Loogle!
Result
Found 17556 definitions mentioning HAdd.hAdd. Of these, 466 have a name containing "comm". Of these, 449 match your pattern(s). Of these, only the first 200 are shown.
- Nat.instCommutativeHAdd Init.Data.Nat.Basic
Std.Commutative fun x x_1 => x + x_1 - Nat.add_comm Init.Data.Nat.Basic
∀ (n m : ℕ), n + m = m + n - Nat.add_left_comm Init.Data.Nat.Basic
∀ (n m k : ℕ), n + (m + k) = m + (n + k) - Nat.add_right_comm Init.Data.Nat.Basic
∀ (n m k : ℕ), n + m + k = n + k + m - Int.instCommutativeHAdd Init.Data.Int.Lemmas
Std.Commutative fun x x_1 => x + x_1 - Int.add_comm Init.Data.Int.Lemmas
∀ (a b : ℤ), a + b = b + a - Int.add_left_comm Init.Data.Int.Lemmas
∀ (a b c : ℤ), a + (b + c) = b + (a + c) - Int.add_right_comm Init.Data.Int.Lemmas
∀ (a b c : ℤ), a + b + c = a + c + b - Nat.sub_add_comm Init.Data.Nat.Lemmas
∀ {n m k : ℕ}, k ≤ n → n + m - k = n - k + m - Nat.add_add_add_comm Init.Data.Nat.Lemmas
∀ (a b c d : ℕ), a + b + (c + d) = a + c + (b + d) - Nat.Simproc.sub_add_eq_comm Init.Data.Nat.Simproc
∀ (a b c : ℕ), a - (b + c) = a - c - b - BitVec.instCommutativeHAdd Init.Data.BitVec.Lemmas
∀ {n : ℕ}, Std.Commutative fun x x_1 => x + x_1 - BitVec.add_comm Init.Data.BitVec.Lemmas
∀ {n : ℕ} (x y : BitVec n), x + y = y + x - ShareCommon.StateFactoryImpl.mk.sizeOf_spec Init.ShareCommon
∀ (Map Set : Type) (mkState : Unit → Map × Set) (mapFind? : Map → ShareCommon.Object → Option ShareCommon.Object) (mapInsert : Map → ShareCommon.Object → ShareCommon.Object → Map) (setFind? : Set → ShareCommon.Object → Option ShareCommon.Object) (setInsert : Set → ShareCommon.Object → Set), sizeOf { Map := Map, Set := Set, mkState := mkState, mapFind? := mapFind?, mapInsert := mapInsert, setFind? := setFind?, setInsert := setInsert } = 1 + sizeOf Map + sizeOf Set + sizeOf mkState - Lean.Elab.CommandInfo.mk.sizeOf_spec Lean.Elab.InfoTree.Types
∀ (toElabInfo : Lean.Elab.ElabInfo), sizeOf { toElabInfo := toElabInfo } = 1 + sizeOf toElabInfo - Lean.Elab.Info.ofCommandInfo.sizeOf_spec Lean.Elab.InfoTree.Types
∀ (i : Lean.Elab.CommandInfo), sizeOf (Lean.Elab.Info.ofCommandInfo i) = 1 + sizeOf i - Lean.Elab.PartialContextInfo.commandCtx.sizeOf_spec Lean.Elab.InfoTree.Types
∀ (info : Lean.Elab.CommandContextInfo), sizeOf (Lean.Elab.PartialContextInfo.commandCtx info) = 1 + sizeOf info - Lean.Elab.CommandContextInfo.mk.sizeOf_spec Lean.Elab.InfoTree.Types
∀ (env : Lean.Environment) (fileMap : Lean.FileMap) (mctx : Lean.MetavarContext) (options : Lean.Options) (currNamespace : Lean.Name) (openDecls : List Lean.OpenDecl) (ngen : Lean.NameGenerator), sizeOf { env := env, fileMap := fileMap, mctx := mctx, options := options, currNamespace := currNamespace, openDecls := openDecls, ngen := ngen } = 1 + sizeOf env + sizeOf fileMap + sizeOf mctx + sizeOf options + sizeOf currNamespace + sizeOf openDecls + sizeOf ngen - Lean.Elab.Command.Linter.mk.sizeOf_spec Lean.Elab.Command
∀ (run : Lean.Syntax → Lean.Elab.Command.CommandElabM Unit) (name : autoParam Lean.Name _auto✝), sizeOf { run := run, name := name } = 1 + sizeOf name - Lean.Elab.Command.State.mk.sizeOf_spec Lean.Elab.Command
∀ (env : Lean.Environment) (messages : Lean.MessageLog) (scopes : List Lean.Elab.Command.Scope) (nextMacroScope maxRecDepth : ℕ) (ngen : Lean.NameGenerator) (infoState : Lean.Elab.InfoState) (traceState : Lean.TraceState), sizeOf { env := env, messages := messages, scopes := scopes, nextMacroScope := nextMacroScope, maxRecDepth := maxRecDepth, ngen := ngen, infoState := infoState, traceState := traceState } = 1 + sizeOf env + sizeOf messages + sizeOf scopes + sizeOf nextMacroScope + sizeOf maxRecDepth + sizeOf ngen + sizeOf infoState + sizeOf traceState - Lean.Elab.Command.Context.mk.sizeOf_spec Lean.Elab.Command
∀ (fileName : String) (fileMap : Lean.FileMap) (currRecDepth : ℕ) (cmdPos : String.Pos) (macroStack : Lean.Elab.MacroStack) (currMacroScope : Lean.MacroScope) (ref : Lean.Syntax) (tacticCache? : Option (IO.Ref Lean.Elab.Tactic.Cache)) (snap? : Option (Lean.Language.SnapshotBundle Lean.Language.DynamicSnapshot)), sizeOf { fileName := fileName, fileMap := fileMap, currRecDepth := currRecDepth, cmdPos := cmdPos, macroStack := macroStack, currMacroScope := currMacroScope, ref := ref, tacticCache? := tacticCache?, snap? := snap? } = 1 + sizeOf fileName + sizeOf fileMap + sizeOf currRecDepth + sizeOf cmdPos + sizeOf macroStack + sizeOf currMacroScope + sizeOf ref + sizeOf tacticCache? + sizeOf snap? - Lean.Elab.Command.Scope.mk.sizeOf_spec Lean.Elab.Command
∀ (header : String) (opts : Lean.Options) (currNamespace : Lean.Name) (openDecls : List Lean.OpenDecl) (levelNames : List Lean.Name) (varDecls : Array (Lean.TSyntax `Lean.Parser.Term.bracketedBinder)) (varUIds : Array Lean.Name) (isNoncomputable : Bool), sizeOf { header := header, opts := opts, currNamespace := currNamespace, openDecls := openDecls, levelNames := levelNames, varDecls := varDecls, varUIds := varUIds, isNoncomputable := isNoncomputable } = 1 + sizeOf header + sizeOf opts + sizeOf currNamespace + sizeOf openDecls + sizeOf levelNames + sizeOf varDecls + sizeOf varUIds + sizeOf isNoncomputable - Lean.Lsp.Command.mk.sizeOf_spec Lean.Data.Lsp.Basic
