module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.Module.Defs | {
"line": 1320,
"column": 4
} | {
"line": 1320,
"column": 64
} | {
"line": 1321,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁷ : Preorder α\ninst✝⁶ : Preorder β\ninst✝⁵ : Preorder γ\ninst✝⁴ : SMul α β\ninst✝³ : SMul α γ\nf : β → γ\ninst✝² : Zero β\ninst✝¹ : Zero γ\ninst✝ : SMulPosStrictMono α γ\nhf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂\nsmul : ∀ (a : α) (b : β), f (a • b) = a •... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁷ : Preorder α\ninst✝⁶ : Preorder β\ninst✝⁵ : Preorder γ\ninst✝⁴ : SMul α β\ninst✝³ : SMul α γ\nf : β → γ\ninst✝² : Zero β\ninst✝¹ : Zero γ\ninst✝ : SMulPosStrictMono α γ\nhf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂\nb : β\na₁ a₂ : α\nha : a₁ < a₂\nsmul : ∀ (a : α) (b :... | simp only [← lt_iff_lt_of_le_iff_le' hf hf, zero, smul] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Ordinal.Univ | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 54
} | {
"line": 87,
"column": 6
} | [
{
"pp": "case mpr.type.refine_1\nb : Ordinal.{max (u + 1) v}\nβ : Type (max (u + 1) v)\ns : β → β → Prop\ninst✝¹ : IsWellOrder β s\nh : lift.{max (u + 1) v, max (u + 1) v} (type s) < lift.{max (u + 1) v, u + 1} (typeLT Ordinal.{u})\nf : s ↪r fun x1 x2 ↦ x1 < x2\nα : Type u\nr : α → α → Prop\ninst✝ : IsWellOrder... | [
"case mpr.type.refine_2\nb : Ordinal.{max (u + 1) v}\nβ : Type (max (u + 1) v)\ns : β → β → Prop\ninst✝¹ : IsWellOrder β s\nh : lift.{max (u + 1) v, max (u + 1) v} (type s) < lift.{max (u + 1) v, u + 1} (typeLT Ordinal.{u})\nf : s ↪r fun x1 x2 ↦ x1 < x2\nα : Type u\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhf : ∀... | · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Family | {
"line": 230,
"column": 2
} | {
"line": 232,
"column": 74
} | {
"line": 233,
"column": 2
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nf : α ⊕ β → Ordinal.{u}\ninst✝¹ : Small.{u, u_3} α\ninst✝ : Small.{u, u_4} β\n⊢ ∀ (i : α ⊕ β), f i ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Ordinal.partialOrd... | [
"α : Type u_3\nβ : Type u_4\nf : α ⊕ β → Ordinal.{u}\ninst✝¹ : Small.{u, u_3} α\ninst✝ : Small.{u, u_4} β\n⊢ ⨆ a, f (Sum.inl a) ≤ ⨆ i, f i",
"α : Type u_3\nβ : Type u_4\nf : α ⊕ β → Ordinal.{u}\ninst✝¹ : Small.{u, u_3} α\ninst✝ : Small.{u, u_4} β\n⊢ ⨆ b, f (Sum.inr b) ≤ ⨆ i, f i"
] | · rintro (i | i)
· exact le_max_of_le_left (Ordinal.le_iSup (fun x ↦ f (Sum.inl x)) i)
· exact le_max_of_le_right (Ordinal.le_iSup (fun x ↦ f (Sum.inr x)) i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 450,
"column": 76
} | {
"line": 451,
"column": 37
} | {
"line": 453,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_4}\n⊢ a - b = 0 ↔ a ≤ b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Ordinal.partialOrder",
"congrArg",
"_private.Mathlib.SetTheory.Ordinal.Arithmetic.0.Ordinal.sub_eq_zero_iff_le._simp_1_2",
"instIsBotZeroClass",
"AddMonoid.toAdd... | [] | by
simp [← nonpos_iff_eq_zero, sub_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Cofinality.Enum | {
"line": 130,
"column": 4
} | {
"line": 132,
"column": 8
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nf : α → α\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs))\n⊢ Subtype.val ∘ ⇑(enum s hs) = f → StrictMono f ∧ range f = s",
"ppTerm": "?mp",
"assigned": true,
... | [] | rintro rfl
use (Subtype.strictMono_coe _).comp (enum s hs).strictMono
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Cofinality.Enum | {
"line": 130,
"column": 4
} | {
"line": 132,
"column": 8
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nf : α → α\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs))\n⊢ Subtype.val ∘ ⇑(enum s hs) = f → StrictMono f ∧ range f = s",
"ppTerm": "?mp",
"assigned": true,
... | [] | rintro rfl
use (Subtype.strictMono_coe _).comp (enum s hs).strictMono
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1307,
"column": 49
} | {
"line": 1308,
"column": 34
} | {
"line": 1310,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ 0 < o.card ↔ 0 < o",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAdd... | [] | by
simpa using nat_lt_card (n := 0) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 817,
"column": 2
} | {
"line": 821,
"column": 80
} | {
"line": 823,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_4}\n⊢ IsSuccLimit (a + b) ↔ IsSuccLimit b ∨ b = 0 ∧ IsSuccLimit a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"lt_add_of_pos_right",
"Ne.pos",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZe... | [] | refine ⟨fun h ↦ ?_, by grind [isSuccLimit_add]⟩
rcases eq_or_ne b 0 with (rfl | h')
· grind
rw [← add_sub_cancel a b]
exact .inl <| isSuccLimit_sub h.isSuccPrelimit <| lt_add_of_pos_right a h'.pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 817,
"column": 2
} | {
"line": 821,
"column": 80
} | {
"line": 823,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_4}\n⊢ IsSuccLimit (a + b) ↔ IsSuccLimit b ∨ b = 0 ∧ IsSuccLimit a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"lt_add_of_pos_right",
"Ne.pos",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZe... | [] | refine ⟨fun h ↦ ?_, by grind [isSuccLimit_add]⟩
rcases eq_or_ne b 0 with (rfl | h')
· grind
rw [← add_sub_cancel a b]
exact .inl <| isSuccLimit_sub h.isSuccPrelimit <| lt_add_of_pos_right a h'.pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Log | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 64
} | {
"line": 110,
"column": 2
} | [
{
"pp": "b : ℕ\nhb : 1 < b\nx y : ℕ\nhy : y ≠ 0\n⊢ log b y < x ↔ y < b ^ x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Nat.lt_pow_self",
"instPowNat",
"instHDiv",
"HDiv.hDiv",
"instOfNatNat",
"LE.le",
"instLENat",
"Prod.fst",
"And.... | [
"b : ℕ\nhb : 1 < b\nx y : ℕ\nhy : y ≠ 0\nH₁ : b ^ (log.go y b y).snd ≤ y\nH₂ : y < b ^ ((log.go y b y).snd + 1)\n⊢ log b y < x ↔ y < b ^ x"
] | rcases log.go_spec hb hy (Nat.lt_pow_self hb) with ⟨-, H₁, H₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 71,
"column": 11
} | {
"line": 71,
"column": 30
} | {
"line": 71,
"column": 31
} | [
{
"pp": "a b : Ordinal.{u_1}\nha : a ≠ 0\nhb : IsSuccLimit b\n⊢ a ^ b = ⨆ x, a ^ ↑x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"_private.Mathlib.SetTheory.Ordinal.Exponential.0.Ordinal.opow_of_ne_zero",
