✝ : ℕ\n⊢ 0 < a ^ Int.ofNat a✝",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"DivInvMonoid.toZPow",
"id",
... | [
"case ofNat\na : ℝ≥0∞\nha : a ≠ 0\nh'a : a ≠ ∞\na✝ : ℕ\n⊢ 0 < a ^ a✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 693,
"column": 4
} | {
"line": 693,
"column": 15
} | {
"line": 693,
"column": 16
} | [
{
"pp": "case ofNat\na : ℝ≥0∞\nha : a ≠ 0\nh'a : a ≠ ∞\na✝ : ℕ\n⊢ a ^ Int.ofNat a✝ < ∞",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"DivInvMonoid.toZPow",
"id",
... | [
"case ofNat\na : ℝ≥0∞\nha : a ≠ 0\nh'a : a ≠ ∞\na✝ : ℕ\n⊢ a < ∞ ∨ a✝ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 704,
"column": 23
} | {
"line": 704,
"column": 57
} | {
"line": 704,
"column": 58
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ y ≠ 0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
],
"usedFVars": [
"y"
],
"usedGoals": [
{
... | [
"x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ ¬y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 38
} | {
"line": 707,
"column": 39
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ x ≠ 0",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
],
"usedFVars": [
"x"
],
"usedGoals": [... | [
"x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 716,
"column": 23
} | {
"line": 716,
"column": 57
} | {
"line": 716,
"column": 58
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ y ≠ 0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
],
"usedFVars": [
"y"
],
"usedGoals": [
{
... | [
"x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ ¬y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 719,
"column": 4
} | {
"line": 719,
"column": 38
} | {
"line": 719,
"column": 39
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ x ≠ 0",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
],
"usedFVars": [
"x"
],
"usedGoals": [... | [
"x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Defs | {
"line": 458,
"column": 33
} | {
"line": 458,
"column": 44
} | {
"line": 458,
"column": 45
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx y : α\nε : ℝ≥0∞\nh : y ∈ eball x ε\n⊢ 0 < ε - edist y x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"tsub_pos_iff_lt._simp_1",
"Preorder.toLT",
"ENNReal.ins... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nx y : α\nε : ℝ≥0∞\nh : y ∈ eball x ε\n⊢ edist y x < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 757,
"column": 27
} | {
"line": 757,
"column": 61
} | {
"line": 757,
"column": 62
} | [
{
"pp": "m n : ℤ\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ x ≠ 0",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
],
"usedFVars": [
"x"
],
"usedGoals": [
{
"new": t... | [
"m n : ℤ\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 36
} | {
"line": 93,
"column": 37
} | [
{
"pp": "case inr.inr\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\nα : Type u_3\ndist : α → α → ℝ\ndist_comm : ∀ (x y : α), dist x y = dist y x\ndist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\ns : Set α\nhs : s ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}\nt : Set α\nh... | [
"case inr.inr\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\nα : Type u_3\ndist : α → α → ℝ\ndist_comm : ∀ (x y : α), dist x y = dist y x\ndist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\ns : Set α\nhs : s ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}\nt : Set α\nht : t ∈ {s |... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 781,
"column": 23
} | {
"line": 781,
"column": 34
} | {
"line": 781,
"column": 35
} | [
{
"pp": "x : ℝ≥0∞\nhx : 1 ≤ x\nm : ℤ\nhn : -m ≤ 0\n⊢ 0 ≤ m",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ≥0∞\nhx : 1 ≤ x\nm : ℤ\nhn : -m ≤ 0\n⊢ 0 ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 814,
"column": 4
} | {
"line": 815,
"column": 11
} | {
"line": 815,
"column": 12
} | [
{
"pp": "case neg.inr.inl\nι : Sort u_1\nf : ι → ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\nha₀ : ∞ ≠ 0\ni : ι\nhi : ¬f i = 0\n⊢ ∞ * ⨆ i, f i = ⨆ i, ∞ * f i",
"ppTerm": "?neg.inr.inl✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