∀ (title command : String) (arguments? : Option (Array Lean.Json)), sizeOf { title := title, command := command, arguments? := arguments? } = 1 + sizeOf title + sizeOf command + sizeOf arguments? - Lean.Language.Lean.CommandFinishedSnapshot.mk.sizeOf_spec Lean.Language.Lean
∀ (toSnapshot : Lean.Language.Snapshot) (cmdState : Lean.Elab.Command.State), sizeOf { toSnapshot := toSnapshot, cmdState := cmdState } = 1 + sizeOf toSnapshot + sizeOf cmdState - Lean.Language.Lean.CommandParsedSnapshot.mk.sizeOf_spec Lean.Language.Lean
∀ (data : Lean.Language.Lean.CommandParsedSnapshotData) (nextCmdSnap? : Option (Lean.Language.SnapshotTask Lean.Language.Lean.CommandParsedSnapshot)), sizeOf (Lean.Language.Lean.CommandParsedSnapshot.mk data nextCmdSnap?) = 1 + sizeOf data + sizeOf nextCmdSnap? - Lean.Language.Lean.CommandParsedSnapshotData.mk.sizeOf_spec Lean.Language.Lean
∀ (toSnapshot : Lean.Language.Snapshot) (stx : Lean.Syntax) (parserState : Lean.Parser.ModuleParserState) (elabSnap : Lean.Language.SnapshotTask Lean.Language.DynamicSnapshot) (finishedSnap : Lean.Language.SnapshotTask Lean.Language.Lean.CommandFinishedSnapshot) (tacticCache : IO.Ref Lean.Elab.Tactic.Cache), sizeOf { toSnapshot := toSnapshot, stx := stx, parserState := parserState, elabSnap := elabSnap, finishedSnap := finishedSnap, tacticCache := tacticCache } = 1 + sizeOf toSnapshot + sizeOf stx + sizeOf parserState + sizeOf elabSnap + sizeOf finishedSnap + sizeOf tacticCache - Lean.CodeAction.CommandCodeActionEntry.mk.sizeOf_spec Lean.Server.CodeActions.Attr
∀ (declName : Lean.Name) (cmdKinds : Array Lean.Name), sizeOf { declName := declName, cmdKinds := cmdKinds } = 1 + sizeOf declName + sizeOf cmdKinds - Lean.CodeAction.CommandCodeActions.mk.sizeOf_spec Lean.Server.CodeActions.Attr
∀ (onAnyCmd : Array Lean.CodeAction.CommandCodeAction) (onCmd : Lean.NameMap (Array Lean.CodeAction.CommandCodeAction)), sizeOf { onAnyCmd := onAnyCmd, onCmd := onCmd } = 1 + sizeOf onAnyCmd + sizeOf onCmd - AddCommMagma.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toAdd : Add G] → (∀ (a b : G), a + b = b + a) → AddCommMagma G - add_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a - AddCommMagma.add_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : AddCommMagma G] (a b : G), a + b = b + a - AddCommSemigroup.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toAddSemigroup : AddSemigroup G] → (∀ (a b : G), a + b = b + a) → AddCommSemigroup G - AddCommSemigroup.add_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : AddCommSemigroup G] (a b : G), a + b = b + a - AddCommMonoid.mk Mathlib.Algebra.Group.Defs
{M : Type u} → [toAddMonoid : AddMonoid M] → (∀ (a b : M), a + b = b + a) → AddCommMonoid M - AddCommMonoid.add_comm Mathlib.Algebra.Group.Defs
∀ {M : Type u} [self : AddCommMonoid M] (a b : M), a + b = b + a - AddCancelCommMonoid.mk Mathlib.Algebra.Group.Defs
{M : Type u} → [toAddLeftCancelMonoid : AddLeftCancelMonoid M] → (∀ (a b : M), a + b = b + a) → AddCancelCommMonoid M - AddCancelCommMonoid.add_comm Mathlib.Algebra.Group.Defs
∀ {M : Type u} [self : AddCancelCommMonoid M] (a b : M), a + b = b + a - AddCommGroup.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toAddGroup : AddGroup G] → (∀ (a b : G), a + b = b + a) → AddCommGroup G - SubtractionCommMonoid.mk Mathlib.Algebra.Group.Defs
{G : Type u} → [toSubtractionMonoid : SubtractionMonoid G] → (∀ (a b : G), a + b = b + a) → SubtractionCommMonoid G - AddCommGroup.add_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : AddCommGroup G] (a b : G), a + b = b + a - SubtractionCommMonoid.add_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u} [self : SubtractionCommMonoid G] (a b : G), a + b = b + a - add_neg_cancel_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : AddCommGroup G] (a b : G), a + b + -a = b - add_neg_cancel_comm_assoc Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : AddCommGroup G] (a b : G), a + (b + -a) = b - neg_add_cancel_comm Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : AddCommGroup G] (a b : G), -a + b + a = b - neg_add_cancel_comm_assoc Mathlib.Algebra.Group.Defs
∀ {G : Type u_1} [inst : AddCommGroup G] (a b : G), -a + (b + a) = b - nsmul_add_comm' Mathlib.Algebra.Group.Defs
∀ {M : Type u_2} [inst : AddMonoid M] (a : M) (n : ℕ), n • a + a = a + n • a - nsmul_add_comm Mathlib.Algebra.Group.Defs
∀ {M : Type u_2} [inst : AddMonoid M] (a : M) (m n : ℕ), m • a + n • a = n • a + m • a - Lean.Xml.Content.Comment.sizeOf_spec Lean.Data.Xml.Basic
∀ (comment : String), sizeOf (Lean.Xml.Content.Comment comment) = 1 + sizeOf comment - Lean.Compiler.LCNF.JoinPointCommonArgs.AnalysisCtx.mk.sizeOf_spec Lean.Compiler.LCNF.JoinPoints
