Add files using upload-large-folder tool

README.md

@@ -0,0 +1,19 @@
+# MATHLIB4 DEPENDENCY GRAPH
+
+make dependency graph for [mathlib4](https://github.com/leanprover-community/mathlib4) using [jixia](https://github.com/frenzymath/jixia_py)
+
+## HOW TO
+
+0. Clone the repository at [https://github.com/fbundle/mathlib4_dependency_graph](https://github.com/fbundle/mathlib4_dependency_graph)
+
+1. Check out your favorite mathlib version
+
+2. Use `build` script to build mathlib and jixia
+
+3. Extract dependency graph using `jixia_export.py`
+
+4. Upload to huggingface using `upload_huggingface.py`
+
+## PREBUILT GRAPH
+
+- [https://huggingface.co/khanh2023/mathlib4_dependency_graph](https://huggingface.co/khanh2023/mathlib4_dependency_graph)

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.ContinuedFractions.Basic.sym.json

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["GenContFract","convs'Aux"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.4 : DivisionRing.{u_2} K], (Stream'.Seq.{u_2} (GenContFract.Pair.{u_2} K)) -> Nat -> K","typeFull":"{K : Type u_2} → [DivisionRing K] → Stream'.Seq (GenContFract.Pair K) → ℕ → K","typeReadable":"{K : Type u_2} → [DivisionRing K] → Stream'.Seq (GenContFract.Pair K) → ℕ → K","typeReferences":[["Nat"],["Stream'","Seq"],["GenContFract","Pair"],["DivisionRing"]],"valueReferences":[["Ring","toNonAssocRing"],["Stream'","Seq","head"],["instHDiv"],["HDiv","hDiv"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Stream'","Seq"],["Stream'","Seq","tail"],["Zero","toOfNat0"],["NonAssocRing","toNonUnitalNonAssocRing"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["GenContFract","contsAux","match_1"],["GenContFract","Pair","a"],["DivisionRing","toRing"],["Nat","brecOn"],["GenContFract","Pair","b"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["DivisionRing","toDivInvMonoid"],["HAdd","hAdd"],["GenContFract","convs'Aux","match_1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Nat","below"],["GenContFract","Pair"]]},{"isProp":false,"kind":"definition","name":["SimpContFract","IsContFract"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1552000515._hygCtx._hyg.3 : One.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1552000515._hygCtx._hyg.6 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1552000515._hygCtx._hyg.9 : LT.{u_1} α], (SimpContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1552000515._hygCtx._hyg.3) -> Prop","typeFull":"{α : Type u_1} → [inst : One α] → [Zero α] → [LT α] → SimpContFract α → Prop","typeReadable":"{α : Type u_1} → [inst : One α] → [Zero α] → [LT α] → SimpContFract α → Prop","typeReferences":[["One"],["SimpContFract"],["Zero"],["LT"]],"valueReferences":[["LT","lt"],["GenContFract","IsSimpContFract"],["Nat"],["Option"],["Zero","toOfNat0"],["Option","some"],["GenContFract"],["Eq"],["Subtype","val"],["OfNat","ofNat"],["GenContFract","partDens"],["Stream'","Seq","get?"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","ext_iff"],"typeFallback":"forall {α : Type.{u_1}} {x : GenContFract.{u_1} α} {y : GenContFract.{u_1} α}, Iff (Eq.{succ u_1} (GenContFract.{u_1} α) x y) (And (Eq.{succ u_1} α (GenContFract.h.{u_1} α x) (GenContFract.h.{u_1} α y)) (Eq.{succ u_1} (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) (GenContFract.s.{u_1} α x) (GenContFract.s.{u_1} α y)))","typeFull":"∀ {α : Type u_1} {x y : GenContFract α}, x = y ↔ x.h = y.h ∧ x.s = y.s","typeReadable":"∀ {α : Type u_1} {x y : GenContFract α}, x = y ↔ x.h = y.h ∧ x.s = y.s","typeReferences":[["Stream'","Seq"],["GenContFract","h"],["GenContFract","s"],["Iff"],["And"],["GenContFract","Pair"],["Eq"],["GenContFract"]],"valueReferences":[["HEq","refl"],["GenContFract","h"],["GenContFract","s"],["And"],["GenContFract","ext"],["GenContFract"],["Eq","casesOn"],["Iff","intro"],["And","intro"],["Stream'","Seq"],["Eq","refl"],["Eq","symm"],["HEq"],["GenContFract","Pair"],["Eq"],["Eq","ndrec"],["And","casesOn"]]},{"isProp":false,"kind":"definition","name":["SimpContFract","instCoeGenContFract"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105233._hygCtx._hyg.3 : One.{u_1} α], Coe.{succ u_1, succ u_1} (SimpContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105233._hygCtx._hyg.3) (GenContFract.{u_1} α)","typeFull":"{α : Type u_1} → [inst : One α] → Coe (SimpContFract α) (GenContFract α)","typeReadable":"{α : Type u_1} → [inst : One α] → Coe (SimpContFract α) (GenContFract α)","typeReferences":[["Coe"],["One"],["SimpContFract"],["GenContFract"]],"valueReferences":[["GenContFract","IsSimpContFract"],["Coe","mk"],["SimpContFract"],["GenContFract"],["Subtype","val"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","_sizeOf_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.9 : SizeOf.{succ u_1} α], (GenContFract.Pair.{u_1} α) -> Nat","typeFull":"{α : Type u_1} → [SizeOf α] → GenContFract.Pair α → ℕ","typeReadable":"{α : Type u_1} → [SizeOf α] → GenContFract.Pair α → ℕ","typeReferences":[["Nat"],["GenContFract","Pair"],["SizeOf"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Nat"],["instOfNatNat"],["instHAdd"],["GenContFract","Pair","rec"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","instRepr"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.437227469._hygCtx._hyg.3 : Repr.{u_1} α], Repr.{u_1} (GenContFract.Pair.{u_1} α)","typeFull":"{α : Type u_1} → [Repr α] → Repr (GenContFract.Pair α)","typeReadable":"{α : Type u_1} → [Repr α] → Repr (GenContFract.Pair α)","typeReferences":[["Repr"],["GenContFract","Pair"]],"valueReferences":[["repr"],["instHAppendOfAppend"],["HAppend","hAppend"],["Std","Format","text"],["GenContFract","Pair","a"],["Std","Format"],["Std","Format","instAppend"],["Repr","mk"],["GenContFract","Pair"],["GenContFract","Pair","b"]]},{"isProp":false,"kind":"definition","name":["GenContFract","convs'Aux","_unsafe_rec"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.4 : DivisionRing.{u_2} K], (Stream'.Seq.{u_2} (GenContFract.Pair.{u_2} K)) -> Nat -> K","typeFull":"{K : Type u_2} → [DivisionRing K] → Stream'.Seq (GenContFract.Pair K) → ℕ → K","typeReadable":"{K : Type u_2} → [DivisionRing K] → Stream'.Seq (GenContFract.Pair K) → ℕ → K","typeReferences":[["Nat"],["Stream'","Seq"],["GenContFract","Pair"],["DivisionRing"]],"valueReferences":[["Distrib","toAdd"],["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["GenContFract","Pair","a"],["GenContFract","contsAux","match_1"],["DivisionRing","toRing"],["Stream'","Seq","head"],["GenContFract","Pair","b"],["instHDiv"],["DivisionRing","toDivInvMonoid"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["HAdd","hAdd"],["HDiv","hDiv"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["GenContFract","convs'Aux","match_1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Stream'","Seq","tail"],["Zero","toOfNat0"],["GenContFract","Pair"],["GenContFract","convs'Aux","_unsafe_rec"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":false,"kind":"definition","name":["GenContFract","casesOn"],"typeFallback":"forall {α : Type.{u_1}} {motive : (GenContFract.{u_1} α) -> Sort.{u}} (t : GenContFract.{u_1} α), (forall (h : α) (s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)), motive (GenContFract.mk.{u_1} α h s)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {motive : GenContFract α → Sort u} →\n    (t : GenContFract α) → ((h : α) → (s : Stream'.Seq (GenContFract.Pair α)) → motive { h := h, s := s }) → motive t","typeReadable":"{α : Type u_1} →\n  {motive : GenContFract α → Sort u} →\n    (t : GenContFract α) → ((h : α) → (s : Stream'.Seq (GenContFract.Pair α)) → motive { h := h, s := s }) → motive t","typeReferences":[["Stream'","Seq"],["GenContFract","Pair"],["GenContFract","mk"],["GenContFract"]],"valueReferences":[["GenContFract","rec"]]},{"isProp":false,"kind":"definition","name":["GenContFract","ctorIdx"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> Nat","typeFull":"{α : Type u_1} → GenContFract α → ℕ","typeReadable":"{α : Type u_1} → GenContFract α → ℕ","typeReferences":[["Nat"],["GenContFract"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["GenContFract","mk","_flat_ctor"],"typeFallback":"forall {α : Type.{u_1}}, α -> (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) -> (GenContFract.{u_1} α)","typeFull":"{α : Type u_1} → α → Stream'.Seq (GenContFract.Pair α) → GenContFract α","typeReadable":"{α : Type u_1} → α → Stream'.Seq (GenContFract.Pair α) → GenContFract α","typeReferences":[["Stream'","Seq"],["GenContFract","Pair"],["GenContFract"]],"valueReferences":[["GenContFract","mk"]]},{"isProp":false,"kind":"definition","name":["ContFract","instCoeSimpContFract"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3755639240._hygCtx._hyg.3 : One.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3755639240._hygCtx._hyg.6 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3755639240._hygCtx._hyg.9 : LT.{u_1} α], Coe.{succ u_1, succ u_1} (ContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3755639240._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3755639240._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3755639240._hygCtx._hyg.9) (SimpContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3755639240._hygCtx._hyg.3)","typeFull":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → Coe (ContFract α) (SimpContFract α)","typeReadable":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → Coe (ContFract α) (SimpContFract α)","typeReferences":[["Coe"],["ContFract"],["One"],["SimpContFract"],["Zero"],["LT"]],"valueReferences":[["ContFract"],["Coe","mk"],["SimpContFract","IsContFract"],["SimpContFract"],["Subtype","val"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","recOn"],"typeFallback":"forall {α : Type.{u_1}} {motive : (GenContFract.Pair.{u_1} α) -> Sort.{u}} (t : GenContFract.Pair.{u_1} α), (forall (a : α) (b : α), motive (GenContFract.Pair.mk.{u_1} α a b)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {motive : GenContFract.Pair α → Sort u} →\n    (t : GenContFract.Pair α) → ((a b : α) → motive { a := a, b := b }) → motive t","typeReadable":"{α : Type u_1} →\n  {motive : GenContFract.Pair α → Sort u} →\n    (t : GenContFract.Pair α) → ((a b : α) → motive { a := a, b := b }) → motive t","typeReferences":[["GenContFract","Pair"],["GenContFract","Pair","mk"]],"valueReferences":[["GenContFract","Pair","rec"]]},{"isProp":false,"kind":"definition","name":["SimpContFract"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1084397055._hygCtx._hyg.3 : One.{u_1} α], Sort.{max 1 (succ u_1)}","typeFull":"(α : Type u_1) → [One α] → Type (max 0 u_1)","typeReadable":"(α : Type u_1) → [One α] → Type (max 0 u_1)","typeReferences":[["One"]],"valueReferences":[["GenContFract","IsSimpContFract"],["Subtype"],["GenContFract"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","noConfusionType"],"typeFallback":"Sort.{u} -> (forall {α : Type.{u_1}}, (GenContFract.Pair.{u_1} α) -> (forall {α' : Type.{u_1}}, (GenContFract.Pair.{u_1} α') -> Sort.{u}))","typeFull":"Sort u → {α : Type u_1} → GenContFract.Pair α → {α' : Type u_1} → GenContFract.Pair α' → Sort u","typeReadable":"Sort u → {α : Type u_1} → GenContFract.Pair α → {α' : Type u_1} → GenContFract.Pair α' → Sort u","typeReferences":[["GenContFract","Pair"]],"valueReferences":[["GenContFract","Pair","casesOn"],["HEq"]]},{"isProp":false,"kind":"definition","name":["GenContFract","partNums"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> (Stream'.Seq.{u_1} α)","typeFull":"{α : Type u_1} → GenContFract α → Stream'.Seq α","typeReadable":"{α : Type u_1} → GenContFract α → Stream'.Seq α","typeReferences":[["Stream'","Seq"],["GenContFract"]],"valueReferences":[["GenContFract","s"],["GenContFract","Pair","a"],["GenContFract","Pair"],["Stream'","Seq","map"]]},{"isProp":false,"kind":"definition","name":["GenContFract","ofInteger"],"typeFallback":"forall {α : Type.{u_1}}, α -> (GenContFract.{u_1} α)","typeFull":"{α : Type u_1} → α → GenContFract α","typeReadable":"{α : Type u_1} → α → GenContFract α","typeReferences":[["GenContFract"]],"valueReferences":[["GenContFract","Pair"],["GenContFract","mk"],["Stream'","Seq","nil"]]},{"isProp":false,"kind":"definition","name":["GenContFract","conts"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.70041594._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> (Stream'.{u_2} (GenContFract.Pair.{u_2} K))","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReferences":[["GenContFract","Pair"],["Stream'"],["GenContFract"],["DivisionRing"]],"valueReferences":[["Stream'","tail"],["GenContFract","contsAux"],["GenContFract","Pair"]]},{"isProp":false,"kind":"definition","name":["GenContFract","nextConts"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.241040215._hygCtx._hyg.4 : DivisionRing.{u_2} K], K -> K -> (GenContFract.Pair.{u_2} K) -> (GenContFract.Pair.{u_2} K) -> (GenContFract.Pair.{u_2} K)","typeFull":"{K : Type u_2} → [DivisionRing K] → K → K → GenContFract.Pair K → GenContFract.Pair K → GenContFract.Pair K","typeReadable":"{K : Type u_2} → [DivisionRing K] → K → K → GenContFract.Pair K → GenContFract.Pair K → GenContFract.Pair K","typeReferences":[["GenContFract","Pair"],["DivisionRing"]],"valueReferences":[["GenContFract","Pair","a"],["GenContFract","Pair","b"],["GenContFract","nextNum"],["GenContFract","nextDen"],["GenContFract","Pair","mk"]]},{"isProp":false,"kind":"definition","name":["ContFract","instInhabited"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808126._hygCtx._hyg.3 : One.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808126._hygCtx._hyg.6 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808126._hygCtx._hyg.9 : LT.{u_1} α], Inhabited.{succ u_1} (ContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808126._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808126._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808126._hygCtx._hyg.9)","typeFull":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → Inhabited (ContFract α)","typeReadable":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → Inhabited (ContFract α)","typeReferences":[["ContFract"],["One"],["Zero"],["Inhabited"],["LT"]],"valueReferences":[["Inhabited","mk"],["ContFract","ofInteger"],["ContFract"],["Zero","toOfNat0"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["GenContFract","contsAux","_unsafe_rec"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> (Stream'.{u_2} (GenContFract.Pair.{u_2} K))","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReferences":[["GenContFract","Pair"],["Stream'"],["GenContFract"],["DivisionRing"]],"valueReferences":[["instAddNat"],["Ring","toNonAssocRing"],["GenContFract","h"],["AddGroupWithOne","toAddMonoidWithOne"],["GenContFract","nextConts"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ring","toAddGroupWithOne"],["instOfNatNat"],["Zero","toOfNat0"],["NonAssocRing","toNonUnitalNonAssocRing"],["instHAdd"],["GenContFract","Pair","a"],["GenContFract","contsAux","match_1"],["GenContFract","s"],["DivisionRing","toRing"],["GenContFract","Pair","b"],["OfNat","ofNat"],["GenContFract","contsAux","match_3"],["HAdd","hAdd"],["GenContFract","contsAux","_unsafe_rec"],["Nat"],["One","toOfNat1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddMonoidWithOne","toOne"],["GenContFract","Pair"],["GenContFract","Pair","mk"],["Stream'","Seq","get?"]]},{"isProp":false,"kind":"definition","name":["GenContFract","s"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α))","typeFull":"{α : Type u_1} → GenContFract α → Stream'.Seq (GenContFract.Pair α)","typeReadable":"{α : Type u_1} → GenContFract α → Stream'.Seq (GenContFract.Pair α)","typeReferences":[["Stream'","Seq"],["GenContFract","Pair"],["GenContFract"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["GenContFract","convs'Aux","_sunfold"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.4 : DivisionRing.{u_2} K], (Stream'.Seq.{u_2} (GenContFract.Pair.{u_2} K)) -> Nat -> K","typeFull":"{K : Type u_2} → [DivisionRing K] → Stream'.Seq (GenContFract.Pair K) → ℕ → K","typeReadable":"{K : Type u_2} → [DivisionRing K] → Stream'.Seq (GenContFract.Pair K) → ℕ → K","typeReferences":[["Nat"],["Stream'","Seq"],["GenContFract","Pair"],["DivisionRing"]],"valueReferences":[["Distrib","toAdd"],["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toDistrib"],["GenContFract","convs'Aux"],["instHAdd"],["GenContFract","Pair","a"],["GenContFract","contsAux","match_1"],["DivisionRing","toRing"],["Stream'","Seq","head"],["GenContFract","Pair","b"],["instHDiv"],["DivisionRing","toDivInvMonoid"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["HAdd","hAdd"],["HDiv","hDiv"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["GenContFract","convs'Aux","match_1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Stream'","Seq","tail"],["Zero","toOfNat0"],["GenContFract","Pair"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":false,"kind":"definition","name":["GenContFract","noConfusionType"],"typeFallback":"Sort.{u} -> (forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> (forall {α' : Type.{u_1}}, (GenContFract.{u_1} α') -> Sort.{u}))","typeFull":"Sort u → {α : Type u_1} → GenContFract α → {α' : Type u_1} → GenContFract α' → Sort u","typeReadable":"Sort u → {α : Type u_1} → GenContFract α → {α' : Type u_1} → GenContFract α' → Sort u","typeReferences":[["GenContFract"]],"valueReferences":[["Stream'","Seq"],["HEq"],["GenContFract","casesOn"],["GenContFract","Pair"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","_sizeOf_inst"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.10 : SizeOf.{succ u_1} α], SizeOf.{succ u_1} (GenContFract.Pair.{u_1} α)","typeFull":"(α : Type u_1) → [SizeOf α] → SizeOf (GenContFract.Pair α)","typeReadable":"(α : Type u_1) → [SizeOf α] → SizeOf (GenContFract.Pair α)","typeReferences":[["GenContFract","Pair"],["SizeOf"]],"valueReferences":[["GenContFract","Pair","_sizeOf_1"],["GenContFract","Pair"],["SizeOf","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","h"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> α","typeFull":"{α : Type u_1} → GenContFract α → α","typeReadable":"{α : Type u_1} → GenContFract α → α","typeReferences":[["GenContFract"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["GenContFract","instDecidableTerminatedAt"],"typeFallback":"forall {α : Type.{u_1}} (g : GenContFract.{u_1} α) (n : Nat), Decidable (GenContFract.TerminatedAt.{u_1} α g n)","typeFull":"{α : Type u_1} → (g : GenContFract α) → (n : ℕ) → Decidable (g.TerminatedAt n)","typeReadable":"{α : Type u_1} → (g : GenContFract α) → (n : ℕ) → Decidable (g.TerminatedAt n)","typeReferences":[["Nat"],["Decidable"],["GenContFract","TerminatedAt"],["GenContFract"]],"valueReferences":[["GenContFract","instDecidableTerminatedAt","_aux_1"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","mk","inj"],"typeFallback":"forall {α : Type.{u_1}} {h : α} {s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)} {h_1 : α} {s_1 : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)}, (Eq.{succ u_1} (GenContFract.{u_1} α) (GenContFract.mk.{u_1} α h s) (GenContFract.mk.{u_1} α h_1 s_1)) -> (And (Eq.{succ u_1} α h h_1) (Eq.{succ u_1} (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) s s_1))","typeFull":"∀ {α : Type u_1} {h : α} {s : Stream'.Seq (GenContFract.Pair α)} {h_1 : α} {s_1 : Stream'.Seq (GenContFract.Pair α)},\n  { h := h, s := s } = { h := h_1, s := s_1 } → h = h_1 ∧ s = s_1","typeReadable":"∀ {α : Type u_1} {h : α} {s : Stream'.Seq (GenContFract.Pair α)} {h_1 : α} {s_1 : Stream'.Seq (GenContFract.Pair α)},\n  { h := h, s := s } = { h := h_1, s := s_1 } → h = h_1 ∧ s = s_1","typeReferences":[["Stream'","Seq"],["And"],["GenContFract","Pair"],["GenContFract","mk"],["GenContFract"],["Eq"]],"valueReferences":[["And","intro"],["GenContFract","mk","noConfusion"],["Stream'","Seq"],["eq_of_heq"],["And"],["GenContFract","Pair"],["Eq"]]},{"isProp":false,"kind":"definition","name":["SimpContFract","ofInteger"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022417._hygCtx._hyg.3 : One.{u_1} α], α -> (SimpContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022417._hygCtx._hyg.3)","typeFull":"{α : Type u_1} → [inst : One α] → α → SimpContFract α","typeReadable":"{α : Type u_1} → [inst : One α] → α → SimpContFract α","typeReferences":[["One"],["SimpContFract"]],"valueReferences":[["GenContFract","IsSimpContFract"],["SimpContFract","ofInteger","_proof_1"],["Subtype","mk"],["GenContFract"],["GenContFract","ofInteger"]]},{"isProp":false,"kind":"definition","name":["GenContFract","_sizeOf_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1831630741._hygCtx._hyg.13 : SizeOf.{succ u_1} α], (GenContFract.{u_1} α) -> Nat","typeFull":"{α : Type u_1} → [SizeOf α] → GenContFract α → ℕ","typeReadable":"{α : Type u_1} → [SizeOf α] → GenContFract α → ℕ","typeReferences":[["Nat"],["GenContFract"],["SizeOf"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Nat"],["Stream'","Seq"],["instOfNatNat"],["instHAdd"],["instSizeOfDefault"],["GenContFract","Pair"],["OfNat","ofNat"],["GenContFract","rec"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","instCoe"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3123013781._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} α β], Coe.{succ u_1, succ u_2} (GenContFract.Pair.{u_1} α) (GenContFract.Pair.{u_2} β)","typeFull":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → Coe (GenContFract.Pair α) (GenContFract.Pair β)","typeReadable":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → Coe (GenContFract.Pair α) (GenContFract.Pair β)","typeReferences":[["Coe"],["GenContFract","Pair"]],"valueReferences":[["GenContFract","Pair","coeFn"],["Coe","mk"],["GenContFract","Pair"]]},{"isProp":false,"kind":"definition","name":["GenContFract","instInhabitedPair","default"],"typeFallback":"forall {a._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.21 : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1498887226._hygCtx._hyg.4 : Inhabited.{succ u_2} a._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.21], GenContFract.Pair.{u_2} a._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.21","typeFull":"{a : Type u_2} → [Inhabited a] → GenContFract.Pair a","typeReadable":"{a : Type u_2} → [Inhabited a] → GenContFract.Pair a","typeReferences":[["GenContFract","Pair"],["Inhabited"]],"valueReferences":[["Inhabited","default"],["GenContFract","Pair","mk"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","ext"],"typeFallback":"forall {α : Type.{u_1}} {x : GenContFract.{u_1} α} {y : GenContFract.{u_1} α}, (Eq.{succ u_1} α (GenContFract.h.{u_1} α x) (GenContFract.h.{u_1} α y)) -> (Eq.{succ u_1} (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) (GenContFract.s.{u_1} α x) (GenContFract.s.{u_1} α y)) -> (Eq.{succ u_1} (GenContFract.{u_1} α) x y)","typeFull":"∀ {α : Type u_1} {x y : GenContFract α}, x.h = y.h → x.s = y.s → x = y","typeReadable":"∀ {α : Type u_1} {x y : GenContFract α}, x.h = y.h → x.s = y.s → x = y","typeReferences":[["Stream'","Seq"],["GenContFract","h"],["GenContFract","s"],["GenContFract","Pair"],["Eq"],["GenContFract"]],"valueReferences":[["HEq","refl"],["_private","Mathlib","Algebra","ContinuedFractions","Basic",0,"GenContFract","ext","match_1"],["GenContFract","h"],["GenContFract","s"],["GenContFract"],["GenContFract","mk"],["Eq","casesOn"],["Stream'","Seq"],["Eq","refl"],["HEq"],["Eq","symm"],["GenContFract","Pair"],["Eq"],["Eq","ndrec"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IsSimpContFract"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> (forall [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.2734341015._hygCtx._hyg.5 : One.{u_1} α], Prop)","typeFull":"{α : Type u_1} → GenContFract α → [One α] → Prop","typeReadable":"{α : Type u_1} → GenContFract α → [One α] → Prop","typeReferences":[["One"],["GenContFract"]],"valueReferences":[["Nat"],["One","toOfNat1"],["GenContFract","partNums"],["Option"],["Option","some"],["Eq"],["OfNat","ofNat"],["Stream'","Seq","get?"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","mk","_flat_ctor"],"typeFallback":"forall {α : Type.{u_1}}, α -> α -> (GenContFract.Pair.{u_1} α)","typeFull":"{α : Type u_1} → α → α → GenContFract.Pair α","typeReadable":"{α : Type u_1} → α → α → GenContFract.Pair α","typeReferences":[["GenContFract","Pair"]],"valueReferences":[["GenContFract","Pair","mk"]]},{"isProp":false,"kind":"definition","name":["SimpContFract","instInhabited"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808125._hygCtx._hyg.3 : One.{u_1} α], Inhabited.{succ u_1} (SimpContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808125._hygCtx._hyg.3)","typeFull":"{α : Type u_1} → [inst : One α] → Inhabited (SimpContFract α)","typeReadable":"{α : Type u_1} → [inst : One α] → Inhabited (SimpContFract α)","typeReferences":[["One"],["SimpContFract"],["Inhabited"]],"valueReferences":[["Inhabited","mk"],["One","toOfNat1"],["SimpContFract","ofInteger"],["SimpContFract"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["GenContFract","mk","noConfusion"],"typeFallback":"forall {α : Type.{u_1}} {P : Sort.{u}} {h : α} {s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)} {h' : α} {s' : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)}, (Eq.{succ u_1} (GenContFract.{u_1} α) (GenContFract.mk.{u_1} α h s) (GenContFract.mk.{u_1} α h' s')) -> ((HEq.{succ u_1} α h α h') -> (HEq.{succ u_1} (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) s (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) s') -> P) -> P","typeFull":"{α : Type u_1} →\n  {P : Sort u} →\n    {h : α} →\n      {s : Stream'.Seq (GenContFract.Pair α)} →\n        {h' : α} →\n          {s' : Stream'.Seq (GenContFract.Pair α)} →\n            { h := h, s := s } = { h := h', s := s' } → (h ≍ h' → s ≍ s' → P) → P","typeReadable":"{α : Type u_1} →\n  {P : Sort u} →\n    {h : α} →\n      {s : Stream'.Seq (GenContFract.Pair α)} →\n        {h' : α} →\n          {s' : Stream'.Seq (GenContFract.Pair α)} →\n            { h := h, s := s } = { h := h', s := s' } → (h ≍ h' → s ≍ s' → P) → P","typeReferences":[["Stream'","Seq"],["HEq"],["GenContFract","Pair"],["GenContFract","mk"],["GenContFract"],["Eq"]],"valueReferences":[["GenContFract","noConfusion"],["Eq","refl"],["id"],["heq_of_eq"],["GenContFract"],["GenContFract","mk"]]},{"isProp":false,"kind":"definition","name":["ContFract","instCoeGenContFract"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105234._hygCtx._hyg.3 : One.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105234._hygCtx._hyg.6 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105234._hygCtx._hyg.9 : LT.{u_1} α], Coe.{succ u_1, succ u_1} (ContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105234._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105234._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4034105234._hygCtx._hyg.9) (GenContFract.{u_1} α)","typeFull":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → Coe (ContFract α) (GenContFract α)","typeReadable":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → Coe (ContFract α) (GenContFract α)","typeReferences":[["Coe"],["ContFract"],["One"],["Zero"],["GenContFract"],["LT"]],"valueReferences":[["GenContFract","IsSimpContFract"],["ContFract"],["Coe","mk"],["SimpContFract","IsContFract"],["SimpContFract"],["GenContFract"],["Subtype","val"]]},{"isProp":false,"kind":"definition","name":["GenContFract","nextNum"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1479143419._hygCtx._hyg.4 : DivisionRing.{u_2} K], K -> K -> K -> K -> K","typeFull":"{K : Type u_2} → [DivisionRing K] → K → K → K → K → K","typeReadable":"{K : Type u_2} → [DivisionRing K] → K → K → K → K → K","typeReferences":[["DivisionRing"]],"valueReferences":[["HAdd","hAdd"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Distrib","toAdd"],["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Distrib","toMul"],["DivisionRing","toRing"],["instHMul"],["HMul","hMul"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":false,"kind":"definition","name":["GenContFract","convs'Aux","match_1"],"typeFallback":"forall {K : Type.{u_1}} (motive : (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} K)) -> Nat -> Sort.{u_2}) (x._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx.18.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.38 : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} K)) (x._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx.19.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.41 : Nat), (forall (x._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.48 : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} K)), motive x._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.48 (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall (s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} K)) (n : Nat), motive s (Nat.succ n)) -> (motive x._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx.18.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.38 x._@.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx.19.Mathlib.Algebra.ContinuedFractions.Basic.490387607._hygCtx._hyg.41)","typeFull":"{K : Type u_1} →\n  (motive : Stream'.Seq (GenContFract.Pair K) → ℕ → Sort u_2) →\n    (x : Stream'.Seq (GenContFract.Pair K)) →\n      (x_1 : ℕ) →\n        ((x : Stream'.Seq (GenContFract.Pair K)) → motive x 0) →\n          ((s : Stream'.Seq (GenContFract.Pair K)) → (n : ℕ) → motive s n.succ) → motive x x_1","typeReadable":"{K : Type u_1} →\n  (motive : Stream'.Seq (GenContFract.Pair K) → ℕ → Sort u_2) →\n    (x : Stream'.Seq (GenContFract.Pair K)) →\n      (x_1 : ℕ) →\n        ((x : Stream'.Seq (GenContFract.Pair K)) → motive x 0) →\n          ((s : Stream'.Seq (GenContFract.Pair K)) → (n : ℕ) → motive s n.succ) → motive x x_1","typeReferences":[["Nat"],["Stream'","Seq"],["Nat","succ"],["instOfNatNat"],["GenContFract","Pair"],["OfNat","ofNat"]],"valueReferences":[["Nat","casesOn"]]},{"isProp":false,"kind":"definition","name":["GenContFract","coeFn"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1174359231._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} α β], (GenContFract.{u_1} α) -> (GenContFract.{u_2} β)","typeFull":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → GenContFract α → GenContFract β","typeReadable":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → GenContFract α → GenContFract β","typeReferences":[["Coe"],["GenContFract"]],"valueReferences":[["GenContFract","Pair","coeFn"],["GenContFract","h"],["GenContFract","s"],["Coe","coe"],["GenContFract","Pair"],["Stream'","Seq","map"],["GenContFract","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","casesOn"],"typeFallback":"forall {α : Type.{u_1}} {motive : (GenContFract.Pair.{u_1} α) -> Sort.{u}} (t : GenContFract.Pair.{u_1} α), (forall (a : α) (b : α), motive (GenContFract.Pair.mk.{u_1} α a b)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {motive : GenContFract.Pair α → Sort u} →\n    (t : GenContFract.Pair α) → ((a b : α) → motive { a := a, b := b }) → motive t","typeReadable":"{α : Type u_1} →\n  {motive : GenContFract.Pair α → Sort u} →\n    (t : GenContFract.Pair α) → ((a b : α) → motive { a := a, b := b }) → motive t","typeReferences":[["GenContFract","Pair"],["GenContFract","Pair","mk"]],"valueReferences":[["GenContFract","Pair","rec"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","map"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}}, (α -> β) -> (GenContFract.Pair.{u_1} α) -> (GenContFract.Pair.{u_2} β)","typeFull":"{α : Type u_1} → {β : Type u_2} → (α → β) → GenContFract.Pair α → GenContFract.Pair β","typeReadable":"{α : Type u_1} → {β : Type u_2} → (α → β) → GenContFract.Pair α → GenContFract.Pair β","typeReferences":[["GenContFract","Pair"]],"valueReferences":[["GenContFract","Pair","a"],["GenContFract","Pair","b"],["GenContFract","Pair","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","contsAux","_sunfold"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> (Stream'.{u_2} (GenContFract.Pair.{u_2} K))","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReferences":[["GenContFract","Pair"],["Stream'"],["GenContFract"],["DivisionRing"]],"valueReferences":[["instAddNat"],["Ring","toNonAssocRing"],["GenContFract","h"],["AddGroupWithOne","toAddMonoidWithOne"],["GenContFract","nextConts"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ring","toAddGroupWithOne"],["instOfNatNat"],["Zero","toOfNat0"],["NonAssocRing","toNonUnitalNonAssocRing"],["instHAdd"],["GenContFract","Pair","a"],["GenContFract","contsAux","match_1"],["GenContFract","s"],["DivisionRing","toRing"],["GenContFract","Pair","b"],["OfNat","ofNat"],["GenContFract","contsAux","match_3"],["HAdd","hAdd"],["Nat"],["GenContFract","contsAux"],["One","toOfNat1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddMonoidWithOne","toOne"],["GenContFract","Pair"],["GenContFract","Pair","mk"],["Stream'","Seq","get?"]]},{"isProp":false,"kind":"definition","name":["GenContFract","partDens"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> (Stream'.Seq.{u_1} α)","typeFull":"{α : Type u_1} → GenContFract α → Stream'.Seq α","typeReadable":"{α : Type u_1} → GenContFract α → Stream'.Seq α","typeReferences":[["Stream'","Seq"],["GenContFract"]],"valueReferences":[["GenContFract","s"],["GenContFract","Pair"],["GenContFract","Pair","b"],["Stream'","Seq","map"]]},{"isProp":false,"kind":"definition","name":["GenContFract","instInhabitedPair"],"typeFallback":"forall {a._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.21 : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.956473056._hygCtx._hyg.4 : Inhabited.{succ u_2} a._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.21], Inhabited.{succ u_2} (GenContFract.Pair.{u_2} a._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.21)","typeFull":"{a : Type u_2} → [Inhabited a] → Inhabited (GenContFract.Pair a)","typeReadable":"{a : Type u_2} → [Inhabited a] → Inhabited (GenContFract.Pair a)","typeReferences":[["GenContFract","Pair"],["Inhabited"]],"valueReferences":[["Inhabited","mk"],["GenContFract","instInhabitedPair","default"],["GenContFract","Pair"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","ContinuedFractions","Basic",0,"GenContFract","ext","match_1"],"typeFallback":"forall {α._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.23 : Type.{u_1}} (motive : (GenContFract.{u_1} α._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.23) -> Prop) (h._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.24.37 : GenContFract.{u_1} α._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.23), (forall (h._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.46 : α._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.23) (s._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.47 : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.23)), motive (GenContFract.mk.{u_1} α._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.23 h._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.46 s._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.47)) -> (motive h._@.Mathlib.Algebra.ContinuedFractions.Basic._hyg.24.37)","typeFull":"∀ {α : Type u_1} (motive : GenContFract α → Prop) (h : GenContFract α),\n  (∀ (h : α) (s : Stream'.Seq (GenContFract.Pair α)), motive { h := h, s := s }) → motive h","typeReadable":"∀ {α : Type u_1} (motive : GenContFract α → Prop) (h : GenContFract α),\n  (∀ (h : α) (s : Stream'.Seq (GenContFract.Pair α)), motive { h := h, s := s }) → motive h","typeReferences":[["Stream'","Seq"],["GenContFract","Pair"],["GenContFract","mk"],["GenContFract"]],"valueReferences":[["GenContFract","casesOn"]]},{"isProp":true,"kind":"theorem","name":["ContFract","ofInteger","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.3 : One.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.6 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.9 : LT.{u_1} α] (a : α) (n : Nat) (bₙ : α), (Eq.{succ u_1} (Option.{u_1} α) (Stream'.Seq.get?.{u_1} α (GenContFract.partDens.{u_1} α (Subtype.val.{succ u_1} (GenContFract.{u_1} α) (fun (g : GenContFract.{u_1} α) => GenContFract.IsSimpContFract.{u_1} α g inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.3) (SimpContFract.ofInteger.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.3 a))) n) (Option.some.{u_1} α bₙ)) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.9 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.6)) bₙ)","typeFull":"∀ {α : Type u_1} [inst : One α] [inst_1 : Zero α] [inst_2 : LT α] (a : α) (n : ℕ) (bₙ : α),\n  (↑(SimpContFract.ofInteger a)).partDens.get? n = some bₙ → 0 < bₙ","typeReadable":"∀ {α : Type u_1} [inst : One α] [inst_1 : Zero α] [inst_2 : LT α] (a : α) (n : ℕ) (bₙ : α),\n  (↑(SimpContFract.ofInteger a)).partDens.get? n = some bₙ → 0 < bₙ","typeReferences":[["GenContFract","IsSimpContFract"],["SimpContFract","ofInteger"],["Option","some"],["GenContFract"],["Subtype","val"],["OfNat","ofNat"],["GenContFract","partDens"],["LT","lt"],["Nat"],["Option"],["One"],["Zero","toOfNat0"],["Zero"],["Eq"],["LT"],["Stream'","Seq","get?"]],"valueReferences":[["GenContFract","IsSimpContFract"],["HEq","refl"],["SimpContFract","ofInteger"],["noConfusion_of_Nat"],["Option","some"],["GenContFract"],["Eq","casesOn"],["Subtype","val"],["GenContFract","partDens"],["OfNat","ofNat"],["LT","lt"],["Option","ctorIdx"],["Option","none"],["False","elim"],["Eq","refl"],["Option"],["HEq"],["Zero","toOfNat0"],["Eq"],["Stream'","Seq","get?"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","coe_toGenContFract"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4060579844._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} α β] {g : GenContFract.{u_1} α}, Eq.{succ u_2} (GenContFract.{u_2} β) (GenContFract.coeFn.{u_1, u_2} α β inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4060579844._hygCtx._hyg.4 g) (GenContFract.mk.{u_2} β (Coe.coe.{succ u_1, succ u_2} α β inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4060579844._hygCtx._hyg.4 (GenContFract.h.{u_1} α g)) (Stream'.Seq.map.{u_1, u_2} (GenContFract.Pair.{u_1} α) (GenContFract.Pair.{u_2} β) (GenContFract.Pair.coeFn.{u_1, u_2} α β inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4060579844._hygCtx._hyg.4) (GenContFract.s.{u_1} α g)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Coe α β] {g : GenContFract α},\n  ↑g = { h := Coe.coe g.h, s := Stream'.Seq.map GenContFract.Pair.coeFn g.s }","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Coe α β] {g : GenContFract α},\n  ↑g = { h := Coe.coe g.h, s := Stream'.Seq.map GenContFract.Pair.coeFn g.s }","typeReferences":[["GenContFract","coeFn"],["GenContFract","Pair","coeFn"],["GenContFract","h"],["GenContFract","s"],["Coe"],["Coe","coe"],["GenContFract","Pair"],["Stream'","Seq","map"],["GenContFract","mk"],["Eq"],["GenContFract"]],"valueReferences":[["GenContFract","coeFn"],["rfl"],["GenContFract"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","Pair","mk","inj"],"typeFallback":"forall {α : Type.{u_1}} {a : α} {b : α} {a_1 : α} {b_1 : α}, (Eq.{succ u_1} (GenContFract.Pair.{u_1} α) (GenContFract.Pair.mk.{u_1} α a b) (GenContFract.Pair.mk.{u_1} α a_1 b_1)) -> (And (Eq.{succ u_1} α a a_1) (Eq.{succ u_1} α b b_1))","typeFull":"∀ {α : Type u_1} {a b a_1 b_1 : α}, { a := a, b := b } = { a := a_1, b := b_1 } → a = a_1 ∧ b = b_1","typeReadable":"∀ {α : Type u_1} {a b a_1 b_1 : α}, { a := a, b := b } = { a := a_1, b := b_1 } → a = a_1 ∧ b = b_1","typeReferences":[["And"],["GenContFract","Pair"],["Eq"],["GenContFract","Pair","mk"]],"valueReferences":[["And","intro"],["eq_of_heq"],["And"],["GenContFract","Pair","mk","noConfusion"],["Eq"]]},{"isProp":false,"kind":"definition","name":["GenContFract","instDecidableTerminatedAt","_aux_1"],"typeFallback":"forall {α : Type.{u_1}} (g : GenContFract.{u_1} α) (n : Nat), Decidable (GenContFract.TerminatedAt.{u_1} α g n)","typeFull":"{α : Type u_1} → (g : GenContFract α) → (n : ℕ) → Decidable (g.TerminatedAt n)","typeReadable":"{α : Type u_1} → (g : GenContFract α) → (n : ℕ) → Decidable (g.TerminatedAt n)","typeReferences":[["Nat"],["Decidable"],["GenContFract","TerminatedAt"],["GenContFract"]],"valueReferences":[["decidable_of_iff'"],["instDecidableEqBool"],["Bool"],["GenContFract","s"],["Stream'","Seq","TerminatedAt"],["Stream'","Seq","terminatedAtDecidable","_proof_1"],["GenContFract","Pair"],["Eq"],["Bool","true"],["Option","isNone"],["Stream'","Seq","get?"]]},{"isProp":false,"kind":"constructor","name":["GenContFract","Pair","mk"],"typeFallback":"forall {α : Type.{u_1}}, α -> α -> (GenContFract.Pair.{u_1} α)","typeFull":"{α : Type u_1} → α → α → GenContFract.Pair α","typeReadable":"{α : Type u_1} → α → α → GenContFract.Pair α","typeReferences":[["GenContFract","Pair"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["GenContFract","contsAux","match_1"],"typeFallback":"forall {K : Type.{u_1}} (motive : (Option.{u_1} (GenContFract.Pair.{u_1} K)) -> Sort.{u_2}) (x._@.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx._hyg.59 : Option.{u_1} (GenContFract.Pair.{u_1} K)), (Unit -> (motive (Option.none.{u_1} (GenContFract.Pair.{u_1} K)))) -> (forall (gp : GenContFract.Pair.{u_1} K), motive (Option.some.{u_1} (GenContFract.Pair.{u_1} K) gp)) -> (motive x._@.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx._hyg.59)","typeFull":"{K : Type u_1} →\n  (motive : Option (GenContFract.Pair K) → Sort u_2) →\n    (x : Option (GenContFract.Pair K)) →\n      (Unit → motive none) → ((gp : GenContFract.Pair K) → motive (some gp)) → motive x","typeReadable":"{K : Type u_1} →\n  (motive : Option (GenContFract.Pair K) → Sort u_2) →\n    (x : Option (GenContFract.Pair K)) →\n      (Unit → motive none) → ((gp : GenContFract.Pair K) → motive (some gp)) → motive x","typeReferences":[["Option","none"],["Option"],["GenContFract","Pair"],["Option","some"],["Unit"]],"valueReferences":[["Option","casesOn"],["GenContFract","Pair"],["Unit","unit"]]},{"isProp":false,"kind":"inductive","name":["GenContFract"],"typeFallback":"Type.{u_1} -> Type.{u_1}","typeFull":"Type u_1 → Type u_1","typeReadable":"Type u_1 → Type u_1","typeReferences":[],"valueReferences":null},{"isProp":false,"kind":"definition","name":["GenContFract","nextDen"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1436138081._hygCtx._hyg.4 : DivisionRing.{u_2} K], K -> K -> K -> K -> K","typeFull":"{K : Type u_2} → [DivisionRing K] → K → K → K → K → K","typeReadable":"{K : Type u_2} → [DivisionRing K] → K → K → K → K → K","typeReferences":[["DivisionRing"]],"valueReferences":[["HAdd","hAdd"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Distrib","toAdd"],["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Distrib","toMul"],["DivisionRing","toRing"],["instHMul"],["HMul","hMul"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","mk","noConfusion"],"typeFallback":"forall {α : Type.{u_1}} {P : Sort.{u}} {a : α} {b : α} {a' : α} {b' : α}, (Eq.{succ u_1} (GenContFract.Pair.{u_1} α) (GenContFract.Pair.mk.{u_1} α a b) (GenContFract.Pair.mk.{u_1} α a' b')) -> ((HEq.{succ u_1} α a α a') -> (HEq.{succ u_1} α b α b') -> P) -> P","typeFull":"{α : Type u_1} → {P : Sort u} → {a b a' b' : α} → { a := a, b := b } = { a := a', b := b' } → (a ≍ a' → b ≍ b' → P) → P","typeReadable":"{α : Type u_1} → {P : Sort u} → {a b a' b' : α} → { a := a, b := b } = { a := a', b := b' } → (a ≍ a' → b ≍ b' → P) → P","typeReferences":[["HEq"],["GenContFract","Pair"],["Eq"],["GenContFract","Pair","mk"]],"valueReferences":[["GenContFract","Pair","noConfusion"],["Eq","refl"],["id"],["GenContFract","Pair"],["heq_of_eq"],["GenContFract","Pair","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","recOn"],"typeFallback":"forall {α : Type.{u_1}} {motive : (GenContFract.{u_1} α) -> Sort.{u}} (t : GenContFract.{u_1} α), (forall (h : α) (s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)), motive (GenContFract.mk.{u_1} α h s)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {motive : GenContFract α → Sort u} →\n    (t : GenContFract α) → ((h : α) → (s : Stream'.Seq (GenContFract.Pair α)) → motive { h := h, s := s }) → motive t","typeReadable":"{α : Type u_1} →\n  {motive : GenContFract α → Sort u} →\n    (t : GenContFract α) → ((h : α) → (s : Stream'.Seq (GenContFract.Pair α)) → motive { h := h, s := s }) → motive t","typeReferences":[["Stream'","Seq"],["GenContFract","Pair"],["GenContFract","mk"],["GenContFract"]],"valueReferences":[["GenContFract","rec"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","noConfusion"],"typeFallback":"forall {P : Sort.{u}} {α : Type.{u_1}} {t : GenContFract.Pair.{u_1} α} {α' : Type.{u_1}} {t' : GenContFract.Pair.{u_1} α'}, (Eq.{succ (succ u_1)} Type.{u_1} α α') -> (HEq.{succ u_1} (GenContFract.Pair.{u_1} α) t (GenContFract.Pair.{u_1} α') t') -> (GenContFract.Pair.noConfusionType.{u, u_1} P α t α' t')","typeFull":"{P : Sort u} →\n  {α : Type u_1} →\n    {t : GenContFract.Pair α} →\n      {α' : Type u_1} → {t' : GenContFract.Pair α'} → α = α' → t ≍ t' → GenContFract.Pair.noConfusionType P t t'","typeReadable":"{P : Sort u} →\n  {α : Type u_1} →\n    {t : GenContFract.Pair α} →\n      {α' : Type u_1} → {t' : GenContFract.Pair α'} → α = α' → t ≍ t' → GenContFract.Pair.noConfusionType P t t'","typeReferences":[["HEq"],["GenContFract","Pair","noConfusionType"],["GenContFract","Pair"],["Eq"]],"valueReferences":[["GenContFract","Pair","casesOn"],["HEq","refl"],["eq_of_heq"],["HEq"],["GenContFract","Pair","noConfusionType"],["GenContFract","Pair"],["Eq","ndrec"]]},{"isProp":false,"kind":"inductive","name":["GenContFract","Pair"],"typeFallback":"Type.{u_1} -> Type.{u_1}","typeFull":"Type u_1 → Type u_1","typeReadable":"Type u_1 → Type u_1","typeReferences":[],"valueReferences":null},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","a"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.Pair.{u_1} α) -> α","typeFull":"{α : Type u_1} → GenContFract.Pair α → α","typeReadable":"{α : Type u_1} → GenContFract.Pair α → α","typeReferences":[["GenContFract","Pair"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["GenContFract","Pair","mk","injEq"],"typeFallback":"forall {α : Type.{u_1}} (a : α) (b : α) (a_1 : α) (b_1 : α), Eq.{1} Prop (Eq.{succ u_1} (GenContFract.Pair.{u_1} α) (GenContFract.Pair.mk.{u_1} α a b) (GenContFract.Pair.mk.{u_1} α a_1 b_1)) (And (Eq.{succ u_1} α a a_1) (Eq.{succ u_1} α b b_1))","typeFull":"∀ {α : Type u_1} (a b a_1 b_1 : α), ({ a := a, b := b } = { a := a_1, b := b_1 }) = (a = a_1 ∧ b = b_1)","typeReadable":"∀ {α : Type u_1} (a b a_1 b_1 : α), ({ a := a, b := b } = { a := a_1, b := b_1 }) = (a = a_1 ∧ b = b_1)","typeReferences":[["And"],["GenContFract","Pair"],["Eq"],["GenContFract","Pair","mk"]],"valueReferences":[["Eq","propIntro"],["GenContFract","Pair","mk","inj"],["Eq","refl"],["And"],["GenContFract","Pair"],["Eq","ndrec"],["Lean","injEq_helper"],["Eq"],["GenContFract","Pair","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","contsAux"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> (Stream'.{u_2} (GenContFract.Pair.{u_2} K))","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' (GenContFract.Pair K)","typeReferences":[["GenContFract","Pair"],["Stream'"],["GenContFract"],["DivisionRing"]],"valueReferences":[["Ring","toNonAssocRing"],["GenContFract","h"],["GenContFract","Pair","a"],["GenContFract","s"],["GenContFract","contsAux","match_1"],["DivisionRing","toRing"],["AddGroupWithOne","toAddMonoidWithOne"],["GenContFract","nextConts"],["Nat","brecOn"],["GenContFract","Pair","b"],["GenContFract","contsAux","match_3"],["OfNat","ofNat"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["One","toOfNat1"],["Ring","toAddGroupWithOne"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Nat","below"],["AddMonoidWithOne","toOne"],["GenContFract","Pair"],["Zero","toOfNat0"],["NonAssocRing","toNonUnitalNonAssocRing"],["Stream'","Seq","get?"],["GenContFract","Pair","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Terminates"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> Prop","typeFull":"{α : Type u_1} → GenContFract α → Prop","typeReadable":"{α : Type u_1} → GenContFract α → Prop","typeReferences":[["GenContFract"]],"valueReferences":[["GenContFract","s"],["GenContFract","Pair"],["Stream'","Seq","Terminates"]]},{"isProp":false,"kind":"definition","name":["GenContFract","TerminatedAt"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.{u_1} α) -> Nat -> Prop","typeFull":"{α : Type u_1} → GenContFract α → ℕ → Prop","typeReadable":"{α : Type u_1} → GenContFract α → ℕ → Prop","typeReferences":[["Nat"],["GenContFract"]],"valueReferences":[["GenContFract","s"],["Stream'","Seq","TerminatedAt"],["GenContFract","Pair"]]},{"isProp":false,"kind":"definition","name":["GenContFract","convs'"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.2683677832._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> Nat -> K","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → ℕ → K","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → ℕ → K","typeReferences":[["Nat"],["GenContFract"],["DivisionRing"]],"valueReferences":[["HAdd","hAdd"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Distrib","toAdd"],["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toDistrib"],["GenContFract","convs'Aux"],["GenContFract","h"],["instHAdd"],["GenContFract","s"],["DivisionRing","toRing"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","Pair","mk","sizeOf_spec"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.11 : SizeOf.{succ u_1} α] (a : α) (b : α), Eq.{1} Nat (SizeOf.sizeOf.{succ u_1} (GenContFract.Pair.{u_1} α) (GenContFract.Pair._sizeOf_inst.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.11) (GenContFract.Pair.mk.{u_1} α a b)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (SizeOf.sizeOf.{succ u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.11 a)) (SizeOf.sizeOf.{succ u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.487281005._hygCtx._hyg.11 b))","typeFull":"∀ {α : Type u_1} [inst : SizeOf α] (a b : α), sizeOf { a := a, b := b } = 1 + sizeOf a + sizeOf b","typeReadable":"∀ {α : Type u_1} [inst : SizeOf α] (a b : α), sizeOf { a := a, b := b } = 1 + sizeOf a + sizeOf b","typeReferences":[["instAddNat"],["HAdd","hAdd"],["GenContFract","Pair","_sizeOf_inst"],["SizeOf","sizeOf"],["Nat"],["instOfNatNat"],["instHAdd"],["GenContFract","Pair"],["Eq"],["SizeOf"],["OfNat","ofNat"],["GenContFract","Pair","mk"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Nat"],["instOfNatNat"],["instHAdd"],["Eq","refl"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","Pair","coe_toPair"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1287474635._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} α β] {a : α} {b : α}, Eq.{succ u_2} (GenContFract.Pair.{u_2} β) (GenContFract.Pair.coeFn.{u_1, u_2} α β inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1287474635._hygCtx._hyg.4 (GenContFract.Pair.mk.{u_1} α a b)) (GenContFract.Pair.mk.{u_2} β (Coe.coe.{succ u_1, succ u_2} α β inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1287474635._hygCtx._hyg.4 a) (Coe.coe.{succ u_1, succ u_2} α β inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1287474635._hygCtx._hyg.4 b))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Coe α β] {a b : α}, ↑{ a := a, b := b } = { a := Coe.coe a, b := Coe.coe b }","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Coe α β] {a b : α}, ↑{ a := a, b := b } = { a := Coe.coe a, b := Coe.coe b }","typeReferences":[["GenContFract","Pair","coeFn"],["Coe"],["Coe","coe"],["GenContFract","Pair"],["Eq"],["GenContFract","Pair","mk"]],"valueReferences":[["rfl"],["GenContFract","Pair","coeFn"],["GenContFract","Pair"],["GenContFract","Pair","mk"]]},{"isProp":true,"kind":"theorem","name":["SimpContFract","ofInteger","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022417._hygCtx._hyg.3 : One.{u_1} α] (a : α) (n : Nat) (aₙ : α), (Eq.{succ u_1} (Option.{u_1} α) (Stream'.Seq.get?.{u_1} α (GenContFract.partNums.{u_1} α (GenContFract.ofInteger.{u_1} α a)) n) (Option.some.{u_1} α aₙ)) -> (Eq.{succ u_1} α aₙ (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022417._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : One α] (a : α) (n : ℕ) (aₙ : α), (GenContFract.ofInteger a).partNums.get? n = some aₙ → aₙ = 1","typeReadable":"∀ {α : Type u_1} [inst : One α] (a : α) (n : ℕ) (aₙ : α), (GenContFract.ofInteger a).partNums.get? n = some aₙ → aₙ = 1","typeReferences":[["Nat"],["One","toOfNat1"],["GenContFract","partNums"],["Option"],["One"],["Option","some"],["Eq"],["GenContFract","ofInteger"],["OfNat","ofNat"],["Stream'","Seq","get?"]],"valueReferences":[["HEq","refl"],["GenContFract","partNums"],["noConfusion_of_Nat"],["Option","some"],["Eq","casesOn"],["OfNat","ofNat"],["Option","ctorIdx"],["Option","none"],["False","elim"],["One","toOfNat1"],["Option"],["Eq","refl"],["HEq"],["Eq"],["GenContFract","ofInteger"],["Stream'","Seq","get?"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","coeFn"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1174359230._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} α β], (GenContFract.Pair.{u_1} α) -> (GenContFract.Pair.{u_2} β)","typeFull":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → GenContFract.Pair α → GenContFract.Pair β","typeReadable":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → GenContFract.Pair α → GenContFract.Pair β","typeReferences":[["Coe"],["GenContFract","Pair"]],"valueReferences":[["GenContFract","Pair","map"],["Coe","coe"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","b"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.Pair.{u_1} α) -> α","typeFull":"{α : Type u_1} → GenContFract.Pair α → α","typeReadable":"{α : Type u_1} → GenContFract.Pair α → α","typeReferences":[["GenContFract","Pair"]],"valueReferences":[]},{"isProp":false,"kind":"recursor","name":["GenContFract","rec"],"typeFallback":"forall {α : Type.{u_1}} {motive : (GenContFract.{u_1} α) -> Sort.{u}}, (forall (h : α) (s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)), motive (GenContFract.mk.{u_1} α h s)) -> (forall (t : GenContFract.{u_1} α), motive t)","typeFull":"{α : Type u_1} →\n  {motive : GenContFract α → Sort u} →\n    ((h : α) → (s : Stream'.Seq (GenContFract.Pair α)) → motive { h := h, s := s }) → (t : GenContFract α) → motive t","typeReadable":"{α : Type u_1} →\n  {motive : GenContFract α → Sort u} →\n    ((h : α) → (s : Stream'.Seq (GenContFract.Pair α)) → motive { h := h, s := s }) → (t : GenContFract α) → motive t","typeReferences":[["Stream'","Seq"],["GenContFract","Pair"],["GenContFract","mk"],["GenContFract"]],"valueReferences":null},{"isProp":false,"kind":"recursor","name":["GenContFract","Pair","rec"],"typeFallback":"forall {α : Type.{u_1}} {motive : (GenContFract.Pair.{u_1} α) -> Sort.{u}}, (forall (a : α) (b : α), motive (GenContFract.Pair.mk.{u_1} α a b)) -> (forall (t : GenContFract.Pair.{u_1} α), motive t)","typeFull":"{α : Type u_1} →\n  {motive : GenContFract.Pair α → Sort u} →\n    ((a b : α) → motive { a := a, b := b }) → (t : GenContFract.Pair α) → motive t","typeReadable":"{α : Type u_1} →\n  {motive : GenContFract.Pair α → Sort u} →\n    ((a b : α) → motive { a := a, b := b }) → (t : GenContFract.Pair α) → motive t","typeReferences":[["GenContFract","Pair"],["GenContFract","Pair","mk"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["GenContFract","instCoe"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3123013782._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} α β], Coe.{succ u_1, succ u_2} (GenContFract.{u_1} α) (GenContFract.{u_2} β)","typeFull":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → Coe (GenContFract α) (GenContFract β)","typeReadable":"{α : Type u_1} → {β : Type u_2} → [Coe α β] → Coe (GenContFract α) (GenContFract β)","typeReferences":[["Coe"],["GenContFract"]],"valueReferences":[["GenContFract","coeFn"],["Coe","mk"],["GenContFract"]]},{"isProp":false,"kind":"definition","name":["GenContFract","convs"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.2290856867._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> (Stream'.{u_2} K)","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' K","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' K","typeReferences":[["Stream'"],["GenContFract"],["DivisionRing"]],"valueReferences":[["HDiv","hDiv"],["GenContFract","nums"],["GenContFract","dens"],["instHDiv"],["DivisionRing","toDivInvMonoid"],["DivInvMonoid","toDiv"]]},{"isProp":false,"kind":"definition","name":["ContFract"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.95993801._hygCtx._hyg.3 : One.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.95993801._hygCtx._hyg.6 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.95993801._hygCtx._hyg.9 : LT.{u_1} α], Sort.{max 1 (succ u_1)}","typeFull":"(α : Type u_1) → [One α] → [Zero α] → [LT α] → Type (max 0 u_1)","typeReadable":"(α : Type u_1) → [One α] → [Zero α] → [LT α] → Type (max 0 u_1)","typeReferences":[["One"],["Zero"],["LT"]],"valueReferences":[["Subtype"],["SimpContFract","IsContFract"],["SimpContFract"]]},{"isProp":false,"kind":"definition","name":["GenContFract","Pair","ctorIdx"],"typeFallback":"forall {α : Type.{u_1}}, (GenContFract.Pair.{u_1} α) -> Nat","typeFull":"{α : Type u_1} → GenContFract.Pair α → ℕ","typeReadable":"{α : Type u_1} → GenContFract.Pair α → ℕ","typeReferences":[["Nat"],["GenContFract","Pair"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["GenContFract","contsAux","match_3"],"typeFallback":"forall (motive : Nat -> Sort.{u_1}) (x._@.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx.14.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx._hyg.26 : Nat), (Unit -> (motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (Unit -> (motive (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> (forall (n : Nat), motive (Nat.succ (Nat.succ n))) -> (motive x._@.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx.14.Mathlib.Algebra.ContinuedFractions.Basic.2368582518._hygCtx._hyg.26)","typeFull":"(motive : ℕ → Sort u_1) → (x : ℕ) → (Unit → motive 0) → (Unit → motive 1) → ((n : ℕ) → motive n.succ.succ) → motive x","typeReadable":"(motive : ℕ → Sort u_1) → (x : ℕ) → (Unit → motive 0) → (Unit → motive 1) → ((n : ℕ) → motive n.succ.succ) → motive x","typeReferences":[["Nat"],["Nat","succ"],["instOfNatNat"],["OfNat","ofNat"],["Unit"]],"valueReferences":[["Nat","casesOn"],["Nat","succ"],["Unit","unit"]]},{"isProp":false,"kind":"constructor","name":["GenContFract","mk"],"typeFallback":"forall {α : Type.{u_1}}, α -> (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) -> (GenContFract.{u_1} α)","typeFull":"{α : Type u_1} → α → Stream'.Seq (GenContFract.Pair α) → GenContFract α","typeReadable":"{α : Type u_1} → α → Stream'.Seq (GenContFract.Pair α) → GenContFract α","typeReferences":[["Stream'","Seq"],["GenContFract","Pair"],["GenContFract"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["GenContFract","dens"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3837166550._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> (Stream'.{u_2} K)","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' K","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' K","typeReferences":[["Stream'"],["GenContFract"],["DivisionRing"]],"valueReferences":[["GenContFract","conts"],["GenContFract","Pair"],["GenContFract","Pair","b"],["Stream'","map"]]},{"isProp":false,"kind":"definition","name":["ContFract","ofInteger"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.3 : One.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.6 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.9 : LT.{u_1} α], α -> (ContFract.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Basic.3386022418._hygCtx._hyg.9)","typeFull":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → α → ContFract α","typeReadable":"{α : Type u_1} → [inst : One α] → [inst_1 : Zero α] → [inst_2 : LT α] → α → ContFract α","typeReferences":[["ContFract"],["One"],["Zero"],["LT"]],"valueReferences":[["ContFract","ofInteger","_proof_1"],["SimpContFract","ofInteger"],["SimpContFract","IsContFract"],["SimpContFract"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","mk","injEq"],"typeFallback":"forall {α : Type.{u_1}} (h : α) (s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) (h_1 : α) (s_1 : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)), Eq.{1} Prop (Eq.{succ u_1} (GenContFract.{u_1} α) (GenContFract.mk.{u_1} α h s) (GenContFract.mk.{u_1} α h_1 s_1)) (And (Eq.{succ u_1} α h h_1) (Eq.{succ u_1} (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) s s_1))","typeFull":"∀ {α : Type u_1} (h : α) (s : Stream'.Seq (GenContFract.Pair α)) (h_1 : α) (s_1 : Stream'.Seq (GenContFract.Pair α)),\n  ({ h := h, s := s } = { h := h_1, s := s_1 }) = (h = h_1 ∧ s = s_1)","typeReadable":"∀ {α : Type u_1} (h : α) (s : Stream'.Seq (GenContFract.Pair α)) (h_1 : α) (s_1 : Stream'.Seq (GenContFract.Pair α)),\n  ({ h := h, s := s } = { h := h_1, s := s_1 }) = (h = h_1 ∧ s = s_1)","typeReferences":[["Stream'","Seq"],["And"],["GenContFract","Pair"],["GenContFract","mk"],["GenContFract"],["Eq"]],"valueReferences":[["GenContFract","mk","inj"],["Eq","propIntro"],["Stream'","Seq"],["Eq","refl"],["And"],["GenContFract","Pair"],["Eq","ndrec"],["Lean","injEq_helper"],["GenContFract","mk"],["GenContFract"],["Eq"]]},{"isProp":false,"kind":"definition","name":["GenContFract","instInhabited"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.4037808124._hygCtx._hyg.3 : Inhabited.{succ u_1} α], Inhabited.{succ u_1} (GenContFract.{u_1} α)","typeFull":"{α : Type u_1} → [Inhabited α] → Inhabited (GenContFract α)","typeReadable":"{α : Type u_1} → [Inhabited α] → Inhabited (GenContFract α)","typeReferences":[["GenContFract"],["Inhabited"]],"valueReferences":[["Inhabited","mk"],["GenContFract"],["GenContFract","ofInteger"],["Inhabited","default"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","mk","sizeOf_spec"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1831630741._hygCtx._hyg.15 : SizeOf.{succ u_1} α] (h : α) (s : Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)), Eq.{1} Nat (SizeOf.sizeOf.{succ u_1} (GenContFract.{u_1} α) (GenContFract._sizeOf_inst.{u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1831630741._hygCtx._hyg.15) (GenContFract.mk.{u_1} α h s)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (SizeOf.sizeOf.{succ u_1} α inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1831630741._hygCtx._hyg.15 h)) (SizeOf.sizeOf.{succ u_1} (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α)) (instSizeOfDefault.{succ u_1} (Stream'.Seq.{u_1} (GenContFract.Pair.{u_1} α))) s))","typeFull":"∀ {α : Type u_1} [inst : SizeOf α] (h : α) (s : Stream'.Seq (GenContFract.Pair α)),\n  sizeOf { h := h, s := s } = 1 + sizeOf h + sizeOf s","typeReadable":"∀ {α : Type u_1} [inst : SizeOf α] (h : α) (s : Stream'.Seq (GenContFract.Pair α)),\n  sizeOf { h := h, s := s } = 1 + sizeOf h + sizeOf s","typeReferences":[["instAddNat"],["instHAdd"],["GenContFract"],["GenContFract","mk"],["SizeOf"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["SizeOf","sizeOf"],["GenContFract","_sizeOf_inst"],["Stream'","Seq"],["instOfNatNat"],["instSizeOfDefault"],["GenContFract","Pair"],["Eq"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Nat"],["Stream'","Seq"],["instOfNatNat"],["instHAdd"],["Eq","refl"],["instSizeOfDefault"],["GenContFract","Pair"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["GenContFract","noConfusion"],"typeFallback":"forall {P : Sort.{u}} {α : Type.{u_1}} {t : GenContFract.{u_1} α} {α' : Type.{u_1}} {t' : GenContFract.{u_1} α'}, (Eq.{succ (succ u_1)} Type.{u_1} α α') -> (HEq.{succ u_1} (GenContFract.{u_1} α) t (GenContFract.{u_1} α') t') -> (GenContFract.noConfusionType.{u, u_1} P α t α' t')","typeFull":"{P : Sort u} →\n  {α : Type u_1} →\n    {t : GenContFract α} →\n      {α' : Type u_1} → {t' : GenContFract α'} → α = α' → t ≍ t' → GenContFract.noConfusionType P t t'","typeReadable":"{P : Sort u} →\n  {α : Type u_1} →\n    {t : GenContFract α} →\n      {α' : Type u_1} → {t' : GenContFract α'} → α = α' → t ≍ t' → GenContFract.noConfusionType P t t'","typeReferences":[["HEq"],["Eq"],["GenContFract"],["GenContFract","noConfusionType"]],"valueReferences":[["Stream'","Seq"],["HEq","refl"],["eq_of_heq"],["GenContFract","casesOn"],["HEq"],["GenContFract","Pair"],["GenContFract"],["Eq","ndrec"],["GenContFract","noConfusionType"]]},{"isProp":false,"kind":"definition","name":["GenContFract","_sizeOf_inst"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.1831630741._hygCtx._hyg.14 : SizeOf.{succ u_1} α], SizeOf.{succ u_1} (GenContFract.{u_1} α)","typeFull":"(α : Type u_1) → [SizeOf α] → SizeOf (GenContFract α)","typeReadable":"(α : Type u_1) → [SizeOf α] → SizeOf (GenContFract α)","typeReferences":[["GenContFract"],["SizeOf"]],"valueReferences":[["GenContFract","_sizeOf_1"],["GenContFract"],["SizeOf","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","nums"],"typeFallback":"forall {K : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Basic.247368621._hygCtx._hyg.4 : DivisionRing.{u_2} K], (GenContFract.{u_2} K) -> (Stream'.{u_2} K)","typeFull":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' K","typeReadable":"{K : Type u_2} → [DivisionRing K] → GenContFract K → Stream' K","typeReferences":[["Stream'"],["GenContFract"],["DivisionRing"]],"valueReferences":[["GenContFract","conts"],["GenContFract","Pair","a"],["GenContFract","Pair"],["Stream'","map"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Ring.Subring.Units.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","_proof_7"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))] {x._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.52 : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))}, (LT.lt.{u_1} R (Preorder.toLT.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))))) (Units.val.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)) x._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.52)) -> (LT.lt.{u_1} R (Preorder.toLT.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))))) (Units.val.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)) (Inv.inv.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instInv.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) x._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.52)))","typeFull":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {x : Rˣ}, 0 < ↑x → 0 < ↑x⁻¹","typeReadable":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {x : Rˣ}, 0 < ↑x → 0 < ↑x⁻¹","typeReferences":[["Inv","inv"],["IsStrictOrderedRing"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Units","instInv"],["LinearOrder"],["Preorder","toLT"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["LT","lt"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["DistribLattice","toLattice"],["Units","val"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Units"],["Semiring"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["PartialOrder","toPreorder"],["Preorder","toLT"],["Units","instInv"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Units"],["SemilatticeInf","toPartialOrder"],["IsStrictOrderedRing","toPosMulStrictMono"],["IsStrictOrderedRing","toCharZero"],["Inv","inv"],["Lattice","toSemilatticeInf"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["LT","lt"],["DistribLattice","toLattice"],["MulZeroClass","toZero"],["Units","val"],["Iff","mpr"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NeZero","charZero_one"],["Units","inv_pos"],["IsStrictOrderedRing","toZeroLEOneClass"]]},{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","_proof_4"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))], NeZero.{u_1} R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))))","typeFull":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R], NeZero 1","typeReadable":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R], NeZero 1","typeReferences":[["IsStrictOrderedRing"],["Lattice","toSemilatticeInf"],["LinearOrder"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["OfNat","ofNat"],["NeZero"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["DistribLattice","toLattice"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddZero","toZero"],["Semiring"],["SemilatticeInf","toPartialOrder"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsStrictOrderedRing","toCharZero"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["Semiring","toNonAssocSemiring"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NeZero","charZero_one"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["Units","mem_posSubgroup"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6))))] (u : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))), Iff (Membership.mem.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Subgroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)))) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)))) (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Subgroup.instSetLike.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))))) (Units.posSubgroup.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.9) u) (LT.lt.{u_1} R (Preorder.toLT.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)))))) (Units.val.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)) u))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] (u : Rˣ),\n  u ∈ Units.posSubgroup R ↔ 0 < ↑u","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] (u : Rˣ),\n  u ∈ Units.posSubgroup R ↔ 0 < ↑u","typeReferences":[["Units","instGroup"],["PartialOrder","toPreorder"],["Subgroup","instSetLike"],["Membership","mem"],["Preorder","toLT"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Units","posSubgroup"],["Zero","toOfNat0"],["Units"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearOrder"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["LT","lt"],["DistribLattice","toLattice"],["Units","val"],["MulZeroClass","toZero"],["Iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Subgroup"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["Units","instGroup"],["Subgroup","instSetLike"],["MonoidWithZero","toMonoid"],["Membership","mem"],["Semiring","toMonoidWithZero"],["Units","posSubgroup"],["Subgroup"],["Iff","rfl"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","_proof_3"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))], ZeroLEOneClass.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))))","typeFull":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R], ZeroLEOneClass R","typeReadable":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R], ZeroLEOneClass R","typeReferences":[["PartialOrder","toPreorder"],["IsStrictOrderedRing"],["Lattice","toSemilatticeInf"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearOrder"],["ZeroLEOneClass"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddMonoidWithOne","toOne"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Semiring"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["SemilatticeInf","toPartialOrder"],["IsStrictOrderedRing","toZeroLEOneClass"]]},{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","_proof_5"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))] {a : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))} {b : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))}, (Membership.mem.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Set.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Set.instMembership.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Subsemigroup.carrier.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOne.toMul.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Submonoid.toSubsemigroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.comap.{u_1, u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (MonoidHom.instFunLike.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Units.posSubgroup._proof_1.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3) (Units.coeHom.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.pos.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6)))) (Units.posSubgroup._proof_2.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_3.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_4.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9))))) a) -> (Membership.mem.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Set.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Set.instMembership.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Subsemigroup.carrier.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOne.toMul.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Submonoid.toSubsemigroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.comap.{u_1, u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (MonoidHom.instFunLike.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Units.posSubgroup._proof_1.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3) (Units.coeHom.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.pos.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6)))) (Units.posSubgroup._proof_2.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_3.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_4.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9))))) b) -> (Membership.mem.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Set.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Set.instMembership.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Subsemigroup.carrier.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOne.toMul.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Submonoid.toSubsemigroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.comap.{u_1, u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (MonoidHom.instFunLike.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Units.posSubgroup._proof_1.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3) (Units.coeHom.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.pos.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6)))) (Units.posSubgroup._proof_2.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_3.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_4.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9))))) (HMul.hMul.{u_1, u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (instHMul.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOne.toMul.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))))) a b))","typeFull":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] {a b : Rˣ},\n  a ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier →\n    b ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier →\n      a * b ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier","typeReadable":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] {a b : Rˣ},\n  a ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier →\n    b ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier →\n      a * b ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier","typeReferences":[["MulOneClass","toMulOne"],["Membership","mem"],["MonoidHom"],["Units","posSubgroup","_proof_1"],["MonoidHom","instFunLike"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["instDistribLatticeOfLinearOrder"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Submonoid","pos"],["Units"],["NonAssocSemiring","toMulZeroOneClass"],["SemilatticeInf","toPartialOrder"],["Units","posSubgroup","_proof_3"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["Set"],["LinearOrder"],["Semiring","toMonoidWithZero"],["Submonoid","comap"],["Units","coeHom"],["Units","instMulOneClass"],["Submonoid","toSubsemigroup"],["Units","posSubgroup","_proof_4"],["Set","instMembership"],["Subsemigroup","carrier"],["Units","posSubgroup","_proof_2"],["DistribLattice","toLattice"],["instHMul"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Subsemigroup","mul_mem'"],["MonoidHom"],["MonoidHom","instFunLike"],["Units","posSubgroup","_proof_1"],["MulZeroOneClass","toMulOneClass"],["instDistribLatticeOfLinearOrder"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["NonAssocSemiring","toMulZeroOneClass"],["Units"],["Submonoid","pos"],["SemilatticeInf","toPartialOrder"],["Units","posSubgroup","_proof_3"],["Lattice","toSemilatticeInf"],["Submonoid","comap"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["Units","coeHom"],["Submonoid","toSubsemigroup"],["Units","posSubgroup","_proof_4"],["DistribLattice","toLattice"],["Units","posSubgroup","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","_proof_1"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R], MonoidHomClass.{u_1, u_1, u_1} (MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MonoidHom.instFunLike.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))))","typeFull":"∀ (R : Type u_1) [inst : Semiring R], MonoidHomClass (Rˣ →* R) Rˣ R","typeReadable":"∀ (R : Type u_1) [inst : Semiring R], MonoidHomClass (Rˣ →* R) Rˣ R","typeReferences":[["MulOneClass","toMulOne"],["MonoidHomClass"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["MonoidHom"],["MonoidHom","instFunLike"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["Units"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["Units"],["MonoidHom","instMonoidHomClass"]]},{"isProp":false,"kind":"definition","name":["Units","posSubgroup"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))], Subgroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))","typeFull":"(R : Type u_1) → [inst : Semiring R] → [inst_1 : LinearOrder R] → [IsStrictOrderedRing R] → Subgroup Rˣ","typeReadable":"(R : Type u_1) → [inst : Semiring R] → [inst_1 : LinearOrder R] → [IsStrictOrderedRing R] → Subgroup Rˣ","typeReferences":[["instDistribLatticeOfLinearOrder"],["Units","instGroup"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["MonoidWithZero","toMonoid"],["LinearOrder"],["Semiring","toMonoidWithZero"],["Subgroup"],["Units"],["SemilatticeInf","toPartialOrder"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["PartialOrder","toPreorder"],["Units","instGroup"],["Units","posSubgroup","_proof_5"],["MonoidHom"],["Preorder","toLT"],["Units","posSubgroup","_proof_1"],["MonoidHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["Subsemigroup","mk"],["instDistribLatticeOfLinearOrder"],["MulOne","toMul"],["Submonoid","mk"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Group","toDivInvMonoid"],["NonAssocSemiring","toMulZeroOneClass"],["Units"],["Submonoid","pos"],["SemilatticeInf","toPartialOrder"],["Units","posSubgroup","_proof_3"],["Lattice","toSemilatticeInf"],["Units","posSubgroup","_proof_7"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Submonoid","comap"],["Units","instMulOneClass"],["Units","coeHom"],["Units","posSubgroup","_proof_6"],["OfNat","ofNat"],["Units","posSubgroup","_proof_4"],["LT","lt"],["DistribLattice","toLattice"],["Units","posSubgroup","_proof_2"],["DivInvMonoid","toMonoid"],["Units","val"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["setOf"],["Subgroup","mk"]]},{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","_proof_2"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))], PosMulStrictMono.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6)))))","typeFull":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R], PosMulStrictMono R","typeReadable":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R], PosMulStrictMono R","typeReferences":[["PartialOrder","toPreorder"],["IsStrictOrderedRing"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearOrder"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["PosMulStrictMono"],["Semiring"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["IsStrictOrderedRing","toPosMulStrictMono"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","_proof_6"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))], Membership.mem.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Set.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Set.instMembership.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Subsemigroup.carrier.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOne.toMul.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Submonoid.toSubsemigroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.comap.{u_1, u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (MonoidHom.instFunLike.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) R (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))))) (Units.posSubgroup._proof_1.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3) (Units.coeHom.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Submonoid.pos.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6)))) (Units.posSubgroup._proof_2.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_3.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup._proof_4.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9))))) (OfNat.ofNat.