"Ordinal.par... | [
"a b : Ordinal.{u_1}\nha : a ≠ 0\nhb : IsSuccLimit b\n⊢ (limitRecOn b 1 (fun x x_1 ↦ x_1 * a) fun o x f ↦ ⨆ x, f ↑x ⋯) =\n ⨆ x, limitRecOn (↑x) 1 (fun x x_1 ↦ x_1 * a) fun o x f ↦ ⨆ x, f ↑x ⋯"
] | opow_of_ne_zero ha, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Log | {
"line": 265,
"column": 39
} | {
"line": 265,
"column": 76
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case inl\nb c : ℕ\nhc : 1 < c\nhb : c ≤ b\n⊢ log b 0 ≤ log c 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"Nat.log_zero_right",
"id",
"instOfNatNat",
"LE.le",
"instLENat",
"Nat.instPreorder"... | [
"case inr\nb c n : ℕ\nhc : 1 < c\nhb : c ≤ b\nhn : n ≠ 0\n⊢ log b n ≤ log c n"
] | · rw [log_zero_right, log_zero_right] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Log | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 25
} | {
"line": 307,
"column": 26
} | [
{
"pp": "b : Bool\nn : ℕ\nhn : n ≠ 0\n⊢ log 2 (bit b n) = log 2 n + 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Nat.bit",
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"id",
"HDiv.hDiv",
"instMulNat",
"instOfNatNat",
"inst... | [
"b : Bool\nn : ℕ\nhn : n ≠ 0\n⊢ log 2 (bit b n / 2 * 2) = log 2 n + 1"
] | ← log_div_mul_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 957,
"column": 35
} | {
"line": 957,
"column": 47
} | {
"line": 957,
"column": 48
} | [
{
"pp": "case neg\no : Ordinal.{u_1}\ns : Set Ordinal.{u_1}\nhs : s.Nonempty\nho : 0 < o\nbdd : ¬BddAbove s\n⊢ o * sSup ∅ = sSup ((fun x ↦ o * x) '' s)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"MulZeroClass.t... | [
"case neg\no : Ordinal.{u_1}\ns : Set Ordinal.{u_1}\nhs : s.Nonempty\nho : 0 < o\nbdd : ¬BddAbove s\n⊢ o * ⊥ = sSup ((fun x ↦ o * x) '' s)"
] | csSup_empty, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 503,
"column": 2
} | {
"line": 507,
"column": 18
} | {
"line": 509,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_1}\n⊢ a < ω ^ succ b ↔ ∃ n, a < ω ^ b * ↑n",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"le_refl",
"Ordinal.instLinearOrder",
"Ordinal.mulRightMono",
"Ordinal.instAddRightMono",
"Preorder.toLT",
"HMul.hMul",
"Order.succ... | [] | refine ⟨fun ha ↦ ?_, fun ⟨n, hn⟩ ↦ hn.trans (opow_mul_lt_opow (natCast_lt_omega0 n) (lt_succ b))⟩
obtain ⟨c, hc, n, hn⟩ := (lt_omega0_opow (add_pos_of_right zero_lt_one b).ne').1 ha
refine ⟨n, hn.trans_le ?_⟩
grw [lt_succ_iff.1 hc]
exact omega0_pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 503,
"column": 2
} | {
"line": 507,
"column": 18
} | {
"line": 509,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_1}\n⊢ a < ω ^ succ b ↔ ∃ n, a < ω ^ b * ↑n",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"le_refl",
"Ordinal.instLinearOrder",
"Ordinal.mulRightMono",
"Ordinal.instAddRightMono",
"Preorder.toLT",
"HMul.hMul",
"Order.succ... | [] | refine ⟨fun ha ↦ ?_, fun ⟨n, hn⟩ ↦ hn.trans (opow_mul_lt_opow (natCast_lt_omega0 n) (lt_succ b))⟩
obtain ⟨c, hc, n, hn⟩ := (lt_omega0_opow (add_pos_of_right zero_lt_one b).ne').1 ha
refine ⟨n, hn.trans_le ?_⟩
grw [lt_succ_iff.1 hc]
exact omega0_pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 168,
"column": 4
} | {
"line": 172,
"column": 45
} | {
"line": 173,
"column": 2
} | [
{
"pp": "case inl\nop : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}\no a b : Ordinal.{u_1}\nm : ℕ\nha : a < (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[m] o\nn : ℕ\nhb : b < (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[n] o\nh : (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[m] o ≤ (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[n] ... | [] | use n + 1
rw [Function.iterate_succ']
apply (lt_succ _).trans_le
exact Ordinal.le_iSup (fun y : Set.Iio _ ×ˢ Set.Iio _ ↦ succ (op y.1.1 y.1.2))
⟨_, Set.mk_mem_prod (ha.trans_le h) hb⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 168,
"column": 4
} | {
"line": 172,
"column": 45
} | {
"line": 173,
"column": 2
} | [
{
"pp": "case inl\nop : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}\no a b : Ordinal.{u_1}\nm : ℕ\nha : a < (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[m] o\nn : ℕ\nhb : b < (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[n] o\nh : (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[m] o ≤ (fun x ↦ ⨆ y, succ (op (↑y).1 (↑y).2))^[n] ... | [] | use n + 1
rw [Function.iterate_succ']
apply (lt_succ _).trans_le
exact Ordinal.le_iSup (fun y : Set.Iio _ ×ˢ Set.Iio _ ↦ succ (op y.1.1 y.1.2))
⟨_, Set.mk_mem_prod (ha.trans_le h) hb⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 495,
"column": 31
} | {
"line": 497,
"column": 62
} | {
"line": 499,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_1}\nha : 0 < a\n⊢ a * b ≤ b ↔ a ^ ω ∣ b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"StrictMono.le_apply",
"Eq.mpr",
"Ordinal.isNormal_mul_right",
"Ordinal.instLinearOrder",
"Dvd.dvd",
"HMul.hMul",
"Ordinal.omega0",
... | [] | by
rw [← mul_eq_right_iff_opow_omega0_dvd]
exact (isNormal_mul_right ha).strictMono.le_apply.ge_iff_eq' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 36
} | {
"line": 384,
"column": 0
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ preAleph c.ord",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"Ordinal.card_le_preAleph",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Eq.mp",
"OrderIso",
... | [] | simpa using c.ord.card_le_preAleph | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 36
} | {
"line": 384,
"column": 0
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ preAleph c.ord",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"Ordinal.card_le_preAleph",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Eq.mp",
"OrderIso",
... | [] | simpa using c.ord.card_le_preAleph | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 36
} | {
"line": 384,
"column": 0
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ preAleph c.ord",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"Ordinal.card_le_preAleph",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Eq.mp",
"OrderIso",
... | [] | simpa using c.ord.card_le_preAleph | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 17
} | {
"line": 125,
"column": 17