"HMul.hMul",
"eq_false",
... | [
"case neg.inr.inl\nι : Sort u_1\nf : ι → ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\nha₀ : ∞ ≠ 0\ni : ι\nhi : ¬f i = 0\n⊢ ⨆ i, ∞ * f i = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 851,
"column": 40
} | {
"line": 851,
"column": 69
} | {
"line": 851,
"column": 70
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0\nh₀ : a = 0 → Nonempty ι\n⊢ (⨅ i, f i) * a = ⨅ i, f i * a",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0\nh₀ : a = 0 → Nonempty ι\n⊢ (⨅ i, f i) * a = ⨅ i, f i * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 914,
"column": 2
} | {
"line": 914,
"column": 31
} | {
"line": 915,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nhg : ∃ j, g j ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\n⊢ a ≤ (⨅ i, f ↑i) * ⨅ j, g j",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Exists",
"Subtype",
"Ne",... | [
"ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nhg : ∃ j, g j ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\nthis : Nonempty { a // f a ≠ ∞ }\n⊢ a ≤ (⨅ i, f ↑i) * ⨅ j, g j"
] | have := nonempty_subtype.2 hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.ENNReal.Inv | {
"line": 916,
"column": 34
} | {
"line": 916,
"column": 45
} | {
"line": 916,
"column": 46
} | [
{
"pp": "ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nhg : ∃ j, g j ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\nthis✝ : Nonempty { a // f a ≠ ∞ }\nthis : Nonempty κ\n⊢ ⨅ j, g j ≠ ∞",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"iInf_eq_top._s... | [
"ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nhg : ∃ j, g j ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\nthis✝ : Nonempty { a // f a ≠ ∞ }\nthis : Nonempty κ\n⊢ ∃ x, ¬g x = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Defs | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 32
} | {
"line": 104,
"column": 33
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ dist x y ≠ 0 ↔ x ≠ y",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"id",
"Ne",
"Iff",
"Zero.toOfNat0",
"MetricSpace.toPseudoMetricSpace",
"Dist.d... | [
"γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ dist x y = 0 ↔ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Defs | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 44
} | {
"line": 108,
"column": 45
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ dist x y ≤ 0 ↔ x = y",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ dist x y ≤ 0 ↔ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Defs | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 27
} | {
"line": 112,
"column": 28
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ 0 < dist x y ↔ x ≠ y",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ 0 < dist x y ↔ x ≠ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 919,
"column": 2
} | {
"line": 919,
"column": 44
} | {
"line": 919,
"column": 45
} | [
{
"pp": "ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\nthis✝ : Nonempty { a // f a ≠ ∞ }\nthis : Nonempty κ\nhg : ⨅ j, g j ≠ ∞\ni : ι\nhi : f i ≠ ∞\n⊢ a ≤ f ↑⟨i, hi⟩ * ⨅ j, g j",
"ppTerm": "?m.89",
"assigned": true,
"usedC... | [
"ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\nthis✝ : Nonempty { a // f a ≠ ∞ }\nthis : Nonempty κ\nhg : ⨅ j, g j ≠ ∞\ni : ι\nhi : f i ≠ ∞\n⊢ ∀ (i_1 : κ), a ≤ f i * g i_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 325,
"column": 6
} | {
"line": 325,
"column": 59
} | {
"line": 325,
"column": 60
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)\nx y : E\n⊢ dist x y ≤ dist 1 (x⁻¹ * y)",
"p... | [
"case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)\nx y : E\n⊢ dist x y ≤ dist (x⁻¹ * x) (x⁻¹ * y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 70
} | {
"line": 326,
"column": 71
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)\nx y : E\n⊢ dist 1 (x⁻¹ * y) ≤ dist x y",
"p... | [
"case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)\nx y : E\n⊢ dist 1 (x⁻¹ * y) ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 70
} | {
"line": 337,
"column": 71