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) 1 (One.toOfNat1.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOne.toOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))))))","typeFull":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R],\n  1 ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier","typeReadable":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R],\n  1 ∈ (Submonoid.comap (Units.coeHom R) (Submonoid.pos R)).carrier","typeReferences":[["MulOneClass","toMulOne"],["Membership","mem"],["MonoidHom"],["Units","posSubgroup","_proof_1"],["MonoidHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["instDistribLatticeOfLinearOrder"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Submonoid","pos"],["Units"],["NonAssocSemiring","toMulZeroOneClass"],["SemilatticeInf","toPartialOrder"],["Units","posSubgroup","_proof_3"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["MulOne","toOne"],["Set"],["LinearOrder"],["Semiring","toMonoidWithZero"],["Submonoid","comap"],["Units","coeHom"],["Units","instMulOneClass"],["Submonoid","toSubsemigroup"],["OfNat","ofNat"],["Units","posSubgroup","_proof_4"],["Set","instMembership"],["Subsemigroup","carrier"],["Units","posSubgroup","_proof_2"],["DistribLattice","toLattice"],["One","toOfNat1"],["Semiring"]],"valueReferences":[["Units","posSubgroup","_proof_3"],["MulOneClass","toMulOne"],["Lattice","toSemilatticeInf"],["MonoidHom"],["Units","posSubgroup","_proof_1"],["MonoidHom","instFunLike"],["Semiring","toMonoidWithZero"],["Submonoid","comap"],["Units","coeHom"],["Units","instMulOneClass"],["MulZeroOneClass","toMulOneClass"],["Submonoid","one_mem'"],["Units","posSubgroup","_proof_4"],["instDistribLatticeOfLinearOrder"],["Units","posSubgroup","_proof_2"],["DistribLattice","toLattice"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Submonoid","pos"],["Units"],["NonAssocSemiring","toMulZeroOneClass"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["Units","mem_posSubgroup","_simp_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6))))] (u : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))), Eq.{1} Prop (Membership.mem.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Subgroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)))) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)))) (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Subgroup.instSetLike.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3))))) (Units.posSubgroup.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.9) u) (LT.lt.{u_1} R (Preorder.toLT.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)))))) (Units.val.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.1940047970._hygCtx._hyg.3)) u))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] (u : Rˣ),\n  (u ∈ Units.posSubgroup R) = (0 < ↑u)","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] (u : Rˣ),\n  (u ∈ Units.posSubgroup R) = (0 < ↑u)","typeReferences":[["Units","instGroup"],["PartialOrder","toPreorder"],["Subgroup","instSetLike"],["Membership","mem"],["Preorder","toLT"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Units","posSubgroup"],["Zero","toOfNat0"],["Eq"],["Units"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearOrder"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["LT","lt"],["DistribLattice","toLattice"],["Units","val"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Subgroup"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["Units","instGroup"],["Subgroup","instSetLike"],["Membership","mem"],["Preorder","toLT"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Units","posSubgroup"],["Zero","toOfNat0"],["Units"],["propext"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["Lattice","toSemilatticeInf"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["LT","lt"],["DistribLattice","toLattice"],["Units","mem_posSubgroup"],["MulZeroClass","toZero"],["Units","val"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Subgroup"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.AlgebraicTopology.SimplicialSet.Monomorphisms.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["SSet","instIsStableUnderFilteredColimitsMonomorphisms"],"typeFallback":"CategoryTheory.MorphismProperty.IsStableUnderFilteredColimits.{u, u, u, succ u} SSet.{u} (CategoryTheory.Functor.category.{0, u, 0, succ u} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u} CategoryTheory.types.{u}) (CategoryTheory.MorphismProperty.monomorphisms.{u, succ u} SSet.{u} (CategoryTheory.Functor.category.{0, u, 0, succ u} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u} CategoryTheory.types.{u}))","typeFull":"CategoryTheory.MorphismProperty.IsStableUnderFilteredColimits.{u, u, u, u + 1}\n  (CategoryTheory.MorphismProperty.monomorphisms SSet)","typeReadable":"CategoryTheory.MorphismProperty.IsStableUnderFilteredColimits.{u, u, u, u + 1}\n  (CategoryTheory.MorphismProperty.monomorphisms SSet)","typeReferences":[["SSet"],["CategoryTheory","Category","opposite"],["SimplexCategory"],["Opposite"],["CategoryTheory","MorphismProperty","IsStableUnderFilteredColimits"],["CategoryTheory","MorphismProperty","monomorphisms"],["CategoryTheory","types"],["SimplexCategory","smallCategory"],["CategoryTheory","Functor","category"]],"valueReferences":[["SimplexCategory"],["CategoryTheory","MorphismProperty","IsStableUnderColimitsOfShape","functorCategory"],["CategoryTheory","Limits","Types","instHasPullbacksType"],["Opposite"],["UnivLE","small"],["congrArg"],["CategoryTheory","MorphismProperty","IsStableUnderColimitsOfShape"],["Eq","symm"],["Eq"],["CategoryTheory","Functor"],["CategoryTheory","Category","opposite"],["CategoryTheory","MorphismProperty","IsStableUnderFilteredColimits","mk"],["CategoryTheory","Types","instIsStableUnderFilteredColimitsMonomorphismsType"],["CategoryTheory","Limits","Types","hasColimitsOfShape"],["CategoryTheory","MorphismProperty","functorCategory"],["CategoryTheory","MorphismProperty","monomorphisms"],["CategoryTheory","types"],["CategoryTheory","Category","toCategoryStruct"],["SSet"],["UnivLE","self"],["CategoryTheory","MorphismProperty","functorCategory_monomorphisms"],["id"],["CategoryTheory","MorphismProperty"],["CategoryTheory","MorphismProperty","IsStableUnderFilteredColimits","isStableUnderColimitsOfShape"],["inferInstance"],["Eq","mpr"],["SimplexCategory","smallCategory"],["CategoryTheory","Functor","category"]]},{"isProp":true,"kind":"theorem","name":["SSet","instIsStableUnderCobaseChangeMonomorphisms"],"typeFallback":"CategoryTheory.MorphismProperty.IsStableUnderCobaseChange.{u_1, succ u_1} SSet.{u_1} (CategoryTheory.Functor.category.{0, u_1, 0, succ u_1} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u_1} CategoryTheory.types.{u_1}) (CategoryTheory.MorphismProperty.monomorphisms.{u_1, succ u_1} SSet.{u_1} (CategoryTheory.Functor.category.{0, u_1, 0, succ u_1} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u_1} CategoryTheory.types.{u_1}))","typeFull":"(CategoryTheory.MorphismProperty.monomorphisms SSet).IsStableUnderCobaseChange","typeReadable":"(CategoryTheory.MorphismProperty.monomorphisms SSet).IsStableUnderCobaseChange","typeReferences":[["SSet"],["CategoryTheory","Category","opposite"],["SimplexCategory"],["Opposite"],["CategoryTheory","MorphismProperty","monomorphisms"],["CategoryTheory","types"],["SimplexCategory","smallCategory"],["CategoryTheory","Functor","category"],["CategoryTheory","MorphismProperty","IsStableUnderCobaseChange"]],"valueReferences":[["SimplexCategory"],["CategoryTheory","Category","opposite"],["CategoryTheory","Functor"],["CategoryTheory","Limits","Types","instHasPullbacksType"],["Opposite"],["CategoryTheory","MorphismProperty","functorCategory"],["CategoryTheory","types"],["CategoryTheory","MorphismProperty","monomorphisms"],["CategoryTheory","Limits","Types","instHasPushoutsType"],["CategoryTheory","Types","instIsStableUnderCobaseChangeMonomorphismsType"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","MorphismProperty","IsStableUnderCobaseChange"],["congrArg"],["SSet"],["CategoryTheory","MorphismProperty","instIsStableUnderCobaseChangeFunctorFunctorCategoryOfHasPushouts"],["CategoryTheory","MorphismProperty","functorCategory_monomorphisms"],["Eq","symm"],["CategoryTheory","MorphismProperty"],["id"],["inferInstance"],["Eq","mpr"],["SimplexCategory","smallCategory"],["Eq"],["CategoryTheory","Functor","category"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","AlgebraicTopology","SimplicialSet","Monomorphisms",0,"SSet","instIsStableUnderCoproductsMonomorphismsOfHasCoproductsType","_proof_1"],"typeFallback":"forall [inst._@.Mathlib.AlgebraicTopology.SimplicialSet.Monomorphisms.1641045791._hygCtx._hyg.2 : CategoryTheory.Limits.HasCoproducts.{u_1, u_2, succ u_2} Type.{u_2} CategoryTheory.types.{u_2}], CategoryTheory.MorphismProperty.IsStableUnderCoproducts.{u_1, u_2, succ u_2} SSet.{u_2} (CategoryTheory.Functor.category.{0, u_2, 0, succ u_2} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u_2} CategoryTheory.types.{u_2}) (CategoryTheory.MorphismProperty.monomorphisms.{u_2, succ u_2} SSet.{u_2} (CategoryTheory.Functor.category.{0, u_2, 0, succ u_2} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u_2} CategoryTheory.types.{u_2}))","typeFull":"∀ [CategoryTheory.Limits.HasCoproducts (Type u_2)],\n  CategoryTheory.MorphismProperty.IsStableUnderCoproducts.{u_1, u_2, u_2 + 1}\n    (CategoryTheory.MorphismProperty.monomorphisms SSet)","typeReadable":"∀ [CategoryTheory.Limits.HasCoproducts (Type u_2)],\n  CategoryTheory.MorphismProperty.IsStableUnderCoproducts.{u_1, u_2, u_2 + 1}\n    (CategoryTheory.MorphismProperty.monomorphisms SSet)","typeReferences":[["SSet"],["CategoryTheory","Category","opposite"],["SimplexCategory"],["CategoryTheory","Limits","HasCoproducts"],["Opposite"],["CategoryTheory","MorphismProperty","IsStableUnderCoproducts"],["CategoryTheory","MorphismProperty","monomorphisms"],["CategoryTheory","types"],["SimplexCategory","smallCategory"],["CategoryTheory","Functor","category"]],"valueReferences":[["CategoryTheory","Category","opposite"],["SimplexCategory"],["CategoryTheory","MorphismProperty","instIsStableUnderCoproductsFunctorMonomorphismsOfHasCoproductsOfHasPullbacks"],["CategoryTheory","Limits","Types","instHasPullbacksType"],["Opposite"],["CategoryTheory","types"],["SimplexCategory","smallCategory"],["CategoryTheory","Types","instIsStableUnderCoproductsMonomorphismsType"]]},{"isProp":true,"kind":"theorem","name":["SSet","instIsStableUnderCoproductsMonomorphismsOfHasCoproductsType"],"typeFallback":"forall [inst._@.Mathlib.AlgebraicTopology.SimplicialSet.Monomorphisms.1641045791._hygCtx._hyg.2 : CategoryTheory.Limits.HasCoproducts.{v', u, succ u} Type.{u} CategoryTheory.types.{u}], CategoryTheory.MorphismProperty.IsStableUnderCoproducts.{v', u, succ u} SSet.{u} (CategoryTheory.Functor.category.{0, u, 0, succ u} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u} CategoryTheory.types.{u}) (CategoryTheory.MorphismProperty.monomorphisms.{u, succ u} SSet.{u} (CategoryTheory.Functor.category.{0, u, 0, succ u} (Opposite.{1} SimplexCategory) (CategoryTheory.Category.opposite.{0, 0} SimplexCategory SimplexCategory.smallCategory) Type.{u} CategoryTheory.types.{u}))","typeFull":"∀ [CategoryTheory.Limits.HasCoproducts (Type u)],\n  CategoryTheory.MorphismProperty.IsStableUnderCoproducts.{v', u, u + 1}\n    (CategoryTheory.MorphismProperty.monomorphisms SSet)","typeReadable":"∀ [CategoryTheory.Limits.HasCoproducts (Type u)],\n  CategoryTheory.MorphismProperty.IsStableUnderCoproducts.{v', u, u + 1}\n    (CategoryTheory.MorphismProperty.monomorphisms SSet)","typeReferences":[["SSet"],["CategoryTheory","Category","opposite"],["SimplexCategory"],["CategoryTheory","Limits","HasCoproducts"],["Opposite"],["CategoryTheory","MorphismProperty","IsStableUnderCoproducts"],["CategoryTheory","MorphismProperty","monomorphisms"],["CategoryTheory","types"],["SimplexCategory","smallCategory"],["CategoryTheory","Functor","category"]],"valueReferences":[["_private","Mathlib","AlgebraicTopology","SimplicialSet","Monomorphisms",0,"SSet","instIsStableUnderCoproductsMonomorphismsOfHasCoproductsType","_proof_1"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Calculus.TangentCone.ProperSpace.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["tangentConeAt_nonempty_of_properSpace"],"typeFallback":"forall {𝕜 : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3 : NontriviallyNormedField.{u_1} 𝕜] {E : Type.{u_2}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7 : NormedAddCommGroup.{u_2} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.10 : NormedSpace.{u_1, u_2} 𝕜 E (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3) (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.14 : ProperSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7))] {s : Set.{u_2} E} {x : E}, (AccPt.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) x (Filter.principal.{u_2} E s)) -> (Set.Nonempty.{u_2} E (Inter.inter.{u_2} (Set.{u_2} E) (Set.instInter.{u_2} E) (tangentConeAt.{u_1, u_2} 𝕜 E (NormedAddCommGroup.toAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7) (SMulZeroClass.toSMul.{u_1, u_2} 𝕜 E (AddZero.toZero.{u_2} E (AddZeroClass.toAddZero.{u_2} E (AddMonoid.toAddZeroClass.{u_2} E (ESeminormedAddMonoid.toAddMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7))))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} 𝕜 E (AddMonoid.toAddZeroClass.{u_2} E (ESeminormedAddMonoid.toAddMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7))))) (DistribMulAction.toDistribSMul.{u_1, u_2} 𝕜 E (MonoidWithZero.toMonoid.{u_1} 𝕜 (Semiring.toMonoidWithZero.{u_1} 𝕜 (DivisionSemiring.toSemiring.{u_1} 𝕜 (Semifield.toDivisionSemiring.{u_1} 𝕜 (Field.toSemifield.{u_1} 𝕜 (NormedField.toField.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3))))))) (ESeminormedAddMonoid.toAddMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (ESeminormedAddCommMonoid.toESeminormedAddMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (Module.toDistribMulAction.{u_1, u_2} 𝕜 E (DivisionSemiring.toSemiring.{u_1} 𝕜 (Semifield.toDivisionSemiring.{u_1} 𝕜 (Field.toSemifield.{u_1} 𝕜 (NormedField.toField.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3))))) (ESeminormedAddCommMonoid.toAddCommMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{u_2} E (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) (NormedAddCommGroup.toENormedAddCommMonoid.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7))) (NormedSpace.toModule.{u_1, u_2} 𝕜 E (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3) (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7) inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.10))))) (UniformSpace.toTopologicalSpace.{u_2} E (PseudoMetricSpace.toUniformSpace.{u_2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u_2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7)))) s x) (Compl.compl.{u_2} (Set.{u_2} E) (Set.instCompl.{u_2} E) (Singleton.singleton.{u_2, u_2} E (Set.{u_2} E) (Set.instSingletonSet.{u_2} E) (OfNat.ofNat.{u_2} E 0 (Zero.toOfNat0.{u_2} E (NegZeroClass.toZero.{u_2} E (SubNegZeroMonoid.toNegZeroClass.{u_2} E (SubtractionMonoid.toSubNegZeroMonoid.{u_2} E (SubtractionCommMonoid.toSubtractionMonoid.{u_2} E (AddCommGroup.toDivisionAddCommMonoid.{u_2} E (NormedAddCommGroup.toAddCommGroup.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.7))))))))))))","typeFull":"∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]\n  [inst_2 : NormedSpace 𝕜 E] [ProperSpace E] {s : Set E} {x : E},\n  AccPt x (Filter.principal s) → (tangentConeAt 𝕜 s x ∩ {0}ᶜ).Nonempty","typeReadable":"∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]\n  [inst_2 : NormedSpace 𝕜 E] [ProperSpace E] {s : Set E} {x : E},\n  AccPt x (Filter.principal s) → (tangentConeAt 𝕜 s x ∩ {0}ᶜ).Nonempty","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["PseudoMetricSpace","toUniformSpace"],["Singleton","singleton"],["Compl","compl"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["UniformSpace","toTopologicalSpace"],["Inter","inter"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["SubNegZeroMonoid","toNegZeroClass"],["ProperSpace"],["NormedAddCommGroup","toAddCommGroup"],["NontriviallyNormedField","toNormedField"],["NormedField","toField"],["NontriviallyNormedField"],["NormedSpace"],["MonoidWithZero","toMonoid"],["NormedAddCommGroup"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Set","instInter"],["DistribSMul","toSMulZeroClass"],["ESeminormedAddMonoid","toAddMonoid"],["Set","Nonempty"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["Set","instSingletonSet"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Filter","principal"],["Set","instCompl"],["Module","toDistribMulAction"],["AccPt"],["AddCommGroup","toDivisionAddCommMonoid"],["NegZeroClass","toZero"],["Field","toSemifield"],["tangentConeAt"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["implies_congr"],["not_lt","_simp_1"],["SeminormedAddGroup","toAddGroup"],["MulZeroClass","toMul"],["Real","normedCommRing"],["eq_true"],["AddGroupWithOne","toAddMonoidWithOne"],["MonoidWithZero","toMulZeroOneClass"],["SMulZeroClass","toSMul"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_8"],["AddGroup","toSubtractionMonoid"],["NonUnitalNormedCommRing","toNonUnitalNormedRing"],["norm_zero"],["NormedField","toNormedCommRing"],["squeeze_zero_norm"],["Exists"],["NormedSpace","toModule"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["LinearOrderedSemiField","toDenselyOrdered"],["DivisionSemiring","toSemiring"],["true_and"],["Ring","toSemiring"],["Filter","atTop"],["MulZeroOneClass","toMulZeroClass"],["NormedAddCommGroup","toNormedAddGroup"],["eq_false"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_5"],["Eq","mpr"],["StrictMono"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["Real","instIsStrictOrderedRing"],["Nat","instSemiring"],["NormedAddGroup","toAddGroup"],["Eq"],["Filter","instMembership"],["Metric","isOpen_ball"],["Set"],["Filter","atTop_neBot"],["SeminormedAddCommGroup","toSeminormedAddGroup"],["Module","toDistribMulAction"],["and_self"],["Real","instZero"],["SDiff","sdiff"],["Ne"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_9"],["instHSub"],["Real","instDivInvMonoid"],["PartialOrder","toPreorder"],["AddLeftCancelSemigroup","toIsLeftCancelAdd"],["Membership","mem"],["Inter","inter"],["Preorder","toLT"],["Metric","mem_closedBall","_simp_1"],["Real","instNontrivial"],["not_false_eq_true"],["forall_congr"],["AddLeftCancelMonoid","toAddLeftCancelSemigroup"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Filter","Tendsto"],["DistribSMul","toSMulZeroClass"],["And","left"],["IsStrictOrderedRing","toPosMulStrictMono"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["Mathlib","Tactic","Contrapose","contrapose₃"],["Real"],["NormedAddCommGroup","toMetricSpace"],["Nat","instPartialOrder"],["And","right"],["IsCompact","tendsto_subseq"],["AddZeroClass","toAddZero"],["Filter","principal"],["Exists","casesOn"],["PseudoEMetricSpace","pseudoMetrizableSpace"],["Nat"],["AccPt"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","match_1_1"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["Real","instOne"],["NormedAddCommGroup","toNorm"],["instArchimedeanNat"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_7"],["Eq","mp"],["Compl","compl"],["UniformSpace","toTopologicalSpace"],["instIsDirectedOrder"],["PosMulReflectLE","toPosMulReflectLT"],["add_sub_cancel"],["Set","mem_compl_iff","_simp_1"],["NormedField","toField"],["GroupWithZero","toMonoidWithZero"],["Dist","dist"],["AddCommSemigroup","toAddCommMagma"],["NormedField","toNorm"],["Real","partialOrder"],["Not"],["IsStrictOrderedRing","toCharZero"],["Real","instMul"],["dist_self"],["Set","Nonempty"],["IsCompact","diff"],["Inv","inv"],["instHAdd"],["Real","instZeroLEOneClass"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_4"],["NormedField","exists_one_lt_norm"],["zero_lt_one"],["Function","comp"],["Set","instSingletonSet"],["Real","semiring"],["Set","instCompl"],["LT","lt"],["of_eq_true"],["One","toOfNat1"],["Field","toSemifield"],["False"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["NormedCommRing","toNonUnitalNormedCommRing"],["rescale_to_shell"],["Classical","choose_spec"],["DivInvMonoid","toInv"],["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["Singleton","singleton"],["Exists","intro"],["Classical","choose"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["SubNegMonoid","toSub"],["nhds"],["Set","instInter"],["Set","instSDiff"],["PseudoMetricSpace","toDist"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Norm","norm"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionSemiring","toGroupWithZero"],["TopologicalSpace","PseudoMetrizableSpace","firstCountableTopology"],["zero_le_one","_simp_1"],["NonUnitalNormedRing","toNormedAddCommGroup"],["Filter"],["Set","instMembership"],["add_eq_left","_simp_1"],["instNonemptyOfInhabited"],["Eq","refl"],["instNoMaxOrderOfNontrivial"],["AddCommGroup","toAddCommMonoid"],["tangentConeAt"],["instInhabitedNat"],["AddMonoid","toAddZeroClass"],["dist_eq_norm"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["one_div"],["Metric","ball"],["SubtractionCommMonoid","toSubtractionMonoid"],["instHDiv"],["Real","instField"],["Nat","instPreorder"],["NontriviallyNormedField","toNormedField"],["accPt_iff_nhds"],["congr"],["AddCommMagma","toAdd"],["Preorder","toLE"],["ESeminormedAddMonoid","toAddMonoid"],["IsStrictOrderedRing","toIsOrderedRing"],["StrictMono","tendsto_atTop"],["dist_zero_right"],["OfNat","ofNat"],["HAdd","hAdd"],["Metric","closedBall"],["Real","instInv"],["LinearOrder","toPartialOrder"],["Filter","Tendsto","comp"],["exists_seq_strictAnti_tendsto"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Nat","instIsStrictOrderedRing"],["EMetricSpace","toPseudoEMetricSpace"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["And","casesOn"],["AddCommGroup","toAddCancelCommMonoid"],["Iff","mp"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_6"],["SeminormedAddGroup","toNorm"],["HMul","hMul"],["Filter","Eventually","of_forall"],["NormedAddCommGroup","toAddCommGroup"],["HDiv","hDiv"],["And","intro"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["And"],["DistribMulAction","toDistribSMul"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_3"],["Real","instLE"],["Real","pseudoMetricSpace"],["HSMul","hSMul"],["NegZeroClass","toZero"],["id"],["mem_tangentConeAt_of_seq"],["instHMul"],["Real","linearOrder"],["NeZero","charZero_one"],["Real","instRing"],["SubNegZeroMonoid","toNegZeroClass"],["Real","instIsOrderedRing"],["congrArg"],["instOrderTopologyReal"],["AddCancelCommMonoid","toAddLeftCancelMonoid"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instLT"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["congrFun'"],["MetricSpace","toEMetricSpace"],["ProperSpace","isCompact_closedBall"],["LT","lt","le"],["SeminormedAddCommGroup","toNorm"],["Metric","closedBall_mem_nhds"],["Real","metricSpace"],["True"],["Semiring","toMonoidWithZero"],["DivInvMonoid","toDiv"],["Set","mem_diff","_simp_1"],["StrictAnti"],["AddCommMonoid","toAddCommSemigroup"],["LE","le"],["NormedCommRing","toSeminormedCommRing"],["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_10"],["Set","mem_singleton_iff","_simp_1"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","match_1_1"],"typeFallback":"forall {𝕜 : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3 : NontriviallyNormedField.{u_1} 𝕜] (motive : (Exists.{succ u_1} 𝕜 (fun (x : 𝕜) => LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne)) (Norm.norm.{u_1} 𝕜 (NormedField.toNorm.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3)) x))) -> Prop) (x._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.262 : Exists.{succ u_1} 𝕜 (fun (x : 𝕜) => LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne)) (Norm.norm.{u_1} 𝕜 (NormedField.toNorm.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3)) x))), (forall (r : 𝕜) (hr : LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne)) (Norm.norm.{u_1} 𝕜 (NormedField.toNorm.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3)) r)), motive (Exists.intro.{succ u_1} 𝕜 (fun (x : 𝕜) => LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne)) (Norm.norm.{u_1} 𝕜 (NormedField.toNorm.{u_1} 𝕜 (NontriviallyNormedField.toNormedField.{u_1} 𝕜 inst._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.3)) x)) r hr)) -> (motive x._@.Mathlib.Analysis.Calculus.TangentCone.ProperSpace.4010270802._hygCtx._hyg.262)","typeFull":"∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] (motive : (∃ x, 1 < ‖x‖) → Prop) (x : ∃ x, 1 < ‖x‖),\n  (∀ (r : 𝕜) (hr : 1 < ‖r‖), motive ⋯) → motive x","typeReadable":"∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] (motive : (∃ x, 1 < ‖x‖) → Prop) (x : ∃ x, 1 < ‖x‖),\n  (∀ (r : 𝕜) (hr : 1 < ‖r‖), motive ⋯) → motive x","typeReferences":[["LT","lt"],["Exists"],["NontriviallyNormedField"],["Norm","norm"],["One","toOfNat1"],["Real"],["Real","instLT"],["Exists","intro"],["Real","instOne"],["NormedField","toNorm"],["OfNat","ofNat"],["NontriviallyNormedField","toNormedField"]],"valueReferences":[["LT","lt"],["Exists","casesOn"],["Norm","norm"],["One","toOfNat1"],["Real"],["Real","instLT"],["Real","instOne"],["NormedField","toNorm"],["OfNat","ofNat"],["NontriviallyNormedField","toNormedField"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_5"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Topology.MetricSpace.Pseudo.Defs.3112555092._hygCtx._hyg.6 : PseudoMetricSpace.{u} α] {x : α} {y : α} {ε : Real}, Eq.{1} Prop (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (Metric.ball.{u} α inst._@.Mathlib.Topology.MetricSpace.Pseudo.Defs.3112555092._hygCtx._hyg.6 x ε) y) (LT.lt.{0} Real Real.instLT (Dist.dist.{u} α (PseudoMetricSpace.toDist.{u} α inst._@.Mathlib.Topology.MetricSpace.Pseudo.Defs.3112555092._hygCtx._hyg.6) y x) ε)","typeFull":"∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, (y ∈ Metric.ball x ε) = (dist y x < ε)","typeReadable":"∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, (y ∈ Metric.ball x ε) = (dist y x < ε)","typeReferences":[["PseudoMetricSpace","toDist"],["LT","lt"],["Real"],["Set"],["Real","instLT"],["Dist","dist"],["Membership","mem"],["PseudoMetricSpace"],["Metric","ball"],["Eq"],["Set","instMembership"]],"valueReferences":[["PseudoMetricSpace","toDist"],["LT","lt"],["Real"],["Metric","mem_ball"],["Set"],["Real","instLT"],["Dist","dist"],["Membership","mem"],["Metric","ball"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_3"],"typeFallback":"forall {α : Type.{u}} {s : Set.{u} α} {t : Set.{u} α} (x : α), Eq.{1} Prop (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (SDiff.sdiff.{u} (Set.{u} α) (Set.instSDiff.{u} α) s t) x) (And (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) s x) (Not (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) t x)))","typeFull":"∀ {α : Type u} {s t : Set α} (x : α), (x ∈ s \\ t) = (x ∈ s ∧ x ∉ t)","typeReadable":"∀ {α : Type u} {s t : Set α} (x : α), (x ∈ s \\ t) = (x ∈ s ∧ x ∉ t)","typeReferences":[["Not"],["Set"],["Membership","mem"],["And"],["SDiff","sdiff"],["Eq"],["Set","instSDiff"],["Set","instMembership"]],"valueReferences":[["Not"],["Set"],["Membership","mem"],["And"],["SDiff","sdiff"],["Set","mem_diff"],["propext"],["Set","instSDiff"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_8"],"typeFallback":"forall {G₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.5 : GroupWithZero.{u_3} G₀] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.8 : PartialOrder.{u_3} G₀] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.11 : PosMulReflectLT.{u_3} G₀ (MulZeroClass.toMul.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.5)))) (PartialOrder.toPreorder.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.8)] {a : G₀}, Eq.{1} Prop (LT.lt.{u_3} G₀ (Preorder.toLT.{u_3} G₀ (PartialOrder.toPreorder.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.8)) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.5)))))) (Inv.inv.{u_3} G₀ (InvOneClass.toInv.{u_3} G₀ (DivInvOneMonoid.toInvOneClass.{u_3} G₀ (DivisionMonoid.toDivInvOneMonoid.{u_3} G₀ (GroupWithZero.toDivisionMonoid.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.5)))) a)) (LT.lt.{u_3} G₀ (Preorder.toLT.{u_3} G₀ (PartialOrder.toPreorder.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.8)) (OfNat.ofNat.{u_3} G₀ 0 (Zero.toOfNat0.{u_3} G₀ (MulZeroClass.toZero.{u_3} G₀ (MulZeroOneClass.toMulZeroClass.{u_3} G₀ (MonoidWithZero.toMulZeroOneClass.{u_3} G₀ (GroupWithZero.toMonoidWithZero.{u_3} G₀ inst._@.Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic.3871353733._hygCtx._hyg.5)))))) a)","typeFull":"∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : PartialOrder G₀] [PosMulReflectLT G₀] {a : G₀},\n  (0 < a⁻¹) = (0 < a)","typeReadable":"∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] [inst_1 : PartialOrder G₀] [PosMulReflectLT G₀] {a : G₀},\n  (0 < a⁻¹) = (0 < a)","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["PosMulReflectLT"],["PartialOrder","toPreorder"],["MulZeroClass","toMul"],["Preorder","toLT"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["LT","lt"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["PartialOrder"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["Eq"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Inv","inv"],["PartialOrder","toPreorder"],["InvOneClass","toInv"],["Preorder","toLT"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["LT","lt"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["inv_pos"],["DivisionMonoid","toDivInvOneMonoid"],["propext"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_10"],"typeFallback":"forall {a : Prop}, Eq.{1} Prop (Not (Not a)) a","typeFull":"∀ {a : Prop}, (¬¬a) = a","typeReadable":"∀ {a : Prop}, (¬¬a) = a","typeReferences":[["Not"],["Eq"]],"valueReferences":[["Not"],["Classical","not_not"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_7"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.3 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.6 : One.{u_1} α] [inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.9 : LE.{u_1} α] [inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.12 : ZeroLEOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.9], Eq.{1} Prop (LE.le.{u_1} α inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.9 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.3)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Order.ZeroLEOne.1467683948._hygCtx._hyg.6))) True","typeFull":"∀ {α : Type u_1} [inst : Zero α] [inst_1 : One α] [inst_2 : LE α] [ZeroLEOneClass α], (0 ≤ 1) = True","typeReadable":"∀ {α : Type u_1} [inst : Zero α] [inst_1 : One α] [inst_2 : LE α] [ZeroLEOneClass α], (0 ≤ 1) = True","typeReferences":[["True"],["One","toOfNat1"],["LE","le"],["LE"],["One"],["Zero","toOfNat0"],["Zero"],["Eq"],["OfNat","ofNat"],["ZeroLEOneClass"]],"valueReferences":[["One","toOfNat1"],["LE","le"],["zero_le_one"],["eq_true"],["Zero","toOfNat0"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_9"],"typeFallback":"forall {E : Type.{u_5}} [inst._@.Mathlib.Analysis.Normed.Group.Basic.1207578777._hygCtx._hyg.9 : NormedAddGroup.{u_5} E] {a : E}, Eq.{1} Prop (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (Norm.norm.{u_5} E (NormedAddGroup.toNorm.{u_5} E inst._@.Mathlib.Analysis.Normed.Group.Basic.1207578777._hygCtx._hyg.9) a)) (Ne.{succ u_5} E a (OfNat.ofNat.{u_5} E 0 (Zero.toOfNat0.{u_5} E (NegZeroClass.toZero.{u_5} E (SubNegZeroMonoid.toNegZeroClass.{u_5} E (SubtractionMonoid.toSubNegZeroMonoid.{u_5} E (AddGroup.toSubtractionMonoid.{u_5} E (NormedAddGroup.toAddGroup.{u_5} E inst._@.Mathlib.Analysis.Normed.Group.Basic.1207578777._hygCtx._hyg.9))))))))","typeFull":"∀ {E : Type u_5} [inst : NormedAddGroup E] {a : E}, (0 < ‖a‖) = (a ≠ 0)","typeReadable":"∀ {E : Type u_5} [inst : NormedAddGroup E] {a : E}, (0 < ‖a‖) = (a ≠ 0)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Real"],["Norm","norm"],["NormedAddGroup","toNorm"],["NormedAddGroup"],["SubNegZeroMonoid","toNegZeroClass"],["AddGroup","toSubtractionMonoid"],["OfNat","ofNat"],["LT","lt"],["Real","instZero"],["Real","instLT"],["NormedAddGroup","toAddGroup"],["NegZeroClass","toZero"],["Zero","toOfNat0"],["Ne"],["Eq"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Real"],["Norm","norm"],["NormedAddGroup","toNorm"],["SubNegZeroMonoid","toNegZeroClass"],["AddGroup","toSubtractionMonoid"],["OfNat","ofNat"],["LT","lt"],["norm_pos_iff"],["Real","instZero"],["Real","instLT"],["NormedAddGroup","toAddGroup"],["NegZeroClass","toZero"],["Zero","toOfNat0"],["Ne"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_4"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Topology.MetricSpace.Pseudo.Defs.1390266476._hygCtx._hyg.6 : PseudoMetricSpace.{u} α] {x : α} {y : α} {ε : Real}, Eq.{1} Prop (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (Metric.closedBall.{u} α inst._@.Mathlib.Topology.MetricSpace.Pseudo.Defs.1390266476._hygCtx._hyg.6 x ε) y) (LE.le.{0} Real Real.instLE (Dist.dist.{u} α (PseudoMetricSpace.toDist.{u} α inst._@.Mathlib.Topology.MetricSpace.Pseudo.Defs.1390266476._hygCtx._hyg.6) y x) ε)","typeFull":"∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, (y ∈ Metric.closedBall x ε) = (dist y x ≤ ε)","typeReadable":"∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, (y ∈ Metric.closedBall x ε) = (dist y x ≤ ε)","typeReferences":[["PseudoMetricSpace","toDist"],["Metric","closedBall"],["Real"],["Set"],["Dist","dist"],["LE","le"],["Membership","mem"],["PseudoMetricSpace"],["Eq"],["Real","instLE"],["Set","instMembership"]],"valueReferences":[["PseudoMetricSpace","toDist"],["Metric","closedBall"],["Real"],["Set"],["Metric","mem_closedBall"],["Dist","dist"],["LE","le"],["Membership","mem"],["propext"],["Real","instLE"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","ProperSpace",0,"tangentConeAt_nonempty_of_properSpace","_simp_1_6"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Defs.LinearOrder.4272507027._hygCtx._hyg.3 : LinearOrder.{u_1} α] {a : α} {b : α}, Eq.{1} Prop (Not (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (LinearOrder.toPartialOrder.{u_1} α inst._@.Mathlib.Order.Defs.LinearOrder.4272507027._hygCtx._hyg.3))) a b)) (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α (LinearOrder.toPartialOrder.{u_1} α inst._@.Mathlib.Order.Defs.LinearOrder.4272507027._hygCtx._hyg.3))) b a)","typeFull":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, (¬a < b) = (b ≤ a)","typeReadable":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : α}, (¬a < b) = (b ≤ a)","typeReferences":[["LT","lt"],["Not"],["LinearOrder","toPartialOrder"],["PartialOrder","toPreorder"],["LE","le"],["Preorder","toLT"],["LinearOrder"],["Preorder","toLE"],["Eq"]],"valueReferences":[["LT","lt"],["Not"],["LinearOrder","toPartialOrder"],["not_lt"],["PartialOrder","toPreorder"],["LE","le"],["Preorder","toLT"],["Preorder","toLE"],["propext"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Oscillation.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["IsCompact","uniform_oscillationWithin"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7 : PseudoEMetricSpace.{u} E] {K : Set.{u} E} {f : E -> F} {D : Set.{u} E} {ε : ENNReal}, (IsCompact.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoEMetricSpace.toUniformSpace.{u} E inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7)) K) -> (forall (x : E), (Membership.mem.{u, u} E (Set.{u} E) (Set.instMembership.{u} E) K x) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (oscillationWithin.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 (UniformSpace.toTopologicalSpace.{u} E (PseudoEMetricSpace.toUniformSpace.{u} E inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7)) f D x) ε)) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLT δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero))) (forall (x : E), (Membership.mem.{u, u} E (Set.{u} E) (Set.instMembership.{u} E) K x) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (Metric.ediam.{v} F inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 (Set.image.{u, v} E F f (Inter.inter.{u} (Set.{u} E) (Set.instInter.{u} E) (Metric.eball.{u} E inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7 x (ENNReal.ofReal δ)) D))) ε))))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : PseudoEMetricSpace E] {K : Set E} {f : E → F}\n  {D : Set E} {ε : ENNReal},\n  IsCompact K →\n    (∀ x ∈ K, oscillationWithin f D x < ε) →\n      ∃ δ > 0, ∀ x ∈ K, Metric.ediam (f '' (Metric.eball x (ENNReal.ofReal δ) ∩ D)) ≤ ε","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : PseudoEMetricSpace E] {K : Set E} {f : E → F}\n  {D : Set E} {ε : ENNReal},\n  IsCompact K →\n    (∀ x ∈ K, oscillationWithin f D x < ε) →\n      ∃ δ > 0, ∀ x ∈ K, Metric.ediam (f '' (Metric.eball x (ENNReal.ofReal δ) ∩ D)) ≤ ε","typeReferences":[["PartialOrder","toPreorder"],["Inter","inter"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["Preorder","toLT"],["GT","gt"],["PseudoEMetricSpace"],["oscillationWithin"],["PseudoEMetricSpace","toUniformSpace"],["Real","instLT"],["Zero","toOfNat0"],["Preorder","toLE"],["Set","instInter"],["Metric","ediam"],["Exists"],["Real"],["Set"],["And"],["OfNat","ofNat"],["Set","instMembership"],["LT","lt"],["ENNReal","ofReal"],["Metric","eball"],["ENNReal"],["Set","image"],["IsCompact"],["Real","instZero"],["LE","le"],["ENNReal","instPartialOrder"]],"valueReferences":[["Set","IsWF","min"],["Mathlib","Tactic","CancelDenoms","sub_subst"],["Ring","toNonAssocRing"],["sub_add_cancel"],["Real","instAddGroup"],["Set","instIsTransSubset"],["Bool","false"],["MulZeroClass","toMul"],["Classical","propDecidable"],["eq_true"],["AddGroupWithOne","toAddMonoidWithOne"],["AddGroup","toSubtractionMonoid"],["ENNReal","instLinearOrder"],["Set","instReflSubset"],["Real","commRing"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_4"],["le_of_not_gt"],["Eq","symm"],["ConditionallyCompletePartialOrderInf","toInfSet"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Metric","ediam"],["Nat","ofNat_pos","_simp_1"],["Exists"],["Real","instCommSemiring"],["Nat","instCommSemiring"],["Mathlib","Meta","NormNum","IsNNRat","to_isNat"],["ne_of_gt"],["Filter","map"],["LinearOrderedSemiField","toDenselyOrdered"],["DivisionSemiring","toSemiring"],["Ring","toSemiring"],["ENNReal","add_lt_add"],["AddMonoid","toAddSemigroup"],["Mathlib","Meta","NormNum","isNat_lt_true"],["gt_iff_lt","_simp_1"],["instIsLeftCancelAddOfAddLeftReflectLE"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["Top","top"],["Int","negOfNat"],["Eq","mpr"],["Set","Finite"],["covariant_swap_add_of_covariant_add"],["Real","instIsStrictOrderedRing"],["Mathlib","Tactic","Ring","add_mul"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Real","instNatCast"],["Mathlib","Tactic","Linarith","lt_irrefl"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_9"],["Mathlib","Meta","NormNum","isNat_eq_true"],["ENNReal","ofReal_toReal_le"],["Nat","instNeZeroSucc"],["CompleteLinearOrder","toConditionallyCompleteLinearOrderBot"],["Mathlib","Tactic","Ring","neg_one_mul"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_5"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["Eq"],["ENNReal","instOrderTop"],["Set","IsWF","min_le"],["Filter","instMembership"],["Mathlib","Tactic","Ring","neg_zero"],["Set"],["DivisionRing","toRing"],["ENNReal","instCompleteLinearOrder"],["instOfNatAtLeastTwo"],["EMetric","mem_nhdsWithin_iff"],["Field","toDivisionRing"],["Real","instAdd"],["AddZero","toAdd"],["HPow","hPow"],["Mathlib","Tactic","Ring","mul_congr"],["Set","Finite","isWF"],["Metric","eball_subset_eball"],["IsOpen"],["Real","instZero"],["Mathlib","Meta","NormNum","instAtLeastTwo"],["Set","mem_empty_iff_false","_simp_1"],["CompleteLinearOrder","toCompleteLattice"],["Mathlib","Meta","NormNum","instAddMonoidWithOne'"],["CompleteLattice","toCompleteSemilatticeInf"],["instHSub"],["Real","instDivInvMonoid"],["Nat","instAtLeastTwoHAddOfNat"],["Mathlib","Meta","NormNum","IsNat","to_isNNRat"],["contravariant_swap_add_of_contravariant_add"],["instIsRightCancelAddOfAddRightReflectLE"],["PartialOrder","toPreorder"],["lt_of_lt_of_eq"],["Mathlib","Meta","NormNum","IsNat","of_raw"],["Mathlib","Meta","NormNum","IsInt","of_raw"],["Mathlib","Tactic","Ring","add_pf_zero_add"],["Membership","mem"],["Inter","inter"],["Preorder","toLT"],["Nat","one_lt_ofNat","_simp_1"],["Real","instNontrivial"],["GT","gt"],["subset_trans"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Mathlib","Tactic","Ring","add_pf_add_overlap"],["one_pos"],["ENNReal","toReal"],["Mathlib","Tactic","Linarith","add_neg"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["Int","ofNat"],["SemilatticeInf","toPartialOrder"],["And","left"],["IsStrictOrderedRing","toPosMulStrictMono"],["Nat","mul"],["Real"],["Real","instIsOrderedAddMonoid"],["And","right"],["lt_of_le_of_lt"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Mathlib","Tactic","Ring","mul_zero"],["EMetric","isOpen_iff"],["AddZeroClass","toAddZero"],["Exists","casesOn"],["Mathlib","Tactic","Linarith","mul_neg"],["Nat"],["div_pos_iff_of_pos_left","_simp_1"],["AddMonoidWithOne","toNatCast"],["Mathlib","Tactic","Ring","atom_pf"],["Iff","of_eq"],["HasSubset","Subset"],["Real","instOne"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["Mathlib","Tactic","Linarith","mul_nonpos"],["PosMulStrictMono","toPosMulReflectLE"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["Mathlib","Tactic","Ring","mul_one"],["Eq","mp"],["le_trans"],["UniformSpace","toTopologicalSpace"],["CommRing","toNonUnitalCommRing"],["PosMulReflectLE","toPosMulReflectLT"],["instDistribLatticeOfLinearOrder"],["PseudoEMetricSpace","toUniformSpace"],["Mathlib","Tactic","Ring","add_pf_add_lt"],["Set","iUnion_congr_Prop"],["Mathlib","Meta","NormNum","isNNRat_inv_pos"],["AddCommSemigroup","toAddCommMagma"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["sub_pos","_simp_1"],["Real","partialOrder"],["Not"],["IsStrictOrderedRing","toCharZero"],["Set","Nonempty"],["Set","preimage"],["Inv","inv"],["Field","toCommRing"],["instHAdd"],["Distrib","toMul"],["Real","instZeroLEOneClass"],["Mathlib","Tactic","Ring","cast_pos"],["Mathlib","Tactic","Ring","add_congr"],["Real","semiring"],["instIsEmptyFalse"],["LT","lt"],["Mathlib","Tactic","CancelDenoms","div_subst"],["of_eq_true"],["One","toOfNat1"],["Mathlib","Tactic","Ring","neg_add"],["Mathlib","Tactic","Ring","add_pf_add_zero"],["le_of_lt"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","match_1_1"],["subset_refl"],["Mathlib","Tactic","Linarith","sub_nonpos_of_le"],["Field","toSemifield"],["False"],["_private","Mathlib","Data","Real","Basic",0,"Real","lt"],["le_imp_le_of_le_of_le"],["iInf_congr_Prop"],["instAddNat"],["Mathlib","Tactic","Linarith","add_lt_of_neg_of_le"],["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["Eq","trans"],["instZeroENNReal"],["Exists","intro"],["Real","instIsOrderedCancelAddMonoid"],["Mathlib","Meta","NormNum","IsInt","to_raw_eq"],["Mathlib","Meta","NormNum","isNNRat_div"],["Mathlib","Tactic","Linarith","sub_neg_of_lt"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["EDist","edist"],["Mathlib","Tactic","Ring","one_mul"],["SubNegMonoid","toSub"],["ENNReal","ofReal_le_ofReal"],["Mathlib","Tactic","Ring","add_overlap_pf_zero"],["Mathlib","Tactic","Ring","sub_pf"],["Set","instInter"],["rfl"],["Set","image_mono"],["Real","instAddCommSemigroup"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionSemiring","toGroupWithZero"],["instNeZeroNatHAdd_1"],["Filter"],["Set","instMembership"],["ENNReal","ofReal_add"],["Eq","refl"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_6"],["AddMonoidWithOne","toOne"],["Nat","rawCast"],["AddMonoid","toAddZeroClass"],["setOf"],["Mathlib","Meta","NormNum","IsNat","to_isInt"],["Mathlib","Meta","NormNum","isNat_mul"],["Bool"],["EmptyCollection","emptyCollection"],["instHDiv"],["Real","instField"],["Mathlib","Tactic","Ring","add_pf_add_overlap_zero"],["instOfNatNat"],["contravariant_lt_of_covariant_le"],["iInf"],["AddCommMagma","toAdd"],["Set","iUnion_empty"],["IsLeftCancelAdd","addLeftReflectLE_of_addLeftReflectLT"],["Preorder","toLE"],["Mathlib","Tactic","Ring","mul_add"],["AddMonoidWithOne"],["propext"],["lt_of_not_ge"],["Distrib","toAdd"],["Mul"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["Metric","eball"],["CommRing","toRing"],["LinearOrder","toDecidableEq"],["ENNReal"],["AddGroupWithOne","toAddGroup"],["IsCompact","elim_finite_subcover_image"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["PseudoEMetricSpace","toEDist"],["inferInstance"],["dite"],["Mathlib","Meta","NormNum","isNNRat_mul"],["Real","instAddMonoid"],["And","casesOn"],["Mathlib","Tactic","Ring","zero_mul"],["Nat","cast_one"],["Set","IsWF","min_mem"],["le_refl"],["Mathlib","Meta","NormNum","IsNat","to_raw_eq"],["Set","iUnion_of_empty"],["GroupWithZero","toDivInvMonoid"],["Iff","mp"],["HMul","hMul"],["Int","rawCast"],["ENNReal","toReal_pos"],["Set","iUnion"],["IsRightCancelAdd","addRightStrictMono_of_addRightMono"],["HDiv","hDiv"],["And","intro"],["Set","not_nonempty_iff_eq_empty"],["Ring","toAddGroupWithOne"],["funext"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["ENNReal","ofReal_pos","_simp_1"],["HSub","hSub"],["Mathlib","Meta","NormNum","IsInt","to_isNat"],["Real","instAddCommMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["PseudoEMetricSpace","edist_triangle"],["AddSemigroup","toAdd"],["instHPow"],["InvOneClass","toInv"],["NonUnitalNonAssocSemiring","toDistrib"],["And"],["Real","instSub"],["IsOrderedAddMonoid","toAddLeftMono"],["DivisionRing","toDivInvMonoid"],["nhdsWithin"],["Set","image"],["Nat","cast_zero"],["Iff","mpr"],["Mathlib","Meta","NormNum","instAddMonoidWithOne"],["Mathlib","Tactic","Ring","mul_pf_right"],["id"],["NegZeroClass","toZero"],["instHMul"],["Real","linearOrder"],["NeZero","charZero_one"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_7"],["Mathlib","Meta","NormNum","isNat_ofNat"],["Set","empty_subset","_simp_1"],["ENNReal","instPartialOrder"],["Metric","ediam_mono"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_8"],["Mathlib","Tactic","Ring","neg_mul"],["Mathlib","Meta","NormNum","isInt_add"],["Real","instRing"],["Set","instEmptyCollection"],["CompleteLattice","toCompleteSemilatticeSup"],["SubNegZeroMonoid","toNegZeroClass"],["Real","instIsOrderedRing"],["instTopENNReal"],["congrArg"],["oscillationWithin"],["Real","instLT"],["MonoidWithZero","toMonoid"],["Set","image_subset_iff"],["Set","inter_subset_inter"],["congrFun'"],["Zero","toOfNat0"],["Mathlib","Tactic","Ring","sub_congr"],["Mathlib","Tactic","Ring","cast_zero"],["Mathlib","Meta","NormNum","isInt_mul"],["Mathlib","Tactic","Ring","add_overlap_pf"],["Mathlib","Tactic","Ring","of_eq"],["Lattice","toSemilatticeInf"],["True"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["exists_between"],["DivInvMonoid","toDiv"],["ENNReal","instCommSemiring"],["CompleteSemilatticeInf","toInfSet"],["Set","instHasSubset"],["ENNReal","ofReal"],["Set","mem_iUnion"],["DistribLattice","toLattice"],["DivInvOneMonoid","toInvOneClass"],["SubNegMonoid","toAddMonoid"],["LE","le"],["CompleteSemilatticeSup","toPartialOrder"],["Mathlib","Tactic","Ring","add_pf_add_gt"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_9"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Order.BoundedOrder.Basic.4046263676._hygCtx._hyg.4 : LE.{u} α] [inst._@.Mathlib.Order.BoundedOrder.Basic.4046263676._hygCtx._hyg.7 : OrderTop.{u} α inst._@.Mathlib.Order.BoundedOrder.Basic.4046263676._hygCtx._hyg.4] {a : α}, Eq.{1} Prop (LE.le.{u} α inst._@.Mathlib.Order.BoundedOrder.Basic.4046263676._hygCtx._hyg.4 a (Top.top.{u} α (OrderTop.toTop.{u} α inst._@.Mathlib.Order.BoundedOrder.Basic.4046263676._hygCtx._hyg.4 inst._@.Mathlib.Order.BoundedOrder.Basic.4046263676._hygCtx._hyg.7))) True","typeFull":"∀ {α : Type u} [inst : LE α] [inst_1 : OrderTop α] {a : α}, (a ≤ ⊤) = True","typeReadable":"∀ {α : Type u} [inst : LE α] [inst_1 : OrderTop α] {a : α}, (a ≤ ⊤) = True","typeReferences":[["True"],["LE","le"],["Top","top"],["LE"],["OrderTop"],["Eq"],["OrderTop","toTop"]],"valueReferences":[["le_top"],["LE","le"],["Top","top"],["eq_true"],["OrderTop","toTop"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_5"],"typeFallback":"forall {α : Type.{u_1}} {ι : Sort.{u_4}} [inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13 : CompleteLinearOrder.{u_1} α] {a : α} {f : ι -> α}, Eq.{1} Prop (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeSup.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeSup.{u_1} α (CompleteLinearOrder.toCompleteLattice.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13))))) (iInf.{u_1, u_4} α ι (CompleteSemilatticeInf.toInfSet.{u_1} α (CompleteLattice.toCompleteSemilatticeInf.{u_1} α (CompleteLinearOrder.toCompleteLattice.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13))) f) a) (Exists.{u_4} ι (fun (i : ι) => LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeSup.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeSup.{u_1} α (CompleteLinearOrder.toCompleteLattice.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13))))) (f i) a))","typeFull":"∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLinearOrder α] {a : α} {f : ι → α}, (iInf f < a) = ∃ i, f i < a","typeReadable":"∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLinearOrder α] {a : α} {f : ι → α}, (iInf f < a) = ∃ i, f i < a","typeReferences":[["CompleteSemilatticeInf","toInfSet"],["LT","lt"],["Exists"],["PartialOrder","toPreorder"],["CompleteLinearOrder","toCompleteLattice"],["CompleteSemilatticeSup","toPartialOrder"],["Preorder","toLT"],["iInf"],["CompleteLinearOrder"],["CompleteLattice","toCompleteSemilatticeInf"],["CompleteLattice","toCompleteSemilatticeSup"],["Eq"]],"valueReferences":[["CompleteSemilatticeInf","toInfSet"],["LT","lt"],["iInf_lt_iff"],["Exists"],["PartialOrder","toPreorder"],["CompleteLinearOrder","toCompleteLattice"],["CompleteSemilatticeSup","toPartialOrder"],["Preorder","toLT"],["iInf"],["CompleteLattice","toCompleteSemilatticeInf"],["CompleteLattice","toCompleteSemilatticeSup"],["propext"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","match_1_1"],"typeFallback":"forall {E : Type.{u_1}} {F : Type.{u_2}} [inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 : PseudoEMetricSpace.{u_2} F] [inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7 : PseudoEMetricSpace.{u_1} E] {f : E -> F} {D : Set.{u_1} E} {ε : ENNReal} (r : Real) (x : E), let S : Real -> (Set.{u_1} E) := fun (r : Real) => setOf.{u_1} E (fun (x : E) => Exists.{1} Real (fun (a : Real) => And (GT.gt.{0} Real Real.instLT a r) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (Metric.ediam.{u_2} F inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 (Set.image.{u_1, u_2} E F f (Inter.inter.{u_1} (Set.{u_1} E) (Set.instInter.{u_1} E) (Metric.eball.{u_1} E inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7 x (ENNReal.ofReal a)) D))) ε))); forall (motive : (Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (S r) x) -> Prop) (x._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx.240.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.249 : Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (S r) x), (forall (a : Real) (ar : GT.gt.{0} Real Real.instLT a r) (ha : LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (Metric.ediam.{u_2} F inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 (Set.image.{u_1, u_2} E F f (Inter.inter.{u_1} (Set.{u_1} E) (Set.instInter.{u_1} E) (Metric.eball.{u_1} E inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7 x (ENNReal.ofReal a)) D))) ε), motive (Exists.intro.{1} Real (fun (a : Real) => And (GT.gt.{0} Real Real.instLT a r) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (Metric.ediam.{u_2} F inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 (Set.image.{u_1, u_2} E F f (Inter.inter.{u_1} (Set.{u_1} E) (Set.instInter.{u_1} E) (Metric.eball.{u_1} E inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7 x (ENNReal.ofReal a)) D))) ε)) a (And.intro (GT.gt.{0} Real Real.instLT a r) (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (Metric.ediam.{u_2} F inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.4 (Set.image.{u_1, u_2} E F f (Inter.inter.{u_1} (Set.{u_1} E) (Set.instInter.{u_1} E) (Metric.eball.{u_1} E inst._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.7 x (ENNReal.ofReal a)) D))) ε) ar ha))) -> (motive x._@.Mathlib.Analysis.Oscillation.3164812897._hygCtx.240.Mathlib.Analysis.Oscillation.3164812897._hygCtx._hyg.249)","typeFull":"∀ {E : Type u_1} {F : Type u_2} [inst : PseudoEMetricSpace F] [inst_1 : PseudoEMetricSpace E] {f : E → F} {D : Set E}\n  {ε : ENNReal} (r : ℝ) (x : E),\n  let S := fun r => {x | ∃ a > r, Metric.ediam (f '' (Metric.eball x (ENNReal.ofReal a) ∩ D)) ≤ ε};\n  ∀ (motive : x ∈ S r → Prop) (x_1 : x ∈ S r),\n    (∀ (a : ℝ) (ar : a > r) (ha : Metric.ediam (f '' (Metric.eball x (ENNReal.ofReal a) ∩ D)) ≤ ε), motive ⋯) →\n      motive x_1","typeReadable":"∀ {E : Type u_1} {F : Type u_2} [inst : PseudoEMetricSpace F] [inst_1 : PseudoEMetricSpace E] {f : E → F} {D : Set E}\n  {ε : ENNReal} (r : ℝ) (x : E),\n  let S := fun r => {x | ∃ a > r, Metric.ediam (f '' (Metric.eball x (ENNReal.ofReal a) ∩ D)) ≤ ε};\n  ∀ (motive : x ∈ S r → Prop) (x_1 : x ∈ S r),\n    (∀ (a : ℝ) (ar : a > r) (ha : Metric.ediam (f '' (Metric.eball x (ENNReal.ofReal a) ∩ D)) ≤ ε), motive ⋯) →\n      motive x_1","typeReferences":[["Metric","ediam"],["Exists"],["PartialOrder","toPreorder"],["Real"],["Set"],["Membership","mem"],["Inter","inter"],["And"],["Exists","intro"],["GT","gt"],["PseudoEMetricSpace"],["Set","instMembership"],["And","intro"],["ENNReal","ofReal"],["Metric","eball"],["Set","image"],["ENNReal"],["Real","instLT"],["LE","le"],["Preorder","toLE"],["ENNReal","instPartialOrder"],["Set","instInter"],["setOf"]],"valueReferences":[["Metric","ediam"],["Exists"],["PartialOrder","toPreorder"],["Real"],["Set"],["Inter","inter"],["And"],["Exists","intro"],["GT","gt"],["Exists","casesOn"],["ENNReal","ofReal"],["Metric","eball"],["ENNReal"],["Set","image"],["Real","instLT"],["LE","le"],["Preorder","toLE"],["ENNReal","instPartialOrder"],["Set","instInter"],["And","casesOn"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["OscillationWithin","eq_zero_iff_continuousWithinAt"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.733780586._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.733780586._hygCtx._hyg.7 : TopologicalSpace.{u} E] (f : E -> F) {D : Set.{u} E} {x : E}, (Membership.mem.{u, u} E (Set.{u} E) (Set.instMembership.{u} E) D x) -> (Iff (Eq.{1} ENNReal (oscillationWithin.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.733780586._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.733780586._hygCtx._hyg.7 f D x) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (ContinuousWithinAt.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.733780586._hygCtx._hyg.7 (UniformSpace.toTopologicalSpace.{v} F (PseudoEMetricSpace.toUniformSpace.{v} F inst._@.Mathlib.Analysis.Oscillation.733780586._hygCtx._hyg.4)) f D x))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) {D : Set E} {x : E},\n  x ∈ D → (oscillationWithin f D x = 0 ↔ ContinuousWithinAt f D x)","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) {D : Set E} {x : E},\n  x ∈ D → (oscillationWithin f D x = 0 ↔ ContinuousWithinAt f D x)","typeReferences":[["Set"],["instZeroENNReal"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["PseudoEMetricSpace"],["OfNat","ofNat"],["Set","instMembership"],["ContinuousWithinAt"],["TopologicalSpace"],["oscillationWithin"],["PseudoEMetricSpace","toUniformSpace"],["ENNReal"],["Iff"],["Zero","toOfNat0"],["Eq"]],"valueReferences":[["Filter","Eventually"],["PartialOrder","toPreorder"],["Eq","trans"],["instZeroENNReal"],["Membership","mem"],["Iff","mp"],["Preorder","toLT"],["EMetric","tendsto_nhds"],["GT","gt"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["EDist","edist"],["funext"],["Eq","symm"],["ConditionallyCompletePartialOrderInf","toInfSet"],["nhds"],["Filter","Tendsto"],["ContinuousWithinAt","oscillationWithin_eq_zero"],["Metric","ediam"],["Exists"],["lt_of_le_of_lt"],["Filter","map"],["_private","Mathlib","Analysis","Oscillation",0,"OscillationWithin","eq_zero_iff_continuousWithinAt","_simp_1_2"],["mem_of_mem_nhdsWithin"],["Filter"],["Set","instMembership"],["Exists","casesOn"],["nhdsWithin"],["Iff","mpr"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["ENNReal","instPartialOrder"],["setOf"],["Eq","mp"],["UniformSpace","toTopologicalSpace"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["CompleteLattice","toCompleteSemilatticeSup"],["Iff","intro"],["ContinuousWithinAt"],["congrArg"],["oscillationWithin"],["PseudoEMetricSpace","toUniformSpace"],["CompleteLinearOrder","toConditionallyCompleteLinearOrderBot"],["Metric","edist_le_ediam_of_mem"],["iInf"],["Zero","toOfNat0"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["Eq"],["Filter","instMembership"],["Set","preimage"],["Set"],["ENNReal","instCompleteLinearOrder"],["OfNat","ofNat"],["LT","lt"],["CompleteSemilatticeInf","toInfSet"],["ENNReal"],["Filter","mem_of_superset"],["CompleteSemilatticeSup","toPartialOrder"],["CompleteLinearOrder","toCompleteLattice"],["PseudoEMetricSpace","toEDist"],["Set","mem_preimage"],["CompleteLattice","toCompleteSemilatticeInf"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"OscillationWithin","eq_zero_iff_continuousWithinAt","_simp_1_2"],"typeFallback":"forall {α : Type.{u_1}} {ι : Sort.{u_4}} [inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13 : CompleteLinearOrder.{u_1} α] {a : α} {f : ι -> α}, Eq.{1} Prop (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeSup.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeSup.{u_1} α (CompleteLinearOrder.toCompleteLattice.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13))))) (iInf.{u_1, u_4} α ι (CompleteSemilatticeInf.toInfSet.{u_1} α (CompleteLattice.toCompleteSemilatticeInf.{u_1} α (CompleteLinearOrder.toCompleteLattice.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13))) f) a) (Exists.{u_4} ι (fun (i : ι) => LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeSup.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeSup.{u_1} α (CompleteLinearOrder.toCompleteLattice.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Defs.1048642406._hygCtx._hyg.13))))) (f i) a))","typeFull":"∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLinearOrder α] {a : α} {f : ι → α}, (iInf f < a) = ∃ i, f i < a","typeReadable":"∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLinearOrder α] {a : α} {f : ι → α}, (iInf f < a) = ∃ i, f i < a","typeReferences":[["CompleteSemilatticeInf","toInfSet"],["LT","lt"],["Exists"],["PartialOrder","toPreorder"],["CompleteLinearOrder","toCompleteLattice"],["CompleteSemilatticeSup","toPartialOrder"],["Preorder","toLT"],["iInf"],["CompleteLinearOrder"],["CompleteLattice","toCompleteSemilatticeInf"],["CompleteLattice","toCompleteSemilatticeSup"],["Eq"]],"valueReferences":[["CompleteSemilatticeInf","toInfSet"],["LT","lt"],["iInf_lt_iff"],["Exists"],["PartialOrder","toPreorder"],["CompleteLinearOrder","toCompleteLattice"],["CompleteSemilatticeSup","toPartialOrder"],["Preorder","toLT"],["iInf"],["CompleteLattice","toCompleteSemilatticeInf"],["CompleteLattice","toCompleteSemilatticeSup"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_7"],"typeFallback":"forall {α : Type.{u}} {ι : Sort.{v}} {x : α} {s : ι -> (Set.{u} α)}, Eq.{1} Prop (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (Set.iUnion.{u, v} α ι (fun (i : ι) => s i)) x) (Exists.{v} ι (fun (i : ι) => Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (s i) x))","typeFull":"∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋃ i, s i) = ∃ i, x ∈ s i","typeReadable":"∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋃ i, s i) = ∃ i, x ∈ s i","typeReferences":[["Exists"],["Set"],["Membership","mem"],["Set","iUnion"],["Eq"],["Set","instMembership"]],"valueReferences":[["Set","mem_iUnion"],["Exists"],["Set"],["Membership","mem"],["Set","iUnion"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["ContinuousAt","oscillation_eq_zero"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.3241985246._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.3241985246._hygCtx._hyg.7 : TopologicalSpace.{u} E] {f : E -> F} {x : E}, (ContinuousAt.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.3241985246._hygCtx._hyg.7 (UniformSpace.toTopologicalSpace.{v} F (PseudoEMetricSpace.toUniformSpace.{v} F inst._@.Mathlib.Analysis.Oscillation.3241985246._hygCtx._hyg.4)) f x) -> (Eq.{1} ENNReal (oscillation.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.3241985246._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.3241985246._hygCtx._hyg.7 f x) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal)))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] {f : E → F} {x : E},\n  ContinuousAt f x → oscillation f x = 0","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] {f : E → F} {x : E},\n  ContinuousAt f x → oscillation f x = 0","typeReferences":[["TopologicalSpace"],["PseudoEMetricSpace","toUniformSpace"],["ENNReal"],["instZeroENNReal"],["UniformSpace","toTopologicalSpace"],["oscillation"],["Zero","toOfNat0"],["ContinuousAt"],["Eq"],["PseudoEMetricSpace"],["OfNat","ofNat"]],"valueReferences":[["continuousWithinAt_univ"],["instZeroENNReal"],["Eq","mp"],["UniformSpace","toTopologicalSpace"],["OfNat","ofNat"],["congrArg"],["ContinuousWithinAt"],["oscillationWithin"],["Set","univ"],["PseudoEMetricSpace","toUniformSpace"],["ENNReal"],["Eq","symm"],["oscillation"],["oscillationWithin_univ_eq_oscillation"],["Zero","toOfNat0"],["ContinuousAt"],["Eq","rec"],["Eq"],["propext"],["ContinuousWithinAt","oscillationWithin_eq_zero"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_8"],"typeFallback":"forall {b : Prop} {a : Prop}, Eq.{1} Prop (Exists.{0} a (fun (_h : a) => b)) (And a b)","typeFull":"∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b)","typeReadable":"∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b)","typeReferences":[["Exists"],["And"],["Eq"]],"valueReferences":[["exists_prop"],["Exists"],["And"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_4"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {f : Filter.{u_1} α} {m : α -> β} {t : Set.{u_2} β}, Eq.{1} Prop (Membership.mem.{u_2, u_2} (Set.{u_2} β) (Filter.{u_2} β) (Filter.instMembership.{u_2} β) (Filter.map.{u_1, u_2} α β m f) t) (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Filter.{u_1} α) (Filter.instMembership.{u_1} α) f (Set.preimage.{u_1, u_2} α β m t))","typeFull":"∀ {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {t : Set β}, (t ∈ Filter.map m f) = (m ⁻¹' t ∈ f)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {t : Set β}, (t ∈ Filter.map m f) = (m ⁻¹' t ∈ f)","typeReferences":[["Filter","instMembership"],["Set","preimage"],["Set"],["Membership","mem"],["Filter","map"],["Eq"],["Filter"]],"valueReferences":[["Filter","instMembership"],["Set","preimage"],["Set"],["Membership","mem"],["Filter","map"],["Filter","mem_map"],["Filter"],["propext"]]},{"isProp":true,"kind":"theorem","name":["oscillationWithin_univ_eq_oscillation"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.7 : TopologicalSpace.{u} E] (f : E -> F) (x : E), Eq.{1} ENNReal (oscillationWithin.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.7 f (Set.univ.{u} E) x) (oscillation.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.7 f x)","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillationWithin f Set.univ x = oscillation f x","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillationWithin f Set.univ x = oscillation f x","typeReferences":[["Set","univ"],["oscillationWithin"],["TopologicalSpace"],["ENNReal"],["oscillation"],["Eq"],["PseudoEMetricSpace"]],"valueReferences":[["Set","univ"],["oscillationWithin_nhds_eq_oscillation"],["Filter","univ_mem"],["nhds"]]},{"isProp":true,"kind":"theorem","name":["oscillationWithin_nhds_eq_oscillation"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.76823848._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.76823848._hygCtx._hyg.7 : TopologicalSpace.{u} E] (f : E -> F) (D : Set.{u} E) (x : E), (Membership.mem.{u, u} (Set.{u} E) (Filter.{u} E) (Filter.instMembership.{u} E) (nhds.{u} E inst._@.Mathlib.Analysis.Oscillation.76823848._hygCtx._hyg.7 x) D) -> (Eq.{1} ENNReal (oscillationWithin.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.76823848._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.76823848._hygCtx._hyg.7 f D x) (oscillation.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.76823848._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.76823848._hygCtx._hyg.7 f x))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (D : Set E) (x : E),\n  D ∈ nhds x → oscillationWithin f D x = oscillation f x","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (D : Set E) (x : E),\n  D ∈ nhds x → oscillationWithin f D x = oscillation f x","typeReferences":[["oscillationWithin"],["Filter","instMembership"],["TopologicalSpace"],["ENNReal"],["Set"],["Membership","mem"],["oscillation"],["Eq"],["nhds"],["PseudoEMetricSpace"],["Filter"]],"valueReferences":[["nhdsWithin_eq_nhds"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Membership","mem"],["congrArg"],["oscillationWithin"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["CompleteLinearOrder","toConditionallyCompleteLinearOrderBot"],["iInf"],["oscillation"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["Eq"],["nhds"],["Filter","instMembership"],["Metric","ediam"],["oscillationWithin","eq_1"],["Set"],["ENNReal","instCompleteLinearOrder"],["Filter","map"],["oscillation","eq_1"],["Filter"],["nhdsWithin"],["ENNReal"],["Eq","refl"],["Iff","mpr"],["id"],["Eq","mpr"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"]]},{"isProp":true,"kind":"theorem","name":["oscillation","eq_1"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.7 : TopologicalSpace.{u} E] (f : E -> F) (x : E), Eq.{1} ENNReal (oscillation.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.7 f x) (iInf.{0, succ v} ENNReal (Set.{v} F) (ConditionallyCompletePartialOrderInf.toInfSet.{0} ENNReal (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{0} ENNReal (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrder)))))) (fun (S : Set.{v} F) => iInf.{0, 0} ENNReal (Membership.mem.{v, v} (Set.{v} F) (Filter.{v} F) (Filter.instMembership.{v} F) (Filter.map.{u, v} E F f (nhds.{u} E inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.7 x)) S) (ConditionallyCompletePartialOrderInf.toInfSet.{0} ENNReal (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{0} ENNReal (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrder)))))) (fun (h._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.21 : Membership.mem.{v, v} (Set.{v} F) (Filter.{v} F) (Filter.instMembership.{v} F) (Filter.map.{u, v} E F f (nhds.{u} E inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.7 x)) S) => Metric.ediam.{v} F inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.4 S)))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillation f x = ⨅ S ∈ Filter.map f (nhds x), Metric.ediam S","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillation f x = ⨅ S ∈ Filter.map f (nhds x), Metric.ediam S","typeReferences":[["Metric","ediam"],["Filter","instMembership"],["Set"],["Membership","mem"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Filter","map"],["ENNReal","instCompleteLinearOrder"],["PseudoEMetricSpace"],["Filter"],["TopologicalSpace"],["ENNReal"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["CompleteLinearOrder","toConditionallyCompleteLinearOrderBot"],["oscillation"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["Eq"],["nhds"]],"valueReferences":[["ENNReal"],["Eq","refl"],["oscillation"]]},{"isProp":true,"kind":"theorem","name":["oscillationWithin","eq_1"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.7 : TopologicalSpace.{u} E] (f : E -> F) (D : Set.{u} E) (x : E), Eq.{1} ENNReal (oscillationWithin.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.7 f D x) (iInf.{0, succ v} ENNReal (Set.{v} F) (ConditionallyCompletePartialOrderInf.toInfSet.{0} ENNReal (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{0} ENNReal (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrder)))))) (fun (S : Set.{v} F) => iInf.{0, 0} ENNReal (Membership.mem.{v, v} (Set.{v} F) (Filter.{v} F) (Filter.instMembership.{v} F) (Filter.map.{u, v} E F f (nhdsWithin.{u} E inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.7 x D)) S) (ConditionallyCompletePartialOrderInf.toInfSet.{0} ENNReal (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{0} ENNReal (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{0} ENNReal (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{0} ENNReal (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{0} ENNReal (CompleteLinearOrder.toConditionallyCompleteLinearOrderBot.{0} ENNReal ENNReal.instCompleteLinearOrder)))))) (fun (h._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.23 : Membership.mem.{v, v} (Set.{v} F) (Filter.{v} F) (Filter.instMembership.{v} F) (Filter.map.{u, v} E F f (nhdsWithin.{u} E inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.7 x D)) S) => Metric.ediam.{v} F inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.4 S)))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (D : Set E) (x : E),\n  oscillationWithin f D x = ⨅ S ∈ Filter.map f (nhdsWithin x D), Metric.ediam S","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (D : Set E) (x : E),\n  oscillationWithin f D x = ⨅ S ∈ Filter.map f (nhdsWithin x D), Metric.ediam S","typeReferences":[["Metric","ediam"],["Filter","instMembership"],["Set"],["Membership","mem"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Filter","map"],["ENNReal","instCompleteLinearOrder"],["PseudoEMetricSpace"],["Filter"],["TopologicalSpace"],["oscillationWithin"],["nhdsWithin"],["ENNReal"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["CompleteLinearOrder","toConditionallyCompleteLinearOrderBot"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["Eq"]],"valueReferences":[["oscillationWithin"],["ENNReal"],["Eq","refl"]]},{"isProp":true,"kind":"theorem","name":["ContinuousWithinAt","oscillationWithin_eq_zero"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.4078862168._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.4078862168._hygCtx._hyg.7 : TopologicalSpace.{u} E] {f : E -> F} {D : Set.{u} E} {x : E}, (ContinuousWithinAt.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.4078862168._hygCtx._hyg.7 (UniformSpace.toTopologicalSpace.{v} F (PseudoEMetricSpace.toUniformSpace.{v} F inst._@.Mathlib.Analysis.Oscillation.4078862168._hygCtx._hyg.4)) f D x) -> (Eq.{1} ENNReal (oscillationWithin.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.4078862168._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.4078862168._hygCtx._hyg.7 f D x) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal)))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] {f : E → F} {D : Set E} {x : E},\n  ContinuousWithinAt f D x → oscillationWithin f D x = 0","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] {f : E → F} {D : Set E} {x : E},\n  ContinuousWithinAt f D x → oscillationWithin f D x = 0","typeReferences":[["oscillationWithin"],["TopologicalSpace"],["PseudoEMetricSpace","toUniformSpace"],["ENNReal"],["instZeroENNReal"],["Set"],["UniformSpace","toTopologicalSpace"],["Zero","toOfNat0"],["Eq"],["PseudoEMetricSpace"],["OfNat","ofNat"],["ContinuousWithinAt"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["instZeroENNReal"],["le_of_le_of_eq"],["Membership","mem"],["Preorder","toLT"],["HMul","hMul"],["ENNReal","instCharZero"],["instAddCommMonoidWithOneENNReal"],["AddMonoidWithOne","toAddMonoid"],["ENNReal","instLinearOrder"],["HDiv","hDiv"],["ENNReal","div_pos_iff","_simp_1"],["Semiring","toNonAssocSemiring"],["not_false_eq_true"],["zero_le"],["ENNReal","mul_div_cancel"],["SemilatticeInf","toPartialOrder"],["Metric","ediam"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Filter","map"],["ne_of_gt"],["And"],["AddZeroClass","toAddZero"],["Set","instMembership"],["nhdsWithin"],["zero_add"],["Nat"],["AddMonoidWithOne","toNatCast"],["eq_false"],["Top","top"],["id"],["instHMul"],["Eq","mpr"],["ENNReal","instPartialOrder"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["ENNReal","instDivInvMonoid"],["ENNReal","instCanonicallyOrderedAdd"],["CompleteLinearOrder","toCompletelyDistribLattice"],["le_of_forall_pos_le_add"],["instHDiv"],["LE","le","trans"],["instTopENNReal"],["ENNReal","addLeftReflectLT"],["congrArg"],["oscillationWithin"],["Nat","instNeZeroSucc"],["instDistribLatticeOfLinearOrder"],["instOfNatNat"],["congr"],["iInf"],["Zero","toOfNat0"],["ENNReal","instDenselyOrdered"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["Eq"],["le_antisymm"],["Not"],["Lattice","toSemilatticeInf"],["True"],["Set"],["instHAdd"],["CommSemiring","toSemiring"],["Distrib","toMul"],["ENNReal","instCompleteLinearOrder"],["instOfNatAtLeastTwo"],["Filter","sets"],["AddZero","toAdd"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["ENNReal","instCommSemiring"],["Metric","ediam_eball_le"],["CompletelyDistribLattice","toCompleteLattice"],["CompleteSemilatticeInf","toInfSet"],["HAdd","hAdd"],["LT","lt"],["Metric","eball"],["and_self"],["ENNReal"],["DistribLattice","toLattice"],["of_eq_true"],["OfNat","ofNat_ne_zero","_simp_1"],["CanonicallyOrderedAdd","toExistsAddOfLE"],["biInf_le"],["ENNReal","ofNat_ne_top","_simp_1"],["LE","le"],["False"],["Ne"],["CompleteLattice","toCompleteSemilatticeInf"],["Metric","eball_mem_nhds"],["Nat","instAtLeastTwoHAddOfNat"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillationWithin","_simp_1_6"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Init.Core.3961863493._hygCtx._hyg.9 : LT.{u_1} α] {x : α} {y : α}, Eq.{1} Prop (GT.gt.{u_1} α inst._@.Init.Core.3961863493._hygCtx._hyg.9 x y) (LT.lt.{u_1} α inst._@.Init.Core.3961863493._hygCtx._hyg.9 y x)","typeFull":"∀ {α : Type u_1} [inst : LT α] {x y : α}, (x > y) = (y < x)","typeReadable":"∀ {α : Type u_1} [inst : LT α] {x y : α}, (x > y) = (y < x)","typeReferences":[["LT","lt"],["GT","gt"],["Eq"],["LT"]],"valueReferences":[["LT","lt"],["gt_iff_lt"],["GT","gt"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillation","_simp_1_1"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.7 : TopologicalSpace.{u} E] (f : E -> F) (x : E), Eq.{1} ENNReal (oscillation.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.7 f x) (oscillationWithin.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.3687463397._hygCtx._hyg.7 f (Set.univ.{u} E) x)","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillation f x = oscillationWithin f Set.univ x","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillation f x = oscillationWithin f Set.univ x","typeReferences":[["Set","univ"],["oscillationWithin"],["TopologicalSpace"],["ENNReal"],["oscillation"],["Eq"],["PseudoEMetricSpace"]],"valueReferences":[["Set","univ"],["oscillationWithin"],["ENNReal"],["Eq","symm"],["oscillation"],["oscillationWithin_univ_eq_oscillation"]]},{"isProp":true,"kind":"theorem","name":["IsCompact","uniform_oscillation"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.1118989316._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.1118989316._hygCtx._hyg.7 : PseudoEMetricSpace.{u} E] {K : Set.{u} E}, (IsCompact.{u} E (UniformSpace.toTopologicalSpace.{u} E (PseudoEMetricSpace.toUniformSpace.{u} E inst._@.Mathlib.Analysis.Oscillation.1118989316._hygCtx._hyg.7)) K) -> (forall {f : E -> F} {ε : ENNReal}, (forall (x : E), (Membership.mem.{u, u} E (Set.{u} E) (Set.instMembership.{u} E) K x) -> (LT.lt.{0} ENNReal (Preorder.toLT.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (oscillation.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.1118989316._hygCtx._hyg.4 (UniformSpace.toTopologicalSpace.{u} E (PseudoEMetricSpace.toUniformSpace.{u} E inst._@.Mathlib.Analysis.Oscillation.1118989316._hygCtx._hyg.7)) f x) ε)) -> (Exists.{1} Real (fun (δ : Real) => And (GT.gt.{0} Real Real.instLT δ (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero))) (forall (x : E), (Membership.mem.{u, u} E (Set.{u} E) (Set.instMembership.{u} E) K x) -> (LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal ENNReal.instPartialOrder)) (Metric.ediam.{v} F inst._@.Mathlib.Analysis.Oscillation.1118989316._hygCtx._hyg.4 (Set.image.{u, v} E F f (Metric.eball.{u} E inst._@.Mathlib.Analysis.Oscillation.1118989316._hygCtx._hyg.7 x (ENNReal.ofReal δ)))) ε)))))","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : PseudoEMetricSpace E] {K : Set E},\n  IsCompact K →\n    ∀ {f : E → F} {ε : ENNReal},\n      (∀ x ∈ K, oscillation f x < ε) → ∃ δ > 0, ∀ x ∈ K, Metric.ediam (f '' Metric.eball x (ENNReal.ofReal δ)) ≤ ε","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : PseudoEMetricSpace E] {K : Set E},\n  IsCompact K →\n    ∀ {f : E → F} {ε : ENNReal},\n      (∀ x ∈ K, oscillation f x < ε) → ∃ δ > 0, ∀ x ∈ K, Metric.ediam (f '' Metric.eball x (ENNReal.ofReal δ)) ≤ ε","typeReferences":[["PartialOrder","toPreorder"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["Preorder","toLT"],["GT","gt"],["PseudoEMetricSpace"],["PseudoEMetricSpace","toUniformSpace"],["Real","instLT"],["oscillation"],["Zero","toOfNat0"],["Preorder","toLE"],["Metric","ediam"],["Exists"],["Real"],["Set"],["And"],["OfNat","ofNat"],["Set","instMembership"],["LT","lt"],["ENNReal","ofReal"],["Metric","eball"],["ENNReal"],["Set","image"],["IsCompact"],["Real","instZero"],["LE","le"],["ENNReal","instPartialOrder"]],"valueReferences":[["implies_congr"],["pi_congr"],["PartialOrder","toPreorder"],["Eq","mp"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["Inter","inter"],["Preorder","toLT"],["GT","gt"],["congrArg"],["oscillationWithin"],["PseudoEMetricSpace","toUniformSpace"],["Real","instLT"],["forall_congr"],["funext"],["eq_of_heq"],["Eq","symm"],["oscillation"],["congrFun'"],["Zero","toOfNat0"],["Eq","ndrec"],["Eq"],["Preorder","toLE"],["Set","inter_univ"],["Set","instInter"],["Metric","ediam"],["Exists"],["IsCompact","uniform_oscillationWithin"],["HEq","refl"],["Real"],["Set"],["And"],["_private","Mathlib","Analysis","Oscillation",0,"IsCompact","uniform_oscillation","_simp_1_1"],["Eq","casesOn"],["OfNat","ofNat"],["Set","instMembership"],["LT","lt"],["Set","univ"],["Metric","eball"],["ENNReal","ofReal"],["ENNReal"],["Set","image"],["Real","instZero"],["Eq","refl"],["LE","le"],["HEq"],["ENNReal","instPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["Oscillation","eq_zero_iff_continuousAt"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.2245224934._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.2245224934._hygCtx._hyg.7 : TopologicalSpace.{u} E] (f : E -> F) (x : E), Iff (Eq.{1} ENNReal (oscillation.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.2245224934._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Oscillation.2245224934._hygCtx._hyg.7 f x) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (ContinuousAt.{u, v} E F inst._@.Mathlib.Analysis.Oscillation.2245224934._hygCtx._hyg.7 (UniformSpace.toTopologicalSpace.{v} F (PseudoEMetricSpace.toUniformSpace.{v} F inst._@.Mathlib.Analysis.Oscillation.2245224934._hygCtx._hyg.4)) f x)","typeFull":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillation f x = 0 ↔ ContinuousAt f x","typeReadable":"∀ {E : Type u} {F : Type v} [inst : PseudoEMetricSpace F] [inst_1 : TopologicalSpace E] (f : E → F) (x : E),\n  oscillation f x = 0 ↔ ContinuousAt f x","typeReferences":[["TopologicalSpace"],["PseudoEMetricSpace","toUniformSpace"],["ENNReal"],["instZeroENNReal"],["Iff"],["UniformSpace","toTopologicalSpace"],["oscillation"],["ContinuousAt"],["Zero","toOfNat0"],["Eq"],["PseudoEMetricSpace"],["OfNat","ofNat"]],"valueReferences":[["continuousWithinAt_univ"],["instZeroENNReal"],["UniformSpace","toTopologicalSpace"],["OfNat","ofNat"],["Set","mem_univ"],["ContinuousWithinAt"],["congrArg"],["oscillationWithin"],["Set","univ"],["OscillationWithin","eq_zero_iff_continuousWithinAt"],["PseudoEMetricSpace","toUniformSpace"],["ENNReal"],["Iff"],["Eq","symm"],["id"],["oscillation"],["oscillationWithin_univ_eq_oscillation"],["Eq","mpr"],["Zero","toOfNat0"],["ContinuousAt"],["Eq"],["propext"]]},{"isProp":false,"kind":"definition","name":["oscillationWithin"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.1413229780._hygCtx._hyg.7 : TopologicalSpace.{u} E], (E -> F) -> (Set.