} | [
{
"pp": "case e'_4\na b : Cardinal.{u_1}\nh : ℵ₀ ≤ a\n⊢ max a b = max a b * max a b",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"Cardinal",
"congrArg",
"Cardinal.mul_eq_self",
"SemilatticeSup.t... | [
"case e'_4\na b : Cardinal.{u_1}\nh : ℵ₀ ≤ a\n⊢ max a b = max a b",
"case e'_4\na b : Cardinal.{u_1}\nh : ℵ₀ ≤ a\n⊢ ℵ₀ ≤ max a b"
] | mul_eq_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 53
} | {
"line": 204,
"column": 4
} | [
{
"pp": "case neg\na b : Cardinal.{u_1}\nh : a * b = a\nha : a = 0 ∨ b = 0 ∨ (∃ n, a = ↑n) ∧ ∃ n, b = ↑n\nh2a : ¬a = 0\nhb : b ≠ 0\n⊢ b = 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Cardinal.instOne",
"Cardinal",
"Exists",
"Cardinal.instMul... | [
"case neg.inl\nb : Cardinal.{u_1}\nhb : b ≠ 0\nh : 0 * b = 0\nh2a : ¬0 = 0\n⊢ b = 1",
"case neg.inr.inl\na : Cardinal.{u_1}\nh2a : ¬a = 0\nh : a * 0 = a\nhb : 0 ≠ 0\n⊢ 0 = 1",
"case neg.inr.inr\nn : ℕ\nh2a : ¬↑n = 0\nm : ℕ\nhb : ↑m ≠ 0\nh : ↑n * ↑m = ↑n\n⊢ ↑m = 1"
] | rcases ha with (rfl | rfl | ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 30
} | {
"line": 94,
"column": 30
} | [
{
"pp": "case neg\nι : Type u\nc : Cardinal.{v}\nf : ι → Ordinal.{v}\nhι : Cardinal.lift.{v, u} #ι ≤ Cardinal.lift.{u, v} c\nhf : ∀ (i : ι), (f i).card ≤ c\nhc : ℵ₀ ≤ c\n⊢ (⨆ i, f i).card ≤ c",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg"... | [
"case neg\nι : Type u\nc : Cardinal.{v}\nf : ι → Ordinal.{v}\nhι : Cardinal.lift.{v, u} #ι ≤ Cardinal.lift.{u, v} c\nhf : ∀ (i : ι), (f i).card ≤ c\nhc : ℵ₀ ≤ c\n⊢ Cardinal.lift.{u, v} (⨆ i, f i).card ≤ Cardinal.lift.{u, v} c"
] | ← Cardinal.lift_le.{u} | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 258,
"column": 20
} | {
"line": 258,
"column": 34
} | {
"line": 258,
"column": 35
} | [
{
"pp": "case inr.inl\na b : Cardinal.{u_1}\nha : a < ℵ₀\nhb : ℵ₀ ≤ b\n⊢ b + a ≤ max (max a b) ℵ₀",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSe... | [
"case inr.inl\na b : Cardinal.{u_1}\nha : a < ℵ₀\nhb : ℵ₀ ≤ b\n⊢ max b a ≤ max (max a b) ℵ₀"
] | add_eq_max hb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 152,
"column": 2
} | {
"line": 157,
"column": 72
} | {
"line": 159,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_1}\n⊢ (a ^ b).card ≤ max ℵ₀ (max a.card b.card)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"le_max_right",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Ordinal.monoid",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Cardinal",
... | [] | obtain ⟨n, rfl⟩ | ha := eq_natCast_or_omega0_le a
· obtain ⟨m, rfl⟩ | hb := eq_natCast_or_omega0_le b
· rw [opow_natCast, ← natCast_pow, card_nat]
exact le_max_of_le_left natCast_le_aleph0
· exact (card_opow_le_of_omega0_le_right _ hb).trans (le_max_right _ _)
· exact (card_opow_le_of_omega0_le_left h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 152,
"column": 2
} | {
"line": 157,
"column": 72
} | {
"line": 159,
"column": 0
} | [
{
"pp": "a b : Ordinal.{u_1}\n⊢ (a ^ b).card ≤ max ℵ₀ (max a.card b.card)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"le_max_right",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Ordinal.monoid",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Cardinal",
... | [] | obtain ⟨n, rfl⟩ | ha := eq_natCast_or_omega0_le a
· obtain ⟨m, rfl⟩ | hb := eq_natCast_or_omega0_le b
· rw [opow_natCast, ← natCast_pow, card_nat]
exact le_max_of_le_left natCast_le_aleph0
· exact (card_opow_le_of_omega0_le_right _ hb).trans (le_max_right _ _)
· exact (card_opow_le_of_omega0_le_left h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 18
} | {
"line": 123,
"column": 18
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ (ℵ_ (o + 1)).IsRegular",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Order.succ",
"Cardinal.aleph",
"Cardinal.IsRegular",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
... | [
"o : Ordinal.{u_1}\n⊢ (succ (ℵ_ o)).IsRegular"
] | ← succ_aleph | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 33
} | {
"line": 438,
"column": 2
} | [
{
"pp": "ι : Type u\nf : ι → Cardinal.{max u v}\nhι : ℵ₀ ≤ #ι\nh : lift.{v, u} #ι ≤ ⨆ i, f i\n⊢ sum f = ⨆ i, f i",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"iSup",
"Cardinal.lift",
"Cardinal.lift_id'",
"id",
... | [
"ι : Type u\nf : ι → Cardinal.{max u v}\nhι : ℵ₀ ≤ #ι\nh : lift.{v, u} #ι ≤ ⨆ i, f i\n⊢ sum f = lift.{u, max u v} (⨆ i, f i)"
] | rw [← lift_id'.{u, v} (iSup _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 471,
"column": 85
} | {
"line": 473,
"column": 27
} | {
"line": 475,
"column": 0
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u}\n⊢ (o.blsub f).cof ≤ o.card",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Ordinal.cof_blsub_le_lift",
"Cardinal.lift",
"id",
"LE.le",
"... | [] | by
rw [← o.card.lift_id]
exact cof_blsub_le_lift f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 28
} | {
"line": 731,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : Nonempty α\n⊢ #(List α) = max ℵ₀ #α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal",
"Finite",
"Cardinal.mk",
"finite_or_infinite",
"SemilatticeSup.toMax",
"Cardinal.aleph0",
... | [
"case inl\nα : Type u\ninst✝ : Nonempty α\nh✝ : Finite α\n⊢ #(List α) = max ℵ₀ #α",
"case inr\nα : Type u\ninst✝ : Nonempty α\nh✝ : Infinite α\n⊢ #(List α) = max ℵ₀ #α"
] | cases finite_or_infinite α | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 745,
"column": 2
} | {
"line": 745,
"column": 28
} | {
"line": 746,
"column": 2
} | [
{
"pp": "α : Type u\n⊢ #(List α) ≤ max ℵ₀ #α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal",
"Finite",
"Cardinal.mk",
"finite_or_infinite",
"SemilatticeSup.toMax",
"Cardinal.aleph0",
"LE.le",
"Cond... | [
"case inl\nα : Type u\nh✝ : Finite α\n⊢ #(List α) ≤ max ℵ₀ #α",
"case inr\nα : Type u\nh✝ : Infinite α\n⊢ #(List α) ≤ max ℵ₀ #α"
] | cases finite_or_infinite α | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 816,
"column": 4
} | {
"line": 816,
"column": 44
} | {
"line": 816,
"column": 45
} | [
{
"pp": "case inj'\nα : Type u\ns : Set α\nc : Cardinal.{u}\nt : Set α\nht1 : t ⊆ s\nht2 : #↑t ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : #↑t' ≤ c\nh : Subtype.val ⁻¹' t = Subtype.val ⁻¹' t'\n⊢ t = t'",
"ppTerm": "?inj'",
"assigned": true,
"usedConstants": [