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y\nx y : E\n⊢ dist x y ≤ dist 1 (x⁻¹ * y)",
"p... | [
"case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y\nx y : E\n⊢ dist x y ≤ dist 1 (x⁻¹ * y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 338,
"column": 6
} | {
"line": 338,
"column": 59
} | {
"line": 338,
"column": 60
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y\nx y : E\n⊢ dist 1 (x⁻¹ * y) ≤ dist x y",
"p... | [
"case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y\nx y : E\n⊢ dist (x⁻¹ * x) (x⁻¹ * y) ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 640,
"column": 27
} | {
"line": 640,
"column": 64
} | {
"line": 640,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \\ {p}, q.asI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 24
} | {
"line": 401,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\nh : x ∈ ball y ε\n⊢ ∃ ε' < ε, x ∈ ball y ε'",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instLT",
"Membership.mem",
"Exists",
"id",
"Metr... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\nh : x ∈ ball y ε\n⊢ ∃ ε' < ε, dist x y < ε'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 439,
"column": 31
} | {
"line": 439,
"column": 42
} | {
"line": 439,
"column": 43
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\nh : y ∈ sphere x ε\nhε : ε ≠ 0\n⊢ x ∉ sphere x ε",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Membership.mem",
"id",
"dist_self",
... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\nh : y ∈ sphere x ε\nhε : ε ≠ 0\n⊢ ¬0 = ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 647,
"column": 63
} | {
"line": 647,
"column": 81
} | {
"line": 647,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \\ {p}, q.asI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 742,
"column": 27
} | {
"line": 742,
"column": 38
} | {
"line": 742,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n_ε : ℝ\nε0 : 0 < _ε\nn : ℕ\nhn : 1 / (↑n + 1) < _ε\n⊢ 1 / ↑(n + 1) ≤ _ε",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"AddMonoid.toAddS... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\n_ε : ℝ\nε0 : 0 < _ε\nn : ℕ\nhn : 1 / (↑n + 1) < _ε\n⊢ (↑n + 1)⁻¹ ≤ _ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 660,
"column": 62
} | {
"line": 660,
"column": 78
} | {
"line": 660,
"column": 79
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis✝ : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s ... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis✝ : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \\ {p}, q.as... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 931,
"column": 2
} | {
"line": 931,
"column": 30
} | {
"line": 931,
"column": 31
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : Dense s\nx : α\nε : ℝ\nhε : 0 < ε\nthis : (ball x ε).Nonempty\n⊢ ∃ y ∈ s, dist x y < ε",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : Dense s\nx : α\nε : ℝ\nhε : 0 < ε\nthis : (ball x ε).Nonempty\n⊢ ∃ y ∈ s, dist x y < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 984,
"column": 8
} | {
"line": 984,
"column": 19
} | {
"line": 984,
"column": 20
} | [
{
"pp": "α : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nx y z : α\n⊢ 0 ≤ dist x y + dist y z",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nx y z : α\n⊢ 0 ≤ dist x y + dist y z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1079,
"column": 20
} | {
"line": 1079,
"column": 78
} | {
"line": 1079,
"column": 79
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx : X\n⊢ dist x x = 0",
"ppTerm": "?m.44",
"assigned": fal... | [
"α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx : X\n⊢ dist x x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1080,
"column": 22
} | {
"line": 1080,
"column": 50
} | {
"line": 1080,
"column": 51
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx y : X\n⊢ dist x y = dist y x",
"ppTerm": "?m.56",
"assig... | [
"α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx y : X\n⊢ dist x y = dist y x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1082,
"column": 4
} | {
"line": 1082,
"column": 66