∀ (jpScopes : Lean.FVarIdMap Lean.FVarIdSet), sizeOf { jpScopes := jpScopes } = 1 + sizeOf jpScopes - Lean.Compiler.LCNF.JoinPointCommonArgs.AnalysisState.mk.sizeOf_spec Lean.Compiler.LCNF.JoinPoints
∀ (jpJmpArgs : Lean.FVarIdMap Lean.Compiler.LCNF.FVarSubst), sizeOf { jpJmpArgs := jpJmpArgs } = 1 + sizeOf jpJmpArgs - Lean.Elab.Command.CtorView.mk.sizeOf_spec Lean.Elab.Inductive
∀ (ref : Lean.Syntax) (modifiers : Lean.Elab.Modifiers) (declName : Lean.Name) (binders : Lean.Syntax) (type? : Option Lean.Syntax), sizeOf { ref := ref, modifiers := modifiers, declName := declName, binders := binders, type? := type? } = 1 + sizeOf ref + sizeOf modifiers + sizeOf declName + sizeOf binders + sizeOf type? - Lean.Elab.Command.ElabHeaderResult.mk.sizeOf_spec Lean.Elab.Inductive
∀ (view : Lean.Elab.Command.InductiveView) (lctx : Lean.LocalContext) (localInsts : Lean.LocalInstances) (params : Array Lean.Expr) (type : Lean.Expr), sizeOf { view := view, lctx := lctx, localInsts := localInsts, params := params, type := type } = 1 + sizeOf view + sizeOf lctx + sizeOf localInsts + sizeOf params + sizeOf type - Lean.Elab.Command.ComputedFieldView.mk.sizeOf_spec Lean.Elab.Inductive
∀ (ref modifiers : Lean.Syntax) (fieldId : Lean.Name) (type : Lean.Term) (matchAlts : Lean.TSyntax `Lean.Parser.Term.matchAlts), sizeOf { ref := ref, modifiers := modifiers, fieldId := fieldId, type := type, matchAlts := matchAlts } = 1 + sizeOf ref + sizeOf modifiers + sizeOf fieldId + sizeOf type + sizeOf matchAlts - Lean.Elab.Command.InductiveView.mk.sizeOf_spec Lean.Elab.Inductive
∀ (ref declId : Lean.Syntax) (modifiers : Lean.Elab.Modifiers) (shortDeclName declName : Lean.Name) (levelNames : List Lean.Name) (binders : Lean.Syntax) (type? : Option Lean.Syntax) (ctors : Array Lean.Elab.Command.CtorView) (derivingClasses : Array Lean.Elab.DerivingClassView) (computedFields : Array Lean.Elab.Command.ComputedFieldView), sizeOf { ref := ref, declId := declId, modifiers := modifiers, shortDeclName := shortDeclName, declName := declName, levelNames := levelNames, binders := binders, type? := type?, ctors := ctors, derivingClasses := derivingClasses, computedFields := computedFields } = 1 + sizeOf ref + sizeOf declId + sizeOf modifiers + sizeOf shortDeclName + sizeOf declName + sizeOf levelNames + sizeOf binders + sizeOf type? + sizeOf ctors + sizeOf derivingClasses + sizeOf computedFields - Lean.Elab.Command.StructCtorView.mk.sizeOf_spec Lean.Elab.Structure
∀ (ref : Lean.Syntax) (modifiers : Lean.Elab.Modifiers) (name declName : Lean.Name), sizeOf { ref := ref, modifiers := modifiers, name := name, declName := declName } = 1 + sizeOf ref + sizeOf modifiers + sizeOf name + sizeOf declName - Lean.Elab.Command.StructFieldInfo.mk.sizeOf_spec Lean.Elab.Structure
∀ (name declName : Lean.Name) (fvar : Lean.Expr) (kind : Lean.Elab.Command.StructFieldKind) (value? : Option Lean.Expr), sizeOf { name := name, declName := declName, fvar := fvar, kind := kind, value? := value? } = 1 + sizeOf name + sizeOf declName + sizeOf fvar + sizeOf kind + sizeOf value? - Lean.Elab.Command.StructFieldView.mk.sizeOf_spec Lean.Elab.Structure
∀ (ref : Lean.Syntax) (modifiers : Lean.Elab.Modifiers) (binderInfo : Lean.BinderInfo) (declName name rawName : Lean.Name) (binders : Lean.Syntax) (type? value? : Option Lean.Syntax), sizeOf { ref := ref, modifiers := modifiers, binderInfo := binderInfo, declName := declName, name := name, rawName := rawName, binders := binders, type? := type?, value? := value? } = 1 + sizeOf ref + sizeOf modifiers + sizeOf binderInfo + sizeOf declName + sizeOf name + sizeOf rawName + sizeOf binders + sizeOf type? + sizeOf value? - Lean.Elab.Command.StructView.mk.sizeOf_spec Lean.Elab.Structure
∀ (ref : Lean.Syntax) (modifiers : Lean.Elab.Modifiers) (scopeLevelNames allUserLevelNames : List Lean.Name) (isClass : Bool) (declName : Lean.Name) (scopeVars params : Array Lean.Expr) (parents : Array Lean.Syntax) (type : Lean.Syntax) (ctor : Lean.Elab.Command.StructCtorView) (fields : Array Lean.Elab.Command.StructFieldView), sizeOf { ref := ref, modifiers := modifiers, scopeLevelNames := scopeLevelNames, allUserLevelNames := allUserLevelNames, isClass := isClass, declName := declName, scopeVars := scopeVars, params := params, parents := parents, type := type, ctor := ctor, fields := fields } = 1 + sizeOf ref + sizeOf modifiers + sizeOf scopeLevelNames + sizeOf allUserLevelNames + sizeOf isClass + sizeOf declName + sizeOf scopeVars + sizeOf params + sizeOf parents + sizeOf type + sizeOf ctor + sizeOf fields - Lean.Elab.Command.CollectAxioms.State.mk.sizeOf_spec Lean.Elab.Print