"Set.preimage_eq_preimage'",
"Mem... | [
"case inj'.refine_1\nα : Type u\ns : Set α\nc : Cardinal.{u}\nt : Set α\nht1 : t ⊆ s\nht2 : #↑t ≤ c\nt' : Set α\nh1t' : t' ⊆ s\nh2t' : #↑t' ≤ c\nh : Subtype.val ⁻¹' t = Subtype.val ⁻¹' t'\n⊢ t ⊆ range Subtype.val",
"case inj'.refine_2\nα : Type u\ns : Set α\nc : Cardinal.{u}\nt : Set α\nht1 : t ⊆ s\nht2 : #↑t ≤ c... | refine (preimage_eq_preimage' ?_ ?_).1 h | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.DFinsupp.Defs | {
"line": 740,
"column": 4
} | {
"line": 740,
"column": 79
} | {
"line": 741,
"column": 4
} | [
{
"pp": "case h.empty\nι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\nH : ∀ (i : ι), i ∈ 0 ∨ f i = 0\n⊢ p { toFun := f... | [
"case h.empty\nι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddZeroClass (β i)\np : (Π₀ (i : ι), β i) → Prop\nh0 : p 0\nha : ∀ (i : ι) (b : β i) (f : Π₀ (i : ι), β i), f i = 0 → b ≠ 0 → p f → p (single i b + f)\nf : (i : ι) → β i\nH : ∀ (i : ι), i ∈ 0 ∨ f i = 0\nthis : f = 0\n⊢ p { toFun :=... | have : f = 0 := funext fun i => (H i).resolve_left (Multiset.notMem_zero _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 74
} | {
"line": 259,
"column": 2
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : IsReflexive R M\nM' : Type u_6\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nf g : M →ₗ[R] M'\nh : Injective ⇑(Dual.eval R M')\nhfg : Dual.eval R M' ∘ₗ f ∘ₗ ↑(evalEquiv R M).symm = Dual.eval ... | [
"R : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : IsReflexive R M\nM' : Type u_6\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nf g : M →ₗ[R] M'\nh : Injective ⇑(Dual.eval R M')\nhfg : f = g\n⊢ f = g"
] | rw [propext (cancel_left h), LinearEquiv.eq_comp_toLinearMap_iff] at hfg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 441,
"column": 6
} | {
"line": 441,
"column": 10
} | {
"line": 442,
"column": 6
} | [
{
"pp": "case h\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ns : Finset ι\np : ι → Submodule R N\na : N\nμ : Π₀ (i : ι), ↥(⨆ (_ : i ∈ s), p i)\nhμ : (μ.sum fun x xi ↦ ↑xi) = a\n⊢ ∑ i ∈ s, ↑((fun i ↦ ⟨↑(μ i), ⋯⟩) i) = μ.sum fun x xi ↦ ↑xi",
"pp... | [
"case h\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ns : Finset ι\np : ι → Submodule R N\na : N\nμ : Π₀ (i : ι), ↥(⨆ (_ : i ∈ s), p i)\nhμ : (μ.sum fun x xi ↦ ↑xi) = a\n⊢ (μ.sum fun x xi ↦ ↑xi) = ∑ i ∈ s, ↑((fun i ↦ ⟨↑(μ i), ⋯⟩) i)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 489,
"column": 11
} | {
"line": 489,
"column": 32
} | {
"line": 489,
"column": 32
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : ι → Submodule R N\ni : ι\n⊢ (∀ (x : ↥(p i)) (x_1 : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i x_1) = ↑x → ↑x = 0) ↔\n ∀ (x : ↥(p i)) (v : Π₀ (i... | [] | Submodule.coe_eq_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 553,
"column": 2
} | {
"line": 553,
"column": 12
} | {
"line": 554,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i v) = ↑x → x = 0\n⊢ ∀ (m : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) ... | [
"ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i v) = ↑x → x = 0\nm : Π₀ (i : ι), ↥(p i)\nhm : ((lsum ℕ) fun i ↦ (p i)... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 27
} | {
"line": 559,
"column": 28
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i v) = ↑x → x = 0\nm : Π₀ (i : ι), ↥(p i)\ni : ι\nhm : ((ls... | [
"ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i v) = ↑x → x = 0\nm : Π₀ (i : ι), ↥(p i)\ni : ι\nhm : ((lsum ℕ) fun i ... | add_eq_zero_iff_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 25
} | {
"line": 287,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\n⊢ LinearIndependent R ![-x, y] ↔ LinearIndependent R ![x, y]",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"LinearIndependent.pair_iff",
"Eq.mpr",
"NegZeroClass.t... | [
"R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\n⊢ (∀ (s t : R), s • -x + t • y = 0 → s = 0 ∧ t = 0) ↔ ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0"
] | rw [pair_iff, pair_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.BilinearMap | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 53
} | {
"line": 116,
"column": 53
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nS : Type u_3\nS₂ : Type u_4\ninst✝²⁹ : Semiring R\ninst✝²⁸ : Semiring R₂\ninst✝²⁷ : Semiring S\ninst✝²⁶ : Semiring S₂\nM : Type u_5\nM₂ : Type u_6\nN : Type u_7\nN₂ : Type u_8\nP : Type u_9\nP₂ : Type u_10\nPₗ : Type u_11\ninst✝²⁵ : AddCommMonoid M\ninst✝²⁴ : AddCommMonoid ... | [] | simp only [map_add, add_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.BilinearMap | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 53
} | {
"line": 116,
"column": 53
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nS : Type u_3\nS₂ : Type u_4\ninst✝²⁹ : Semiring R\ninst✝²⁸ : Semiring R₂\ninst✝²⁷ : Semiring S\ninst✝²⁶ : Semiring S₂\nM : Type u_5\nM₂ : Type u_6\nN : Type u_7\nN₂ : Type u_8\nP : Type u_9\nP₂ : Type u_10\nPₗ : Type u_11\ninst✝²⁵ : AddCommMonoid M\ninst✝²⁴ : AddCommMonoid ... | [] | simp only [map_add, add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.BilinearMap | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 53
} | {
"line": 116,
"column": 53
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nS : Type u_3\nS₂ : Type u_4\ninst✝²⁹ : Semiring R\ninst✝²⁸ : Semiring R₂\ninst✝²⁷ : Semiring S\ninst✝²⁶ : Semiring S₂\nM : Type u_5\nM₂ : Type u_6\nN : Type u_7\nN₂ : Type u_8\nP : Type u_9\nP₂ : Type u_10\nPₗ : Type u_11\ninst✝²⁵ : AddCommMonoid M\ninst✝²⁴ : AddCommMonoid ... | [] | simp only [map_add, add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Isomorphisms | {
"line": 78,
"column": 75
} | {
"line": 78,
"column": 92
} | {
"line": 78,
"column": 92
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : Submodule R M\n⊢ comap p.subtype (p ⊓ p') ≤ comap p.subtype (map (p ⊔ p').subtype (comap (p ⊔ p').subtype p'))",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"S... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : Submodule R M\n⊢ comap p.subtype (p ⊓ p') ≤ comap p.subtype ((p ⊔ p') ⊓ p')"
] | map_comap_subtype | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.Module | {
"line": 164,
"column": 2
} | {
"line": 168,
"column": 17