{u} E) -> E -> ENNReal","typeFull":"{E : Type u} → {F : Type v} → [PseudoEMetricSpace F] → [TopologicalSpace E] → (E → F) → Set E → E → ENNReal","typeReadable":"{E : Type u} → {F : Type v} → [PseudoEMetricSpace F] → [TopologicalSpace E] → (E → F) → Set E → E → ENNReal","typeReferences":[["TopologicalSpace"],["ENNReal"],["Set"],["PseudoEMetricSpace"]],"valueReferences":[["Metric","ediam"],["Filter","instMembership"],["Set"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Membership","mem"],["Filter","map"],["ENNReal","instCompleteLinearOrder"],["Filter"],["nhdsWithin"],["ENNReal"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["CompleteLinearOrder","toConditionallyCompleteLinearOrderBot"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"]]},{"isProp":false,"kind":"definition","name":["oscillation"],"typeFallback":"forall {E : Type.{u}} {F : Type.{v}} [inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.4 : PseudoEMetricSpace.{v} F] [inst._@.Mathlib.Analysis.Oscillation.2969598370._hygCtx._hyg.7 : TopologicalSpace.{u} E], (E -> F) -> E -> ENNReal","typeFull":"{E : Type u} → {F : Type v} → [PseudoEMetricSpace F] → [TopologicalSpace E] → (E → F) → E → ENNReal","typeReadable":"{E : Type u} → {F : Type v} → [PseudoEMetricSpace F] → [TopologicalSpace E] → (E → F) → E → ENNReal","typeReferences":[["TopologicalSpace"],["ENNReal"],["PseudoEMetricSpace"]],"valueReferences":[["Metric","ediam"],["Filter","instMembership"],["Set"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Membership","mem"],["Filter","map"],["ENNReal","instCompleteLinearOrder"],["Filter"],["ENNReal"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["CompleteLinearOrder","toConditionallyCompleteLinearOrderBot"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["nhds"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.SpecialFunctions.Pow.Integral.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["MeasureTheory","lintegral_rpow_eq_lintegral_meas_lt_mul"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] (μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3) {f : α -> Real}, (Filter.EventuallyLE.{u_1, 0} α Real Real.instLE (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3) μ) (OfNat.ofNat.{u_1} (α -> Real) 0 (Zero.toOfNat0.{u_1} (α -> Real) (Pi.instZero.{u_1, 0} α (fun (a._@._internal._hyg.0 : α) => Real) (fun (i : α) => Real.instZero)))) f) -> (AEMeasurable.{u_1, 0} α Real Real.measurableSpace inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3 f μ) -> (forall {p : Real}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) p) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3 μ (fun (ω : α) => ENNReal.ofReal (HPow.hPow.{0, 0, 0} Real Real Real (instHPow.{0, 0} Real Real Real.instPow) (f ω) p))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal ENNReal.instCommSemiring)))))) (ENNReal.ofReal p) (MeasureTheory.lintegral.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace) (MeasureTheory.Measure.restrict.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace) (Set.Ioi.{0} Real Real.instPreorder (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) (fun (t : Real) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal ENNReal.instCommSemiring)))))) (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.891005133._hygCtx._hyg.3) μ (setOf.{u_1} α (fun (a : α) => LT.lt.{0} Real Real.instLT t (f a)))) (ENNReal.ofReal (HPow.hPow.{0, 0, 0} Real Real Real (instHPow.{0, 0} Real Real Real.instPow) t (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSub) p (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne))))))))))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {f : α → ℝ},\n  0 ≤ᵐ[μ] f →\n    AEMeasurable f μ →\n      ∀ {p : ℝ},\n        0 < p →\n          ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =\n            ENNReal.ofReal p * ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {f : α → ℝ},\n  0 ≤ᵐ[μ] f →\n    AEMeasurable f μ →\n      ∀ {p : ℝ},\n        0 < p →\n          ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =\n            ENNReal.ofReal p * ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))","typeReferences":[["Real","instPreorder"],["AEMeasurable"],["HMul","hMul"],["DFunLike","coe"],["Real","instPow"],["Real","measureSpace"],["MeasureTheory","Measure"],["MeasureTheory","Measure","restrict"],["Semiring","toNonAssocSemiring"],["Real","instLT"],["HSub","hSub"],["Set","Ioi"],["Zero","toOfNat0"],["MeasurableSpace"],["Eq"],["MeasureTheory","ae"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["Real"],["MeasureTheory","lintegral"],["Set"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["Real","instSub"],["HPow","hPow"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["ENNReal","instCommSemiring"],["Real","instLE"],["LT","lt"],["ENNReal","ofReal"],["ENNReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["One","toOfNat1"],["Real","instZero"],["MeasureTheory","MeasureSpace","volume"],["Real","measurableSpace"],["instHMul"],["Real","instOne"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["instHSub"],["setOf"],["Pi","instZero"],["Filter","EventuallyLE"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["Real","noAtoms_volume"],["Membership","mem"],["Preorder","toLT"],["HMul","hMul"],["Filter","univ_mem'"],["MeasureTheory","lintegral_congr_ae"],["Semiring","toNonAssocSemiring"],["HSub","hSub"],["Set","Ioi"],["MeasureTheory","Measure","restrict","instNoAtoms"],["SemilatticeInf","toPartialOrder"],["instHPow"],["Real"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["congr_arg"],["Real","instSub"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["Real","instLE"],["Eq","refl"],["MeasureTheory","lintegral_rpow_eq_lintegral_meas_le_mul"],["id"],["instHMul"],["Real","linearOrder"],["Eq","mpr"],["Real","instOne"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["setOf"],["MeasureTheory","meas_le_ae_eq_meas_lt"],["Filter","mp_mem"],["DFunLike","coe"],["congrArg"],["Real","instPow"],["instDistribLatticeOfLinearOrder"],["Real","measureSpace"],["MeasureTheory","Measure"],["MeasureTheory","Measure","restrict"],["Real","instLT"],["Zero","toOfNat0"],["Eq"],["Preorder","toLE"],["MeasureTheory","ae"],["Lattice","toSemilatticeInf"],["MeasureTheory","lintegral"],["Set"],["CommSemiring","toSemiring"],["Distrib","toMul"],["HPow","hPow"],["OfNat","ofNat"],["ENNReal","instCommSemiring"],["LT","lt"],["ENNReal","ofReal"],["ENNReal"],["DistribLattice","toLattice"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","instZero"],["One","toOfNat1"],["MeasureTheory","MeasureSpace","volume"],["LE","le"],["instHSub"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","lintegral_rpow_eq_lintegral_meas_le_mul"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] (μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3) {f : α -> Real}, (Filter.EventuallyLE.{u_1, 0} α Real Real.instLE (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3) μ) (OfNat.ofNat.{u_1} (α -> Real) 0 (Zero.toOfNat0.{u_1} (α -> Real) (Pi.instZero.{u_1, 0} α (fun (a._@._internal._hyg.0 : α) => Real) (fun (i : α) => Real.instZero)))) f) -> (AEMeasurable.{u_1, 0} α Real Real.measurableSpace inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3 f μ) -> (forall {p : Real}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) p) -> (Eq.{1} ENNReal (MeasureTheory.lintegral.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3 μ (fun (ω : α) => ENNReal.ofReal (HPow.hPow.{0, 0, 0} Real Real Real (instHPow.{0, 0} Real Real Real.instPow) (f ω) p))) (HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal ENNReal.instCommSemiring)))))) (ENNReal.ofReal p) (MeasureTheory.lintegral.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace) (MeasureTheory.Measure.restrict.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace) (Set.Ioi.{0} Real Real.instPreorder (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))) (fun (t : Real) => HMul.hMul.{0, 0, 0} ENNReal ENNReal ENNReal (instHMul.{0} ENNReal (Distrib.toMul.{0} ENNReal (NonUnitalNonAssocSemiring.toDistrib.{0} ENNReal (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} ENNReal (Semiring.toNonAssocSemiring.{0} ENNReal (CommSemiring.toSemiring.{0} ENNReal ENNReal.instCommSemiring)))))) (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Analysis.SpecialFunctions.Pow.Integral.4199220926._hygCtx._hyg.3) μ (setOf.{u_1} α (fun (a : α) => LE.le.{0} Real Real.instLE t (f a)))) (ENNReal.ofReal (HPow.hPow.{0, 0, 0} Real Real Real (instHPow.{0, 0} Real Real Real.instPow) t (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSub) p (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne))))))))))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {f : α → ℝ},\n  0 ≤ᵐ[μ] f →\n    AEMeasurable f μ →\n      ∀ {p : ℝ},\n        0 < p →\n          ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =\n            ENNReal.ofReal p * ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1))","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) {f : α → ℝ},\n  0 ≤ᵐ[μ] f →\n    AEMeasurable f μ →\n      ∀ {p : ℝ},\n        0 < p →\n          ∫⁻ (ω : α), ENNReal.ofReal (f ω ^ p) ∂μ =\n            ENNReal.ofReal p * ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1))","typeReferences":[["Real","instPreorder"],["AEMeasurable"],["HMul","hMul"],["DFunLike","coe"],["Real","instPow"],["Real","measureSpace"],["MeasureTheory","Measure"],["MeasureTheory","Measure","restrict"],["Semiring","toNonAssocSemiring"],["Real","instLT"],["HSub","hSub"],["Set","Ioi"],["Zero","toOfNat0"],["MeasurableSpace"],["Eq"],["MeasureTheory","ae"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["Real"],["MeasureTheory","lintegral"],["Set"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["Real","instSub"],["HPow","hPow"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["ENNReal","instCommSemiring"],["Real","instLE"],["LT","lt"],["ENNReal","ofReal"],["ENNReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["One","toOfNat1"],["Real","instZero"],["MeasureTheory","MeasureSpace","volume"],["Real","measurableSpace"],["LE","le"],["instHMul"],["Real","instOne"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["instHSub"],["setOf"],["Pi","instZero"],["Filter","EventuallyLE"]],"valueReferences":[["Ring","toNonAssocRing"],["sub_add_cancel"],["Real","instAddGroup"],["MulZeroClass","toMul"],["Real","normedCommRing"],["AddGroupWithOne","toAddMonoidWithOne"],["MonoidWithZero","toMulZeroOneClass"],["sub_zero"],["AEMeasurable","pow"],["Real","commRing"],["intervalIntegral","intervalIntegrable_rpow'"],["Measurable","ennreal_ofReal"],["Eq","symm"],["Real","lattice"],["Measurable","aemeasurable"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Real","normedAddCommGroup"],["Real","instCommSemiring"],["MeasureTheory","lintegral_const_mul''"],["ContinuousMul","measurableMul"],["Nat","instCommSemiring"],["MulZeroOneClass","toMulZeroClass"],["AddMonoid","toAddSemigroup"],["IsTopologicalRing","toIsSemitopologicalRing"],["Int","negOfNat"],["Eq","mpr"],["autoParam"],["MulOneClass","toMulOne"],["Real","instIsStrictOrderedRing"],["Mathlib","Tactic","Linarith","lt_irrefl"],["MulOne","toMul"],["Or","inl"],["Mathlib","Tactic","Ring","neg_one_mul"],["Eq"],["MeasureTheory","ae"],["Mathlib","Tactic","Ring","neg_zero"],["Set"],["Real","rpow_nonneg"],["Real","instAdd"],["HPow","hPow"],["RCLike","toInnerProductSpaceReal"],["eq_self"],["Real","instZero"],["measurable_const"],["Ne"],["instHSub"],["Real","instDivInvMonoid"],["ENNReal","ofReal_mul"],["PartialOrder","toPreorder"],["Mathlib","Meta","NormNum","IsNat","of_raw"],["Mathlib","Tactic","Ring","add_pf_zero_add"],["Mathlib","Meta","NormNum","IsInt","of_raw"],["Membership","mem"],["Preorder","toLT"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Set","Ioi"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Eq","rec"],["Int","ofNat"],["MeasureTheory","lintegral_comp_eq_lintegral_meas_le_mul"],["SemilatticeInf","toPartialOrder"],["Real","hasMeasurablePow"],["instIsTopologicalDivisionRingReal"],["Real"],["Real","instIsOrderedAddMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["ContinuousInv₀","measurableInv"],["Nat"],["Mathlib","Tactic","Ring","atom_pf"],["Real","measurableSpace"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["Real","instOne"],["CommGroupWithZero","toGroupWithZero"],["measurableDiv_of_mul_inv"],["Eq","mp"],["UniformSpace","toTopologicalSpace"],["DFunLike","coe"],["MeasureTheory","Measure"],["instDistribLatticeOfLinearOrder"],["MeasureTheory","Measure","restrict"],["T5Space","toT1Space"],["Real","instNeg"],["GroupWithZero","toMonoidWithZero"],["Monoid","toMulOneClass"],["Measurable","comp_aemeasurable"],["IsTopologicalDivisionRing","toContinuousInv₀"],["Real","instDivisionRing"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["MeasureTheory","self_mem_ae_restrict"],["AEMeasurable","_auto_1"],["Real","partialOrder"],["Real","instMul"],["Not"],["instClosedIicTopology"],["MeasureTheory","lintegral"],["instHAdd"],["Distrib","toMul"],["Mathlib","Tactic","Ring","cast_pos"],["mul_div_cancel₀"],["Mathlib","Tactic","Ring","add_congr"],["Real","semiring"],["LT","lt"],["Mathlib","Tactic","Ring","add_pf_add_zero"],["Mathlib","Tactic","Ring","neg_add"],["One","toOfNat1"],["of_eq_true"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Mathlib","Tactic","Ring","neg_congr"],["MeasureTheory","MeasureSpace","volume"],["Mathlib","Tactic","Linarith","sub_nonpos_of_le"],["Field","toSemifield"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Real","instAddCommGroup"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["NormedCommRing","toNonUnitalNormedCommRing"],["Mathlib","Tactic","Linarith","add_lt_of_neg_of_le"],["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["IsSemitopologicalSemiring","toSeparatelyContinuousMul"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["Real","zero_rpow"],["Filter","univ_mem'"],["Mathlib","Meta","NormNum","IsInt","to_raw_eq"],["Mathlib","Tactic","Linarith","sub_neg_of_lt"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["measurable_id"],["SubNegMonoid","toSub"],["Mathlib","Tactic","Ring","add_overlap_pf_zero"],["Mathlib","Tactic","Ring","sub_pf"],["rfl"],["EMetricSpace","metrizableSpace"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionSemiring","toGroupWithZero"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["DivInvMonoid","toMonoid"],["Eq","refl"],["Nat","rawCast"],["measurableSet_Ioi"],["Mathlib","Meta","NormNum","IsNat","to_isInt"],["setOf"],["Ne","symm"],["SubtractionCommMonoid","toSubtractionMonoid"],["instHDiv"],["Real","instField"],["Mathlib","Tactic","Ring","add_pf_add_overlap_zero"],["HasSolidNorm","orderClosedTopology"],["congr"],["lt_of_not_ge"],["Distrib","toAdd"],["T6Space","toT5Space"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["CommRing","toRing"],["ENNReal"],["AddGroupWithOne","toAddGroup"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Set","mem_Ioi"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Real","borelSpace"],["Real","instAddMonoid"],["Semifield","toCommGroupWithZero"],["Nat","cast_one"],["Mathlib","Meta","NormNum","IsNat","to_raw_eq"],["GroupWithZero","toDivInvMonoid"],["AEMeasurable"],["Iff","mp"],["Int","rawCast"],["HMul","hMul"],["HDiv","hDiv"],["Real","instRCLike"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["LT","lt","ne"],["funext"],["HSub","hSub"],["Mathlib","Meta","NormNum","IsInt","to_isNat"],["BorelSpace","opensMeasurable"],["Semiring","toNonUnitalSemiring"],["NonAssocRing","toNonUnitalNonAssocRing"],["AddSemigroup","toAdd"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["Set","uIcc"],["And"],["intervalIntegral"],["Real","instSub"],["instIsTopologicalRingReal"],["instT6SpaceOfMetrizableSpace"],["Real","instLE"],["Real","pseudoMetricSpace"],["Mathlib","Meta","NormNum","instAddMonoidWithOne"],["Nat","cast_zero"],["id"],["instHMul"],["Real","linearOrder"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["Mathlib","Meta","NormNum","isNat_ofNat"],["Filter","mp_mem"],["AEMeasurable","div_const"],["Mathlib","Meta","NormNum","isInt_add"],["Mathlib","Tactic","Ring","neg_mul"],["Real","instRing"],["Real","instIsOrderedRing"],["integral_rpow"],["congrArg"],["Real","instPow"],["Real","measureSpace"],["Real","instLT"],["MonoidWithZero","toMonoid"],["Mathlib","Tactic","Ring","sub_congr"],["Zero","toOfNat0"],["congrFun'"],["MetricSpace","toEMetricSpace"],["Mathlib","Tactic","Ring","cast_zero"],["Mathlib","Meta","NormNum","isInt_mul"],["Mathlib","Tactic","Ring","of_eq"],["LT","lt","le"],["Lattice","toSemilatticeInf"],["Real","metricSpace"],["True"],["CommSemiring","toSemiring"],["ENNReal","measurableSpace"],["Semiring","toMonoidWithZero"],["DivInvMonoid","toDiv"],["instHasSolidNormReal"],["ENNReal","instCommSemiring"],["ENNReal","ofReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["DistribLattice","toLattice"],["LE","le"],["Mathlib","Tactic","Ring","add_pf_add_gt"],["InnerProductSpace","toNormedSpace"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Localization.Equivalence.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","IsLocalization","of_equivalence_source"],"typeFallback":"forall {C₁ : Type.{u_1}} {C₂ : Type.{u_2}} {D : Type.{u_3}} [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 : CategoryTheory.Category.{v_1, u_1} C₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 : CategoryTheory.Category.{v_2, u_2} C₂] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13 : CategoryTheory.Category.{v_3, u_3} D] (L₁ : CategoryTheory.Functor.{v_1, v_3, u_1, u_3} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13) (W₁ : CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7)) (L₂ : CategoryTheory.Functor.{v_2, v_3, u_2, u_3} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13) (W₂ : CategoryTheory.MorphismProperty.{v_2, u_2} C₂ (CategoryTheory.Category.toCategoryStruct.{v_2, u_2} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10)) (E : CategoryTheory.Equivalence.{v_1, v_2, u_1, u_2} C₁ C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10), (LE.le.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7)) (Preorder.toLE.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7)) (PartialOrder.toPreorder.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7)) (CompleteSemilatticeInf.toPartialOrder.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7)) (CompleteLattice.toCompleteSemilatticeInf.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7)) (CompleteBooleanAlgebra.toCompleteLattice.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7)) (CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7))))))) W₁ (CategoryTheory.MorphismProperty.inverseImage.{v_1, u_1, v_2, u_2} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 (CategoryTheory.MorphismProperty.isoClosure.{v_2, u_2} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 W₂) (CategoryTheory.Equivalence.functor.{v_1, v_2, u_1, u_2} C₁ C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 E))) -> (CategoryTheory.MorphismProperty.IsInvertedBy.{v_2, v_3, u_2, u_3} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13 W₂ L₂) -> (forall [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.49 : CategoryTheory.Functor.IsLocalization.{v_1, u_1, v_3, u_3} C₁ D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13 L₁ W₁], (CategoryTheory.Iso.{max u_1 v_3, max (max (max u_3 u_1) v_3) v_1} (CategoryTheory.Functor.{v_1, v_3, u_1, u_3} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13) (CategoryTheory.Functor.category.{v_1, v_3, u_1, u_3} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13) (CategoryTheory.Functor.comp.{v_1, v_2, v_3, u_1, u_2, u_3} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13 (CategoryTheory.Equivalence.functor.{v_1, v_2, u_1, u_2} C₁ C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 E) L₂) L₁) -> (CategoryTheory.Functor.IsLocalization.{v_2, u_2, v_3, u_3} C₂ D inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.3609333666._hygCtx._hyg.13 L₂ W₂))","typeFull":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_3, u_3} D]\n  (L₁ : CategoryTheory.Functor C₁ D) (W₁ : CategoryTheory.MorphismProperty C₁) (L₂ : CategoryTheory.Functor C₂ D)\n  (W₂ : CategoryTheory.MorphismProperty C₂) (E : C₁ ≌ C₂),\n  W₁ ≤ W₂.isoClosure.inverseImage E.functor →\n    W₂.IsInvertedBy L₂ → ∀ [L₁.IsLocalization W₁] (iso : E.functor.comp L₂ ≅ L₁), L₂.IsLocalization W₂","typeReadable":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_3, u_3} D]\n  (L₁ : CategoryTheory.Functor C₁ D) (W₁ : CategoryTheory.MorphismProperty C₁) (L₂ : CategoryTheory.Functor C₂ D)\n  (W₂ : CategoryTheory.MorphismProperty C₂) (E : C₁ ≌ C₂),\n  W₁ ≤ W₂.isoClosure.inverseImage E.functor →\n    W₂.IsInvertedBy L₂ → ∀ [L₁.IsLocalization W₁] (iso : E.functor.comp L₂ ≅ L₁), L₂.IsLocalization W₂","typeReferences":[["CategoryTheory","MorphismProperty","instCompleteBooleanAlgebra"],["CategoryTheory","Functor"],["CategoryTheory","Iso"],["PartialOrder","toPreorder"],["CategoryTheory","MorphismProperty","isoClosure"],["CategoryTheory","Category"],["CategoryTheory","Functor","comp"],["CompleteBooleanAlgebra","toCompleteLattice"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","MorphismProperty","IsInvertedBy"],["CategoryTheory","Equivalence"],["LE","le"],["CompleteSemilatticeInf","toPartialOrder"],["CategoryTheory","MorphismProperty"],["CategoryTheory","Equivalence","functor"],["CategoryTheory","Functor","category"],["Preorder","toLE"],["CompleteLattice","toCompleteSemilatticeInf"],["CategoryTheory","Functor","IsLocalization"],["CategoryTheory","MorphismProperty","inverseImage"]],"valueReferences":[["CategoryTheory","Functor","id"],["Trans","trans"],["CategoryTheory","Localization","fac"],["Iff","mp"],["CategoryTheory","Equivalence","inverse"],["CategoryTheory","IsIso"],["CategoryTheory","Functor","map"],["CategoryTheory","Iso","instTransIso"],["CategoryTheory","Functor","mapIso"],["CategoryTheory","Localization","Construction","lift"],["CategoryTheory","Functor","leftUnitor"],["CategoryTheory","MorphismProperty","Localization"],["CategoryTheory","Functor","IsLocalization","mk"],["CategoryTheory","Comma","right"],["CategoryTheory","Functor"],["CategoryTheory","Iso"],["Exists"],["CategoryTheory","Iso","symm"],["CategoryTheory","Localization","liftingConstructionLift"],["Nonempty"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","Functor","isoWhiskerLeft"],["CategoryTheory","Localization","Lifting","iso"],["Exists","casesOn"],["Nonempty","casesOn"],["CategoryTheory","instCategoryArrow"],["CategoryTheory","Localization","inverts"],["CategoryTheory","Functor","associator"],["CategoryTheory","Comma","hom"],["CategoryTheory","Comma","left"],["CategoryTheory","Functor","comp"],["CategoryTheory","Equivalence","counitIso"],["CategoryTheory","Functor","obj"],["Quiver","Hom"],["CategoryTheory","Equivalence","functor"],["CategoryTheory","Functor","q_isLocalization"],["CategoryTheory","Arrow"],["CategoryTheory","MorphismProperty","instCategoryLocalization"],["CategoryTheory","Localization","isEquivalence"],["CategoryTheory","MorphismProperty","Q"],["CategoryTheory","Iso","trans"],["CategoryTheory","Functor","mapArrow"],["CategoryTheory","Localization","lift"],["CategoryTheory","Functor","isoWhiskerRight"],["CategoryTheory","Localization","liftingLift"],["CategoryTheory","CategoryStruct","toQuiver"],["CategoryTheory","MorphismProperty","RespectsIso","isomorphisms"],["CategoryTheory","MorphismProperty","isomorphisms"],["CategoryTheory","Functor","category"],["CategoryTheory","Arrow","mk"],["CategoryTheory","MorphismProperty","arrow_mk_iso_iff"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Localization","isEquivalence"],"typeFallback":"forall {C₁ : Type.{u_1}} {C₂ : Type.{u_2}} {D₁ : Type.{u_4}} {D₂ : Type.{u_5}} [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 : CategoryTheory.Category.{v_1, u_1} C₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 : CategoryTheory.Category.{v_2, u_2} C₂] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 : CategoryTheory.Category.{v_4, u_4} D₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16) (W₁ : CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7)) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.29 : CategoryTheory.Functor.IsLocalization.{v_1, u_1, v_4, u_4} C₁ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 L₁ W₁] (L₂ : CategoryTheory.Functor.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19) (W₂ : CategoryTheory.MorphismProperty.{v_2, u_2} C₂ (CategoryTheory.Category.toCategoryStruct.{v_2, u_2} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10)) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.39 : CategoryTheory.Functor.IsLocalization.{v_2, u_2, v_5, u_5} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 L₂ W₂] (G : CategoryTheory.Functor.{v_1, v_5, u_1, u_5} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19) (G' : CategoryTheory.Functor.{v_4, v_5, u_4, u_5} D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.52 : CategoryTheory.Localization.Lifting.{v_1, u_1, v_4, u_4, v_5, u_5} C₁ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 L₁ W₁ G G'] (F : CategoryTheory.Functor.{v_2, v_4, u_2, u_4} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16) (F' : CategoryTheory.Functor.{v_5, v_4, u_5, u_4} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.68 : CategoryTheory.Localization.Lifting.{v_2, u_2, v_5, u_5, v_4, u_4} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 L₂ W₂ F F'], (CategoryTheory.Iso.{max u_1 v_4, max (max (max u_4 u_1) v_4) v_1} (CategoryTheory.Functor.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16) (CategoryTheory.Functor.category.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16) (CategoryTheory.Functor.comp.{v_1, v_5, v_4, u_1, u_5, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.7 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 G F') L₁) -> (CategoryTheory.Iso.{max u_2 v_5, max (max (max u_5 u_2) v_5) v_2} (CategoryTheory.Functor.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19) (CategoryTheory.Functor.category.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19) (CategoryTheory.Functor.comp.{v_2, v_4, v_5, u_2, u_4, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 F G') L₂) -> (CategoryTheory.Functor.IsEquivalence.{v_4, v_5, u_4, u_5} D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.2224916322._hygCtx._hyg.19 G')","typeFull":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_4} {D₂ : Type u_5} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁]\n  [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor C₁ D₁)\n  (W₁ : CategoryTheory.MorphismProperty C₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂)\n  (W₂ : CategoryTheory.MorphismProperty C₂) [L₂.IsLocalization W₂] (G : CategoryTheory.Functor C₁ D₂)\n  (G' : CategoryTheory.Functor D₁ D₂) [CategoryTheory.Localization.Lifting L₁ W₁ G G']\n  (F : CategoryTheory.Functor C₂ D₁) (F' : CategoryTheory.Functor D₂ D₁)\n  [CategoryTheory.Localization.Lifting L₂ W₂ F F'] (α : G.comp F' ≅ L₁) (β : F.comp G' ≅ L₂), G'.IsEquivalence","typeReadable":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_4} {D₂ : Type u_5} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁]\n  [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor C₁ D₁)\n  (W₁ : CategoryTheory.MorphismProperty C₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂)\n  (W₂ : CategoryTheory.MorphismProperty C₂) [L₂.IsLocalization W₂] (G : CategoryTheory.Functor C₁ D₂)\n  (G' : CategoryTheory.Functor D₁ D₂) [CategoryTheory.Localization.Lifting L₁ W₁ G G']\n  (F : CategoryTheory.Functor C₂ D₁) (F' : CategoryTheory.Functor D₂ D₁)\n  [CategoryTheory.Localization.Lifting L₂ W₂ F F'] (α : G.comp F' ≅ L₁) (β : F.comp G' ≅ L₂), G'.IsEquivalence","typeReferences":[["CategoryTheory","Functor","IsEquivalence"],["CategoryTheory","Functor"],["CategoryTheory","Iso"],["CategoryTheory","Category"],["CategoryTheory","MorphismProperty"],["CategoryTheory","Functor","comp"],["CategoryTheory","Functor","category"],["CategoryTheory","Localization","Lifting"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","Functor","IsLocalization"]],"valueReferences":[["CategoryTheory","Localization","equivalence"],["CategoryTheory","Equivalence","isEquivalence_functor"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Localization","equivalence_counitIso_app"],"typeFallback":"forall {C₁ : Type.{u_1}} {C₂ : Type.{u_2}} {D₁ : Type.{u_4}} {D₂ : Type.{u_5}} [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 : CategoryTheory.Category.{v_1, u_1} C₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 : CategoryTheory.Category.{v_2, u_2} C₂] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 : CategoryTheory.Category.{v_4, u_4} D₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16) (W₁ : CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7)) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.29 : CategoryTheory.Functor.IsLocalization.{v_1, u_1, v_4, u_4} C₁ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 L₁ W₁] (L₂ : CategoryTheory.Functor.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19) (W₂ : CategoryTheory.MorphismProperty.{v_2, u_2} C₂ (CategoryTheory.Category.toCategoryStruct.{v_2, u_2} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10)) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.39 : CategoryTheory.Functor.IsLocalization.{v_2, u_2, v_5, u_5} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ W₂] (G : CategoryTheory.Functor.{v_1, v_5, u_1, u_5} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19) (G' : CategoryTheory.Functor.{v_4, v_5, u_4, u_5} D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.52 : CategoryTheory.Localization.Lifting.{v_1, u_1, v_4, u_4, v_5, u_5} C₁ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₁ W₁ G G'] (F : CategoryTheory.Functor.{v_2, v_4, u_2, u_4} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16) (F' : CategoryTheory.Functor.{v_5, v_4, u_5, u_4} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.68 : CategoryTheory.Localization.Lifting.{v_2, u_2, v_5, u_5, v_4, u_4} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 L₂ W₂ F F'] (α : CategoryTheory.Iso.{max u_1 v_4, max (max (max u_4 u_1) v_4) v_1} (CategoryTheory.Functor.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16) (CategoryTheory.Functor.category.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16) (CategoryTheory.Functor.comp.{v_1, v_5, v_4, u_1, u_5, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 G F') L₁) (β : CategoryTheory.Iso.{max u_2 v_5, max (max (max u_5 u_2) v_5) v_2} (CategoryTheory.Functor.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19) (CategoryTheory.Functor.category.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19) (CategoryTheory.Functor.comp.{v_2, v_4, v_5, u_2, u_4, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F G') L₂) (X : C₂), Eq.{succ v_5} (CategoryTheory.Iso.{v_5, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.obj.{v_5, v_5, u_5, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.comp.{v_5, v_4, v_5, u_5, u_4, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Equivalence.inverse.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Localization.equivalence.{v_1, u_1, v_2, u_2, v_4, u_4, v_5, u_5} C₁ C₂ D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₁ W₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.29 L₂ W₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.39 G G' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.52 F F' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.68 α β)) (CategoryTheory.Equivalence.functor.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Localization.equivalence.{v_1, u_1, v_2, u_2, v_4, u_4, v_5, u_5} C₁ C₂ D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₁ W₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.29 L₂ W₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.39 G G' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.52 F F' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.68 α β))) (CategoryTheory.Functor.obj.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ X)) (CategoryTheory.Functor.obj.{v_5, v_5, u_5, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.id.{v_5, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19) (CategoryTheory.Functor.obj.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ X))) (CategoryTheory.Iso.app.{v_5, v_5, u_5, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.comp.{v_5, v_4, v_5, u_5, u_4, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Equivalence.inverse.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Localization.equivalence.{v_1, u_1, v_2, u_2, v_4, u_4, v_5, u_5} C₁ C₂ D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₁ W₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.29 L₂ W₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.39 G G' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.52 F F' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.68 α β)) (CategoryTheory.Equivalence.functor.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Localization.equivalence.{v_1, u_1, v_2, u_2, v_4, u_4, v_5, u_5} C₁ C₂ D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₁ W₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.29 L₂ W₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.39 G G' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.52 F F' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.68 α β))) (CategoryTheory.Functor.id.{v_5, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19) (CategoryTheory.Equivalence.counitIso.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Localization.equivalence.{v_1, u_1, v_2, u_2, v_4, u_4, v_5, u_5} C₁ C₂ D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₁ W₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.29 L₂ W₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.39 G G' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.52 F F' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.68 α β)) (CategoryTheory.Functor.obj.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ X)) (CategoryTheory.Iso.trans.{v_5, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.obj.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.comp.{v_2, v_5, v_5, u_2, u_5, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ (CategoryTheory.Functor.comp.{v_5, v_4, v_5, u_5, u_4, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F' G')) X) (CategoryTheory.Functor.obj.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.comp.{v_2, v_4, v_5, u_2, u_4, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F G') X) (CategoryTheory.Functor.obj.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ X) (CategoryTheory.Iso.app.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.comp.{v_2, v_5, v_5, u_2, u_5, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ (CategoryTheory.Functor.comp.{v_5, v_4, v_5, u_5, u_4, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F' G')) (CategoryTheory.Functor.comp.{v_2, v_4, v_5, u_2, u_4, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F G') (CategoryTheory.Localization.Lifting.iso.{v_2, u_2, v_5, u_5, v_5, u_5} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ W₂ (CategoryTheory.Functor.comp.{v_2, v_4, v_5, u_2, u_4, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F G') (CategoryTheory.Functor.comp.{v_5, v_4, v_5, u_5, u_4, u_5} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F' G') (CategoryTheory.Localization.Lifting.compRight.{v_2, u_2, v_5, u_5, v_4, u_4, v_5, u_5} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 L₂ W₂ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F F' inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.68 G')) X) (CategoryTheory.Iso.app.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 (CategoryTheory.Functor.comp.{v_2, v_4, v_5, u_2, u_4, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.1233525960._hygCtx._hyg.19 F G') L₂ β X))","typeFull":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_4} {D₂ : Type u_5} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁]\n  [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor C₁ D₁)\n  (W₁ : CategoryTheory.MorphismProperty C₁) [inst_4 : L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂)\n  (W₂ : CategoryTheory.MorphismProperty C₂) [inst_5 : L₂.IsLocalization W₂] (G : CategoryTheory.Functor C₁ D₂)\n  (G' : CategoryTheory.Functor D₁ D₂) [inst_6 : CategoryTheory.Localization.Lifting L₁ W₁ G G']\n  (F : CategoryTheory.Functor C₂ D₁) (F' : CategoryTheory.Functor D₂ D₁)\n  [inst_7 : CategoryTheory.Localization.Lifting L₂ W₂ F F'] (α : G.comp F' ≅ L₁) (β : F.comp G' ≅ L₂) (X : C₂),\n  (CategoryTheory.Localization.equivalence L₁ W₁ L₂ W₂ G G' F F' α β).counitIso.app (L₂.obj X) =\n    (CategoryTheory.Localization.Lifting.iso L₂ W₂ (F.comp G') (F'.comp G')).app X ≪≫ β.app X","typeReadable":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_4} {D₂ : Type u_5} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁]\n  [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor C₁ D₁)\n  (W₁ : CategoryTheory.MorphismProperty C₁) [inst_4 : L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂)\n  (W₂ : CategoryTheory.MorphismProperty C₂) [inst_5 : L₂.IsLocalization W₂] (G : CategoryTheory.Functor C₁ D₂)\n  (G' : CategoryTheory.Functor D₁ D₂) [inst_6 : CategoryTheory.Localization.Lifting L₁ W₁ G G']\n  (F : CategoryTheory.Functor C₂ D₁) (F' : CategoryTheory.Functor D₂ D₁)\n  [inst_7 : CategoryTheory.Localization.Lifting L₂ W₂ F F'] (α : G.comp F' ≅ L₁) (β : F.comp G' ≅ L₂) (X : C₂),\n  (CategoryTheory.Localization.equivalence L₁ W₁ L₂ W₂ G G' F F' α β).counitIso.app (L₂.obj X) =\n    (CategoryTheory.Localization.Lifting.iso L₂ W₂ (F.comp G') (F'.comp G')).app X ≪≫ β.app X","typeReferences":[["CategoryTheory","Functor"],["CategoryTheory","Iso","trans"],["CategoryTheory","Functor","id"],["CategoryTheory","Iso"],["CategoryTheory","Localization","equivalence"],["CategoryTheory","Category"],["CategoryTheory","Equivalence","inverse"],["CategoryTheory","Functor","comp"],["CategoryTheory","Equivalence","counitIso"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","Functor","obj"],["CategoryTheory","Localization","Lifting","iso"],["CategoryTheory","Iso","app"],["CategoryTheory","MorphismProperty"],["CategoryTheory","Localization","Lifting","compRight"],["CategoryTheory","Equivalence","functor"],["CategoryTheory","Localization","Lifting"],["CategoryTheory","Functor","category"],["Eq"],["CategoryTheory","Functor","IsLocalization"]],"valueReferences":[["CategoryTheory","Functor","id"],["CategoryTheory","Localization","equivalence"],["CategoryTheory","Localization","Lifting","id"],["CategoryTheory","Equivalence","inverse"],["CategoryTheory","Functor","comp"],["CategoryTheory","Equivalence","counitIso"],["CategoryTheory","Functor","obj"],["congrArg"],["CategoryTheory","CategoryStruct","id"],["CategoryTheory","Functor","map"],["Quiver","Hom"],["CategoryTheory","Localization","Lifting","compRight"],["CategoryTheory","Equivalence","functor"],["Eq"],["CategoryTheory","NatTrans","app"],["CategoryTheory","Iso","hom"],["CategoryTheory","Functor"],["CategoryTheory","Iso","trans"],["CategoryTheory","Iso","inv"],["CategoryTheory","Category","comp_id"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","Localization","Lifting","iso"],["CategoryTheory","Localization","liftNatTrans_app"],["CategoryTheory","Iso","app"],["CategoryTheory","CategoryStruct","toQuiver"],["CategoryTheory","CategoryStruct","comp"],["Eq","refl"],["CategoryTheory","Localization","liftNatTrans"],["id"],["Eq","mpr"],["CategoryTheory","Iso","ext"],["CategoryTheory","Functor","category"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","IsLocalization","of_equivalences"],"typeFallback":"forall {C₁ : Type.{u_1}} {C₂ : Type.{u_2}} {D₁ : Type.{u_4}} {D₂ : Type.{u_5}} [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 : CategoryTheory.Category.{v_1, u_1} C₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 : CategoryTheory.Category.{v_2, u_2} C₂] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.16 : CategoryTheory.Category.{v_4, u_4} D₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.19 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.16) (W₁ : CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7)) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.29 : CategoryTheory.Functor.IsLocalization.{v_1, u_1, v_4, u_4} C₁ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.16 L₁ W₁] (L₂ : CategoryTheory.Functor.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.19) (W₂ : CategoryTheory.MorphismProperty.{v_2, u_2} C₂ (CategoryTheory.Category.toCategoryStruct.{v_2, u_2} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10)) (E : CategoryTheory.Equivalence.{v_1, v_2, u_1, u_2} C₁ C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10) (E' : CategoryTheory.Equivalence.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.19) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.49 : CategoryTheory.CatCommSq.{v_1, u_1, v_2, u_2, v_4, u_4, v_5, u_5} C₁ C₂ D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.19 (CategoryTheory.Equivalence.functor.{v_1, v_2, u_1, u_2} C₁ C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 E) L₁ L₂ (CategoryTheory.Equivalence.functor.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.19 E')], (LE.le.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7)) (Preorder.toLE.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7)) (PartialOrder.toPreorder.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7)) (CompleteSemilatticeInf.toPartialOrder.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7)) (CompleteLattice.toCompleteSemilatticeInf.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7)) (CompleteBooleanAlgebra.toCompleteLattice.{max u_1 v_1} (CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7)) (CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7))))))) W₁ (CategoryTheory.MorphismProperty.inverseImage.{v_1, u_1, v_2, u_2} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 (CategoryTheory.MorphismProperty.isoClosure.{v_2, u_2} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 W₂) (CategoryTheory.Equivalence.functor.{v_1, v_2, u_1, u_2} C₁ C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 E))) -> (CategoryTheory.MorphismProperty.IsInvertedBy.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.19 W₂ L₂) -> (CategoryTheory.Functor.IsLocalization.{v_2, u_2, v_5, u_5} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.4078045214._hygCtx._hyg.19 L₂ W₂)","typeFull":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_4} {D₂ : Type u_5} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁]\n  [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor C₁ D₁)\n  (W₁ : CategoryTheory.MorphismProperty C₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂)\n  (W₂ : CategoryTheory.MorphismProperty C₂) (E : C₁ ≌ C₂) (E' : D₁ ≌ D₂)\n  [CategoryTheory.CatCommSq E.functor L₁ L₂ E'.functor],\n  W₁ ≤ W₂.isoClosure.inverseImage E.functor → W₂.IsInvertedBy L₂ → L₂.IsLocalization W₂","typeReadable":"∀ {C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_4} {D₂ : Type u_5} [inst : CategoryTheory.Category.{v_1, u_1} C₁]\n  [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁]\n  [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor C₁ D₁)\n  (W₁ : CategoryTheory.MorphismProperty C₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂)\n  (W₂ : CategoryTheory.MorphismProperty C₂) (E : C₁ ≌ C₂) (E' : D₁ ≌ D₂)\n  [CategoryTheory.CatCommSq E.functor L₁ L₂ E'.functor],\n  W₁ ≤ W₂.isoClosure.inverseImage E.functor → W₂.IsInvertedBy L₂ → L₂.IsLocalization W₂","typeReferences":[["CategoryTheory","MorphismProperty","instCompleteBooleanAlgebra"],["CategoryTheory","Functor"],["PartialOrder","toPreorder"],["CategoryTheory","MorphismProperty","isoClosure"],["CategoryTheory","Category"],["CompleteBooleanAlgebra","toCompleteLattice"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","MorphismProperty","IsInvertedBy"],["CategoryTheory","Equivalence"],["LE","le"],["CompleteSemilatticeInf","toPartialOrder"],["CategoryTheory","MorphismProperty"],["CategoryTheory","Equivalence","functor"],["CategoryTheory","CatCommSq"],["Preorder","toLE"],["CompleteLattice","toCompleteSemilatticeInf"],["CategoryTheory","Functor","IsLocalization"],["CategoryTheory","MorphismProperty","inverseImage"]],"valueReferences":[["CategoryTheory","Functor","IsLocalization","of_equivalence_target"],["CategoryTheory","Functor"],["CategoryTheory","Functor","IsLocalization","of_equivalence_source"],["CategoryTheory","CatCommSq","iso"],["CategoryTheory","Iso","symm"],["CategoryTheory","Iso","refl"],["CategoryTheory","Equivalence","functor"],["CategoryTheory","Functor","comp"],["CategoryTheory","Functor","category"]]},{"isProp":false,"kind":"definition","name":["CategoryTheory","Localization","equivalence"],"typeFallback":"forall {C₁ : Type.{u_1}} {C₂ : Type.{u_2}} {D₁ : Type.{u_4}} {D₂ : Type.{u_5}} [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 : CategoryTheory.Category.{v_1, u_1} C₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 : CategoryTheory.Category.{v_2, u_2} C₂] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 : CategoryTheory.Category.{v_4, u_4} D₁] [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19 : CategoryTheory.Category.{v_5, u_5} D₂] (L₁ : CategoryTheory.Functor.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16) (W₁ : CategoryTheory.MorphismProperty.{v_1, u_1} C₁ (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7)) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.29 : CategoryTheory.Functor.IsLocalization.{v_1, u_1, v_4, u_4} C₁ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 L₁ W₁] (L₂ : CategoryTheory.Functor.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19) (W₂ : CategoryTheory.MorphismProperty.{v_2, u_2} C₂ (CategoryTheory.Category.toCategoryStruct.{v_2, u_2} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10)) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.39 : CategoryTheory.Functor.IsLocalization.{v_2, u_2, v_5, u_5} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19 L₂ W₂] (G : CategoryTheory.Functor.{v_1, v_5, u_1, u_5} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19) (G' : CategoryTheory.Functor.{v_4, v_5, u_4, u_5} D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.52 : CategoryTheory.Localization.Lifting.{v_1, u_1, v_4, u_4, v_5, u_5} C₁ D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19 L₁ W₁ G G'] (F : CategoryTheory.Functor.{v_2, v_4, u_2, u_4} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16) (F' : CategoryTheory.Functor.{v_5, v_4, u_5, u_4} D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16) [inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.68 : CategoryTheory.Localization.Lifting.{v_2, u_2, v_5, u_5, v_4, u_4} C₂ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 L₂ W₂ F F'], (CategoryTheory.Iso.{max u_1 v_4, max (max (max u_4 u_1) v_4) v_1} (CategoryTheory.Functor.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16) (CategoryTheory.Functor.category.{v_1, v_4, u_1, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16) (CategoryTheory.Functor.comp.{v_1, v_5, v_4, u_1, u_5, u_4} C₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.7 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 G F') L₁) -> (CategoryTheory.Iso.{max u_2 v_5, max (max (max u_5 u_2) v_5) v_2} (CategoryTheory.Functor.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19) (CategoryTheory.Functor.category.{v_2, v_5, u_2, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19) (CategoryTheory.Functor.comp.{v_2, v_4, v_5, u_2, u_4, u_5} C₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.10 D₁ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19 F G') L₂) -> (CategoryTheory.Equivalence.{v_4, v_5, u_4, u_5} D₁ D₂ inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.16 inst._@.Mathlib.CategoryTheory.Localization.Equivalence.8919254._hygCtx._hyg.19)","typeFull":"{C₁ : Type u_1} →\n  {C₂ : Type u_2} →\n    {D₁ : Type u_4} →\n      {D₂ : Type u_5} →\n        [inst : CategoryTheory.Category.{v_1, u_1} C₁] →\n          [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] →\n            [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁] →\n              [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] →\n                (L₁ : CategoryTheory.Functor C₁ D₁) →\n                  (W₁ : CategoryTheory.MorphismProperty C₁) →\n                    [L₁.IsLocalization W₁] →\n                      (L₂ : CategoryTheory.Functor C₂ D₂) →\n                        (W₂ : CategoryTheory.MorphismProperty C₂) →\n                          [L₂.IsLocalization W₂] →\n                            (G : CategoryTheory.Functor C₁ D₂) →\n                              (G' : CategoryTheory.Functor D₁ D₂) →\n                                [CategoryTheory.Localization.Lifting L₁ W₁ G G'] →\n                                  (F : CategoryTheory.Functor C₂ D₁) →\n                                    (F' : CategoryTheory.Functor D₂ D₁) →\n                                      [CategoryTheory.Localization.Lifting L₂ W₂ F F'] →\n                                        (G.comp F' ≅ L₁) → (F.comp G' ≅ L₂) → (D₁ ≌ D₂)","typeReadable":"{C₁ : Type u_1} →\n  {C₂ : Type u_2} →\n    {D₁ : Type u_4} →\n      {D₂ : Type u_5} →\n        [inst : CategoryTheory.Category.{v_1, u_1} C₁] →\n          [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] →\n            [inst_2 : CategoryTheory.Category.{v_4, u_4} D₁] →\n              [inst_3 : CategoryTheory.Category.{v_5, u_5} D₂] →\n                (L₁ : CategoryTheory.Functor C₁ D₁) →\n                  (W₁ : CategoryTheory.MorphismProperty C₁) →\n                    [L₁.IsLocalization W₁] →\n                      (L₂ : CategoryTheory.Functor C₂ D₂) →\n                        (W₂ : CategoryTheory.MorphismProperty C₂) →\n                          [L₂.IsLocalization W₂] →\n                            (G : CategoryTheory.Functor C₁ D₂) →\n                              (G' : CategoryTheory.Functor D₁ D₂) →\n                                [CategoryTheory.Localization.Lifting L₁ W₁ G G'] →\n                                  (F : CategoryTheory.Functor C₂ D₁) →\n                                    (F' : CategoryTheory.Functor D₂ D₁) →\n                                      [CategoryTheory.Localization.Lifting L₂ W₂ F F'] →\n                                        (G.comp F' ≅ L₁) → (F.comp G' ≅ L₂) → (D₁ ≌ D₂)","typeReferences":[["CategoryTheory","Functor"],["CategoryTheory","Equivalence"],["CategoryTheory","Iso"],["CategoryTheory","Category"],["CategoryTheory","MorphismProperty"],["CategoryTheory","Functor","comp"],["CategoryTheory","Functor","category"],["CategoryTheory","Localization","Lifting"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","Functor","IsLocalization"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Functor","id"],["CategoryTheory","Localization","liftNatIso"],["CategoryTheory","Equivalence","mk"],["CategoryTheory","Iso","symm"],["CategoryTheory","Localization","Lifting","id"],["CategoryTheory","Localization","Lifting","compRight"],["CategoryTheory","Functor","comp"],["CategoryTheory","Functor","category"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.sym.json

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["CategoryTheory","Functor","functorCategoryClosed"],"typeFallback":"forall (I : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 : CategoryTheory.Category.{v₂, u₂} I] (C : Type.{u₁}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 : CategoryTheory.Category.{v₁, u₁} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10 : CategoryTheory.MonoidalCategory.{v₁, u₁} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.13 : CategoryTheory.MonoidalClosed.{v₁, u₁} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.16 : forall (F : CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7), CategoryTheory.Functor.HasRightKanExtension.{u₂, u₂, v₁, u₁, v₂, u₂} (CategoryTheory.Discrete.{u₂} I) C I (CategoryTheory.discreteCategory.{u₂} I) inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 (CategoryTheory.Discrete.functor.{v₂, u₂, u₂} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 I (id.{succ u₂} I)) F] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.33 : CategoryTheory.Limits.HasLimitsOfShape.{0, 0, v₁, u₁} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7] (F : CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7), CategoryTheory.Closed.{max u₂ v₁, max (max (max u₁ u₂) v₁) v₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Monoidal.functorCategoryMonoidal.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10) F","typeFull":"(I : Type u₂) →\n  [inst : CategoryTheory.Category.{v₂, u₂} I] →\n    (C : Type u₁) →\n      [inst_1 : CategoryTheory.Category.{v₁, u₁} C] →\n        [inst_2 : CategoryTheory.MonoidalCategory C] →\n          [CategoryTheory.MonoidalClosed C] →\n            [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n                  (CategoryTheory.Discrete.functor id).HasRightKanExtension F] →\n              [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] →\n                (F : CategoryTheory.Functor I C) → CategoryTheory.Closed F","typeReadable":"(I : Type u₂) →\n  [inst : CategoryTheory.Category.{v₂, u₂} I] →\n    (C : Type u₁) →\n      [inst_1 : CategoryTheory.Category.{v₁, u₁} C] →\n        [inst_2 : CategoryTheory.MonoidalCategory C] →\n          [CategoryTheory.MonoidalClosed C] →\n            [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n                  (CategoryTheory.Discrete.functor id).HasRightKanExtension F] →\n              [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] →\n                (F : CategoryTheory.Functor I C) → CategoryTheory.Closed F","typeReferences":[["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Category"],["CategoryTheory","Closed"],["CategoryTheory","Discrete"],["CategoryTheory","Limits","HasLimitsOfShape"],["CategoryTheory","Functor","HasRightKanExtension"],["CategoryTheory","Limits","walkingParallelPairHomCategory"],["CategoryTheory","discreteCategory"],["CategoryTheory","MonoidalCategory"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["id"],["CategoryTheory","Functor","category"],["CategoryTheory","MonoidalClosed"],["CategoryTheory","Discrete","functor"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","Functor","functorCategoryClosed","_proof_1"],["CategoryTheory","Adjunction","ofIsLeftAdjoint"],["CategoryTheory","Functor","rightAdjoint"],["CategoryTheory","MonoidalCategory","tensorLeft"],["CategoryTheory","Functor","functorCategoryClosed","_proof_2"],["CategoryTheory","Functor","category"],["CategoryTheory","Closed","mk"]]},{"isProp":false,"kind":"definition","name":["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],"typeFallback":"forall (I : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2345969726._hygCtx._hyg.3 : CategoryTheory.Category.{v₂, u₂} I], CategoryTheory.Functor.{u₂, v₂, u₂, u₂} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2345969726._hygCtx._hyg.3","typeFull":"(I : Type u₂) → [inst : CategoryTheory.Category.{v₂, u₂} I] → CategoryTheory.Functor (CategoryTheory.Discrete I) I","typeReadable":"(I : Type u₂) → [inst : CategoryTheory.Category.{v₂, u₂} I] → CategoryTheory.Functor (CategoryTheory.Discrete I) I","typeReferences":[["CategoryTheory","discreteCategory"],["CategoryTheory","Functor"],["CategoryTheory","Category"],["CategoryTheory","Discrete"]],"valueReferences":[["id"],["CategoryTheory","Discrete","functor"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","instReflectsIsomorphismsDiscreteObjWhiskeringLeftIncl"],"typeFallback":"forall (I : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 : CategoryTheory.Category.{v₂, u₂} I] (C : Type.{u₁}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7 : CategoryTheory.Category.{v₁, u₁} C], CategoryTheory.Functor.ReflectsIsomorphisms.{max v₁ u₂, max (max (max u₁ v₁) v₂) u₂, max v₁ u₂, max (max u₁ v₁) u₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.obj.{max v₂ u₂, max (max (max u₁ v₁) v₂) u₂, max v₂ u₂, max (max (max u₁ v₁) v₂) u₂} (CategoryTheory.Functor.{u₂, v₂, u₂, u₂} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3) (CategoryTheory.Functor.category.{u₂, v₂, u₂, u₂} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3) (CategoryTheory.Functor.{max u₂ v₁, max u₂ v₁, max (max (max u₁ u₂) v₁) v₂, max (max (max u₁ u₂) v₁) u₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7)) (CategoryTheory.Functor.category.{max u₂ v₁, max u₂ v₁, max (max (max u₂ u���) v₂) v₁, max (max u₂ u₁) v₁} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7)) (CategoryTheory.Functor.whiskeringLeft.{u₂, u₂, u₂, v₂, u₁, v₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.7) (_private.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.0.CategoryTheory.Functor.incl.{v₂, u₂} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.1431268458._hygCtx._hyg.3))","typeFull":"∀ (I : Type u₂) [inst : CategoryTheory.Category.{v₂, u₂} I] (C : Type u₁) [inst_1 : CategoryTheory.Category.{v₁, u₁} C],\n  ((CategoryTheory.Functor.whiskeringLeft (CategoryTheory.Discrete I) I C).obj\n      (CategoryTheory.Functor.incl✝ I)).ReflectsIsomorphisms","typeReadable":"∀ (I : Type u₂) [inst : CategoryTheory.Category.{v₂, u₂} I] (C : Type u₁) [inst_1 : CategoryTheory.Category.{v₁, u₁} C],\n  ((CategoryTheory.Functor.whiskeringLeft (CategoryTheory.Discrete I) I C).obj\n      (CategoryTheory.Functor.incl✝ I)).ReflectsIsomorphisms","typeReferences":[["CategoryTheory","discreteCategory"],["CategoryTheory","Functor","ReflectsIsomorphisms"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","Functor"],["CategoryTheory","Category"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","whiskeringLeft"],["CategoryTheory","Functor","category"],["CategoryTheory","Functor","obj"]],"valueReferences":[["CategoryTheory","Functor"],["Eq","mp"],["CategoryTheory","Discrete"],["CategoryTheory","IsIso"],["CategoryTheory","Functor","whiskeringLeft"],["CategoryTheory","Functor","obj"],["CategoryTheory","Functor","map"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","instReflectsIsomorphismsDiscreteObjWhiskeringLeftIncl","_simp_1"],["CategoryTheory","discreteCategory"],["CategoryTheory","Discrete","mk"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["id"],["CategoryTheory","Functor","ReflectsIsomorphisms","mk"],["Eq","mpr"],["CategoryTheory","Functor","category"],["Eq"],["CategoryTheory","NatTrans","app"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","functorCategoryClosed","_proof_1"],"typeFallback":"forall (I : Type.{u_1}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 : CategoryTheory.Category.{u_4, u_1} I] (C : Type.{u_3}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 : CategoryTheory.Category.{u_2, u_3} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10 : CategoryTheory.MonoidalCategory.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.13 : CategoryTheory.MonoidalClosed.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10] (F : CategoryTheory.Functor.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7), CategoryTheory.Functor.IsLeftAdjoint.{max u_1 u_2, max u_1 u_2, max (max u_3 u_1) u_2, max (max u_3 u_1) u_2} (CategoryTheory.Functor.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.MonoidalCategory.tensorLeft.{max u_1 u_2, max (max (max u_3 u_1) u_2) u_1} (CategoryTheory.Functor.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Monoidal.functorCategoryMonoidal.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10) (CategoryTheory.Functor.comp.{u_1, u_4, u_2, u_1, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 (_private.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.0.CategoryTheory.Functor.incl.{u_4, u_1} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3) F))","typeFull":"∀ (I : Type u_1) [inst : CategoryTheory.Category.{u_4, u_1} I] (C : Type u_3)\n  [inst_1 : CategoryTheory.Category.{u_2, u_3} C] [inst_2 : CategoryTheory.MonoidalCategory C]\n  [CategoryTheory.MonoidalClosed C] (F : CategoryTheory.Functor I C),\n  (CategoryTheory.MonoidalCategory.tensorLeft ((CategoryTheory.Functor.incl✝ I).comp F)).IsLeftAdjoint","typeReadable":"∀ (I : Type u_1) [inst : CategoryTheory.Category.{u_4, u_1} I] (C : Type u_3)\n  [inst_1 : CategoryTheory.Category.{u_2, u_3} C] [inst_2 : CategoryTheory.MonoidalCategory C]\n  [CategoryTheory.MonoidalClosed C] (F : CategoryTheory.Functor I C),\n  (CategoryTheory.MonoidalCategory.tensorLeft ((CategoryTheory.Functor.incl✝ I).comp F)).IsLeftAdjoint","typeReferences":[["CategoryTheory","discreteCategory"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","Functor"],["CategoryTheory","MonoidalCategory"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","Functor","IsLeftAdjoint"],["CategoryTheory","Category"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","comp"],["CategoryTheory","MonoidalCategory","tensorLeft"],["CategoryTheory","Functor","category"],["CategoryTheory","MonoidalClosed"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Functor","closed"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","comp"],["CategoryTheory","MonoidalCategory","tensorLeft"],["CategoryTheory","ihom"],["CategoryTheory","discreteCategory"],["CategoryTheory","instGroupoidDiscrete"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","Adjunction","isLeftAdjoint"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","ihom","adjunction"],["CategoryTheory","Functor","category"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","functorCategoryClosed","_proof_2"],"typeFallback":"forall (I : Type.{u_1}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 : CategoryTheory.Category.{u_4, u_1} I] (C : Type.{u_3}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 : CategoryTheory.Category.{u_2, u_3} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10 : CategoryTheory.MonoidalCategory.{u_2, u_3} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.16 : forall (F : CategoryTheory.Functor.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7), CategoryTheory.Functor.HasRightKanExtension.{u_1, u_1, u_2, u_3, u_4, u_1} (CategoryTheory.Discrete.{u_1} I) C I (CategoryTheory.discreteCategory.{u_1} I) inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 (CategoryTheory.Discrete.functor.{u_4, u_1, u_1} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 I (id.{succ u_1} I)) F] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.33 : CategoryTheory.Limits.HasLimitsOfShape.{0, 0, u_2, u_3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7] (F : CategoryTheory.Functor.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7), (CategoryTheory.Functor.IsLeftAdjoint.{max u_1 u_2, max u_1 u_2, max (max u_3 u_1) u_2, max (max u_3 u_1) u_2} (CategoryTheory.Functor.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.MonoidalCategory.tensorLeft.{max u_1 u_2, max (max (max u_3 u_1) u_2) u_1} (CategoryTheory.Functor.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Monoidal.functorCategoryMonoidal.{u_1, u_2, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10) (CategoryTheory.Functor.comp.{u_1, u_4, u_2, u_1, u_1, u_3} (CategoryTheory.Discrete.{u_1} I) (CategoryTheory.discreteCategory.{u_1} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 (_private.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.0.CategoryTheory.Functor.incl.{u_4, u_1} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3) F))) -> (CategoryTheory.Functor.IsLeftAdjoint.{max u_1 u_2, max u_1 u_2, max (max (max u_3 u_1) u_2) u_4, max (max (max u_3 u_1) u_2) u_4} (CategoryTheory.Functor.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.MonoidalCategory.tensorLeft.{max u_1 u_2, max (max (max u_3 u_1) u_2) u_4} (CategoryTheory.Functor.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7) (CategoryTheory.Monoidal.functorCategoryMonoidal.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2383383900._hygCtx._hyg.10) F))","typeFull":"∀ (I : Type u_1) [inst : CategoryTheory.Category.{u_4, u_1} I] (C : Type u_3)\n  [inst_1 : CategoryTheory.Category.{u_2, u_3} C] [inst_2 : CategoryTheory.MonoidalCategory C]\n  [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n      (CategoryTheory.Discrete.functor id).HasRightKanExtension F]\n  [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] (F : CategoryTheory.Functor I C),\n  (CategoryTheory.MonoidalCategory.tensorLeft ((CategoryTheory.Functor.incl✝ I).comp F)).IsLeftAdjoint →\n    (CategoryTheory.MonoidalCategory.tensorLeft F).IsLeftAdjoint","typeReadable":"∀ (I : Type u_1) [inst : CategoryTheory.Category.{u_4, u_1} I] (C : Type u_3)\n  [inst_1 : CategoryTheory.Category.{u_2, u_3} C] [inst_2 : CategoryTheory.MonoidalCategory C]\n  [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n      (CategoryTheory.Discrete.functor id).HasRightKanExtension F]\n  [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] (F : CategoryTheory.Functor I C),\n  (CategoryTheory.MonoidalCategory.tensorLeft ((CategoryTheory.Functor.incl✝ I).comp F)).IsLeftAdjoint →\n    (CategoryTheory.MonoidalCategory.tensorLeft F).IsLeftAdjoint","typeReferences":[["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Category"],["CategoryTheory","Discrete"],["CategoryTheory","Limits","HasLimitsOfShape"],["CategoryTheory","Functor","HasRightKanExtension"],["CategoryTheory","Functor","comp"],["CategoryTheory","MonoidalCategory","tensorLeft"],["CategoryTheory","Limits","walkingParallelPairHomCategory"],["CategoryTheory","discreteCategory"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","MonoidalCategory"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","Functor","IsLeftAdjoint"],["id"],["CategoryTheory","Functor","category"],["CategoryTheory","Discrete","functor"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Limits","functorCategoryHasLimitsOfShape"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","whiskeringLeft"],["CategoryTheory","Functor","comp"],["CategoryTheory","MonoidalCategory","tensorLeft"],["CategoryTheory","Limits","walkingParallelPairHomCategory"],["CategoryTheory","Limits","hasCoreflexiveEqualizers_of_hasEqualizers"],["CategoryTheory","Functor","obj"],["CategoryTheory","isLeftAdjoint_square_lift_comonadic"],["CategoryTheory","discreteCategory"],["CategoryTheory","Functor","instComonadicLeftAdjointDiscreteObjWhiskeringLeftIncl"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","instIsLeftAdjointOfComonadicLeftAdjoint"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","Iso","refl"],["CategoryTheory","Functor","category"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","instPreservesLimitOfIsCoreflexivePairDiscreteObjWhiskeringLeftIncl"],"typeFallback":"forall (I : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 : CategoryTheory.Category.{v₂, u₂} I] (C : Type.{u₁}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7 : CategoryTheory.Category.{v₁, u₁} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.33 : CategoryTheory.Limits.HasLimitsOfShape.{0, 0, v₁, u₁} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7], CategoryTheory.Comonad.PreservesLimitOfIsCoreflexivePair.{max v₁ u₂, max (max (max u₁ v₁) v₂) u₂, max (max u₁ v₁) u₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.obj.{max v₂ u₂, max (max (max u₁ v₁) v₂) u₂, max v₂ u₂, max (max (max u₁ v₁) v₂) u₂} (CategoryTheory.Functor.{u₂, v₂, u₂, u₂} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3) (CategoryTheory.Functor.category.{u₂, v₂, u₂, u₂} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3) (CategoryTheory.Functor.{max u₂ v₁, max u₂ v₁, max (max (max u₁ u₂) v₁) v₂, max (max (max u₁ u₂) v₁) u₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7)) (CategoryTheory.Functor.category.{max u₂ v₁, max u₂ v₁, max (max (max u₂ u₁) v₂) v₁, max (max u₂ u₁) v₁} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7)) (CategoryTheory.Functor.whiskeringLeft.{u₂, u₂, u₂, v₂, u₁, v₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.7) (_private.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.0.CategoryTheory.Functor.incl.{v₂, u₂} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.657749614._hygCtx._hyg.3))","typeFull":"∀ (I : Type u₂) [inst : CategoryTheory.Category.{v₂, u₂} I] (C : Type u₁) [inst_1 : CategoryTheory.Category.{v₁, u₁} C]\n  [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C],\n  CategoryTheory.Comonad.PreservesLimitOfIsCoreflexivePair\n    ((CategoryTheory.Functor.whiskeringLeft (CategoryTheory.Discrete I) I C).obj (CategoryTheory.Functor.incl✝ I))","typeReadable":"∀ (I : Type u₂) [inst : CategoryTheory.Category.{v₂, u₂} I] (C : Type u₁) [inst_1 : CategoryTheory.Category.{v₁, u₁} C]\n  [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C],\n  CategoryTheory.Comonad.PreservesLimitOfIsCoreflexivePair\n    ((CategoryTheory.Functor.whiskeringLeft (CategoryTheory.Discrete I) I C).obj (CategoryTheory.Functor.incl✝ I))","typeReferences":[["CategoryTheory","discreteCategory"],["CategoryTheory","Comonad","PreservesLimitOfIsCoreflexivePair"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Category"],["CategoryTheory","Discrete"],["CategoryTheory","Limits","HasLimitsOfShape"],["CategoryTheory","Functor","whiskeringLeft"],["CategoryTheory","Functor","category"],["CategoryTheory","Functor","obj"],["CategoryTheory","Limits","walkingParallelPairHomCategory"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Limits","PreservesLimit"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","whiskeringLeft"],["CategoryTheory","Limits","walkingParallelPairHomCategory"],["CategoryTheory","Functor","obj"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","Comonad","PreservesLimitOfIsCoreflexivePair","mk"],["CategoryTheory","discreteCategory"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","CategoryStruct","toQuiver"],["CategoryTheory","Limits","PreservesLimitsOfShape","preservesLimit"],["Quiver","Hom"],["CategoryTheory","whiskeringLeft_preservesLimitsOfShape"],["inferInstance"],["CategoryTheory","Functor","category"],["CategoryTheory","Limits","parallelPair"],["CategoryTheory","IsCoreflexivePair"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","instIsLeftAdjointDiscreteTensorLeftCompIncl"],"typeFallback":"forall (I : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.3 : CategoryTheory.Category.{v₂, u₂} I] (C : Type.{u₁}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7 : CategoryTheory.Category.{v₁, u₁} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.10 : CategoryTheory.MonoidalCategory.{v₁, u₁} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.13 : CategoryTheory.MonoidalClosed.{v₁, u₁} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.10] (F : CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7), CategoryTheory.Functor.IsLeftAdjoint.{max u₂ v₁, max u₂ v₁, max (max (max u₁ u₂) v₁) u₂, max (max (max u₁ u₂) v₁) u₂} (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7) (CategoryTheory.MonoidalCategory.tensorLeft.{max u₂ v₁, max (max (max u₁ u₂) v₁) u₂} (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7) (CategoryTheory.Monoidal.functorCategoryMonoidal.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.10) (CategoryTheory.Functor.comp.{u₂, v₂, v₁, u₂, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.7 (_private.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.0.CategoryTheory.Functor.incl.{v₂, u₂} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.3557172746._hygCtx._hyg.3) F))","typeFull":"∀ (I : Type u₂) [inst : CategoryTheory.Category.{v₂, u₂} I] (C : Type u₁) [inst_1 : CategoryTheory.Category.{v₁, u₁} C]\n  [inst_2 : CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F : CategoryTheory.Functor I C),\n  (CategoryTheory.MonoidalCategory.tensorLeft ((CategoryTheory.Functor.incl✝ I).comp F)).IsLeftAdjoint","typeReadable":"∀ (I : Type u₂) [inst : CategoryTheory.Category.{v₂, u₂} I] (C : Type u₁) [inst_1 : CategoryTheory.Category.{v₁, u₁} C]\n  [inst_2 : CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F : CategoryTheory.Functor I C),\n  (CategoryTheory.MonoidalCategory.tensorLeft ((CategoryTheory.Functor.incl✝ I).comp F)).IsLeftAdjoint","typeReferences":[["CategoryTheory","discreteCategory"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","Functor"],["CategoryTheory","MonoidalCategory"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","Functor","IsLeftAdjoint"],["CategoryTheory","Category"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","comp"],["CategoryTheory","MonoidalCategory","tensorLeft"],["CategoryTheory","Functor","category"],["CategoryTheory","MonoidalClosed"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Functor","closed"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","comp"],["CategoryTheory","MonoidalCategory","tensorLeft"],["CategoryTheory","ihom"],["CategoryTheory","discreteCategory"],["CategoryTheory","instGroupoidDiscrete"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","Adjunction","isLeftAdjoint"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","ihom","adjunction"],["CategoryTheory","Functor","category"]]},{"isProp":false,"kind":"definition","name":["CategoryTheory","Functor","instComonadicLeftAdjointDiscreteObjWhiskeringLeftIncl"],"typeFallback":"forall (I : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 : CategoryTheory.Category.{v₂, u₂} I] (C : Type.{u₁}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7 : CategoryTheory.Category.{v₁, u₁} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.16 : forall (F : CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7), CategoryTheory.Functor.HasRightKanExtension.{u₂, u₂, v₁, u₁, v₂, u₂} (CategoryTheory.Discrete.{u₂} I) C I (CategoryTheory.discreteCategory.{u₂} I) inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 (CategoryTheory.Discrete.functor.{v₂, u₂, u₂} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 I (id.{succ u₂} I)) F] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.33 : CategoryTheory.Limits.HasLimitsOfShape.{0, 0, v₁, u₁} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7], CategoryTheory.ComonadicLeftAdjoint.{max v₁ u₂, max v₁ u₂, max (max (max u₁ v₁) v₂) u₂, max (max u₁ v₁) u₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.obj.{max v₂ u₂, max (max (max u₁ v₁) v₂) u₂, max v₂ u₂, max (max (max u₁ v₁) v₂) u₂} (CategoryTheory.Functor.{u₂, v₂, u₂, u₂} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3) (CategoryTheory.Functor.category.{u₂, v₂, u₂, u₂} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3) (CategoryTheory.Functor.{max u₂ v₁, max u₂ v₁, max (max (max u₁ u₂) v₁) v₂, max (max (max u₁ u₂) v₁) u₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7)) (CategoryTheory.Functor.category.{max u₂ v₁, max u₂ v₁, max (max (max u₂ u₁) v₂) v₁, max (max u₂ u₁) v₁} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7)) (CategoryTheory.Functor.whiskeringLeft.{u₂, u₂, u₂, v₂, u₁, v₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (_private.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.0.CategoryTheory.Functor.incl.{v₂, u₂} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3))","typeFull":"(I : Type u₂) →\n  [inst : CategoryTheory.Category.{v₂, u₂} I] →\n    (C : Type u₁) →\n      [inst_1 : CategoryTheory.Category.{v₁, u₁} C] →\n        [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n              (CategoryTheory.Discrete.functor id).HasRightKanExtension F] →\n          [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] →\n            CategoryTheory.ComonadicLeftAdjoint\n              ((CategoryTheory.Functor.whiskeringLeft (CategoryTheory.Discrete I) I C).obj\n                (CategoryTheory.Functor.incl✝ I))","typeReadable":"(I : Type u₂) →\n  [inst : CategoryTheory.Category.{v₂, u₂} I] →\n    (C : Type u₁) →\n      [inst_1 : CategoryTheory.Category.{v₁, u₁} C] →\n        [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n              (CategoryTheory.Discrete.functor id).HasRightKanExtension F] →\n          [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] →\n            CategoryTheory.ComonadicLeftAdjoint\n              ((CategoryTheory.Functor.whiskeringLeft (CategoryTheory.Discrete I) I C).obj\n                (CategoryTheory.Functor.incl✝ I))","typeReferences":[["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Category"],["CategoryTheory","Discrete"],["CategoryTheory","Limits","HasLimitsOfShape"],["CategoryTheory","Functor","HasRightKanExtension"],["CategoryTheory","Functor","whiskeringLeft"],["CategoryTheory","Limits","walkingParallelPairHomCategory"],["CategoryTheory","Functor","obj"],["CategoryTheory","discreteCategory"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","ComonadicLeftAdjoint"],["id"],["CategoryTheory","Functor","category"],["CategoryTheory","Discrete","functor"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Discrete"],["CategoryTheory","Functor","whiskeringLeft"],["CategoryTheory","Functor","obj"],["CategoryTheory","discreteCategory"],["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","incl"],["CategoryTheory","Functor","instComonadicLeftAdjointDiscreteObjWhiskeringLeftIncl","_proof_1"],["CategoryTheory","Comonad","comonadicOfHasPreservesCoreflexiveEqualizersOfReflectsIsomorphisms"],["CategoryTheory","Functor","ran"],["CategoryTheory","Functor","ranAdjunction"],["CategoryTheory","Functor","category"],["CategoryTheory","Functor","instPreservesLimitOfIsCoreflexivePairDiscreteObjWhiskeringLeftIncl"],["CategoryTheory","Functor","instReflectsIsomorphismsDiscreteObjWhiskeringLeftIncl"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","CategoryTheory","Monoidal","Closed","FunctorCategory","Complete",0,"CategoryTheory","Functor","instReflectsIsomorphismsDiscreteObjWhiskeringLeftIncl","_simp_1"],"typeFallback":"forall {C : Type.{u₁}} [inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 : CategoryTheory.Category.{v₁, u₁} C] {D : Type.{u₂}} [inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7 : CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7} {G : CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7} (τ : Quiver.Hom.{max u₁ v₂, max (max (max u₁ u₂) v₁) v₂} (CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7) (CategoryTheory.CategoryStruct.toQuiver.{max u₁ v₂, max (max (max u₁ u₂) v₁) v₂} (CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7) (CategoryTheory.Category.toCategoryStruct.{max u₁ v₂, max (max (max u₁ u₂) v₁) v₂} (CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7))) F G), Eq.{1} Prop (CategoryTheory.IsIso.{max u₁ v₂, max (max (max u₁ u₂) v₁) v₂} (CategoryTheory.Functor.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7) F G τ) (forall (X : C), CategoryTheory.IsIso.{v₂, u₂} D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7 (CategoryTheory.Functor.obj.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7 F X) (CategoryTheory.Functor.obj.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7 G X) (CategoryTheory.NatTrans.app.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.NatIso.1553492190._hygCtx._hyg.7 F G τ X))","typeFull":"∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]\n  {F G : CategoryTheory.Functor C D} (τ : F ⟶ G), CategoryTheory.IsIso τ = ∀ (X : C), CategoryTheory.IsIso (τ.app X)","typeReadable":"∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]\n  {F G : CategoryTheory.Functor C D} (τ : F ⟶ G), CategoryTheory.IsIso τ = ∀ (X : C), CategoryTheory.IsIso (τ.app X)","typeReferences":[["CategoryTheory","Functor"],["CategoryTheory","CategoryStruct","toQuiver"],["Quiver","Hom"],["CategoryTheory","Category"],["CategoryTheory","IsIso"],["Eq"],["CategoryTheory","Functor","category"],["CategoryTheory","NatTrans","app"],["CategoryTheory","Functor","obj"],["CategoryTheory","Category","toCategoryStruct"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","NatTrans","isIso_iff_isIso_app"],["CategoryTheory","IsIso"],["CategoryTheory","Functor","category"],["CategoryTheory","NatTrans","app"],["CategoryTheory","Functor","obj"],["propext"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","Functor","instComonadicLeftAdjointDiscreteObjWhiskeringLeftIncl","_proof_1"],"typeFallback":"forall (I : Type.{u_1}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 : CategoryTheory.Category.{u_4, u_1} I] (C : Type.{u_3}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7 : CategoryTheory.Category.{u_2, u_3} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.33 : CategoryTheory.Limits.HasLimitsOfShape.{0, 0, u_2, u_3} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7], CategoryTheory.Limits.HasCoreflexiveEqualizers.{max u_1 u_2, max (max (max u_3 u_1) u_2) u_4} (CategoryTheory.Functor.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7) (CategoryTheory.Functor.category.{u_4, u_2, u_1, u_3} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2975377582._hygCtx._hyg.7)","typeFull":"∀ (I : Type u_1) [inst : CategoryTheory.Category.