} | {
"line": 1082,
"column": 67
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx y z : X\n⊢ dist x z ≤ dist x y + dist y z",
"ppTerm": "?m.68... | [
"α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx y z : X\n⊢ dist x z ≤ dist x y + dist y z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1087,
"column": 4
} | {
"line": 1088,
"column": 11
} | {
"line": 1088,
"column": 12
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\n⊢ ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε} = ⨅ ε, ⨅ (_ : ε >... | [
"α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\n⊢ ⨅ ε, ⨅ (_ : 0 < ε), 𝓟 {p | ENNReal.ofReal (dist p.1 p.2) < ε} = ⨅ ε, ⨅ (_ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1147,
"column": 6
} | {
"line": 1147,
"column": 17
} | {
"line": 1147,
"column": 17
} | [
{
"pp": "x r y : ℝ\n⊢ x - y < r ∧ y - x < r ↔ x - r < y ∧ y - x < r",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"sub_lt_comm",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"R... | [
"x r y : ℝ\n⊢ x - r < y ∧ y - x < r ↔ x - r < y ∧ y - x < r"
] | sub_lt_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1202,
"column": 2
} | {
"line": 1202,
"column": 32
} | {
"line": 1202,
"column": 33
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y z : α\n⊢ dist (dist x y) (dist x z) ≤ dist y z",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"id",
"dist_comm",
"LE.le",
"congr",
"cong... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nx y z : α\n⊢ dist (dist y x) (dist z x) ≤ dist y z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1230,
"column": 4
} | {
"line": 1230,
"column": 82
} | {
"line": 1230,
"column": 83
} | [
{
"pp": "α : Type u\ninst✝¹ : PseudoMetricSpace α\nβ : Type u_3\np : β → Prop\nf : β → ℝ\nl : Filter β\ninst✝ : l.NeBot\nhf₀ : ∀ (i : β), p i → 0 < f i\nhf₁ : ∀ᶠ (i : β) in l, p i\nhf : ∀ (i : ℝ), 0 < i → ∀ᶠ (x : β) in l, f x ∈ closedBall 0 i\nε : ℝ\nhε : 0 < ε\ni : β\nhp : p i\nhi : f i ∈ closedBall 0 ε\n⊢ f i... | [
"α : Type u\ninst✝¹ : PseudoMetricSpace α\nβ : Type u_3\np : β → Prop\nf : β → ℝ\nl : Filter β\ninst✝ : l.NeBot\nhf₀ : ∀ (i : β), p i → 0 < f i\nhf₁ : ∀ᶠ (i : β) in l, p i\nhf : ∀ (i : ℝ), 0 < i → ∀ᶠ (x : β) in l, f x ∈ closedBall 0 i\nε : ℝ\nhε : 0 < ε\ni : β\nhp : p i\nhi : f i ∈ closedBall 0 ε\n⊢ f i ≤ ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 13
} | {
"line": 55,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\na : α\nb : ℝ≥0\nc : α\n⊢ ↑(Pi.mulSingle a b c) = Pi.mulSingle a (↑b) c",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : DecidableEq α\na : α\nb : ℝ≥0\nc : α\n⊢ ↑(Pi.mulSingle a b c) = Pi.mulSingle a (↑b) c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Basic | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 13
} | {
"line": 60,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\na : α\nb : ℝ≥0\nc : α\n⊢ ↑(Pi.single a b c) = Pi.single a (↑b) c",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : DecidableEq α\na : α\nb : ℝ≥0\nc : α\n⊢ ↑(Pi.single a b c) = Pi.single a (↑b) c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1250,
"column": 2
} | {
"line": 1250,
"column": 34
} | {
"line": 1250,
"column": 35
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\na : α\n⊢ a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\na : α\n⊢ a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Basic | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 29
} | {
"line": 172,
"column": 30
} | [
{
"pp": "ι : Sort u_3\nf : ι → ℝ≥0\na : ℝ≥0\n⊢ a * iInf f = ⨅ i, a * f i",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Sort u_3\nf : ι → ℝ≥0\na : ℝ≥0\n⊢ a * iInf f = ⨅ i, a * f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Basic | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 13
} | {
"line": 279,
"column": 14
} | [
{
"pp": "X : Type u_2\ninst✝ : PseudoMetricSpace X\ns : Set X\nε : ℝ\nhs : IsCompact (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s)\nhε : 0 < ε\nt : Finset ↑s\nhst : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s ⊆ ⋃ i ∈ t, ball (↑i) ε\n⊢ s ⊆ ⋃ x ∈ ↑(Finset.map { toFun := Subty... | [