} | {
"line": 170,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : Semiring R\ninst✝ : Module R M\nl : NF R M\nx : M\nh : x = l.eval\nr : R\n⊢ (r • l).eval = r • x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Mathlib.Tactic.Module.NF",
... | [] | unfold NF.eval at h ⊢
simp only [h, smul_sum, map_map, NF.smul_apply]
congr
ext p
simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.Module | {
"line": 164,
"column": 2
} | {
"line": 168,
"column": 17
} | {
"line": 170,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : Semiring R\ninst✝ : Module R M\nl : NF R M\nx : M\nh : x = l.eval\nr : R\n⊢ (r • l).eval = r • x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Mathlib.Tactic.Module.NF",
... | [] | unfold NF.eval at h ⊢
simp only [h, smul_sum, map_map, NF.smul_apply]
congr
ext p
simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.FreeAbelianGroup.Finsupp | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 28
} | {
"line": 66,
"column": 29
} | [
{
"pp": "X : Type u_1\nx✝ : X\n⊢ (liftAddHom fun x ↦ (smulAddHom ℤ (FreeAbelianGroup X)).flip (of x)) (single x✝ 1) =\n (AddMonoidHom.id (FreeAbelianGroup X)) (of x✝)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"congrArg",
... | [
"X : Type u_1\nx✝ : X\n⊢ ((smulAddHom ℤ (FreeAbelianGroup X)).flip (of x✝)) 1 = (AddMonoidHom.id (FreeAbelianGroup X)) (of x✝)"
] | liftAddHom_apply_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 463,
"column": 4
} | {
"line": 463,
"column": 14
} | {
"line": 464,
"column": 4
} | [
{
"pp": "case right\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nI : Set ι\nhIlinind : LinearIndepOn R v I\ni : ι\nhi : i ∉ I\nJ : Set ι := I ∪ {i}\nhImaximal : LinearIndepOn R v J → I = J\nhJ : J = I ∪ {i}\nmemJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ ... | [
"case right\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nI : Set ι\nhIlinind : LinearIndepOn R v I\ni : ι\nhi : i ∉ I\nJ : Set ι := I ∪ {i}\nhImaximal : LinearIndepOn R v J → I = J\nhJ : J = I ∪ {i}\nmemJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I\nhiJ :... | rw [sum_f] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 910,
"column": 26
} | {
"line": 910,
"column": 53
} | {
"line": 912,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝¹ : Ring R\ninst✝ : MulOneClass M\nn : ℕ\n⊢ single 1 ↑(Int.negSucc n) = -↑(n + 1)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtracti... | [] | simp [natCast_def, one_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 910,
"column": 26
} | {
"line": 910,
"column": 53
} | {
"line": 912,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝¹ : Ring R\ninst✝ : MulOneClass M\nn : ℕ\n⊢ single 1 ↑(Int.negSucc n) = -↑(n + 1)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtracti... | [] | simp [natCast_def, one_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 910,
"column": 26
} | {
"line": 910,
"column": 53
} | {
"line": 912,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝¹ : Ring R\ninst✝ : MulOneClass M\nn : ℕ\n⊢ single 1 ↑(Int.negSucc n) = -↑(n + 1)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtracti... | [] | simp [natCast_def, one_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 864,
"column": 13
} | {
"line": 864,
"column": 50
} | {
"line": 866,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\na : α\nn : ℕ\n⊢ {a} ^ (n + 1) = {a ^ (n + 1)}",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | [] | by simp [pow_succ, singleton_pow _ n] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 988,
"column": 2
} | {
"line": 988,
"column": 21
} | {
"line": 988,
"column": 21
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns t : Finset α\nht : 1 ∈ t\n⊢ s ⊆ s / t",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Finset.divisionMonoid",
"Monoid.toM... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns t : Finset α\nht : 1 ∈ t\n⊢ s ⊆ s * t⁻¹"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 991,
"column": 2
} | {
"line": 991,
"column": 21
} | {
"line": 991,
"column": 21
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns t : Finset α\nhs : 1 ∈ s\n⊢ t⁻¹ ⊆ s / t",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns t : Finset α\nhs : 1 ∈ s\n⊢ t⁻¹ ⊆ s * t⁻¹"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 31
} | {
"line": 422,
"column": 6
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhcard : 1 < #C ∨ 1 < #D\nhC : 1 ∈ C\nhD : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nhc1 : 1 ∈... | [
"case neg.inl\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhC✝ : 1 ∈ C\nhD : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nhc1 : 1 ∈ C\nhd2 : 1 ∈ D\nhc2 : 1 ∈ C\nhd... | rcases hcard with hC | hD | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.StdBasis | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 8
} | {
"line": 94,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_2\nη : Type u_4\nιs : η → Type u_5\nMs : η → Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : (i : η) → AddCommMonoid (Ms i)\ninst✝² : (i : η) → Module R (Ms i)\ninst✝¹ : Fintype η\ninst✝ : DecidableEq η\ns : (j : η) → Basis (ιs j) R (Ms j)\nj : η\ni i' : ιs j\n⊢ ({ toFun := fun f i ↦ (s i)... | [
"case pos\nR : Type u_2\nη : Type u_4\nιs : η → Type u_5\nMs : η → Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : (i : η) → AddCommMonoid (Ms i)\ninst✝² : (i : η) → Module R (Ms i)\ninst✝¹ : Fintype η\ninst✝ : DecidableEq η\ns : (j : η) → Basis (ιs j) R (Ms j)\nj : η\ni i' : ιs j\n⊢ (Finsupp.single ⟨j, i⟩ 1) ⟨j, i'⟩ =\n ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.StdBasis | {
"line": 89,
"column": 4
} | {
"line": 94,
"column": 31
} | {
"line": 95,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_2\nη : Type u_4\nιs : η → Type u_5\nMs : η → Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : (i : η) → AddCommMonoid (Ms i)\ninst✝² : (i : η) → Module R (Ms i)\ninst✝¹ : Fintype η\ninst✝ : DecidableEq η\ns : (j : η) → Basis (ιs j) R (Ms j)\nj : η\ni : ιs j\nj' : η\ni' : ιs j'\nhj : j = j'\... | [] | subst hj
simp only [Pi.basis, LinearEquiv.trans_apply,
LinearEquiv.piCongrRight, Finsupp.sigmaFinsuppLEquivPiFinsupp_symm_apply,
Basis.repr_symm_apply, LinearEquiv.coe_mk]
symm