∀ (visited : Lean.NameSet) (axioms : Array Lean.Name), sizeOf { visited := visited, axioms := axioms } = 1 + sizeOf visited + sizeOf axioms - AddCommute.eq Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Add S] {a b : S}, AddCommute a b → a + b = b + a - addCommute_iff_eq Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : Add S] (a b : S), AddCommute a b ↔ a + b = b + a - AddCommute.add_left Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : AddSemigroup S] {a b c : S}, AddCommute a c → AddCommute b c → AddCommute (a + b) c - AddCommute.add_right Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : AddSemigroup S] {a b c : S}, AddCommute a b → AddCommute a c → AddCommute a (b + c) - AddCommute.left_comm Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : AddSemigroup S] {a b : S}, AddCommute a b → ∀ (c : S), a + (b + c) = b + (a + c) - AddCommute.right_comm Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : AddSemigroup S] {b c : S}, AddCommute b c → ∀ (a : S), a + b + c = a + c + b - AddCommute.add_neg_cancel Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : AddGroup G] {a b : G}, AddCommute a b → a + b + -a = b - AddCommute.add_neg_cancel_assoc Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : AddGroup G] {a b : G}, AddCommute a b → a + (b + -a) = b - AddCommute.add_neg Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : SubtractionMonoid G] {a b : G}, AddCommute a b → -(a + b) = -a + -b - AddCommute.neg Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : SubtractionMonoid G] {a b : G}, AddCommute a b → -(a + b) = -a + -b - AddCommute.add_nsmul Mathlib.Algebra.Group.Commute.Defs
∀ {M : Type u_2} [inst : AddMonoid M] {a b : M}, AddCommute a b → ∀ (n : ℕ), n • (a + b) = n • a + n • b - AddCommute.add_add_add_comm Mathlib.Algebra.Group.Commute.Defs
∀ {S : Type u_3} [inst : AddSemigroup S] {b c : S}, AddCommute b c → ∀ (a d : S), a + b + (c + d) = a + c + (b + d) - AddCommute.zsmul_add Mathlib.Algebra.Group.Commute.Defs
∀ {G : Type u_1} [inst : SubtractionMonoid G] {a b : G}, AddCommute a b → ∀ (n : ℤ), n • (a + b) = n • a + n • b - String.Pos.addChar_right_comm Batteries.Data.String.Lemmas
∀ (p : String.Pos) (c₁ c₂ : Char), p + c₁ + c₂ = p + c₂ + c₁ - AddCommMagma.to_isCommutative Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : AddCommMagma G], Std.Commutative fun x x_1 => x + x_1 - add_left_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : AddCommSemigroup G] (a b c : G), a + (b + c) = b + (a + c) - add_right_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : AddCommSemigroup G] (a b c : G), a + b + c = a + c + b - add_comm_sub Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = a + (c - b) - add_sub_left_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a + (b - c) = b + (a - c) - add_sub_right_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a + b - c = a - c + b - sub_add_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c : α), a - b + c = c - b + a - add_add_add_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : AddCommSemigroup G] (a b c d : G), a + b + (c + d) = a + c + (b + d) - zsmul_add_comm Mathlib.Algebra.Group.Basic
∀ {G : Type u_3} [inst : AddGroup G] (a : G) (m n : ℤ), m • a + n • a = n • a + m • a - add_sub_add_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c d : α), a + b - (c + d) = a - c + (b - d) - sub_add_sub_comm Mathlib.Algebra.Group.Basic
∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c d : α), a - b + (c - d) = a + c - (b + d) - AddCommMonoidWithOne.mk Mathlib.Data.Nat.Cast.Defs
{R : Type u_2} → [toAddMonoidWithOne : AddMonoidWithOne R] → (∀ (a b : R), a + b = b + a) → AddCommMonoidWithOne R - AddCommMonoidWithOne.add_comm Mathlib.Data.Nat.Cast.Defs
∀ {R : Type u_2} [self : AddCommMonoidWithOne R] (a b : R), a + b = b + a - AddCommGroupWithOne.intCast_negSucc Mathlib.Data.Int.Cast.Defs
∀ {R : Type u} [self : AddCommGroupWithOne R] (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1) - AddCommGroupWithOne.natCast_succ Mathlib.Data.Int.Cast.Defs
∀ {R : Type u} [self : AddCommGroupWithOne R] (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1 - AddCommGroupWithOne.mk Mathlib.Data.Int.Cast.Defs
{R : Type u} → [toAddCommGroup : AddCommGroup R] → [toIntCast : IntCast R] → [toNatCast : NatCast R] → [toOne : One R] → autoParam (NatCast.natCast 0 = 0) _auto✝ → autoParam (∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1) _auto✝¹ → autoParam (∀ (n : ℕ), IntCast.intCast ↑n = ↑n) _auto✝² → autoParam (∀ (n : ℕ), IntCast.intCast (Int.negSucc n) = -↑(n + 1)) _auto✝³ → AddCommGroupWithOne R - Function.Injective.addCommMagma Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₁] → [inst_1 : AddCommMagma M₂] → (f : M₁ → M₂) → Function.Injective f → (∀ (x y : M₁), f (x + y) = f x + f y) → AddCommMagma M₁ - Function.Surjective.addCommMagma Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₂] → [inst_1 : AddCommMagma M₁] → (f : M₁ → M₂) → Function.Surjective f → (∀ (x y : M₁), f (x + y) = f x + f y) → AddCommMagma M₂ - Function.Injective.addCommSemigroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₁] → [inst_1 : AddCommSemigroup M₂] → (f : M₁ → M₂) → Function.Injective f → (∀ (x y : M₁), f (x + y) = f x + f y) → AddCommSemigroup M₁ - Function.Surjective.addCommSemigroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₂] → [inst_1 : AddCommSemigroup M₁] → (f : M₁ → M₂) → Function.Surjective f → (∀ (x y : M₁), f (x + y) = f x + f y) → AddCommSemigroup M₂ - Function.Injective.addCommMagma.