"X : Type u_2\ninst✝ : PseudoMetricSpace X\ns : Set X\nε : ℝ\nhs : IsCompact (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s)\nhε : 0 < ε\nt : Finset ↑s\nhst : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s ⊆ ⋃ i ∈ t, ball (↑i) ε\n⊢ s ⊆ ⋃ x, ⋃ (i : x ∈ s), ⋃ (_ : ⟨x, ⋯⟩ ∈ t), ball x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 47
} | {
"line": 169,
"column": 48
} | [
{
"pp": "α : Type u_2\ninst✝ : EMetricSpace α\n⊢ Subsingleton α ↔ IndiscreteTopology α",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : EMetricSpace α\n⊢ Subsingleton α ↔ IndiscreteTopology α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 38
} | {
"line": 251,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nhs : ∀ ε > 0, ∃ t, t.Countable ∧ ⋃ x ∈ t, closedEBall x ε = univ\n⊢ ∀ ε > 0, ∃ t, t.Countable ∧ univ ⊆ ⋃ x ∈ t, closedEBall x ε",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"_private.Mathlib.... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nhs : ∀ ε > 0, ∃ t, t.Countable ∧ ⋃ x ∈ t, closedEBall x ε = univ\n⊢ ∀ ε > 0, ∃ t, t.Countable ∧ ⋃ x ∈ t, closedEBall x ε = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 325,
"column": 2
} | {
"line": 326,
"column": 9
} | {
"line": 326,
"column": 10
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nι : Sort u_2\nc : ι → Set α\nhs : IsCompact s\nhc₁ : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (c i)\nhc₂ : s ⊆ ⋃ i, c i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ i, eball x δ ⊆ c i",
"ppTerm": "?m.30",
"assigned": true,
"usedC... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nι : Sort u_2\nc : ι → Set α\nhs : IsCompact s\nhc₁ : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (c i)\nhc₂ : s ⊆ ⋃ i, c i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ i, {y | edist x y < δ} ⊆ c i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 330,
"column": 2
} | {
"line": 331,
"column": 9
} | {
"line": 331,
"column": 10
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : (x : α) → x ∈ s → Set α\nhs : IsCompact s\nhc : ∀ (x : α) (hx : x ∈ s), c x hx ∈ 𝓝 x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ⊆ c ↑y ⋯",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.edist_comm",
... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : (x : α) → x ∈ s → Set α\nhs : IsCompact s\nhc : ∀ (x : α) (hx : x ∈ s), c x hx ∈ 𝓝 x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, {y | edist x y < δ} ⊆ c ↑y ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 9
} | {
"line": 336,
"column": 10
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : α → Set α\nhs : IsCompact s\nhc : ∀ x ∈ s, c x ∈ 𝓝 x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ⊆ c y",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"PseudoEMetricSpace... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : α → Set α\nhs : IsCompact s\nhc : ∀ x ∈ s, c x ∈ 𝓝 x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, {y | edist x y < δ} ⊆ c y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 341,
"column": 2
} | {
"line": 342,
"column": 9
} | {
"line": 342,
"column": 10
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : (x : α) → x ∈ s → Set α\nhs : IsCompact s\nhc : ∀ (x : α) (hx : x ∈ s), c x hx ∈ 𝓝[s] x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ∩ s ⊆ c ↑y ⋯",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.edist_comm",... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : (x : α) → x ∈ s → Set α\nhs : IsCompact s\nhc : ∀ (x : α) (hx : x ∈ s), c x hx ∈ 𝓝[s] x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, {y | edist x y < δ} ∩ s ⊆ c ↑y ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 346,
"column": 2
} | {
"line": 347,
"column": 9
} | {
"line": 347,
"column": 10
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : α → Set α\nhs : IsCompact s\nhc : ∀ x ∈ s, c x ∈ 𝓝[s] x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ∩ s ⊆ c y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"PseudoEMetr... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : α → Set α\nhs : IsCompact s\nhc : ∀ x ∈ s, c x ∈ 𝓝[s] x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, {y | edist x y < δ} ∩ s ⊆ c y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 351,