simp [Finsupp.single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.StdBasis | {
"line": 89,
"column": 4
} | {
"line": 94,
"column": 31
} | {
"line": 95,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_2\nη : Type u_4\nιs : η → Type u_5\nMs : η → Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : (i : η) → AddCommMonoid (Ms i)\ninst✝² : (i : η) → Module R (Ms i)\ninst✝¹ : Fintype η\ninst✝ : DecidableEq η\ns : (j : η) → Basis (ιs j) R (Ms j)\nj : η\ni : ιs j\nj' : η\ni' : ιs j'\nhj : j = j'\... | [] | subst hj
simp only [Pi.basis, LinearEquiv.trans_apply,
LinearEquiv.piCongrRight, Finsupp.sigmaFinsuppLEquivPiFinsupp_symm_apply,
Basis.repr_symm_apply, LinearEquiv.coe_mk]
symm
simp [Finsupp.single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 532,
"column": 14
} | {
"line": 532,
"column": 60
} | {
"line": 533,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\... | [
"case refine_2.refine_1.inr\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Fin... | rcases hc with hc | hc; · exact ihA _ (hc.2 _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 532,
"column": 14
} | {
"line": 532,
"column": 60
} | {
"line": 533,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\... | [
"case refine_2.refine_1.inr\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Fin... | rcases hc with hc | hc; · exact ihA _ (hc.2 _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 532,
"column": 14
} | {
"line": 532,
"column": 60
} | {
"line": 533,
"column": 4
} | [
{
"pp": "case refine_1\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Fins... | [
"case refine_1.inr\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Finset ((i :... | rcases hc with hc | hc; · exact ihA _ (hc.2 _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 532,
"column": 14
} | {
"line": 532,
"column": 60
} | {
"line": 533,
"column": 4
} | [
{
"pp": "case refine_1\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Fins... | [
"case refine_1.inr\nG✝ : Type u\nH : Type v\ninst✝³ : Mul G✝\ninst✝² : Mul H\nι : Type u_2\nG : ι → Type u_1\ninst✝¹ : (i : ι) → Mul (G i)\ninst✝ : ∀ (i : ι), TwoUniqueProds (G i)\nA✝ : Finset ((i : ι) → G i)\nx✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x1 x2 ↦ x1 ⊂ x2 := isWellFounded_ssubset\nA : Finset ((i :... | rcases hc with hc | hc; · exact ihA _ (hc.2 _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Finiteness.Finsupp | {
"line": 76,
"column": 2
} | {
"line": 80,
"column": 27
} | {
"line": 81,
"column": 2
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nf : M →ₗ[R] P\ns : Submodule R M\nthis✝¹ : DecidableEq R\nthis✝ : DecidableEq M\nthis : DecidableEq P\nt1 : Finset P\nht1 : span R ↑t1 = map f s\... | [
"case a.refine_1\nR : Type u_1\nM : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nf : M →ₗ[R] P\ns : Submodule R M\nthis✝¹ : DecidableEq R\nthis✝ : DecidableEq M\nthis : DecidableEq P\nt1 : Finset P\nht1 : span R ↑t1 = map f s\nt2... | refine
mem_sup.2
⟨(linearCombination R id).toFun ((lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
x - linearCombination R id ((lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
add_sub_cancel _ _⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.RelSeries | {
"line": 438,
"column": 10
} | {
"line": 438,
"column": 41
} | {
"line": 439,
"column": 10
} | [
{
"pp": "case e'_4.h\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\np : RelSeries r\ni : Fin p.length\na : α\nprev_connect : (p.toFun i.castSucc, a) ∈ r\nconnect_next : (a, p.toFun i.succ) ∈ r\nm : Fin (p.length + 1)\nx : α := i.succ.castSucc.insertNth a p.toFun m.castSucc\ny : α := i.succ.castSuc... | [
"case e'_4\nα : Type u_1\nr : SetRel α α\nβ : Type u_2\ns : SetRel β β\np : RelSeries r\ni : Fin p.length\na : α\nprev_connect : (p.toFun i.castSucc, a) ∈ r\nconnect_next : (a, p.toFun i.succ) ∈ r\nm : Fin (p.length + 1)\nx : α := i.succ.castSucc.insertNth a p.toFun m.castSucc\ny : α := i.succ.castSucc.insertNth a ... | · change i.1 + 1 < m.1 + 1; lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Exact.Basic | {
"line": 443,
"column": 4
} | {
"line": 443,
"column": 87
} | {
"line": 444,
"column": 2
} | [
{
"pp": "case refine_3\nR✝ : Type u_1\nM✝ : Type u_2\nM' : Type u_3\nN✝ : Type u_4\nN' : Type u_5\nP✝ : Type u_6\nP' : Type u_7\ninst✝¹³ : Semiring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup N✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ N✝\ninst✝⁷ : Module R✝ P✝\nf✝ : M✝ →ₗ... | [] | rw [LinearMap.comp_assoc, (LinearEquiv.eq_toLinearMap_symm_comp _ _).mp e.2.1]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Exact.Basic | {
"line": 443,
"column": 4
} | {
"line": 443,
"column": 87
} | {
"line": 444,
"column": 2
} | [
{
"pp": "case refine_3\nR✝ : Type u_1\nM✝ : Type u_2\nM' : Type u_3\nN✝ : Type u_4\nN' : Type u_5\nP✝ : Type u_6\nP' : Type u_7\ninst✝¹³ : Semiring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup N✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ N✝\ninst✝⁷ : Module R✝ P✝\nf✝ : M✝ →ₗ... | [] | rw [LinearMap.comp_assoc, (LinearEquiv.eq_toLinearMap_symm_comp _ _).mp e.2.1]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 32
} | {
"line": 295,
"column": 4
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : Preorder α\na b : α\nhab : a < b\nn : ℕ\nhfin : height a = ↑n\n⊢ ↑n + 1 ≤ height b",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"instAddENat",
"instPreorderENat",
"Nat.c... | [
"case coe\nα : Type u_1\ninst✝ : Preorder α\na b : α\nhab : a < b\nn : ℕ\nhfin : height a = ↑n\n⊢ ↑n < height b"
] | apply Order.add_one_le_of_lt | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.KrullDimension | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 32
} | {
"line": 313,
"column": 4
} | [
{
"pp": "case coe\nα : Type u_1\ninst✝ : Preorder α\na b : α\nhab : b < a\nn : ℕ\nhfin : coheight a = ↑n\n⊢ ↑n + 1 ≤ coheight b",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"instAddENat",
"instPreorderENat",
"N... | [
"case coe\nα : Type u_1\ninst✝ : Preorder α\na b : α\nhab : b < a\nn : ℕ\nhfin : coheight a = ↑n\n⊢ ↑n < coheight b"
] | apply Order.add_one_le_of_lt | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.KrullDimension | {
"line": 1071,
"column": 2
} | {
"line": 1071,
"column": 6
} | {