{u_4, u_1} I] (C : Type u_3)\n  [inst_1 : CategoryTheory.Category.{u_2, u_3} C]\n  [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C],\n  CategoryTheory.Limits.HasCoreflexiveEqualizers (CategoryTheory.Functor I C)","typeReadable":"∀ (I : Type u_1) [inst : CategoryTheory.Category.{u_4, u_1} I] (C : Type u_3)\n  [inst_1 : CategoryTheory.Category.{u_2, u_3} C]\n  [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C],\n  CategoryTheory.Limits.HasCoreflexiveEqualizers (CategoryTheory.Functor I C)","typeReferences":[["CategoryTheory","Limits","HasCoreflexiveEqualizers"],["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Category"],["CategoryTheory","Limits","HasLimitsOfShape"],["CategoryTheory","Functor","category"],["CategoryTheory","Limits","walkingParallelPairHomCategory"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Limits","functorCategoryHasLimitsOfShape"],["CategoryTheory","Functor","category"],["CategoryTheory","Limits","walkingParallelPairHomCategory"],["CategoryTheory","Limits","hasCoreflexiveEqualizers_of_hasEqualizers"]]},{"isProp":false,"kind":"definition","name":["CategoryTheory","Functor","functorCategoryMonoidalClosed"],"typeFallback":"forall (I : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.3 : CategoryTheory.Category.{v₂, u₂} I] (C : Type.{u₁}) [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7 : CategoryTheory.Category.{v₁, u₁} C] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.10 : CategoryTheory.MonoidalCategory.{v₁, u₁} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.13 : CategoryTheory.MonoidalClosed.{v₁, u₁} C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.10] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.16 : forall (F : CategoryTheory.Functor.{u₂, v₁, u₂, u₁} (CategoryTheory.Discrete.{u₂} I) (CategoryTheory.discreteCategory.{u₂} I) C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7), CategoryTheory.Functor.HasRightKanExtension.{u₂, u₂, v₁, u₁, v₂, u₂} (CategoryTheory.Discrete.{u₂} I) C I (CategoryTheory.discreteCategory.{u₂} I) inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.3 (CategoryTheory.Discrete.functor.{v₂, u₂, u₂} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.3 I (id.{succ u₂} I)) F] [inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.33 : CategoryTheory.Limits.HasLimitsOfShape.{0, 0, v₁, u₁} CategoryTheory.Limits.WalkingParallelPair CategoryTheory.Limits.walkingParallelPairHomCategory C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7], CategoryTheory.MonoidalClosed.{max u₂ v₁, max (max (max u₁ u₂) v₁) v₂} (CategoryTheory.Functor.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7) (CategoryTheory.Functor.category.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7) (CategoryTheory.Monoidal.functorCategoryMonoidal.{v₂, v₁, u₂, u₁} I inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.3 C inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.7 inst._@.Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Complete.2463007374._hygCtx._hyg.10)","typeFull":"(I : Type u₂) →\n  [inst : CategoryTheory.Category.{v₂, u₂} I] →\n    (C : Type u₁) →\n      [inst_1 : CategoryTheory.Category.{v₁, u₁} C] →\n        [inst_2 : CategoryTheory.MonoidalCategory C] →\n          [CategoryTheory.MonoidalClosed C] →\n            [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n                  (CategoryTheory.Discrete.functor id).HasRightKanExtension F] →\n              [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] →\n                CategoryTheory.MonoidalClosed (CategoryTheory.Functor I C)","typeReadable":"(I : Type u₂) →\n  [inst : CategoryTheory.Category.{v₂, u₂} I] →\n    (C : Type u₁) →\n      [inst_1 : CategoryTheory.Category.{v₁, u₁} C] →\n        [inst_2 : CategoryTheory.MonoidalCategory C] →\n          [CategoryTheory.MonoidalClosed C] →\n            [∀ (F : CategoryTheory.Functor (CategoryTheory.Discrete I) C),\n                  (CategoryTheory.Discrete.functor id).HasRightKanExtension F] →\n              [CategoryTheory.Limits.HasLimitsOfShape CategoryTheory.Limits.WalkingParallelPair C] →\n                CategoryTheory.MonoidalClosed (CategoryTheory.Functor I C)","typeReferences":[["CategoryTheory","Functor"],["CategoryTheory","Limits","WalkingParallelPair"],["CategoryTheory","Category"],["CategoryTheory","Discrete"],["CategoryTheory","Limits","HasLimitsOfShape"],["CategoryTheory","Functor","HasRightKanExtension"],["CategoryTheory","Limits","walkingParallelPairHomCategory"],["CategoryTheory","discreteCategory"],["CategoryTheory","MonoidalCategory"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["id"],["CategoryTheory","Functor","category"],["CategoryTheory","MonoidalClosed"],["CategoryTheory","Discrete","functor"]],"valueReferences":[["CategoryTheory","Functor"],["CategoryTheory","Monoidal","functorCategoryMonoidal"],["CategoryTheory","Functor","functorCategoryClosed"],["CategoryTheory","MonoidalClosed","mk"],["CategoryTheory","Functor","category"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Fintype.Pigeonhole.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Finite","exists_ne_map_eq_of_infinite"],"typeFallback":"forall {α : Sort.{u_4}} {β : Sort.{u_5}} [inst._@.Mathlib.Data.Fintype.Pigeonhole.1769385511._hygCtx._hyg.7 : Infinite.{u_4} α] [inst._@.Mathlib.Data.Fintype.Pigeonhole.1769385511._hygCtx._hyg.10 : Finite.{u_5} β] (f : α -> β), Exists.{u_4} α (fun (x : α) => Exists.{u_4} α (fun (y : α) => And (Ne.{u_4} α x y) (Eq.{u_5} β (f x) (f y))))","typeFull":"∀ {α : Sort u_4} {β : Sort u_5} [Infinite α] [Finite β] (f : α → β), ∃ x y, x ≠ y ∧ f x = f y","typeReadable":"∀ {α : Sort u_4} {β : Sort u_5} [Infinite α] [Finite β] (f : α → β), ∃ x y, x ≠ y ∧ f x = f y","typeReferences":[["Exists"],["And"],["Ne"],["Eq"],["Finite"],["Infinite"]],"valueReferences":[["Not"],["Exists"],["Eq","trans"],["Eq","mp"],["And"],["Classical","not_forall","_simp_1"],["congrArg"],["not_injective_infinite_finite"],["_private","Mathlib","Data","Fintype","Pigeonhole",0,"Finite","exists_ne_map_eq_of_infinite","_simp_1_1"],["funext"],["id"],["Classical","not_imp","_simp_1"],["Eq","mpr"],["Ne"],["Eq"],["Function","Injective"]]},{"isProp":true,"kind":"theorem","name":["Finite","exists_infinite_fiber"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Data.Fintype.Pigeonhole.4238776531._hygCtx._hyg.5 : Infinite.{succ u_1} α] [inst._@.Mathlib.Data.Fintype.Pigeonhole.4238776531._hygCtx._hyg.8 : Finite.{succ u_2} β] (f : α -> β), Exists.{succ u_2} β (fun (y : β) => Infinite.{succ u_1} (Set.Elem.{u_1} α (Set.preimage.{u_1, u_2} α β f (Singleton.singleton.{u_2, u_2} β (Set.{u_2} β) (Set.instSingletonSet.{u_2} β) y))))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [Infinite α] [Finite β] (f : α → β), ∃ y, Infinite ↑(f ⁻¹' {y})","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [Infinite α] [Finite β] (f : α → β), ∃ y, Infinite ↑(f ⁻¹' {y})","typeReferences":[["Set","preimage"],["Exists"],["Set"],["Singleton","singleton"],["Set","instSingletonSet"],["Finite"],["Infinite"],["Set","Elem"]],"valueReferences":[["Finset"],["Finset","mem_biUnion","_simp_1"],["Eq","trans"],["Singleton","singleton"],["Membership","mem"],["Classical","propDecidable"],["Set","Elem"],["forall_congr"],["funext"],["Finite"],["Fintype","false"],["binderNameHint"],["Exists"],["SetLike","instMembership"],["And"],["Nonempty"],["Fintype"],["true_and"],["Set","instMembership"],["implies_true"],["Decidable","byContradiction"],["Set","mem_toFinset","_simp_1"],["Eq","refl"],["Nonempty","casesOn"],["exists_eq'","_simp_1"],["not_infinite_iff_finite","_simp_1"],["Finset","instSetLike"],["Set","mem_preimage","_simp_1"],["Fintype","decidableExistsFintype"],["Eq","mp"],["congrArg"],["Fintype","ofFinite"],["Mathlib","Tactic","Push","not_exists","_simp_1"],["congr"],["Eq"],["nonempty_fintype"],["Infinite"],["Not"],["Finset","univ"],["Set","preimage"],["True"],["Set"],["Set","toFinset"],["Set","instSingletonSet"],["Finset","biUnion"],["Finset","mem_univ","_simp_1"],["Fintype","mk"],["of_eq_true"],["False"],["Set","mem_singleton_iff","_simp_1"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Data","Fintype","Pigeonhole",0,"Finite","exists_ne_map_eq_of_infinite","_simp_1_1"],"typeFallback":"forall {a : Prop} {b : Prop}, Eq.{1} Prop (And a b) (And b a)","typeFull":"∀ {a b : Prop}, (a ∧ b) = (b ∧ a)","typeReadable":"∀ {a b : Prop}, (a ∧ b) = (b ∧ a)","typeReferences":[["And"],["Eq"]],"valueReferences":[["And"],["and_comm"],["propext"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Data","Fintype","Pigeonhole",0,"Function","Embedding","isEmpty_of_card_lt","match_1_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} (f : Function.Embedding.{succ u_1, succ u_2} α β) (motive : (Exists.{succ u_1} α (fun (x : α) => Exists.{succ u_1} α (fun (y : α) => And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f x) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f y))))) -> Prop) (x._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.44 : Exists.{succ u_1} α (fun (x : α) => Exists.{succ u_1} α (fun (y : α) => And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f x) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f y))))), (forall (_x : α) (_y : α) (ne : Ne.{succ u_1} α _x _y) (feq : Eq.{succ u_2} β (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f _x) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f _y)), motive (Exists.intro.{succ u_1} α (fun (x : α) => Exists.{succ u_1} α (fun (y : α) => And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f x) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f y)))) _x (Exists.intro.{succ u_1} α (fun (y : α) => And (Ne.{succ u_1} α _x y) (Eq.{succ u_2} β (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f _x) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f y))) _y (And.intro (Ne.{succ u_1} α _x _y) (Eq.{succ u_2} β (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f _x) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (Function.Embedding.{succ u_1, succ u_2} α β) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (Function.instFunLikeEmbedding.{succ u_1, succ u_2} α β) f _y)) ne feq)))) -> (motive x._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.44)","typeFull":"∀ {α : Type u_1} {β : Type u_2} (f : α ↪ β) (motive : (∃ x y, x ≠ y ∧ f x = f y) → Prop) (x : ∃ x y, x ≠ y ∧ f x = f y),\n  (∀ (_x _y : α) (ne : _x ≠ _y) (feq : f _x = f _y), motive ⋯) → motive x","typeReadable":"∀ {α : Type u_1} {β : Type u_2} (f : α ↪ β) (motive : (∃ x y, x ≠ y ∧ f x = f y) → Prop) (x : ∃ x y, x ≠ y ∧ f x = f y),\n  (∀ (_x _y : α) (ne : _x ≠ _y) (feq : f _x = f _y), motive ⋯) → motive x","typeReferences":[["And","intro"],["Exists"],["Function","Embedding"],["And"],["Exists","intro"],["Ne"],["Function","instFunLikeEmbedding"],["Eq"],["DFunLike","coe"]],"valueReferences":[["Exists","casesOn"],["Exists"],["Function","Embedding"],["And"],["Exists","intro"],["Ne"],["Function","instFunLikeEmbedding"],["Eq"],["DFunLike","coe"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Function","Embedding","isEmpty_of_card_lt","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.5 : Fintype.{u_1} α] [inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.8 : Fintype.{u_2} β], (LT.lt.{0} Nat instLTNat (Fintype.card.{u_2} β inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.8) (Fintype.card.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.5)) -> (Eq.{1} Prop (IsEmpty.{max (succ u_2) (succ u_1)} (Function.Embedding.{succ u_1, succ u_2} α β)) True)","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β],\n  Fintype.card β < Fintype.card α → IsEmpty (α ↪ β) = True","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β],\n  Fintype.card β < Fintype.card α → IsEmpty (α ↪ β) = True","typeReferences":[["IsEmpty"],["Fintype","card"],["LT","lt"],["instLTNat"],["Nat"],["True"],["Function","Embedding"],["Eq"],["Fintype"]],"valueReferences":[["IsEmpty"],["Function","Embedding","isEmpty_of_card_lt"],["Function","Embedding"],["eq_true"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Data","Fintype","Pigeonhole",0,"Fintype","exists_ne_map_eq_of_card_lt","match_1_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5 : Fintype.{u_1} α] (f : α -> β) (motive : (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} ��)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) x) (Exists.{succ u_1} α (fun (y : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) y) (And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y))))))) -> Prop) (x._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.67 : Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) x) (Exists.{succ u_1} α (fun (y : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) y) (And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y))))))), (forall (x : α) (left._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.89 : Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) x) (y : α) (left._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.90 : Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) y) (h : And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y))), motive (Exists.intro.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) x) (Exists.{succ u_1} α (fun (y : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) y) (And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y)))))) x (And.intro (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) x) (Exists.{succ u_1} α (fun (y : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) y) (And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y))))) left._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.89 (Exists.intro.{succ u_1} α (fun (y : α) => And (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) y) (And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y)))) y (And.intro (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.univ.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5) y) (And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y))) left._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.90 h))))) -> (motive x._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.67)","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] (f : α → β)\n  (motive : (∃ x ∈ Finset.univ, ∃ y ∈ Finset.univ, x ≠ y ∧ f x = f y) → Prop)\n  (x : ∃ x ∈ Finset.univ, ∃ y ∈ Finset.univ, x ≠ y ∧ f x = f y),\n  (∀ (x : α) (left : x ∈ Finset.univ) (y : α) (left_1 : y ∈ Finset.univ) (h : x ≠ y ∧ f x = f y), motive ⋯) → motive x","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] (f : α → β)\n  (motive : (∃ x ∈ Finset.univ, ∃ y ∈ Finset.univ, x ≠ y ∧ f x = f y) → Prop)\n  (x : ∃ x ∈ Finset.univ, ∃ y ∈ Finset.univ, x ≠ y ∧ f x = f y),\n  (∀ (x : α) (left : x ∈ Finset.univ) (y : α) (left_1 : y ∈ Finset.univ) (h : x ≠ y ∧ f x = f y), motive ⋯) → motive x","typeReferences":[["And","intro"],["Finset","univ"],["Finset","instSetLike"],["SetLike","instMembership"],["Finset"],["Exists"],["Membership","mem"],["And"],["Exists","intro"],["Ne"],["Eq"],["Fintype"]],"valueReferences":[["Finset","instSetLike"],["Finset","univ"],["Finset"],["SetLike","instMembership"],["Exists"],["Membership","mem"],["And"],["Exists","intro"],["Exists","casesOn"],["And","intro"],["Ne"],["Eq"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Fintype","exists_ne_map_eq_of_card_lt"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5 : Fintype.{u_1} α] [inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.8 : Fintype.{u_2} β] (f : α -> β), (LT.lt.{0} Nat instLTNat (Fintype.card.{u_2} β inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.8) (Fintype.card.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.3324101907._hygCtx._hyg.5)) -> (Exists.{succ u_1} α (fun (x : α) => Exists.{succ u_1} α (fun (y : α) => And (Ne.{succ u_1} α x y) (Eq.{succ u_2} β (f x) (f y)))))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] (f : α → β),\n  Fintype.card β < Fintype.card α → ∃ x y, x ≠ y ∧ f x = f y","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β] (f : α → β),\n  Fintype.card β < Fintype.card α → ∃ x y, x ≠ y ∧ f x = f y","typeReferences":[["Fintype","card"],["LT","lt"],["instLTNat"],["Nat"],["Exists"],["And"],["Ne"],["Eq"],["Fintype"]],"valueReferences":[["Finset","exists_ne_map_eq_of_card_lt_of_maps_to"],["Finset","univ"],["Exists"],["_private","Mathlib","Data","Fintype","Pigeonhole",0,"Fintype","exists_ne_map_eq_of_card_lt","match_1_1"],["And"],["Exists","intro"],["Finset","mem_univ"],["Ne"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["Function","Embedding","isEmpty_of_card_lt"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.5 : Fintype.{u_1} α] [inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.8 : Fintype.{u_2} β], (LT.lt.{0} Nat instLTNat (Fintype.card.{u_2} β inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.8) (Fintype.card.{u_1} α inst._@.Mathlib.Data.Fintype.Pigeonhole.2508162688._hygCtx._hyg.5)) -> (IsEmpty.{max (max 1 (succ u_2)) (succ u_1)} (Function.Embedding.{succ u_1, succ u_2} α β))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β],\n  Fintype.card β < Fintype.card α → IsEmpty (α ↪ β)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst_1 : Fintype β],\n  Fintype.card β < Fintype.card α → IsEmpty (α ↪ β)","typeReferences":[["IsEmpty"],["Fintype","card"],["LT","lt"],["instLTNat"],["Nat"],["Function","Embedding"],["Fintype"]],"valueReferences":[["Fintype","exists_ne_map_eq_of_card_lt"],["Function","Embedding","injective"],["IsEmpty","mk"],["Function","Embedding"],["False"],["_private","Mathlib","Data","Fintype","Pigeonhole",0,"Function","Embedding","isEmpty_of_card_lt","match_1_1"],["Function","instFunLikeEmbedding"],["DFunLike","coe"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.SProd.sym.json

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["SProd","casesOn"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {motive : (SProd.{u, v, w} α β γ) -> Sort.{u_1}} (t : SProd.{u, v, w} α β γ), (forall (sprod : α -> β -> γ), motive (SProd.mk.{u, v, w} α β γ sprod)) -> (motive t)","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : SProd α β γ → Sort u_1} →\n        (t : SProd α β γ) → ((sprod : α → β → γ) → motive { sprod := sprod }) → motive t","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : SProd α β γ → Sort u_1} →\n        (t : SProd α β γ) → ((sprod : α → β → γ) → motive { sprod := sprod }) → motive t","typeReferences":[["SProd","mk"],["SProd"]],"valueReferences":[["SProd","rec"]]},{"isProp":false,"kind":"definition","name":["SProd","noConfusionType"],"typeFallback":"Sort.{u_1} -> (forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}}, (SProd.{u, v, w} α β γ) -> (forall {α' : Type.{u}} {β' : Type.{v}} {γ' : Type.{w}}, (SProd.{u, v, w} α' β' γ') -> Sort.{u_1}))","typeFull":"Sort u_1 →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} → SProd α β γ → {α' : Type u} → {β' : Type v} → {γ' : Type w} → SProd α' β' γ' → Sort u_1","typeReadable":"Sort u_1 →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} → SProd α β γ → {α' : Type u} → {β' : Type v} → {γ' : Type w} → SProd α' β' γ' → Sort u_1","typeReferences":[["SProd"]],"valueReferences":[["HEq"],["SProd","casesOn"]]},{"isProp":false,"kind":"definition","name":["SProd","mk","noConfusion"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}} {P : Sort.{u_1}} {sprod : α -> β -> γ} {sprod' : α -> β -> γ}, (Eq.{max (max (succ u) (succ v)) (succ w)} (SProd.{u, v, w} α β γ) (SProd.mk.{u, v, w} α β γ sprod) (SProd.mk.{u, v, w} α β γ sprod')) -> ((HEq.{max (max (succ u) (succ v)) (succ w)} (α -> β -> γ) sprod (α -> β -> γ) sprod') -> P) -> P","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : outParam (Type w)} →\n      {P : Sort u_1} → {sprod sprod' : α → β → γ} → { sprod := sprod } = { sprod := sprod' } → (sprod ≍ sprod' → P) → P","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : outParam (Type w)} →\n      {P : Sort u_1} → {sprod sprod' : α → β → γ} → { sprod := sprod } = { sprod := sprod' } → (sprod ≍ sprod' → P) → P","typeReferences":[["SProd","mk"],["outParam"],["HEq"],["Eq"],["SProd"]],"valueReferences":[["SProd","mk"],["outParam"],["Eq","refl"],["SProd","noConfusion"],["id"],["heq_of_eq"],["SProd"]]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Data_SProd___unexpand_SProd_sprod_1"],"typeFallback":"Lean.PrettyPrinter.Unexpander","typeFull":"Lean.PrettyPrinter.Unexpander","typeReadable":"Lean.PrettyPrinter.Unexpander","typeReferences":[["Lean","PrettyPrinter","Unexpander"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Bool","false"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","MonadQuotation","toMonadRef"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["List","cons"],["Bool","true"],["Unit","unit"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","PrettyPrinter","instMonadQuotationUnexpandM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["Bool","or"],["cond"],["Unit"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","withRef"],["Bool"],["Applicative","toPure"],["Lean","PrettyPrinter","UnexpandM"],["ReaderT","instMonadExceptOf"],["EStateM"],["Lean","Syntax","isOfKind"],["Lean","Syntax","matchesNull"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Eq"],["List","nil"],["Bind","bind"],["EStateM","instMonadExceptOfOfBacktrackable"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","node3"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["term_×ˢ_"],"typeFallback":"Lean.TrailingParserDescr","typeFull":"Lean.TrailingParserDescr","typeReadable":"Lean.TrailingParserDescr","typeReferences":[["Lean","TrailingParserDescr"]],"valueReferences":[["Nat"],["Lean","ParserDescr","cat"],["Lean","Name","mkStr1"],["instOfNatNat"],["Lean","ParserDescr","trailingNode"],["Lean","ParserDescr","symbol"],["OfNat","ofNat"],["Lean","ParserDescr","binary"]]},{"isProp":false,"kind":"constructor","name":["SProd","mk"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}}, (α -> β -> γ) -> (SProd.{u, v, w} α β γ)","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → (α → β → γ) → SProd α β γ","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} ��� (α → β → γ) → SProd α β γ","typeReferences":[["outParam"],["SProd"]],"valueReferences":null},{"isProp":false,"kind":"recursor","name":["SProd","rec"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {motive : (SProd.{u, v, w} α β γ) -> Sort.{u_1}}, (forall (sprod : α -> β -> γ), motive (SProd.mk.{u, v, w} α β γ sprod)) -> (forall (t : SProd.{u, v, w} α β γ), motive t)","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : SProd α β γ → Sort u_1} →\n        ((sprod : α → β → γ) → motive { sprod := sprod }) → (t : SProd α β γ) → motive t","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : SProd α β γ → Sort u_1} →\n        ((sprod : α → β → γ) → motive { sprod := sprod }) → (t : SProd α β γ) → motive t","typeReferences":[["SProd","mk"],["SProd"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["SProd","ctorIdx"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}}, (SProd.{u, v, w} α β γ) -> Nat","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → SProd α β γ → Nat","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → SProd α β γ → Nat","typeReferences":[["Nat"],["outParam"],["SProd"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["SProd","recOn"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {motive : (SProd.{u, v, w} α β γ) -> Sort.{u_1}} (t : SProd.{u, v, w} α β γ), (forall (sprod : α -> β -> γ), motive (SProd.mk.{u, v, w} α β γ sprod)) -> (motive t)","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : SProd α β γ → Sort u_1} →\n        (t : SProd α β γ) → ((sprod : α → β → γ) → motive { sprod := sprod }) → motive t","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : SProd α β γ → Sort u_1} →\n        (t : SProd α β γ) → ((sprod : α → β → γ) → motive { sprod := sprod }) → motive t","typeReferences":[["SProd","mk"],["SProd"]],"valueReferences":[["SProd","rec"]]},{"isProp":false,"kind":"definition","name":["SProd","sprod"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}} [self : SProd.{u, v, w} α β γ], α -> β -> γ","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : SProd α β γ] → α → β → γ","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : SProd α β γ] → α → β → γ","typeReferences":[["outParam"],["SProd"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Data_SProd___macroRules_term_×ˢ__1"],"typeFallback":"Lean.Macro","typeFull":"Lean.Macro","typeReadable":"Lean.Macro","typeReferences":[["Lean","Macro"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Lean","Macro","State"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","Syntax","ident"],["String"],["Lean","Macro","Context"],["Lean","Macro","Exception","unsupportedSyntax"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["Lean","Syntax","Preresolved"],["List","cons"],["Bool","true"],["Lean","Syntax","Preresolved","decl"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","Macro","instMonadQuotationMacroM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Bool"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["EStateM"],["String","toRawSubstring'"],["Lean","Syntax","isOfKind"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Lean","addMacroScope"],["Lean","MacroM"],["Eq"],["Lean","Name","mkStr2"],["Lean","Syntax","node2"],["EStateM","instMonadExceptOfOfBacktrackable"],["Bind","bind"],["List","nil"],["Lean","Name","mkStr1"],["ite"],["Lean","Macro","instMonadRefMacroM"],["Lean","Macro","Exception"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Data_SProd___macroRules_term_×ˢ__2"],"typeFallback":"Lean.Macro","typeFull":"Lean.Macro","typeReadable":"Lean.Macro","typeReferences":[["Lean","Macro"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Lean","Macro","State"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","Syntax","ident"],["String"],["Lean","Macro","Context"],["Lean","Syntax","node4"],["Lean","Macro","Exception","unsupportedSyntax"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["Lean","Syntax","Preresolved"],["List","cons"],["Bool","true"],["Lean","Syntax","Preresolved","decl"],["MonadExcept","throw"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","Macro","instMonadQuotationMacroM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Bool"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["EStateM"],["String","toRawSubstring'"],["Lean","Syntax","isOfKind"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Lean","addMacroScope"],["Lean","MacroM"],["Eq"],["Lean","Name","mkStr2"],["EStateM","instMonadExceptOfOfBacktrackable"],["Bind","bind"],["List","nil"],["Lean","Name","mkStr1"],["ite"],["Lean","Macro","instMonadRefMacroM"],["Lean","Macro","Exception"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["SProd","noConfusion"],"typeFallback":"forall {P : Sort.{u_1}} {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {t : SProd.{u, v, w} α β γ} {α' : Type.{u}} {β' : Type.{v}} {γ' : Type.{w}} {t' : SProd.{u, v, w} α' β' γ'}, (Eq.{succ (succ u)} Type.{u} α α') -> (Eq.{succ (succ v)} Type.{v} β β') -> (Eq.{succ (succ w)} Type.{w} γ γ') -> (HEq.{max (max (succ u) (succ v)) (succ w)} (SProd.{u, v, w} α β γ) t (SProd.{u, v, w} α' β' γ') t') -> (SProd.noConfusionType.{u_1, u, v, w} P α β γ t α' β' γ' t')","typeFull":"{P : Sort u_1} →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} →\n        {t : SProd α β γ} →\n          {α' : Type u} →\n            {β' : Type v} →\n              {γ' : Type w} → {t' : SProd α' β' γ'} → α = α' → β = β' → γ = γ' → t ≍ t' → SProd.noConfusionType P t t'","typeReadable":"{P : Sort u_1} →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} →\n        {t : SProd α β γ} →\n          {α' : Type u} →\n            {β' : Type v} →\n              {γ' : Type w} → {t' : SProd α' β' γ'} → α = α' → β = β' → γ = γ' → t ≍ t' → SProd.noConfusionType P t t'","typeReferences":[["HEq"],["Eq"],["SProd"],["SProd","noConfusionType"]],"valueReferences":[["HEq","refl"],["eq_of_heq"],["HEq"],["SProd","casesOn"],["Eq","ndrec"],["SProd"],["SProd","noConfusionType"]]},{"isProp":false,"kind":"definition","name":["SProd","mk","_flat_ctor"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}}, (α -> β -> γ) -> (SProd.{u, v, w} α β γ)","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → (α → β → γ) → SProd α β γ","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → (α → β → γ) → SProd α β γ","typeReferences":[["outParam"],["SProd"]],"valueReferences":[["SProd","mk"]]},{"isProp":false,"kind":"inductive","name":["SProd"],"typeFallback":"Type.{u} -> Type.{v} -> (outParam.{succ (succ w)} Type.{w}) -> Sort.{max (max (succ u) (succ v)) (succ w)}","typeFull":"Type u → Type v → outParam (Type w) → Type (max (max u v) w)","typeReadable":"Type u → Type v → outParam (Type w) → Type (max (max u v) w)","typeReferences":[["outParam"]],"valueReferences":null}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Set.Constructions.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"recursor","name":["FiniteInter","finiteInterClosure","rec"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop}, (forall {s : Set.{u_1} α} (a._@._internal._hyg.0 : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s), motive s (FiniteInter.finiteInterClosure.basic.{u_1} α S s a._@._internal._hyg.0)) -> (motive (Set.univ.{u_1} α) (FiniteInter.finiteInterClosure.univ.{u_1} α S)) -> (forall {s : Set.{u_1} α} {t : Set.{u_1} α} (a._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S s) (a_1._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S t), (motive s a._@._internal._hyg.0) -> (motive t a_1._@._internal._hyg.0) -> (motive (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t) (FiniteInter.finiteInterClosure.inter.{u_1} α S s t a._@._internal._hyg.0 a_1._@._internal._hyg.0))) -> (forall {a._@._internal._hyg.0 : Set.{u_1} α} (t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0), motive a._@._internal._hyg.0 t)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop},\n  (∀ {s : Set α} (a : s ∈ S), motive s ⋯) →\n    motive Set.univ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n          motive s a → motive t a_1 → motive (s ∩ t) ⋯) →\n        ∀ {a : Set α} (t : FiniteInter.finiteInterClosure S a), motive a t","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop},\n  (∀ {s : Set α} (a : s ∈ S), motive s ⋯) →\n    motive Set.univ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n          motive s a → motive t a_1 → motive (s ∩ t) ⋯) →\n        ∀ {a : Set α} (t : FiniteInter.finiteInterClosure S a), motive a t","typeReferences":[["Set","univ"],["FiniteInter","finiteInterClosure","basic"],["FiniteInter","finiteInterClosure"],["Set"],["Inter","inter"],["Membership","mem"],["FiniteInter","finiteInterClosure","inter"],["Set","instInter"],["FiniteInter","finiteInterClosure","univ"],["Set","instMembership"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["FiniteInter","recOn"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : (FiniteInter.{u_1} α S) -> Sort.{u}} (t : FiniteInter.{u_1} α S), (forall (univ_mem : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Set.univ.{u_1} α)) (inter_mem : forall {{s : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (forall {{t : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S t) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t)))), motive (FiniteInter.mk.{u_1} α S univ_mem inter_mem)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : FiniteInter S → Sort u} →\n      (t : FiniteInter S) →\n        ((univ_mem : Set.univ ∈ S) → (inter_mem : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → motive ⋯) →\n          motive t","typeReadable":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : FiniteInter S → Sort u} →\n      (t : FiniteInter S) →\n        ((univ_mem : Set.univ ∈ S) → (inter_mem : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → motive ⋯) →\n          motive t","typeReferences":[["Set","univ"],["Set"],["FiniteInter","mk"],["Inter","inter"],["Membership","mem"],["FiniteInter"],["Set","instInter"],["Set","instMembership"]],"valueReferences":[["FiniteInter","rec"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Data","Set","Constructions",0,"FiniteInter","finiteInterClosure_insert","_simp_1_1"],"typeFallback":"forall {α : Type.{u}} (a : Set.{u} α) (b : Set.{u} α) (c : Set.{u} α), Eq.{succ u} (Set.{u} α) (Inter.inter.{u} (Set.{u} α) (Set.instInter.{u} α) a (Inter.inter.{u} (Set.{u} α) (Set.instInter.{u} α) b c)) (Inter.inter.{u} (Set.{u} α) (Set.instInter.{u} α) (Inter.inter.{u} (Set.{u} α) (Set.instInter.{u} α) a b) c)","typeFull":"∀ {α : Type u} (a b c : Set α), a ∩ (b ∩ c) = a ∩ b ∩ c","typeReadable":"∀ {α : Type u} (a b c : Set α), a ∩ (b ∩ c) = a ∩ b ∩ c","typeReferences":[["Set"],["Inter","inter"],["Eq"],["Set","instInter"]],"valueReferences":[["Set"],["Inter","inter"],["Eq","symm"],["Set","instInter"],["Set","inter_assoc"]]},{"isProp":true,"kind":"theorem","name":["Set","biUnion_empty_finset"],"typeFallback":"forall {ι : Type.{u_2}} {X : Type.{u_3}} {s : ι -> (Set.{u_3} X)}, Eq.{succ u_3} (Set.{u_3} X) (Set.iUnion.{u_3, succ u_2} X ι (fun (i : ι) => Set.iUnion.{u_3, 0} X (Membership.mem.{u_2, u_2} ι (Finset.{u_2} ι) (SetLike.instMembership.{u_2, u_2} (Finset.{u_2} ι) ι (Finset.instSetLike.{u_2} ι)) (EmptyCollection.emptyCollection.{u_2} (Finset.{u_2} ι) (Finset.instEmptyCollection.{u_2} ι)) i) (fun (h._@.Mathlib.Data.Set.Constructions.3801755705._hygCtx._hyg.20 : Membership.mem.{u_2, u_2} ι (Finset.{u_2} ι) (SetLike.instMembership.{u_2, u_2} (Finset.{u_2} ι) ι (Finset.instSetLike.{u_2} ι)) (EmptyCollection.emptyCollection.{u_2} (Finset.{u_2} ι) (Finset.instEmptyCollection.{u_2} ι)) i) => s i))) (EmptyCollection.emptyCollection.{u_3} (Set.{u_3} X) (Set.instEmptyCollection.{u_3} X))","typeFull":"∀ {ι : Type u_2} {X : Type u_3} {s : ι → Set X}, ⋃ i ∈ ∅, s i = ∅","typeReadable":"∀ {ι : Type u_2} {X : Type u_3} {s : ι → Set X}, ⋃ i ∈ ∅, s i = ∅","typeReferences":[["Finset","instSetLike"],["SetLike","instMembership"],["Finset"],["Set"],["Finset","instEmptyCollection"],["Membership","mem"],["Set","iUnion"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["Eq"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["Eq","trans"],["Set","iUnion_of_empty"],["Finset","instEmptyCollection"],["Membership","mem"],["Set","iUnion"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["congrArg"],["_private","Mathlib","Data","Set","Constructions",0,"Set","biUnion_empty_finset","_simp_1_1"],["funext"],["Set","iUnion_congr_Prop"],["congrFun'"],["Eq"],["Set","iUnion_empty"],["SetLike","instMembership"],["True"],["Set"],["instIsEmptyFalse"],["eq_self"],["of_eq_true"],["Iff","of_eq"],["Eq","refl"],["False"]]},{"isProp":true,"kind":"theorem","name":["FiniteInter","finiteInter_mem"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)}, (FiniteInter.{u_1} α S) -> (forall (F : Finset.{u_1} (Set.{u_1} α)), (HasSubset.Subset.{u_1} (Set.{u_1} (Set.{u_1} α)) (Set.instHasSubset.{u_1} (Set.{u_1} α)) (SetLike.coe.{u_1, u_1} (Finset.{u_1} (Set.{u_1} α)) (Set.{u_1} α) (Finset.instSetLike.{u_1} (Set.{u_1} α)) F) S) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Set.sInter.{u_1} α (SetLike.coe.{u_1, u_1} (Finset.{u_1} (Set.{u_1} α)) (Set.{u_1} α) (Finset.instSetLike.{u_1} (Set.{u_1} α)) F))))","typeFull":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter S → ∀ (F : Finset (Set α)), ↑F ⊆ S → ⋂₀ ↑F ∈ S","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter S → ∀ (F : Finset (Set α)), ↑F ⊆ S → ⋂₀ ↑F ∈ S","typeReferences":[["Set","instHasSubset"],["Finset","instSetLike"],["Set","sInter"],["SetLike","coe"],["Finset"],["HasSubset","Subset"],["Set"],["Membership","mem"],["FiniteInter"],["Set","instMembership"]],"valueReferences":[["Finset","induction_on"],["Finset","instSetLike"],["Set","sInter_empty"],["FiniteInter","univ_mem"],["Finset"],["Finset","instInsert"],["Eq","trans"],["Inter","inter"],["Set","decidableEq"],["Membership","mem"],["Finset","instEmptyCollection"],["eq_true"],["Set","instInsert"],["EmptyCollection","emptyCollection"],["Set","instEmptyCollection"],["congrArg"],["Finset","coe_empty"],["Finset","coe_insert"],["Set","sInter_insert"],["Eq"],["Set","instInter"],["FiniteInter","inter_mem"],["Set","sInter"],["True"],["Set"],["Insert","insert"],["Set","instMembership"],["Set","instHasSubset"],["Set","univ"],["SetLike","coe"],["of_eq_true"],["HasSubset","Subset"],["id"],["Eq","mpr"],["Finset","mem_insert_self"],["Finset","mem_insert_of_mem"]]},{"isProp":true,"kind":"constructor","name":["FiniteInter","finiteInterClosure","below","univ"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop}, FiniteInter.finiteInterClosure.below.{u_1} α S motive (Set.univ.{u_1} α) (FiniteInter.finiteInterClosure.univ.{u_1} α S)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop},\n  FiniteInter.finiteInterClosure.below ⋯","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop},\n  FiniteInter.finiteInterClosure.below ⋯","typeReferences":[["Set","univ"],["FiniteInter","finiteInterClosure"],["Set"],["FiniteInter","finiteInterClosure","univ"],["FiniteInter","finiteInterClosure","below"]],"valueReferences":null},{"isProp":true,"kind":"recursor","name":["FiniteInter","finiteInterClosure","below","rec"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {motive_1 : forall {a._@._internal._hyg.0 : Set.{u_1} α} (t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0), (FiniteInter.finiteInterClosure.below.{u_1} α S motive a._@._internal._hyg.0 t) -> Prop}, (forall {s : Set.{u_1} α} (a._@._internal._hyg.0 : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s), motive_1 s (FiniteInter.finiteInterClosure.basic.{u_1} α S s a._@._internal._hyg.0) (FiniteInter.finiteInterClosure.below.basic.{u_1} α S motive s a._@._internal._hyg.0)) -> (motive_1 (Set.univ.{u_1} α) (FiniteInter.finiteInterClosure.univ.{u_1} α S) (FiniteInter.finiteInterClosure.below.univ.{u_1} α S motive)) -> (forall {s : Set.{u_1} α} {t : Set.{u_1} α} (a._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S s) (a_1._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S t) (ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.46 : FiniteInter.finiteInterClosure.below.{u_1} α S motive s a._@._internal._hyg.0) (a_ih._@._internal._hyg.0 : motive s a._@._internal._hyg.0) (ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.47 : FiniteInter.finiteInterClosure.below.{u_1} α S motive t a_1._@._internal._hyg.0) (a_ih_1._@._internal._hyg.0 : motive t a_1._@._internal._hyg.0), (motive_1 s a._@._internal._hyg.0 ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.46) -> (motive_1 t a_1._@._internal._hyg.0 ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.47) -> (motive_1 (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t) (FiniteInter.finiteInterClosure.inter.{u_1} α S s t a._@._internal._hyg.0 a_1._@._internal._hyg.0) (FiniteInter.finiteInterClosure.below.inter.{u_1} α S motive s t a._@._internal._hyg.0 a_1._@._internal._hyg.0 ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.46 a_ih._@._internal._hyg.0 ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.47 a_ih_1._@._internal._hyg.0))) -> (forall {a._@._internal._hyg.0 : Set.{u_1} α} {t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0} (t_1 : FiniteInter.finiteInterClosure.below.{u_1} α S motive a._@._internal._hyg.0 t), motive_1 a._@._internal._hyg.0 t t_1)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop}\n  {motive_1 : {a : Set α} → (t : FiniteInter.finiteInterClosure S a) → FiniteInter.finiteInterClosure.below t → Prop},\n  (∀ {s : Set α} (a : s ∈ S), motive_1 ⋯ ⋯) →\n    motive_1 ⋯ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t)\n          (ih : FiniteInter.finiteInterClosure.below a) (a_ih : motive s a)\n          (ih_1 : FiniteInter.finiteInterClosure.below a_1) (a_ih_1 : motive t a_1),\n          motive_1 a ih → motive_1 a_1 ih_1 → motive_1 ⋯ ⋯) →\n        ∀ {a : Set α} {t : FiniteInter.finiteInterClosure S a} (t_1 : FiniteInter.finiteInterClosure.below t),\n          motive_1 t t_1","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop}\n  {motive_1 : {a : Set α} → (t : FiniteInter.finiteInterClosure S a) → FiniteInter.finiteInterClosure.below t → Prop},\n  (∀ {s : Set α} (a : s ∈ S), motive_1 ⋯ ⋯) →\n    motive_1 ⋯ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t)\n          (ih : FiniteInter.finiteInterClosure.below a) (a_ih : motive s a)\n          (ih_1 : FiniteInter.finiteInterClosure.below a_1) (a_ih_1 : motive t a_1),\n          motive_1 a ih → motive_1 a_1 ih_1 → motive_1 ⋯ ⋯) →\n        ∀ {a : Set α} {t : FiniteInter.finiteInterClosure S a} (t_1 : FiniteInter.finiteInterClosure.below t),\n          motive_1 t t_1","typeReferences":[["FiniteInter","finiteInterClosure","basic"],["Set"],["Membership","mem"],["Inter","inter"],["Set","instMembership"],["FiniteInter","finiteInterClosure","below","univ"],["Set","univ"],["FiniteInter","finiteInterClosure","below","inter"],["FiniteInter","finiteInterClosure"],["FiniteInter","finiteInterClosure","below","basic"],["FiniteInter","finiteInterClosure","inter"],["FiniteInter","finiteInterClosure","below"],["FiniteInter","finiteInterClosure","univ"],["Set","instInter"]],"valueReferences":null},{"isProp":true,"kind":"definition","name":["FiniteInter","finiteInterClosure","recOn"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {a._@._internal._hyg.0 : Set.{u_1} α} (t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0), (forall {s : Set.{u_1} α} (a._@._internal._hyg.0 : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s), motive s (FiniteInter.finiteInterClosure.basic.{u_1} α S s a._@._internal._hyg.0)) -> (motive (Set.univ.{u_1} α) (FiniteInter.finiteInterClosure.univ.{u_1} α S)) -> (forall {s : Set.{u_1} α} {t : Set.{u_1} α} (a._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S s) (a_1._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S t), (motive s a._@._internal._hyg.0) -> (motive t a_1._@._internal._hyg.0) -> (motive (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t) (FiniteInter.finiteInterClosure.inter.{u_1} α S s t a._@._internal._hyg.0 a_1._@._internal._hyg.0))) -> (motive a._@._internal._hyg.0 t)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {a : Set α}\n  (t : FiniteInter.finiteInterClosure S a),\n  (∀ {s : Set α} (a : s ∈ S), motive s ⋯) →\n    motive Set.univ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n          motive s a → motive t a_1 → motive (s ∩ t) ⋯) →\n        motive a t","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {a : Set α}\n  (t : FiniteInter.finiteInterClosure S a),\n  (∀ {s : Set α} (a : s ∈ S), motive s ⋯) →\n    motive Set.univ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n          motive s a → motive t a_1 → motive (s ∩ t) ⋯) →\n        motive a t","typeReferences":[["Set","univ"],["FiniteInter","finiteInterClosure","basic"],["FiniteInter","finiteInterClosure"],["Set"],["Inter","inter"],["Membership","mem"],["FiniteInter","finiteInterClosure","inter"],["Set","instInter"],["FiniteInter","finiteInterClosure","univ"],["Set","instMembership"]],"valueReferences":[["FiniteInter","finiteInterClosure","rec"]]},{"isProp":true,"kind":"theorem","name":["FiniteInter","mk₂"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)}, (forall {{s : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (forall {{t : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S t) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t)))) -> (FiniteInter.{u_1} α (Insert.insert.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instInsert.{u_1} (Set.{u_1} α)) (Set.univ.{u_1} α) S))","typeFull":"∀ {α : Type u_1} {S : Set (Set α)},\n  (∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → FiniteInter (insert Set.univ S)","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)},\n  (∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → FiniteInter (insert Set.univ S)","typeReferences":[["Set","univ"],["Set"],["Inter","inter"],["Membership","mem"],["Insert","insert"],["Set","instInsert"],["FiniteInter"],["Set","instInter"],["Set","instMembership"]],"valueReferences":[["Eq","trans"],["Eq","mp"],["Inter","inter"],["Membership","mem"],["eq_true"],["or_true"],["Set","instInsert"],["congrArg"],["Or"],["congr"],["Eq","symm"],["congrFun'"],["Eq"],["Eq","ndrec"],["Set","inter_univ"],["Set","instInter"],["Set","inter_self"],["true_or"],["True"],["Set"],["Insert","insert"],["Set","instMembership"],["Set","univ"],["Or","casesOn"],["eq_self"],["of_eq_true"],["Set","univ_inter"],["FiniteInter","mk"],["Set","mem_insert_iff","_simp_1"],["id"],["Eq","mpr"],["Set","mem_insert"]]},{"isProp":true,"kind":"constructor","name":["FiniteInter","finiteInterClosure","basic"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {s : Set.{u_1} α}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (FiniteInter.finiteInterClosure.{u_1} α S s)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {s : Set α}, s ∈ S → FiniteInter.finiteInterClosure S s","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {s : Set α}, s ∈ S → FiniteInter.finiteInterClosure S s","typeReferences":[["FiniteInter","finiteInterClosure"],["Set"],["Membership","mem"],["Set","instMembership"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Data","Set","Constructions",0,"Set","biUnion_empty_finset","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} (a : α), Eq.{1} Prop (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (Finset.instEmptyCollection.{u_1} α)) a) False","typeFull":"∀ {α : Type u_1} (a : α), (a ∈ ∅) = False","typeReadable":"∀ {α : Type u_1} (a : α), (a ∈ ∅) = False","typeReferences":[["Finset","instSetLike"],["SetLike","instMembership"],["Finset"],["Finset","instEmptyCollection"],["Membership","mem"],["False"],["EmptyCollection","emptyCollection"],["Eq"]],"valueReferences":[["Finset","instSetLike"],["SetLike","instMembership"],["Finset"],["eq_false"],["Finset","instEmptyCollection"],["Membership","mem"],["EmptyCollection","emptyCollection"],["Finset","notMem_empty"]]},{"isProp":false,"kind":"definition","name":["FiniteInter","casesOn"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : (FiniteInter.{u_1} α S) -> Sort.{u}} (t : FiniteInter.{u_1} α S), (forall (univ_mem : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Set.univ.{u_1} α)) (inter_mem : forall {{s : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (forall {{t : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S t) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t)))), motive (FiniteInter.mk.{u_1} α S univ_mem inter_mem)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : FiniteInter S → Sort u} →\n      (t : FiniteInter S) →\n        ((univ_mem : Set.univ ∈ S) → (inter_mem : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → motive ⋯) →\n          motive t","typeReadable":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : FiniteInter S → Sort u} →\n      (t : FiniteInter S) →\n        ((univ_mem : Set.univ ∈ S) → (inter_mem : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → motive ⋯) →\n          motive t","typeReferences":[["Set","univ"],["Set"],["FiniteInter","mk"],["Inter","inter"],["Membership","mem"],["FiniteInter"],["Set","instInter"],["Set","instMembership"]],"valueReferences":[["FiniteInter","rec"]]},{"isProp":false,"kind":"inductive","name":["FiniteInter"],"typeFallback":"forall {α : Type.{u_1}}, (Set.{u_1} (Set.{u_1} α)) -> Prop","typeFull":"{α : Type u_1} → Set (Set α) → Prop","typeReadable":"{α : Type u_1} → Set (Set α) → Prop","typeReferences":[["Set"]],"valueReferences":null},{"isProp":true,"kind":"constructor","name":["FiniteInter","finiteInterClosure","univ"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)}, FiniteInter.finiteInterClosure.{u_1} α S (Set.univ.{u_1} α)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter.finiteInterClosure S Set.univ","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter.finiteInterClosure S Set.univ","typeReferences":[["Set","univ"],["FiniteInter","finiteInterClosure"],["Set"]],"valueReferences":null},{"isProp":true,"kind":"definition","name":["FiniteInter","finiteInterClosure","below","casesOn"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {motive_1 : forall {a._@._internal._hyg.0 : Set.{u_1} α} (t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0), (FiniteInter.finiteInterClosure.below.{u_1} α S motive a._@._internal._hyg.0 t) -> Prop} {a._@._internal._hyg.0 : Set.{u_1} α} {t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0} (t_1 : FiniteInter.finiteInterClosure.below.{u_1} α S motive a._@._internal._hyg.0 t), (forall {s : Set.{u_1} α} (a._@._internal._hyg.0 : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s), motive_1 s (FiniteInter.finiteInterClosure.basic.{u_1} α S s a._@._internal._hyg.0) (FiniteInter.finiteInterClosure.below.basic.{u_1} α S motive s a._@._internal._hyg.0)) -> (motive_1 (Set.univ.{u_1} α) (FiniteInter.finiteInterClosure.univ.{u_1} α S) (FiniteInter.finiteInterClosure.below.univ.{u_1} α S motive)) -> (forall {s : Set.{u_1} α} {t : Set.{u_1} α} (a._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S s) (a_1._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S t) (ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.46 : FiniteInter.finiteInterClosure.below.{u_1} α S motive s a._@._internal._hyg.0) (a_ih._@._internal._hyg.0 : motive s a._@._internal._hyg.0) (ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.47 : FiniteInter.finiteInterClosure.below.{u_1} α S motive t a_1._@._internal._hyg.0) (a_ih_1._@._internal._hyg.0 : motive t a_1._@._internal._hyg.0), motive_1 (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t) (FiniteInter.finiteInterClosure.inter.{u_1} α S s t a._@._internal._hyg.0 a_1._@._internal._hyg.0) (FiniteInter.finiteInterClosure.below.inter.{u_1} α S motive s t a._@._internal._hyg.0 a_1._@._internal._hyg.0 ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.46 a_ih._@._internal._hyg.0 ih._@.Mathlib.Data.Set.Constructions.2405406905._hygCtx._hyg.47 a_ih_1._@._internal._hyg.0)) -> (motive_1 a._@._internal._hyg.0 t t_1)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop}\n  {motive_1 : {a : Set α} → (t : FiniteInter.finiteInterClosure S a) → FiniteInter.finiteInterClosure.below t → Prop}\n  {a : Set α} {t : FiniteInter.finiteInterClosure S a} (t_1 : FiniteInter.finiteInterClosure.below t),\n  (∀ {s : Set α} (a : s ∈ S), motive_1 ⋯ ⋯) →\n    motive_1 ⋯ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t)\n          (ih : FiniteInter.finiteInterClosure.below a) (a_ih : motive s a)\n          (ih_1 : FiniteInter.finiteInterClosure.below a_1) (a_ih_1 : motive t a_1), motive_1 ⋯ ⋯) →\n        motive_1 t t_1","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop}\n  {motive_1 : {a : Set α} → (t : FiniteInter.finiteInterClosure S a) → FiniteInter.finiteInterClosure.below t → Prop}\n  {a : Set α} {t : FiniteInter.finiteInterClosure S a} (t_1 : FiniteInter.finiteInterClosure.below t),\n  (∀ {s : Set α} (a : s ∈ S), motive_1 ⋯ ⋯) →\n    motive_1 ⋯ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t)\n          (ih : FiniteInter.finiteInterClosure.below a) (a_ih : motive s a)\n          (ih_1 : FiniteInter.finiteInterClosure.below a_1) (a_ih_1 : motive t a_1), motive_1 ⋯ ⋯) →\n        motive_1 t t_1","typeReferences":[["FiniteInter","finiteInterClosure","basic"],["Set"],["Membership","mem"],["Inter","inter"],["Set","instMembership"],["FiniteInter","finiteInterClosure","below","univ"],["Set","univ"],["FiniteInter","finiteInterClosure","below","inter"],["FiniteInter","finiteInterClosure"],["FiniteInter","finiteInterClosure","below","basic"],["FiniteInter","finiteInterClosure","inter"],["FiniteInter","finiteInterClosure","below"],["FiniteInter","finiteInterClosure","univ"],["Set","instInter"]],"valueReferences":[["FiniteInter","finiteInterClosure","below","rec"]]},{"isProp":false,"kind":"inductive","name":["FiniteInter","finiteInterClosure","below"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {a._@._internal._hyg.0 : Set.{u_1} α}, (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop","typeFull":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} →\n      {a : Set α} → FiniteInter.finiteInterClosure S a → Prop","typeReadable":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} →\n      {a : Set α} → FiniteInter.finiteInterClosure S a → Prop","typeReferences":[["FiniteInter","finiteInterClosure"],["Set"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["FiniteInter","univ_mem"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)}, (FiniteInter.{u_1} α S) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Set.univ.{u_1} α))","typeFull":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter S → Set.univ ∈ S","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter S → Set.univ ∈ S","typeReferences":[["Set","univ"],["Set"],["Membership","mem"],["FiniteInter"],["Set","instMembership"]],"valueReferences":[]},{"isProp":true,"kind":"constructor","name":["FiniteInter","finiteInterClosure","inter"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {s : Set.{u_1} α} {t : Set.{u_1} α}, (FiniteInter.finiteInterClosure.{u_1} α S s) -> (FiniteInter.finiteInterClosure.{u_1} α S t) -> (FiniteInter.finiteInterClosure.{u_1} α S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t))","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {s t : Set α},\n  FiniteInter.finiteInterClosure S s → FiniteInter.finiteInterClosure S t → FiniteInter.finiteInterClosure S (s ∩ t)","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {s t : Set α},\n  FiniteInter.finiteInterClosure S s → FiniteInter.finiteInterClosure S t → FiniteInter.finiteInterClosure S (s ∩ t)","typeReferences":[["FiniteInter","finiteInterClosure"],["Set"],["Inter","inter"],["Set","instInter"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["FiniteInter","finiteInterClosure","brecOn"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {a._@._internal._hyg.0 : Set.{u_1} α} (t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0), (forall (a._@._internal._hyg.0 : Set.{u_1} α) (t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0), (FiniteInter.finiteInterClosure.below.{u_1} α S motive a._@._internal._hyg.0 t) -> (motive a._@._internal._hyg.0 t)) -> (motive a._@._internal._hyg.0 t)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {a : Set α}\n  (t : FiniteInter.finiteInterClosure S a),\n  (∀ (a : Set α) (t : FiniteInter.finiteInterClosure S a), FiniteInter.finiteInterClosure.below t → motive a t) →\n    motive a t","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {a : Set α}\n  (t : FiniteInter.finiteInterClosure S a),\n  (∀ (a : Set α) (t : FiniteInter.finiteInterClosure S a), FiniteInter.finiteInterClosure.below t → motive a t) →\n    motive a t","typeReferences":[["FiniteInter","finiteInterClosure"],["Set"],["FiniteInter","finiteInterClosure","below"]],"valueReferences":[["FiniteInter","finiteInterClosure","below","inter"],["FiniteInter","finiteInterClosure","rec"],["FiniteInter","finiteInterClosure","below","basic"],["FiniteInter","finiteInterClosure","below"],["FiniteInter","finiteInterClosure","below","univ"]]},{"isProp":true,"kind":"definition","name":["FiniteInter","mk","_flat_ctor"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Set.univ.{u_1} α)) -> (forall {{s : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (forall {{t : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S t) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t)))) -> (FiniteInter.{u_1} α S)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)},\n  Set.univ ∈ S → (∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → FiniteInter S","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)},\n  Set.univ ∈ S → (∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → FiniteInter S","typeReferences":[["Set","univ"],["Set"],["Inter","inter"],["Membership","mem"],["FiniteInter"],["Set","instInter"],["Set","instMembership"]],"valueReferences":[["FiniteInter","mk"]]},{"isProp":true,"kind":"constructor","name":["FiniteInter","finiteInterClosure","below","basic"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {s : Set.{u_1} α} (a._@._internal._hyg.0 : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s), FiniteInter.finiteInterClosure.below.{u_1} α S motive s (FiniteInter.finiteInterClosure.basic.{u_1} α S s a._@._internal._hyg.0)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {s : Set α}\n  (a : s ∈ S), FiniteInter.finiteInterClosure.below ⋯","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {s : Set α}\n  (a : s ∈ S), FiniteInter.finiteInterClosure.below ⋯","typeReferences":[["FiniteInter","finiteInterClosure","basic"],["FiniteInter","finiteInterClosure"],["Set"],["Membership","mem"],["FiniteInter","finiteInterClosure","below"],["Set","instMembership"]],"valueReferences":null},{"isProp":false,"kind":"inductive","name":["FiniteInter","finiteInterClosure"],"typeFallback":"forall {α : Type.{u_1}}, (Set.{u_1} (Set.{u_1} α)) -> (Set.{u_1} (Set.{u_1} α))","typeFull":"{α : Type u_1} → Set (Set α) → Set (Set α)","typeReadable":"{α : Type u_1} → Set (Set α) → Set (Set α)","typeReferences":[["Set"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["FiniteInter","finiteInterClosure_insert"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {A : Set.{u_1} α}, (FiniteInter.{u_1} α S) -> (forall (P : Set.{u_1} α), (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) (FiniteInter.finiteInterClosure.{u_1} α (Insert.insert.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instInsert.{u_1} (Set.{u_1} α)) A S)) P) -> (Or (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S P) (Exists.{succ u_1} (Set.{u_1} α) (fun (Q : Set.{u_1} α) => And (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S Q) (Eq.{succ u_1} (Set.{u_1} α) P (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) A Q))))))","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {A : Set α},\n  FiniteInter S → ∀ P ∈ FiniteInter.finiteInterClosure (insert A S), P ∈ S ∨ ∃ Q ∈ S, P = A ∩ Q","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {A : Set α},\n  FiniteInter S → ∀ P ∈ FiniteInter.finiteInterClosure (insert A S), P ∈ S ∨ ∃ Q ∈ S, P = A ∩ Q","typeReferences":[["Exists"],["Set"],["Membership","mem"],["Inter","inter"],["Insert","insert"],["And"],["FiniteInter"],["Set","instInsert"],["Set","instMembership"],["FiniteInter","finiteInterClosure"],["Or"],["Eq"],["Set","instInter"]],"valueReferences":[["Or","inr"],["FiniteInter","univ_mem"],["Eq","trans"],["Inter","inter"],["Membership","mem"],["eq_true"],["Exists","intro"],["Set","instInsert"],["Iff","intro"],["congrArg"],["And","intro"],["Or"],["Or","inl"],["congr"],["Eq","symm"],["congrFun'"],["Eq","ndrec"],["Eq"],["Set","instInter"],["Set","inter_univ"],["Set","inter_assoc"],["FiniteInter","inter_mem"],["Exists"],["True"],["Set"],["And"],["Insert","insert"],["Set","inter_comm"],["Set","instMembership"],["Exists","casesOn"],["Or","casesOn"],["Set","univ"],["implies_true"],["eq_self"],["and_self"],["FiniteInter","finiteInterClosure"],["of_eq_true"],["FiniteInter","finiteInterClosure","rec"],["Eq","refl"],["_private","Mathlib","Data","Set","Constructions",0,"FiniteInter","finiteInterClosure_insert","_simp_1_1"],["id"],["Set","ext"],["Eq","mpr"],["implies_congr_ctx"],["Set","mem_inter_iff","_simp_1"],["And","casesOn"]]},{"isProp":false,"kind":"recursor","name":["FiniteInter","rec"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : (FiniteInter.{u_1} α S) -> Sort.{u}}, (forall (univ_mem : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Set.univ.{u_1} α)) (inter_mem : forall {{s : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (forall {{t : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S t) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t)))), motive (FiniteInter.mk.{u_1} α S univ_mem inter_mem)) -> (forall (t : FiniteInter.{u_1} α S), motive t)","typeFull":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : FiniteInter S → Sort u} →\n      ((univ_mem : Set.univ ∈ S) → (inter_mem : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → motive ⋯) →\n        (t : FiniteInter S) → motive t","typeReadable":"{α : Type u_1} →\n  {S : Set (Set α)} →\n    {motive : FiniteInter S → Sort u} →\n      ((univ_mem : Set.univ ∈ S) → (inter_mem : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → motive ⋯) →\n        (t : FiniteInter S) → motive t","typeReferences":[["Set","univ"],["Set"],["FiniteInter","mk"],["Inter","inter"],["Membership","mem"],["FiniteInter"],["Set","instInter"],["Set","instMembership"]],"valueReferences":null},{"isProp":true,"kind":"definition","name":["FiniteInter","finiteInterClosure","casesOn"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {a._@._internal._hyg.0 : Set.{u_1} α} (t : FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0), (forall {s : Set.{u_1} α} (a._@._internal._hyg.0 : Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s), motive s (FiniteInter.finiteInterClosure.basic.{u_1} α S s a._@._internal._hyg.0)) -> (motive (Set.univ.{u_1} α) (FiniteInter.finiteInterClosure.univ.{u_1} α S)) -> (forall {s : Set.{u_1} α} {t : Set.{u_1} α} (a._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S s) (a_1._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S t), motive (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t) (FiniteInter.finiteInterClosure.inter.{u_1} α S s t a._@._internal._hyg.0 a_1._@._internal._hyg.0)) -> (motive a._@._internal._hyg.0 t)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {a : Set α}\n  (t : FiniteInter.finiteInterClosure S a),\n  (∀ {s : Set α} (a : s ∈ S), motive s ⋯) →\n    motive Set.univ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n          motive (s ∩ t) ⋯) →\n        motive a t","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {a : Set α}\n  (t : FiniteInter.finiteInterClosure S a),\n  (∀ {s : Set α} (a : s ∈ S), motive s ⋯) →\n    motive Set.univ ⋯ →\n      (∀ {s t : Set α} (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n          motive (s ∩ t) ⋯) →\n        motive a t","typeReferences":[["Set","univ"],["FiniteInter","finiteInterClosure","basic"],["FiniteInter","finiteInterClosure"],["Set"],["Inter","inter"],["Membership","mem"],["FiniteInter","finiteInterClosure","inter"],["Set","instInter"],["FiniteInter","finiteInterClosure","univ"],["Set","instMembership"]],"valueReferences":[["FiniteInter","finiteInterClosure","rec"]]},{"isProp":true,"kind":"constructor","name":["FiniteInter","finiteInterClosure","below","inter"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)} {motive : forall (a._@._internal._hyg.0 : Set.{u_1} α), (FiniteInter.finiteInterClosure.{u_1} α S a._@._internal._hyg.0) -> Prop} {s : Set.{u_1} α} {t : Set.{u_1} α} (a._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S s) (a_1._@._internal._hyg.0 : FiniteInter.finiteInterClosure.{u_1} α S t), (FiniteInter.finiteInterClosure.below.{u_1} α S motive s a._@._internal._hyg.0) -> (motive s a._@._internal._hyg.0) -> (FiniteInter.finiteInterClosure.below.{u_1} α S motive t a_1._@._internal._hyg.0) -> (motive t a_1._@._internal._hyg.0) -> (FiniteInter.finiteInterClosure.below.{u_1} α S motive (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t) (FiniteInter.finiteInterClosure.inter.{u_1} α S s t a._@._internal._hyg.0 a_1._@._internal._hyg.0))","typeFull":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {s t : Set α}\n  (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n  FiniteInter.finiteInterClosure.below a →\n    motive s a → FiniteInter.finiteInterClosure.below a_1 → motive t a_1 → FiniteInter.finiteInterClosure.below ⋯","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)} {motive : (a : Set α) → FiniteInter.finiteInterClosure S a → Prop} {s t : Set α}\n  (a : FiniteInter.finiteInterClosure S s) (a_1 : FiniteInter.finiteInterClosure S t),\n  FiniteInter.finiteInterClosure.below a →\n    motive s a → FiniteInter.finiteInterClosure.below a_1 → motive t a_1 → FiniteInter.finiteInterClosure.below ⋯","typeReferences":[["FiniteInter","finiteInterClosure"],["Set"],["Inter","inter"],["FiniteInter","finiteInterClosure","inter"],["Set","instInter"],["FiniteInter","finiteInterClosure","below"]],"valueReferences":null},{"isProp":true,"kind":"constructor","name":["FiniteInter","mk"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Set.univ.{u_1} α)) -> (forall {{s : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (forall {{t : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S t) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t)))) -> (FiniteInter.{u_1} α S)","typeFull":"∀ {α : Type u_1} {S : Set (Set α)},\n  Set.univ ∈ S → (∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → FiniteInter S","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)},\n  Set.univ ∈ S → (∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) → FiniteInter S","typeReferences":[["Set","univ"],["Set"],["Inter","inter"],["Membership","mem"],["FiniteInter"],["Set","instInter"],["Set","instMembership"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["FiniteInter","inter_mem"],"typeFallback":"forall {α : Type.{u_1}} {S : Set.{u_1} (Set.{u_1} α)}, (FiniteInter.{u_1} α S) -> (forall {{s : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S s) -> (forall {{t : Set.{u_1} α}}, (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S t) -> (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Set.{u_1} (Set.{u_1} α)) (Set.instMembership.{u_1} (Set.{u_1} α)) S (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s t))))","typeFull":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter S → ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S","typeReadable":"∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter S → ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S","typeReferences":[["Set"],["Inter","inter"],["Membership","mem"],["FiniteInter"],["Set","instInter"],["Set","instMembership"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["FiniteInter","finiteInterClosure_finiteInter"],"typeFallback":"forall {α : Type.{u_1}} (S : Set.{u_1} (Set.{u_1} α)), FiniteInter.{u_1} α (FiniteInter.finiteInterClosure.{u_1} α S)","typeFull":"∀ {α : Type u_1} (S : Set (Set α)), FiniteInter (FiniteInter.finiteInterClosure S)","typeReadable":"∀ {α : Type u_1} (S : Set (Set α)), FiniteInter (FiniteInter.finiteInterClosure S)","typeReferences":[["FiniteInter","finiteInterClosure"],["Set"],["FiniteInter"]],"valueReferences":[["FiniteInter","finiteInterClosure"],["FiniteInter","mk"],["FiniteInter","finiteInterClosure","inter"],["FiniteInter","finiteInterClosure","univ"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Dynamics.Ergodic.Conservative.sym.json

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["MeasureTheory","Conservative","casesOn"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14} {motive : (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ) -> Sort.{u}} (t : MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ), (forall (toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ μ) (exists_mem_iterate_mem' : forall {{s : Set.{u_1} α}}, (MeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 s) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Exists.{1} Nat (fun (m : Nat) => And (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x))))))), motive (MeasureTheory.Conservative.mk.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ toQuasiMeasurePreserving exists_mem_iterate_mem')) -> (motive t)","typeFull":"{α : Type u_1} →\n  [inst : MeasurableSpace α] →\n    {f : α → α} →\n      {μ : MeasureTheory.Measure α} →\n        {motive : MeasureTheory.Conservative f μ → Sort u} →\n          (t : MeasureTheory.Conservative f μ) →\n            ((toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) →\n                (exists_mem_iterate_mem' :\n                    ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) →\n                  motive ⋯) →\n              motive t","typeReadable":"{α : Type u_1} →\n  [inst : MeasurableSpace α] →\n    {f : α → α} →\n      {μ : MeasureTheory.Measure α} →\n        {motive : MeasureTheory.Conservative f μ → Sort u} →\n          (t : MeasureTheory.Conservative f μ) →\n            ((toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) →\n                (exists_mem_iterate_mem' :\n                    ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) →\n                  motive ⋯) →\n              motive t","typeReferences":[["Exists"],["Set"],["instZeroENNReal"],["Membership","mem"],["And"],["MeasureTheory","Measure","QuasiMeasurePreserving"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Conservative","mk"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["instOfNatNat"],["MeasurableSpace"],["MeasurableSet"],["Ne"],["Zero","toOfNat0"]],"valueReferences":[["MeasureTheory","Conservative","rec"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","exists_mem_iterate_mem'"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ) -> (forall {{s : Set.{u_1} α}}, (MeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 s) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Exists.{1} Nat (fun (m : Nat) => And (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x)))))))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s","typeReferences":[["Exists"],["Set"],["instZeroENNReal"],["Membership","mem"],["And"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["instOfNatNat"],["MeasurableSpace"],["MeasurableSet"],["Ne"],["Zero","toOfNat0"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","toQuasiMeasurePreserving"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ) -> (MeasureTheory.Measure.QuasiMeasurePreserving.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ μ)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → MeasureTheory.Measure.QuasiMeasurePreserving f μ μ","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → MeasureTheory.Measure.QuasiMeasurePreserving f μ μ","typeReferences":[["MeasureTheory","Measure"],["MeasureTheory","Measure","QuasiMeasurePreserving"],["MeasurableSpace"],["MeasureTheory","Conservative"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","measureRestrict"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2764716636._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2764716636._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2764716636._hygCtx._hyg.3 f μ) -> (Set.MapsTo.{u_1, u_1} α α f s s) -> (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2764716636._hygCtx._hyg.3 f (MeasureTheory.Measure.restrict.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2764716636._hygCtx._hyg.3 μ s))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → Set.MapsTo f s s → MeasureTheory.Conservative f (μ.restrict s)","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → Set.MapsTo f s s → MeasureTheory.Conservative f (μ.restrict s)","typeReferences":[["MeasureTheory","Measure"],["MeasureTheory","Measure","restrict"],["Set","MapsTo"],["Set"],["MeasurableSpace"],["MeasureTheory","Conservative"]],"valueReferences":[["MeasureTheory","Conservative","toQuasiMeasurePreserving"],["MeasureTheory","Measure","restrict"],["MeasureTheory","Conservative","of_absolutelyContinuous"],["MeasureTheory","Measure","absolutelyContinuous_of_le"],["MeasureTheory","Measure","restrict_le_self"],["MeasureTheory","Measure","QuasiMeasurePreserving","restrict"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","exists_gt_measure_inter_ne_zero","match_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3} (N : Nat) (motive : (Exists.{1} Nat (fun (x : Nat) => And (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (Set.preimage.{u_1, u_1} α α (Nat.iterate.{succ u_1} α f x) s))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (LT.lt.{0} Nat (Preorder.toLT.{0} Nat Nat.instPreorder) N x))) -> Prop) (x._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.100 : Exists.{1} Nat (fun (x : Nat) => And (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (Set.preimage.{u_1, u_1} α α (Nat.iterate.{succ u_1} α f x) s))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (LT.lt.{0} Nat (Preorder.toLT.{0} Nat Nat.instPreorder) N x))), (forall (m : Nat) (hm : Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (Set.preimage.{u_1, u_1} α α (Nat.iterate.{succ u_1} α f m) s))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (hmN : LT.lt.{0} Nat (Preorder.toLT.{0} Nat Nat.instPreorder) N m), motive (Exists.intro.{1} Nat (fun (x : Nat) => And (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (Set.preimage.{u_1, u_1} α α (Nat.iterate.{succ u_1} α f x) s))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (LT.lt.{0} Nat (Preorder.toLT.{0} Nat Nat.instPreorder) N x)) m (And.intro (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (Set.preimage.{u_1, u_1} α α (Nat.iterate.{succ u_1} α f m) s))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (LT.lt.{0} Nat (Preorder.toLT.{0} Nat Nat.instPreorder) N m) hm hmN))) -> (motive x._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.100)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (N : ℕ)\n  (motive : (∃ x, μ (s ∩ f^[x] ⁻¹' s) ≠ 0 ∧ N < x) → Prop) (x : ∃ x, μ (s ∩ f^[x] ⁻¹' s) ≠ 0 ∧ N < x),\n  (∀ (m : ℕ) (hm : μ (s ∩ f^[m] ⁻¹' s) ≠ 0) (hmN : N < m), motive ⋯) → motive x","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (N : ℕ)\n  (motive : (∃ x, μ (s ∩ f^[x] ⁻¹' s) ≠ 0 ∧ N < x) → Prop) (x : ∃ x, μ (s ∩ f^[x] ⁻¹' s) ≠ 0 ∧ N < x),\n  (∀ (m : ℕ) (hm : μ (s ∩ f^[m] ⁻¹' s) ≠ 0) (hmN : N < m), motive ⋯) → motive x","typeReferences":[["Set","preimage"],["Exists"],["instZeroENNReal"],["Set"],["Inter","inter"],["Preorder","toLT"],["And"],["Exists","intro"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Nat","instPreorder"],["LT","lt"],["And","intro"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["Zero","toOfNat0"],["MeasurableSpace"],["Ne"],["Set","instInter"]],"valueReferences":[["Set","preimage"],["instZeroENNReal"],["Set"],["Inter","inter"],["Preorder","toLT"],["And"],["Exists","intro"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Nat","instPreorder"],["LT","lt"],["Exists","casesOn"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["Zero","toOfNat0"],["Ne"],["Set","instInter"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","measure_mem_forall_ge_image_notMem_eq_zero","_simp_1_1"],"typeFallback":"forall {α : Type.{u_3}} (p : α -> Prop), Eq.{succ u_3} (Set.{u_3} α) (setOf.{u_3} α (fun (a : α) => Not (p a))) (Compl.compl.{u_3} (Set.{u_3} α) (Set.instCompl.{u_3} α) (setOf.{u_3} α (fun (a : α) => p a)))","typeFull":"∀ {α : Type u_3} (p : α → Prop), {a | ¬p a} = {a | p a}ᶜ","typeReadable":"∀ {α : Type u_3} (p : α → Prop), {a | ¬p a} = {a | p a}ᶜ","typeReferences":[["Not"],["Set"],["Compl","compl"],["Eq"],["Set","instCompl"],["setOf"]],"valueReferences":[["Not"],["Set"],["Compl","compl"],["Set","compl_setOf"],["Eq","symm"],["setOf"],["Set","instCompl"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","id","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Logic.Nontrivial.Defs.637383090._hygCtx._hyg.4 : Nontrivial.{u_1} α] (x : α), Eq.{1} Prop (Exists.{succ u_1} α (fun (y : α) => Ne.{succ u_1} α y x)) True","typeFull":"∀ {α : Type u_1} [Nontrivial α] (x : α), (∃ y, y ≠ x) = True","typeReadable":"∀ {α : Type u_1} [Nontrivial α] (x : α), (∃ y, y ≠ x) = True","typeReferences":[["Exists"],["True"],["Nontrivial"],["Ne"],["Eq"]],"valueReferences":[["Exists"],["exists_ne"],["eq_true"],["Ne"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","of_absolutelyContinuous"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3} {ν : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3 f μ) -> (MeasureTheory.Measure.AbsolutelyContinuous.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3 ν μ) -> (MeasureTheory.Measure.QuasiMeasurePreserving.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3 inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3 f ν ν) -> (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3198101372._hygCtx._hyg.3 f ν)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ ν : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    ν.AbsolutelyContinuous μ → MeasureTheory.Measure.QuasiMeasurePreserving f ν ν → MeasureTheory.Conservative f ν","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ ν : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    ν.AbsolutelyContinuous μ → MeasureTheory.Measure.QuasiMeasurePreserving f ν ν → MeasureTheory.Conservative f ν","typeReferences":[["MeasureTheory","Measure"],["MeasureTheory","Measure","QuasiMeasurePreserving"],["MeasurableSpace"],["MeasureTheory","Measure","AbsolutelyContinuous"],["MeasureTheory","Conservative"]],"valueReferences":[["MeasureTheory","Conservative","mk"],["MeasureTheory","Measure"],["MeasureTheory","Conservative","exists_mem_iterate_mem'"],["ENNReal"],["mt"],["instZeroENNReal"],["Set"],["Zero","toOfNat0"],["Eq"],["MeasureTheory","Measure","instFunLike"],["DFunLike","coe"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","congr_ae"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3} {ν : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3 f μ) -> (Eq.{succ u_1} (Filter.{u_1} α) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3) μ) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3) ν)) -> (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.735651560._hygCtx._hyg.3 f ν)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ ν : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → MeasureTheory.ae μ = MeasureTheory.ae ν → MeasureTheory.Conservative f ν","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ ν : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → MeasureTheory.ae μ = MeasureTheory.ae ν → MeasureTheory.Conservative f ν","typeReferences":[["MeasureTheory","Measure"],["MeasureTheory","Measure","instOuterMeasureClass"],["MeasurableSpace"],["Eq"],["MeasureTheory","Measure","instFunLike"],["MeasureTheory","ae"],["Filter"],["MeasureTheory","Conservative"]],"valueReferences":[["PartialOrder","toPreorder"],["MeasureTheory","Measure","QuasiMeasurePreserving","mono"],["LE","le","absolutelyContinuous_of_ae"],["Filter","instPartialOrder"],["Eq","ge"],["MeasureTheory","Measure","instFunLike"],["Filter"],["MeasureTheory","Measure"],["MeasureTheory","Conservative","toQuasiMeasurePreserving"],["MeasureTheory","Conservative","of_absolutelyContinuous"],["MeasureTheory","Measure","instOuterMeasureClass"],["MeasureTheory","ae"],["Eq","le"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","id"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1952539257._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] (μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1952539257._hygCtx._hyg.3), MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1952539257._hygCtx._hyg.3 (id.{succ u_1} α) μ","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α), MeasureTheory.Conservative id μ","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α), MeasureTheory.Conservative id μ","typeReferences":[["MeasureTheory","Measure"],["id"],["MeasurableSpace"],["MeasureTheory","Conservative"]],"valueReferences":[["Eq","trans"],["Membership","mem"],["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","id","_simp_1_1"],["congrArg"],["Function","iterate_id"],["instOfNatNat"],["congrFun"],["funext"],["MeasureTheory","Measure","QuasiMeasurePreserving","id"],["congrFun'"],["MeasureTheory","nonempty_of_measure_ne_zero"],["Eq"],["Not"],["Exists"],["exists_and_right","_simp_1"],["True"],["Set"],["And"],["Nat","instNontrivial"],["true_and"],["OfNat","ofNat"],["Set","instMembership"],["MeasureTheory","Conservative","mk"],["and_self"],["Nat"],["Nat","iterate"],["id"],["Ne"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","inter_frequently_image_mem_ae_eq"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1232004143._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1232004143._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1232004143._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1232004143._hygCtx._hyg.3 s μ) -> (Filter.EventuallyEq.{u_1, 0} α Prop (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1232004143._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1232004143._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1232004143._hygCtx._hyg.3) μ) (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (setOf.{u_1} α (fun (x : α) => Filter.Frequently.{0} Nat (fun (n : Nat) => Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f n x)) (Filter.atTop.