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₁] [inst_1 : AddCommMagma M₂] (f : M₁ → M₂) (hf : Function.Injective f) (mul : ∀ (x y : M₁), f (x + y) = f x + f y), Function.Injective.addCommMagma f hf mul = AddCommMagma.mk ⋯ - Function.Surjective.addCommMagma.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₂] [inst_1 : AddCommMagma M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (mul : ∀ (x y : M₁), f (x + y) = f x + f y), Function.Surjective.addCommMagma f hf mul = AddCommMagma.mk ⋯ - Function.Injective.addCommSemigroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₁] [inst_1 : AddCommSemigroup M₂] (f : M₁ → M₂) (hf : Function.Injective f) (mul : ∀ (x y : M₁), f (x + y) = f x + f y), Function.Injective.addCommSemigroup f hf mul = let __spread.0 := Function.Injective.addCommMagma f hf mul; AddCommSemigroup.mk ⋯ - Function.Surjective.addCommSemigroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₂] [inst_1 : AddCommSemigroup M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (mul : ∀ (x y : M₁), f (x + y) = f x + f y), Function.Surjective.addCommSemigroup f hf mul = let __spread.0 := Function.Surjective.addCommMagma f hf mul; AddCommSemigroup.mk ⋯ - Function.Injective.addCommMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₁] → [inst_1 : Zero M₁] → [inst_2 : SMul ℕ M₁] → [inst_3 : AddCommMonoid M₂] → (f : M₁ → M₂) → Function.Injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) → AddCommMonoid M₁ - Function.Surjective.addCommMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₂] → [inst_1 : Zero M₂] → [inst_2 : SMul ℕ M₂] → [inst_3 : AddCommMonoid M₁] → (f : M₁ → M₂) → Function.Surjective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) → AddCommMonoid M₂ - Function.Injective.addCancelCommMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₁] → [inst_1 : Zero M₁] → [inst_2 : SMul ℕ M₁] → [inst_3 : AddCancelCommMonoid M₂] → (f : M₁ → M₂) → Function.Injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) → AddCancelCommMonoid M₁ - Function.Injective.addCommMonoidWithOne Mathlib.Algebra.Group.InjSurj
{M₂ : Type u_2} → {M₁ : Type u_3} → [inst : Zero M₁] → [inst_1 : One M₁] → [inst_2 : Add M₁] → [inst_3 : SMul ℕ M₁] → [inst_4 : NatCast M₁] → [inst_5 : AddCommMonoidWithOne M₂] → (f : M₁ → M₂) → Function.Injective f → f 0 = 0 → f 1 = 1 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (n : ℕ) (x : M₁), f (n • x) = n • f x) → (∀ (n : ℕ), f ↑n = ↑n) → AddCommMonoidWithOne M₁ - Function.Surjective.addCommMonoidWithOne Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_3} → [inst : Zero M₂] → [inst_1 : One M₂] → [inst_2 : Add M₂] → [inst_3 : SMul ℕ M₂] → [inst_4 : NatCast M₂] → [inst_5 : AddCommMonoidWithOne M₁] → (f : M₁ → M₂) → Function.Surjective f → f 0 = 0 → f 1 = 1 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (n : ℕ) (x : M₁), f (n • x) = n • f x) → (∀ (n : ℕ), f ↑n = ↑n) → AddCommMonoidWithOne M₂ - Function.Injective.addCommMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₁] [inst_1 : Zero M₁] [inst_2 : SMul ℕ M₁] [inst_3 : AddCommMonoid M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : M₁), f (x + y) = f x + f y) (npow : ∀ (x : M₁) (n : ℕ), f (n • x) = n • f x), Function.Injective.addCommMonoid f hf one mul npow = let __src := Function.Injective.addMonoid f hf one mul npow; let __src_1 := Function.Injective.addCommSemigroup f hf mul; AddCommMonoid.mk ⋯ - Function.Surjective.addCommMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₂] [inst_1 : Zero M₂] [inst_2 : SMul ℕ M₂] [inst_3 : AddCommMonoid M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (one : f 0 = 0) (mul : ∀ (x y : M₁), f (x + y) = f x + f y) (npow : ∀ (x : M₁) (n : ℕ), f (n • x) = n • f x), Function.Surjective.addCommMonoid f hf one mul npow = let __src := Function.Surjective.addCommSemigroup f hf mul; let __src_1 := Function.Surjective.addMonoid f hf one mul npow; AddCommMonoid.mk ⋯ - Function.Injective.subtractionCommMonoid Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₁] → [inst_1 : Zero M₁] → [inst_2 : SMul ℕ M₁] → [inst_3 : Neg M₁] → [inst_4 : Sub M₁] → [inst_5 : SMul ℤ M₁] → [inst_6 : SubtractionCommMonoid M₂] → (f : M₁ → M₂) → Function.Injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → (∀ (x y : M₁), f (x - y) = f x - f y) → (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) → (∀ (x : M₁) (n : ℤ), f (n • x) = n • f x) → SubtractionCommMonoid M₁ - Function.Injective.addCommGroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₁] → [inst_1 : Zero M₁] → [inst_2 : SMul ℕ M₁] → [inst_3 : Neg M₁] → [inst_4 : Sub M₁] → [inst_5 : SMul ℤ M₁] → [inst_6 : AddCommGroup M₂] → (f : M₁ → M₂) → Function.Injective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → (∀ (x y : M₁), f (x - y) = f x - f y) → (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) → (∀ (x : M₁) (n : ℤ), f (n • x) = n • f x) → AddCommGroup M₁ - Function.Surjective.addCommGroup Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_2} → [inst : Add M₂] → [inst_1 : Zero M₂] → [inst_2 : SMul ℕ M₂] → [inst_3 : Neg M₂] → [inst_4 : Sub M₂] → [inst_5 : SMul ℤ M₂] → [inst_6 : AddCommGroup M₁] → (f : M₁ → M₂) → Function.Surjective f → f 0 = 0 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → (∀ (x y : M₁), f (x - y) = f x - f y) → (∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) → (∀ (x : M₁) (n : ℤ), f (n • x) = n • f x) → AddCommGroup M₂ - Function.Injective.addCancelCommMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₁] [inst_1 : Zero M₁] [inst_2 : SMul ℕ M₁] [inst_3 : AddCancelCommMonoid M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : M₁), f (x + y) = f x + f y) (npow : ∀ (x : M₁) (n : ℕ), f (n • x) = n • f x), Function.Injective.addCancelCommMonoid f hf one mul npow = let __src := Function.Injective.addLeftCancelSemigroup f hf mul; let __src_1 := Function.Injective.addCommMonoid f hf one mul npow; AddCancelCommMonoid.mk ⋯ - Function.Injective.addCommGroupWithOne Mathlib.Algebra.Group.InjSurj
{M₂ : Type u_2} → {M₁ : Type u_3} → [inst : Zero M₁] → [inst_1 : One M₁] → [inst_2 : Add M₁] → [inst_3 : SMul ℕ M₁] → [inst_4 : Neg M₁] → [inst_5 : Sub M₁] → [inst_6 : SMul ℤ M₁] → [inst_7 : NatCast M₁] → [inst_8 : IntCast M₁] → [inst_9 : AddCommGroupWithOne M₂] → (f : M₁ → M₂) → Function.Injective f → f 0 = 0 → f 1 = 1 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → (∀ (x y : M₁), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : M₁), f (n • x) = n • f x) → (∀ (n : ℤ) (x : M₁), f (n • x) = n • f x) → (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → AddCommGroupWithOne M₁ - Function.Surjective.addCommGroupWithOne Mathlib.Algebra.Group.InjSurj
{M₁ : Type u_1} → {M₂ : Type u_3} → [inst : Zero M₂] → [inst_1 : One M₂] → [inst_2 : Add M₂] → [inst_3 : Neg M₂] → [inst_4 : Sub M₂] → [inst_5 : SMul ℕ M₂] → [inst_6 : SMul ℤ M₂] → [inst_7 : NatCast M₂] → [inst_8 : IntCast M₂] → [inst_9 : AddCommGroupWithOne M₁] → (f : M₁ → M₂) → Function.Surjective f → f 0 = 0 → f 1 = 1 → (∀ (x y : M₁), f (x + y) = f x + f y) → (∀ (x : M₁), f (-x) = -f x) → (∀ (x y : M₁), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : M₁), f (n • x) = n • f x) → (∀ (n : ℤ) (x : M₁), f (n • x) = n • f x) → (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → AddCommGroupWithOne M₂ - Function.Injective.subtractionCommMonoid.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₁] [inst_1 : Zero M₁] [inst_2 : SMul ℕ M₁] [inst_3 : Neg M₁] [inst_4 : Sub M₁] [inst_5 : SMul ℤ M₁] [inst_6 : SubtractionCommMonoid M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : M₁), f (x + y) = f x + f y) (inv : ∀ (x : M₁), f (-x) = -f x) (div : ∀ (x y : M₁), f (x - y) = f x - f y) (npow : ∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) (zpow : ∀ (x : M₁) (n : ℤ), f (n • x) = n • f x), Function.Injective.subtractionCommMonoid f hf one mul inv div npow zpow = let __src := Function.Injective.subtractionMonoid f hf one mul inv div npow zpow; let __src_1 := Function.Injective.addCommSemigroup f hf mul; SubtractionCommMonoid.mk ⋯ - Function.Injective.addCommGroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₁] [inst_1 : Zero M₁] [inst_2 : SMul ℕ M₁] [inst_3 : Neg M₁] [inst_4 : Sub M₁] [inst_5 : SMul ℤ M₁] [inst_6 : AddCommGroup M₂] (f : M₁ → M₂) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : M₁), f (x + y) = f x + f y) (inv : ∀ (x : M₁), f (-x) = -f x) (div : ∀ (x y : M₁), f (x - y) = f x - f y) (npow : ∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) (zpow : ∀ (x : M₁) (n : ℤ), f (n • x) = n • f x), Function.Injective.addCommGroup f hf one mul inv div npow zpow = let __src := Function.Injective.addCommMonoid f hf one mul npow; let __src_1 := Function.Injective.addGroup f hf one mul inv div npow zpow; AddCommGroup.mk ⋯ - Function.Surjective.addCommGroup.eq_1 Mathlib.Algebra.Group.InjSurj
∀ {M₁ : Type u_1} {M₂ : Type u_2} [inst : Add M₂] [inst_1 : Zero M₂] [inst_2 : SMul ℕ M₂] [inst_3 : Neg M₂] [inst_4 : Sub M₂] [inst_5 : SMul ℤ M₂] [inst_6 : AddCommGroup M₁] (f : M₁ → M₂) (hf : Function.Surjective f) (one : f 0 = 0) (mul : ∀ (x y : M₁), f (x + y) = f x + f y) (inv : ∀ (x : M₁), f (-x) = -f x) (div : ∀ (x y : M₁), f (x - y) = f x - f y) (npow : ∀ (x : M₁) (n : ℕ), f (n • x) = n • f x) (zpow : ∀ (x : M₁) (n : ℤ), f (n • x) = n • f x), Function.Surjective.addCommGroup f hf one mul inv div npow zpow = let __src := Function.Surjective.addCommMonoid f hf one mul npow; let __src_1 := Function.Surjective.addGroup f hf one mul inv div npow zpow; AddCommGroup.mk ⋯ - add_vadd_comm Mathlib.GroupTheory.GroupAction.Defs