"line": 1072,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ⨆ x, height ↑x + 1 = ⨆ i, height i + 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"WithTop.instPreorder",
"iSup",
"CompletelyDistr... | [
"α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Nonempty α\n⊢ ⨆ i, height i + 1 = ⨆ x, height ↑x + 1"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.TensorProduct.Defs | {
"line": 432,
"column": 70
} | {
"line": 434,
"column": 74
} | {
"line": 436,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_7\nN : Type u_8\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\n⊢ Submodule.map₂ (mk R M N) ⊤ ⊤ = ⊤",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
... | [] | by
rw [← top_le_iff, ← span_tmul_eq_top, Submodule.map₂_eq_span_image2]
exact Submodule.span_mono fun _ ⟨m, n, h⟩ => ⟨m, trivial, n, trivial, h⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.NonZeroDivisors | {
"line": 99,
"column": 28
} | {
"line": 99,
"column": 39
} | {
"line": 99,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nS : Submonoid R\n⊢ S ≤ nonZeroDivisorsLeft R ⊓ nonZeroDivisorsRight R ↔ ∀ (s : ↥S), IsRegular ↑s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"nonZeroDivisorsRight",
"CompleteLattice.toLattice",
"congrArg",
"P... | [
"R : Type u_1\ninst✝ : Ring R\nS : Submonoid R\n⊢ S ≤ nonZeroDivisorsLeft R ∧ S ≤ nonZeroDivisorsRight R ↔ ∀ (s : ↥S), IsRegular ↑s"
] | le_inf_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 28
} | {
"line": 139,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH : IsCoprime x (y * z)\n⊢ IsCoprime x y",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"isCoprime_comm",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"id",
"in... | [
"R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH : IsCoprime (y * z) x\n⊢ IsCoprime y x"
] | rw [isCoprime_comm] at H ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 180,
"column": 16
} | {
"line": 180,
"column": 72
} | {
"line": 180,
"column": 72
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nx y : R\nH : IsCoprime x y\nS : Type v\ninst✝ : CommSemiring S\nf : R →+* S\na b : R\nh : a * x + b * y = 1\n⊢ f a * f x + f b * f y = 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWith... | [] | rw [← f.map_mul, ← f.map_mul, ← f.map_add, h, f.map_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 180,
"column": 16
} | {
"line": 180,
"column": 72
} | {
"line": 180,
"column": 72
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nx y : R\nH : IsCoprime x y\nS : Type v\ninst✝ : CommSemiring S\nf : R →+* S\na b : R\nh : a * x + b * y = 1\n⊢ f a * f x + f b * f y = 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWith... | [] | rw [← f.map_mul, ← f.map_mul, ← f.map_add, h, f.map_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coprime.Basic | {
"line": 180,
"column": 16
} | {
"line": 180,
"column": 72
} | {
"line": 180,
"column": 72
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nx y : R\nH : IsCoprime x y\nS : Type v\ninst✝ : CommSemiring S\nf : R →+* S\na b : R\nh : a * x + b * y = 1\n⊢ f a * f x + f b * f y = 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWith... | [] | rw [← f.map_mul, ← f.map_mul, ← f.map_add, h, f.map_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 31
} | {
"line": 341,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (z * x + y)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Distrib.toAdd",
"IsCoprime.ad... | [] | rw [add_comm]
exact h.add_mul_right_right z | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coprime.Basic | {
"line": 338,
"column": 2
} | {
"line": 339,
"column": 31
} | {
"line": 341,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (z * x + y)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Distrib.toAdd",
"IsCoprime.ad... | [] | rw [add_comm]
exact h.add_mul_right_right z | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coprime.Basic | {
"line": 469,
"column": 4
} | {
"line": 471,
"column": 31
} | {
"line": 473,
"column": 0
} | [
{
"pp": "case refine_2\nm n : ℕ\nh : m = 1 ∨ n = 1\n⊢ IsCoprime m n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Or.casesOn",
"Nat",
"isCoprime_one_right",
"Nat.instCommSemiring",
"Eq.ndrec",
"Or",
"OfNat.ofNat",
... | [] | obtain rfl | rfl := h
· exact isCoprime_one_left
· exact isCoprime_one_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coprime.Basic | {
"line": 469,
"column": 4
} | {
"line": 471,
"column": 31
} | {
"line": 473,
"column": 0
} | [
{
"pp": "case refine_2\nm n : ℕ\nh : m = 1 ∨ n = 1\n⊢ IsCoprime m n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Or.casesOn",
"Nat",
"isCoprime_one_right",
"Nat.instCommSemiring",
"Eq.ndrec",
"Or",
"OfNat.ofNat",
... | [] | obtain rfl | rfl := h
· exact isCoprime_one_left
· exact isCoprime_one_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 196,
"column": 55
} | {
"line": 196,
"column": 91
} | {
"line": 196,
"column": 91
} | [
{
"pp": "R : Type u\ninst✝⁶ : Semiring R\nA : Type v\ninst✝⁵ : Semiring A\ninst✝⁴ : Module R A\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nι : Sort u_2\nt✝ : ι → Submodule R A\nN : Submodule R M\nt : A\nht : t ∈ ⨆ i, t✝ i\ns : M\nhs : s ∈ N\nx ... | [] | by simp_rw [add_smul]; apply add_mem | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Operations | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 38
} | {
"line": 355,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nn : ℕ\nhn : n ≠ 0\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Submodule",
... | [] | exact (pow_toAddSubmonoid M hn).ge | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Algebra.Operations | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 38
} | {
"line": 355,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nn : ℕ\nhn : n ≠ 0\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Submodule",
... | [] | exact (pow_toAddSubmonoid M hn).ge | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Operations | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 38
} | {
"line": 355,
"column": 0
} | [
{
"pp": "case inr\nR : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nn : ℕ\nhn : n ≠ 0\n⊢ M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Submodule",