{0} Nat Nat.instPreorder)))) s)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → s ∩ {x | ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s} =ᵐ[μ] s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → s ∩ {x | ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s} =ᵐ[μ] s","typeReferences":[["Set"],["Filter","Frequently"],["Membership","mem"],["Inter","inter"],["MeasureTheory","Measure","instFunLike"],["Nat","instPreorder"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["Filter","atTop"],["Nat"],["Nat","iterate"],["MeasureTheory","Measure","instOuterMeasureClass"],["Filter","EventuallyEq"],["MeasurableSpace"],["MeasureTheory","ae"],["MeasureTheory","NullMeasurableSet"],["Set","instInter"],["setOf"]],"valueReferences":[["Filter","Eventually"],["Set"],["Filter","Frequently"],["Filter","inter_eventuallyEq_left"],["Inter","inter"],["Membership","mem"],["MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem"],["MeasureTheory","Measure","instFunLike"],["Nat","instPreorder"],["Set","instMembership"],["MeasureTheory","Measure"],["Filter","atTop"],["Nat"],["Nat","iterate"],["MeasureTheory","Measure","instOuterMeasureClass"],["Iff","mpr"],["Filter","EventuallyEq"],["MeasureTheory","ae"],["Set","instInter"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","measure_inter_frequently_image_mem_eq"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3 s μ) -> (Eq.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (setOf.{u_1} α (fun (x : α) => Filter.Frequently.{0} Nat (fun (n : Nat) => Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f n x)) (Filter.atTop.{0} Nat Nat.instPreorder))))) (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3787533842._hygCtx._hyg.3) μ s))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ (s ∩ {x | ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s}) = μ s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ (s ∩ {x | ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s}) = μ s","typeReferences":[["Set"],["Filter","Frequently"],["Membership","mem"],["Inter","inter"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["Nat","instPreorder"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["Filter","atTop"],["Nat"],["ENNReal"],["Nat","iterate"],["MeasurableSpace"],["Eq"],["MeasureTheory","NullMeasurableSet"],["Set","instInter"],["setOf"]],"valueReferences":[["Set"],["Filter","Frequently"],["Inter","inter"],["Membership","mem"],["MeasureTheory","Conservative","inter_frequently_image_mem_ae_eq"],["MeasureTheory","Measure","instFunLike"],["Nat","instPreorder"],["Set","instMembership"],["MeasureTheory","Measure"],["Filter","atTop"],["Nat"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","iterate"],["Set","instInter"],["MeasureTheory","measure_congr"],["setOf"]]},{"isProp":true,"kind":"definition","name":["MeasureTheory","Conservative","mk","_flat_ctor"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14}, (Measurable.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f) -> (MeasureTheory.Measure.AbsolutelyContinuous.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 (MeasureTheory.Measure.map.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ) μ) -> (forall {{s : Set.{u_1} α}}, (MeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 s) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Exists.{1} Nat (fun (m : Nat) => And (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x))))))) -> (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  Measurable f →\n    (MeasureTheory.Measure.map f μ).AbsolutelyContinuous μ →\n      (∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) → MeasureTheory.Conservative f μ","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  Measurable f →\n    (MeasureTheory.Measure.map f μ).AbsolutelyContinuous μ →\n      (∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) → MeasureTheory.Conservative f μ","typeReferences":[["Measurable"],["Exists"],["instZeroENNReal"],["Set"],["Membership","mem"],["And"],["MeasureTheory","Measure","map"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["MeasureTheory","Conservative"],["Set","instMembership"],["MeasureTheory","Measure","AbsolutelyContinuous"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["instOfNatNat"],["MeasurableSpace"],["MeasurableSet"],["Ne"],["Zero","toOfNat0"]],"valueReferences":[["MeasureTheory","Conservative","mk"],["MeasureTheory","Measure","QuasiMeasurePreserving","mk"]]},{"isProp":false,"kind":"inductive","name":["MeasureTheory","Conservative"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α], (α -> α) -> (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) -> Prop","typeFull":"{α : Type u_1} → [inst : MeasurableSpace α] → (α → α) → MeasureTheory.Measure α → Prop","typeReadable":"{α : Type u_1} → [inst : MeasurableSpace α] → (α → α) → MeasureTheory.Measure α → Prop","typeReferences":[["MeasureTheory","Measure"],["MeasurableSpace"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1725580302._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1725580302._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1725580302._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1725580302._hygCtx._hyg.3 s μ) -> (Filter.Eventually.{u_1} α (fun (x : α) => (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) -> (Filter.Frequently.{0} Nat (fun (n : Nat) => Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f n x)) (Filter.atTop.{0} Nat Nat.instPreorder))) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1725580302._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1725580302._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1725580302._hygCtx._hyg.3) μ))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → ∀ᵐ (x : α) ∂μ, x ∈ s → ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → ∀ᵐ (x : α) ∂μ, x ∈ s → ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReferences":[["Filter","Eventually"],["Set"],["Filter","Frequently"],["Membership","mem"],["MeasureTheory","Measure","instFunLike"],["MeasureTheory","Conservative"],["Set","instMembership"],["Nat","instPreorder"],["MeasureTheory","Measure"],["Nat"],["Filter","atTop"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","iterate"],["MeasurableSpace"],["MeasureTheory","ae"],["MeasureTheory","NullMeasurableSet"]],"valueReferences":[["Filter","Eventually"],["implies_congr"],["Eq","trans"],["instZeroENNReal"],["Membership","mem"],["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem","_simp_1_2"],["Iff","mp"],["Filter","univ_mem'"],["GE","ge"],["forall_comm"],["forall_congr"],["funext"],["Classical","not_imp","_simp_1"],["not_and","_simp_1"],["Exists"],["Filter","Frequently"],["Nat","instPartialOrder"],["And"],["Classical","not_forall","_simp_1"],["MeasureTheory","Measure","instFunLike"],["instCountableNat"],["Filter"],["Set","instMembership"],["implies_true"],["Nat"],["Filter","atTop"],["instNonemptyOfInhabited"],["Eq","refl"],["id"],["Eq","mpr"],["instInhabitedNat"],["instArchimedeanNat"],["setOf"],["imp_self","_simp_1"],["Filter","mp_mem"],["instIsDirectedOrder"],["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem","_simp_1_1"],["DFunLike","coe"],["congrArg"],["Nat","instPreorder"],["MeasureTheory","Measure"],["Nat","instSemiring"],["congrFun'"],["Zero","toOfNat0"],["MeasureTheory","Conservative","measure_mem_forall_ge_image_notMem_eq_zero"],["Eq"],["Preorder","toLE"],["MeasureTheory","ae"],["ge_iff_le","_simp_1"],["propext"],["Not"],["True"],["IsStrictOrderedRing","toIsOrderedRing"],["Set"],["OfNat","ofNat"],["ENNReal"],["Classical","not_not","_simp_1"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","iterate"],["of_eq_true"],["MeasureTheory","measure_eq_zero_iff_ae_notMem"],["LE","le"],["Nat","instIsStrictOrderedRing"],["instLENat"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","ae_frequently_mem_of_mem_nhds"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.13 : TopologicalSpace.{u_1} α] [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.16 : SecondCountableTopology.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.13] [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.19 : OpensMeasurableSpace.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.13 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.3] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.3 f μ) -> (Filter.Eventually.{u_1} α (fun (x : α) => forall (s : Set.{u_1} α), (Membership.mem.{u_1, u_1} (Set.{u_1} α) (Filter.{u_1} α) (Filter.instMembership.{u_1} α) (nhds.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.13 x) s) -> (Filter.Frequently.{0} Nat (fun (n : Nat) => Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f n x)) (Filter.atTop.{0} Nat Nat.instPreorder))) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2169850569._hygCtx._hyg.3) μ))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : TopologicalSpace α] [SecondCountableTopology α]\n  [OpensMeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → ∀ᵐ (x : α) ∂μ, ∀ s ∈ nhds x, ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : TopologicalSpace α] [SecondCountableTopology α]\n  [OpensMeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → ∀ᵐ (x : α) ∂μ, ∀ s ∈ nhds x, ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReferences":[["SecondCountableTopology"],["Filter","Eventually"],["Filter","instMembership"],["Set"],["Filter","Frequently"],["Membership","mem"],["MeasureTheory","Measure","instFunLike"],["Filter"],["Nat","instPreorder"],["Set","instMembership"],["MeasureTheory","Conservative"],["TopologicalSpace"],["MeasureTheory","Measure"],["Filter","atTop"],["Nat"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","iterate"],["MeasurableSpace"],["MeasureTheory","ae"],["OpensMeasurableSpace"],["nhds"]],"valueReferences":[["Filter","Eventually"],["Membership","mem"],["Iff","mp"],["MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem"],["Nat","instPreorder"],["MeasureTheory","Measure"],["TopologicalSpace","countableBasis"],["TopologicalSpace","isBasis_countableBasis"],["nhds"],["MeasureTheory","ae"],["Filter","instMembership"],["Exists"],["Filter","Frequently"],["Set"],["And"],["Filter","Frequently","mono"],["MeasureTheory","ae_ball_iff"],["TopologicalSpace","IsTopologicalBasis","mem_nhds_iff"],["IsOpen","nullMeasurableSet"],["MeasureTheory","Measure","instFunLike"],["TopologicalSpace","isOpen_of_mem_countableBasis"],["Filter"],["Set","instMembership"],["TopologicalSpace","countable_countableBasis"],["Set","instHasSubset"],["Exists","casesOn"],["Filter","atTop"],["Nat"],["Nat","iterate"],["MeasureTheory","Measure","instOuterMeasureClass"],["HasSubset","Subset"],["Iff","mpr"],["And","casesOn"],["Filter","Eventually","mono"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","measure_mem_forall_ge_image_notMem_eq_zero"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1926982340._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1926982340._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1926982340._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1926982340._hygCtx._hyg.3 s μ) -> (forall (n : Nat), Eq.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1926982340._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1926982340._hygCtx._hyg.3) μ (setOf.{u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (forall (m : Nat), (GE.ge.{0} Nat instLENat m n) -> (Not (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x))))))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal)))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → ∀ (n : ℕ), μ {x | x ∈ s ∧ ∀ m ≥ n, f^[m] x ∉ s} = 0","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → ∀ (n : ℕ), μ {x | x ∈ s ∧ ∀ m ≥ n, f^[m] x ∉ s} = 0","typeReferences":[["Not"],["instZeroENNReal"],["Set"],["Membership","mem"],["And"],["GE","ge"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["Zero","toOfNat0"],["MeasurableSpace"],["Eq"],["instLENat"],["MeasureTheory","NullMeasurableSet"],["setOf"]],"valueReferences":[["Eq","trans"],["instZeroENNReal"],["Inter","inter"],["Membership","mem"],["MeasureTheory","NullMeasurableSet","preimage"],["MeasureTheory","NullMeasurableSet","compl"],["GT","gt"],["Set","iInter"],["GE","ge"],["ENNReal","instLinearOrder"],["MeasureTheory","Conservative","toQuasiMeasurePreserving"],["funext"],["Set","instInter"],["Set","iInter_congr_Prop"],["instLTNat"],["Set","to_countable"],["And"],["GT","gt","lt"],["MeasureTheory","Measure","instFunLike"],["instCountableNat"],["Set","instMembership"],["Exists","casesOn"],["Decidable","byContradiction"],["Nat"],["Iff","of_eq"],["Eq","refl"],["id"],["Eq","mpr"],["autoParam"],["setOf"],["Compl","compl"],["MeasureTheory","Measure","QuasiMeasurePreserving","iterate"],["DFunLike","coe"],["MeasureTheory","NullMeasurableSet","_auto_1"],["Nat","instPreorder"],["congrArg"],["MeasureTheory","Measure"],["Set","setOf_forall"],["SetCoe","countable"],["Zero","toOfNat0"],["congrFun'"],["MeasureTheory","nonempty_of_measure_ne_zero"],["Eq"],["MeasureTheory","NullMeasurableSet","inter"],["LT","lt","le"],["Nat","le"],["Not"],["Set","preimage"],["Set"],["OfNat","ofNat"],["Set","instCompl"],["LinearOrder","toDecidableEq"],["ENNReal"],["Nat","iterate"],["False"],["Ne"],["MeasureTheory","NullMeasurableSet","biInter"],["instLENat"],["MeasureTheory","NullMeasurableSet"],["And","casesOn"],["MeasureTheory","Conservative","exists_gt_measure_inter_ne_zero"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem","_simp_1_2"],"typeFallback":"forall {α : Type.{u_1}} {F : Type.{u_3}} [inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.5 : FunLike.{succ u_3, succ u_1, 1} F (Set.{u_1} α) ENNReal] [inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.15 : MeasureTheory.OuterMeasureClass.{u_3, u_1} F α inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.5] {μ : F} {ι : Sort.{u_4}} [inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.25 : Countable.{u_4} ι] {p : α -> ι -> Prop}, Eq.{1} Prop (Filter.Eventually.{u_1} α (fun (a : α) => forall (i : ι), p a i) (MeasureTheory.ae.{u_1, u_3} α F inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.5 inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.15 μ)) (forall (i : ι), Filter.Eventually.{u_1} α (fun (a : α) => p a i) (MeasureTheory.ae.{u_1, u_3} α F inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.5 inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3764960985._hygCtx._hyg.15 μ))","typeFull":"∀ {α : Type u_1} {F : Type u_3} [inst : FunLike F (Set α) ENNReal] [inst_1 : MeasureTheory.OuterMeasureClass F α]\n  {μ : F} {ι : Sort u_4} [Countable ι] {p : α → ι → Prop},\n  (∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i) = ∀ (i : ι), ∀ᵐ (a : α) ∂μ, p a i","typeReadable":"∀ {α : Type u_1} {F : Type u_3} [inst : FunLike F (Set α) ENNReal] [inst_1 : MeasureTheory.OuterMeasureClass F α]\n  {μ : F} {ι : Sort u_4} [Countable ι] {p : α → ι → Prop},\n  (∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i) = ∀ (i : ι), ∀ᵐ (a : α) ∂μ, p a i","typeReferences":[["Filter","Eventually"],["FunLike"],["ENNReal"],["Set"],["MeasureTheory","OuterMeasureClass"],["Countable"],["Eq"],["MeasureTheory","ae"]],"valueReferences":[["Filter","Eventually"],["MeasureTheory","ae_all_iff"],["MeasureTheory","ae"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","frequently_measure_inter_ne_zero","_simp_1_1"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.3 : AddZeroClass.{u} α] [inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.6 : PartialOrder.{u} α] [inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.9 : CanonicallyOrderedAdd.{u} α (AddZero.toAdd.{u} α (AddZeroClass.toAddZero.{u} α inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.3)) (Preorder.toLE.{u} α (PartialOrder.toPreorder.{u} α inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.6))] {a : α}, Eq.{1} Prop (LT.lt.{u} α (Preorder.toLT.{u} α (PartialOrder.toPreorder.{u} α inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.6)) (OfNat.ofNat.{u} α 0 (Zero.toOfNat0.{u} α (AddZero.toZero.{u} α (AddZeroClass.toAddZero.{u} α inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.3)))) a) (Ne.{succ u} α a (OfNat.ofNat.{u} α 0 (Zero.toOfNat0.{u} α (AddZero.toZero.{u} α (AddZeroClass.toAddZero.{u} α inst._@.Mathlib.Algebra.Order.Monoid.Canonical.Defs.3779110990._hygCtx._hyg.3)))))","typeFull":"∀ {α : Type u} [inst : AddZeroClass α] [inst_1 : PartialOrder α] [CanonicallyOrderedAdd α] {a : α}, (0 < a) = (a ≠ 0)","typeReadable":"∀ {α : Type u} [inst : AddZeroClass α] [inst_1 : PartialOrder α] [CanonicallyOrderedAdd α] {a : α}, (0 < a) = (a ≠ 0)","typeReferences":[["PartialOrder","toPreorder"],["AddZeroClass"],["Preorder","toLT"],["CanonicallyOrderedAdd"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["LT","lt"],["PartialOrder"],["Ne"],["Zero","toOfNat0"],["Preorder","toLE"],["Eq"],["AddZero","toZero"]],"valueReferences":[["LT","lt"],["PartialOrder","toPreorder"],["Preorder","toLT"],["Ne"],["Zero","toOfNat0"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["OfNat","ofNat"],["propext"],["pos_iff_ne_zero"]]},{"isProp":true,"kind":"constructor","name":["MeasureTheory","Conservative","mk"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14}, (MeasureTheory.Measure.QuasiMeasurePreserving.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ μ) -> (forall {{s : Set.{u_1} α}}, (MeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 s) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Exists.{1} Nat (fun (m : Nat) => And (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x))))))) -> (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Measure.QuasiMeasurePreserving f μ μ →\n    (∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) → MeasureTheory.Conservative f μ","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Measure.QuasiMeasurePreserving f μ μ →\n    (∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) → MeasureTheory.Conservative f μ","typeReferences":[["Exists"],["Set"],["instZeroENNReal"],["Membership","mem"],["And"],["MeasureTheory","Measure","QuasiMeasurePreserving"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["MeasureTheory","Conservative"],["Set","instMembership"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["instOfNatNat"],["MeasurableSpace"],["MeasurableSet"],["Ne"],["Zero","toOfNat0"]],"valueReferences":null},{"isProp":false,"kind":"recursor","name":["MeasureTheory","Conservative","rec"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14} {motive : (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ) -> Sort.{u}}, (forall (toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ μ) (exists_mem_iterate_mem' : forall {{s : Set.{u_1} α}}, (MeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 s) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Exists.{1} Nat (fun (m : Nat) => And (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x))))))), motive (MeasureTheory.Conservative.mk.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ toQuasiMeasurePreserving exists_mem_iterate_mem')) -> (forall (t : MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ), motive t)","typeFull":"{α : Type u_1} →\n  [inst : MeasurableSpace α] →\n    {f : α → α} →\n      {μ : MeasureTheory.Measure α} →\n        {motive : MeasureTheory.Conservative f μ → Sort u} →\n          ((toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) →\n              (exists_mem_iterate_mem' : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) →\n                motive ⋯) →\n            (t : MeasureTheory.Conservative f μ) → motive t","typeReadable":"{α : Type u_1} →\n  [inst : MeasurableSpace α] →\n    {f : α → α} →\n      {μ : MeasureTheory.Measure α} →\n        {motive : MeasureTheory.Conservative f μ → Sort u} →\n          ((toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) →\n              (exists_mem_iterate_mem' : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) →\n                motive ⋯) →\n            (t : MeasureTheory.Conservative f μ) → motive t","typeReferences":[["Exists"],["Set"],["instZeroENNReal"],["Membership","mem"],["And"],["MeasureTheory","Measure","QuasiMeasurePreserving"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Conservative","mk"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["instOfNatNat"],["MeasurableSpace"],["MeasurableSet"],["Ne"],["Zero","toOfNat0"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["MeasureTheory","conservative_congr"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3} {ν : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3}, (Eq.{succ u_1} (Filter.{u_1} α) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3) μ) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3) ν)) -> (Iff (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3 f μ) (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1671960449._hygCtx._hyg.3 f ν))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ ν : MeasureTheory.Measure α},\n  MeasureTheory.ae μ = MeasureTheory.ae ν → (MeasureTheory.Conservative f μ ↔ MeasureTheory.Conservative f ν)","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ ν : MeasureTheory.Measure α},\n  MeasureTheory.ae μ = MeasureTheory.ae ν → (MeasureTheory.Conservative f μ ↔ MeasureTheory.Conservative f ν)","typeReferences":[["MeasureTheory","Measure"],["MeasureTheory","Measure","instOuterMeasureClass"],["Iff"],["MeasurableSpace"],["Eq"],["MeasureTheory","Measure","instFunLike"],["MeasureTheory","ae"],["Filter"],["MeasureTheory","Conservative"]],"valueReferences":[["MeasureTheory","Measure"],["MeasureTheory","Measure","instOuterMeasureClass"],["Eq","symm"],["MeasureTheory","Measure","instFunLike"],["MeasureTheory","ae"],["Filter"],["MeasureTheory","Conservative","congr_ae"],["MeasureTheory","Conservative"],["Iff","intro"]]},{"isProp":false,"kind":"definition","name":["MeasureTheory","Conservative","recOn"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14} {motive : (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ) -> Sort.{u}} (t : MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ), (forall (toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ μ) (exists_mem_iterate_mem' : forall {{s : Set.{u_1} α}}, (MeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 s) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Exists.{1} Nat (fun (m : Nat) => And (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x))))))), motive (MeasureTheory.Conservative.mk.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2034020969._hygCtx._hyg.14 f μ toQuasiMeasurePreserving exists_mem_iterate_mem')) -> (motive t)","typeFull":"{α : Type u_1} →\n  [inst : MeasurableSpace α] →\n    {f : α → α} →\n      {μ : MeasureTheory.Measure α} →\n        {motive : MeasureTheory.Conservative f μ → Sort u} →\n          (t : MeasureTheory.Conservative f μ) →\n            ((toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) →\n                (exists_mem_iterate_mem' :\n                    ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) →\n                  motive ⋯) →\n              motive t","typeReadable":"{α : Type u_1} →\n  [inst : MeasurableSpace α] →\n    {f : α → α} →\n      {μ : MeasureTheory.Measure α} →\n        {motive : MeasureTheory.Conservative f μ → Sort u} →\n          (t : MeasureTheory.Conservative f μ) →\n            ((toQuasiMeasurePreserving : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) →\n                (exists_mem_iterate_mem' :\n                    ∀ ⦃s : Set α⦄, MeasurableSet s → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s) →\n                  motive ⋯) →\n              motive t","typeReferences":[["Exists"],["Set"],["instZeroENNReal"],["Membership","mem"],["And"],["MeasureTheory","Measure","QuasiMeasurePreserving"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Conservative","mk"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["instOfNatNat"],["MeasurableSpace"],["MeasurableSet"],["Ne"],["Zero","toOfNat0"]],"valueReferences":[["MeasureTheory","Conservative","rec"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","ae_forall_image_mem_imp_frequently_image_mem"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.3201498236._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3201498236._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3201498236._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3201498236._hygCtx._hyg.3 s μ) -> (Filter.Eventually.{u_1} α (fun (x : α) => forall (k : Nat), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f k x)) -> (Filter.Frequently.{0} Nat (fun (n : Nat) => Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f n x)) (Filter.atTop.{0} Nat Nat.instPreorder))) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3201498236._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3201498236._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3201498236._hygCtx._hyg.3) μ))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ →\n      ∀ᵐ (x : α) ∂μ, ∀ (k : ℕ), f^[k] x ∈ s → ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ →\n      ∀ᵐ (x : α) ∂μ, ∀ (k : ℕ), f^[k] x ∈ s → ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReferences":[["Filter","Eventually"],["Set"],["Filter","Frequently"],["Membership","mem"],["MeasureTheory","Measure","instFunLike"],["MeasureTheory","Conservative"],["Set","instMembership"],["Nat","instPreorder"],["MeasureTheory","Measure"],["Nat"],["Filter","atTop"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","iterate"],["MeasurableSpace"],["MeasureTheory","ae"],["MeasureTheory","NullMeasurableSet"]],"valueReferences":[["instAddNat"],["Filter","Eventually"],["Filter","map_add_atTop_eq_nat"],["Membership","mem"],["Nat","instAddCommSemigroup"],["MeasureTheory","NullMeasurableSet","preimage"],["MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem"],["MeasureTheory","Measure","QuasiMeasurePreserving","iterate"],["MeasureTheory","ae_all_iff"],["congrArg"],["Nat","instPreorder"],["MeasureTheory","Measure"],["MeasureTheory","Conservative","toQuasiMeasurePreserving"],["Eq","symm"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Function","iterate_add_apply"],["Eq"],["add_comm"],["MeasureTheory","ae"],["propext"],["Filter","frequently_map"],["Set","preimage"],["instHAdd"],["Filter","Frequently"],["Set"],["Filter","map"],["Filter","Frequently","mono"],["MeasureTheory","Measure","instFunLike"],["instCountableNat"],["Filter"],["Set","instMembership"],["HAdd","hAdd"],["Filter","atTop"],["Nat"],["Nat","iterate"],["MeasureTheory","Measure","instOuterMeasureClass"],["Iff","mpr"],["id"],["Eq","mpr"],["Filter","Eventually","mono"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","MeasurePreserving","conservative"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2117481248._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2117481248._hygCtx._hyg.3} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2117481248._hygCtx._hyg.13 : MeasureTheory.IsFiniteMeasure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2117481248._hygCtx._hyg.3 μ], (MeasureTheory.MeasurePreserving.{u_1, u_1} α α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2117481248._hygCtx._hyg.3 inst._@.Mathlib.Dynamics.Ergodic.Conservative.2117481248._hygCtx._hyg.3 f μ μ) -> (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2117481248._hygCtx._hyg.3 f μ)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ],\n  MeasureTheory.MeasurePreserving f μ μ → MeasureTheory.Conservative f μ","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ],\n  MeasureTheory.MeasurePreserving f μ μ → MeasureTheory.Conservative f μ","typeReferences":[["MeasureTheory","Measure"],["MeasureTheory","MeasurePreserving"],["MeasurableSpace"],["MeasureTheory","IsFiniteMeasure"],["MeasureTheory","Conservative"]],"valueReferences":[["MeasureTheory","Conservative","mk"],["MeasurableSet","nullMeasurableSet"],["MeasureTheory","MeasurePreserving","exists_mem_iterate_mem"],["MeasureTheory","MeasurePreserving","quasiMeasurePreserving"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","iterate"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1202121004._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1202121004._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1202121004._hygCtx._hyg.3 f μ) -> (forall (n : Nat), MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1202121004._hygCtx._hyg.3 (Nat.iterate.{succ u_1} α f n) μ)","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → ∀ (n : ℕ), MeasureTheory.Conservative f^[n] μ","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → ∀ (n : ℕ), MeasureTheory.Conservative f^[n] μ","typeReferences":[["MeasureTheory","Measure"],["Nat"],["Nat","iterate"],["MeasurableSpace"],["MeasureTheory","Conservative"]],"valueReferences":[["instAddNat"],["tsub_eq_zero_iff_le"],["PartialOrder","toPreorder"],["MulZeroClass","toMul"],["Membership","mem"],["Iff","mp"],["Nat","instAddCommSemigroup"],["HMul","hMul"],["Exists","intro"],["Nat","instAddCommMonoid"],["MeasureTheory","Conservative"],["HDiv","hDiv"],["And","intro"],["MeasureTheory","Conservative","toQuasiMeasurePreserving"],["MeasurableSet","nullMeasurableSet"],["Nat","instOrderedSub"],["Eq","symm"],["HSub","hSub"],["tsub_add_cancel_of_le"],["Nat","instMulZeroClass"],["Nat","exists_lt_modEq_of_infinite"],["Nat","instIsOrderedAddMonoid"],["Nat","mul_div_cancel'"],["Nat","casesAuxOn"],["instLTNat"],["Exists"],["Dvd","dvd"],["Nat","instPartialOrder"],["Filter","Frequently"],["eq_comm"],["MeasureTheory","Conservative","frequently_ae_mem_and_frequently_image_mem"],["And"],["Set","Infinite"],["IsOrderedAddMonoid","toAddLeftMono"],["MulZeroClass","mul_zero"],["AddZeroClass","toAddZero"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["Nat","instCanonicallyOrderedAdd"],["Exists","casesOn"],["MeasureTheory","Conservative","mk"],["Filter","atTop"],["Nat"],["id"],["instHMul"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Nat","instDvd"],["setOf"],["Eq","mp"],["Nat","instDiv"],["MeasureTheory","Conservative","id"],["AddCommMonoid","toAddMonoid"],["MeasureTheory","Measure","QuasiMeasurePreserving","iterate"],["Nat","ModEq"],["instHDiv"],["Nat","instPreorder"],["congrArg"],["MeasureTheory","Measure"],["LE","le","not_gt"],["instOfNatNat"],["Nat","succ_pos"],["instMulNat"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Zero","toOfNat0"],["Function","iterate_add_apply"],["Preorder","toLE"],["Eq"],["MeasureTheory","ae"],["propext"],["LT","lt","le"],["Set"],["instHAdd"],["Filter","Frequently","exists"],["OfNat","ofNat"],["HAdd","hAdd"],["LT","lt"],["Nat","iterate"],["MeasureTheory","Measure","instOuterMeasureClass"],["instSubNat"],["CanonicallyOrderedAdd","toExistsAddOfLE"],["MulZeroClass","toZero"],["Nat","succ"],["Function","iterate_mul"],["LE","le"],["Nat","frequently_atTop_iff_infinite"],["Nat","modEq_iff_dvd'"],["Ne"],["instHSub"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem","_simp_1_1"],"typeFallback":"forall {α : Type.{u_3}} [inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2533177475._hygCtx._hyg.7 : Preorder.{u_3} α] [inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2533177475._hygCtx._hyg.10 : IsDirectedOrder.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2533177475._hygCtx._hyg.7)] {p : α -> Prop} [inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2533177475._hygCtx._hyg.16 : Nonempty.{succ u_3} α], Eq.{1} Prop (Filter.Frequently.{u_3} α (fun (x : α) => p x) (Filter.atTop.{u_3} α inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2533177475._hygCtx._hyg.7)) (forall (a : α), Exists.{succ u_3} α (fun (b : α) => And (GE.ge.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2533177475._hygCtx._hyg.7) b a) (p b)))","typeFull":"∀ {α : Type u_3} [inst : Preorder α] [IsDirectedOrder α] {p : α → Prop} [Nonempty α],\n  (∃ᶠ (x : α) in Filter.atTop, p x) = ∀ (a : α), ∃ b ≥ a, p b","typeReadable":"∀ {α : Type u_3} [inst : Preorder α] [IsDirectedOrder α] {p : α → Prop} [Nonempty α],\n  (∃ᶠ (x : α) in Filter.atTop, p x) = ∀ (a : α), ∃ b ≥ a, p b","typeReferences":[["Filter","atTop"],["Preorder"],["Exists"],["Filter","Frequently"],["And"],["IsDirectedOrder"],["Nonempty"],["Eq"],["Preorder","toLE"],["GE","ge"]],"valueReferences":[["Filter","atTop"],["Exists"],["Filter","Frequently"],["And"],["Filter","frequently_atTop"],["Preorder","toLE"],["GE","ge"],["propext"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","frequently_ae_mem_and_frequently_image_mem"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3 s μ) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Filter.Frequently.{u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Filter.Frequently.{0} Nat (fun (n : Nat) => Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f n x)) (Filter.atTop.{0} Nat Nat.instPreorder))) (MeasureTheory.ae.{u_1, u_1} α (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3) (MeasureTheory.Measure.instOuterMeasureClass.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1445401240._hygCtx._hyg.3) μ))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∃ᵐ (x : α) ∂μ, x ∈ s ∧ ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∃ᵐ (x : α) ∂μ, x ∈ s ∧ ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s","typeReferences":[["Set"],["instZeroENNReal"],["Filter","Frequently"],["Membership","mem"],["And"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Nat","instPreorder"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["Filter","atTop"],["Nat"],["ENNReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","iterate"],["MeasurableSpace"],["Ne"],["Zero","toOfNat0"],["MeasureTheory","ae"],["MeasureTheory","NullMeasurableSet"]],"valueReferences":[["instZeroENNReal"],["Membership","mem"],["MeasureTheory","Conservative","ae_mem_imp_frequently_image_mem"],["DFunLike","coe"],["Nat","instPreorder"],["MeasureTheory","Measure"],["And","intro"],["Zero","toOfNat0"],["MeasureTheory","ae"],["And","left"],["Filter","Frequently"],["Set"],["And","right"],["And"],["Filter","Frequently","mono"],["Filter","Frequently","and_eventually"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["Filter","atTop"],["Nat"],["ENNReal"],["Nat","iterate"],["MeasureTheory","Measure","instOuterMeasureClass"],["MeasureTheory","frequently_ae_mem_iff"],["Iff","mpr"],["Ne"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","frequently_measure_inter_ne_zero"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3 s μ) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Filter.Frequently.{0} Nat (fun (m : Nat) => Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.3658841341._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (Set.preimage.{u_1, u_1} α α (Nat.iterate.{succ u_1} α f m) s))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) (Filter.atTop.{0} Nat Nat.instPreorder))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∃ᶠ (m : ℕ) in Filter.atTop, μ (s ∩ f^[m] ⁻¹' s) ≠ 0","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∃ᶠ (m : ℕ) in Filter.atTop, μ (s ∩ f^[m] ⁻¹' s) ≠ 0","typeReferences":[["Set","preimage"],["Set"],["instZeroENNReal"],["Filter","Frequently"],["Inter","inter"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Nat","instPreorder"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["Filter","atTop"],["Nat"],["ENNReal"],["Nat","iterate"],["MeasurableSpace"],["Ne"],["Zero","toOfNat0"],["Set","instInter"],["MeasureTheory","NullMeasurableSet"]],"valueReferences":[["instAddNat"],["lt_add_iff_pos_right","_simp_1"],["Eq","trans"],["instZeroENNReal"],["Exists","intro"],["MeasureTheory","Conservative","toQuasiMeasurePreserving"],["Eq","symm"],["Eq","ndrec"],["Set","instInter"],["Set","instSDiff"],["Set","inter_self"],["rfl"],["instLTNat"],["Exists"],["Set","to_countable"],["Filter","Frequently"],["Set","Infinite"],["MeasureTheory","Measure","instFunLike"],["instCountableNat"],["Set","instMembership"],["Filter","atTop"],["Iff"],["Eq","refl"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["instIsLeftCancelAddOfAddLeftReflectLE"],["gt_iff_lt","_simp_1"],["Classical","byContradiction"],["HEq"],["Eq","mpr"],["Set","not_infinite"],["Set","Finite"],["MeasureTheory","NullMeasurableSet","diff"],["AddMonoid","toAddZeroClass"],["setOf"],["gt_iff_lt"],["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","frequently_measure_inter_ne_zero","_simp_1_1"],["pi_congr"],["not_imp_comm"],["MeasureTheory","measure_diff_null"],["Nat","instPreorder"],["contravariant_lt_of_covariant_le"],["instOfNatNat"],["Preorder","toLE"],["Eq"],["propext"],["MeasureTheory","NullMeasurableSet","inter"],["Nat","le"],["Nat","instAddMonoid"],["Set"],["AddZero","toAdd"],["OfNat","ofNat"],["HAdd","hAdd"],["LinearOrder","toPartialOrder"],["MeasureTheory","NullMeasurableSet","biUnion"],["ENNReal"],["Nat","iterate"],["Nat","frequently_atTop_iff_infinite"],["SDiff","sdiff"],["Ne"],["MeasureTheory","Conservative","exists_mem_iterate_mem"],["IsLeftCancelAdd","addLeftStrictMono_of_addLeftMono"],["And","casesOn"],["Set","mem_iUnion₂"],["PartialOrder","toPreorder"],["Membership","mem"],["Inter","inter"],["Preorder","toLT"],["MeasureTheory","NullMeasurableSet","preimage"],["Set","iUnion"],["GT","gt"],["Nat","instAddCommMonoid"],["And","intro"],["funext"],["eq_of_heq"],["Nat","instIsOrderedAddMonoid"],["Nat","instLinearOrder"],["And","left"],["SemilatticeInf","toPartialOrder"],["Nat","instPartialOrder"],["Set","exists_max_image"],["And","right"],["And"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["Nat","instCanonicallyOrderedAdd"],["Exists","casesOn"],["Nat"],["Iff","mpr"],["id"],["AddZero","toZero"],["Eq","mp"],["MeasureTheory","Measure","QuasiMeasurePreserving","iterate"],["DFunLike","coe"],["congrArg"],["Function","iterate_add"],["Set","mem_setOf"],["instDistribLatticeOfLinearOrder"],["MeasureTheory","Measure"],["SetCoe","countable"],["Nat","instIsOrderedCancelAddMonoid"],["Zero","toOfNat0"],["congrFun'"],["Not"],["not_le"],["Set","preimage"],["Lattice","toSemilatticeInf"],["MeasureTheory","measure_biUnion_null_iff"],["HEq","refl"],["instHAdd"],["Function","comp"],["Iff","rfl"],["Eq","casesOn"],["LT","lt"],["DistribLattice","toLattice"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","succ"],["LE","le"],["False"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","exists_mem_iterate_mem"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.2256215287._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2256215287._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2256215287._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2256215287._hygCtx._hyg.3 s μ) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2256215287._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.2256215287._hygCtx._hyg.3) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (Exists.{succ u_1} α (fun (x : α) => And (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) (Exists.{1} Nat (fun (m : Nat) => And (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s (Nat.iterate.{succ u_1} α f m x))))))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ → MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s","typeReferences":[["Exists"],["Set"],["instZeroENNReal"],["Membership","mem"],["And"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["Nat"],["ENNReal"],["Nat","iterate"],["instOfNatNat"],["MeasurableSpace"],["Ne"],["Zero","toOfNat0"],["MeasureTheory","NullMeasurableSet"]],"valueReferences":[["instZeroENNReal"],["Membership","mem"],["Exists","intro"],["DFunLike","coe"],["congrArg"],["And","intro"],["MeasureTheory","Measure"],["instOfNatNat"],["Filter","EventuallyEq"],["Zero","toOfNat0"],["Eq"],["MeasureTheory","ae"],["MeasureTheory","Conservative","exists_mem_iterate_mem'"],["Exists"],["Set"],["And"],["MeasureTheory","NullMeasurableSet","exists_measurable_subset_ae_eq"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["Set","instMembership"],["Set","instHasSubset"],["Exists","casesOn"],["Nat"],["ENNReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Nat","iterate"],["HasSubset","Subset"],["id"],["MeasurableSet"],["Ne"],["Eq","mpr"],["MeasureTheory","measure_congr"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["MeasureTheory","Conservative","exists_gt_measure_inter_ne_zero"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3 : MeasurableSpace.{u_1} α] {f : α -> α} {s : Set.{u_1} α} {μ : MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3}, (MeasureTheory.Conservative.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3 f μ) -> (MeasureTheory.NullMeasurableSet.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3 s μ) -> (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) μ s) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal))) -> (forall (N : Nat), Exists.{1} Nat (fun (m : Nat) => And (GT.gt.{0} Nat instLTNat m N) (Ne.{1} ENNReal (DFunLike.coe.{succ u_1, succ u_1, 1} (MeasureTheory.Measure.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) (MeasureTheory.Measure.instFunLike.{u_1} α inst._@.Mathlib.Dynamics.Ergodic.Conservative.1759747795._hygCtx._hyg.3) μ (Inter.inter.{u_1} (Set.{u_1} α) (Set.instInter.{u_1} α) s (Set.preimage.{u_1, u_1} α α (Nat.iterate.{succ u_1} α f m) s))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal)))))","typeFull":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∀ (N : ℕ), ∃ m > N, μ (s ∩ f^[m] ⁻¹' s) ≠ 0","typeReadable":"∀ {α : Type u_1} [inst : MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α},\n  MeasureTheory.Conservative f μ →\n    MeasureTheory.NullMeasurableSet s μ → μ s ≠ 0 → ∀ (N : ℕ), ∃ m > N, μ (s ∩ f^[m] ⁻¹' s) ≠ 0","typeReferences":[["instLTNat"],["Set","preimage"],["Exists"],["Set"],["instZeroENNReal"],["Inter","inter"],["And"],["GT","gt"],["DFunLike","coe"],["MeasureTheory","Measure","instFunLike"],["OfNat","ofNat"],["MeasureTheory","Conservative"],["MeasureTheory","Measure"],["ENNReal"],["Nat"],["Nat","iterate"],["MeasurableSpace"],["Ne"],["Zero","toOfNat0"],["Set","instInter"],["MeasureTheory","NullMeasurableSet"]],"valueReferences":[["instZeroENNReal"],["Inter","inter"],["Preorder","toLT"],["Exists","intro"],["GT","gt"],["DFunLike","coe"],["Nat","instPreorder"],["And","intro"],["MeasureTheory","Measure"],["Filter","eventually_gt_atTop"],["Zero","toOfNat0"],["Set","instInter"],["instLTNat"],["instNoTopOrderOfNoMaxOrder"],["Set","preimage"],["Nat","instNoMaxOrder"],["Exists"],["Set"],["And"],["MeasureTheory","Conservative","frequently_measure_inter_ne_zero"],["Filter","Frequently","exists"],["Filter","Frequently","and_eventually"],["MeasureTheory","Measure","instFunLike"],["_private","Mathlib","Dynamics","Ergodic","Conservative",0,"MeasureTheory","Conservative","exists_gt_measure_inter_ne_zero","match_1_1"],["OfNat","ofNat"],["LT","lt"],["ENNReal"],["Nat"],["Filter","atTop"],["Nat","iterate"],["Ne"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.FieldTheory.RatFunc.Degree.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_div","_proof_1_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3 : Field.{u_1} K] {x : RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))}, (Ne.{succ u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) x (OfNat.ofNat.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instZero.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))))) -> (Not (Eq.{succ u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) x (OfNat.ofNat.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instZero.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))))))","typeFull":"∀ {K : Type u_1} [inst : Field K] {x : RatFunc K}, x ≠ 0 → ¬x = 0","typeReadable":"∀ {K : Type u_1} [inst : Field K] {x : RatFunc K}, x ≠ 0 → ¬x = 0","typeReferences":[["Not"],["RatFunc","instZero"],["RatFunc"],["EuclideanDomain","toCommRing"],["Field","toEuclideanDomain"],["Field"],["Zero","toOfNat0"],["Ne"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Not"],["EuclideanDomain","toCommRing"],["Eq","trans"],["True"],["Eq","mp"],["eq_true"],["Lean","Grind","not_not"],["True","intro"],["OfNat","ofNat"],["RatFunc"],["RatFunc","instZero"],["Field","toEuclideanDomain"],["eq_false"],["Classical","byContradiction"],["Eq","symm"],["id"],["False"],["Zero","toOfNat0"],["Lean","Grind","intro_with_eq"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_neg"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.174166442._hygCtx._hyg.3 : Field.{u} K] (x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.174166442._hygCtx._hyg.3))), Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.174166442._hygCtx._hyg.3 (Neg.neg.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.174166442._hygCtx._hyg.3))) (RatFunc.instNeg.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.174166442._hygCtx._hyg.3))) x)) (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.174166442._hygCtx._hyg.3 x)","typeFull":"∀ {K : Type u} [inst : Field K] (x : RatFunc K), (-x).intDegree = x.intDegree","typeReadable":"∀ {K : Type u} [inst : Field K] (x : RatFunc K), (-x).intDegree = x.intDegree","typeReferences":[["RatFunc"],["EuclideanDomain","toCommRing"],["Field","toEuclideanDomain"],["Neg","neg"],["Field"],["RatFunc","instNeg"],["Eq"],["RatFunc","intDegree"],["Int"]],"valueReferences":[["RatFunc","denom_ne_zero"],["SubtractionMonoid","toSubNegZeroMonoid"],["Int","instSub"],["Classical","propDecidable"],["RatFunc","denom"],["HSub","hSub"],["Eq","symm"],["RatFunc","num_ne_zero"],["neg_zero"],["Semifield","toDivisionSemiring"],["Neg","neg"],["DivisionSemiring","toSemiring"],["Ring","toSemiring"],["Polynomial"],["Nat"],["Iff","mpr"],["Eq","refl"],["NegZeroClass","toZero"],["id"],["RatFunc","num"],["Eq","mpr"],["Polynomial","natDegree_sub_eq_of_prod_eq"],["EuclideanDomain","toCommRing"],["Nat","cast"],["IsDomain","to_noZeroDivisors"],["RatFunc","instNeg"],["Polynomial","natDegree"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["RatFunc"],["RatFunc","instZero"],["Field","toEuclideanDomain"],["Zero","toOfNat0"],["Eq"],["Polynomial","instNeg"],["instNatCastInt"],["Polynomial","ring"],["instIsDomain"],["DivisionRing","toRing"],["Field","toDivisionRing"],["RatFunc","instField"],["RatFunc","num_denom_neg"],["OfNat","ofNat"],["Int"],["Ring","toAddCommGroup"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["RatFunc","intDegree","eq_1"],["neg_ne_zero"],["Field","toSemifield"],["Ne"],["dite"],["instHSub"],["Polynomial","natDegree_neg"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_polynomial"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3 : Field.{u} K] {p : Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))}, Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3 (DFunLike.coe.{succ u, succ u, succ u} (RingHom.{u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (CommSemiring.toSemiring.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (Polynomial.commSemiring.{u} K (Semifield.toCommSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))))) (Semiring.toNonAssocSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (DivisionSemiring.toSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Semifield.toDivisionSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Field.toSemifield.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (RatFunc.instField.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))))))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) => RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (RingHom.instFunLike.{u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (CommSemiring.toSemiring.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (Polynomial.commSemiring.{u} K (Semifield.toCommSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))))) (Semiring.toNonAssocSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (DivisionSemiring.toSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Semifield.toDivisionSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Field.toSemifield.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (RatFunc.instField.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))))))) (algebraMap.{u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Polynomial.commSemiring.{u} K (Semifield.toCommSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (DivisionSemiring.toSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Semifield.toDivisionSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Field.toSemifield.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (RatFunc.instField.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))))) (RatFunc.instAlgebraOfPolynomial.{u, u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (Polynomial.commSemiring.{u} K (Semifield.toCommSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) (Algebra.id.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))) (Polynomial.commSemiring.{u} K (Semifield.toCommSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3)))))) p)) (Nat.cast.{0} Int instNatCastInt (Polynomial.natDegree.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2021812206._hygCtx._hyg.3))) p))","typeFull":"∀ {K : Type u} [inst : Field K] {p : Polynomial K}, ((algebraMap (Polynomial K) (RatFunc K)) p).intDegree = ↑p.natDegree","typeReadable":"∀ {K : Type u} [inst : Field K] {p : Polynomial K}, ((algebraMap (Polynomial K) (RatFunc K)) p).intDegree = ↑p.natDegree","typeReferences":[["Semifield","toCommSemiring"],["RingHom"],["EuclideanDomain","toCommRing"],["Nat","cast"],["Field"],["Polynomial","natDegree"],["RingHom","instFunLike"],["DFunLike","coe"],["Algebra","id"],["Polynomial","commSemiring"],["RatFunc"],["Semiring","toNonAssocSemiring"],["Field","toEuclideanDomain"],["Semifield","toDivisionSemiring"],["Eq"],["instNatCastInt"],["instIsDomain"],["CommSemiring","toSemiring"],["RatFunc","instField"],["DivisionSemiring","toSemiring"],["Int"],["Polynomial"],["Field","toSemifield"],["RatFunc","instAlgebraOfPolynomial"],["RatFunc","intDegree"],["algebraMap"]],"valueReferences":[["Polynomial","natDegree_one"],["RingHom"],["Int","instSub"],["SubtractionMonoid","toSubNegZeroMonoid"],["sub_zero"],["Algebra","id"],["Polynomial","commSemiring"],["Semiring","toNonAssocSemiring"],["RatFunc","denom"],["SubNegMonoid","toSub"],["HSub","hSub"],["Int","ofNat_zero"],["Semifield","toDivisionSemiring"],["DivisionSemiring","toSemiring"],["Polynomial","instOne"],["Polynomial"],["Nat"],["instOfNat"],["Eq","refl"],["NegZeroClass","toZero"],["RatFunc","num"],["id"],["RatFunc","num_algebraMap"],["RatFunc","instAlgebraOfPolynomial"],["Eq","mpr"],["algebraMap"],["Semifield","toCommSemiring"],["Nat","cast"],["EuclideanDomain","toCommRing"],["Polynomial","natDegree"],["RingHom","instFunLike"],["SubNegZeroMonoid","toSubNegMonoid"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["congrArg"],["RatFunc"],["Int","instAddCommGroup"],["Field","toEuclideanDomain"],["instOfNatNat"],["Zero","toOfNat0"],["Eq"],["instNatCastInt"],["instIsDomain"],["CommSemiring","toSemiring"],["RatFunc","instField"],["OfNat","ofNat"],["Int"],["RatFunc","denom_algebraMap"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["RatFunc","intDegree","eq_1"],["Field","toSemifield"],["instHSub"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_add"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 : Field.{u} K] {x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))} {y : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))}, (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (HAdd.hAdd.{u, u, u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (instHAdd.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (RatFunc.instAdd.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) x y) (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))))) -> (Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 (HAdd.hAdd.{u, u, u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (instHAdd.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (RatFunc.instAdd.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) x y)) (HSub.hSub.{0, 0, 0} Int Int Int (instHSub.{0} Int Int.instSub) (Nat.cast.{0} Int instNatCastInt (Polynomial.natDegree.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (HAdd.hAdd.{u, u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (instHAdd.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.instAdd.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))))) (HMul.hMul.{u, u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (instHMul.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.instMul.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))))) (RatFunc.num.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 x) (RatFunc.denom.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 y)) (HMul.hMul.{u, u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (instHMul.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.instMul.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))))) (RatFunc.denom.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 x) (RatFunc.num.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 y))))) (Nat.cast.{0} Int instNatCastInt (Polynomial.natDegree.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))) (HMul.hMul.{u, u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (instHMul.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3)))) (Polynomial.instMul.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3))))) (RatFunc.denom.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 x) (RatFunc.denom.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2639073805._hygCtx._hyg.3 y))))))","typeFull":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K},\n  x + y ≠ 0 → (x + y).intDegree = ↑(x.num * y.denom + x.denom * y.num).natDegree - ↑(x.denom * y.denom).natDegree","typeReadable":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K},\n  x + y ≠ 0 → (x + y).intDegree = ↑(x.num * y.denom + x.denom * y.num).natDegree - ↑(x.denom * y.denom).natDegree","typeReferences":[["Polynomial","instAdd"],["Int","instSub"],["EuclideanDomain","toCommRing"],["Nat","cast"],["Field"],["Polynomial","natDegree"],["HMul","hMul"],["RatFunc","instZero"],["RatFunc"],["RatFunc","denom"],["Field","toEuclideanDomain"],["Polynomial","instMul"],["HSub","hSub"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Eq"],["instNatCastInt"],["instHAdd"],["DivisionSemiring","toSemiring"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["Polynomial"],["RatFunc","num"],["Field","toSemifield"],["instHMul"],["Ne"],["instHSub"],["RatFunc","instAdd"],["RatFunc","intDegree"]],"valueReferences":[["Polynomial","natDegree_sub_eq_of_prod_eq"],["Polynomial","instAdd"],["RatFunc","denom_ne_zero"],["EuclideanDomain","toCommRing"],["IsDomain","to_noZeroDivisors"],["Polynomial","instNoZeroDivisors"],["HMul","hMul"],["mul_ne_zero"],["RatFunc"],["RatFunc","denom"],["Field","toEuclideanDomain"],["RatFunc","num_mul_denom_add_denom_mul_num_ne_zero"],["Polynomial","instMul"],["RatFunc","num_ne_zero"],["Semifield","toDivisionSemiring"],["instIsDomain"],["instHAdd"],["Polynomial","instZero"],["DivisionSemiring","toSemiring"],["RatFunc","num_denom_add"],["HAdd","hAdd"],["Polynomial"],["RatFunc","num"],["Field","toSemifield"],["instHMul"],["RatFunc","instAdd"]]},{"isProp":false,"kind":"definition","name":["RatFunc","intDegree"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3 : Field.{u} K], (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3))) -> Int","typeFull":"{K : Type u} → [inst : Field K] → RatFunc K → ℤ","typeReadable":"{K : Type u} → [inst : Field K] → RatFunc K → ℤ","typeReferences":[["RatFunc"],["EuclideanDomain","toCommRing"],["Field","toEuclideanDomain"],["Field"],["Int"]],"valueReferences":[["Int","instSub"],["RatFunc","denom"],["Nat","cast"],["RatFunc","num"],["Field","toSemifield"],["Polynomial","natDegree"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["DivisionSemiring","toSemiring"],["instHSub"],["instNatCastInt"],["Int"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_add_le"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3 : Field.{u} K] {x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))} {y : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))}, (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) y (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3)))))) -> (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (HAdd.hAdd.{u, u, u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (instHAdd.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.instAdd.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3)))) x y) (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3)))))) -> (LE.le.{0} Int Int.instLEInt (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3 (HAdd.hAdd.{u, u, u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (instHAdd.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3))) (RatFunc.instAdd.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3)))) x y)) (Max.max.{0} Int Int.instMax (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3 x) (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2683369634._hygCtx._hyg.3 y)))","typeFull":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K}, y ≠ 0 → x + y ≠ 0 → (x + y).intDegree ≤ max x.intDegree y.intDegree","typeReadable":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K}, y ≠ 0 → x + y ≠ 0 → (x + y).intDegree ≤ max x.intDegree y.intDegree","typeReferences":[["EuclideanDomain","toCommRing"],["Field"],["instHAdd"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["Max","max"],["RatFunc"],["RatFunc","instZero"],["Field","toEuclideanDomain"],["LE","le"],["Int","instLEInt"],["Ne"],["Zero","toOfNat0"],["RatFunc","instAdd"],["Int","instMax"],["RatFunc","intDegree"]],"valueReferences":[["RatFunc","denom_ne_zero"],["Polynomial","instAdd"],["Lattice","toSemilatticeSup"],["Int","instSub"],["PartialOrder","toPreorder"],["Eq","trans"],["Classical","propDecidable"],["le_max_iff"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","natDegree_add_le"],["RatFunc","denom"],["Or"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocCommSemiring"],["mul_comm"],["Eq","symm"],["Polynomial","instMul"],["Int","instLEInt"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["Nat","instLinearOrder"],["Int","instAddGroup"],["Int","instMax"],["SemilatticeInf","toPartialOrder"],["instLatticeInt"],["CommMagma","toMul"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["RatFunc","intDegree_zero"],["DivisionSemiring","toSemiring"],["Polynomial","commRing"],["Polynomial"],["Nat"],["zero_add"],["instOfNat"],["id"],["RatFunc","num"],["instHMul"],["Eq","mpr"],["covariant_swap_add_of_covariant_add"],["RatFunc","instAdd"],["AddMonoid","toAddZeroClass"],["le_sup_right","_simp_2"],["Int","instAddCommSemigroup"],["EuclideanDomain","toCommRing"],["Nat","cast"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["Polynomial","natDegree"],["CommRing","toNonUnitalCommRing"],["congrArg"],["instDistribLatticeOfLinearOrder"],["RatFunc","instZero"],["RatFunc"],["Field","toEuclideanDomain"],["RatFunc","natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree"],["congr"],["Int","instAddLeftMono"],["congrFun'"],["Zero","toOfNat0"],["Preorder","toLE"],["Eq"],["propext"],["Int","instLinearOrder"],["instNatCastInt"],["Lattice","toSemilatticeInf"],["True"],["instIsDomain"],["Int","ofNat_le"],["RatFunc","intDegree_add"],["instHAdd"],["DivisionRing","toRing"],["Field","toDivisionRing"],["RatFunc","instField"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["Max","max"],["DistribLattice","toLattice"],["of_eq_true"],["SemilatticeSup","toMax"],["LE","le"],["Field","toSemifield"],["dite"],["instHSub"],["instLENat"],["sub_le_sub_iff_right"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree","eq_1"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3 : Field.{u} K] (x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3))), Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3 x) (HSub.hSub.{0, 0, 0} Int Int Int (instHSub.{0} Int Int.instSub) (Nat.cast.{0} Int instNatCastInt (Polynomial.natDegree.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3))) (RatFunc.num.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3 x))) (Nat.cast.{0} Int instNatCastInt (Polynomial.natDegree.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3))) (RatFunc.denom.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1930628275._hygCtx._hyg.3 x))))","typeFull":"∀ {K : Type u} [inst : Field K] (x : RatFunc K), x.intDegree = ↑x.num.natDegree - ↑x.denom.natDegree","typeReadable":"∀ {K : Type u} [inst : Field K] (x : RatFunc K), x.intDegree = ↑x.num.natDegree - ↑x.denom.natDegree","typeReferences":[["Int","instSub"],["EuclideanDomain","toCommRing"],["Nat","cast"],["Field"],["Polynomial","natDegree"],["DivisionSemiring","toSemiring"],["Int"],["RatFunc"],["RatFunc","denom"],["Field","toEuclideanDomain"],["RatFunc","num"],["Field","toSemifield"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["Eq"],["instHSub"],["RatFunc","intDegree"],["instNatCastInt"]],"valueReferences":[["Eq","refl"],["RatFunc","intDegree"],["Int"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_div","_proof_1_2"],"typeFallback":"forall {K : Type.{u_1}} [inst : Field.{u_1} K], IsDomain.{u_1} K (DivisionSemiring.toSemiring.{u_1} K (Semifield.toDivisionSemiring.{u_1} K (Field.toSemifield.{u_1} K inst)))","typeFull":"∀ {K : Type u_1} [inst : Field K], IsDomain K","typeReadable":"∀ {K : Type u_1} [inst : Field K], IsDomain K","typeReferences":[["Field"],["Field","toSemifield"],["IsDomain"],["Semifield","toDivisionSemiring"],["DivisionSemiring","toSemiring"]],"valueReferences":[["instIsDomain"],["Field","toSemifield"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_inv","_simp_1_1"],"typeFallback":"forall {G : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Basic.125726414._hygCtx._hyg.6 : AddGroup.{u_3} G] {a : G} {b : G}, Eq.{1} Prop (Eq.{succ u_3} G a (Neg.neg.{u_3} G (NegZeroClass.toNeg.{u_3} G (SubNegZeroMonoid.toNegZeroClass.{u_3} G (SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (AddGroup.toSubtractionMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.125726414._hygCtx._hyg.6)))) b)) (Eq.{succ u_3} G (HAdd.hAdd.{u_3, u_3, u_3} G G G (instHAdd.{u_3} G (AddZero.toAdd.{u_3} G (AddZeroClass.toAddZero.{u_3} G (AddMonoid.toAddZeroClass.{u_3} G (SubNegMonoid.toAddMonoid.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.125726414._hygCtx._hyg.6)))))) a b) (OfNat.ofNat.{u_3} G 0 (Zero.toOfNat0.{u_3} G (NegZeroClass.toZero.{u_3} G (SubNegZeroMonoid.toNegZeroClass.{u_3} G (SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (AddGroup.toSubtractionMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.125726414._hygCtx._hyg.6)))))))","typeFull":"∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, (a = -b) = (a + b = 0)","typeReadable":"∀ {G : Type u_3} [inst : AddGroup G] {a b : G}, (a = -b) = (a + b = 0)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Neg","neg"],["instHAdd"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["SubNegZeroMonoid","toNegZeroClass"],["OfNat","ofNat"],["AddGroup","toSubtractionMonoid"],["HAdd","hAdd"],["NegZeroClass","toNeg"],["SubNegMonoid","toAddMonoid"],["NegZeroClass","toZero"],["AddGroup"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Neg","neg"],["instHAdd"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["SubNegZeroMonoid","toNegZeroClass"],["OfNat","ofNat"],["AddGroup","toSubtractionMonoid"],["eq_neg_iff_add_eq_zero"],["HAdd","hAdd"],["NegZeroClass","toNeg"],["SubNegMonoid","toAddMonoid"],["NegZeroClass","toZero"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["propext"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_C"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3 : Field.{u} K] (k : K), Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3 (DFunLike.coe.{succ u, succ u, succ u} (RingHom.{u, u} K (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (Semiring.toNonAssocSemiring.{u} K (CommSemiring.toSemiring.{u} K (CommRing.toCommSemiring.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))))) (Semiring.toNonAssocSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (DivisionSemiring.toSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (Semifield.toDivisionSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (Field.toSemifield.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (RatFunc.instField.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3)))))))) K (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : K) => RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (RingHom.instFunLike.{u, u} K (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (Semiring.toNonAssocSemiring.{u} K (CommSemiring.toSemiring.{u} K (CommRing.toCommSemiring.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))))) (Semiring.toNonAssocSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (DivisionSemiring.toSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (Semifield.toDivisionSemiring.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (Field.toSemifield.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) (RatFunc.instField.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3)))))))) (RatFunc.C.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1525076466._hygCtx._hyg.3))) k)) (OfNat.ofNat.{0} Int 0 (instOfNat 0))","typeFull":"∀ {K : Type u} [inst : Field K] (k : K), (RatFunc.C k).intDegree = 0","typeReadable":"∀ {K : Type u} [inst : Field K] (k : K), (RatFunc.C k).intDegree = 0","typeReferences":[["RingHom"],["CommRing","toCommSemiring"],["EuclideanDomain","toCommRing"],["instIsDomain"],["Field"],["CommSemiring","toSemiring"],["RingHom","instFunLike"],["RatFunc","instField"],["RatFunc","C"],["DivisionSemiring","toSemiring"],["DFunLike","coe"],["OfNat","ofNat"],["Int"],["RatFunc"],["Semiring","toNonAssocSemiring"],["instOfNat"],["Field","toEuclideanDomain"],["Field","toSemifield"],["Semifield","toDivisionSemiring"],["Eq"],["RatFunc","intDegree"]],"valueReferences":[["Polynomial","natDegree_one"],["Int","instSub"],["RingHom"],["Semiring","toNonAssocSemiring"],["RatFunc","denom"],["SubNegMonoid","toSub"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["RatFunc","num_C"],["Int","instAddGroup"],["sub_self"],["Polynomial","natDegree_C"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Polynomial","instOne"],["Polynomial"],["Nat"],["instOfNat"],["Eq","refl"],["RatFunc","num"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Nat","cast"],["EuclideanDomain","toCommRing"],["Polynomial","natDegree"],["RingHom","instFunLike"],["RatFunc","C"],["DFunLike","coe"],["congrArg"],["RatFunc"],["RatFunc","denom_C"],["Field","toEuclideanDomain"],["instOfNatNat"],["Zero","toOfNat0"],["Eq"],["instNatCastInt"],["CommRing","toCommSemiring"],["instIsDomain"],["Polynomial","C"],["CommSemiring","toSemiring"],["RatFunc","instField"],["Polynomial","semiring"],["OfNat","ofNat"],["Int"],["One","toOfNat1"],["SubNegMonoid","toAddMonoid"],["RatFunc","intDegree","eq_1"],["Field","toSemifield"],["instHSub"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_X"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.2115570599._hygCtx._hyg.3 : Field.{u} K], Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2115570599._hygCtx._hyg.3 (RatFunc.X.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2115570599._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2115570599._hygCtx._hyg.3)))) (OfNat.ofNat.{0} Int 1 (instOfNat 1))","typeFull":"∀ {K : Type u} [inst : Field K], RatFunc.X.intDegree = 1","typeReadable":"∀ {K : Type u} [inst : Field K], RatFunc.X.intDegree = 1","typeReferences":[["EuclideanDomain","toCommRing"],["instOfNat"],["instIsDomain"],["Field","toEuclideanDomain"],["RatFunc","X"],["Field"],["Field","toSemifield"],["Eq"],["OfNat","ofNat"],["RatFunc","intDegree"],["Int"]],"valueReferences":[["Polynomial","natDegree_one"],["Int","instSub"],["SubtractionMonoid","toSubNegZeroMonoid"],["sub_zero"],["RatFunc","num_X"],["RatFunc","denom"],["SubNegMonoid","toSub"],["HSub","hSub"],["Int","ofNat_zero"],["Semifield","toDivisionSemiring"],["RatFunc","X"],["DivisionSemiring","toSemiring"],["Polynomial","instOne"],["Polynomial"],["Nat"],["instOfNat"],["Eq","refl"],["id"],["RatFunc","num"],["NegZeroClass","toZero"],["Eq","mpr"],["Field","instIsLocalRing"],["Nat","cast"],["EuclideanDomain","toCommRing"],["Polynomial","natDegree_X"],["Polynomial","natDegree"],["SubNegZeroMonoid","toSubNegMonoid"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["Field","toEuclideanDomain"],["Int","instAddCommGroup"],["instOfNatNat"],["RatFunc","denom_X"],["Zero","toOfNat0"],["Eq"],["Int","ofNat_one"],["instNatCastInt"],["Polynomial","X"],["instIsDomain"],["IsLocalRing","toNontrivial"],["OfNat","ofNat"],["Int"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["RatFunc","intDegree","eq_1"],["Field","toSemifield"],["instHSub"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_mul"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3 : Field.{u} K] {x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))} {y : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))}, (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) x (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3)))))) -> (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) y (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3)))))) -> (Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3 (HMul.hMul.{u, u, u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) (instHMul.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3))) (RatFunc.instMul.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3)))) x y)) (HAdd.hAdd.{0, 0, 0} Int Int Int (instHAdd.{0} Int Int.instAdd) (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3 x) (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1639364214._hygCtx._hyg.3 y)))","typeFull":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K}, x ≠ 0 → y ≠ 0 → (x * y).intDegree = x.intDegree + y.intDegree","typeReadable":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K}, x ≠ 0 → y ≠ 0 → (x * y).intDegree = x.intDegree + y.intDegree","typeReferences":[["EuclideanDomain","toCommRing"],["instHAdd"],["Field"],["HMul","hMul"],["RatFunc","instMul"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["RatFunc"],["RatFunc","instZero"],["Field","toEuclideanDomain"],["Int","instAdd"],["instHMul"],["Ne"],["Zero","toOfNat0"],["Eq"],["RatFunc","intDegree"]],"valueReferences":[["RatFunc","denom_ne_zero"],["instAddNat"],["Int","instSub"],["Eq","trans"],["AddCommGroup","toAddGroup"],["Polynomial","instNoZeroDivisors"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["RatFunc","denom"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["Eq","symm"],["Polynomial","instMul"],["HSub","hSub"],["RatFunc","num_ne_zero"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Int","instAddGroup"],["sub_sub_eq_add_sub"],["Polynomial","natDegree_mul"],["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_mul","_simp_1_2"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Polynomial"],["Nat"],["AddMonoidWithOne","toNatCast"],["SubtractionMonoid","toSubNegMonoid"],["Eq","refl"],["RatFunc","num"],["id"],["Int","instCharZero"],["instHMul"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["Nat","cast_inj","_simp_1"],["IsDomain","to_noZeroDivisors"],["Nat","cast"],["EuclideanDomain","toCommRing"],["Nat","cast_add","_simp_1"],["sub_add"],["Polynomial","natDegree"],["mul_ne_zero"],["SubtractionCommMonoid","toSubtractionMonoid"],["RatFunc","instMul"],["Int","instRing"],["congrArg"],["add_sub"],["RatFunc","instZero"],["RatFunc"],["Int","instAddCommGroup"],["Field","toEuclideanDomain"],["congr"],["RatFunc","num_denom_mul"],["Int","instAdd"],["congrFun'"],["Eq"],["instNatCastInt"],["instIsDomain"],["instHAdd"],["Polynomial","instZero"],["RatFunc","instField"],["AddZero","toAdd"],["Int"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["sub_sub"],["Field","toSemifield"],["instHSub"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_one"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.2299563304._hygCtx._hyg.3 : Field.{u} K], Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2299563304._hygCtx._hyg.3 (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2299563304._hygCtx._hyg.3))) 1 (One.toOfNat1.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2299563304._hygCtx._hyg.3))) (RatFunc.instOne.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.2299563304._hygCtx._hyg.3)))))) (OfNat.ofNat.{0} Int 0 (instOfNat 0))","typeFull":"∀ {K : Type u} [inst : Field K], RatFunc.intDegree 1 = 0","typeReadable":"∀ {K : Type u} [inst : Field K], RatFunc.intDegree 1 = 0","typeReferences":[["RatFunc"],["EuclideanDomain","toCommRing"],["instOfNat"],["One","toOfNat1"],["Field","toEuclideanDomain"],["Field"],["Eq"],["OfNat","ofNat"],["RatFunc","instOne"],["RatFunc","intDegree"],["Int"]],"valueReferences":[["Int","instSub"],["EuclideanDomain","toCommRing"],["Nat","cast"],["Polynomial","natDegree"],["congrArg"],["RatFunc"],["RatFunc","denom"],["Field","toEuclideanDomain"],["SubNegMonoid","toSub"],["HSub","hSub"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["Eq"],["Int","instAddGroup"],["instNatCastInt"],["sub_self"],["RatFunc","num_one"],["RatFunc","denom_one"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Polynomial","instOne"],["OfNat","ofNat"],["RatFunc","instOne"],["Int"],["Polynomial"],["SubNegMonoid","toAddMonoid"],["instOfNat"],["One","toOfNat1"],["Eq","refl"],["RatFunc","intDegree","eq_1"],["Field","toSemifield"],["RatFunc","num"],["id"],["Eq","mpr"],["instHSub"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_inv"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.1248326587._hygCtx._hyg.3 : Field.{u} K] (x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1248326587._hygCtx._hyg.3))), Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1248326587._hygCtx._hyg.3 (Inv.inv.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1248326587._hygCtx._hyg.3))) (RatFunc.instInv.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1248326587._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1248326587._hygCtx._hyg.3))) x)) (Neg.neg.{0} Int Int.instNegInt (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.1248326587._hygCtx._hyg.3 x))","typeFull":"∀ {K : Type u} [inst : Field K] (x : RatFunc K), x⁻¹.intDegree = -x.intDegree","typeReadable":"∀ {K : Type u} [inst : Field K] (x : RatFunc K), x⁻¹.intDegree = -x.intDegree","typeReferences":[["Inv","inv"],["EuclideanDomain","toCommRing"],["instIsDomain"],["Field"],["Neg","neg"],["RatFunc","instInv"],["Int","instNegInt"],["Int"],["RatFunc"],["Field","toEuclideanDomain"],["Field","toSemifield"],["Eq"],["RatFunc","intDegree"]],"valueReferences":[["DivInvMonoid","toInv"],["SubtractionMonoid","toSubNegZeroMonoid"],["inv_zero"],["Eq","trans"],["RatFunc","intDegree_one"],["GroupWithZero","toDivInvMonoid"],["Classical","propDecidable"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["AddGroup","toSubtractionMonoid"],["not_false_eq_true"],["Eq","symm"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["neg_zero"],["inv_mul_cancel₀"],["inv_ne_zero"],["Int","instAddGroup"],["MulOne","toOne"],["Neg","neg"],["RatFunc","intDegree_zero"],["DivisionSemiring","toGroupWithZero"],["AddZeroClass","toAddZero"],["Int","instNegInt"],["instOfNat"],["MulZeroOneClass","toMulZeroClass"],["eq_false"],["NegZeroClass","toZero"],["instHMul"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["RatFunc","intDegree_mul"],["EuclideanDomain","toCommRing"],["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_inv","_simp_1_1"],["MulZeroOneClass","toMulOneClass"],["SubtractionCommMonoid","toSubtractionMonoid"],["RatFunc","instMul"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["RatFunc","instZero"],["RatFunc"],["Field","toEuclideanDomain"],["Int","instAddCommGroup"],["congr"],["Int","instAdd"],["GroupWithZero","toMonoidWithZero"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["Not"],["Inv","inv"],["instIsDomain"],["True"],["instHAdd"],["RatFunc","instInv"],["RatFunc","instField"],["AddZero","toAdd"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["eq_self"],["NegZeroClass","toNeg"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["of_eq_true"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Field","toSemifield"],["False"],["dite"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3 : Field.{u} K] {x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))}, (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))) x (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))))) -> (forall {s : Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))}, (Ne.{succ u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) s (OfNat.ofNat.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) 0 (Zero.toOfNat0.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (Polynomial.instZero.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))))))) -> (Eq.{1} Int (HSub.hSub.{0, 0, 0} Int Int Int (instHSub.{0} Int Int.instSub) (Nat.cast.{0} Int instNatCastInt (Polynomial.natDegree.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))) (HMul.hMul.{u, u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (instHMul.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (Polynomial.instMul.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))))) (RatFunc.num.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3 x) s))) (Nat.cast.{0} Int instNatCastInt (Polynomial.natDegree.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))) (HMul.hMul.{u, u, u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (instHMul.{u} (Polynomial.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3)))) (Polynomial.instMul.{u} K (DivisionSemiring.toSemiring.{u} K (Semifield.toDivisionSemiring.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3))))) s (RatFunc.denom.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3 x))))) (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.689123384._hygCtx._hyg.3 x)))","typeFull":"∀ {K : Type u} [inst : Field K] {x : RatFunc K},\n  x ≠ 0 → ∀ {s : Polynomial K}, s ≠ 0 → ↑(x.num * s).natDegree - ↑(s * x.denom).natDegree = x.intDegree","typeReadable":"∀ {K : Type u} [inst : Field K] {x : RatFunc K},\n  x ≠ 0 → ∀ {s : Polynomial K}, s ≠ 0 → ↑(x.num * s).natDegree - ↑(s * x.denom).natDegree = x.intDegree","typeReferences":[["Int","instSub"],["EuclideanDomain","toCommRing"],["Nat","cast"],["Field"],["Polynomial","natDegree"],["HMul","hMul"],["RatFunc","instZero"],["RatFunc"],["RatFunc","denom"],["Field","toEuclideanDomain"],["HSub","hSub"],["Polynomial","instMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Eq"],["instNatCastInt"],["Polynomial","instZero"],["DivisionSemiring","toSemiring"],["OfNat","ofNat"],["Int"],["Polynomial"],["Field","toSemifield"],["RatFunc","num"],["instHMul"],["Ne"],["instHSub"],["RatFunc","intDegree"]],"valueReferences":[["Polynomial","natDegree_sub_eq_of_prod_eq"],["RatFunc","denom_ne_zero"],["EuclideanDomain","toCommRing"],["IsDomain","to_noZeroDivisors"],["Polynomial","instNoZeroDivisors"],["HMul","hMul"],["NonUnitalSemiring","toSemigroupWithZero"],["mul_assoc"],["mul_ne_zero"],["CommRing","toNonUnitalCommRing"],["congrArg"],["Semigroup","toMul"],["RatFunc","denom"],["Field","toEuclideanDomain"],["Polynomial","instMul"],["RatFunc","num_ne_zero"],["Semifield","toDivisionSemiring"],["Eq"],["instIsDomain"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["Polynomial","instZero"],["DivisionSemiring","toSemiring"],["NonUnitalCommRing","toNonUnitalCommSemiring"],["Polynomial","commRing"],["Polynomial"],["Eq","refl"],["id"],["RatFunc","num"],["Field","toSemifield"],["instHMul"],["Eq","mpr"],["SemigroupWithZero","toSemigroup"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_mul","_simp_1_2"],"typeFallback":"forall {G : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Basic.3686488003._hygCtx._hyg.6 : AddCommGroup.{u_3} G] {a : G} {b : G} {c : G} {d : G}, Eq.{1} Prop (Eq.{succ u_3} G (HSub.hSub.{u_3, u_3, u_3} G G G (instHSub.{u_3} G (SubNegMonoid.toSub.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G (AddCommGroup.toAddGroup.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3686488003._hygCtx._hyg.6)))) a b) (HSub.hSub.{u_3, u_3, u_3} G G G (instHSub.{u_3} G (SubNegMonoid.toSub.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G (AddCommGroup.toAddGroup.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3686488003._hygCtx._hyg.6)))) c d)) (Eq.{succ u_3} G (HAdd.hAdd.{u_3, u_3, u_3} G G G (instHAdd.{u_3} G (AddZero.toAdd.{u_3} G (AddZeroClass.toAddZero.{u_3} G (AddMonoid.toAddZeroClass.{u_3} G (SubNegMonoid.toAddMonoid.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G (AddCommGroup.toAddGroup.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3686488003._hygCtx._hyg.6))))))) a d) (HAdd.hAdd.{u_3, u_3, u_3} G G G (instHAdd.{u_3} G (AddZero.toAdd.{u_3} G (AddZeroClass.toAddZero.{u_3} G (AddMonoid.toAddZeroClass.{u_3} G (SubNegMonoid.toAddMonoid.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G (AddCommGroup.toAddGroup.