∀ {α : Type u_6} {β : Type u_7} [inst : Add β] [inst_1 : VAdd α β] [inst_2 : VAddCommClass α β β] (s : α) (x y : β), x + (s +ᵥ y) = s +ᵥ (x + y) - VAddCommClass.of_add_vadd_zero Mathlib.GroupTheory.GroupAction.Defs
∀ {M : Type u_10} {N : Type u_11} [inst : AddMonoid N] [inst_1 : VAdd M N], (∀ (x : M) (y : N), y + (x +ᵥ 0) = x +ᵥ y) → VAddCommClass M N N - AddCommute.function_commute_add_left Mathlib.Algebra.GroupPower.IterateHom
∀ {G : Type u_3} [inst : AddSemigroup G] {a b : G}, AddCommute a b → Function.Commute (fun x => a + x) fun x => b + x - AddCommute.function_commute_add_right Mathlib.Algebra.GroupPower.IterateHom
∀ {G : Type u_3} [inst : AddSemigroup G] {a b : G}, AddCommute a b → Function.Commute (fun x => x + a) fun x => x + b - AddMonoidHom.addCommMonoid.eq_1 Mathlib.Algebra.Group.Hom.Instances
∀ {M : Type uM} {N : Type uN} [inst : AddZeroClass M] [inst_1 : AddCommMonoid N], AddMonoidHom.addCommMonoid = AddCommMonoid.mk ⋯ - AddCommute.isAddUnit_add_iff Mathlib.Algebra.Group.Commute.Units
∀ {M : Type u_1} [inst : AddMonoid M] {a b : M}, AddCommute a b → (IsAddUnit (a + b) ↔ IsAddUnit a ∧ IsAddUnit b) - AddCommute.sub_eq_sub_iff_of_isAddUnit Mathlib.Algebra.Group.Commute.Units
∀ {M : Type u_1} [inst : SubtractionMonoid M] {a b c d : M}, AddCommute b d → IsAddUnit b → IsAddUnit d → (a - b = c - d ↔ a + d = c + b) - Function.Injective.nonUnitalNonAssocCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → (f : β → α) → Function.Injective f → [inst_4 : NonUnitalNonAssocCommSemiring α] → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → NonUnitalNonAssocCommSemiring β - Function.Surjective.nonUnitalNonAssocCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → (f : α → β) → Function.Surjective f → [inst_4 : NonUnitalNonAssocCommSemiring α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → NonUnitalNonAssocCommSemiring β - Function.Injective.nonUnitalCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → [inst_4 : NonUnitalCommSemiring α] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → NonUnitalCommSemiring β - Function.Surjective.nonUnitalCommSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : SMul ℕ β] → (f : α → β) → Function.Surjective f → [inst_4 : NonUnitalCommSemiring α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → NonUnitalCommSemiring β - Function.Injective.commSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : SMul ℕ β] → [inst_5 : Pow β ℕ] → [inst_6 : NatCast β] → (f : β → α) → Function.Injective f → [inst_7 : CommSemiring α] → f 0 = 0 → f 1 = 1 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → CommSemiring β - Function.Surjective.commSemiring Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : SMul ℕ β] → [inst_5 : Pow β ℕ] → [inst_6 : NatCast β] → (f : α → β) → Function.Surjective f → [inst_7 : CommSemiring α] → f 0 = 0 → f 1 = 1 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → CommSemiring β - Function.Injective.nonUnitalNonAssocCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → [inst_7 : NonUnitalNonAssocCommRing α] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (n : ℤ) (x : β), f (n • x) = n • f x) → NonUnitalNonAssocCommRing β - Function.Surjective.nonUnitalNonAssocCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → (f : α → β) → Function.Surjective f → [inst_7 : NonUnitalNonAssocCommRing α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (x : α), f (-x) = -f x) → (∀ (x y : α), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (n : ℤ) (x : α), f (n • x) = n • f x) → NonUnitalNonAssocCommRing β - Function.Injective.nonUnitalCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → [inst_7 : NonUnitalCommRing α] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (n : ℤ) (x : β), f (n • x) = n • f x) → NonUnitalCommRing β - Function.Surjective.nonUnitalCommRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : Add β] → [inst_2 : Mul β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → (f : α → β) → Function.Surjective f → [inst_7 : NonUnitalCommRing α] → f 0 = 0 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (x : α), f (-x) = -f x) → (∀ (x y : α), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (n : ℤ) (x : α), f (n • x) = n • f x) → NonUnitalCommRing β - Function.Injective.commRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : Neg β] → [inst_5 : Sub β] → [inst_6 : SMul ℕ β] → [inst_7 : SMul ℤ β] → [inst_8 : Pow β ℕ] → [inst_9 : NatCast β] → [inst_10 : IntCast β] → (f : β → α) → Function.Injective f → [inst_11 : CommRing α] → f 0 = 0 → f 1 = 1 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x y : β), f (x * y) = f x * f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : β), f (n • x) = n • f x) → (∀ (n : ℤ) (x : β), f (n • x) = n • f x) → (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → CommRing β - Function.Surjective.commRing Mathlib.Algebra.Ring.InjSurj