"column": 32
} | {
"line": 351,
"column": 43
} | {
"line": 351,
"column": 44
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ t ∈ c, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] t\nhc₂ : s ⊆ ⋃ i, ↑i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, eball x δ ⊆ t",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ t ∈ c, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] t\nhc₂ : s ⊆ ⋃ i, ↑i\n⊢ ∃ δ, 0 < δ ∧ ∀ x ∈ s, ∃ t ∈ c, eball x δ ⊆ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 677,
"column": 4
} | {
"line": 677,
"column": 75
} | {
"line": 677,
"column": 76
} | [
{
"pp": "case h\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\na b c : FractionalIdeal R⁰ K\nhac : a ≤ c\nha : a ≠ 0\nhb : b ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_1 : Field K] [... | [
"case h\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\na b c : FractionalIdeal R⁰ K\nhac : a ≤ c\nha : a ≠ 0\nhb : b ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_1 : Field K] [inst_2 : Alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 686,
"column": 8
} | {
"line": 686,
"column": 19
} | {
"line": 686,
"column": 20
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁸ : CommRing R✝\nK✝ : Type u_2\ninst✝⁷ : Field K✝\ninst✝⁶ : Algebra R✝ K✝\ninst✝⁵ : IsFractionRing R✝ K✝\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\na c : FractionalIdeal R⁰ K\nhac... | [
"R✝ : Type u_1\ninst✝⁸ : CommRing R✝\nK✝ : Type u_2\ninst✝⁷ : Field K✝\ninst✝⁶ : Algebra R✝ K✝\ninst✝⁵ : IsFractionRing R✝ K✝\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\na c : FractionalIdeal R⁰ K\nhac : a ≤ c\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 23
} | {
"line": 31,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na : α\nh : (Ioi a).Nonempty\n⊢ closure[inst✝³] (Ioi a) = Ici a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Set.Ioi",
... | [
"case h₁\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na : α\nh : (Ioi a).Nonempty\n⊢ closure[inst✝³] (Ioi a) ⊆ Ici a",
"case h₂\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOr... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 23
} | {
"line": 74,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ioo a b) = Icc a b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"PartialOrder.toPre... | [
"case h₁\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ioo a b) ⊆ Icc a b",
"case h₂\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrder... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 23
} | {
"line": 91,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ioc a b) = Icc a b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Set.Ioc",
"P... | [
"case h₁\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ioc a b) ⊆ Icc a b",
"case h₂\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrder... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 23
} | {
"line": 103,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ico a b) = Icc a b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"PartialOrder.toPre... | [
"case h₁\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ico a b) ⊆ Icc a b",
"case h₂\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrder... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Pseudo.Lemmas | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 29
} | {
"line": 98,
"column": 30
} | [
{
"pp": "α : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : WeaklyLocallyCompactSpace α\nx : α\nthis : ∀ᶠ (r : ℝ) in 𝓝[>] 0, IsCompact (closedBall x r)\n⊢ ∃ r, 0 < r ∧ IsCompact (closedBall x r)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.ins... | [
"α : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : WeaklyLocallyCompactSpace α\nx : α\nthis : ∀ᶠ (r : ℝ) in 𝓝[>] 0, IsCompact (closedBall x r)\n⊢ ∃ r, IsCompact (closedBall x r) ∧ 0 < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Lemmas | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 9