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3686488003._hygCtx._hyg.6))))))) c b))","typeFull":"∀ {G : Type u_3} [inst : AddCommGroup G] {a b c d : G}, (a - b = c - d) = (a + d = c + b)","typeReadable":"∀ {G : Type u_3} [inst : AddCommGroup G] {a b c d : G}, (a - b = c - d) = (a + d = c + b)","typeReferences":[["instHAdd"],["AddCommGroup","toAddGroup"],["AddCommGroup"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["sub_eq_sub_iff_add_eq_add"],["instHAdd"],["AddCommGroup","toAddGroup"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["propext"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_zero"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.3663947919._hygCtx._hyg.3 : Field.{u} K], Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.3663947919._hygCtx._hyg.3 (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.3663947919._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.3663947919._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.3663947919._hygCtx._hyg.3)))))) (OfNat.ofNat.{0} Int 0 (instOfNat 0))","typeFull":"∀ {K : Type u} [inst : Field K], RatFunc.intDegree 0 = 0","typeReadable":"∀ {K : Type u} [inst : Field K], RatFunc.intDegree 0 = 0","typeReferences":[["RatFunc","instZero"],["RatFunc"],["EuclideanDomain","toCommRing"],["instOfNat"],["Field","toEuclideanDomain"],["Field"],["Zero","toOfNat0"],["Eq"],["OfNat","ofNat"],["RatFunc","intDegree"],["Int"]],"valueReferences":[["Polynomial","natDegree_one"],["Int","instSub"],["EuclideanDomain","toCommRing"],["Nat","cast"],["Polynomial","natDegree"],["congrArg"],["RatFunc","instZero"],["RatFunc"],["RatFunc","denom"],["RatFunc","num_zero"],["Field","toEuclideanDomain"],["instOfNatNat"],["SubNegMonoid","toSub"],["HSub","hSub"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["Eq"],["Int","instAddGroup"],["instNatCastInt"],["sub_self"],["Polynomial","instZero"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Polynomial","instOne"],["OfNat","ofNat"],["Int"],["Polynomial"],["Nat"],["SubNegMonoid","toAddMonoid"],["One","toOfNat1"],["instOfNat"],["Eq","refl"],["RatFunc","intDegree","eq_1"],["Field","toSemifield"],["RatFunc","num"],["id"],["Polynomial","natDegree_zero"],["Eq","mpr"],["instHSub"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["RatFunc","denom_zero"],["RatFunc","intDegree"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_div","_proof_1_3"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3 : Field.{u_1} K] {y : RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))}, (Ne.{succ u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) y (OfNat.ofNat.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instZero.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))))) -> (Not (Eq.{succ u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (Inv.inv.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (DivInvMonoid.toInv.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (DivisionRing.toDivInvMonoid.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (Field.toDivisionRing.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instField.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)) (instIsDomain.{u_1} K (Field.toSemifield.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))))) y) (OfNat.ofNat.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u_1} (RatFunc.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instZero.{u_1} K (EuclideanDomain.toCommRing.{u_1} K (Field.toEuclideanDomain.{u_1} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))))))","typeFull":"∀ {K : Type u_1} [inst : Field K] {y : RatFunc K}, y ≠ 0 → ¬y⁻¹ = 0","typeReadable":"∀ {K : Type u_1} [inst : Field K] {y : RatFunc K}, y ≠ 0 → ¬y⁻¹ = 0","typeReferences":[["Not"],["DivInvMonoid","toInv"],["Inv","inv"],["EuclideanDomain","toCommRing"],["instIsDomain"],["Field"],["Field","toDivisionRing"],["RatFunc","instField"],["OfNat","ofNat"],["DivisionRing","toDivInvMonoid"],["RatFunc"],["RatFunc","instZero"],["Field","toEuclideanDomain"],["Field","toSemifield"],["Ne"],["Zero","toOfNat0"],["Eq"]],"valueReferences":[["DivInvMonoid","toInv"],["Eq","trans"],["Lean","Grind","Semiring","toMul"],["Lean","Grind","CommRing","Stepwise","simp"],["Classical","propDecidable"],["eq_true"],["HMul","hMul"],["eagerReduce"],["IsDomain"],["Lean","Grind","CommRing","Mon","mult"],["Lean","Grind","Field","toCommRing"],["Eq","symm"],["Semifield","toDivisionSemiring"],["Lean","Grind","CommRing","Power","mk"],["Bool","true"],["Lean","Grind","CommSemiring","toSemiring"],["Neg","neg"],["Lean","Grind","Semiring","ofNat"],["Lean","Grind","not_not"],["DivisionSemiring","toSemiring"],["ite_cond_eq_false"],["Lean","Grind","nestedProof"],["DivisionRing","toDivInvMonoid"],["Int","instNegInt"],["Lean","Grind","Semiring","toNatModule"],["Lean","Grind","CommRing","inv_split"],["Nat"],["Lean","Grind","NatModule","toAddCommMonoid"],["instOfNat"],["eq_false"],["Lean","Grind","CommRing","Stepwise","core"],["Lean","Grind","CommRing","Poly","num"],["Eq","refl"],["Classical","byContradiction"],["id"],["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_div","_proof_1_2"],["instHMul"],["Lean","Grind","CommRing","Expr","var"],["EuclideanDomain","toCommRing"],["Bool"],["Lean","Grind","CommRing","Expr","mul"],["Lean","Grind","CommRing","one_eq_zero_unsat"],["Lean","RArray","leaf"],["RatFunc","instZero"],["RatFunc"],["Lean","Grind","nestedDecidable"],["Field","toEuclideanDomain"],["instOfNatNat"],["Lean","RArray","branch"],["Zero","toOfNat0"],["Lean","Grind","AddCommMonoid","toZero"],["Eq"],["Not"],["Lean","Grind","Field","toInv"],["Inv","inv"],["True"],["instIsDomain"],["ite"],["Field","toDivisionRing"],["RatFunc","instField"],["Lean","Grind","CommRing","Mon","unit"],["OfNat","ofNat"],["Int"],["Lean","Grind","CommRing","Expr","num"],["Lean","Grind","CommRing","toCommSemiring"],["of_eq_true"],["Field","toGrindField"],["Field","toSemifield"],["False"],["Lean","Grind","intro_with_eq"],["Lean","Grind","CommRing","Poly","add"]]},{"isProp":true,"kind":"theorem","name":["RatFunc","intDegree_div"],"typeFallback":"forall {K : Type.{u}} [inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3 : Field.{u} K] {x : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))} {y : RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))}, (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) x (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))))) -> (Ne.{succ u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) y (OfNat.ofNat.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) 0 (Zero.toOfNat0.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instZero.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))))) -> (Eq.{1} Int (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3 (HDiv.hDiv.{u, u, u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (instHDiv.{u} (RatFunc.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3))) (RatFunc.instDiv.{u} K (EuclideanDomain.toCommRing.{u} K (Field.toEuclideanDomain.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)) (instIsDomain.{u} K (Field.toSemifield.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3)))) x y)) (HSub.hSub.{0, 0, 0} Int Int Int (instHSub.{0} Int Int.instSub) (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3 x) (RatFunc.intDegree.{u} K inst._@.Mathlib.FieldTheory.RatFunc.Degree.186764456._hygCtx._hyg.3 y)))","typeFull":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K}, x ≠ 0 → y ≠ 0 → (x / y).intDegree = x.intDegree - y.intDegree","typeReadable":"∀ {K : Type u} [inst : Field K] {x y : RatFunc K}, x ≠ 0 → y ≠ 0 → (x / y).intDegree = x.intDegree - y.intDegree","typeReferences":[["Int","instSub"],["EuclideanDomain","toCommRing"],["instIsDomain"],["Field"],["instHDiv"],["RatFunc","instDiv"],["OfNat","ofNat"],["Int"],["HDiv","hDiv"],["RatFunc"],["RatFunc","instZero"],["Field","toEuclideanDomain"],["HSub","hSub"],["Field","toSemifield"],["Ne"],["Zero","toOfNat0"],["instHSub"],["Eq"],["RatFunc","intDegree"]],"valueReferences":[["DivInvMonoid","toInv"],["div_eq_mul_inv"],["Int","instSub"],["HMul","hMul"],["sub_eq_add_neg"],["HDiv","hDiv"],["SubNegMonoid","toSub"],["Eq","symm"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Int","instAddGroup"],["RatFunc","intDegree_inv"],["Neg","neg"],["AddZeroClass","toAddZero"],["DivisionRing","toDivInvMonoid"],["Int","instNegInt"],["DivInvMonoid","toMonoid"],["Eq","refl"],["id"],["instHMul"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["RatFunc","intDegree_mul"],["EuclideanDomain","toCommRing"],["RatFunc","instMul"],["instHDiv"],["RatFunc","instDiv"],["congrArg"],["MulOne","toMul"],["RatFunc"],["Field","toEuclideanDomain"],["Monoid","toMulOneClass"],["Int","instAdd"],["Eq"],["Inv","inv"],["instIsDomain"],["instHAdd"],["SubNegMonoid","toNeg"],["Field","toDivisionRing"],["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_div","_proof_1_1"],["RatFunc","instField"],["RatFunc","instInv"],["_private","Mathlib","FieldTheory","RatFunc","Degree",0,"RatFunc","intDegree_div","_proof_1_3"],["AddZero","toAdd"],["DivInvMonoid","toDiv"],["Int"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["Field","toSemifield"],["instHSub"],["RatFunc","intDegree"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.GroupTheory.Schreier.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","card_commutator_dvd_index_center_pow","_simp_1_3"],"typeFallback":"forall {α : Sort.{u}} {p : α -> Prop} {a1 : Subtype.{u} α (fun (x : α) => p x)} {a2 : Subtype.{u} α (fun (x : α) => p x)}, Eq.{1} Prop (Eq.{max 1 u} (Subtype.{u} α (fun (x : α) => p x)) a1 a2) (Eq.{u} α (Subtype.val.{u} α (fun (x : α) => p x) a1) (Subtype.val.{u} α (fun (x : α) => p x) a2))","typeFull":"∀ {α : Sort u} {p : α → Prop} {a1 a2 : { x // p x }}, (a1 = a2) = (↑a1 = ↑a2)","typeReadable":"∀ {α : Sort u} {p : α → Prop} {a1 a2 : { x // p x }}, (a1 = a2) = (↑a1 = ↑a2)","typeReferences":[["Subtype"],["Eq"],["Subtype","val"]],"valueReferences":[["Subtype"],["Subtype","ext_iff"],["Eq"],["Subtype","val"],["propext"]]},{"isProp":true,"kind":"theorem","name":["card_dvd_exponent_pow_rank"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3 : CommGroup.{u_1} G] [inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.6 : Group.FG.{u_1} G (CommGroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3)], Dvd.dvd.{0} Nat Nat.instDvd (Nat.card.{u_1} G) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.toPow.{0} Nat Nat.instMonoid)) (Monoid.exponent.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G (CommGroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3)))) (Group.rank.{u_1} G (CommGroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3) inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.6))","typeFull":"∀ (G : Type u_1) [inst : CommGroup G] [inst_1 : Group.FG G], Nat.card G ∣ Monoid.exponent G ^ Group.rank G","typeReadable":"∀ (G : Type u_1) [inst : CommGroup G] [inst_1 : Group.FG G], Nat.card G ∣ Monoid.exponent G ^ Group.rank G","typeReferences":[["instHPow"],["Dvd","dvd"],["CommGroup"],["Nat","card"],["HPow","hPow"],["Nat"],["Nat","instMonoid"],["Monoid","toPow"],["DivInvMonoid","toMonoid"],["Group","FG"],["Group","rank"],["Group","toDivInvMonoid"],["CommGroup","toGroup"],["Monoid","exponent"],["Nat","instDvd"]],"valueReferences":[["MonoidHom","range"],["semigroupDvd"],["PartialOrder","toPreorder"],["Finset"],["Nat","instCommMonoid"],["Membership","mem"],["Classical","propDecidable"],["MonoidHom","instFunLike"],["Exists","intro"],["Nat","card_zpowers"],["Subtype","val"],["Monoid","order_dvd_exponent"],["CommGroup","toCommMonoid"],["eq_top_iff"],["Finset","card_univ"],["Monoid","toPow"],["Finset","prod_const"],["Eq","symm"],["mul_comm"],["Monoid","toSemigroup"],["Finset","prod_dvd_prod_of_dvd"],["Group","toDivInvMonoid"],["CommGroup","toGroup"],["instHPow"],["SetLike","instMembership"],["Group","rank_spec"],["Dvd","dvd"],["And"],["Nat","card"],["Exists","casesOn"],["Nat"],["Nat","instMonoid"],["SetLike","coe"],["HasSubset","Subset"],["DivInvMonoid","toMonoid"],["Nat","card_pi"],["Nat","instSemigroup"],["Subgroup","noncommPiCoprod_mulSingle"],["Top","top"],["id"],["Pi","mulOneClass"],["Eq","mpr"],["CompleteLattice","toBoundedOrder"],["Subgroup","instCompleteLattice"],["Nat","instDvd"],["MulOneClass","toMulOne"],["Finset","instSetLike"],["Subtype"],["CommMonoid","toMonoid"],["Eq","mp"],["Pi","mulSingle"],["Subgroup","instSetLike"],["MonoidHom"],["Subgroup","one"],["DFunLike","coe"],["Dvd","dvd","trans"],["congrArg"],["Fintype","card"],["Subgroup","instTop"],["BoundedOrder","toOrderTop"],["Subgroup","toGroup"],["Monoid","toMulOneClass"],["Subgroup","noncommPiCoprod"],["Preorder","toLE"],["Eq"],["Monoid","exponent"],["Group","rank"],["propext"],["CommSemigroup","toCommMagma"],["Finset","univ"],["Set"],["Pi","group"],["Subgroup","instPartialOrder"],["Finset","card"],["orderOf"],["MonoidHom","range_eq_top"],["HPow","hPow"],["Function","Surjective"],["Subgroup","zpowers"],["Set","instHasSubset"],["Finset","prod"],["Subgroup","mem_zpowers"],["Finset","Subtype","fintype"],["CommMonoid","toCommSemigroup"],["LE","le"],["Subtype","instDecidableEq"],["Subgroup"],["Subtype","mk"],["Fintype","card_coe"],["Subgroup","closure_le"],["OrderTop","toTop"],["Subgroup","closure"],["Subgroup","card_dvd_of_surjective"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_3"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Basic.4103067159._hygCtx._hyg.6 : DivisionMonoid.{u_1} α] (a : α) (m : Int) (n : Int), Eq.{succ u_1} α (HPow.hPow.{u_1, 0, u_1} α Int α (instHPow.{u_1, 0} α Int (DivInvMonoid.toZPow.{u_1} α (DivisionMonoid.toDivInvMonoid.{u_1} α inst._@.Mathlib.Algebra.Group.Basic.4103067159._hygCtx._hyg.6))) (HPow.hPow.{u_1, 0, u_1} α Int α (instHPow.{u_1, 0} α Int (DivInvMonoid.toZPow.{u_1} α (DivisionMonoid.toDivInvMonoid.{u_1} α inst._@.Mathlib.Algebra.Group.Basic.4103067159._hygCtx._hyg.6))) a m) n) (HPow.hPow.{u_1, 0, u_1} α Int α (instHPow.{u_1, 0} α Int (DivInvMonoid.toZPow.{u_1} α (DivisionMonoid.toDivInvMonoid.{u_1} α inst._@.Mathlib.Algebra.Group.Basic.4103067159._hygCtx._hyg.6))) a (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.instMul) m n))","typeFull":"∀ {α : Type u_1} [inst : DivisionMonoid α] (a : α) (m n : ℤ), (a ^ m) ^ n = a ^ (m * n)","typeReadable":"∀ {α : Type u_1} [inst : DivisionMonoid α] (a : α) (m n : ℤ), (a ^ m) ^ n = a ^ (m * n)","typeReferences":[["instHPow"],["DivisionMonoid","toDivInvMonoid"],["DivisionMonoid"],["DivInvMonoid","toZPow"],["instHMul"],["HMul","hMul"],["HPow","hPow"],["Eq"],["Int","instMul"],["Int"]],"valueReferences":[["instHPow"],["DivisionMonoid","toDivInvMonoid"],["DivInvMonoid","toZPow"],["Eq","symm"],["instHMul"],["HMul","hMul"],["zpow_mul"],["HPow","hPow"],["Int","instMul"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","rank_le_index_mul_rank"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3 : Group.{u_1} G] (H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3) [hG : Group.FG.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3] [inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.14 : Subgroup.FiniteIndex.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3 H], LE.le.{0} Nat instLENat (Group.rank.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3 H) (Subgroup.fg_of_index_ne_zero.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3 H hG inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.14)) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) (Subgroup.index.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3 H) (Group.rank.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2496859529._hygCtx._hyg.3 hG))","typeFull":"∀ {G : Type u_1} [inst : Group G] (H : Subgroup G) [hG : Group.FG G] [inst_1 : H.FiniteIndex],\n  Group.rank ↥H ≤ H.index * Group.rank G","typeReadable":"∀ {G : Type u_1} [inst : Group G] (H : Subgroup G) [hG : Group.FG G] [inst_1 : H.FiniteIndex],\n  Group.rank ↥H ≤ H.index * Group.rank G","typeReferences":[["Group"],["Subtype"],["SetLike","instMembership"],["Subgroup","instSetLike"],["Membership","mem"],["HMul","hMul"],["Subgroup","fg_of_index_ne_zero"],["Subgroup","FiniteIndex"],["Nat"],["Subgroup","index"],["LE","le"],["Subgroup","toGroup"],["instMulNat"],["instHMul"],["Subgroup"],["Group","FG"],["instLENat"],["Group","rank"]],"valueReferences":[["Finset","instSetLike"],["Trans","trans"],["Finset"],["Subtype"],["Subgroup","instSetLike"],["Membership","mem"],["HMul","hMul"],["Subgroup","fg_of_index_ne_zero"],["Subgroup","instTop"],["Subgroup","toGroup"],["Subgroup","index"],["instMulNat"],["Eq"],["Group","rank"],["SetLike","instMembership"],["Group","rank_spec"],["congr_arg"],["Finset","card"],["And"],["Subgroup","exists_finset_card_le_mul"],["Exists","casesOn"],["Nat"],["Nat","instTransLe"],["SetLike","coe"],["LE","le"],["Top","top"],["Group","rank_le"],["instHMul"],["Subgroup"],["instTransEq_1"],["Subgroup","closure"],["instLENat"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","card_commutator_le_of_finite_commutatorSet"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.3 : Group.{u_1} G] [inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.12 : Finite.{succ u_1} (Set.Elem.{u_1} G (commutatorSet.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.3))], LE.le.{0} Nat instLENat (Nat.card.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.3)) (commutator.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.3) x))) (Subgroup.cardCommutatorBound (Nat.card.{u_1} (Set.Elem.{u_1} G (commutatorSet.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.656232834._hygCtx._hyg.3))))","typeFull":"∀ (G : Type u_1) [inst : Group G] [Finite ↑(commutatorSet G)],\n  Nat.card ↥(commutator G) ≤ Subgroup.cardCommutatorBound (Nat.card ↑(commutatorSet G))","typeReadable":"∀ (G : Type u_1) [inst : Group G] [Finite ↑(commutatorSet G)],\n  Nat.card ↥(commutator G) ≤ Subgroup.cardCommutatorBound (Nat.card ↑(commutatorSet G))","typeReferences":[["Group"],["commutatorSet"],["Subtype"],["SetLike","instMembership"],["Subgroup","cardCommutatorBound"],["Subgroup","instSetLike"],["Membership","mem"],["Nat","card"],["commutator"],["Set","Elem"],["Nat"],["LE","le"],["Subgroup"],["Finite"],["instLENat"]],"valueReferences":[["instAddNat"],["semigroupDvd"],["Subgroup","index_center_le_pow"],["Membership","mem"],["Nat","instAddCommSemigroup"],["Subgroup","center"],["instFiniteElemSubtypeMemSubgroupClosureCommutatorRepresentativesCommutatorSet"],["HMul","hMul"],["Nat","instZeroLEOneClass"],["Nat","instAddCommMonoid"],["Set","Elem"],["instNonemptyElemCommutatorSet"],["pow_succ"],["Monoid","toPow"],["Subgroup","card_commutator_dvd_index_center_pow"],["Eq","symm"],["closureCommutatorRepresentatives"],["Nat","pos_of_ne_zero"],["Nat","instIsOrderedAddMonoid"],["instHPow"],["Subgroup","FiniteIndex","index_ne_zero"],["SetLike","instMembership"],["Subgroup","cardCommutatorBound"],["covariant_swap_mul_of_covariant_mul"],["Dvd","dvd"],["Nat","instPartialOrder"],["Nat","le_of_dvd"],["card_commutatorSet_closureCommutatorRepresentatives"],["IsOrderedAddMonoid","toAddLeftMono"],["Nat","card"],["instNatPowNat"],["pow_dvd_pow"],["commutator"],["Subgroup","finiteIndex_center"],["Nat"],["Nat","instMonoid"],["Nat","instSemigroup"],["instHMul"],["covariant_swap_add_of_covariant_add"],["closureCommutatorRepresentatives_fg"],["Nat","instMulLeftMono"],["Nat","instDvd"],["MulOneClass","toMulOne"],["rank_closureCommutatorRepresentatives_le"],["commutatorSet"],["Subtype"],["pow_pos"],["Eq","mp"],["Subgroup","instSetLike"],["Nat","pow_le_pow_left"],["instPowNat"],["mul_le_mul_left"],["Nat","instMonoidWithZero"],["Nat","instPreorder"],["LE","le","trans"],["congrArg"],["Dvd","dvd","trans"],["MulOne","toMul"],["instOfNatNat"],["Monoid","toMulOneClass"],["Subgroup","index"],["Subgroup","toGroup"],["instMulNat"],["Nat","pow_le_pow_right"],["Preorder","toLE"],["Group","rank"],["instHAdd"],["LinearOrderedCommMonoidWithZero","toPosMulStrictMono"],["HPow","hPow"],["OfNat","ofNat"],["card_commutator_closureCommutatorRepresentatives"],["HAdd","hAdd"],["Nat","instLinearOrderedCommMonoidWithZero"],["Nat","instCommSemigroup"],["LE","le"],["Subgroup"],["add_le_add_left"],["instLENat"],["Finite","card_pos"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","instFiniteSubtypeMemCommutatorOfElemCommutatorSet"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Schreier.1933782359._hygCtx._hyg.3 : Group.{u_1} G] [inst._@.Mathlib.GroupTheory.Schreier.1933782359._hygCtx._hyg.12 : Finite.{succ u_1} (Set.Elem.{u_1} G (commutatorSet.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1933782359._hygCtx._hyg.3))], Finite.{succ u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1933782359._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1933782359._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1933782359._hygCtx._hyg.3)) (commutator.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1933782359._hygCtx._hyg.3) x))","typeFull":"∀ (G : Type u_1) [inst : Group G] [Finite ↑(commutatorSet G)], Finite ↥(commutator G)","typeReadable":"∀ (G : Type u_1) [inst : Group G] [Finite ↑(commutatorSet G)], Finite ↥(commutator G)","typeReferences":[["commutatorSet"],["Group"],["SetLike","instMembership"],["Subtype"],["Subgroup","instSetLike"],["Membership","mem"],["Subgroup"],["Finite"],["commutator"],["Set","Elem"]],"valueReferences":[["eq_zero_of_pow_eq_zero"],["instAddNat"],["semigroupDvd"],["Membership","mem"],["instFiniteElemSubtypeMemSubgroupClosureCommutatorRepresentativesCommutatorSet"],["Subgroup","center"],["HMul","hMul"],["Set","Elem"],["Monoid","toPow"],["Subgroup","card_commutator_dvd_index_center_pow"],["closureCommutatorRepresentatives"],["Nat","instMulZeroClass"],["instHPow"],["SetLike","instMembership"],["Subgroup","FiniteIndex","index_ne_zero"],["Dvd","dvd"],["Nat","card"],["commutator"],["Subgroup","finiteIndex_center"],["Nat"],["Nat","instMonoid"],["instHMul"],["closureCommutatorRepresentatives_fg"],["Nat","instDvd"],["commutatorSet"],["SemigroupWithZero","toMulZeroClass"],["IsDomain","to_noZeroDivisors"],["Subtype"],["Eq","mp"],["Subgroup","instSetLike"],["Nat","instMonoidWithZero"],["congrArg"],["Nat","instSemigroupWithZero"],["Nat","finite_of_card_ne_zero"],["Nat","instSemiring"],["isReduced_of_noZeroDivisors"],["instOfNatNat"],["zero_dvd_iff"],["Subgroup","index"],["Subgroup","toGroup"],["instMulNat"],["Zero","toOfNat0"],["Eq"],["propext"],["instHAdd"],["HPow","hPow"],["card_commutator_closureCommutatorRepresentatives"],["OfNat","ofNat"],["HAdd","hAdd"],["MulZeroClass","toZero"],["Subgroup"],["SemigroupWithZero","toSemigroup"],["Nat","instIsDomain"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","card_commutator_dvd_index_center_pow","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.101068378._hygCtx._hyg.3 : SemigroupWithZero.{u_1} α] (a : α), Eq.{1} Prop (Dvd.dvd.{u_1} α (semigroupDvd.{u_1} α (SemigroupWithZero.toSemigroup.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.101068378._hygCtx._hyg.3)) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (SemigroupWithZero.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.101068378._hygCtx._hyg.3))))) True","typeFull":"∀ {α : Type u_1} [inst : SemigroupWithZero α] (a : α), (a ∣ 0) = True","typeReadable":"∀ {α : Type u_1} [inst : SemigroupWithZero α] (a : α), (a ∣ 0) = True","typeReferences":[["SemigroupWithZero","toMulZeroClass"],["semigroupDvd"],["True"],["Dvd","dvd"],["MulZeroClass","toZero"],["Zero","toOfNat0"],["SemigroupWithZero","toSemigroup"],["Eq"],["SemigroupWithZero"],["OfNat","ofNat"]],"valueReferences":[["SemigroupWithZero","toMulZeroClass"],["semigroupDvd"],["dvd_zero"],["Dvd","dvd"],["MulZeroClass","toZero"],["eq_true"],["Zero","toOfNat0"],["SemigroupWithZero","toSemigroup"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["card_dvd_exponent_nsmul_rank'"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3 : AddCommGroup.{u_1} G] [inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.6 : AddGroup.FG.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3)] {n : Nat}, (forall (g : G), Eq.{succ u_1} G (HSMul.hSMul.{0, u_1, u_1} Nat G G (instHSMul.{0, u_1} Nat G (AddMonoid.toNSMul.{u_1} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3))))) n g) (OfNat.ofNat.{u_1} G 0 (Zero.toOfNat0.{u_1} G (NegZeroClass.toZero.{u_1} G (SubNegZeroMonoid.toNegZeroClass.{u_1} G (SubtractionMonoid.toSubNegZeroMonoid.{u_1} G (SubtractionCommMonoid.toSubtractionMonoid.{u_1} G (AddCommGroup.toDivisionAddCommMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3)))))))) -> (Dvd.dvd.{0} Nat Nat.instDvd (Nat.card.{u_1} G) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.toPow.{0} Nat Nat.instMonoid)) n (AddGroup.rank.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3) inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.6)))","typeFull":"∀ (G : Type u_1) [inst : AddCommGroup G] [inst_1 : AddGroup.FG G] {n : ℕ},\n  (∀ (g : G), n • g = 0) → Nat.card G ∣ n ^ AddGroup.rank G","typeReadable":"∀ (G : Type u_1) [inst : AddCommGroup G] [inst_1 : AddGroup.FG G] {n : ℕ},\n  (∀ (g : G), n • g = 0) → Nat.card G ∣ n ^ AddGroup.rank G","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["AddCommGroup","toAddGroup"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Monoid","toPow"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["AddGroup","FG"],["instHPow"],["Dvd","dvd"],["AddGroup","rank"],["AddCommGroup"],["Nat","card"],["HPow","hPow"],["OfNat","ofNat"],["Nat"],["Nat","instMonoid"],["SubNegMonoid","toAddMonoid"],["AddMonoid","toNSMul"],["AddCommGroup","toDivisionAddCommMonoid"],["HSMul","hSMul"],["NegZeroClass","toZero"],["Nat","instDvd"]],"valueReferences":[["instHPow"],["Nat","instCommMonoid"],["AddCommGroup","toAddGroup"],["AddGroup","rank"],["Nat","card"],["HPow","hPow"],["AddMonoid","exponent"],["Dvd","dvd","trans"],["card_dvd_exponent_nsmul_rank"],["Nat"],["Nat","instMonoid"],["Monoid","toPow"],["SubNegMonoid","toAddMonoid"],["AddMonoid","exponent_dvd_of_forall_nsmul_eq_zero"],["Nat","instSemigroup"],["pow_dvd_pow_of_dvd"],["AddGroup","toSubNegMonoid"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","exists_finset_card_le_mul"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 : Group.{u_1} G] (H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) [inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.12 : Subgroup.FiniteIndex.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 H] {S : Finset.{u_1} G}, (Eq.{succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (Subgroup.closure.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Finset.{u_1} G) G (Finset.instSetLike.{u_1} G) S)) (Top.top.{u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (Subgroup.instTop.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3))) -> (Exists.{succ u_1} (Finset.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x))) (fun (T : Finset.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x))) => And (LE.le.{0} Nat instLENat (Finset.card.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x)) T) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) (Subgroup.index.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 H) (Finset.card.{u_1} G S))) (Eq.{succ u_1} (Subgroup.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 H)) (Subgroup.closure.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 H) (SetLike.coe.{u_1, u_1} (Finset.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x))) (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x)) (Finset.instSetLike.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x))) T)) (Top.top.{u_1} (Subgroup.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 H)) (Subgroup.instTop.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2077639025._hygCtx._hyg.3 H))))))","typeFull":"∀ {G : Type u_1} [inst : Group G] (H : Subgroup G) [H.FiniteIndex] {S : Finset G},\n  Subgroup.closure ↑S = ⊤ → ∃ T, T.card ≤ H.index * S.card ∧ Subgroup.closure ↑T = ⊤","typeReadable":"∀ {G : Type u_1} [inst : Group G] (H : Subgroup G) [H.FiniteIndex] {S : Finset G},\n  Subgroup.closure ↑S = ⊤ → ∃ T, T.card ≤ H.index * S.card ∧ Subgroup.closure ↑T = ⊤","typeReferences":[["Finset","instSetLike"],["Subtype"],["Finset"],["Subgroup","instSetLike"],["Membership","mem"],["HMul","hMul"],["Subgroup","instTop"],["Subgroup","index"],["Subgroup","toGroup"],["instMulNat"],["Eq"],["Group"],["Exists"],["SetLike","instMembership"],["Finset","card"],["And"],["Subgroup","FiniteIndex"],["Nat"],["SetLike","coe"],["LE","le"],["Top","top"],["instHMul"],["Subgroup"],["Subgroup","closure"],["instLENat"]],"valueReferences":[["Trans","trans"],["Finset"],["Subgroup","IsComplement"],["QuotientGroup","rightRel"],["Membership","mem"],["Exists","intro"],["HMul","hMul"],["Subtype","val"],["Set","Elem"],["QuotientGroup","fintypeQuotientRightRel"],["And","intro"],["Eq","symm"],["Group","toDivInvMonoid"],["Subgroup","exists_isComplement_right"],["Finset","mul"],["instTransEq"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["SetLike","instMembership"],["Exists"],["congr_arg"],["InvOneClass","toOne"],["And"],["Set","coe_toFinset"],["Nat","card"],["FinsetCoe","fintype"],["Set","instMembership"],["QuotientGroup","instHasQuotientSubgroup"],["Exists","casesOn"],["Nat"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["Subgroup","fintypeQuotientOfFiniteIndex"],["Eq","refl"],["Nat","card_eq_fintype_card"],["Top","top"],["id"],["instHMul"],["Eq","mpr"],["instTransEq_1"],["Subgroup","IsComplement","toRightFun"],["DivisionMonoid","toDivInvOneMonoid"],["MulOneClass","toMulOne"],["Quotient"],["Classical","decEq"],["Finset","instSetLike"],["Subtype"],["HasQuotient","Quotient"],["Subgroup","instSetLike"],["Fintype","ofEquiv"],["QuotientGroup","card_quotient_rightRel"],["congrArg"],["Fintype","card"],["MulOne","toMul"],["Subgroup","instTop"],["Finset","card_image_le"],["Subgroup","IsComplement","mul_inv_toRightFun_mem"],["Monoid","toMulOneClass"],["Subgroup","index"],["Subgroup","toGroup"],["Fintype","card_congr"],["instMulNat"],["Subgroup","index_eq_card"],["Eq"],["propext"],["Finset","card_mul_le"],["Inv","inv"],["Set","mem_toFinset"],["Set"],["Finset","card"],["Set","toFinset"],["OfNat","ofNat"],["Nat","instTransLe"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["Subgroup","IsComplement","rightQuotientEquiv"],["Finset","Subtype","fintype"],["Finset","image"],["LE","le"],["Subtype","instDecidableEq"],["Subgroup"],["Fintype","card_coe"],["Subtype","mk"],["instLENat"],["Subgroup","closure"],["Subgroup","closure_mul_image_eq_top'"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_4"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3 : Semigroup.{u_1} G] (a : G) (b : G) (c : G), Eq.{succ u_1} G (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) b c)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a b) c)","typeFull":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReadable":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReferences":[["instHMul"],["HMul","hMul"],["Semigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["Eq","symm"],["instHMul"],["HMul","hMul"],["mul_assoc"],["Semigroup","toMul"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_7"],"typeFallback":"forall {G : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Basic.3787184031._hygCtx._hyg.6 : Group.{u_3} G] (a : G) (n : Int), Eq.{succ u_3} G (HMul.hMul.{u_3, u_3, u_3} G G G (instHMul.{u_3} G (MulOne.toMul.{u_3} G (MulOneClass.toMulOne.{u_3} G (Monoid.toMulOneClass.{u_3} G (DivInvMonoid.toMonoid.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3787184031._hygCtx._hyg.6)))))) a (HPow.hPow.{u_3, 0, u_3} G Int G (instHPow.{u_3, 0} G Int (DivInvMonoid.toZPow.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3787184031._hygCtx._hyg.6))) a n)) (HPow.hPow.{u_3, 0, u_3} G Int G (instHPow.{u_3, 0} G Int (DivInvMonoid.toZPow.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3787184031._hygCtx._hyg.6))) a (HAdd.hAdd.{0, 0, 0} Int Int Int (instHAdd.{0} Int Int.instAdd) (OfNat.ofNat.{0} Int 1 (instOfNat 1)) n))","typeFull":"∀ {G : Type u_3} [inst : Group G] (a : G) (n : ℤ), a * a ^ n = a ^ (1 + n)","typeReadable":"∀ {G : Type u_3} [inst : Group G] (a : G) (n : ℤ), a * a ^ n = a ^ (1 + n)","typeReferences":[["MulOneClass","toMulOne"],["instHPow"],["Group"],["instHAdd"],["DivInvMonoid","toZPow"],["HMul","hMul"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["MulOne","toMul"],["instOfNat"],["DivInvMonoid","toMonoid"],["Int","instAdd"],["Monoid","toMulOneClass"],["instHMul"],["Eq"],["Group","toDivInvMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["instHPow"],["DivInvMonoid","toZPow"],["instHAdd"],["HMul","hMul"],["HPow","hPow"],["zpow_one_add"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["instOfNat"],["Monoid","toMulOneClass"],["Int","instAdd"],["Eq","symm"],["instHMul"],["Group","toDivInvMonoid"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_2"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.1851416777._hygCtx._hyg.3 : DivInvMonoid.{u_1} G] (a : G) (n : Nat), Eq.{succ u_1} G (HPow.hPow.{u_1, 0, u_1} G Nat G (instHPow.{u_1, 0} G Nat (Monoid.toPow.{u_1} G (DivInvMonoid.toMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1851416777._hygCtx._hyg.3))) a n) (HPow.hPow.{u_1, 0, u_1} G Int G (instHPow.{u_1, 0} G Int (DivInvMonoid.toZPow.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1851416777._hygCtx._hyg.3)) a (Nat.cast.{0} Int instNatCastInt n))","typeFull":"∀ {G : Type u_1} [inst : DivInvMonoid G] (a : G) (n : ℕ), a ^ n = a ^ ↑n","typeReadable":"∀ {G : Type u_1} [inst : DivInvMonoid G] (a : G) (n : ℕ), a ^ n = a ^ ↑n","typeReferences":[["instHPow"],["DivInvMonoid"],["Nat"],["Nat","cast"],["DivInvMonoid","toMonoid"],["Monoid","toPow"],["DivInvMonoid","toZPow"],["HPow","hPow"],["Eq"],["instNatCastInt"],["Int"]],"valueReferences":[["instHPow"],["Nat"],["Nat","cast"],["DivInvMonoid","toMonoid"],["Monoid","toPow"],["DivInvMonoid","toZPow"],["Eq","symm"],["zpow_natCast"],["HPow","hPow"],["instNatCastInt"],["Int"]]},{"isProp":true,"kind":"theorem","name":["card_dvd_exponent_pow_rank'"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3 : CommGroup.{u_1} G] [inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.6 : Group.FG.{u_1} G (CommGroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3)] {n : Nat}, (forall (g : G), Eq.{succ u_1} G (HPow.hPow.{u_1, 0, u_1} G Nat G (instHPow.{u_1, 0} G Nat (Monoid.toPow.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G (CommGroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3))))) g n) (OfNat.ofNat.{u_1} G 1 (One.toOfNat1.{u_1} G (InvOneClass.toOne.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (DivisionCommMonoid.toDivisionMonoid.{u_1} G (CommGroup.toDivisionCommMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3)))))))) -> (Dvd.dvd.{0} Nat Nat.instDvd (Nat.card.{u_1} G) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.toPow.{0} Nat Nat.instMonoid)) n (Group.rank.{u_1} G (CommGroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.3) inst._@.Mathlib.GroupTheory.Schreier.380918151._hygCtx._hyg.6)))","typeFull":"∀ (G : Type u_1) [inst : CommGroup G] [inst_1 : Group.FG G] {n : ℕ},\n  (∀ (g : G), g ^ n = 1) → Nat.card G ∣ n ^ Group.rank G","typeReadable":"∀ (G : Type u_1) [inst : CommGroup G] [inst_1 : Group.FG G] {n : ℕ},\n  (∀ (g : G), g ^ n = 1) → Nat.card G ∣ n ^ Group.rank G","typeReferences":[["instHPow"],["Dvd","dvd"],["InvOneClass","toOne"],["CommGroup"],["Nat","card"],["HPow","hPow"],["CommGroup","toDivisionCommMonoid"],["OfNat","ofNat"],["Nat"],["Nat","instMonoid"],["DivInvOneMonoid","toInvOneClass"],["Monoid","toPow"],["DivInvMonoid","toMonoid"],["One","toOfNat1"],["Group","FG"],["Eq"],["Group","rank"],["DivisionCommMonoid","toDivisionMonoid"],["CommGroup","toGroup"],["Group","toDivInvMonoid"],["DivisionMonoid","toDivInvOneMonoid"],["Nat","instDvd"]],"valueReferences":[["instHPow"],["Nat","instCommMonoid"],["Nat","card"],["HPow","hPow"],["Dvd","dvd","trans"],["Nat"],["Nat","instMonoid"],["Monoid","toPow"],["DivInvMonoid","toMonoid"],["Nat","instSemigroup"],["pow_dvd_pow_of_dvd"],["card_dvd_exponent_pow_rank"],["Monoid","exponent_dvd_of_forall_pow_eq_one"],["Monoid","exponent"],["Group","toDivInvMonoid"],["CommGroup","toGroup"],["Group","rank"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","card_commutator_dvd_index_center_pow"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3 : Group.{u_1} G] [inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.12 : Finite.{succ u_1} (Set.Elem.{u_1} G (commutatorSet.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3))], Dvd.dvd.{0} Nat Nat.instDvd (Nat.card.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3)) (commutator.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3) x))) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.toPow.{0} Nat Nat.instMonoid)) (Subgroup.index.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3 (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) (Subgroup.index.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3 (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3)) (Nat.card.{u_1} (Set.Elem.{u_1} G (commutatorSet.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3126650769._hygCtx._hyg.3)))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))","typeFull":"∀ (G : Type u_1) [inst : Group G] [Finite ↑(commutatorSet G)],\n  Nat.card ↥(commutator G) �� (Subgroup.center G).index ^ ((Subgroup.center G).index * Nat.card ↑(commutatorSet G) + 1)","typeReadable":"∀ (G : Type u_1) [inst : Group G] [Finite ↑(commutatorSet G)],\n  Nat.card ↥(commutator G) ∣ (Subgroup.center G).index ^ ((Subgroup.center G).index * Nat.card ↑(commutatorSet G) + 1)","typeReferences":[["instAddNat"],["commutatorSet"],["Subtype"],["Subgroup","instSetLike"],["Membership","mem"],["Subgroup","center"],["HMul","hMul"],["Set","Elem"],["Monoid","toPow"],["instOfNatNat"],["Subgroup","index"],["instMulNat"],["Finite"],["instHPow"],["Group"],["SetLike","instMembership"],["Dvd","dvd"],["instHAdd"],["Nat","card"],["HPow","hPow"],["OfNat","ofNat"],["commutator"],["HAdd","hAdd"],["Nat"],["Nat","instMonoid"],["instHMul"],["Subgroup"],["Nat","instDvd"]],"valueReferences":[["instAddNat"],["ne_zero_of_dvd_ne_zero"],["Eq","trans"],["Abelianization","commutator_subset_ker"],["MonoidHom","instFunLike"],["Subgroup","center"],["Subtype","val"],["Set","Elem"],["Eq","symm"],["Group","toDivInvMonoid"],["CommGroup","toGroup"],["Dvd","dvd"],["MulZeroClass","zero_mul"],["commutator"],["Nat","mul_le_mul"],["DivInvMonoid","toMonoid"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["commutatorSet"],["MonoidHom"],["Nat","instMonoidWithZero"],["Nat","instPreorder"],["LE","le","trans"],["Nat","instSemigroupWithZero"],["pow_one"],["Subgroup","fg_of_index_ne_zero"],["MulOne","toMul"],["instOfNatNat"],["Subgroup","relIndex_dvd_index_of_normal"],["congr"],["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","card_commutator_dvd_index_center_pow","_simp_1_1"],["Subtype","property"],["Eq"],["Group","rank"],["Nat","instAddMonoid"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","card_commutator_dvd_index_center_pow","_simp_1_3"],["MulZeroClass","toZero"],["CommSemigroup","toSemigroup"],["instDecidableEqNat"],["Subgroup"],["dvd_trans"],["dite"],["Subgroup","rank_commutator_le_card"],["MonoidHom","transferCenterPow"],["semigroupDvd"],["Subgroup","relIndex"],["Membership","mem"],["HMul","hMul"],["Subgroup","center","isMulCommutative"],["CommGroup","toDivisionCommMonoid"],["Monoid","toPow"],["pow_succ"],["Nat","instMulZeroClass"],["Nat","pos_of_ne_zero"],["instHPow"],["SetLike","instMembership"],["MulOne","toOne"],["Nat","le_of_dvd"],["InvOneClass","toOne"],["Nat","card"],["pow_dvd_pow"],["Subgroup","card_mul_index"],["zero_add"],["Nat"],["Nat","instMonoid"],["id"],["instHMul"],["DivisionMonoid","toDivInvOneMonoid"],["Nat","instDvd"],["Subgroup","instFGSubtypeMemCommutator"],["CommGroup","ofIsMulCommutative"],["SemigroupWithZero","toMulZeroClass"],["Subtype"],["Eq","mp"],["Subgroup","instSetLike"],["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","card_commutator_dvd_index_center_pow","_simp_1_2"],["DFunLike","coe"],["congrArg"],["Subgroup","instNormalCenter"],["mul_dvd_mul"],["MonoidHom","ker"],["Subgroup","toGroup"],["Subgroup","index"],["Monoid","toMulOneClass"],["Subgroup","rank_le_index_mul_rank"],["Subgroup","subgroupOf_isMulCommutative"],["instMulNat"],["Subgroup","subgroupOf"],["Zero","toOfNat0"],["congrFun'"],["DivisionCommMonoid","toDivisionMonoid"],["instHAdd"],["Subgroup","FiniteIndex","mk"],["Nat","instCommSemigroup"],["DivInvOneMonoid","toInvOneClass"],["of_eq_true"],["One","toOfNat1"],["card_dvd_exponent_pow_rank'"],["SemigroupWithZero","toSemigroup"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_6"],"typeFallback":"forall {G : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Basic.601486174._hygCtx._hyg.6 : Group.{u_3} G] (a : G) (n : Int), Eq.{succ u_3} G (HMul.hMul.{u_3, u_3, u_3} G G G (instHMul.{u_3} G (MulOne.toMul.{u_3} G (MulOneClass.toMulOne.{u_3} G (Monoid.toMulOneClass.{u_3} G (DivInvMonoid.toMonoid.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.601486174._hygCtx._hyg.6)))))) (HPow.hPow.{u_3, 0, u_3} G Int G (instHPow.{u_3, 0} G Int (DivInvMonoid.toZPow.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.601486174._hygCtx._hyg.6))) a n) a) (HPow.hPow.{u_3, 0, u_3} G Int G (instHPow.{u_3, 0} G Int (DivInvMonoid.toZPow.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.601486174._hygCtx._hyg.6))) a (HAdd.hAdd.{0, 0, 0} Int Int Int (instHAdd.{0} Int Int.instAdd) n (OfNat.ofNat.{0} Int 1 (instOfNat 1))))","typeFull":"∀ {G : Type u_3} [inst : Group G] (a : G) (n : ℤ), a ^ n * a = a ^ (n + 1)","typeReadable":"∀ {G : Type u_3} [inst : Group G] (a : G) (n : ℤ), a ^ n * a = a ^ (n + 1)","typeReferences":[["MulOneClass","toMulOne"],["instHPow"],["Group"],["instHAdd"],["DivInvMonoid","toZPow"],["HMul","hMul"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["MulOne","toMul"],["instOfNat"],["DivInvMonoid","toMonoid"],["Int","instAdd"],["Monoid","toMulOneClass"],["instHMul"],["Eq"],["Group","toDivInvMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["instHPow"],["DivInvMonoid","toZPow"],["instHAdd"],["HMul","hMul"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["instOfNat"],["Monoid","toMulOneClass"],["Int","instAdd"],["Eq","symm"],["instHMul"],["zpow_add_one"],["Group","toDivInvMonoid"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","card_commutator_dvd_index_center_pow","_simp_1_2"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5 : Group.{u_1} G] {M : Type.{u_7}} [inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.29 : MulOneClass.{u_7} M] {f : MonoidHom.{u_1, u_7} G M (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5)))) (MulOneClass.toMulOne.{u_7} M inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.29)} {x : G}, Eq.{1} Prop (Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5)) (MonoidHom.ker.{u_1, u_7} G inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5 M inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.29 f) x) (Eq.{succ u_7} M (DFunLike.coe.{max (succ u_1) (succ u_7), succ u_1, succ u_7} (MonoidHom.{u_1, u_7} G M (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5)))) (MulOneClass.toMulOne.{u_7} M inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.29)) G (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : G) => M) (MonoidHom.instFunLike.{u_1, u_7} G M (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.5)))) (MulOneClass.toMulOne.{u_7} M inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.29)) f x) (OfNat.ofNat.{u_7} M 1 (One.toOfNat1.{u_7} M (MulOne.toOne.{u_7} M (MulOneClass.toMulOne.{u_7} M inst._@.Mathlib.Algebra.Group.Subgroup.Ker.2335166255._hygCtx._hyg.29)))))","typeFull":"∀ {G : Type u_1} [inst : Group G] {M : Type u_7} [inst_1 : MulOneClass M] {f : G →* M} {x : G}, (x ∈ f.ker) = (f x = 1)","typeReadable":"∀ {G : Type u_1} [inst : Group G] {M : Type u_7} [inst_1 : MulOneClass M] {f : G →* M} {x : G}, (x ∈ f.ker) = (f x = 1)","typeReferences":[["MulOneClass","toMulOne"],["Group"],["SetLike","instMembership"],["MulOne","toOne"],["Subgroup","instSetLike"],["MonoidHom"],["Membership","mem"],["MonoidHom","instFunLike"],["MulOneClass"],["DFunLike","coe"],["OfNat","ofNat"],["MonoidHom","ker"],["One","toOfNat1"],["DivInvMonoid","toMonoid"],["Monoid","toMulOneClass"],["Subgroup"],["Eq"],["Group","toDivInvMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["SetLike","instMembership"],["MulOne","toOne"],["Subgroup","instSetLike"],["Membership","mem"],["MonoidHom"],["MonoidHom","instFunLike"],["DFunLike","coe"],["OfNat","ofNat"],["MonoidHom","ker"],["One","toOfNat1"],["DivInvMonoid","toMonoid"],["Monoid","toMulOneClass"],["MonoidHom","mem_ker"],["Subgroup"],["Eq"],["Group","toDivInvMonoid"],["propext"]]},{"isProp":true,"kind":"theorem","name":["MonoidHom","transferCenterPow","congr_simp"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3 : Group.{u_1} G] [inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.18 : Subgroup.FiniteIndex.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3 (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3)], Eq.{succ u_1} (MonoidHom.{u_1, u_1} G (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3)) (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) x)) (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3)) (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) x)) (Monoid.toMulOneClass.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3)) (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) x)) (DivInvMonoid.toMonoid.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3)) (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) x)) (Group.toDivInvMonoid.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3)) (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3) x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3 (Subgroup.center.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3))))))) (MonoidHom.transferCenterPow.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3 inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.18) (MonoidHom.transferCenterPow.{u_1} G inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.3 inst._@.Mathlib.GroupTheory.Transfer.489320021._hygCtx._hyg.18)","typeFull":"∀ (G : Type u_1) [inst : Group G] [inst_1 : (Subgroup.center G).FiniteIndex],\n  MonoidHom.transferCenterPow G = MonoidHom.transferCenterPow G","typeReadable":"∀ (G : Type u_1) [inst : Group G] [inst_1 : (Subgroup.center G).FiniteIndex],\n  MonoidHom.transferCenterPow G = MonoidHom.transferCenterPow G","typeReferences":[["MonoidHom","transferCenterPow"],["MulOneClass","toMulOne"],["Group"],["Subtype"],["SetLike","instMembership"],["Subgroup","instSetLike"],["MonoidHom"],["Membership","mem"],["Subgroup","center"],["Subgroup","FiniteIndex"],["DivInvMonoid","toMonoid"],["Subgroup","toGroup"],["Monoid","toMulOneClass"],["Subgroup"],["Eq"],["Group","toDivInvMonoid"]],"valueReferences":[["MonoidHom","transferCenterPow"],["MulOneClass","toMulOne"],["Subtype"],["SetLike","instMembership"],["Subgroup","instSetLike"],["MonoidHom"],["Membership","mem"],["Subgroup","center"],["DivInvMonoid","toMonoid"],["Eq","refl"],["Monoid","toMulOneClass"],["Subgroup","toGroup"],["Subgroup"],["Group","toDivInvMonoid"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","closure_mul_image_eq_top'"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 : Group.{u_1} G] {H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3} [inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.12 : DecidableEq.{succ u_1} G] {R : Finset.{u_1} G} {S : Finset.{u_1} G} (hR : Subgroup.IsComplement.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) H) (SetLike.coe.{u_1, u_1} (Finset.{u_1} G) G (Finset.instSetLike.{u_1} G) R)), (Membership.mem.{u_1, u_1} G (Finset.{u_1} G) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} G) G (Finset.instSetLike.{u_1} G)) R (OfNat.ofNat.{u_1} G 1 (One.toOfNat1.{u_1} G (InvOneClass.toOne.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3))))))) -> (Eq.{succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (Subgroup.closure.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Finset.{u_1} G) G (Finset.instSetLike.{u_1} G) S)) (Top.top.{u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (Subgroup.instTop.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3))) -> (Eq.{succ u_1} (Subgroup.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 H)) (Subgroup.closure.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 H) (SetLike.coe.{u_1, u_1} (Finset.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x))) (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) (Finset.instSetLike.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x))) (Finset.image.{u_1, u_1} G (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) (fun (a : Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) (b : Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) => Subtype.instDecidableEq.{succ u_1} G (Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H) inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.12 a b) (fun (g : G) => Subtype.mk.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)))))) g (Inv.inv.{u_1} G (InvOneClass.toInv.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)))) (Subtype.val.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Set.{u_1} G) (Set.instMembership.{u_1} G) (SetLike.coe.{u_1, u_1} (Finset.{u_1} G) G (Finset.instSetLike.{u_1} G) R) x) (Subgroup.IsComplement.toRightFun.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 H (SetLike.coe.{u_1, u_1} (Finset.{u_1} G) G (Finset.instSetLike.{u_1} G) R) hR g)))) (Subgroup.IsComplement.mul_inv_toRightFun_mem.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 H (SetLike.coe.{u_1, u_1} (Finset.{u_1} G) G (Finset.instSetLike.{u_1} G) R) hR g)) (HMul.hMul.{u_1, u_1, u_1} (Finset.{u_1} G) (Finset.{u_1} G) (Finset.{u_1} G) (instHMul.{u_1} (Finset.{u_1} G) (Finset.mul.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.12 (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3))))))) R S)))) (Top.top.{u_1} (Subgroup.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 H)) (Subgroup.instTop.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2732716266._hygCtx._hyg.3 H))))","typeFull":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} [inst_1 : DecidableEq G] {R S : Finset G}\n  (hR : Subgroup.IsComplement ↑H ↑R),\n  1 ∈ R →\n    Subgroup.closure ↑S = ⊤ → Subgroup.closure ↑(Finset.image (fun g => ⟨g * (↑(hR.toRightFun g))⁻¹, ⋯⟩) (R * S)) = ⊤","typeReadable":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} [inst_1 : DecidableEq G] {R S : Finset G}\n  (hR : Subgroup.IsComplement ↑H ↑R),\n  1 ∈ R →\n    Subgroup.closure ↑S = ⊤ → Subgroup.closure ↑(Finset.image (fun g => ⟨g * (↑(hR.toRightFun g))⁻¹, ⋯⟩) (R * S)) = ⊤","typeReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Subgroup","IsComplement"],["Finset"],["Subtype"],["Subgroup","instSetLike"],["DecidableEq"],["Membership","mem"],["HMul","hMul"],["Subtype","val"],["MulOne","toMul"],["Subgroup","instTop"],["Subgroup","IsComplement","mul_inv_toRightFun_mem"],["Monoid","toMulOneClass"],["Subgroup","toGroup"],["Eq"],["Group","toDivInvMonoid"],["Finset","mul"],["Group"],["Inv","inv"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["SetLike","instMembership"],["Set"],["InvOneClass","toOne"],["OfNat","ofNat"],["Set","instMembership"],["SetLike","coe"],["DivInvOneMonoid","toInvOneClass"],["DivInvMonoid","toMonoid"],["One","toOfNat1"],["Finset","image"],["Top","top"],["instHMul"],["Subgroup"],["Subtype","instDecidableEq"],["Subtype","mk"],["Subgroup","IsComplement","toRightFun"],["Subgroup","closure"],["DivisionMonoid","toDivInvOneMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset"],["Subtype"],["Subgroup","instSetLike"],["Membership","mem"],["HMul","hMul"],["Subtype","val"],["congrArg"],["Set","mul"],["MulOne","toMul"],["Subgroup","instTop"],["Subgroup","IsComplement","mul_inv_toRightFun_mem"],["Subgroup","toGroup"],["Monoid","toMulOneClass"],["Eq"],["Group","toDivInvMonoid"],["Finset","mul"],["Subgroup","closure_mul_image_eq_top"],["Group","toDivisionMonoid"],["Inv","inv"],["Finset","coe_mul"],["SetLike","instMembership"],["InvOneClass","toInv"],["Set"],["Set","instMembership"],["Set","image"],["SetLike","coe"],["DivInvOneMonoid","toInvOneClass"],["DivInvMonoid","toMonoid"],["Finset","image"],["Finset","coe_image"],["id"],["Top","top"],["instHMul"],["Subgroup"],["Eq","mpr"],["Subtype","instDecidableEq"],["Subtype","mk"],["Subgroup","IsComplement","toRightFun"],["Subgroup","closure"],["DivisionMonoid","toDivInvOneMonoid"]]},{"isProp":false,"kind":"definition","name":["Subgroup","cardCommutatorBound"],"typeFallback":"Nat -> Nat","typeFull":"ℕ → ℕ","typeReadable":"ℕ → ℕ","typeReferences":[["Nat"]],"valueReferences":[["instHPow"],["instAddNat"],["instHAdd"],["HMul","hMul"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["Nat","instMonoid"],["Monoid","toPow"],["instOfNatNat"],["instMulNat"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["card_dvd_exponent_nsmul_rank"],"typeFallback":"forall (G : Type.{u_1}) [inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3 : AddCommGroup.{u_1} G] [inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.6 : AddGroup.FG.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3)], Dvd.dvd.{0} Nat Nat.instDvd (Nat.card.{u_1} G) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.toPow.{0} Nat Nat.instMonoid)) (AddMonoid.exponent.{u_1} G (SubNegMonoid.toAddMonoid.{u_1} G (AddGroup.toSubNegMonoid.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3)))) (AddGroup.rank.{u_1} G (AddCommGroup.toAddGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.3) inst._@.Mathlib.GroupTheory.Schreier.2770173940._hygCtx._hyg.6))","typeFull":"∀ (G : Type u_1) [inst : AddCommGroup G] [inst_1 : AddGroup.FG G], Nat.card G ∣ AddMonoid.exponent G ^ AddGroup.rank G","typeReadable":"∀ (G : Type u_1) [inst : AddCommGroup G] [inst_1 : AddGroup.FG G], Nat.card G ∣ AddMonoid.exponent G ^ AddGroup.rank G","typeReferences":[["instHPow"],["Dvd","dvd"],["AddGroup","rank"],["AddCommGroup","toAddGroup"],["AddCommGroup"],["Nat","card"],["HPow","hPow"],["AddMonoid","exponent"],["Nat"],["Nat","instMonoid"],["Monoid","toPow"],["SubNegMonoid","toAddMonoid"],["AddGroup","toSubNegMonoid"],["AddGroup","FG"],["Nat","instDvd"]],"valueReferences":[["semigroupDvd"],["PartialOrder","toPreorder"],["AddMonoidHom","range"],["Finset"],["AddGroup","rank_spec"],["AddSubgroup","instTop"],["AddCommGroup","toAddGroup"],["Nat","instCommMonoid"],["Membership","mem"],["Classical","propDecidable"],["AddMonoidHom","range_eq_top"],["Exists","intro"],["AddSubgroup","instCompleteLattice"],["Subtype","val"],["Pi","addZeroClass"],["eq_top_iff"],["Finset","card_univ"],["Monoid","toPow"],["AddSubgroup","mem_zmultiples"],["AddMonoidHom"],["Finset","prod_const"],["AddSubgroup","closure_le"],["Eq","symm"],["AddMonoidHom","instFunLike"],["Monoid","toSemigroup"],["AddGroup","toSubNegMonoid"],["Finset","prod_dvd_prod_of_dvd"],["add_comm"],["instHPow"],["SetLike","instMembership"],["Dvd","dvd"],["AddSubgroup","instSetLike"],["AddSubgroup","instPartialOrder"],["And"],["AddSubgroup","noncommPiCoprod"],["AddZeroClass","toAddZero"],["Nat","card"],["Exists","casesOn"],["AddSubgroup"],["Nat"],["Nat","instMonoid"],["SetLike","coe"],["HasSubset","Subset"],["Nat","card_pi"],["Nat","instSemigroup"],["AddSubgroup","closure"],["Top","top"],["id"],["AddCommGroup","toAddCommMonoid"],["AddSubgroup","toAddGroup"],["Eq","mpr"],["CompleteLattice","toBoundedOrder"],["AddMonoid","toAddZeroClass"],["Nat","instDvd"],["Finset","instSetLike"],["AddSubgroup","noncommPiCoprod_single"],["Subtype"],["CommMonoid","toMonoid"],["Eq","mp"],["DFunLike","coe"],["AddMonoid","exponent"],["Dvd","dvd","trans"],["congrArg"],["Pi","addGroup"],["Fintype","card"],["BoundedOrder","toOrderTop"],["AddSubgroup","zero"],["AddCommSemigroup","toAddCommMagma"],["Preorder","toLE"],["Eq"],["propext"],["Finset","univ"],["addOrderOf"],["Set"],["AddMonoid","addOrder_dvd_exponent"],["AddGroup","rank"],["Finset","card"],["HPow","hPow"],["Function","Surjective"],["Pi","single"],["AddSubgroup","card_dvd_of_surjective"],["Set","instHasSubset"],["Finset","prod"],["SubNegMonoid","toAddMonoid"],["AddCommMonoid","toAddCommSemigroup"],["Finset","Subtype","fintype"],["LE","le"],["AddSubgroup","zmultiples"],["Subtype","instDecidableEq"],["Subtype","mk"],["Fintype","card_coe"],["Nat","card_zmultiples"],["OrderTop","toTop"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","closure_mul_image_eq_top"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 : Group.{u_1} G] {H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3} {R : Set.{u_1} G} {S : Set.{u_1} G} (hR : Subgroup.IsComplement.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) H) R), (Membership.mem.{u_1, u_1} G (Set.{u_1} G) (Set.instMembership.{u_1} G) R (OfNat.ofNat.{u_1} G 1 (One.toOfNat1.{u_1} G (InvOneClass.toOne.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3))))))) -> (Eq.{succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (Subgroup.closure.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 S) (Top.top.{u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (Subgroup.instTop.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3))) -> (Eq.{succ u_1} (Subgroup.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 H)) (Subgroup.closure.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 H) (Set.image.{u_1, u_1} G (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)) H x)) (fun (g : G) => Subtype.mk.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)) H x) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)))))) g (Inv.inv.{u_1} G (InvOneClass.toInv.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)))) (Subtype.val.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Set.{u_1} G) (Set.instMembership.{u_1} G) R x) (Subgroup.IsComplement.toRightFun.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 H R hR g)))) (Subgroup.IsComplement.mul_inv_toRightFun_mem.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 H R hR g)) (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} G) (Set.{u_1} G) (Set.{u_1} G) (instHMul.{u_1} (Set.{u_1} G) (Set.mul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3))))))) R S))) (Top.top.{u_1} (Subgroup.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 H)) (Subgroup.instTop.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3411659243._hygCtx._hyg.3 H))))","typeFull":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} {R S : Set G} (hR : Subgroup.IsComplement (↑H) R),\n  1 ∈ R → Subgroup.closure S = ⊤ → Subgroup.closure ((fun g => ⟨g * (↑(hR.toRightFun g))⁻¹, ⋯⟩) '' (R * S)) = ⊤","typeReadable":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} {R S : Set G} (hR : Subgroup.IsComplement (↑H) R),\n  1 ∈ R → Subgroup.closure S = ⊤ → Subgroup.closure ((fun g => ⟨g * (↑(hR.toRightFun g))⁻¹, ⋯⟩) '' (R * S)) = ⊤","typeReferences":[["MulOneClass","toMulOne"],["Subgroup","IsComplement"],["Subtype"],["Subgroup","instSetLike"],["Membership","mem"],["HMul","hMul"],["Subtype","val"],["Set","mul"],["MulOne","toMul"],["Subgroup","instTop"],["Subgroup","IsComplement","mul_inv_toRightFun_mem"],["Monoid","toMulOneClass"],["Subgroup","toGroup"],["Eq"],["Group","toDivInvMonoid"],["Group"],["Inv","inv"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["SetLike","instMembership"],["Set"],["InvOneClass","toOne"],["OfNat","ofNat"],["Set","instMembership"],["Set","image"],["DivInvOneMonoid","toInvOneClass"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["One","toOfNat1"],["Top","top"],["instHMul"],["Subgroup"],["Subtype","mk"],["Subgroup","IsComplement","toRightFun"],["Subgroup","closure"],["DivisionMonoid","toDivInvOneMonoid"]],"valueReferences":[["PartialOrder","toPreorder"],["Membership","mem"],["MonoidHom","instFunLike"],["HMul","hMul"],["Subtype","val"],["Set","mul"],["eq_top_iff"],["Subgroup","map"],["Eq","symm"],["Group","toDivInvMonoid"],["Subgroup","closure_mul_image_eq"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["SetLike","instMembership"],["Set","instMembership"],["Set","image"],["Subgroup","subtype"],["DivInvMonoid","toMonoid"],["ge_of_eq"],["Top","top"],["id"],["instHMul"],["Eq","mpr"],["Subgroup","IsComplement","toRightFun"],["CompleteLattice","toBoundedOrder"],["DivisionMonoid","toDivInvOneMonoid"],["Subgroup","instCompleteLattice"],["MulOneClass","toMulOne"],["Subtype"],["Subgroup","instSetLike"],["MonoidHom"],["Subgroup","map_subtype_le"],["DFunLike","coe"],["LE","le","trans"],["congrArg"],["MulOne","toMul"],["Subgroup","instTop"],["Subgroup","IsComplement","mul_inv_toRightFun_mem"],["BoundedOrder","toOrderTop"],["Monoid","toMulOneClass"],["Subgroup","toGroup"],["Eq"],["Preorder","toLE"],["propext"],["Inv","inv"],["Set"],["Subgroup","instPartialOrder"],["Set","image_image"],["MonoidHom","map_closure"],["DivInvOneMonoid","toInvOneClass"],["LE","le"],["Subgroup"],["Subtype","mk"],["Subgroup","closure"],["OrderTop","toTop"],["Subgroup","map_subtype_le_map_subtype"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_5"],"typeFallback":"forall {G : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Basic.4111739801._hygCtx._hyg.6 : Group.{u_3} G] (a : G) (m : Int) (n : Int), Eq.{succ u_3} G (HMul.hMul.{u_3, u_3, u_3} G G G (instHMul.{u_3} G (MulOne.toMul.{u_3} G (MulOneClass.toMulOne.{u_3} G (Monoid.toMulOneClass.{u_3} G (DivInvMonoid.toMonoid.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.4111739801._hygCtx._hyg.6)))))) (HPow.hPow.{u_3, 0, u_3} G Int G (instHPow.{u_3, 0} G Int (DivInvMonoid.toZPow.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.4111739801._hygCtx._hyg.6))) a m) (HPow.hPow.{u_3, 0, u_3} G Int G (instHPow.{u_3, 0} G Int (DivInvMonoid.toZPow.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.4111739801._hygCtx._hyg.6))) a n)) (HPow.hPow.{u_3, 0, u_3} G Int G (instHPow.{u_3, 0} G Int (DivInvMonoid.toZPow.{u_3} G (Group.toDivInvMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.4111739801._hygCtx._hyg.6))) a (HAdd.hAdd.{0, 0, 0} Int Int Int (instHAdd.{0} Int Int.instAdd) m n))","typeFull":"∀ {G : Type u_3} [inst : Group G] (a : G) (m n : ℤ), a ^ m * a ^ n = a ^ (m + n)","typeReadable":"∀ {G : Type u_3} [inst : Group G] (a : G) (m n : ℤ), a ^ m * a ^ n = a ^ (m + n)","typeReferences":[["MulOneClass","toMulOne"],["instHPow"],["Group"],["instHAdd"],["DivInvMonoid","toZPow"],["HMul","hMul"],["HPow","hPow"],["Int"],["HAdd","hAdd"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["Int","instAdd"],["Monoid","toMulOneClass"],["instHMul"],["Eq"],["Group","toDivInvMonoid"]],"valueReferences":[["instHPow"],["MulOneClass","toMulOne"],["DivInvMonoid","toZPow"],["instHAdd"],["HMul","hMul"],["HPow","hPow"],["Int"],["HAdd","hAdd"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["Monoid","toMulOneClass"],["Int","instAdd"],["Eq","symm"],["instHMul"],["Group","toDivInvMonoid"],["zpow_add"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","closure_mul_image_eq"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3 : Group.{u_1} G] {H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3} {R : Set.{u_1} G} {S : Set.{u_1} G} (hR : Subgroup.IsComplement.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3) H) R), (Membership.mem.{u_1, u_1} G (Set.{u_1} G) (Set.instMembership.{u_1} G) R (OfNat.ofNat.{u_1} G 1 (One.toOfNat1.{u_1} G (InvOneClass.toOne.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3))))))) -> (Eq.{succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3) (Subgroup.closure.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3 S) (Top.top.{u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3) (Subgroup.instTop.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3))) -> (Eq.{succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3) (Subgroup.closure.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3 (Set.image.{u_1, u_1} G G (fun (g : G) => HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3)))))) g (Inv.inv.{u_1} G (InvOneClass.toInv.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3)))) (Subtype.val.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Set.{u_1} G) (Set.instMembership.{u_1} G) R x) (Subgroup.IsComplement.toRightFun.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3 H R hR g)))) (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} G) (Set.{u_1} G) (Set.{u_1} G) (instHMul.{u_1} (Set.{u_1} G) (Set.mul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.254318856._hygCtx._hyg.3))))))) R S))) H)","typeFull":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} {R S : Set G} (hR : Subgroup.IsComplement (↑H) R),\n  1 ∈ R → Subgroup.closure S = ⊤ → Subgroup.closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) = H","typeReadable":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} {R S : Set G} (hR : Subgroup.IsComplement (↑H) R),\n  1 ∈ R → Subgroup.closure S = ⊤ → Subgroup.closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S)) = H","typeReferences":[["MulOneClass","toMulOne"],["Subgroup","IsComplement"],["Subgroup","instSetLike"],["Membership","mem"],["HMul","hMul"],["Subtype","val"],["Set","mul"],["MulOne","toMul"],["Subgroup","instTop"],["Monoid","toMulOneClass"],["Eq"],["Group","toDivInvMonoid"],["Group"],["Inv","inv"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["Set"],["InvOneClass","toOne"],["OfNat","ofNat"],["Set","instMembership"],["Set","image"],["DivInvOneMonoid","toInvOneClass"],["SetLike","coe"],["One","toOfNat1"],["DivInvMonoid","toMonoid"],["Top","top"],["instHMul"],["Subgroup"],["Subgroup","IsComplement","toRightFun"],["Subgroup","closure"],["DivisionMonoid","toDivInvOneMonoid"]],"valueReferences":[["DivInvMonoid","toInv"],["Subgroup","IsComplement"],["PartialOrder","toPreorder"],["Eq","trans"],["Subgroup","isComplement_iff_existsUnique_mul_inv_mem"],["Membership","mem"],["Iff","mp"],["HMul","hMul"],["CompleteBooleanAlgebra","toCompleteLattice"],["Subtype","val"],["Set","Elem"],["Set","mul"],["eq_top_iff"],["Eq","ndrec"],["Group","toDivInvMonoid"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["Exists"],["SetLike","instMembership"],["MulOne","toOne"],["InvOneClass","toOne"],["And"],["Set","instMembership"],["Exists","casesOn"],["mul_inv_cancel"],["Set","image"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["HasSubset","Subset"],["Set","instCompleteAtomicBooleanAlgebra"],["inv_one"],["id"],["Top","top"],["instHMul"],["Set","instOrderTop"],["Eq","mpr"],["Subgroup","IsComplement","toRightFun"],["DivisionMonoid","toDivInvOneMonoid"],["MulOneClass","toMulOne"],["CompleteLattice","instOmegaCompletePartialOrder"],["Subtype"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["Subgroup","instSetLike"],["OmegaCompletePartialOrder","toPartialOrder"],["congrArg"],["MulOne","toMul"],["Subgroup","mul_mem_cancel_left"],["Subgroup","closure_mul_image_mul_eq_top"],["Subgroup","IsComplement","mul_inv_toRightFun_mem"],["Monoid","toMulOneClass"],["Subtype","coe_mk"],["Preorder","toLE"],["Eq"],["le_antisymm"],["propext"],["Inv","inv"],["Subgroup","mem_top"],["Subtype","ext_iff"],["Set"],["Subgroup","instPartialOrder"],["mul_one"],["OfNat","ofNat"],["Set","instHasSubset"],["Subgroup","one_mem"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["LE","le"],["ExistsUnique"],["ExistsUnique","unique"],["Subgroup"],["Subtype","mk"],["Subgroup","closure_le"],["Subgroup","closure"],["OrderTop","toTop"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","IsComplement","toRightFun","congr_simp"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3 : Group.{u_1} G] {H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3} {H_1 : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3} (e_H : Eq.{succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3) H H_1) {T : Set.{u_1} G} (hT : Subgroup.IsComplement.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3) H) T) (a._@._internal._hyg.0 : G) (a_1._@._internal._hyg.0 : G), (Eq.{succ u_1} G a._@._internal._hyg.0 a_1._@._internal._hyg.0) -> (Eq.{succ u_1} (Set.Elem.{u_1} G T) (Subgroup.IsComplement.toRightFun.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3 H T hT a._@._internal._hyg.0) (Subgroup.IsComplement.toRightFun.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3 H_1 T (Eq.ndrec.{0, succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3) H (fun {H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3} => Subgroup.IsComplement.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Complement.3075208186._hygCtx._hyg.3) H) T) hT H_1 e_H) a_1._@._internal._hyg.0))","typeFull":"∀ {G : Type u_1} [inst : Group G] {H H_1 : Subgroup G} (e_H : H = H_1) {T : Set G} (hT : Subgroup.IsComplement (↑H) T)\n  (a a_1 : G), a = a_1 → hT.toRightFun a = ⋯.toRightFun a_1","typeReadable":"∀ {G : Type u_1} [inst : Group G] {H H_1 : Subgroup G} (e_H : H = H_1) {T : Set G} (hT : Subgroup.IsComplement (↑H) T)\n  (a a_1 : G), a = a_1 → hT.toRightFun a = ⋯.toRightFun a_1","typeReferences":[["Group"],["SetLike","coe"],["Subgroup","IsComplement"],["Set"],["Subgroup","instSetLike"],["Subgroup"],["Eq","ndrec"],["Eq"],["Subgroup","IsComplement","toRightFun"],["Set","Elem"]],"valueReferences":[["SetLike","coe"],["Subgroup","IsComplement"],["Set"],["Eq","refl"],["Subgroup","instSetLike"],["Subgroup"],["Eq","ndrec"],["Eq"],["Eq","rec"],["Subgroup","IsComplement","toRightFun"],["Set","Elem"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","closure_mul_image_mul_eq_top"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3 : Group.{u_1} G] {H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3} {R : Set.{u_1} G} {S : Set.{u_1} G} (hR : Subgroup.IsComplement.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3) H) R), (Membership.mem.{u_1, u_1} G (Set.{u_1} G) (Set.instMembership.{u_1} G) R (OfNat.ofNat.{u_1} G 1 (One.toOfNat1.{u_1} G (InvOneClass.toOne.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3))))))) -> (Eq.{succ u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3) (Subgroup.closure.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3 S) (Top.top.{u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3) (Subgroup.instTop.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3))) -> (Eq.{succ u_1} (Set.{u_1} G) (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} G) (Set.{u_1} G) (Set.{u_1} G) (instHMul.{u_1} (Set.{u_1} G) (Set.mul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3))))))) (SetLike.coe.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3) (Subgroup.closure.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3 (Set.image.{u_1, u_1} G G (fun (g : G) => HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3)))))) g (Inv.inv.{u_1} G (InvOneClass.toInv.{u_1} G (DivInvOneMonoid.toInvOneClass.{u_1} G (DivisionMonoid.toDivInvOneMonoid.{u_1} G (Group.toDivisionMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3)))) (Subtype.val.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Set.{u_1} G) (Set.instMembership.{u_1} G) R x) (Subgroup.IsComplement.toRightFun.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3 H R hR g)))) (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} G) (Set.{u_1} G) (Set.{u_1} G) (instHMul.{u_1} (Set.{u_1} G) (Set.mul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (Group.toDivInvMonoid.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.3144314984._hygCtx._hyg.3))))))) R S)))) R) (Top.top.{u_1} (Set.{u_1} G) (BooleanAlgebra.toTop.{u_1} (Set.{u_1} G) (Set.instBooleanAlgebra.{u_1} G))))","typeFull":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} {R S : Set G} (hR : Subgroup.IsComplement (↑H) R),\n  1 ∈ R → Subgroup.closure S = ⊤ → ↑(Subgroup.closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S))) * R = ⊤","typeReadable":"∀ {G : Type u_1} [inst : Group G] {H : Subgroup G} {R S : Set G} (hR : Subgroup.IsComplement (↑H) R),\n  1 ∈ R → Subgroup.closure S = ⊤ → ↑(Subgroup.closure ((fun g => g * (↑(hR.toRightFun g))⁻¹) '' (R * S))) * R = ⊤","typeReferences":[["MulOneClass","toMulOne"],["Subgroup","IsComplement"],["Subgroup","instSetLike"],["Membership","mem"],["HMul","hMul"],["Subtype","val"],["Set","mul"],["MulOne","toMul"],["Subgroup","instTop"],["Monoid","toMulOneClass"],["BooleanAlgebra","toTop"],["Eq"],["Group","toDivInvMonoid"],["Set","instBooleanAlgebra"],["Group"],["Inv","inv"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["Set"],["InvOneClass","toOne"],["OfNat","ofNat"],["Set","instMembership"],["Set","image"],["DivInvOneMonoid","toInvOneClass"],["SetLike","coe"],["One","toOfNat1"],["DivInvMonoid","toMonoid"],["Top","top"],["instHMul"],["Subgroup"],["Subgroup","IsComplement","toRightFun"],["Subgroup","closure"],["DivisionMonoid","toDivInvOneMonoid"]],"valueReferences":[["DivInvMonoid","toInv"],["neg_add_cancel"],["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_4"],["Subgroup","IsComplement","toRightFun_mul_inv_mem"],["Subgroup","IsComplement"],["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_3"],["Eq","trans"],["Subgroup","isComplement_iff_existsUnique_mul_inv_mem"],["Subgroup","mul_mem"],["Exists","intro"],["Subtype","val"],["Set","Elem"],["Set","mul"],["eq_top_iff"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["DivisionMonoid","toDivInvMonoid"],["mul_right_inj"],["Eq","symm"],["inv_inj"],["Monoid","toSemigroup"],["Eq","ndrec"],["Int","mul_neg"],["Group","toDivInvMonoid"],["Int","instAddGroup"],["rfl"],["Exists"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["InvolutiveInv","toInv"],["Set","instMembership"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["instOfNat"],["Set","instCompleteAtomicBooleanAlgebra"],["Top","top"],["Set","mul_mem_mul"],["Set","instOrderTop"],["Eq","mpr"],["Subgroup","subset_closure"],["CompleteLattice","toBoundedOrder"],["Subgroup","inv_mem"],["Subgroup","IsComplement","toRightFun"],["one_mul"],["Subgroup","instCompleteLattice"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["DivisionMonoid","toInvolutiveInv"],["OmegaCompletePartialOrder","toPartialOrder"],["MulZeroOneClass","toMulOneClass"],["Semigroup","toMul"],["Int","instMul"],["MulOne","toMul"],["congr"],["Subtype","property"],["Int","instAdd"],["BooleanAlgebra","toTop"],["zpow_zero"],["Subtype","coe_mk"],["Preorder","toLE"],["Eq"],["propext"],["zpow_one"],["mul_inv_rev"],["Subgroup","mem_top"],["Mathlib","Tactic","Group","zpow_trick_one'"],["Set"],["Subgroup","instPartialOrder"],["mul_one"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["eq_self"],["inv_inv"],["Subgroup","one_mem"],["ExistsUnique"],["Subgroup"],["And","casesOn"],["Int","instCommRing"],["PartialOrder","toPreorder"],["mul_zpow_neg_one"],["DivInvMonoid","toZPow"],["Membership","mem"],["Iff","mp"],["HMul","hMul"],["CompleteBooleanAlgebra","toCompleteLattice"],["And","intro"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocRing","toHasDistribNeg"],["AddGroup","toSubNegMonoid"],["NonAssocSemiring","toMulZeroOneClass"],["Set","instBooleanAlgebra"],["instHPow"],["Group","toDivisionMonoid"],["InvOneClass","toInv"],["SetLike","instMembership"],["NonUnitalNonAssocSemiring","toDistrib"],["MulOne","toOne"],["Neg","neg"],["InvOneClass","toOne"],["And"],["AddZeroClass","toAddZero"],["Int","instNegInt"],["Exists","casesOn"],["Set","image"],["id"],["instHMul"],["Subgroup","closure_induction_right"],["CancelMonoid","toLeftCancelMonoid"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["CompleteLattice","instOmegaCompletePartialOrder"],["Subtype"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["Subgroup","instSetLike"],["mul_assoc"],["neg_neg"],["CommRing","toNonUnitalCommRing"],["congrArg"],["Group","toCancelMonoid"],["BoundedOrder","toOrderTop"],["Subgroup","IsComplement","mul_inv_toRightFun_mem"],["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_1"],["Monoid","toMulOneClass"],["congrFun'"],["Zero","toOfNat0"],["Inv","inv"],["LeftCancelMonoid","toLeftCancelSemigroup"],["True"],["Subtype","coe_prop"],["Subtype","ext_iff"],["instHAdd"],["Distrib","toMul"],["LeftCancelSemigroup","toIsLeftCancelMul"],["DivInvOneMonoid","toInvOneClass"],["HasDistribNeg","toInvolutiveNeg"],["One","toOfNat1"],["of_eq_true"],["SubNegMonoid","toAddMonoid"],["Int","instSemiring"],["LE","le"],["top_le_iff"],["ExistsUnique","unique"],["Subtype","mk"],["OrderTop","toTop"],["Subgroup","closure"]]},{"isProp":true,"kind":"theorem","name":["Subgroup","fg_of_index_ne_zero"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3 : Group.{u_1} G] (H : Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3) [hG : Group.FG.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3] [inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.14 : Subgroup.FiniteIndex.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3 H], Group.FG.{u_1} (Subtype.{succ u_1} G (fun (x : G) => Membership.mem.{u_1, u_1} G (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (Subgroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3) G (Subgroup.instSetLike.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3)) H x)) (Subgroup.toGroup.{u_1} G inst._@.Mathlib.GroupTheory.Schreier.1663801489._hygCtx._hyg.3 H)","typeFull":"∀ {G : Type u_1} [inst : Group G] (H : Subgroup G) [hG : Group.FG G] [H.FiniteIndex], Group.FG ↥H","typeReadable":"∀ {G : Type u_1} [inst : Group G] (H : Subgroup G) [hG : Group.FG G] [H.FiniteIndex], Group.FG ↥H","typeReferences":[["Subgroup","FiniteIndex"],["Group"],["SetLike","instMembership"],["Subtype"],["Subgroup","instSetLike"],["Subgroup","toGroup"],["Membership","mem"],["Subgroup"],["Group","FG"]],"valueReferences":[["Finset","instSetLike"],["Subtype"],["Finset"],["Group","FG","out"],["Subgroup","instSetLike"],["Membership","mem"],["Exists","intro"],["HMul","hMul"],["Group","FG","mk"],["Subgroup","instTop"],["Subgroup","toGroup"],["Subgroup","index"],["instMulNat"],["Group","FG"],["Eq"],["SetLike","instMembership"],["Finset","card"],["And"],["Subgroup","exists_finset_card_le_mul"],["Exists","casesOn"],["Nat"],["SetLike","coe"],["LE","le"],["Top","top"],["instHMul"],["Subgroup"],["Subgroup","closure"],["instLENat"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","GroupTheory","Schreier",0,"Subgroup","closure_mul_image_mul_eq_top","_simp_1_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.45469925._hygCtx._hyg.3 : DivInvMonoid.{u_1} G] (x : G), Eq.{succ u_1} G (Inv.inv.{u_1} G (DivInvMonoid.toInv.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.45469925._hygCtx._hyg.3) x) (HPow.hPow.{u_1, 0, u_1} G Int G (instHPow.{u_1, 0} G Int (DivInvMonoid.toZPow.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.45469925._hygCtx._hyg.3)) x (Neg.neg.{0} Int Int.instNegInt (OfNat.ofNat.{0} Int 1 (instOfNat 1))))","typeFull":"∀ {G : Type u_1} [inst : DivInvMonoid G] (x : G), x⁻¹ = x ^ (-1)","typeReadable":"∀ {G : Type u_1} [inst : DivInvMonoid G] (x : G), x⁻¹ = x ^ (-1)","typeReferences":[["instHPow"],["DivInvMonoid","toInv"],["DivInvMonoid"],["Inv","inv"],["instOfNat"],["Neg","neg"],["DivInvMonoid","toZPow"],["HPow","hPow"],["Eq"],["OfNat","ofNat"],["Int","instNegInt"],["Int"]],"valueReferences":[["DivInvMonoid","toInv"],["instHPow"],["Inv","inv"],["instOfNat"],["Neg","neg"],["DivInvMonoid","toZPow"],["zpow_neg_one"],["Eq","symm"],["HPow","hPow"],["OfNat","ofNat"],["Int","instNegInt"],["Int"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["hasSum_nat_jacobiTheta"],"typeFallback":"forall {τ : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (Complex.im τ)) -> (HasSum.{0, 0} Complex Nat (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (fun (n : Nat) => Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal Real.pi) Complex.I) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.toPow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)))) (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Nat.cast.{0} Complex Complex.instNatCast n) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) τ)) (HDiv.hDiv.{0, 0, 0} Complex Complex Complex (instHDiv.{0} Complex (DivInvMonoid.toDiv.{0} Complex Complex.instDivInvMonoid)) (HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.instSub) (jacobiTheta τ) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne))) (OfNat.ofNat.{0} Complex 2 (instOfNatAtLeastTwo.{0} Complex 2 Complex.instNatCast (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))) (SummationFilter.unconditional.