... | [] | exact (pow_toAddSubmonoid M hn).ge | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 450,
"column": 31
} | {
"line": 450,
"column": 42
} | {
"line": 450,
"column": 42
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\n| M * (R ∙ 1)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"HMul.hMul",
"IsScalarTower.r... | [
"R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\n| span R ↑M * (R ∙ 1)"
] | ← span_eq M | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 208,
"column": 2
} | {
"line": 210,
"column": 50
} | {
"line": 212,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y : R\nm : ℕ\nhm : 0 < m\n⊢ IsCoprime (x ^ m) y ↔ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"IsCoprime.pow_left",
"Iff.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"Finset.card_... | [] | refine ⟨fun h ↦ ?_, IsCoprime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 208,
"column": 2
} | {
"line": 210,
"column": 50
} | {
"line": 212,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y : R\nm : ℕ\nhm : 0 < m\n⊢ IsCoprime (x ^ m) y ↔ IsCoprime x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"IsCoprime.pow_left",
"Iff.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"Finset.card_... | [] | refine ⟨fun h ↦ ?_, IsCoprime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 695,
"column": 21
} | {
"line": 695,
"column": 61
} | {
"line": 696,
"column": 2
} | [
{
"pp": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\np q : Submodule R Aᵐᵒᵖ\n⊢ op (comap (↑(opLinearEquiv R)) (p + q)) = op (comap (↑(opLinearEquiv R)) p) + op (comap (↑(opLinearEquiv R)) q)",
"ppTerm"... | [] | simp [comap_equiv_eq_map_symm, ← op_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Operations | {
"line": 695,
"column": 21
} | {
"line": 695,
"column": 61
} | {
"line": 696,
"column": 2
} | [
{
"pp": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\np q : Submodule R Aᵐᵒᵖ\n⊢ op (comap (↑(opLinearEquiv R)) (p + q)) = op (comap (↑(opLinearEquiv R)) p) + op (comap (↑(opLinearEquiv R)) q)",
"ppTerm"... | [] | simp [comap_equiv_eq_map_symm, ← op_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Operations | {
"line": 695,
"column": 21
} | {
"line": 695,
"column": 61
} | {
"line": 696,
"column": 2
} | [
{
"pp": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\np q : Submodule R Aᵐᵒᵖ\n⊢ op (comap (↑(opLinearEquiv R)) (p + q)) = op (comap (↑(opLinearEquiv R)) p) + op (comap (↑(opLinearEquiv R)) q)",
"ppTerm"... | [] | simp [comap_equiv_eq_map_symm, ← op_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 848,
"column": 18
} | {
"line": 848,
"column": 29
} | {
"line": 848,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\na : A\nM : Submodule R A\n| Set.up {a} • M",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.instAddCommMonoidWithOne",
"instHSMul",
"Equiv.... | [
"R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\na : A\nM : Submodule R A\n| Set.up {a} • span R ↑M"
] | ← span_eq M | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 544,
"column": 6
} | {
"line": 544,
"column": 17
} | {
"line": 544,
"column": 18
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ map f I = ⊤ ↔ I = ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring... | [
"R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ ⊤ ≤ map f I ↔ I = ⊤"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 544,
"column": 18
} | {
"line": 544,
"column": 45
} | {
"line": 544,
"column": 46
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ ⊤ ≤ map f I ↔ I = ⊤",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring... | [
"R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ comap f ⊤ ≤ I ↔ I = ⊤"
] | ← comap_le_iff_le_map f hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 68
} | {
"line": 546,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ map f I = ⊤ ↔ I = ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring... | [] | rw [eq_top_iff, ← comap_le_iff_le_map f hf, comap_top, top_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 68
} | {
"line": 546,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ map f I = ⊤ ↔ I = ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring... | [] | rw [eq_top_iff, ← comap_le_iff_le_map f hf, comap_top, top_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maps | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 68
} | {
"line": 546,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nhf : Function.Bijective ⇑f\nI : Ideal R\n⊢ map f I = ⊤ ↔ I = ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring... | [] | rw [eq_top_iff, ← comap_le_iff_le_map f hf, comap_top, top_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 600,
"column": 2
} | {
"line": 600,
"column": 98
} | {
"line": 602,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nhf : Function.Surjective ⇑f\nI : Ideal S\nφ : Ideal S ↪o Ideal R := orderEmbeddingOfSurjective f hf\nJ : Ideal S\nK : Ideal R\nh : φ J < K\n⊢ φ (map f K) = K... | [] | · exact (K.comap_map_of_surjective f hf).trans (sup_of_le_left ((comap_mono bot_le).trans h.le)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1004,
"column": 6
} | {
"line": 1004,
"column": 35
} | {
"line": 1006,
"column": 0
} | [
{
"pp": "case mpr.refine_4\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nr : R\nh : ∀ (n : ↑s), r • ↑n = 0\nn : M\nhn : n ∈ span R s\na : R\nx : M\nhx✝ : x ∈ span R s\nhx : r • x = 0\n⊢ r • a • x = 0",
"ppTerm": "?mpr.refine_4",
"assigned"... | [] | rw [smul_comm, hx, smul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 75
} | {
"line": 204,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nI : Ideal R\ninst✝ : I.IsTwoSided\nι : Type u_4\ns : Set ι\nf : ι → M\nx : M\n⊢ x ∈ I • span R (f '' s) ↔ ∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ I), (a.sum fun i c ↦ c • f ↑i) = x",
"ppTerm": "?m.45",
"assigned":... | [] | rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 75
} | {
"line": 204,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nI : Ideal R\ninst✝ : I.IsTwoSided\nι : Type u_4\ns : Set ι\nf : ι → M\nx : M\n⊢ x ∈ I • span R (f '' s) ↔ ∃ a, ∃ (_ : ∀ (i : ↑s), a i ∈ I), (a.sum fun i c ↦ c • f ↑i) = x",
"ppTerm": "?m.45",
"assigned":... | [] | rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.