} | {
"line": 123,
"column": 10
} | [
{
"pp": "α : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Sort u_3\nc : ι → Set α\nhs : IsCompact s\nhc₁ : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (c i)\nhc₂ : s ⊆ ⋃ i, c i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i",
"ppTerm": "?m.30",
"assigned": true,
"usedCo... | [
"α : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Sort u_3\nc : ι → Set α\nhs : IsCompact s\nhc₁ : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (c i)\nhc₂ : s ⊆ ⋃ i, c i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ i, {y | dist x y < δ} ⊆ c i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Lemmas | {
"line": 127,
"column": 32
} | {
"line": 127,
"column": 43
} | {
"line": 127,
"column": 44
} | [
{
"pp": "α : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ t ∈ c, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t\nhc₂ : s ⊆ ⋃ i, ↑i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ t ∈ c, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t\nhc₂ : s ⊆ ⋃ i, ↑i\n⊢ ∃ δ, 0 < δ ∧ ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 13
} | {
"line": 290,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\ninst✝¹ : SeparableSpace α\ninst✝ : Nontrivial α\n⊢ ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∉ s) ∧ ∀ (x : α), IsTop x → x ∉ s",
"ppTerm": "?m.25",
"assigned": fals... | [
"α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\ninst✝¹ : SeparableSpace α\ninst✝ : Nontrivial α\n⊢ ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∉ s) ∧ ∀ (x : α), IsTop x → x ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 49
} | {
"line": 220,
"column": 2
} | [
{
"pp": "γ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous[inst✝, inst✝³] f\n⊢ ⨆ s, f ↑s = ⨆ i, f i",
"ppTerm": "?m.20",
"assigned": true,
... | [
"case pos\nγ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous[inst✝, inst✝³] f\nh : BddAbove (range fun x ↦ f ↑x)\n⊢ ⨆ s, f ↑s = ⨆ i, f i",
"case neg\nγ ... | by_cases h : BddAbove (range (fun x : S ↦ f x)) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.Order.IsLUB | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 49
} | {
"line": 221,
"column": 50
} | [
{
"pp": "case pos\nγ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous[inst✝, inst✝³] f\nh : BddAbove (range fun x ↦ f ↑x)\n⊢ range f ⊆ closure[inst✝³] ... | [
"case pos\nγ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous[inst✝, inst✝³] f\nh : BddAbove (range fun x ↦ f ↑x)\n⊢ range f ⊆ closure[inst✝³] (f '' S)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 255,
"column": 51
} | {
"line": 255,
"column": 92
} | {
"line": 255,
"column": 93
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | [
"α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds s\nha : a ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 255,
"column": 51
} | {
"line": 255,
"column": 100
} | {
"line": 256,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | [] | simpa [nhds_eq_order, eq_true this] using hf c hc | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Order.IsLUB | {
"line": 255,
"column": 51
} | {
"line": 255,
"column": 100
} | {
"line": 256,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | [] | simpa [nhds_eq_order, eq_true this] using hf c hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IsLUB | {
"line": 255,
"column": 51
} | {
"line": 255,
"column": 100
} | {
"line": 256,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | [] | simpa [nhds_eq_order, eq_true this] using hf c hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IsLUB | {
"line": 258,
"column": 27
} | {
"line": 258,
"column": 58
} | {
"line": 258,
"column": 59
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | [
"α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds s\nha : a ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 18
} | {