{0} Nat))","typeFull":"∀ {τ : ℂ}, 0 < τ.im → HasSum (fun n => Complex.exp (↑Real.pi * Complex.I * (↑n + 1) ^ 2 * τ)) ((jacobiTheta τ - 1) / 2)","typeReadable":"∀ {τ : ℂ}, 0 < τ.im → HasSum (fun n => Complex.exp (↑Real.pi * Complex.I * (↑n + 1) ^ 2 * τ)) ((jacobiTheta τ - 1) / 2)","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["HMul","hMul"],["Complex","instSemiring"],["Real","pi"],["Complex","ofReal"],["Complex"],["HDiv","hDiv"],["Monoid","toPow"],["Complex","instAdd"],["HSub","hSub"],["NormedField","toNormedCommRing"],["instHPow"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["Complex","instOne"],["Complex","exp"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["Nat"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["Nat","cast"],["SummationFilter","unconditional"],["UniformSpace","toTopologicalSpace"],["instHDiv"],["Nat","instNeZeroSucc"],["SeminormedRing","toPseudoMetricSpace"],["Complex","instNormedAddCommGroup"],["instOfNatNat"],["Real","instLT"],["MonoidWithZero","toMonoid"],["Complex","instNatCast"],["Zero","toOfNat0"],["Complex","im"],["Complex","I"],["instHAdd"],["Complex","instDivInvMonoid"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["LT","lt"],["HAdd","hAdd"],["jacobiTheta"],["Real","instZero"],["One","toOfNat1"],["HasSum"],["NormedCommRing","toSeminormedCommRing"],["instHSub"],["Complex","instMul"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Complex","instSub"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["AddCommGroup","toAddGroup"],["MulZeroClass","toMul"],["AddGroupWithOne","toAddMonoidWithOne"],["MonoidWithZero","toMulZeroOneClass"],["AddGroup","toSubtractionMonoid"],["Complex"],["AddCommMonoid"],["HasSum","div_const"],["Eq","symm"],["Finset","sum"],["NormedField","toNormedCommRing"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["Mathlib","Meta","NormNum","isInt_eq_true"],["DivisionSemiring","toSemiring"],["Complex","instNormedField"],["Complex","exp_zero"],["MulZeroOneClass","toMulZeroClass"],["instOfNat"],["AddMonoid","toAddSemigroup"],["NormedAddCommGroup","toNormedAddGroup"],["IsTopologicalRing","toIsSemitopologicalRing"],["Int","negOfNat"],["Eq","mpr"],["MulOneClass","toMulOne"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["AddCommMonoid","toAddMonoid"],["Int","cast"],["Mathlib","Meta","NormNum","isInt_neg"],["Nat","instNeZeroSucc"],["SeminormedAddCommGroup","toIsTopologicalAddGroup"],["SeminormedRing","toPseudoMetricSpace"],["MulOne","toMul"],["Field","toEuclideanDomain"],["Complex","instNormedAddCommGroup"],["NormedAddGroup","toAddGroup"],["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"hasSum_nat_jacobiTheta","_simp_1_1"],["Eq"],["SummationFilter"],["Complex","instCharZero"],["instNatCastInt"],["Complex","instRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["instOfNatAtLeastTwo"],["neg_sq"],["HPow","hPow"],["AddZero","toAdd"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Int","cast_neg"],["Nat","cast_add"],["jacobiTheta"],["eq_self"],["Mathlib","Meta","NormNum","instAtLeastTwo"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedDivisionRing","toDivisionRing"],["CharZero","NeZero","two"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"],["jacobiTheta₂_term","_proof_1"],["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"hasSum_nat_jacobiTheta","_simp_1_3"],["two_ne_zero'"],["Semiring","toNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Complex","instAdd"],["Monoid","toPow"],["eq_of_heq"],["jacobiTheta₂"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Int","ofNat"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddZeroClass","toAddZero"],["hasSum_nat_add_iff'"],["Int","instNegInt"],["add_sub_assoc"],["zero_add"],["Nat"],["AddMonoidWithOne","toNatCast"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["Int","cast_natCast"],["AddZero","toZero"],["Nat","cast"],["HasSum","nat_add_neg"],["Eq","mp"],["UniformSpace","toTopologicalSpace"],["EuclideanDomain","toMulDivCancelClass"],["Monoid","toMulOneClass"],["Complex","instNatCast"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Complex","instField"],["instHAdd"],["Complex","instDivInvMonoid"],["Distrib","toMul"],["Ring","toAddCommGroup"],["One","toOfNat1"],["of_eq_true"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Field","toSemifield"],["sq"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["Complex","instMul"],["NormedCommRing","toNonUnitalNormedCommRing"],["instAddNat"],["SubtractionMonoid","toSubNegZeroMonoid"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["Complex","instSemiring"],["Real","pi"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["SubNegMonoid","toSub"],["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"hasSum_nat_jacobiTheta","_simp_1_2"],["Eq","ndrec"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Mathlib","Meta","NormNum","isInt_sub"],["DivisionSemiring","toGroupWithZero"],["MulZeroClass","mul_zero"],["Complex","exp"],["MulZeroClass","zero_mul"],["SeminormedCommRing","toSeminormedRing"],["Eq","refl"],["AddMonoidWithOne","toOne"],["HEq"],["AddCommGroup","toAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Mathlib","Meta","NormNum","IsNat","to_isInt"],["Mathlib","Meta","NormNum","isNat_mul"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["SubtractionCommMonoid","toSubtractionMonoid"],["instHDiv"],["instOfNatNat"],["congr"],["propext"],["ESeminormedAddMonoid","toAddMonoid"],["Complex","I"],["Distrib","toAdd"],["Finset","range"],["Int","cast_zero"],["OfNat","ofNat"],["Int"],["Complex","addCommGroup"],["HAdd","hAdd"],["AddGroupWithOne","toAddGroup"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["optParam"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddGroupWithOne","toIntCast"],["NormedField","toNormedDivisionRing"],["Complex","instIntCast"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Nat","cast_one"],["GroupWithZero","toDivInvMonoid"],["Complex","instNeg"],["HMul","hMul"],["AddMonoidWithOne","toAddMonoid"],["Complex","ofReal"],["HDiv","hDiv"],["Ring","toAddGroupWithOne"],["IsSemitopologicalSemiring","toContinuousAdd"],["funext"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["NonUnitalNonAssocRing","toHasDistribNeg"],["neg_zero"],["hasSum_jacobiTheta₂_term"],["AddSemigroup","toAdd"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["Complex","instOne"],["Neg","neg"],["Complex","addGroupWithOne"],["Complex","instZero"],["Nat","cast_zero"],["NegZeroClass","toZero"],["instHMul"],["Mathlib","Meta","NormNum","isNat_ofNat"],["mul_div_cancel_right₀"],["SummationFilter","unconditional"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["Finset","sum_range_one"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["MonoidWithZero","toMonoid"],["congrFun'"],["Zero","toOfNat0"],["HEq","refl"],["True"],["Semiring","toMonoidWithZero"],["Eq","casesOn"],["DivInvMonoid","toDiv"],["NegZeroClass","toNeg"],["SubNegMonoid","toAddMonoid"],["HasSum"],["NormedCommRing","toSeminormedCommRing"],["Complex","instSub"]]},{"isProp":true,"kind":"theorem","name":["norm_exp_mul_sq_le"],"typeFallback":"forall {τ : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (Complex.im τ)) -> (forall (n : Int), LE.le.{0} Real Real.instLE (Norm.norm.{0} Complex Complex.instNorm (Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal Real.pi) Complex.I) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.toPow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)))) (Int.cast.{0} Complex Complex.instIntCast n) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) τ))) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.toPow.{0} Real Real.instMonoid)) (Real.exp (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) (Neg.neg.{0} Real Real.instNeg Real.pi) (Complex.im τ))) (Int.natAbs n)))","typeFull":"∀ {τ : ℂ},\n  0 < τ.im → ∀ (n : ℤ), ‖Complex.exp (↑Real.pi * Complex.I * ↑n ^ 2 * τ)‖ ≤ Real.exp (-Real.pi * τ.im) ^ n.natAbs","typeReadable":"∀ {τ : ℂ},\n  0 < τ.im → ∀ (n : ℤ), ‖Complex.exp (↑Real.pi * Complex.I * ↑n ^ 2 * τ)‖ ≤ Real.exp (-Real.pi * τ.im) ^ n.natAbs","typeReferences":[["Real","exp"],["HMul","hMul"],["Complex","instSemiring"],["Real","pi"],["Int","cast"],["Complex","ofReal"],["Complex"],["Monoid","toPow"],["instOfNatNat"],["Real","instNeg"],["Real","instLT"],["MonoidWithZero","toMonoid"],["Int","natAbs"],["Zero","toOfNat0"],["Complex","im"],["Real","instMul"],["instHPow"],["Complex","I"],["Norm","norm"],["Real"],["Neg","neg"],["Semiring","toMonoidWithZero"],["Complex","exp"],["HPow","hPow"],["OfNat","ofNat"],["Real","instLE"],["Int"],["LT","lt"],["Complex","instNorm"],["Nat"],["Real","instZero"],["Real","instMonoid"],["LE","le"],["instHMul"],["Complex","instIntCast"],["Complex","instMul"]],"valueReferences":[["Complex","ofReal_mul"],["Int","instMonoid"],["Ring","toNonAssocRing"],["Real","instAddGroup"],["Real","normedCommRing"],["sq_nonneg"],["AddGroup","toSubtractionMonoid"],["Nat","le_mul_self"],["Complex"],["Real","commRing"],["Eq","symm"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddGroup","existsAddOfLE"],["Int","cast_pow"],["Real","instCommSemiring"],["Nat","instCommSemiring"],["Ring","toSemiring"],["instIsLeftCancelAddOfAddLeftReflectLE"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["Int","negOfNat"],["Eq","mpr"],["IsOrderedRing","toPosMulMono"],["MulOneClass","toMulOne"],["Real","instIsStrictOrderedRing"],["Mathlib","Tactic","Ring","add_mul"],["Int","instNatPow"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["Int","cast"],["Int","natAbs_sq"],["LE","le","trans"],["MulOne","toMul"],["Mathlib","Tactic","Ring","neg_one_mul"],["Int","instIsStrictOrderedRing"],["Eq"],["instNatCastInt"],["Int","instLinearOrder"],["Mathlib","Tactic","Ring","neg_zero"],["Nat","cast_pow"],["HPow","hPow"],["AddZero","toAdd"],["Mathlib","Tactic","Ring","mul_congr"],["Real","instZero"],["pow_le_pow_of_le_one"],["IsLeftCancelAdd","addLeftStrictMono_of_addLeftMono"],["Real","instDivInvMonoid"],["Real","exp"],["PartialOrder","toPreorder"],["Mathlib","Meta","NormNum","IsNat","of_raw"],["Real","instMonoidWithZero"],["Mathlib","Meta","NormNum","IsInt","of_raw"],["Preorder","toLT"],["Real","pi_pos"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["AddGroup","toSubNegMonoid"],["Mathlib","Tactic","Ring","single_pow"],["Int","ofNat"],["Eq","rec"],["SemilatticeInf","toPartialOrder"],["Real"],["Real","instIsOrderedAddMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Mathlib","Tactic","Ring","mul_zero"],["AddZeroClass","toAddZero"],["Exists","casesOn"],["Nat"],["Mathlib","Tactic","Ring","atom_pf"],["AddMonoidWithOne","toNatCast"],["Real","instMonoid"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["Real","instOne"],["Mathlib","Tactic","Ring","mul_one"],["Nat","cast"],["Real","exp_pos"],["instPowNat"],["Complex","mul_I_re"],["instDistribLatticeOfLinearOrder"],["Real","instNeg"],["Monoid","toMulOneClass"],["le_of_eq"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Real","partialOrder"],["Real","instMul"],["instHAdd"],["Distrib","toMul"],["Mathlib","Tactic","Ring","cast_pos"],["Complex","re"],["Real","semiring"],["LT","lt"],["One","toOfNat1"],["Mathlib","Tactic","Ring","add_pf_add_zero"],["Mathlib","Tactic","Ring","neg_add"],["Mathlib","Tactic","Ring","neg_congr"],["sq"],["instConditionallyCompleteLinearOrder"],["IsStrictOrderedRing","toMulPosStrictMono"],["Real","exp_mul"],["Complex","instMul"],["NormedCommRing","toNonUnitalNormedCommRing"],["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["Eq","trans"],["Complex","instSemiring"],["Real","pi"],["Real","instIsOrderedCancelAddMonoid"],["Mathlib","Meta","NormNum","IsInt","to_raw_eq"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["Mathlib","Tactic","Ring","one_mul"],["Int","instAddGroup"],["Norm","norm"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Complex","exp"],["DivInvMonoid","toMonoid"],["Eq","refl"],["Real","instIntCast"],["Nat","rawCast"],["AddMonoid","toAddZeroClass"],["Mathlib","Meta","NormNum","IsNat","to_isInt"],["Nat","instAddMonoidWithOne"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["instOfNatNat"],["Mathlib","Tactic","Ring","pow_congr"],["Real","exp_lt_one_iff"],["Int","instAddLeftMono"],["Complex","im"],["Mathlib","Tactic","Ring","mul_add"],["Mathlib","Tactic","Ring","mul_pow"],["Preorder","toLE"],["Mathlib","Tactic","Ring","pow_zero"],["Distrib","toAdd"],["Complex","I"],["IsStrictOrderedRing","toIsOrderedRing"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["Complex","instNorm"],["CommRing","toRing"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddGroupWithOne","toIntCast"],["Complex","instIntCast"],["Mathlib","Tactic","Ring","zero_mul"],["DivInvMonoid","toZPow"],["HMul","hMul"],["Int","rawCast"],["Int","eq_ofNat_of_zero_le"],["Complex","ofReal"],["Complex","re_ofReal_mul"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Real","instAddCommMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["neg_lt_zero"],["Mathlib","Tactic","Ring","one_pow"],["IsOrderedAddMonoid","toAddLeftMono"],["Real","instLE"],["Complex","norm_exp"],["Nat","instMonoid"],["Iff","mpr"],["Mathlib","Tactic","Ring","mul_pf_right"],["id"],["NegZeroClass","toZero"],["instHMul"],["Mathlib","Meta","NormNum","isNat_ofNat"],["Mathlib","Tactic","Ring","neg_mul"],["Real","instRing"],["SubNegZeroMonoid","toNegZeroClass"],["Real","instIsOrderedRing"],["congrArg"],["Real","instPow"],["Real","instLT"],["MonoidWithZero","toMonoid"],["congrFun'"],["Int","natAbs"],["Zero","toOfNat0"],["Real","rpow_intCast"],["Complex","ofReal_pow"],["zpow_natCast"],["Mathlib","Meta","NormNum","isInt_mul"],["Mathlib","Tactic","Ring","of_eq"],["LT","lt","le"],["mul_neg_of_neg_of_pos"],["Lattice","toSemilatticeInf"],["Mathlib","Tactic","Ring","pow_add"],["CommSemiring","toSemiring"],["Complex","instCommSemiring"],["Semiring","toMonoidWithZero"],["NegZeroClass","toNeg"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["Mathlib","Tactic","Ring","mul_pf_left"],["Int","instSemiring"],["LE","le"],["instLENat"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"isBigO_at_im_infty_jacobiTheta_sub_one","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {f : α -> β} {l : Filter.{u_2} β} {p : α -> Prop}, Eq.{1} Prop (Filter.Eventually.{u_1} α (fun (a : α) => p a) (Filter.comap.{u_1, u_2} α β f l)) (Filter.Eventually.{u_2} β (fun (b : β) => forall (a : α), (Eq.{succ u_2} β (f a) b) -> (p a)) l)","typeFull":"∀ {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {p : α → Prop},\n  (∀ᶠ (a : α) in Filter.comap f l, p a) = ∀ᶠ (b : β) in l, ∀ (a : α), f a = b → p a","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {p : α → Prop},\n  (∀ᶠ (a : α) in Filter.comap f l, p a) = ∀ᶠ (b : β) in l, ∀ (a : α), f a = b → p a","typeReferences":[["Filter","Eventually"],["Eq"],["Filter","comap"],["Filter"]],"valueReferences":[["Filter","Eventually"],["Filter","eventually_comap"],["Eq"],["Filter","comap"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"jacobiTheta_S_smul","_simp_1_1"],"typeFallback":"Eq.{1} Complex (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZero)) (Complex.ofReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)))","typeFull":"0 = ↑0","typeReadable":"0 = ↑0","typeReferences":[["Complex"],["Complex","instZero"],["Real","instZero"],["Real"],["Zero","toOfNat0"],["Eq"],["Complex","ofReal"],["OfNat","ofNat"]],"valueReferences":[["Complex"],["Complex","instZero"],["Real","instZero"],["Real"],["Eq","symm"],["Zero","toOfNat0"],["Complex","ofReal_zero"],["OfNat","ofNat"],["Complex","ofReal"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"hasSum_nat_jacobiTheta","_simp_1_1"],"typeFallback":"forall (τ : Complex), Eq.{1} Complex (jacobiTheta₂ (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZero)) τ) (jacobiTheta τ)","typeFull":"∀ (τ : ℂ), jacobiTheta₂ 0 τ = jacobiTheta τ","typeReadable":"∀ (τ : ℂ), jacobiTheta₂ 0 τ = jacobiTheta τ","typeReferences":[["Complex"],["jacobiTheta"],["Complex","instZero"],["Zero","toOfNat0"],["jacobiTheta₂"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Complex"],["Complex","instZero"],["jacobiTheta"],["jacobiTheta_eq_jacobiTheta₂"],["Eq","symm"],["Zero","toOfNat0"],["jacobiTheta₂"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["jacobiTheta_T_sq_smul"],"typeFallback":"forall (τ : UpperHalfPlane), Eq.{1} Complex (jacobiTheta (UpperHalfPlane.coe (HSMul.hSMul.{0, 0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane UpperHalfPlane (instHSMul.{0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane (SemigroupAction.toSMul.{0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane (Monoid.toSemigroup.{0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) (Matrix.SpecialLinearGroup.monoid.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing)) (MulAction.toSemigroupAction.{0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane (Matrix.SpecialLinearGroup.monoid.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) (UpperHalfPlane.SLAction.{0} Int Int.instCommRing (Ring.toIntAlgebra.{0} Real Real.instRing))))) (HPow.hPow.{0, 0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) Nat (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) (instHPow.{0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) Nat (Monoid.toPow.{0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) (Matrix.SpecialLinearGroup.monoid.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing))) ModularGroup.T (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) τ))) (jacobiTheta (UpperHalfPlane.coe τ))","typeFull":"∀ (τ : UpperHalfPlane), jacobiTheta ↑(ModularGroup.T ^ 2 • τ) = jacobiTheta ↑τ","typeReadable":"∀ (τ : UpperHalfPlane), jacobiTheta ↑(ModularGroup.T ^ 2 • τ) = jacobiTheta ↑τ","typeReferences":[["Int","instCommRing"],["Fin"],["SemigroupAction","toSMul"],["Real","instRing"],["UpperHalfPlane"],["Fin","fintype"],["Ring","toIntAlgebra"],["Complex"],["Monoid","toPow"],["instOfNatNat"],["instHSMul"],["Matrix","SpecialLinearGroup","monoid"],["ModularGroup","T"],["Monoid","toSemigroup"],["Eq"],["instHPow"],["instDecidableEqFin"],["Real"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["jacobiTheta"],["Nat"],["HSMul","hSMul"],["UpperHalfPlane","SLAction"],["UpperHalfPlane","coe"],["MulAction","toSemigroupAction"],["Matrix","SpecialLinearGroup"]],"valueReferences":[["Int","instCommRing"],["UpperHalfPlane","modular_T_zpow_smul"],["Eq","trans"],["DivInvMonoid","toZPow"],["AddGroupWithOne","toAddMonoidWithOne"],["SemigroupAction","toSMul"],["Fin"],["Ring","toIntAlgebra"],["Complex","ofReal"],["AddAction","toAddSemigroupAction"],["Complex"],["Ring","toAddGroupWithOne"],["Monoid","toPow"],["Complex","instAdd"],["Monoid","toSemigroup"],["ModularGroup","T"],["Int","cast_ofNat"],["UpperHalfPlane","instAddActionReal"],["Group","toDivInvMonoid"],["rfl"],["instHPow"],["Real"],["instHVAdd"],["jacobiTheta_two_add"],["Nat"],["AddMonoidWithOne","toNatCast"],["instOfNat"],["Eq","refl"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["id"],["Real","instIntCast"],["UpperHalfPlane","SLAction"],["Eq","mpr"],["UpperHalfPlane","coe"],["Nat","cast"],["Matrix","SpecialLinearGroup","instPowNat"],["Real","instRing"],["UpperHalfPlane"],["Int","cast"],["Fin","fintype"],["congrArg"],["Nat","instNeZeroSucc"],["instOfNatNat"],["Complex","instNatCast"],["instHSMul"],["Matrix","SpecialLinearGroup","monoid"],["congrFun'"],["Eq"],["instNatCastInt"],["True"],["instDecidableEqFin"],["instHAdd"],["instOfNatAtLeastTwo"],["HPow","hPow"],["Matrix","SpecialLinearGroup","instGroup"],["OfNat","ofNat"],["HVAdd","hVAdd"],["Int"],["HAdd","hAdd"],["eq_self"],["jacobiTheta"],["of_eq_true"],["AddSemigroupAction","toVAdd"],["Real","instAddMonoid"],["MulAction","toSemigroupAction"],["Nat","instAtLeastTwoHAddOfNat"],["Matrix","SpecialLinearGroup"]]},{"isProp":true,"kind":"theorem","name":["jacobiTheta_eq_jacobiTheta₂"],"typeFallback":"forall (τ : Complex), Eq.{1} Complex (jacobiTheta τ) (jacobiTheta₂ (OfNat.ofNat.{0} Complex 0 (Zero.toOfNat0.{0} Complex Complex.instZero)) τ)","typeFull":"∀ (τ : ℂ), jacobiTheta τ = jacobiTheta₂ 0 τ","typeReadable":"∀ (τ : ℂ), jacobiTheta τ = jacobiTheta₂ 0 τ","typeReferences":[["Complex"],["Complex","instZero"],["jacobiTheta"],["Zero","toOfNat0"],["jacobiTheta₂"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["jacobiTheta₂_term","_proof_1"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["HMul","hMul"],["Complex","instSemiring"],["Real","pi"],["Complex","ofReal"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Monoid","toPow"],["Complex","instAdd"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["forall_congr"],["NormedField","toNormedCommRing"],["jacobiTheta₂_term"],["instHPow"],["NormedAddCommGroup","toENormedAddCommMonoid"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["MulZeroClass","mul_zero"],["Complex","exp"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["implies_true"],["Complex","instZero"],["zero_add"],["Nat"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["AddMonoid","toAddZeroClass"],["ESeminormedAddCommMonoid","toESeminormedAddMonoid"],["SummationFilter","unconditional"],["UniformSpace","toTopologicalSpace"],["Int","cast"],["congrArg"],["SeminormedRing","toPseudoMetricSpace"],["Complex","instNormedAddCommGroup"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Complex","instNatCast"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["ESeminormedAddMonoid","toAddMonoid"],["Complex","I"],["True"],["instHAdd"],["tsum_congr"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["eq_self"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NormedCommRing","toSeminormedCommRing"],["Complex","instIntCast"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["Complex","instMul"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["norm_jacobiTheta_sub_one_le"],"typeFallback":"forall {τ : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (Complex.im τ)) -> (LE.le.{0} Real Real.instLE (Norm.norm.{0} Complex Complex.instNorm (HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.instSub) (jacobiTheta τ) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne)))) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid)) (OfNat.ofNat.{0} Real 2 (instOfNatAtLeastTwo.{0} Real 2 Real.instNatCast (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (HSub.hSub.{0, 0, 0} Real Real Real (instHSub.{0} Real Real.instSub) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne)) (Real.exp (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) (Neg.neg.{0} Real Real.instNeg Real.pi) (Complex.im τ))))) (Real.exp (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) (Neg.neg.{0} Real Real.instNeg Real.pi) (Complex.im τ)))))","typeFull":"∀ {τ : ℂ}, 0 < τ.im → ‖jacobiTheta τ - 1‖ ≤ 2 / (1 - Real.exp (-Real.pi * τ.im)) * Real.exp (-Real.pi * τ.im)","typeReadable":"∀ {τ : ℂ}, 0 < τ.im → ‖jacobiTheta τ - 1‖ ≤ 2 / (1 - Real.exp (-Real.pi * τ.im)) * Real.exp (-Real.pi * τ.im)","typeReferences":[["Real","exp"],["Real","instNatCast"],["HMul","hMul"],["Real","pi"],["instHDiv"],["Nat","instNeZeroSucc"],["Complex"],["HDiv","hDiv"],["instOfNatNat"],["Real","instNeg"],["Real","instLT"],["HSub","hSub"],["Zero","toOfNat0"],["Complex","im"],["Real","instMul"],["Norm","norm"],["Real"],["Neg","neg"],["Complex","instOne"],["instOfNatAtLeastTwo"],["Real","instSub"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["Real","instLE"],["LT","lt"],["Complex","instNorm"],["jacobiTheta"],["Nat"],["Real","instZero"],["One","toOfNat1"],["LE","le"],["instHMul"],["Real","instOne"],["instHSub"],["Complex","instSub"],["Nat","instAtLeastTwoHAddOfNat"],["Real","instDivInvMonoid"]],"valueReferences":[["Real","instAddGroup"],["AddCommGroup","toAddGroup"],["Real","normedCommRing"],["AddGroupWithOne","toAddMonoidWithOne"],["AddGroup","toSubtractionMonoid"],["Complex"],["mul_comm"],["Real","lattice"],["NormedField","toNormedCommRing"],["Nat","ble"],["Real","normedAddCommGroup"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DivisionSemiring","toSemiring"],["TopologicalSpace","t2Space_of_metrizableSpace"],["Complex","instNormedField"],["instOfNat"],["AddMonoid","toAddSemigroup"],["instIsLeftCancelAddOfAddLeftReflectLE"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["Eq","mpr"],["IsOrderedRing","toPosMulMono"],["MulOneClass","toMulOne"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["Real","instIsStrictOrderedRing"],["Real","instNatCast"],["Int","cast"],["LE","le","trans"],["Nat","instNeZeroSucc"],["SeminormedRing","toPseudoMetricSpace"],["MulOne","toMul"],["Complex","instNormedAddCommGroup"],["Summable","tsum_mono"],["Eq"],["SummationFilter"],["instNatCastInt"],["Int","cast_add"],["SeminormedAddCommGroup","toAddCommGroup"],["norm_tsum_le_tsum_norm"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["instOfNatAtLeastTwo"],["HPow","hPow"],["AddZero","toAdd"],["SeminormedAddCommGroup","toSeminormedAddGroup"],["HasSum","tsum_eq"],["jacobiTheta"],["Real","instZero"],["Mathlib","Meta","NormNum","instAtLeastTwo"],["Mathlib","Meta","Positivity","pos_of_isNat"],["instHSub"],["IsLeftCancelAdd","addLeftStrictMono_of_addLeftMono"],["Real","instDivInvMonoid"],["norm_exp_mul_sq_le"],["Nat","instAtLeastTwoHAddOfNat"],["div_eq_mul_inv"],["Real","exp"],["pow_succ'"],["SummationFilter","instNeBotUnconditional"],["PartialOrder","toPreorder"],["Preorder","toLT"],["Real","pi_pos"],["CommGroupWithZero","toDivisionCommMonoid"],["Real","instNontrivial"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Complex","instAdd"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["CommMagma","toMul"],["Real"],["Real","instIsOrderedAddMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["mul_le_mul_of_nonneg_left"],["AddZeroClass","toAddZero"],["Nat"],["AddMonoidWithOne","toNatCast"],["Summable","of_nonneg_of_le"],["Real","instMonoid"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["Real","instOne"],["Nat","cast"],["Eq","mp"],["Real","exp_pos"],["UniformSpace","toTopologicalSpace"],["norm_mul"],["Real","instNeg"],["Monoid","toMulOneClass"],["Complex","instNatCast"],["Real","partialOrder"],["Real","instMul"],["Inv","inv"],["instHAdd"],["div_mul_comm"],["Distrib","toMul"],["jacobiTheta_eq_tsum_nat"],["Real","semiring"],["LT","lt"],["One","toOfNat1"],["le_of_lt"],["Field","toSemifield"],["IsStrictOrderedRing","toMulPosStrictMono"],["Complex","instMul"],["NormedCommRing","toNonUnitalNormedCommRing"],["DivInvMonoid","toInv"],["instAddNat"],["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["Complex","instSemiring"],["Real","pi"],["Real","instIsOrderedCancelAddMonoid"],["DivisionMonoid","toDivInvMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocCommSemiring"],["SubNegMonoid","toSub"],["NormedDivisionRing","toNormMulClass"],["instTransEq"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Norm","norm"],["EMetricSpace","metrizableSpace"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Complex","exp"],["SeminormedCommRing","toSeminormedRing"],["DivInvMonoid","toMonoid"],["Eq","refl"],["AddMonoidWithOne","toOne"],["AddCommGroup","toAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Bool"],["LE","le","trans_eq"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["hasSum_mul_left_iff"],["HasSum","summable"],["instHDiv"],["Real","instField"],["HasSolidNorm","orderClosedTopology"],["instOfNatNat"],["tsum"],["Int","instAdd"],["Real","exp_lt_one_iff"],["Complex","im"],["Preorder","toLE"],["propext"],["sub_eq_iff_eq_add'"],["Complex","I"],["OfNat","ofNat"],["Int"],["Complex","addCommGroup"],["HAdd","hAdd"],["Complex","instNorm"],["optParam"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddGroupWithOne","toIntCast"],["NormedField","toNormedDivisionRing"],["norm_nonneg"],["Complex","instIntCast"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Semifield","toCommGroupWithZero"],["Trans","trans"],["HMul","hMul"],["AddMonoidWithOne","toAddMonoid"],["Complex","ofReal"],["HDiv","hDiv"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["funext"],["HSub","hSub"],["Real","instAddCommMonoid"],["AddSemigroup","toAdd"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["neg_lt_zero"],["Complex","instOne"],["Complex","addGroupWithOne"],["Real","instSub"],["IsOrderedAddMonoid","toAddLeftMono"],["instIsTopologicalRingReal"],["Complex","norm_two"],["hasSum_geometric_of_lt_one"],["Real","instLE"],["Real","pseudoMetricSpace"],["Iff","mpr"],["id"],["NegZeroClass","toZero"],["instHMul"],["Mathlib","Meta","NormNum","isNat_ofNat"],["instTransEq_1"],["SummationFilter","unconditional"],["Real","instRing"],["SubNegZeroMonoid","toNegZeroClass"],["Real","instIsOrderedRing"],["congrArg"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Int","cast_one"],["Real","instLT"],["MonoidWithZero","toMonoid"],["Int","natAbs"],["congrFun'"],["Zero","toOfNat0"],["MetricSpace","toEMetricSpace"],["DivisionCommMonoid","toDivisionMonoid"],["LT","lt","le"],["Real","exp_ne_zero"],["SeminormedAddCommGroup","toNorm"],["mul_neg_of_neg_of_pos"],["Real","metricSpace"],["Semiring","toMonoidWithZero"],["DivInvMonoid","toDiv"],["instHasSolidNormReal"],["NegZeroClass","toNeg"],["SubNegMonoid","toAddMonoid"],["HasSum"],["LE","le"],["NormedCommRing","toSeminormedCommRing"],["Complex","instSub"]]},{"isProp":true,"kind":"theorem","name":["jacobiTheta_eq_tsum_nat"],"typeFallback":"forall {τ : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (Complex.im τ)) -> (Eq.{1} Complex (jacobiTheta τ) (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne)) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (OfNat.ofNat.{0} Complex 2 (instOfNatAtLeastTwo.{0} Complex 2 Complex.instNatCast (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (tsum.{0, 0} Complex Nat (ESeminormedAddCommMonoid.toAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (ENormedAddCommMonoid.toESeminormedAddCommMonoid.{0} Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (NormedAddCommGroup.toENormedAddCommMonoid.{0} Complex Complex.instNormedAddCommGroup))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (fun (n : Nat) => Complex.exp (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Complex.ofReal Real.pi) Complex.I) (HPow.hPow.{0, 0, 0} Complex Nat Complex (instHPow.{0, 0} Complex Nat (Monoid.toPow.{0} Complex (MonoidWithZero.toMonoid.{0} Complex (Semiring.toMonoidWithZero.{0} Complex Complex.instSemiring)))) (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (Nat.cast.{0} Complex Complex.instNatCast n) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne))) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) τ)) (SummationFilter.unconditional.{0} Nat)))))","typeFull":"∀ {τ : ℂ}, 0 < τ.im → jacobiTheta τ = 1 + 2 * ∑' (n : ℕ), Complex.exp (↑Real.pi * Complex.I * (↑n + 1) ^ 2 * τ)","typeReadable":"∀ {τ : ℂ}, 0 < τ.im → jacobiTheta τ = 1 + 2 * ∑' (n : ℕ), Complex.exp (↑Real.pi * Complex.I * (↑n + 1) ^ 2 * τ)","typeReferences":[["Nat","cast"],["PseudoMetricSpace","toUniformSpace"],["SummationFilter","unconditional"],["UniformSpace","toTopologicalSpace"],["HMul","hMul"],["Complex","instSemiring"],["Real","pi"],["Complex","ofReal"],["Complex"],["Nat","instNeZeroSucc"],["SeminormedRing","toPseudoMetricSpace"],["Monoid","toPow"],["Complex","instAdd"],["Complex","instNormedAddCommGroup"],["instOfNatNat"],["tsum"],["Real","instLT"],["MonoidWithZero","toMonoid"],["Complex","instNatCast"],["Zero","toOfNat0"],["Eq"],["Complex","im"],["NormedField","toNormedCommRing"],["instHPow"],["Complex","I"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Real"],["instHAdd"],["Complex","instOne"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["Complex","exp"],["HPow","hPow"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["OfNat","ofNat"],["HAdd","hAdd"],["LT","lt"],["jacobiTheta"],["Nat"],["Real","instZero"],["One","toOfNat1"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["NormedCommRing","toSeminormedCommRing"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Complex","instMul"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["SummationFilter","instNeBotUnconditional"],["PseudoMetricSpace","toUniformSpace"],["AddCommGroup","toAddGroup"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["AddGroupWithOne","toAddMonoidWithOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["Complex","instSemiring"],["Real","pi"],["two_ne_zero'"],["Complex","ofReal"],["Complex"],["HDiv","hDiv"],["Complex","instAdd"],["Monoid","toPow"],["add_sub_cancel_left"],["SubNegMonoid","toSub"],["Eq","symm"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["NormedField","toNormedCommRing"],["instHPow"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Complex","instOne"],["Complex","addGroupWithOne"],["Complex","exp"],["AddZeroClass","toAddZero"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["add_sub_assoc"],["Complex","instZero"],["Nat"],["MulZeroOneClass","toMulZeroClass"],["Eq","refl"],["NormedAddCommGroup","toNormedAddGroup"],["id"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["Eq","mpr"],["CommGroupWithZero","toGroupWithZero"],["AddMonoid","toAddZeroClass"],["Nat","cast"],["UniformSpace","toTopologicalSpace"],["SummationFilter","unconditional"],["instHDiv"],["congrArg"],["Nat","instNeZeroSucc"],["SeminormedRing","toPseudoMetricSpace"],["hasSum_nat_jacobiTheta"],["tsum"],["instOfNatNat"],["Complex","instNormedAddCommGroup"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["NormedAddGroup","toAddGroup"],["Complex","instNatCast"],["Eq"],["Complex","instCharZero"],["Complex","instField"],["Complex","I"],["instHAdd"],["Complex","instDivInvMonoid"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["HPow","hPow"],["mul_div_cancel₀"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["Complex","addCommGroup"],["HasSum","tsum_eq"],["HAdd","hAdd"],["jacobiTheta"],["SubNegMonoid","toAddMonoid"],["One","toOfNat1"],["Field","toSemifield"],["Complex","instT2Space"],["NormedCommRing","toSeminormedCommRing"],["CharZero","NeZero","two"],["instHSub"],["Complex","instSub"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Complex","instMul"],["Nat","instAtLeastTwoHAddOfNat"],["Semifield","toCommGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["isBigO_at_im_infty_jacobiTheta_sub_one"],"typeFallback":"Asymptotics.IsBigO.{0, 0, 0} Complex Complex Real Complex.instNorm Real.norm (Filter.comap.{0, 0} Complex Real Complex.im (Filter.atTop.{0} Real Real.instPreorder)) (fun (τ : Complex) => HSub.hSub.{0, 0, 0} Complex Complex Complex (instHSub.{0} Complex Complex.instSub) (jacobiTheta τ) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne))) (fun (τ : Complex) => Real.exp (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMul) (Neg.neg.{0} Real Real.instNeg Real.pi) (Complex.im τ)))","typeFull":"(fun τ => jacobiTheta τ - 1) =O[Filter.comap Complex.im Filter.atTop] fun τ => Real.exp (-Real.pi * τ.im)","typeReadable":"(fun τ => jacobiTheta τ - 1) =O[Filter.comap Complex.im Filter.atTop] fun τ => Real.exp (-Real.pi * τ.im)","typeReferences":[["Real","instMul"],["Real","exp"],["Real","instPreorder"],["Real"],["Neg","neg"],["Complex","instOne"],["Asymptotics","IsBigO"],["Real","norm"],["HMul","hMul"],["Real","pi"],["OfNat","ofNat"],["Filter","comap"],["Complex"],["Complex","instNorm"],["jacobiTheta"],["Filter","atTop"],["One","toOfNat1"],["Real","instNeg"],["HSub","hSub"],["instHMul"],["Complex","im"],["instHSub"],["Complex","instSub"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["Eq","trans"],["Real","instAddGroup"],["MulZeroClass","toMul"],["Real","normedCommRing"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["MonoidWithZero","toMulZeroOneClass"],["Real","pi"],["AddGroup","toSubtractionMonoid"],["Real","instIsOrderedCancelAddMonoid"],["norm_jacobiTheta_sub_one_le"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["neg_le_neg"],["Eq","symm"],["abs"],["Real","lattice"],["Nat","ble"],["Real","instAddCommSemigroup"],["Exists"],["Norm","norm"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionSemiring","toGroupWithZero"],["Real","instArchimedean"],["Filter","atTop"],["Real","exp_monotone"],["MulZeroOneClass","toMulZeroClass"],["instNonemptyOfInhabited"],["Eq","refl"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["instIsLeftCancelAddOfAddLeftReflectLE"],["Eq","mpr"],["covariant_swap_add_of_covariant_add"],["IsOrderedRing","toPosMulMono"],["AddMonoid","toAddZeroClass"],["Bool"],["Real","instIsStrictOrderedRing"],["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"isBigO_at_im_infty_jacobiTheta_sub_one","_simp_1_2"],["Asymptotics","IsBigOWith_def"],["Real","instNatCast"],["MulZeroOneClass","toMulOneClass"],["Asymptotics","IsBigO_def"],["instHDiv"],["Real","instField"],["LE","le","trans"],["Nat","instNeZeroSucc"],["mul_le_mul_of_nonneg_right"],["contravariant_lt_of_covariant_le"],["instOfNatNat"],["neg_mul"],["AddCommMagma","toAdd"],["IsLeftCancelAdd","addLeftReflectLE_of_addLeftReflectLT"],["Preorder","toLE"],["Eq"],["Complex","im"],["div_le_div₀"],["IsOrderedRing","toMulPosMono"],["Asymptotics","IsBigO"],["instOfNatAtLeastTwo"],["mul_one"],["AddZero","toAdd"],["OfNat","ofNat"],["HAdd","hAdd"],["Real","instInhabited"],["Complex","instNorm"],["jacobiTheta"],["Real","instZero"],["MulZeroClass","toZero"],["Mathlib","Meta","NormNum","instAtLeastTwo"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["IsLeftCancelAdd","addLeftStrictMono_of_addLeftMono"],["Mathlib","Meta","Positivity","pos_of_isNat"],["instHSub"],["Real","instDivInvMonoid"],["Nat","instAtLeastTwoHAddOfNat"],["contravariant_swap_add_of_contravariant_add"],["instIsRightCancelAddOfAddRightReflectLE"],["Filter","Eventually"],["le_refl"],["Real","exp"],["PartialOrder","toPreorder"],["LT","lt","trans_le"],["Preorder","toLT"],["Real","pi_pos"],["HMul","hMul"],["Real","instNontrivial"],["IsRightCancelAdd","addRightStrictMono_of_addRightMono"],["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"isBigO_at_im_infty_jacobiTheta_sub_one","_simp_1_1"],["GE","ge"],["MulPosStrictMono","toMulPosReflectLE"],["HDiv","hDiv"],["Semiring","toNonAssocSemiring"],["Real","instRCLike"],["Ring","toAddGroupWithOne"],["funext"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["NonUnitalNonAssocRing","toHasDistribNeg"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["InvolutiveNeg","toNeg"],["Real","instAddCommMonoid"],["Eq","rec"],["NonAssocSemiring","toMulZeroOneClass"],["RCLike","charZero_rclike"],["IsStrictOrderedRing","toPosMulStrictMono"],["Real"],["NonUnitalNonAssocSemiring","toDistrib"],["Complex","instOne"],["Neg","neg"],["Real","instIsOrderedAddMonoid"],["mul_le_mul_of_nonneg_left"],["IsOrderedAddMonoid","toAddLeftMono"],["Real","instSub"],["AddZeroClass","toAddZero"],["Real","instLE"],["sub_le_sub_left"],["Nat"],["AddMonoidWithOne","toNatCast"],["MulPosReflectLE","toMulPosReflectLT"],["NegZeroClass","toZero"],["id"],["instHMul"],["Real","linearOrder"],["Real","instOne"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["NeZero","charZero_one"],["Mathlib","Meta","NormNum","isNat_ofNat"],["PosMulStrictMono","toPosMulReflectLE"],["Asymptotics","IsBigOWith"],["Real","exp_pos"],["Real","norm_eq_abs"],["instIsDirectedOrder"],["Real","norm"],["Real","instRing"],["SubNegZeroMonoid","toNegZeroClass"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instIsOrderedRing"],["congrArg"],["Real","abs_exp"],["Real","instNeg"],["Real","instLT"],["GroupWithZero","toMonoidWithZero"],["AddCommSemigroup","toAddCommMagma"],["Zero","toOfNat0"],["sub_pos","_simp_1"],["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"isBigO_at_im_infty_jacobiTheta_sub_one","_simp_1_3"],["Real","partialOrder"],["Real","instMul"],["True"],["instHAdd"],["Real","exp_lt_one_iff","_simp_1"],["Distrib","toMul"],["Real","instZeroLEOneClass"],["Left","neg_neg_iff","_simp_1"],["zero_lt_one"],["Real","semiring"],["Filter","comap"],["DivInvMonoid","toDiv"],["LT","lt"],["NegZeroClass","toNeg"],["SubNegMonoid","toAddMonoid"],["of_eq_true"],["One","toOfNat1"],["HasDistribNeg","toInvolutiveNeg"],["le_of_lt"],["LE","le"],["Real","instAddCommGroup"],["Field","toSemifield"],["IsStrictOrderedRing","toMulPosStrictMono"],["Complex","instSub"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"isBigO_at_im_infty_jacobiTheta_sub_one","_simp_1_3"],"typeFallback":"Eq.{1} Prop (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) Real.pi) True","typeFull":"(0 < Real.pi) = True","typeReadable":"(0 < Real.pi) = True","typeReferences":[["LT","lt"],["True"],["Real","instZero"],["Real"],["Real","instLT"],["Zero","toOfNat0"],["Real","pi"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["Real","instZero"],["Real"],["Real","instLT"],["Real","pi_pos"],["eq_true"],["Zero","toOfNat0"],["Real","pi"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"hasSum_nat_jacobiTheta","_simp_1_3"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3 : SubNegMonoid.{u_1} G] (a : G) (b : G), Eq.{succ u_1} G (HAdd.hAdd.{u_1, u_1, u_1} G G G (instHAdd.{u_1} G (AddZero.toAdd.{u_1} G (AddZeroClass.toAddZero.{u_1} G (AddMonoid.toAddZeroClass.{u_1} G (SubNegMonoid.toAddMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3))))) a (Neg.neg.{u_1} G (SubNegMonoid.toNeg.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3) b)) (HSub.hSub.{u_1, u_1, u_1} G G G (instHSub.{u_1} G (SubNegMonoid.toSub.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3)) a b)","typeFull":"∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a + -b = a - b","typeReadable":"∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a + -b = a - b","typeReferences":[["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["Eq"],["instHSub"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["sub_eq_add_neg"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid","toSub"],["Eq","symm"],["HSub","hSub"],["instHSub"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["differentiableAt_jacobiTheta"],"typeFallback":"forall {τ : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (Complex.im τ)) -> (DifferentiableAt.{0, 0, 0} Complex (DenselyNormedField.toNontriviallyNormedField.{0} Complex Complex.instDenselyNormedField) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) Complex Complex.addCommGroup Complex.instModuleSelf (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) jacobiTheta τ)","typeFull":"∀ {τ : ℂ}, 0 < τ.im → DifferentiableAt ℂ jacobiTheta τ","typeReadable":"∀ {τ : ℂ}, 0 < τ.im → DifferentiableAt ℂ jacobiTheta τ","typeReferences":[["DifferentiableAt"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Complex","instDenselyNormedField"],["UniformSpace","toTopologicalSpace"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["OfNat","ofNat"],["Complex","addCommGroup"],["Complex"],["LT","lt"],["jacobiTheta"],["SeminormedRing","toPseudoMetricSpace"],["Real","instZero"],["DenselyNormedField","toNontriviallyNormedField"],["Real","instLT"],["NormedCommRing","toSeminormedCommRing"],["Zero","toOfNat0"],["Complex","im"],["NormedField","toNormedCommRing"],["Complex","instModuleSelf"]],"valueReferences":[["DifferentiableAt"],["PseudoMetricSpace","toUniformSpace"],["Complex","instDenselyNormedField"],["UniformSpace","toTopologicalSpace"],["congrArg"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["DenselyNormedField","toNontriviallyNormedField"],["funext"],["jacobiTheta₂"],["Zero","toOfNat0"],["congrFun'"],["Eq"],["NormedField","toNormedCommRing"],["Complex","instModuleSelf"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["OfNat","ofNat"],["Complex","addCommGroup"],["jacobiTheta"],["Complex","instZero"],["differentiableAt_jacobiTheta₂_snd"],["jacobiTheta_eq_jacobiTheta₂"],["id"],["NormedCommRing","toSeminormedCommRing"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["jacobiTheta_S_smul"],"typeFallback":"forall (τ : UpperHalfPlane), Eq.{1} Complex (jacobiTheta (UpperHalfPlane.coe (HSMul.hSMul.{0, 0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane UpperHalfPlane (instHSMul.{0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane (SemigroupAction.toSMul.{0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane (Monoid.toSemigroup.{0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) (Matrix.SpecialLinearGroup.monoid.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing)) (MulAction.toSemigroupAction.{0, 0} (Matrix.SpecialLinearGroup.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) UpperHalfPlane (Matrix.SpecialLinearGroup.monoid.{0, 0} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (instDecidableEqFin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (Fin.fintype (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) Int Int.instCommRing) (UpperHalfPlane.SLAction.{0} Int Int.instCommRing (Ring.toIntAlgebra.{0} Real Real.instRing))))) ModularGroup.S τ))) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (HPow.hPow.{0, 0, 0} Complex Complex Complex (instHPow.{0, 0} Complex Complex Complex.instPow) (HMul.hMul.{0, 0, 0} Complex Complex Complex (instHMul.{0} Complex Complex.instMul) (Neg.neg.{0} Complex Complex.instNeg Complex.I) (UpperHalfPlane.coe τ)) (HDiv.hDiv.{0, 0, 0} Complex Complex Complex (instHDiv.{0} Complex (DivInvMonoid.toDiv.{0} Complex Complex.instDivInvMonoid)) (OfNat.ofNat.{0} Complex 1 (One.toOfNat1.{0} Complex Complex.instOne)) (OfNat.ofNat.{0} Complex 2 (instOfNatAtLeastTwo.{0} Complex 2 Complex.instNatCast (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))))) (jacobiTheta (UpperHalfPlane.coe τ)))","typeFull":"∀ (τ : UpperHalfPlane), jacobiTheta ↑(ModularGroup.S • τ) = (-Complex.I * ↑τ) ^ (1 / 2) * jacobiTheta ↑τ","typeReadable":"∀ (τ : UpperHalfPlane), jacobiTheta ↑(ModularGroup.S • τ) = (-Complex.I * ↑τ) ^ (1 / 2) * jacobiTheta ↑τ","typeReferences":[["Int","instCommRing"],["Complex","instNeg"],["HMul","hMul"],["Fin"],["SemigroupAction","toSMul"],["Real","instRing"],["UpperHalfPlane"],["instHDiv"],["Fin","fintype"],["Ring","toIntAlgebra"],["Nat","instNeZeroSucc"],["HDiv","hDiv"],["Complex"],["instOfNatNat"],["Complex","instNatCast"],["instHSMul"],["Matrix","SpecialLinearGroup","monoid"],["Monoid","toSemigroup"],["Eq"],["ModularGroup","S"],["instHPow"],["Complex","I"],["instDecidableEqFin"],["Real"],["Complex","instOne"],["Neg","neg"],["Complex","instDivInvMonoid"],["instOfNatAtLeastTwo"],["HPow","hPow"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["Complex","instPow"],["Int"],["jacobiTheta"],["Nat"],["One","toOfNat1"],["HSMul","hSMul"],["instHMul"],["UpperHalfPlane","SLAction"],["UpperHalfPlane","coe"],["Complex","instMul"],["MulAction","toSemigroupAction"],["Nat","instAtLeastTwoHAddOfNat"],["Matrix","SpecialLinearGroup"]],"valueReferences":[["DivInvMonoid","toInv"],["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["UpperHalfPlane","modular_S_smul"],["Eq","trans"],["SemigroupAction","toSMul"],["AddGroupWithOne","toAddMonoidWithOne"],["MonoidWithZero","toMulZeroOneClass"],["Complex","instSemiring"],["Real","pi"],["zero_pow"],["Ring","toIntAlgebra"],["Complex"],["DivisionMonoid","toDivInvMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Eq","symm"],["mul_one_div"],["Ne","eq_1"],["NormedDivisionRing","toNormMulClass"],["Monoid","toSemigroup"],["Complex","cpow_eq_zero_iff"],["ModularGroup","S"],["NormedField","toNormedCommRing"],["Complex","instInv"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["ne_of_gt"],["DivisionSemiring","toGroupWithZero"],["MulZeroClass","mul_zero"],["Complex","exp"],["Complex","instNormedField"],["Complex","exp_zero"],["DivInvMonoid","toMonoid"],["MulZeroOneClass","toMulZeroClass"],["jacobiTheta_eq_jacobiTheta₂"],["AddMonoidWithOne","toOne"],["UpperHalfPlane","SLAction"],["Eq","mpr"],["one_mul"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["UpperHalfPlane","mk"],["one_div"],["Real","instNatCast"],["MulZeroOneClass","toMulOneClass"],["SubtractionCommMonoid","toSubtractionMonoid"],["UpperHalfPlane","im_inv_neg_coe_pos"],["UpperHalfPlane"],["instHDiv"],["Fin","fintype"],["Semigroup","toMul"],["Nat","instNeZeroSucc"],["MulOne","toMul"],["Or","inl"],["instOfNatNat"],["congr"],["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"jacobiTheta_S_smul","_simp_1_2"],["Eq"],["Complex","im"],["propext"],["Complex","I"],["instDecidableEqFin"],["instOfNatAtLeastTwo"],["mul_one"],["HPow","hPow"],["OfNat","ofNat"],["Complex","instPow"],["Complex","addCommGroup"],["Int"],["eq_self"],["jacobiTheta"],["Real","instZero"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["neg_ne_zero"],["Ne"],["NormedField","toNormedDivisionRing"],["Nat","instAtLeastTwoHAddOfNat"],["Semifield","toCommGroupWithZero"],["Matrix","SpecialLinearGroup"],["Nat","cast_one"],["Int","instCommRing"],["Complex","instNeg"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["Fin"],["CommGroupWithZero","toDivisionCommMonoid"],["AddMonoidWithOne","toAddMonoid"],["Complex","ofReal"],["HDiv","hDiv"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Ring","toAddGroupWithOne"],["Or"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocRing","toHasDistribNeg"],["jacobiTheta₂"],["InvolutiveNeg","toNeg"],["Semifield","toDivisionSemiring"],["Nat","instMulZeroClass"],["Eq","rec"],["NonAssocSemiring","toMulZeroOneClass"],["instHPow"],["InvOneClass","toInv"],["div_self"],["Real"],["Complex","instOne"],["ne_of_apply_ne"],["Neg","neg"],["InvOneClass","toOne"],["And"],["Complex","addGroupWithOne"],["NonUnitalCommRing","toNonUnitalCommSemiring"],["AddZeroClass","toAddZero"],["UpperHalfPlane","coe_im_pos"],["Complex","instZero"],["NormMulClass","toNoZeroDivisors"],["Nat"],["AddMonoidWithOne","toNatCast"],["Nat","cast_zero"],["Iff","mpr"],["HSMul","hSMul"],["id"],["NegZeroClass","toZero"],["instHMul"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["UpperHalfPlane","coe"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["Nat","cast"],["inv_neg"],["mul_ne_zero"],["Real","instRing"],["Complex","I_ne_zero"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["jacobiTheta₂_functional_equation"],["Monoid","toMulOneClass"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["Complex","instNatCast"],["NormedCommRing","toNormedRing"],["instHSMul"],["Zero","toOfNat0"],["Matrix","SpecialLinearGroup","monoid"],["congrFun'"],["DivisionCommMonoid","toDivisionMonoid"],["zero_div"],["Complex","instField"],["Not"],["two_ne_zero"],["not_and_or"],["Inv","inv"],["True"],["Complex","instDivInvMonoid"],["Complex","zero_im"],["Semiring","toMonoidWithZero"],["neg_div"],["DivInvMonoid","toDiv"],["DivInvOneMonoid","toInvOneClass"],["NegZeroClass","toNeg"],["of_eq_true"],["HasDistribNeg","toInvolutiveNeg"],["One","toOfNat1"],["Field","toSemifield"],["SemigroupWithZero","toSemigroup"],["Complex","instMul"],["MulAction","toSemigroupAction"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["continuousAt_jacobiTheta"],"typeFallback":"forall {τ : Complex}, (LT.lt.{0} Real Real.instLT (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (Complex.im τ)) -> (ContinuousAt.{0, 0} Complex Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedField)))))) jacobiTheta τ)","typeFull":"∀ {τ : ℂ}, 0 < τ.im → ContinuousAt jacobiTheta τ","typeReadable":"∀ {τ : ℂ}, 0 < τ.im → ContinuousAt jacobiTheta τ","typeReferences":[["Real"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["OfNat","ofNat"],["Complex"],["LT","lt"],["jacobiTheta"],["SeminormedRing","toPseudoMetricSpace"],["Real","instZero"],["Real","instLT"],["NormedCommRing","toSeminormedCommRing"],["Zero","toOfNat0"],["ContinuousAt"],["Complex","im"],["NormedField","toNormedCommRing"]],"valueReferences":[["Complex","instDenselyNormedField"],["PseudoMetricSpace","toUniformSpace"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Complex","instSemiring"],["Complex"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Complex","instAdd"],["IsSemitopologicalSemiring","toContinuousAdd"],["DenselyNormedField","toNontriviallyNormedField"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["NormedField","toNormedCommRing"],["Complex","instModuleSelf"],["DistribSMul","toSMulZeroClass"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["DivisionSemiring","toSemiring"],["SeminormedCommRing","toSeminormedRing"],["AddZeroClass","toAddZero"],["differentiableAt_jacobiTheta"],["Complex","instNormedField"],["IsTopologicalRing","toIsSemitopologicalRing"],["AddCommGroup","toAddCommMonoid"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["UniformSpace","toTopologicalSpace"],["NontriviallyNormedField","toNormedField"],["SeminormedRing","toPseudoMetricSpace"],["IsModuleTopology","toContinuousSMul"],["NormedField","toField"],["MonoidWithZero","toMonoid"],["IsTopologicalSemiring","toIsModuleTopology"],["Semiring","toMonoidWithZero"],["NormedDivisionRing","to_isTopologicalDivisionRing"],["Complex","addCommGroup"],["jacobiTheta"],["DifferentiableAt","continuousAt"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Field","toSemifield"],["IsTopologicalDivisionRing","toIsTopologicalRing"],["NormedField","toNormedDivisionRing"],["NormedDivisionRing","toDivisionRing"],["NormedCommRing","toSeminormedCommRing"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":false,"kind":"definition","name":["jacobiTheta"],"typeFallback":"Complex -> Complex","typeFull":"ℂ → ℂ","typeReadable":"ℂ → ℂ","typeReferences":[["Complex"]],"valueReferences":[["PseudoMetricSpace","toUniformSpace"],["SummationFilter","unconditional"],["UniformSpace","toTopologicalSpace"],["HMul","hMul"],["Complex","instSemiring"],["Real","pi"],["Int","cast"],["Complex","ofReal"],["Complex"],["SeminormedRing","toPseudoMetricSpace"],["Monoid","toPow"],["instOfNatNat"],["tsum"],["Complex","instNormedAddCommGroup"],["MonoidWithZero","toMonoid"],["NormedField","toNormedCommRing"],["instHPow"],["NormedAddCommGroup","toENormedAddCommMonoid"],["Complex","I"],["Semiring","toMonoidWithZero"],["Complex","exp"],["SeminormedCommRing","toSeminormedRing"],["HPow","hPow"],["OfNat","ofNat"],["Complex","instNormedField"],["Int"],["Nat"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["instHMul"],["NormedCommRing","toSeminormedCommRing"],["Complex","instIntCast"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Complex","instMul"]]},{"isProp":true,"kind":"theorem","name":["jacobiTheta_two_add"],"typeFallback":"forall (τ : Complex), Eq.{1} Complex (jacobiTheta (HAdd.hAdd.{0, 0, 0} Complex Complex Complex (instHAdd.{0} Complex Complex.instAdd) (OfNat.ofNat.{0} Complex 2 (instOfNatAtLeastTwo.{0} Complex 2 Complex.instNatCast (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) τ)) (jacobiTheta τ)","typeFull":"∀ (τ : ℂ), jacobiTheta (2 + τ) = jacobiTheta τ","typeReadable":"∀ (τ : ℂ), jacobiTheta (2 + τ) = jacobiTheta τ","typeReferences":[["instHAdd"],["instOfNatAtLeastTwo"],["OfNat","ofNat"],["Complex"],["HAdd","hAdd"],["Nat","instNeZeroSucc"],["jacobiTheta"],["Nat"],["Complex","instAdd"],["instOfNatNat"],["Complex","instNatCast"],["Eq"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["jacobiTheta₂_add_right"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["congrArg"],["Complex"],["Nat","instNeZeroSucc"],["SeminormedRing","toPseudoMetricSpace"],["Complex","instAdd"],["Complex","instNormedAddCommGroup"],["instOfNatNat"],["congr"],["Complex","instNatCast"],["congrFun'"],["jacobiTheta₂"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["add_comm"],["NormedField","toNormedCommRing"],["NormedAddCommGroup","toENormedAddCommMonoid"],["True"],["instHAdd"],["instOfNatAtLeastTwo"],["SeminormedCommRing","toSeminormedRing"],["Complex","instNormedField"],["OfNat","ofNat"],["HAdd","hAdd"],["eq_self"],["jacobiTheta"],["Complex","instZero"],["Nat"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["jacobiTheta_eq_jacobiTheta₂"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["id"],["NormedCommRing","toSeminormedCommRing"],["Eq","mpr"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Nat","instAtLeastTwoHAddOfNat"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"hasSum_nat_jacobiTheta","_simp_1_2"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Defs.2556340985._hygCtx._hyg.4 : NonAssocSemiring.{u} α] (n : α), Eq.{succ u} α (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Defs.2556340985._hygCtx._hyg.4)))) n n) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Defs.2556340985._hygCtx._hyg.4)))) n (OfNat.ofNat.{u} α 2 (instOfNatAtLeastTwo.{u} α 2 (AddMonoidWithOne.toNatCast.{u} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u} α (NonAssocSemiring.toAddCommMonoidWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Defs.2556340985._hygCtx._hyg.4))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))))","typeFull":"∀ {α : Type u} [inst : NonAssocSemiring α] (n : α), n + n = n * 2","typeReadable":"∀ {α : Type u} [inst : NonAssocSemiring α] (n : α), n + n = n * 2","typeReferences":[["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["instOfNatAtLeastTwo"],["HMul","hMul"],["OfNat","ofNat"],["Nat","instNeZeroSucc"],["HAdd","hAdd"],["Nat"],["AddMonoidWithOne","toNatCast"],["instOfNatNat"],["NonAssocSemiring"],["instHMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["instOfNatAtLeastTwo"],["HMul","hMul"],["mul_two"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat","instNeZeroSucc"],["Nat"],["AddMonoidWithOne","toNatCast"],["instOfNatNat"],["Eq","symm"],["instHMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Nat","instAtLeastTwoHAddOfNat"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"jacobiTheta_S_smul","_simp_1_2"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3 : Semigroup.{u_1} G] (a : G) (b : G) (c : G), Eq.{succ u_1} G (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) b c)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a b) c)","typeFull":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReadable":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReferences":[["instHMul"],["HMul","hMul"],["Semigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["Eq","symm"],["instHMul"],["HMul","hMul"],["mul_assoc"],["Semigroup","toMul"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","NumberTheory","ModularForms","JacobiTheta","OneVariable",0,"isBigO_at_im_infty_jacobiTheta_sub_one","_simp_1_2"],"typeFallback":"forall {α : Type.{u_3}} [inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2659569902._hygCtx._hyg.7 : Preorder.{u_3} α] [inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2659569902._hygCtx._hyg.10 : IsDirectedOrder.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2659569902._hygCtx._hyg.7)] {p : α -> Prop} [inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2659569902._hygCtx._hyg.16 : Nonempty.{succ u_3} α], Eq.{1} Prop (Filter.Eventually.{u_3} α (fun (x : α) => p x) (Filter.atTop.{u_3} α inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2659569902._hygCtx._hyg.7)) (Exists.{succ u_3} α (fun (a : α) => forall (b : α), (GE.ge.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Order.Filter.AtTopBot.Basic.2659569902._hygCtx._hyg.7) b a) -> (p b)))","typeFull":"∀ {α : Type u_3} [inst : Preorder α] [IsDirectedOrder α] {p : α → Prop} [Nonempty α],\n  (∀ᶠ (x : α) in Filter.atTop, p x) = ∃ a, ∀ b ≥ a, p b","typeReadable":"∀ {α : Type u_3} [inst : Preorder α] [IsDirectedOrder α] {p : α → Prop} [Nonempty α],\n  (∀ᶠ (x : α) in Filter.atTop, p x) = ∃ a, ∀ b ≥ a, p b","typeReferences":[["Filter","Eventually"],["Filter","atTop"],["Preorder"],["Exists"],["IsDirectedOrder"],["Nonempty"],["Eq"],["Preorder","toLE"],["GE","ge"]],"valueReferences":[["Filter","Eventually"],["Filter","atTop"],["Exists"],["Filter","eventually_atTop"],["Preorder","toLE"],["GE","ge"],["propext"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Order.Filter.Interval.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ico_Ici_Ici"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.1254874558._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1254874558._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Ici.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1254874558._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Ici.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1254874558._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Ici a)) (Filter.principal (Set.Ici a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Ici a)) (Filter.principal (Set.Ici a))","typeReferences":[["Set","Ici"],["Preorder"],["Filter","TendstoIxxClass"],["Set","Ico"],["Filter","principal"]],"valueReferences":[["Filter","OrdConnected","tendsto_Icc"],["Set","Ici"],["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ico_subset_Icc_self"],["Set","ordConnected_Ici"],["Set","Ico"],["Filter","principal"]]},{"isProp":true,"kind":"theorem","name":["Filter","OrdConnected","tendsto_Icc"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.2361556477._hygCtx._hyg.4 : Preorder.{u_1} α] {s : Set.{u_1} α} [hs : Set.OrdConnected.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2361556477._hygCtx._hyg.4 s], Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2361556477._hygCtx._hyg.4) (Filter.principal.{u_1} α s) (Filter.principal.{u_1} α s)","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {s : Set α} [hs : s.OrdConnected],\n  Filter.TendstoIxxClass Set.Icc (Filter.principal s) (Filter.principal s)","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {s : Set α} [hs : s.OrdConnected],\n  Filter.TendstoIxxClass Set.Icc (Filter.principal s) (Filter.principal s)","typeReferences":[["Preorder"],["Set","Icc"],["Set"],["Filter","TendstoIxxClass"],["Set","OrdConnected"],["Filter","principal"]],"valueReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Set"],["Set","Icc"],["Filter","tendstoIxxClass_principal"],["Iff","mpr"],["Membership","mem"],["Filter","TendstoIxxClass"],["Set","OrdConnected","out"],["Filter","principal"],["Set","instMembership"]]},{"isProp":false,"kind":"inductive","name":["Filter","TendstoIxxClass"],"typeFallback":"forall {α : Type.{u_1}}, (α -> α -> (Set.{u_1} α)) -> (Filter.{u_1} α) -> (outParam.{succ u_1} (Filter.{u_1} α)) -> Prop","typeFull":"{α : Type u_1} → (α → α → Set α) → Filter α → outParam (Filter α) → Prop","typeReadable":"{α : Type u_1} → (α → α → Set α) → Filter α → outParam (Filter α) → Prop","typeReferences":[["Set"],["outParam"],["Filter"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ico_Ioi_Ioi"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.3721396647._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3721396647._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3721396647._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3721396647._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Ioi a)) (Filter.principal (Set.Ioi a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Ioi a)) (Filter.principal (Set.Ioi a))","typeReferences":[["Preorder"],["Set","Ioi"],["Filter","TendstoIxxClass"],["Set","Ico"],["Filter","principal"]],"valueReferences":[["Filter","OrdConnected","tendsto_Icc"],["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioi"],["Set","ordConnected_Ioi"],["Set","Ico_subset_Icc_self"],["Set","Ico"],["Filter","principal"]]},{"isProp":true,"kind":"definition","name":["Filter","TendstoIxxClass","mk","_flat_ctor"],"typeFallback":"forall {α : Type.{u_1}} {Ixx : α -> α -> (Set.{u_1} α)} {l₁ : Filter.{u_1} α} {l₂ : outParam.{succ u_1} (Filter.{u_1} α)}, (Filter.Tendsto.{u_1, u_1} (Prod.{u_1, u_1} α α) (Set.{u_1} α) (fun (p : Prod.{u_1, u_1} α α) => Ixx (Prod.fst.{u_1, u_1} α α p) (Prod.snd.{u_1, u_1} α α p)) (SProd.sprod.{u_1, u_1, u_1} (Filter.{u_1} α) (Filter.{u_1} α) (Filter.{u_1} (Prod.{u_1, u_1} α α)) (Filter.instSProd.{u_1, u_1} α α) l₁ l₁) (Filter.smallSets.{u_1} α l₂)) -> (Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂)","typeFull":"∀ {α : Type u_1} {Ixx : α → α → Set α} {l₁ : Filter α} {l₂ : outParam (Filter α)},\n  Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets → Filter.TendstoIxxClass Ixx l₁ l₂","typeReadable":"∀ {α : Type u_1} {Ixx : α → α → Set α} {l₁ : Filter α} {l₂ : outParam (Filter α)},\n  Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets → Filter.TendstoIxxClass Ixx l₁ l₂","typeReferences":[["Filter","smallSets"],["Prod"],["Filter","instSProd"],["Set"],["outParam"],["Filter","TendstoIxxClass"],["SProd","sprod"],["Prod","snd"],["Filter","Tendsto"],["Filter"],["Prod","fst"]],"valueReferences":[["Filter","TendstoIxxClass","mk"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Icc_pure_pure"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.2922361243._hygCtx._hyg.4 : PartialOrder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2922361243._hygCtx._hyg.4)) (Pure.pure.{u_1, u_1} Filter.{u_1} Filter.instPure.{u_1} α a) (Pure.pure.{u_1, u_1} Filter.{u_1} Filter.instPure.{u_1} α a)","typeFull":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Icc (pure a) (pure a)","typeReadable":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Icc (pure a) (pure a)","typeReferences":[["PartialOrder","toPreorder"],["Set","Icc"],["PartialOrder"],["Pure","pure"],["Filter","instPure"],["Filter","TendstoIxxClass"],["Filter"]],"valueReferences":[["PartialOrder","toPreorder"],["Set","Icc"],["Set"],["Filter","tendstoIxxClass_principal"],["Singleton","singleton"],["Membership","mem"],["Set","instSingletonSet"],["Filter"],["Filter","principal"],["Set","instMembership"],["congrArg"],["Set","instHasSubset"],["Set","ordConnected_singleton"],["Filter","principal_singleton"],["HasSubset","Subset"],["Iff","mpr"],["Eq","symm"],["id"],["Pure","pure"],["Eq","mpr"],["Filter","instPure"],["Filter","TendstoIxxClass"],["Eq"],["Set","OrdConnected","out"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Order","Filter","Interval",0,"Filter","tendstoIxxClass_principal","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {f : α -> β} {s : Set.{u_1} α} {t : Set.{u_2} β}, Eq.{1} Prop (Filter.Tendsto.{u_1, u_2} α β f (Filter.principal.{u_1} α s) (Filter.principal.{u_2} β t)) (forall (a : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s a) -> (Membership.mem.{u_2, u_2} β (Set.{u_2} β) (Set.instMembership.{u_2} β) t (f a)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {t : Set β},\n  Filter.Tendsto f (Filter.principal s) (Filter.principal t) = ∀ a ∈ s, f a ∈ t","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set α} {t : Set β},\n  Filter.Tendsto f (Filter.principal s) (Filter.principal t) = ∀ a ∈ s, f a ∈ t","typeReferences":[["Set"],["Membership","mem"],["Eq"],["Filter","Tendsto"],["Filter","principal"],["Set","instMembership"]],"valueReferences":[["Set"],["Membership","mem"],["Filter","tendsto_principal_principal"],["Filter","Tendsto"],["Filter","principal"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioo_pure_bot"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.631482893._hygCtx._hyg.4 : PartialOrder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.631482893._hygCtx._hyg.4)) (Pure.pure.{u_1, u_1} Filter.{u_1} Filter.instPure.{u_1} α a) (Bot.bot.{u_1} (Filter.{u_1} α) (Filter.instBot.{u_1} α))","typeFull":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Ioo (pure a) ⊥","typeReadable":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Ioo (pure a) ⊥","typeReferences":[["PartialOrder","toPreorder"],["PartialOrder"],["Filter","instBot"],["Pure","pure"],["Filter","instPure"],["Filter","TendstoIxxClass"],["Bot","bot"],["Filter"],["Set","Ioo"]],"valueReferences":[["Filter","Eventually"],["PartialOrder","toPreorder"],["Eq","trans"],["Prod","mk"],["Preorder","toLT"],["SProd","sprod"],["EmptyCollection","emptyCollection"],["Set","instEmptyCollection"],["Set","Ioo"],["congrArg"],["Prod","fst"],["not_false_eq_true"],["Filter","prod_pure"],["Filter","instSProd"],["Filter","TendstoIxxClass","mk"],["congr"],["lt_self_iff_false","_simp_1"],["congrFun'"],["Eq"],["Filter","Tendsto"],["Not"],["Filter","eventually_pure","_simp_1"],["True"],["Set"],["Filter","instBot"],["Prod","snd"],["Bot","bot"],["Set","Ioo_eq_empty"],["Filter"],["LT","lt"],["Prod"],["Filter","smallSets"],["eq_self"],["of_eq_true"],["False"],["Pure","pure"],["Filter","smallSets_bot"],["Filter","instPure"],["Filter","tendsto_pure","_simp_1"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Icc_atBot_atBot"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.1188496359._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1188496359._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1188496359._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1188496359._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Icc Filter.atBot Filter.atBot","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Icc Filter.atBot Filter.atBot","typeReferences":[["Preorder"],["Set","Icc"],["Filter","atBot"],["Filter","TendstoIxxClass"]],"valueReferences":[["Filter","hasBasis_iInf_principal_finite"],["Set"],["Set","Icc"],["Membership","mem"],["Set","iInter"],["Set","Iic"],["Filter"],["Filter","principal"],["Set","instMembership"],["iInf"],["Set","ordConnected_Iic"],["Set","Finite"],["Filter","instInfSet"],["Set","OrdConnected","out"],["Filter","HasBasis","tendstoIxxClass"],["Set","ordConnected_biInter"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Icc_Icc_Icc"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.1273976930._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α} {b : α}, Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1273976930._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1273976930._hygCtx._hyg.4 a b)) (Filter.principal.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1273976930._hygCtx._hyg.4 a b))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a b : α},\n  Filter.TendstoIxxClass Set.Icc (Filter.principal (Set.Icc a b)) (Filter.principal (Set.Icc a b))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a b : α},\n  Filter.TendstoIxxClass Set.Icc (Filter.principal (Set.Icc a b)) (Filter.principal (Set.Icc a b))","typeReferences":[["Preorder"],["Set","Icc"],["Filter","TendstoIxxClass"],["Filter","principal"]],"valueReferences":[["Set","Icc"],["Set"],["Filter","tendstoIxxClass_principal"],["And","right"],["Membership","mem"],["Filter","principal"],["Set","instMembership"],["Set","instHasSubset"],["Set","Icc_subset_Icc"],["HasSubset","Subset"],["Iff","mpr"],["LE","le"],["Filter","TendstoIxxClass"],["Preorder","toLE"],["And","left"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_pure_bot"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.1021888327._hygCtx._hyg.4 : PartialOrder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1021888327._hygCtx._hyg.4)) (Pure.pure.{u_1, u_1} Filter.{u_1} Filter.instPure.{u_1} α a) (Bot.bot.{u_1} (Filter.{u_1} α) (Filter.instBot.{u_1} α))","typeFull":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Ioc (pure a) ⊥","typeReadable":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Ioc (pure a) ⊥","typeReferences":[["PartialOrder","toPreorder"],["Set","Ioc"],["PartialOrder"],["Filter","instBot"],["Pure","pure"],["Filter","instPure"],["Filter","TendstoIxxClass"],["Bot","bot"],["Filter"]],"valueReferences":[["Filter","Eventually"],["PartialOrder","toPreorder"],["Eq","trans"],["Set","Ioc_eq_empty"],["Prod","mk"],["Preorder","toLT"],["SProd","sprod"],["EmptyCollection","emptyCollection"],["Set","instEmptyCollection"],["congrArg"],["Prod","fst"],["not_false_eq_true"],["Filter","prod_pure"],["Filter","instSProd"],["Filter","TendstoIxxClass","mk"],["congr"],["lt_self_iff_false","_simp_1"],["congrFun'"],["Eq"],["Filter","Tendsto"],["Not"],["Filter","eventually_pure","_simp_1"],["True"],["Set"],["Filter","instBot"],["Prod","snd"],["Bot","bot"],["Filter"],["LT","lt"],["Prod"],["Filter","smallSets"],["eq_self"],["of_eq_true"],["Set","Ioc"],["False"],["Pure","pure"],["Filter","smallSets_bot"],["Filter","instPure"],["Filter","tendsto_pure","_simp_1"]]},{"isProp":true,"kind":"theorem","name":["Filter","HasBasis","tendstoIxxClass"],"typeFallback":"forall {α : Type.{u_1}} {ι : Type.{u_3}} {p : ι -> Prop} {s : ι -> (Set.{u_1} α)} {l : Filter.{u_1} α}, (Filter.HasBasis.{u_1, succ u_3} α ι l p s) -> (forall {Ixx : α -> α -> (Set.{u_1} α)}, (forall (i : ι), (p i) -> (forall (x : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) (s i) x) -> (forall (y : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) (s i) y) -> (HasSubset.Subset.{u_1} (Set.{u_1} α) (Set.instHasSubset.{u_1} α) (Ixx x y) (s i))))) -> (Filter.TendstoIxxClass.{u_1} α Ixx l l))","typeFull":"∀ {α : Type u_1} {ι : Type u_3} {p : ι → Prop} {s : ι → Set α} {l : Filter α},\n  l.HasBasis p s →\n    ∀ {Ixx : α → α → Set α}, (∀ (i : ι), p i → ∀ x ∈ s i, ∀ y ∈ s i, Ixx x y ⊆ s i) → Filter.TendstoIxxClass Ixx l l","typeReadable":"∀ {α : Type u_1} {ι : Type u_3} {p : ι → Prop} {s : ι → Set α} {l : Filter α},\n  l.HasBasis p s →\n    ∀ {Ixx : α → α → Set α}, (∀ (i : ι), p i → ∀ x ∈ s i, ∀ y ∈ s i, Ixx x y ⊆ s i) → Filter.TendstoIxxClass Ixx l l","typeReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Set"],["Membership","mem"],["Filter","HasBasis"],["Filter","TendstoIxxClass"],["Filter"],["Set","instMembership"]],"valueReferences":[["Exists"],["Filter","HasBasis","smallSets"],["Set"],["And","right"],["Membership","mem"],["And"],["Exists","intro"],["Filter","HasBasis","prod_self"],["Prod","snd"],["SProd","sprod"],["Filter"],["Set","instMembership"],["Prod","fst"],["Prod"],["Filter","smallSets"],["And","intro"],["Filter","instSProd"],["Filter","TendstoIxxClass","mk"],["Iff","mpr"],["Set","powerset"],["Filter","HasBasis","tendsto_iff"],["Set","instSProd"],["Filter","Tendsto"],["And","left"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_uIcc_of_Icc"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.25213499._hygCtx._hyg.4 : LinearOrder.{u_1} α] {l : Filter.{u_1} α} [inst._@.Mathlib.Order.Filter.Interval.25213499._hygCtx._hyg.9 : Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.25213499._hygCtx._hyg.4)))))) l l], Filter.TendstoIxxClass.{u_1} α (Set.uIcc.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.25213499._hygCtx._hyg.4))) l l","typeFull":"∀ {α : Type u_1} [inst : LinearOrder α] {l : Filter α} [Filter.TendstoIxxClass Set.Icc l l],\n  Filter.TendstoIxxClass Set.uIcc l l","typeReadable":"∀ {α : Type u_1} [inst : LinearOrder α] {l : Filter α} [Filter.TendstoIxxClass Set.Icc l l],\n  Filter.TendstoIxxClass Set.uIcc l l","typeReferences":[["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["Set","uIcc"],["Set","Icc"],["LinearOrder"],["Filter","TendstoIxxClass"],["Filter"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["PartialOrder","toPreorder"],["Filter","tendsto_snd"],["Prod","mk"],["Membership","mem"],["Iff","mp"],["Exists","intro"],["SProd","sprod"],["Filter","tendsto_fst"],["And","intro"],["Filter","instSProd"],["Filter","TendstoIxxClass","mk"],["Set","uIcc_of_le"],["And","left"],["SemilatticeInf","toPartialOrder"],["Filter","mem_prod_self_iff"],["Exists"],["Set","uIcc"],["And","right"],["And"],["Filter","map"],["Prod","snd"],["Filter"],["Set","instMembership"],["Exists","casesOn"],["Prod"],["HasSubset","Subset"],["Iff","mpr"],["id"],["Eq","mpr"],["le_total"],["Filter","Tendsto","Icc"],["Set","Icc"],["Set","uIcc_of_ge"],["Prod","fst"],["congrArg"],["instDistribLatticeOfLinearOrder"],["Filter","mem_map"],["Set","instSProd"],["Preorder","toLE"],["Eq"],["propext"],["Filter","instMembership"],["Set","preimage"],["Lattice","toSemilatticeInf"],["Set"],["Set","instHasSubset"],["Or","casesOn"],["LinearOrder","toPartialOrder"],["DistribLattice","toLattice"],["LE","le"],["Set","mem_preimage"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioo_Iic_Iio"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.2368380974._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2368380974._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Iic.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2368380974._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2368380974._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Iic a)) (Filter.principal (Set.Iio a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Iic a)) (Filter.principal (Set.Iio a))","typeReferences":[["Preorder"],["Set","Iio"],["Filter","TendstoIxxClass"],["Set","Iic"],["Filter","principal"],["Set","Ioo"]],"valueReferences":[["Filter","tendstoIxxClass_of_subset"],["Filter","tendsto_Ico_Iic_Iio"],["Set","Iio"],["Set","Ico"],["Set","Iic"],["Set","Ioo_subset_Ico_self"],["Set","Ioo"],["Filter","principal"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_Iio_Iio"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.4096313449._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4096313449._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4096313449._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4096313449._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Iio a)) (Filter.principal (Set.Iio a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Iio a)) (Filter.principal (Set.Iio a))","typeReferences":[["Preorder"],["Set","Ioc"],["Set","Iio"],["Filter","TendstoIxxClass"],["Filter","principal"]],"valueReferences":[["Filter","OrdConnected","tendsto_Icc"],["Set","Icc"],["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioc_subset_Icc_self"],["Set","Iio"],["Filter","principal"],["Set","ordConnected_Iio"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ico_pure_bot"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.3889423639._hygCtx._hyg.4 : PartialOrder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3889423639._hygCtx._hyg.4)) (Pure.pure.{u_1, u_1} Filter.{u_1} Filter.instPure.{u_1} α a) (Bot.bot.{u_1} (Filter.{u_1} α) (Filter.instBot.{u_1} α))","typeFull":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Ico (pure a) ⊥","typeReadable":"∀ {α : Type u_1} [inst : PartialOrder α] {a : α}, Filter.TendstoIxxClass Set.Ico (pure a) ⊥","typeReferences":[["PartialOrder","toPreorder"],["PartialOrder"],["Filter","instBot"],["Pure","pure"],["Filter","instPure"],["Filter","TendstoIxxClass"],["Set","Ico"],["Bot","bot"],["Filter"]],"valueReferences":[["Filter","Eventually"],["PartialOrder","toPreorder"],["Eq","trans"],["Prod","mk"],["Preorder","toLT"],["SProd","sprod"],["EmptyCollection","emptyCollection"],["Set","instEmptyCollection"],["congrArg"],["Prod","fst"],["not_false_eq_true"],["Filter","prod_pure"],["Filter","instSProd"],["Filter","TendstoIxxClass","mk"],["congr"],["lt_self_iff_false","_simp_1"],["congrFun'"],["Set","Ico"],["Eq"],["Filter","Tendsto"],["Set","Ico_eq_empty"],["Not"],["Filter","eventually_pure","_simp_1"],["True"],["Set"],["Filter","instBot"],["Prod","snd"],["Bot","bot"],["Filter"],["LT","lt"],["Prod"],["Filter","smallSets"],["eq_self"],["of_eq_true"],["False"],["Pure","pure"],["Filter","smallSets_bot"],["Filter","instPure"],["Filter","tendsto_pure","_simp_1"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Order","Filter","Interval",0,"Filter","tendstoIxxClass_principal","_simp_1_2"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {s : Set.{u_1} α} {t : Set.{u_2} β} {p : (Prod.{u_1, u_2} α β) -> Prop}, Eq.{1} Prop (forall (x : Prod.{u_1, u_2} α β), (Membership.mem.{max u_1 u_2, max u_1 u_2} (Prod.{u_1, u_2} α β) (Set.{max u_2 u_1} (Prod.{u_1, u_2} α β)) (Set.instMembership.{max u_1 u_2} (Prod.{u_1, u_2} α β)) (SProd.sprod.{u_1, u_2, max u_1 u_2} (Set.{u_1} α) (Set.{u_2} β) (Set.{max u_2 u_1} (Prod.{u_1, u_2} α β)) (Set.instSProd.{u_1, u_2} α β) s t) x) -> (p x)) (forall (x : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) -> (forall (y : β), (Membership.mem.{u_2, u_2} β (Set.{u_2} β) (Set.instMembership.{u_2} β) t y) -> (p (Prod.mk.{u_1, u_2} α β x y))))","typeFull":"∀ {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {p : α × β → Prop},\n  (∀ x ∈ s ×ˢ t, p x) = ∀ x ∈ s, ∀ y ∈ t, p (x, y)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {p : α × β → Prop},\n  (∀ x ∈ s ×ˢ t, p x) = ∀ x ∈ s, ∀ y ∈ t, p (x, y)","typeReferences":[["Prod"],["Prod","mk"],["Set"],["Membership","mem"],["SProd","sprod"],["Set","instSProd"],["Eq"],["Set","instMembership"]],"valueReferences":[["Prod"],["Set","forall_prod_set"],["Prod","mk"],["Set"],["Membership","mem"],["SProd","sprod"],["Set","instSProd"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Icc_atTop_atTop"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.3850998126._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3850998126._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3850998126._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3850998126._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Icc Filter.atTop Filter.atTop","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Icc Filter.atTop Filter.atTop","typeReferences":[["Filter","atTop"],["Preorder"],["Set","Icc"],["Filter","TendstoIxxClass"]],"valueReferences":[["Filter","hasBasis_iInf_principal_finite"],["Set"],["Set","Icc"],["Membership","mem"],["Set","iInter"],["Filter"],["Filter","principal"],["Set","instMembership"],["Set","Ici"],["iInf"],["Set","ordConnected_Ici"],["Set","Finite"],["Filter","instInfSet"],["Set","OrdConnected","out"],["Filter","HasBasis","tendstoIxxClass"],["Set","ordConnected_biInter"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioo_Ioi_Ioi"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.640066888._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.640066888._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.640066888._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.640066888._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Ioi a)) (Filter.principal (Set.Ioi a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Ioi a)) (Filter.principal (Set.Ioi a))","typeReferences":[["Preorder"],["Set","Ioi"],["Filter","TendstoIxxClass"],["Filter","principal"],["Set","Ioo"]],"valueReferences":[["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioo_subset_Ioc_self"],["Set","Ioi"],["Filter","tendsto_Ioc_Ioi_Ioi"],["Set","Ioo"],["Filter","principal"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ico_Iic_Iio"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.651039300._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.651039300._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Iic.{u_1} α inst._@.Mathlib.Order.Filter.Interval.651039300._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.651039300._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Iic a)) (Filter.principal (Set.Iio a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Iic a)) (Filter.principal (Set.Iio a))","typeReferences":[["Preorder"],["Set","Iio"],["Filter","TendstoIxxClass"],["Set","Ico"],["Set","Iic"],["Filter","principal"]],"valueReferences":[["Set"],["Filter","tendstoIxxClass_principal"],["And","right"],["Membership","mem"],["Preorder","toLT"],["Set","Iio"],["Set","Iic"],["Filter","principal"],["Set","instMembership"],["lt_of_lt_of_le"],["LT","lt"],["Set","instHasSubset"],["HasSubset","Subset"],["Iff","mpr"],["LE","le"],["Filter","TendstoIxxClass"],["Preorder","toLE"],["Set","Ico"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioo_atBot_atBot"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.56298823._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.56298823._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.56298823._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.56298823._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioo Filter.atBot Filter.atBot","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioo Filter.atBot Filter.atBot","typeReferences":[["Preorder"],["Filter","atBot"],["Filter","TendstoIxxClass"],["Set","Ioo"]],"valueReferences":[["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Filter","atBot"],["Filter","tendsto_Icc_atBot_atBot"],["Set","Ioo_subset_Icc_self"],["Set","Ioo"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_uIcc_uIcc"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.2765667752._hygCtx._hyg.4 : LinearOrder.{u_1} α] {a : α} {b : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2765667752._hygCtx._hyg.4)))))) (Filter.principal.{u_1} α (Set.uIcc.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2765667752._hygCtx._hyg.4)) a b)) (Filter.principal.{u_1} α (Set.uIcc.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2765667752._hygCtx._hyg.4)) a b))","typeFull":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.uIcc a b)) (Filter.principal (Set.uIcc a b))","typeReadable":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.uIcc a b)) (Filter.principal (Set.uIcc a b))","typeReferences":[["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["Set","uIcc"],["Set","Ioc"],["LinearOrder"],["Filter","TendstoIxxClass"],["Filter","principal"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["Lattice","toSemilatticeSup"],["instDistribLatticeOfLinearOrder"],["Max","max"],["SemilatticeInf","toMin"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["Min","min"],["SemilatticeSup","toMax"],["Filter","tendsto_Ioc_Icc_Icc"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_Ici_Ioi"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.208377295._hygCtx._hyg.4 : Preorder.{u_1} ��] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.208377295._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Ici.{u_1} α inst._@.Mathlib.Order.Filter.Interval.208377295._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.208377295._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Ici a)) (Filter.principal (Set.Ioi a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Ici a)) (Filter.principal (Set.Ioi a))","typeReferences":[["Set","Ici"],["Preorder"],["Set","Ioc"],["Set","Ioi"],["Filter","TendstoIxxClass"],["Filter","principal"]],"valueReferences":[["Set"],["Filter","tendstoIxxClass_principal"],["Membership","mem"],["lt_of_le_of_lt"],["Preorder","toLT"],["Filter","principal"],["Set","instMembership"],["LT","lt"],["Set","instHasSubset"],["Set","Ici"],["HasSubset","Subset"],["Set","Ioc"],["Iff","mpr"],["LE","le"],["Set","Ioi"],["Filter","TendstoIxxClass"],["Preorder","toLE"],["And","left"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ico_Iio_Iio"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.681871212._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.681871212._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.681871212._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.681871212._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Iio a)) (Filter.principal (Set.Iio a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ico (Filter.principal (Set.Iio a)) (Filter.principal (Set.Iio a))","typeReferences":[["Preorder"],["Set","Iio"],["Filter","TendstoIxxClass"],["Set","Ico"],["Filter","principal"]],"valueReferences":[["Filter","OrdConnected","tendsto_Icc"],["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ico_subset_Icc_self"],["Set","Iio"],["Set","Ico"],["Filter","principal"],["Set","ordConnected_Iio"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ico_atTop_atTop"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.3342815798._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3342815798._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3342815798._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3342815798._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ico Filter.atTop Filter.atTop","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ico Filter.atTop Filter.atTop","typeReferences":[["Filter","atTop"],["Preorder"],["Filter","TendstoIxxClass"],["Set","Ico"]],"valueReferences":[["Filter","atTop"],["Filter","tendsto_Icc_atTop_atTop"],["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ico_subset_Icc_self"],["Set","Ico"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioo_Iio_Iio"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.1696192505._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1696192505._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1696192505._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1696192505._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Iio a)) (Filter.principal (Set.Iio a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Iio a)) (Filter.principal (Set.Iio a))","typeReferences":[["Preorder"],["Set","Iio"],["Filter","TendstoIxxClass"],["Filter","principal"],["Set","Ioo"]],"valueReferences":[["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioo_subset_Ioc_self"],["Set","Iio"],["Filter","tendsto_Ioc_Iio_Iio"],["Set","Ioo"],["Filter","principal"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_atTop_atTop"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.3964344151._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3964344151._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3964344151._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3964344151._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioc Filter.atTop Filter.atTop","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioc Filter.atTop Filter.atTop","typeReferences":[["Filter","atTop"],["Preorder"],["Set","Ioc"],["Filter","TendstoIxxClass"]],"valueReferences":[["Filter","atTop"],["Filter","tendsto_Icc_atTop_atTop"],["Set","Icc"],["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioc_subset_Icc_self"]]},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","Icc"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Order.Filter.Interval.3453516106._hygCtx._hyg.4 : Preorder.{u_1} α] {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} [inst._@.Mathlib.Order.Filter.Interval.3453516106._hygCtx._hyg.11 : Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3453516106._hygCtx._hyg.4) l₁ l₂] {lb : Filter.{u_2} β} {u₁ : β -> α} {u₂ : β -> α}, (Filter.Tendsto.{u_2, u_1} β α u₁ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β α u₂ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β (Set.{u_1} α) (fun (x : β) => Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3453516106._hygCtx._hyg.4 (u₁ x) (u₂ x)) lb (Filter.smallSets.{u_1} α l₂))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Icc l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Icc (u₁ x) (u₂ x)) lb l₂.smallSets","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Icc l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Icc (u₁ x) (u₂ x)) lb l₂.smallSets","typeReferences":[["Filter","smallSets"],["Preorder"],["Set"],["Set","Icc"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["Filter"]],"valueReferences":[["Set"],["Prod","mk"],["Set","Icc"],["Filter","TendstoIxxClass","tendsto_Ixx"],["Prod","snd"],["SProd","sprod"],["Filter"],["Filter","Tendsto","prodMk"],["Prod","fst"],["Prod"],["Filter","smallSets"],["Filter","Tendsto","comp"],["Filter","instSProd"]]},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","Ioc"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Order.Filter.Interval.4146628111._hygCtx._hyg.4 : Preorder.{u_1} α] {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} [inst._@.Mathlib.Order.Filter.Interval.4146628111._hygCtx._hyg.11 : Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4146628111._hygCtx._hyg.4) l₁ l₂] {lb : Filter.{u_2} β} {u₁ : β -> α} {u₂ : β -> α}, (Filter.Tendsto.{u_2, u_1} β α u₁ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β α u₂ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β (Set.{u_1} α) (fun (x : β) => Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4146628111._hygCtx._hyg.4 (u₁ x) (u₂ x)) lb (Filter.smallSets.{u_1} α l₂))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Ioc l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Ioc (u₁ x) (u₂ x)) lb l₂.smallSets","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Ioc l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Ioc (u₁ x) (u₂ x)) lb l₂.smallSets","typeReferences":[["Filter","smallSets"],["Preorder"],["Set"],["Set","Ioc"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["Filter"]],"valueReferences":[["Set"],["Prod","mk"],["Filter","TendstoIxxClass","tendsto_Ixx"],["Prod","snd"],["SProd","sprod"],["Filter"],["Filter","Tendsto","prodMk"],["Prod","fst"],["Prod"],["Filter","smallSets"],["Filter","Tendsto","comp"],["Filter","instSProd"],["Set","Ioc"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioo_atTop_atTop"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.1082240635._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1082240635._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1082240635._hygCtx._hyg.4) (Filter.atTop.{u_1} α inst._@.Mathlib.Order.Filter.Interval.1082240635._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioo Filter.atTop Filter.atTop","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioo Filter.atTop Filter.atTop","typeReferences":[["Filter","atTop"],["Preorder"],["Filter","TendstoIxxClass"],["Set","Ioo"]],"valueReferences":[["Filter","atTop"],["Filter","tendsto_Icc_atTop_atTop"],["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioo_subset_Icc_self"],["Set","Ioo"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendstoIxxClass_inf"],"typeFallback":"forall {α : Type.{u_1}} {l₁ : Filter.{u_1} α} {l₁' : Filter.{u_1} α} {l₂ : Filter.{u_1} α} {l₂' : Filter.{u_1} α} {Ixx : α -> α -> (Set.{u_1} α)} [h : Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂] [h' : Filter.TendstoIxxClass.{u_1} α Ixx l₁' l₂'], Filter.TendstoIxxClass.{u_1} α Ixx (Min.min.{u_1} (Filter.{u_1} α) (Filter.instInf.{u_1} α) l₁ l₁') (Min.min.{u_1} (Filter.{u_1} α) (Filter.instInf.{u_1} α) l₂ l₂')","typeFull":"∀ {α : Type u_1} {l₁ l₁' l₂ l₂' : Filter α} {Ixx : α → α → Set α} [h : Filter.TendstoIxxClass Ixx l₁ l₂]\n  [h' : Filter.TendstoIxxClass Ixx l₁' l₂'], Filter.TendstoIxxClass Ixx (l₁ ⊓ l₁') (l₂ ⊓ l₂')","typeReadable":"∀ {α : Type u_1} {l₁ l₁' l₂ l₂' : Filter α} {Ixx : α → α → Set α} [h : Filter.TendstoIxxClass Ixx l₁ l₂]\n  [h' : Filter.TendstoIxxClass Ixx l₁' l₂'], Filter.TendstoIxxClass Ixx (l₁ ⊓ l₁') (l₂ ⊓ l₂')","typeReferences":[["Min","min"],["Set"],["Filter","TendstoIxxClass"],["Filter","instInf"],["Filter"]],"valueReferences":[["Filter","TendstoIxxClass","tendsto_Ixx"],["Eq","mp"],["Set"],["Filter","smallSets_inf"],["Prod","snd"],["SProd","sprod"],["Filter"],["Filter","instInf"],["Filter","Tendsto","inf"],["congrArg"],["Prod","fst"],["Prod"],["Filter","smallSets"],["Filter","prod_inf_prod"],["Filter","instSProd"],["Min","min"],["Filter","TendstoIxxClass","mk"],["id"],["congrFun'"],["Eq","mpr"],["Eq"],["Filter","Tendsto"]]},{"isProp":true,"kind":"constructor","name":["Filter","TendstoIxxClass","mk"],"typeFallback":"forall {α : Type.{u_1}} {Ixx : α -> α -> (Set.{u_1} α)} {l₁ : Filter.{u_1} α} {l₂ : outParam.{succ u_1} (Filter.{u_1} α)}, (Filter.Tendsto.{u_1, u_1} (Prod.{u_1, u_1} α α) (Set.{u_1} α) (fun (p : Prod.{u_1, u_1} α α) => Ixx (Prod.fst.{u_1, u_1} α α p) (Prod.snd.{u_1, u_1} α α p)) (SProd.sprod.{u_1, u_1, u_1} (Filter.{u_1} α) (Filter.{u_1} α) (Filter.{u_1} (Prod.{u_1, u_1} α α)) (Filter.instSProd.{u_1, u_1} α α) l₁ l₁) (Filter.smallSets.{u_1} α l₂)) -> (Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂)","typeFull":"∀ {α : Type u_1} {Ixx : α → α → Set α} {l₁ : Filter α} {l₂ : outParam (Filter α)},\n  Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets → Filter.TendstoIxxClass Ixx l₁ l₂","typeReadable":"∀ {α : Type u_1} {Ixx : α → α → Set α} {l₁ : Filter α} {l₂ : outParam (Filter α)},\n  Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets → Filter.TendstoIxxClass Ixx l₁ l₂","typeReferences":[["Filter","smallSets"],["Prod"],["Filter","instSProd"],["Set"],["outParam"],["Filter","TendstoIxxClass"],["SProd","sprod"],["Prod","snd"],["Filter","Tendsto"],["Filter"],["Prod","fst"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Order","Filter","Interval",0,"Filter","tendstoIxxClass_principal","_simp_1_3"],"typeFallback":"forall {α : Type.{u}} (x : Set.{u} α) (s : Set.{u} α), Eq.{1} Prop (Membership.mem.{u, u} (Set.{u} α) (Set.{u} (Set.{u} α)) (Set.instMembership.{u} (Set.{u} α)) (Set.powerset.{u} α s) x) (HasSubset.Subset.{u} (Set.{u} α) (Set.instHasSubset.{u} α) x s)","typeFull":"∀ {α : Type u} (x s : Set α), (x ∈ 𝒫 s) = (x ⊆ s)","typeReadable":"∀ {α : Type u} (x s : Set α), (x ∈ 𝒫 s) = (x ⊆ s)","typeReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Set"],["Membership","mem"],["Set","powerset"],["Eq"],["Set","instMembership"]],"valueReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Set"],["Membership","mem"],["Set","powerset"],["Set","mem_powerset_iff"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_Ioi_Ioi"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.4039958656._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4039958656._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4039958656._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.4039958656._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Ioi a)) (Filter.principal (Set.Ioi a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Ioi a)) (Filter.principal (Set.Ioi a))","typeReferences":[["Preorder"],["Set","Ioc"],["Set","Ioi"],["Filter","TendstoIxxClass"],["Filter","principal"]],"valueReferences":[["Filter","OrdConnected","tendsto_Icc"],["Set","Icc"],["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioi"],["Set","ordConnected_Ioi"],["Set","Ioc_subset_Icc_self"],["Filter","principal"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendstoIxxClass_principal"],"typeFallback":"forall {α : Type.{u_1}} {s : Set.{u_1} α} {t : Set.{u_1} α} {Ixx : α -> α -> (Set.{u_1} α)}, Iff (Filter.TendstoIxxClass.{u_1} α Ixx (Filter.principal.{u_1} α s) (Filter.principal.{u_1} α t)) (forall (x : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s x) -> (forall (y : α), (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) s y) -> (HasSubset.Subset.{u_1} (Set.{u_1} α) (Set.instHasSubset.{u_1} α) (Ixx x y) t)))","typeFull":"∀ {α : Type u_1} {s t : Set α} {Ixx : α → α → Set α},\n  Filter.TendstoIxxClass Ixx (Filter.principal s) (Filter.principal t) ↔ ∀ x ∈ s, ∀ y ∈ s, Ixx x y ⊆ t","typeReadable":"∀ {α : Type u_1} {s t : Set α} {Ixx : α → α → Set α},\n  Filter.TendstoIxxClass Ixx (Filter.principal s) (Filter.principal t) ↔ ∀ x ∈ s, ∀ y ∈ s, Ixx x y ⊆ t","typeReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Set"],["Iff"],["Membership","mem"],["Filter","TendstoIxxClass"],["Filter","principal"],["Set","instMembership"]],"valueReferences":[["implies_congr"],["Eq","trans"],["_private","Mathlib","Order","Filter","Interval",0,"Filter","tendstoIxxClass_principal","_simp_1_1"],["Prod","mk"],["Membership","mem"],["SProd","sprod"],["Prod","fst"],["Iff","intro"],["congrArg"],["_private","Mathlib","Order","Filter","Interval",0,"Filter","tendstoIxxClass_principal","_simp_1_2"],["iff_self"],["Filter","instSProd"],["congr"],["Filter","TendstoIxxClass","mk"],["forall_congr"],["Filter","prod_principal_principal"],["Set","powerset"],["congrFun'"],["Set","instSProd"],["Filter","Tendsto"],["Iff","trans"],["True"],["Set"],["Filter","TendstoIxxClass","tendsto_Ixx"],["_private","Mathlib","Order","Filter","Interval",0,"Filter","tendstoIxxClass_principal","_simp_1_3"],["Prod","snd"],["Filter"],["Filter","principal"],["Set","instMembership"],["Filter","smallSets"],["Prod"],["Set","instHasSubset"],["HasSubset","Subset"],["of_eq_true"],["Eq","refl"],["Iff"],["Filter","smallSets_principal"],["Filter","TendstoIxxClass"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ico_atBot_atBot"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.912555187._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.912555187._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.912555187._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.912555187._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ico Filter.atBot Filter.atBot","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ico Filter.atBot Filter.atBot","typeReferences":[["Preorder"],["Filter","atBot"],["Filter","TendstoIxxClass"],["Set","Ico"]],"valueReferences":[["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Filter","atBot"],["Set","Ico_subset_Icc_self"],["Filter","tendsto_Icc_atBot_atBot"],["Set","Ico"]]},{"isProp":false,"kind":"definition","name":["Filter","TendstoIxxClass","recOn"],"typeFallback":"forall {α : Type.{u_1}} {Ixx : α -> α -> (Set.{u_1} α)} {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} {motive : (Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂) -> Sort.{u}} (t : Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂), (forall (tendsto_Ixx : Filter.Tendsto.{u_1, u_1} (Prod.{u_1, u_1} α α) (Set.{u_1} α) (fun (p : Prod.{u_1, u_1} α α) => Ixx (Prod.fst.{u_1, u_1} α α p) (Prod.snd.{u_1, u_1} α α p)) (SProd.sprod.{u_1, u_1, u_1} (Filter.{u_1} α) (Filter.{u_1} α) (Filter.{u_1} (Prod.{u_1, u_1} α α)) (Filter.instSProd.{u_1, u_1} α α) l₁ l₁) (Filter.smallSets.{u_1} α l₂)), motive (Filter.TendstoIxxClass.mk.{u_1} α Ixx l₁ l₂ tendsto_Ixx)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {Ixx : α → α → Set α} →\n    {l₁ l₂ : Filter α} →\n      {motive : Filter.TendstoIxxClass Ixx l₁ l₂ → Sort u} →\n        (t : Filter.TendstoIxxClass Ixx l₁ l₂) →\n          ((tendsto_Ixx : Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets) → motive ⋯) → motive t","typeReadable":"{α : Type u_1} →\n  {Ixx : α → α → Set α} →\n    {l₁ l₂ : Filter α} →\n      {motive : Filter.TendstoIxxClass Ixx l₁ l₂ → Sort u} →\n        (t : Filter.TendstoIxxClass Ixx l₁ l₂) →\n          ((tendsto_Ixx : Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets) → motive ⋯) → motive t","typeReferences":[["Filter","smallSets"],["Prod"],["Filter","instSProd"],["Set"],["Filter","TendstoIxxClass","mk"],["SProd","sprod"],["Prod","snd"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["Filter"],["Prod","fst"]],"valueReferences":[["Filter","TendstoIxxClass","rec"]]},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","Ico"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Order.Filter.Interval.2868244452._hygCtx._hyg.4 : Preorder.{u_1} α] {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} [inst._@.Mathlib.Order.Filter.Interval.2868244452._hygCtx._hyg.11 : Filter.TendstoIxxClass.{u_1} α (Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2868244452._hygCtx._hyg.4) l₁ l₂] {lb : Filter.{u_2} β} {u₁ : β -> α} {u₂ : β -> α}, (Filter.Tendsto.{u_2, u_1} β α u₁ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β α u₂ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β (Set.{u_1} α) (fun (x : β) => Set.Ico.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2868244452._hygCtx._hyg.4 (u₁ x) (u₂ x)) lb (Filter.smallSets.{u_1} α l₂))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Ico l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Ico (u₁ x) (u₂ x)) lb l₂.smallSets","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Ico l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Ico (u₁ x) (u₂ x)) lb l₂.smallSets","typeReferences":[["Filter","smallSets"],["Preorder"],["Set"],["Filter","TendstoIxxClass"],["Set","Ico"],["Filter","Tendsto"],["Filter"]],"valueReferences":[["Set"],["Prod","mk"],["Filter","TendstoIxxClass","tendsto_Ixx"],["Prod","snd"],["SProd","sprod"],["Filter"],["Filter","Tendsto","prodMk"],["Prod","fst"],["Prod"],["Filter","smallSets"],["Filter","Tendsto","comp"],["Filter","instSProd"],["Set","Ico"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendstoIxxClass_of_subset"],"typeFallback":"forall {α : Type.{u_1}} {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} {Ixx : α -> α -> (Set.{u_1} α)} {Ixx' : α -> α -> (Set.{u_1} α)}, (forall (a : α) (b : α), HasSubset.Subset.{u_1} (Set.{u_1} α) (Set.instHasSubset.{u_1} α) (Ixx a b) (Ixx' a b)) -> (forall [h' : Filter.TendstoIxxClass.{u_1} α Ixx' l₁ l₂], Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂)","typeFull":"∀ {α : Type u_1} {l₁ l₂ : Filter α} {Ixx Ixx' : α → α → Set α},\n  (∀ (a b : α), Ixx a b ⊆ Ixx' a b) → ∀ [h' : Filter.TendstoIxxClass Ixx' l₁ l₂], Filter.TendstoIxxClass Ixx l₁ l₂","typeReadable":"∀ {α : Type u_1} {l₁ l₂ : Filter α} {Ixx Ixx' : α → α → Set α},\n  (∀ (a b : α), Ixx a b ⊆ Ixx' a b) → ∀ [h' : Filter.TendstoIxxClass Ixx' l₁ l₂], Filter.TendstoIxxClass Ixx l₁ l₂","typeReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Set"],["Filter","TendstoIxxClass"],["Filter"]],"valueReferences":[["Prod","forall"],["Prod","mk"],["Filter","TendstoIxxClass","tendsto_Ixx"],["Set"],["SProd","sprod"],["Prod","snd"],["Filter","Eventually","of_forall"],["Filter"],["Prod","fst"],["Prod"],["Set","instHasSubset"],["Filter","Tendsto","smallSets_mono"],["Filter","instSProd"],["HasSubset","Subset"],["Iff","mpr"],["Filter","TendstoIxxClass","mk"]]},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","Ioo"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Order.Filter.Interval.889005713._hygCtx._hyg.4 : Preorder.{u_1} α] {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} [inst._@.Mathlib.Order.Filter.Interval.889005713._hygCtx._hyg.11 : Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.889005713._hygCtx._hyg.4) l₁ l₂] {lb : Filter.{u_2} β} {u₁ : β -> α} {u₂ : β -> α}, (Filter.Tendsto.{u_2, u_1} β α u₁ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β α u₂ lb l₁) -> (Filter.Tendsto.{u_2, u_1} β (Set.{u_1} α) (fun (x : β) => Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.889005713._hygCtx._hyg.4 (u₁ x) (u₂ x)) lb (Filter.smallSets.{u_1} α l₂))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Ioo l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Ioo (u₁ x) (u₂ x)) lb l₂.smallSets","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] {l₁ l₂ : Filter α} [Filter.TendstoIxxClass Set.Ioo l₁ l₂]\n  {lb : Filter β} {u₁ u₂ : β → α},\n  Filter.Tendsto u₁ lb l₁ → Filter.Tendsto u₂ lb l₁ → Filter.Tendsto (fun x => Set.Ioo (u₁ x) (u₂ x)) lb l₂.smallSets","typeReferences":[["Filter","smallSets"],["Preorder"],["Set"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["Set","Ioo"],["Filter"]],"valueReferences":[["Set"],["Prod","mk"],["Filter","TendstoIxxClass","tendsto_Ixx"],["Prod","snd"],["SProd","sprod"],["Set","Ioo"],["Filter"],["Filter","Tendsto","prodMk"],["Prod","fst"],["Prod"],["Filter","smallSets"],["Filter","Tendsto","comp"],["Filter","instSProd"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_atBot_atBot"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.3456352062._hygCtx._hyg.4 : Preorder.{u_1} α], Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3456352062._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3456352062._hygCtx._hyg.4) (Filter.atBot.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3456352062._hygCtx._hyg.4)","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioc Filter.atBot Filter.atBot","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Filter.TendstoIxxClass Set.Ioc Filter.atBot Filter.atBot","typeReferences":[["Preorder"],["Set","Ioc"],["Filter","atBot"],["Filter","TendstoIxxClass"]],"valueReferences":[["Set","Icc"],["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Filter","atBot"],["Set","Ioc_subset_Icc_self"],["Filter","tendsto_Icc_atBot_atBot"]]},{"isProp":false,"kind":"recursor","name":["Filter","TendstoIxxClass","rec"],"typeFallback":"forall {α : Type.{u_1}} {Ixx : α -> α -> (Set.{u_1} α)} {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} {motive : (Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂) -> Sort.{u}}, (forall (tendsto_Ixx : Filter.Tendsto.{u_1, u_1} (Prod.{u_1, u_1} α α) (Set.{u_1} α) (fun (p : Prod.{u_1, u_1} α α) => Ixx (Prod.fst.{u_1, u_1} α α p) (Prod.snd.{u_1, u_1} α α p)) (SProd.sprod.{u_1, u_1, u_1} (Filter.{u_1} α) (Filter.{u_1} α) (Filter.{u_1} (Prod.{u_1, u_1} α α)) (Filter.instSProd.{u_1, u_1} α α) l₁ l₁) (Filter.smallSets.{u_1} α l₂)), motive (Filter.TendstoIxxClass.mk.{u_1} α Ixx l₁ l₂ tendsto_Ixx)) -> (forall (t : Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂), motive t)","typeFull":"{α : Type u_1} →\n  {Ixx : α → α → Set α} →\n    {l₁ l₂ : Filter α} →\n      {motive : Filter.TendstoIxxClass Ixx l₁ l₂ → Sort u} →\n        ((tendsto_Ixx : Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets) → motive ⋯) →\n          (t : Filter.TendstoIxxClass Ixx l₁ l₂) → motive t","typeReadable":"{α : Type u_1} →\n  {Ixx : α → α → Set α} →\n    {l₁ l₂ : Filter α} →\n      {motive : Filter.TendstoIxxClass Ixx l₁ l₂ → Sort u} →\n        ((tendsto_Ixx : Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets) → motive ⋯) →\n          (t : Filter.TendstoIxxClass Ixx l₁ l₂) → motive t","typeReferences":[["Filter","smallSets"],["Prod"],["Filter","instSProd"],["Set"],["Filter","TendstoIxxClass","mk"],["SProd","sprod"],["Prod","snd"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["Filter"],["Prod","fst"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_Iic_Iic"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.3532795932._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3532795932._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Iic.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3532795932._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Iic.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3532795932._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Iic a)) (Filter.principal (Set.Iic a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Iic a)) (Filter.principal (Set.Iic a))","typeReferences":[["Preorder"],["Set","Ioc"],["Filter","TendstoIxxClass"],["Set","Iic"],["Filter","principal"]],"valueReferences":[["Filter","OrdConnected","tendsto_Icc"],["Set","Icc"],["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioc_subset_Icc_self"],["Set","ordConnected_Iic"],["Set","Iic"],["Filter","principal"]]},{"isProp":true,"kind":"theorem","name":["Filter","Tendsto","uIcc"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Order.Filter.Interval.3013841367._hygCtx._hyg.4 : LinearOrder.{u_1} α] {l : Filter.{u_1} α} [inst._@.Mathlib.Order.Filter.Interval.3013841367._hygCtx._hyg.9 : Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3013841367._hygCtx._hyg.4)))))) l l] {f : β -> α} {g : β -> α} {lb : Filter.{u_2} β}, (Filter.Tendsto.{u_2, u_1} β α f lb l) -> (Filter.Tendsto.{u_2, u_1} β α g lb l) -> (Filter.Tendsto.{u_2, u_1} β (Set.{u_1} α) (fun (x : β) => Set.uIcc.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.3013841367._hygCtx._hyg.4)) (f x) (g x)) lb (Filter.smallSets.{u_1} α l))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {l : Filter α} [Filter.TendstoIxxClass Set.Icc l l] {f g : β → α}\n  {lb : Filter β},\n  Filter.Tendsto f lb l → Filter.Tendsto g lb l → Filter.Tendsto (fun x => Set.uIcc (f x) (g x)) lb l.smallSets","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] {l : Filter α} [Filter.TendstoIxxClass Set.Icc l l] {f g : β → α}\n  {lb : Filter β},\n  Filter.Tendsto f lb l → Filter.Tendsto g lb l → Filter.Tendsto (fun x => Set.uIcc (f x) (g x)) lb l.smallSets","typeReferences":[["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["Set","Icc"],["Set"],["Set","uIcc"],["LinearOrder"],["Filter"],["Filter","smallSets"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["Filter","TendstoIxxClass","tendsto_Ixx"],["Set"],["Prod","mk"],["Set","uIcc"],["Filter","tendsto_uIcc_of_Icc"],["Prod","snd"],["SProd","sprod"],["Filter","Tendsto","prodMk"],["Filter"],["Prod","fst"],["Prod"],["Filter","smallSets"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Filter","Tendsto","comp"],["Filter","instSProd"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Icc_uIcc_uIcc"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.2015571506._hygCtx._hyg.4 : LinearOrder.{u_1} α] {a : α} {b : α}, Filter.TendstoIxxClass.{u_1} α (Set.Icc.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2015571506._hygCtx._hyg.4)))))) (Filter.principal.{u_1} α (Set.uIcc.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2015571506._hygCtx._hyg.4)) a b)) (Filter.principal.{u_1} α (Set.uIcc.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2015571506._hygCtx._hyg.4)) a b))","typeFull":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : α},\n  Filter.TendstoIxxClass Set.Icc (Filter.principal (Set.uIcc a b)) (Filter.principal (Set.uIcc a b))","typeReadable":"∀ {α : Type u_1} [inst : LinearOrder α] {a b : α},\n  Filter.TendstoIxxClass Set.Icc (Filter.principal (Set.uIcc a b)) (Filter.principal (Set.uIcc a b))","typeReferences":[["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["Set","uIcc"],["Set","Icc"],["LinearOrder"],["Filter","TendstoIxxClass"],["Filter","principal"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["Lattice","toSemilatticeSup"],["instDistribLatticeOfLinearOrder"],["Max","max"],["SemilatticeInf","toMin"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["Min","min"],["SemilatticeSup","toMax"],["Filter","tendsto_Icc_Icc_Icc"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioc_Icc_Icc"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.918863124._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α} {b : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.918863124._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.918863124._hygCtx._hyg.4 a b)) (Filter.principal.{u_1} α (Set.Icc.{u_1} α inst._@.Mathlib.Order.Filter.Interval.918863124._hygCtx._hyg.4 a b))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a b : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Icc a b)) (Filter.principal (Set.Icc a b))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a b : α},\n  Filter.TendstoIxxClass Set.Ioc (Filter.principal (Set.Icc a b)) (Filter.principal (Set.Icc a b))","typeReferences":[["Preorder"],["Set","Icc"],["Set","Ioc"],["Filter","TendstoIxxClass"],["Filter","principal"]],"valueReferences":[["Set","Ioc"],["Set","Icc"],["Filter","tendstoIxxClass_of_subset"],["Filter","tendsto_Icc_Icc_Icc"],["Set","Ioc_subset_Icc_self"],["Filter","principal"]]},{"isProp":false,"kind":"definition","name":["Filter","TendstoIxxClass","casesOn"],"typeFallback":"forall {α : Type.{u_1}} {Ixx : α -> α -> (Set.{u_1} α)} {l₁ : Filter.{u_1} α} {l₂ : Filter.{u_1} α} {motive : (Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂) -> Sort.{u}} (t : Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂), (forall (tendsto_Ixx : Filter.Tendsto.{u_1, u_1} (Prod.{u_1, u_1} α α) (Set.{u_1} α) (fun (p : Prod.{u_1, u_1} α α) => Ixx (Prod.fst.{u_1, u_1} α α p) (Prod.snd.{u_1, u_1} α α p)) (SProd.sprod.{u_1, u_1, u_1} (Filter.{u_1} α) (Filter.{u_1} α) (Filter.{u_1} (Prod.{u_1, u_1} α α)) (Filter.instSProd.{u_1, u_1} α α) l₁ l₁) (Filter.smallSets.{u_1} α l₂)), motive (Filter.TendstoIxxClass.mk.{u_1} α Ixx l₁ l₂ tendsto_Ixx)) -> (motive t)","typeFull":"{α : Type u_1} →\n  {Ixx : α → α → Set α} →\n    {l₁ l₂ : Filter α} →\n      {motive : Filter.TendstoIxxClass Ixx l₁ l₂ → Sort u} →\n        (t : Filter.TendstoIxxClass Ixx l₁ l₂) →\n          ((tendsto_Ixx : Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets) → motive ⋯) → motive t","typeReadable":"{α : Type u_1} →\n  {Ixx : α → α → Set α} →\n    {l₁ l₂ : Filter α} →\n      {motive : Filter.TendstoIxxClass Ixx l₁ l₂ → Sort u} →\n        (t : Filter.TendstoIxxClass Ixx l₁ l₂) →\n          ((tendsto_Ixx : Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets) → motive ⋯) → motive t","typeReferences":[["Filter","smallSets"],["Prod"],["Filter","instSProd"],["Set"],["Filter","TendstoIxxClass","mk"],["SProd","sprod"],["Prod","snd"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["Filter"],["Prod","fst"]],"valueReferences":[["Filter","TendstoIxxClass","rec"]]},{"isProp":true,"kind":"theorem","name":["Filter","TendstoIxxClass","tendsto_Ixx"],"typeFallback":"forall {α : Type.{u_1}} {Ixx : α -> α -> (Set.{u_1} α)} {l₁ : Filter.{u_1} α} {l₂ : outParam.{succ u_1} (Filter.{u_1} α)} [self : Filter.TendstoIxxClass.{u_1} α Ixx l₁ l₂], Filter.Tendsto.{u_1, u_1} (Prod.{u_1, u_1} α α) (Set.{u_1} α) (fun (p : Prod.{u_1, u_1} α α) => Ixx (Prod.fst.{u_1, u_1} α α p) (Prod.snd.{u_1, u_1} α α p)) (SProd.sprod.{u_1, u_1, u_1} (Filter.{u_1} α) (Filter.{u_1} α) (Filter.{u_1} (Prod.{u_1, u_1} α α)) (Filter.instSProd.{u_1, u_1} α α) l₁ l₁) (Filter.smallSets.{u_1} α l₂)","typeFull":"∀ {α : Type u_1} {Ixx : α → α → Set α} {l₁ : Filter α} {l₂ : outParam (Filter α)}\n  [self : Filter.TendstoIxxClass Ixx l₁ l₂], Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets","typeReadable":"∀ {α : Type u_1} {Ixx : α → α → Set α} {l₁ : Filter α} {l₂ : outParam (Filter α)}\n  [self : Filter.TendstoIxxClass Ixx l₁ l₂], Filter.Tendsto (fun p => Ixx p.1 p.2) (l₁ ×ˢ l₁) l₂.smallSets","typeReferences":[["Filter","smallSets"],["Prod"],["Filter","instSProd"],["Set"],["outParam"],["SProd","sprod"],["Prod","snd"],["Filter","TendstoIxxClass"],["Filter","Tendsto"],["Filter"],["Prod","fst"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["Filter","tendsto_Ioo_Ici_Ioi"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Filter.Interval.2762517700._hygCtx._hyg.4 : Preorder.{u_1} α] {a : α}, Filter.TendstoIxxClass.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2762517700._hygCtx._hyg.4) (Filter.principal.{u_1} α (Set.Ici.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2762517700._hygCtx._hyg.4 a)) (Filter.principal.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Filter.Interval.2762517700._hygCtx._hyg.4 a))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Ici a)) (Filter.principal (Set.Ioi a))","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α},\n  Filter.TendstoIxxClass Set.Ioo (Filter.principal (Set.Ici a)) (Filter.principal (Set.Ioi a))","typeReferences":[["Set","Ici"],["Preorder"],["Set","Ioi"],["Filter","TendstoIxxClass"],["Filter","principal"],["Set","Ioo"]],"valueReferences":[["Set","Ici"],["Set","Ioc"],["Filter","tendstoIxxClass_of_subset"],["Set","Ioo_subset_Ioc_self"],["Set","Ioi"],["Set","Ioo"],["Filter","principal"],["Filter","tendsto_Ioc_Ici_Ioi"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Topology.Connected.CardComponents.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Connected","CardComponents",0,"IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton","_simp_1_2"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.4 : PartialOrder.{u} α] [inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.7 : OrderTop.{u} α (Preorder.toLE.{u} α (PartialOrder.toPreorder.{u} α inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.4))] {a : α}, Eq.{1} Prop (LE.le.{u} α (Preorder.toLE.{u} α (PartialOrder.toPreorder.{u} α inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.4)) (Top.top.{u} α (OrderTop.toTop.{u} α (Preorder.toLE.{u} α (PartialOrder.toPreorder.{u} α inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.4)) inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.7)) a) (Eq.{succ u} α a (Top.top.{u} α (OrderTop.toTop.{u} α (Preorder.toLE.{u} α (PartialOrder.toPreorder.{u} α inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.4)) inst._@.Mathlib.Order.BoundedOrder.Basic.1640584106._hygCtx._hyg.7)))","typeFull":"∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderTop α] {a : α}, (⊤ ≤ a) = (a = ⊤)","typeReadable":"∀ {α : Type u} [inst : PartialOrder α] [inst_1 : OrderTop α] {a : α}, (⊤ ≤ a) = (a = ⊤)","typeReferences":[["PartialOrder","toPreorder"],["PartialOrder"],["LE","le"],["Top","top"],["OrderTop"],["Eq"],["Preorder","toLE"],["OrderTop","toTop"]],"valueReferences":[["PartialOrder","toPreorder"],["LE","le"],["top_le_iff"],["Top","top"],["Eq"],["Preorder","toLE"],["OrderTop","toTop"],["propext"]]},{"isProp":true,"kind":"theorem","name":["IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton"],"typeFallback":"forall {X : Type.{u_1}} {Y : Type.{u_2}} [inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.4 : TopologicalSpace.{u_1} X] [inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.7 : TopologicalSpace.{u_2} Y] {f : X -> Y}, (IsOpenMap.{u_1, u_2} X Y inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.4 inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.7 f) -> (IsClosedMap.{u_1, u_2} X Y inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.4 inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.7 f) -> (forall [inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.17 : ConnectedSpace.{u_2} Y inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.7] (y : Y), LE.le.{0} ENat instLEENat (ENat.card.{u_1} (ConnectedComponents.{u_1} X inst._@.Mathlib.Topology.Connected.CardComponents.4063866965._hygCtx._hyg.4)) (Set.encard.{u_1} X (Set.preimage.{u_1, u_2} X Y f (Singleton.singleton.{u_2, u_2} Y (Set.{u_2} Y) (Set.instSingletonSet.{u_2} Y) y))))","typeFull":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},\n  IsOpenMap f → IsClosedMap f → ∀ [ConnectedSpace Y] (y : Y), ENat.card (ConnectedComponents X) ≤ (f ⁻¹' {y}).encard","typeReadable":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},\n  IsOpenMap f → IsClosedMap f → ∀ [ConnectedSpace Y] (y : Y), ENat.card (ConnectedComponents X) ≤ (f ⁻¹' {y}).encard","typeReferences":[["ConnectedComponents"],["Set","preimage"],["IsClosedMap"],["Set"],["Singleton","singleton"],["ENat"],["instLEENat"],["IsOpenMap"],["Set","instSingletonSet"],["Set","encard"],["TopologicalSpace"],["ConnectedSpace"],["LE","le"],["ENat","card"]],"valueReferences":[["ConnectedComponents"],["Nat","instSuccAddOrder"],["isClopen_discrete"],["Set","encard_iUnion_of_finite"],["Equiv"],["instOrderTopENat"],["Order","Frame","toHeytingAlgebra"],["_private","Mathlib","Topology","Connected","CardComponents",0,"IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton","_simp_1_1"],["Eq","symm"],["Finset","sum"],["Set","Nonempty","image"],["HeytingAlgebra","toOrderBot"],["NonAssocSemiring","toAddCommMonoidWithOne"],["instLTNat"],["ENat","card_eq_coe_fintype_card"],["Order","lt_add_one_iff","_simp_1"],["instTopENat"],["AddMonoid","toAddSemigroup"],["Set","univ_inter"],["eq_false"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["instIsLeftCancelAddOfAddLeftReflectLE"],["Top","top"],["Eq","mpr"],["covariant_swap_add_of_covariant_add"],["Set","iUnion_singleton_eq_range"],["zero_le","_simp_1"],["lt_add_iff_pos_left","_simp_1"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["ENat"],["ConnectedComponents","exists_fun_isClopen_of_infinite"],["MulZeroOneClass","toMulOneClass"],["instTransLE"],["AddCommMonoid","toAddMonoid"],["Pairwise"],["Fin","fintype"],["Fintype","card"],["Fintype","sum_mono"],["Fintype","equivFin"],["Set","disjoint_singleton_left","_simp_1"],["EquivLike","toFunLike"],["Equiv","symm"],["Eq"],["nonempty_fintype"],["instCanonicallyOrderedAddENat"],["Finset","univ"],["IsClopen","eq_univ"],["Disjoint","inter_left"],["Set"],["mul_one"],["AddZero","toAdd"],["Nat","cast_add"],["Set","univ"],["eq_self"],["IsOpen"],["Set","singleton_nonempty"],["instIsRightCancelAddOfAddRightReflectLE"],["contravariant_swap_add_of_contravariant_add"],["PartialOrder","toPreorder"],["Membership","mem"],["Inter","inter"],["Preorder","toLT"],["Fin"],["Nat","instAddCommMonoid"],["self_le_add_right","_simp_1"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Semiring","toNonAssocSemiring"],["not_false_eq_true"],["eq_of_heq"],["Finset","sum_const"],["Finite"],["Eq","rec"],["IsClosed"],["NonAssocSemiring","toMulZeroOneClass"],["And","left"],["SemilatticeInf","toPartialOrder"],["Nat","instPartialOrder"],["Set","iUnion_inter"],["And","right"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Nonempty"],["AddZeroClass","toAddZero"],["Nat","instCanonicallyOrderedAdd"],["Exists","casesOn"],["Nat"],["AddMonoidWithOne","toNatCast"],["AddMonoid","toNSMul"],["Set","Nonempty","preimage"],["AddZero","toZero"],["IsClopen","preimage"],["_private","Mathlib","Topology","Connected","CardComponents",0,"IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton","_simp_1_2"],["Nat","cast"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["LinearOrderedAddCommMonoidWithTop","toIsOrderedAddMonoid"],["le_trans"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["DFunLike","coe"],["Set","range"],["Set","encard"],["instDistribLatticeOfLinearOrder"],["ConnectedComponents","continuous_coe"],["EquivLike","range_eq_univ"],["AddCommSemigroup","toAddCommMagma"],["finsum"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Not"],["CompleteDistribLattice","toFrame"],["Set","Nonempty"],["ENat","instNatCast"],["Set","preimage"],["ConnectedComponents","instTopologicalSpace"],["instHAdd"],["Distrib","toMul"],["Set","instSingletonSet"],["LT","lt"],["ENat","card_eq_top_of_infinite"],["IsClopen"],["One","toOfNat1"],["of_eq_true"],["instPreorderENat"],["False"],["OrderTop","toTop"],["instAddNat"],["TotallyDisconnectedSpace","t1Space"],["Eq","trans"],["Singleton","singleton"],["Function","onFun"],["Nat","instAddCommSemigroup"],["Finite","of_fintype"],["ConnectedComponents","surjective_coe"],["ConnectedSpace","toPreconnectedSpace"],["Eq","ndrec"],["Set","instInter"],["finite_or_infinite"],["_private","Mathlib","Topology","Connected","CardComponents",0,"IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton","_simp_1_3"],["Set","instMembership"],["instLinearOrderENat"],["BoundedOrder","toOrderBot"],["Set","instCompleteAtomicBooleanAlgebra"],["Eq","refl"],["AddMonoidWithOne","toOne"],["HEq"],["AddMonoid","toAddZeroClass"],["Nat","instPreorder"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["contravariant_lt_of_covariant_le"],["instOfNatNat"],["Std","le_refl","_simp_1"],["AddCommMagma","toAdd"],["Preorder","toLE"],["IsLeftCancelAdd","addLeftReflectLE_of_addLeftReflectLT"],["Infinite"],["CompleteLattice","toConditionallyCompleteLattice"],["Nat","instAddMonoid"],["Set","instDistribLattice"],["Finite","instDiscreteTopology"],["ConnectedComponents","totallyDisconnectedSpace"],["OfNat","ofNat"],["HAdd","hAdd"],["ENat","card"],["And","casesOn"],["Nat","cast_one"],["HMul","hMul"],["Disjoint","preimage"],["instLEENat"],["Set","iUnion"],["CompleteBooleanAlgebra","toCompleteLattice"],["IsRightCancelAdd","addRightStrictMono_of_addRightMono"],["AddMonoidWithOne","toAddMonoid"],["Set","mem_univ"],["finsum_eq_sum_of_fintype"],["And","intro"],["Disjoint"],["instLinearOrderedAddCommMonoidWithTopENat"],["Nat","instIsOrderedAddMonoid"],["Nat","instLinearOrder"],["AddSemigroup","toAdd"],["Nat","instNoMaxOrder"],["NonUnitalNonAssocSemiring","toDistrib"],["LinearOrderedAddCommMonoidWithTop","toAddCommMonoid"],["And"],["IsOrderedAddMonoid","toAddLeftMono"],["Fintype"],["Set","image"],["Iff","mpr"],["HSMul","hSMul"],["Nonempty","casesOn"],["id"],["instHMul"],["EquivLike","toEmbeddingLike"],["Trans","simple"],["Equiv","instEquivLike"],["instCommSemiringENat"],["congrArg"],["Fintype","card_fin"],["Nat","instOne"],["Nat","instIsOrderedCancelAddMonoid"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["instAddMonoidWithOneENat"],["Disjoint","inter_right"],["Lattice","toSemilatticeInf"],["HEq","refl"],["True"],["Set","instBoundedOrder"],["Finset","card"],["CommSemiring","toSemiring"],["instReflLe"],["Eq","casesOn"],["EmbeddingLike","apply_eq_iff_eq","_simp_1"],["Or","casesOn"],["DistribLattice","toLattice"],["Set","one_le_encard_iff_nonempty"],["nsmul_eq_mul"],["LE","le"],["Set","mem_singleton_iff","_simp_1"],["ConnectedComponents","mk"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Connected","CardComponents",0,"IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton","_simp_1_3"],"typeFallback":"forall {n : ENat}, Eq.{1} Prop (Eq.{1} ENat n (Top.top.{0} ENat instTopENat)) (forall (m : Nat), LE.le.{0} ENat instLEENat (Nat.cast.{0} ENat ENat.instNatCast m) n)","typeFull":"∀ {n : ℕ∞}, (n = ⊤) = ∀ (m : ℕ), ↑m ≤ n","typeReadable":"∀ {n : ℕ∞}, (n = ⊤) = ∀ (m : ℕ), ↑m ≤ n","typeReferences":[["ENat","instNatCast"],["Nat"],["Nat","cast"],["instTopENat"],["LE","le"],["Top","top"],["ENat"],["instLEENat"],["Eq"]],"valueReferences":[["ENat","instNatCast"],["Nat"],["Nat","cast"],["instTopENat"],["ENat","eq_top_iff_forall_ge"],["LE","le"],["Top","top"],["ENat"],["instLEENat"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["IsOpenMap","finite_connectedComponents_of_finite_preimage_singleton_of_connectedSpace"],"typeFallback":"forall {X : Type.{u_1}} {Y : Type.{u_2}} [inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.4 : TopologicalSpace.{u_1} X] [inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.7 : TopologicalSpace.{u_2} Y] {f : X -> Y}, (IsOpenMap.{u_1, u_2} X Y inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.4 inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.7 f) -> (IsClosedMap.{u_1, u_2} X Y inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.4 inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.7 f) -> (forall [inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.17 : ConnectedSpace.{u_2} Y inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.7] {y : Y}, (Set.Finite.{u_1} X (Set.preimage.{u_1, u_2} X Y f (Singleton.singleton.{u_2, u_2} Y (Set.{u_2} Y) (Set.instSingletonSet.{u_2} Y) y))) -> (Finite.{succ u_1} (ConnectedComponents.{u_1} X inst._@.Mathlib.Topology.Connected.CardComponents.50337438._hygCtx._hyg.4)))","typeFull":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},\n  IsOpenMap f → IsClosedMap f → ∀ [ConnectedSpace Y] {y : Y}, (f ⁻¹' {y}).Finite → Finite (ConnectedComponents X)","typeReadable":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},\n  IsOpenMap f → IsClosedMap f → ∀ [ConnectedSpace Y] {y : Y}, (f ⁻¹' {y}).Finite → Finite (ConnectedComponents X)","typeReferences":[["TopologicalSpace"],["ConnectedComponents"],["Set","preimage"],["ConnectedSpace"],["IsClosedMap"],["Set"],["Singleton","singleton"],["IsOpenMap"],["Finite"],["Set","instSingletonSet"],["Set","Finite"]],"valueReferences":[["ConnectedComponents"],["Set","preimage"],["Set"],["IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton"],["Singleton","singleton"],["lt_of_le_of_lt"],["ENat"],["Set","Finite","encard_lt_top"],["Set","instSingletonSet"],["instLTENat"],["Set","encard"],["congrArg"],["LT","lt"],["instTopENat"],["instPreorderENat"],["ENat","card_lt_top"],["ENat","card"],["Top","top"],["id"],["Eq","symm"],["Eq","mpr"],["Finite"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Connected","CardComponents",0,"IsOpenMap","enatCard_connectedComponents_le_encard_preimage_singleton","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} {ι : Sort.{u_5}} {f : α -> β} {s : ι -> (Set.{u_2} β)}, Eq.{succ u_1} (Set.{u_1} α) (Set.iUnion.{u_1, u_5} α ι (fun (i : ι) => Set.preimage.{u_1, u_2} α β f (s i))) (Set.preimage.{u_1, u_2} α β f (Set.iUnion.{u_2, u_5} β ι (fun (i : ι) => s i)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {f : α → β} {s : ι → Set β}, ⋃ i, f ⁻¹' s i = f ⁻¹' ⋃ i, s i","typeReadable":"∀ {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {f : α → β} {s : ι → Set β}, ⋃ i, f ⁻¹' s i = f ⁻¹' ⋃ i, s i","typeReferences":[["Set","preimage"],["Set"],["Set","iUnion"],["Eq"]],"valueReferences":[["Set","preimage"],["Set","preimage_iUnion"],["Set"],["Eq","symm"],["Set","iUnion"]]},{"isProp":true,"kind":"theorem","name":["IsOpenMap","finite_connectedComponents_of_finite_preimage_singleton"],"typeFallback":"forall {X : Type.{u_1}} {Y : Type.{u_2}} [inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.4 : TopologicalSpace.{u_1} X] [inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.7 : TopologicalSpace.{u_2} Y] {f : X -> Y}, (IsOpenMap.{u_1, u_2} X Y inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.4 inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.7 f) -> (IsClosedMap.{u_1, u_2} X Y inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.4 inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.7 f) -> (forall [inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.17 : Finite.{succ u_2} (ConnectedComponents.{u_2} Y inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.7)], (Continuous.{u_1, u_2} X Y inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.4 inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.7 f) -> (forall (y : Y), Set.Finite.{u_1} X (Set.preimage.{u_1, u_2} X Y f (Singleton.singleton.{u_2, u_2} Y (Set.{u_2} Y) (Set.instSingletonSet.{u_2} Y) y))) -> (Finite.{succ u_1} (ConnectedComponents.{u_1} X inst._@.Mathlib.Topology.Connected.CardComponents.2864651515._hygCtx._hyg.4)))","typeFull":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},\n  IsOpenMap f →\n    IsClosedMap f →\n      ∀ [Finite (ConnectedComponents Y)],\n        Continuous f → (∀ (y : Y), (f ⁻¹' {y}).Finite) → Finite (ConnectedComponents X)","typeReadable":"∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] {f : X → Y},\n  IsOpenMap f →\n    IsClosedMap f →\n      ∀ [Finite (ConnectedComponents Y)],\n        Continuous f → (∀ (y : Y), (f ⁻¹' {y}).Finite) → Finite (ConnectedComponents X)","typeReferences":[["TopologicalSpace"],["Continuous"],["ConnectedComponents"],["Set","preimage"],["IsClosedMap"],["Set"],["Singleton","singleton"],["IsOpenMap"],["Set","instSingletonSet"],["Set","Finite"],["Finite"]],"valueReferences":[["ConnectedComponents"],["Subtype","mk","injEq"],["TotallyDisconnectedSpace","t1Space"],["Eq","trans"],["isClopen_discrete"],["Singleton","singleton"],["Exists","intro"],["Subtype","val"],["and_iff_left_iff_imp","_simp_1"],["Set","Elem"],["ConnectedComponents","surjective_coe"],["Finite","instSigma"],["IsClosedMap","restrictPreimage"],["Eq","symm"],["Subtype","exists","_simp_1"],["Sigma"],["HeytingAlgebra","toOrderBot"],["Eq","ndrec"],["rfl"],["Exists"],["Set","instMembership"],["ConnectedSpace"],["Function","Injective","injOn"],["BoundedOrder","toOrderBot"],["Iff"],["Eq","refl"],["connectedComponents_preimage_singleton"],["HEq"],["Set","ext"],["Eq","mpr"],["Set","Finite"],["ConnectedComponents","equivOfIsClopen"],["IsOpenMap","finite_connectedComponents_of_finite_preimage_singleton_of_connectedSpace"],["Set","mem_image","_simp_1"],["Set","mem_preimage","_simp_1"],["connectedComponent"],["BooleanAlgebra","toBiheytingAlgebra"],["instTopologicalSpaceSubtype"],["exists_and_left","_simp_1"],["congr"],["Set","disjoint_singleton_left","_simp_1"],["Equiv","symm"],["Eq"],["Preorder","toLE"],["propext"],["Subtype","val_injective"],["BiheytingAlgebra","toHeytingAlgebra"],["Set"],["Set","instDistribLattice"],["IsConnected"],["exists_prop","_simp_1"],["Finite","instDiscreteTopology"],["ConnectedComponents","totallyDisconnectedSpace"],["isConnected_iff_connectedSpace"],["Set","univ"],["PartialOrder","toPreorder"],["Membership","mem"],["Iff","mp"],["Disjoint","preimage"],["Set","iUnion"],["Disjoint"],["eq_of_heq"],["funext"],["Finite"],["BiheytingAlgebra","toCoheytingAlgebra"],["SemilatticeInf","toPartialOrder"],["Set","instBooleanAlgebra"],["CoheytingAlgebra","toGeneralizedCoheytingAlgebra"],["And"],["isConnected_connectedComponent"],["exists_eq_right_right","_simp_1"],["Exists","casesOn"],["Set","image"],["Iff","of_eq"],["id"],["implies_congr_ctx"],["IsClopen","preimage"],["Subtype"],["GeneralizedCoheytingAlgebra","toLattice"],["Finite","of_equiv"],["congrArg"],["Set","iUnion_eq_univ_iff"],["ConnectedComponents","continuous_coe"],["congrFun'"],["Not"],["Set","preimage"],["ConnectedComponents","instTopologicalSpace"],["Lattice","toSemilatticeInf"],["HEq","refl"],["Set","instBoundedOrder"],["mem_connectedComponent"],["Set","instSingletonSet"],["Eq","casesOn"],["exists_prop_congr"],["IsOpenMap","restrictPreimage"],["Set","finite_image_iff"],["DistribLattice","toLattice"],["Set","restrictPreimage"],["Continuous","comp"],["Subtype","mk"],["Set","mem_singleton_iff","_simp_1"],["ConnectedComponents","mk"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Topology.Instances.Irrational.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["eventually_residual_irrational"],"typeFallback":"Filter.Eventually.{0} Real (fun (x : Real) => Irrational x) (residual.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)))","typeFull":"∀ᶠ (x : ℝ) in residual ℝ, Irrational x","typeReadable":"∀ᶠ (x : ℝ) in residual ℝ, Irrational x","typeReferences":[["Filter","Eventually"],["residual"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["Irrational"],["UniformSpace","toTopologicalSpace"]],"valueReferences":[["PseudoMetricSpace","toUniformSpace"],["Real"],["Irrational"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["dense_irrational"],["IsGδ","setOf_irrational"],["residual_of_dense_Gδ"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["Irrational","instNoMinOrderSubtypeReal"],"typeFallback":"NoMinOrder.{0} (Subtype.{1} Real (fun (x : Real) => Irrational x)) (Subtype.instLT.{0} Real Real.instLT (fun (x : Real) => Irrational x))","typeFull":"NoMinOrder { x // Irrational x }","typeReadable":"NoMinOrder { x // Irrational x }","typeReferences":[["Subtype","instLT"],["Subtype"],["Real"],["Irrational"],["Real","instLT"],["NoMinOrder"]],"valueReferences":[["Nat","cast_one"],["SubtractionMonoid","toSubNegZeroMonoid"],["Real","instPreorder"],["Eq","trans"],["Real","instAddGroup"],["AddGroupWithOne","toAddMonoidWithOne"],["Exists","intro"],["Subtype","mk","congr_simp"],["Subtype","mk_lt_mk","_simp_1"],["Real","instIsOrderedCancelAddMonoid"],["AddGroup","toSubtractionMonoid"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Real","instAddCommMonoid"],["Eq","ndrec"],["Subtype","instLT"],["Exists"],["Real"],["Real","instIsOrderedAddMonoid"],["sub_lt_self_iff","_simp_1"],["IsOrderedAddMonoid","toAddLeftMono"],["Real","instSub"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Nat"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["AddMonoidWithOne","toOne"],["instIsLeftCancelAddOfAddLeftReflectLE"],["_private","Mathlib","Topology","Instances","Irrational",0,"Irrational","instNoMaxOrderSubtypeReal","match_1"],["NegZeroClass","toZero"],["NeZero","charZero_one"],["AddMonoid","toAddZeroClass"],["Nat","cast"],["Subtype"],["Real","instIsStrictOrderedRing"],["Real","instNatCast"],["Real","instRing"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["instOfNatNat"],["Irrational"],["Real","instLT"],["congrFun'"],["Zero","toOfNat0"],["zero_lt_one","_simp_1"],["Real","partialOrder"],["IsStrictOrderedRing","toCharZero"],["True"],["Real","instZeroLEOneClass"],["AddZero","toAdd"],["OfNat","ofNat"],["LT","lt"],["One","toOfNat1"],["of_eq_true"],["SubNegMonoid","toAddMonoid"],["NoMinOrder","mk"],["Irrational","sub_natCast"],["Subtype","mk"],["instHSub"],["IsLeftCancelAdd","addLeftStrictMono_of_addLeftMono"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_5"],"typeFallback":"forall {a : Prop} {b : Prop} {c : Prop}, Eq.{1} Prop ((And a b) -> c) (a -> b -> c)","typeFull":"∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c)","typeReadable":"∀ {a b c : Prop}, (a ∧ b → c) = (a → b → c)","typeReferences":[["And"],["Eq"]],"valueReferences":[["and_imp"],["And"],["propext"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Topology","Instances","Irrational",0,"Irrational","instNoMaxOrderSubtypeReal","match_1"],"typeFallback":"forall (motive : (Subtype.{1} Real (fun (x : Real) => Irrational x)) -> Prop) (x._@.Mathlib.Topology.Instances.Irrational.2437527448._hygCtx.20.Mathlib.Topology.Instances.Irrational.2437527448._hygCtx._hyg.28 : Subtype.{1} Real (fun (x : Real) => Irrational x)), (forall (x : Real) (hx : Irrational x), motive (Subtype.mk.{1} Real (fun (x : Real) => Irrational x) x hx)) -> (motive x._@.Mathlib.Topology.Instances.Irrational.2437527448._hygCtx.20.Mathlib.Topology.Instances.Irrational.2437527448._hygCtx._hyg.28)","typeFull":"∀ (motive : { x // Irrational x } → Prop) (x : { x // Irrational x }),\n  (∀ (x : ℝ) (hx : Irrational x), motive ⟨x, hx⟩) → motive x","typeReadable":"∀ (motive : { x // Irrational x } → Prop) (x : { x // Irrational x }),\n  (∀ (x : ℝ) (hx : Irrational x), motive ⟨x, hx⟩) → motive x","typeReferences":[["Subtype"],["Real"],["Irrational"],["Subtype","mk"]],"valueReferences":[["Real"],["Irrational"],["Subtype","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_4"],"typeFallback":"forall {α : Sort.{u_1}} {p : α -> Prop} {q : (Exists.{u_1} α (fun (x : α) => p x)) -> Prop}, Eq.{1} Prop (forall (h : Exists.{u_1} α (fun (x : α) => p x)), q h) (forall (x : α) (h : p x), q (Exists.intro.{u_1} α (fun (x : α) => p x) x h))","typeFull":"∀ {α : Sort u_1} {p : α → Prop} {q : (∃ x, p x) → Prop}, (∀ (h : ∃ x, p x), q h) = ∀ (x : α) (h : p x), q ⋯","typeReadable":"∀ {α : Sort u_1} {p : α → Prop} {q : (∃ x, p x) → Prop}, (∀ (h : ∃ x, p x), q h) = ∀ (x : α) (h : p x), q ⋯","typeReferences":[["Exists"],["Exists","intro"],["Eq"]],"valueReferences":[["Exists"],["Exists","intro"],["forall_exists_index"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_3"],"typeFallback":"forall {b : Prop} {a : Prop}, Eq.{1} Prop (Exists.{0} a (fun (_h : a) => b)) (And a b)","typeFull":"∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b)","typeReadable":"∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b)","typeReferences":[["Exists"],["And"],["Eq"]],"valueReferences":[["exists_prop"],["Exists"],["And"],["propext"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Topology","Instances","Irrational",0,"Irrational","instOrderTopologySubtypeReal","match_1"],"typeFallback":"forall (x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.27 : Real) (x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.29 : Real) (motive : (Exists.{1} Real (fun (r : Real) => And (Irrational r) (And (LT.lt.{0} Real Real.instLT x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.27 r) (LT.lt.{0} Real Real.instLT r x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.29)))) -> Prop) (x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.43 : Exists.{1} Real (fun (r : Real) => And (Irrational r) (And (LT.lt.{0} Real Real.instLT x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.27 r) (LT.lt.{0} Real Real.instLT r x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.29)))), (forall (z : Real) (hz : Irrational z) (hxz : LT.lt.{0} Real Real.instLT x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.27 z) (hzy : LT.lt.{0} Real Real.instLT z x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.29), motive (Exists.intro.{1} Real (fun (r : Real) => And (Irrational r) (And (LT.lt.{0} Real Real.instLT x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.27 r) (LT.lt.{0} Real Real.instLT r x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.29))) z (And.intro (Irrational z) (And (LT.lt.{0} Real Real.instLT x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.27 z) (LT.lt.{0} Real Real.instLT z x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.29)) hz (And.intro (LT.lt.{0} Real Real.instLT x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.27 z) (LT.lt.{0} Real Real.instLT z x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.29) hxz hzy)))) -> (motive x._@.Mathlib.Topology.Instances.Irrational.747933967._hygCtx._hyg.43)","typeFull":"∀ (x x_1 : ℝ) (motive : (∃ r, Irrational r ∧ x < r ∧ r < x_1) → Prop) (x_2 : ∃ r, Irrational r ∧ x < r ∧ r < x_1),\n  (∀ (z : ℝ) (hz : Irrational z) (hxz : x < z) (hzy : z < x_1), motive ⋯) → motive x_2","typeReadable":"∀ (x x_1 : ℝ) (motive : (∃ r, Irrational r ∧ x < r ∧ r < x_1) → Prop) (x_2 : ∃ r, Irrational r ∧ x < r ∧ r < x_1),\n  (∀ (z : ℝ) (hz : Irrational z) (hxz : x < z) (hzy : z < x_1), motive ⋯) → motive x_2","typeReferences":[["LT","lt"],["And","intro"],["Exists"],["Real"],["Irrational"],["Real","instLT"],["And"],["Exists","intro"]],"valueReferences":[["LT","lt"],["Exists","casesOn"],["And","intro"],["Real"],["Irrational"],["Real","instLT"],["And"],["Exists","intro"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["dense_irrational"],"typeFallback":"Dense.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (setOf.{0} Real (fun (x : Real) => Irrational x))","typeFull":"Dense {x | Irrational x}","typeReadable":"Dense {x | Irrational x}","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Real"],["Irrational"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["setOf"],["Dense"]],"valueReferences":[["implies_congr"],["Real","instPreorder"],["PartialOrder","toPreorder"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["Singleton","singleton"],["Inter","inter"],["Membership","mem"],["Preorder","toLT"],["Set","iUnion"],["Set","Ioo"],["Real","isTopologicalBasis_Ioo_rat"],["forall_congr"],["funext"],["Eq","symm"],["Eq","ndrec"],["Set","instInter"],["SemilatticeInf","toPartialOrder"],["Exists"],["Real"],["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_3"],["And"],["Set","inter_comm"],["Set","instMembership"],["DivisionRing","toRatCast"],["Iff","of_eq"],["Real","pseudoMetricSpace"],["Iff","mpr"],["Eq","refl"],["Rat"],["id"],["Real","linearOrder"],["Eq","mpr"],["setOf"],["exists_irrational_btwn"],["Real","instIsStrictOrderedRing"],["UniformSpace","toTopologicalSpace"],["Real","instField"],["Dense"],["congrArg"],["instDistribLatticeOfLinearOrder"],["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_5"],["Real","instRatCast"],["Irrational"],["Rat","cast"],["Rat","cast_lt"],["TopologicalSpace","IsTopologicalBasis","dense_iff"],["Rat","instLT"],["Eq"],["propext"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["Set"],["Field","toDivisionRing"],["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_2"],["Set","instSingletonSet"],["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_1"],["exists_prop_congr"],["LT","lt"],["DistribLattice","toLattice"],["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_4"]]},{"isProp":true,"kind":"theorem","name":["Irrational","eventually_forall_le_dist_cast_div"],"typeFallback":"forall {x : Real}, (Irrational x) -> (forall (n : Nat), Filter.Eventually.{0} Real (fun (ε : Real) => forall (m : Int), LE.le.{0} Real Real.instLE ε (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid)) (Int.cast.{0} Real Real.instIntCast m) (Nat.cast.{0} Real Real.instNatCast n)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero))))","typeFull":"∀ {x : ℝ}, Irrational x → ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in nhds 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)","typeReadable":"∀ {x : ℝ}, Irrational x → ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in nhds 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)","typeReferences":[["Filter","Eventually"],["PseudoMetricSpace","toDist"],["Nat","cast"],["PseudoMetricSpace","toUniformSpace"],["Real"],["UniformSpace","toTopologicalSpace"],["Real","instNatCast"],["Int","cast"],["instHDiv"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["Int"],["Real","instLE"],["HDiv","hDiv"],["Nat"],["Real","instZero"],["Irrational"],["Real","pseudoMetricSpace"],["LE","le"],["Dist","dist"],["Real","instIntCast"],["Zero","toOfNat0"],["nhds"],["Real","instDivInvMonoid"]],"valueReferences":[["Real","instPreorder"],["IsSemitopologicalSemiring","toSeparatelyContinuousMul"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["Exists","intro"],["MonoidWithZero","toMulZeroOneClass"],["SMulZeroClass","toSMul"],["DivisionMonoid","toDivInvMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Real","commRing"],["False","elim"],["Int","isClosedEmbedding_coe_real"],["Eq","ndrec"],["nhds"],["PseudoMetricSpace","toDist"],["Exists"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionSemiring","toGroupWithZero"],["Filter"],["instTopologicalSpaceInt"],["DivInvMonoid","toMonoid"],["MulZeroOneClass","toMulZeroClass"],["instOfNat"],["IsTopologicalRing","toIsSemitopologicalRing"],["Real","instIntCast"],["IsClosedMap","isClosed_range"],["MulOneClass","toMulOne"],["Real","instNatCast"],["Metric","ball"],["irrational_mul_intCast_iff","_simp_1"],["Int","cast"],["instHDiv"],["IsClosed","isOpen_compl"],["Real","instField"],["Nat","not_irrational","_simp_1"],["MulOne","toMul"],["Preorder","toLE"],["Eq"],["MulActionWithZero","toSMulWithZero"],["SeparatelyContinuousMul","to_continuousSMul"],["Filter","instMembership"],["IsClosedMap","comp"],["dist_comm"],["Set"],["OfNat","ofNat"],["Int"],["Metric","mem_nhds_iff"],["eq_self"],["LinearOrder","toPartialOrder"],["Real","instInv"],["MonoidWithZero","toMulActionWithZero"],["Real","instZero"],["MulZeroClass","toZero"],["Ne"],["Real","instDivInvMonoid"],["Semifield","toCommGroupWithZero"],["And","casesOn"],["Filter","Eventually"],["PartialOrder","toPreorder"],["irrational_inv_iff","_simp_1"],["Membership","mem"],["Preorder","toLT"],["Iff","mp"],["HMul","hMul"],["CommGroupWithZero","toDivisionCommMonoid"],["OrderTopology","t5Space"],["GT","gt"],["HDiv","hDiv"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semifield","toDivisionSemiring"],["isClosedMap_smul₀"],["InvOneClass","toInv"],["Real"],["Metric","ball_subset_ball"],["And"],["div_eq_inv_mul"],["instIsTopologicalRingReal"],["SMulWithZero","toSMulZeroClass"],["Real","instLE"],["Exists","casesOn"],["HasSubset","Subset"],["Real","pseudoMetricSpace"],["HSMul","hSMul"],["instHMul"],["Real","linearOrder"],["DivisionMonoid","toDivInvOneMonoid"],["Filter","Eventually","mono"],["ge_mem_nhds"],["Nat","cast"],["Eq","mp"],["Compl","compl"],["UniformSpace","toTopologicalSpace"],["IsOpen","mem_nhds"],["CommRing","toNonUnitalCommRing"],["Set","range"],["congrArg"],["instOrderTopologyReal"],["Topology","IsClosedEmbedding","isClosedMap"],["Irrational"],["T5Space","toT1Space"],["Real","instLT"],["Monoid","toMulOneClass"],["GroupWithZero","toMonoidWithZero"],["Dist","dist"],["instHSMul"],["Zero","toOfNat0"],["DivisionCommMonoid","toDivisionMonoid"],["Real","instMul"],["Not"],["Inv","inv"],["True"],["Function","comp"],["DivInvMonoid","toDiv"],["Set","instCompl"],["LT","lt"],["Set","instHasSubset"],["not_lt"],["DivInvOneMonoid","toInvOneClass"],["of_eq_true"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["LE","le"],["Field","toSemifield"],["False"],["and_false"]]},{"isProp":true,"kind":"theorem","name":["Irrational","eventually_forall_le_dist_cast_div_of_denom_le"],"typeFallback":"forall {x : Real}, (Irrational x) -> (forall (n : Nat), Filter.Eventually.{0} Real (fun (ε : Real) => forall (k : Nat), (LE.le.{0} Nat instLENat k n) -> (forall (m : Int), LE.le.{0} Real Real.instLE ε (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid)) (Int.cast.{0} Real Real.instIntCast m) (Nat.cast.{0} Real Real.instNatCast k))))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero))))","typeFull":"∀ {x : ℝ}, Irrational x → ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in nhds 0, ∀ k ≤ n, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k)","typeReadable":"∀ {x : ℝ}, Irrational x → ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in nhds 0, ∀ k ≤ n, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑k)","typeReferences":[["Filter","Eventually"],["Nat","cast"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["Real","instNatCast"],["instHDiv"],["Int","cast"],["HDiv","hDiv"],["Irrational"],["Dist","dist"],["Zero","toOfNat0"],["nhds"],["PseudoMetricSpace","toDist"],["Real"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["Real","instLE"],["Int"],["Nat"],["Real","instZero"],["Real","pseudoMetricSpace"],["LE","le"],["Real","instIntCast"],["instLENat"],["Real","instDivInvMonoid"]],"valueReferences":[["Set","finite_le_nat"],["Filter","Eventually"],["Nat","cast"],["PseudoMetricSpace","toUniformSpace"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["Real","instNatCast"],["instHDiv"],["Int","cast"],["Set","Finite","eventually_all"],["HDiv","hDiv"],["Irrational","eventually_forall_le_dist_cast_div"],["Dist","dist"],["Zero","toOfNat0"],["nhds"],["PseudoMetricSpace","toDist"],["Real"],["Set"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["Real","instLE"],["Int"],["Set","instMembership"],["Nat"],["Real","instZero"],["Real","pseudoMetricSpace"],["Iff","mpr"],["LE","le"],["Real","instIntCast"],["instLENat"],["Real","instDivInvMonoid"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["IsGδ","setOf_irrational"],"typeFallback":"IsGδ.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (setOf.{0} Real (fun (x : Real) => Irrational x))","typeFull":"IsGδ {x | Irrational x}","typeReadable":"IsGδ {x | Irrational x}","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Real"],["Irrational"],["Real","pseudoMetricSpace"],["IsGδ"],["UniformSpace","toTopologicalSpace"],["setOf"]],"valueReferences":[["Set","countable_range"],["Real"],["PseudoMetricSpace","toUniformSpace"],["Set","Countable","isGδ_compl"],["UniformSpace","toTopologicalSpace"],["OrderTopology","t5Space"],["Set","range"],["instOrderTopologyReal"],["Encodable","countable"],["Real","instRatCast"],["Real","pseudoMetricSpace"],["T5Space","toT1Space"],["Rat","cast"],["Rat"],["Real","linearOrder"],["Rat","instEncodable"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_1"],"typeFallback":"forall {α : Type.{u}} {ι : Sort.{v}} {x : α} {s : ι -> (Set.{u} α)}, Eq.{1} Prop (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (Set.iUnion.{u, v} α ι (fun (i : ι) => s i)) x) (Exists.{v} ι (fun (i : ι) => Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (s i) x))","typeFull":"∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋃ i, s i) = ∃ i, x ∈ s i","typeReadable":"∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋃ i, s i) = ∃ i, x ∈ s i","typeReferences":[["Exists"],["Set"],["Membership","mem"],["Set","iUnion"],["Eq"],["Set","instMembership"]],"valueReferences":[["Set","mem_iUnion"],["Exists"],["Set"],["Membership","mem"],["Set","iUnion"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Irrational","instDenselyOrderedSubtypeReal"],"typeFallback":"DenselyOrdered.{0} (Subtype.{1} Real (fun (x : Real) => Irrational x)) (Subtype.instLT.{0} Real Real.instLT (fun (x : Real) => Irrational x))","typeFull":"DenselyOrdered { x // Irrational x }","typeReadable":"DenselyOrdered { x // Irrational x }","typeReferences":[["Subtype","instLT"],["Subtype"],["Real"],["Irrational"],["Real","instLT"],["DenselyOrdered"]],"valueReferences":[["Subtype","instLT"],["exists_irrational_btwn"],["Subtype"],["Exists"],["Real"],["And"],["Exists","intro"],["DenselyOrdered","mk"],["Subtype","val"],["LT","lt"],["And","intro"],["Irrational"],["Real","instLT"],["_private","Mathlib","Topology","Instances","Irrational",0,"Irrational","instDenselyOrderedSubtypeReal","match_1"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["Irrational","instNoMaxOrderSubtypeReal"],"typeFallback":"NoMaxOrder.{0} (Subtype.{1} Real (fun (x : Real) => Irrational x)) (Subtype.instLT.{0} Real Real.instLT (fun (x : Real) => Irrational x))","typeFull":"NoMaxOrder { x // Irrational x }","typeReadable":"NoMaxOrder { x // Irrational x }","typeReferences":[["NoMaxOrder"],["Subtype","instLT"],["Subtype"],["Real"],["Irrational"],["Real","instLT"]],"valueReferences":[["Nat","cast_one"],["Real","instPreorder"],["lt_add_iff_pos_right","_simp_1"],["Eq","trans"],["Irrational","add_natCast"],["AddGroupWithOne","toAddMonoidWithOne"],["Exists","intro"],["Subtype","mk","congr_simp"],["Subtype","mk_lt_mk","_simp_1"],["Real","instIsOrderedCancelAddMonoid"],["Ring","toAddGroupWithOne"],["Eq","ndrec"],["Real","instAddCommMonoid"],["Subtype","instLT"],["NoMaxOrder","mk"],["Exists"],["Real"],["Real","instIsOrderedAddMonoid"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Nat"],["AddMonoidWithOne","toOne"],["instIsLeftCancelAddOfAddLeftReflectLE"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["_private","Mathlib","Topology","Instances","Irrational",0,"Irrational","instNoMaxOrderSubtypeReal","match_1"],["Real","linearOrder"],["NeZero","charZero_one"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Nat","cast"],["Subtype"],["Real","instIsStrictOrderedRing"],["Real","instNatCast"],["Real","instRing"],["congrArg"],["contravariant_lt_of_covariant_le"],["instOfNatNat"],["Irrational"],["Real","instLT"],["Zero","toOfNat0"],["zero_lt_one","_simp_1"],["Real","partialOrder"],["IsStrictOrderedRing","toCharZero"],["True"],["instHAdd"],["Real","instZeroLEOneClass"],["Real","instAdd"],["AddZero","toAdd"],["OfNat","ofNat"],["LT","lt"],["HAdd","hAdd"],["One","toOfNat1"],["of_eq_true"],["Subtype","mk"],["Real","instAddMonoid"],["IsLeftCancelAdd","addLeftStrictMono_of_addLeftMono"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Topology","Instances","Irrational",0,"dense_irrational","_simp_1_2"],"typeFallback":"forall {α : Type.{u_1}} {a : α} {b : α}, Eq.{1} Prop (Membership.mem.{u_1, u_1} α (Set.{u_1} α) (Set.instMembership.{u_1} α) (Singleton.singleton.{u_1, u_1} α (Set.{u_1} α) (Set.instSingletonSet.{u_1} α) b) a) (Eq.{succ u_1} α a b)","typeFull":"∀ {α : Type u_1} {a b : α}, (a ∈ {b}) = (a = b)","typeReadable":"∀ {α : Type u_1} {a b : α}, (a ∈ {b}) = (a = b)","typeReferences":[["Set"],["Singleton","singleton"],["Membership","mem"],["Set","instSingletonSet"],["Eq"],["Set","instMembership"]],"valueReferences":[["Set","mem_singleton_iff"],["Set"],["Singleton","singleton"],["Membership","mem"],["Eq"],["Set","instSingletonSet"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Irrational","eventually_forall_le_dist_cast_rat_of_den_le"],"typeFallback":"forall {x : Real}, (Irrational x) -> (forall (n : Nat), Filter.Eventually.{0} Real (fun (ε : Real) => forall (r : Rat), (LE.le.{0} Nat instLENat (Rat.den r) n) -> (LE.le.{0} Real Real.instLE ε (Dist.dist.{0} Real (PseudoMetricSpace.toDist.{0} Real Real.pseudoMetricSpace) x (Rat.cast.{0} Real Real.instRatCast r)))) (nhds.{0} Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero))))","typeFull":"∀ {x : ℝ}, Irrational x → ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in nhds 0, ∀ (r : ℚ), r.den ≤ n → ε ≤ dist x ↑r","typeReadable":"∀ {x : ℝ}, Irrational x → ∀ (n : ℕ), ∀ᶠ (ε : ℝ) in nhds 0, ∀ (r : ℚ), r.den ≤ n → ε ≤ dist x ↑r","typeReferences":[["Filter","Eventually"],["PseudoMetricSpace","toDist"],["PseudoMetricSpace","toUniformSpace"],["Real"],["UniformSpace","toTopologicalSpace"],["OfNat","ofNat"],["Real","instLE"],["Nat"],["Real","instZero"],["Real","instRatCast"],["Irrational"],["Real","pseudoMetricSpace"],["LE","le"],["Dist","dist"],["Rat","cast"],["Rat"],["Zero","toOfNat0"],["nhds"],["instLENat"],["Rat","den"]],"valueReferences":[["Nat","cast"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["AddGroupWithOne","toAddMonoidWithOne"],["Real","instNatCast"],["instHDiv"],["Int","cast"],["Rat","num"],["congrArg"],["HDiv","hDiv"],["Ring","toAddGroupWithOne"],["Real","instRatCast"],["Dist","dist"],["Rat","cast"],["Rat","cast_def"],["Zero","toOfNat0"],["Real","instDivisionRing"],["Eq"],["nhds"],["Rat","den"],["PseudoMetricSpace","toDist"],["Real"],["DivisionRing","toRing"],["DivisionRing","toDivInvMonoid"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["Real","instLE"],["Int"],["Nat"],["AddMonoidWithOne","toNatCast"],["Real","instZero"],["Real","pseudoMetricSpace"],["LE","le"],["id"],["AddGroupWithOne","toIntCast"],["Rat"],["Real","instIntCast"],["Eq","mpr"],["Irrational","eventually_forall_le_dist_cast_div_of_denom_le"],["instLENat"],["Real","instDivInvMonoid"],["Filter","Eventually","mono"]]},{"isProp":true,"kind":"theorem","name":["Irrational","instOrderTopologySubtypeReal"],"typeFallback":"OrderTopology.{0} (Subtype.{1} Real (fun (x : Real) => Irrational x)) (instTopologicalSpaceSubtype.{0} Real (fun (x : Real) => Irrational x) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) (Subtype.preorder.{0} Real Real.instPreorder (fun (x : Real) => Irrational x))","typeFull":"OrderTopology { x // Irrational x }","typeReadable":"OrderTopology { x // Irrational x }","typeReferences":[["Real","instPreorder"],["instTopologicalSpaceSubtype"],["Subtype"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["Irrational"],["UniformSpace","toTopologicalSpace"],["Subtype","preorder"],["OrderTopology"]],"valueReferences":[["exists_irrational_btwn"],["Real","instPreorder"],["Exists"],["Subtype"],["Real"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["And"],["Preorder","toLT"],["Subtype","preorder"],["Exists","intro"],["Iff","rfl"],["Subtype","val"],["_private","Mathlib","Topology","Instances","Irrational",0,"Irrational","instOrderTopologySubtypeReal","match_1"],["instOrderTopologyReal"],["LT","lt"],["And","intro"],["induced_orderTopology"],["Irrational"],["Real","pseudoMetricSpace"],["Subtype","mk"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Topology","Instances","Irrational",0,"Irrational","instDenselyOrderedSubtypeReal","match_1"],"typeFallback":"forall (x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.20 : Subtype.{1} Real (fun (x : Real) => Irrational x)) (x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.22 : Subtype.{1} Real (fun (x : Real) => Irrational x)) (motive : (Exists.{1} Real (fun (r : Real) => And (Irrational r) (And (LT.lt.{0} Real Real.instLT (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.20) r) (LT.lt.{0} Real Real.instLT r (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.22))))) -> Prop) (x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.36 : Exists.{1} Real (fun (r : Real) => And (Irrational r) (And (LT.lt.{0} Real Real.instLT (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.20) r) (LT.lt.{0} Real Real.instLT r (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.22))))), (forall (z : Real) (hz : Irrational z) (hxz : LT.lt.{0} Real Real.instLT (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.20) z) (hzy : LT.lt.{0} Real Real.instLT z (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.22)), motive (Exists.intro.{1} Real (fun (r : Real) => And (Irrational r) (And (LT.lt.{0} Real Real.instLT (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.20) r) (LT.lt.{0} Real Real.instLT r (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.22)))) z (And.intro (Irrational z) (And (LT.lt.{0} Real Real.instLT (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.20) z) (LT.lt.{0} Real Real.instLT z (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.22))) hz (And.intro (LT.lt.{0} Real Real.instLT (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.20) z) (LT.lt.{0} Real Real.instLT z (Subtype.val.{1} Real (fun (x : Real) => Irrational x) x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.22)) hxz hzy)))) -> (motive x._@.Mathlib.Topology.Instances.Irrational.2219742453._hygCtx._hyg.36)","typeFull":"∀ (x x_1 : { x // Irrational x }) (motive : (∃ r, Irrational r ∧ ↑x < r ∧ r < ↑x_1) → Prop)\n  (x_2 : ∃ r, Irrational r ∧ ↑x < r ∧ r < ↑x_1),\n  (∀ (z : ℝ) (hz : Irrational z) (hxz : ↑x < z) (hzy : z < ↑x_1), motive ⋯) → motive x_2","typeReadable":"∀ (x x_1 : { x // Irrational x }) (motive : (∃ r, Irrational r ∧ ↑x < r ∧ r < ↑x_1) → Prop)\n  (x_2 : ∃ r, Irrational r ∧ ↑x < r ∧ r < ↑x_1),\n  (∀ (z : ℝ) (hz : Irrational z) (hxz : ↑x < z) (hzy : z < ↑x_1), motive ⋯) → motive x_2","typeReferences":[["LT","lt"],["And","intro"],["Exists"],["Subtype"],["Real"],["Irrational"],["Real","instLT"],["And"],["Exists","intro"],["Subtype","val"]],"valueReferences":[["LT","lt"],["Exists","casesOn"],["And","intro"],["Real"],["Irrational"],["Real","instLT"],["And"],["Exists","intro"],["Subtype","val"],["And","casesOn"]]}]