"line": 46,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\ni : PseudoMetricSpace ((b : β) → X b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g ↦ ↑(Finset.univ.sup fun b ↦ nndist (f b) (g b))) ⋯ ⋯\ns : Set ((b ... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\ni : PseudoMetricSpace ((b : β) → X b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g ↦ ↑(Finset.univ.sup fun b ↦ nndist (f b) (g b))) ⋯ ⋯\ns : Set ((b : β) → X b)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 31
} | {
"line": 97,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Fintype β\ninst✝ : Nonempty β\na b : α\n⊢ (dist (fun x ↦ a) fun x ↦ b) = dist a b",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"pseudoMetricSpacePi",
"congrArg",
"i... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Fintype β\ninst✝ : Nonempty β\na b : α\n⊢ (edist (fun x ↦ a) fun x ↦ b).toReal = (edist a b).toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 19
} | {
"line": 173,
"column": 20
} | [
{
"pp": "case refine_2\nβ : Type u_2\ninst✝³ : Fintype β\nY : Type u_4\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : Zero Y\ninst✝ : DecidableEq β\ni j : β\na b : Y\nh : i ≠ j\n⊢ nndist a 0 ≤ nndist (Pi.single i a) (Pi.single j b)",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedF... | [
"case refine_2\nβ : Type u_2\ninst✝³ : Fintype β\nY : Type u_4\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : Zero Y\ninst✝ : DecidableEq β\ni j : β\na b : Y\nh : i ≠ j\n⊢ nndist a 0 ≤ nndist (Pi.single i a) (Pi.single j b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 32
} | {
"line": 174,
"column": 33
} | [
{
"pp": "case refine_3\nβ : Type u_2\ninst✝³ : Fintype β\nY : Type u_4\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : Zero Y\ninst✝ : DecidableEq β\ni j : β\na b : Y\nh : i ≠ j\n⊢ nndist b 0 ≤ nndist (Pi.single i a) (Pi.single j b)",
"ppTerm": "?refine_3",
"assigned": false,
"usedConstants": [],
"usedF... | [
"case refine_3\nβ : Type u_2\ninst✝³ : Fintype β\nY : Type u_4\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : Zero Y\ninst✝ : DecidableEq β\ni j : β\na b : Y\nh : i ≠ j\n⊢ nndist b 0 ≤ nndist (Pi.single i a) (Pi.single j b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Basic | {
"line": 58,
"column": 56
} | {
"line": 58,
"column": 67
} | {
"line": 58,
"column": 68
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\ns : Set γ\nε : ℝ\nhε : 0 < ε\nhs : s.Pairwise fun x y ↦ ε ≤ dist x y\n⊢ s.Pairwise fun x y ↦ (x, y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
... | [
"γ : Type w\ninst✝ : MetricSpace γ\ns : Set γ\nε : ℝ\nhε : 0 < ε\nhs : s.Pairwise fun x y ↦ ε ≤ dist x y\n⊢ s.Pairwise fun x y ↦ ε ≤ dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 76
} | {
"line": 63,
"column": 77
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstan... | [
"γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ ε ≤ dist (f x) (f y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 77
} | {
"line": 70,
"column": 78
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ ε ≤ dist (f x) (f y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 52
} | {
"line": 454,
"column": 53
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : FirstCountableTopology α\nl : Filter γ\ninst✝ : CountableInterFilter l\nf : γ → α\na d : α\nh : ∀ᶠ (x : γ) in l, f x < d\nhd : IsGLB (Ioi a) d\nH0 : Iic a = Iio d\nx : γ\nhx : f x < d\n⊢ ... | [
"α : Type u_1\nγ : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : FirstCountableTopology α\nl : Filter γ\ninst✝ : CountableInterFilter l\nf : γ → α\na d : α\nh : ∀ᶠ (x : γ) in l, f x < d\nhd : IsGLB (Ioi a) d\nH0 : Iic a = Iio d\nx : γ\nhx : f x < d\n⊢ f x ∈ Iio d"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Real | {
"line": 118,
"column": 61
} | {
"line": 119,
"column": 55
} | {
"line": 121,
"column": 0
} | [
{
"pp": "r : ℝ\n⊢ ENNReal.ofReal r ≤ ‖r‖ₑ",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"le_abs_self",
"Real.lattice",
"ENNReal.ofReal",
"abs",
"congrArg",
"SeminormedAddGroup.toNNNorm",
"ENNReal.ofReal_le_ofReal",
... | [] | by
rw [enorm_eq_ofReal_abs]; gcongr; exact le_abs_self _ | [anonymous] | Lean.Parser.Term.byTactic |
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