{α : Type u_1} → {β : Type u_2} → [inst : Zero β] → [inst_1 : One β] → [inst_2 : Add β] → [inst_3 : Mul β] → [inst_4 : Neg β] → [inst_5 : Sub β] → [inst_6 : SMul ℕ β] → [inst_7 : SMul ℤ β] → [inst_8 : Pow β ℕ] → [inst_9 : NatCast β] → [inst_10 : IntCast β] → (f : α → β) → Function.Surjective f → [inst_11 : CommRing α] → f 0 = 0 → f 1 = 1 → (∀ (x y : α), f (x + y) = f x + f y) → (∀ (x y : α), f (x * y) = f x * f y) → (∀ (x : α), f (-x) = -f x) → (∀ (x y : α), f (x - y) = f x - f y) → (∀ (n : ℕ) (x : α), f (n • x) = n • f x) → (∀ (n : ℤ) (x : α), f (n • x) = n • f x) → (∀ (x : α) (n : ℕ), f (x ^ n) = f x ^ n) → (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → CommRing β - Commute.add_left Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : Distrib R] {a b c : R}, Commute a c → Commute b c → Commute (a + b) c - Commute.add_right Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : Distrib R] {a b c : R}, Commute a b → Commute a c → Commute a (b + c) - Commute.mul_self_sub_mul_self_eq Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : NonUnitalNonAssocRing R] {a b : R}, Commute a b → a * a - b * b = (a + b) * (a - b) - Commute.mul_self_sub_mul_self_eq' Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : NonUnitalNonAssocRing R] {a b : R}, Commute a b → a * a - b * b = (a - b) * (a + b) - Commute.sq_sub_sq Mathlib.Algebra.Ring.Commute
∀ {R : Type x} [inst : Ring R] {a b : R}, Commute a b → a ^ 2 - b ^ 2 = (a + b) * (a - b) - AddCommute.neg_add_cancel Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : AddGroup G] {a b : G}, AddCommute a b → -a + b + a = b - AddCommute.neg_add_cancel_assoc Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : AddGroup G] {a b : G}, AddCommute a b → -a + (b + a) = b - AddCommute.conj Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : AddGroup G] {a b : G}, AddCommute a b → ∀ (h : G), AddCommute (h + a + -h) (h + b + -h) - AddCommute.conj_iff Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : AddGroup G] {a b : G} (h : G), AddCommute (h + a + -h) (h + b + -h) ↔ AddCommute a b - nsmul_neg_comm Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : AddGroup G] (a : G) (m n : ℕ), m • -a + n • a = n • a + m • -a - AddCommute.add_sub_add_comm Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : SubtractionMonoid G] {a b c d : G}, AddCommute c d → AddCommute b (-c) → a + b - (c + d) = a - c + (b - d) - AddCommute.sub_add_sub_comm Mathlib.Algebra.Group.Commute.Basic
∀ {G : Type u_1} [inst : SubtractionMonoid G] {a b c d : G}, AddCommute b d → AddCommute (-b) c → a - b + (c - d) = a + c - (b + d) - OrderedAddCommMonoid.toCovariantClassLeft Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedAddCommMonoid α], CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 - OrderedCancelAddCommMonoid.toContravariantClassLeLeft Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedCancelAddCommMonoid α], ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 - OrderedCancelAddCommMonoid.toContravariantClassLeft Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedCancelAddCommMonoid α], ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1 - OrderedAddCommMonoid.toCovariantClassRight Mathlib.Algebra.Order.Monoid.Defs
∀ (M : Type u_3) [inst : OrderedAddCommMonoid M], CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 - OrderedCancelAddCommMonoid.toContravariantClassRight Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_1} [inst : OrderedCancelAddCommMonoid α], ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1 - OrderedAddCommMonoid.mk Mathlib.Algebra.Order.Monoid.Defs
{α : Type u_3} → [toAddCommMonoid : AddCommMonoid α] → [toPartialOrder : PartialOrder α] → (∀ (a b : α), a ≤ b → ∀ (c : α), c + a ≤ c + b) → OrderedAddCommMonoid α - OrderedAddCommMonoid.add_le_add_left Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_3} [self : OrderedAddCommMonoid α] (a b : α), a ≤ b → ∀ (c : α), c + a ≤ c + b - LinearOrderedAddCommMonoidWithTop.top_add' Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_3} [self : LinearOrderedAddCommMonoidWithTop α] (x : α), ⊤ + x = ⊤ - OrderedCancelAddCommMonoid.mk Mathlib.Algebra.Order.Monoid.Defs
{α : Type u_3} → [toOrderedAddCommMonoid : OrderedAddCommMonoid α] → (∀ (a b c : α), a + b ≤ a + c → b ≤ c) → OrderedCancelAddCommMonoid α - LinearOrderedAddCommMonoidWithTop.mk Mathlib.Algebra.Order.Monoid.Defs
{α : Type u_3} → [toLinearOrderedAddCommMonoid : LinearOrderedAddCommMonoid α] → [toOrderTop : OrderTop α] → (∀ (x : α), ⊤ + x = ⊤) → LinearOrderedAddCommMonoidWithTop α - OrderedCancelAddCommMonoid.le_of_add_le_add_left Mathlib.Algebra.Order.Monoid.Defs
∀ {α : Type u_3} [self : OrderedCancelAddCommMonoid α] (a b c : α), a + b ≤ a + c → b ≤ c - LinearOrderedAddCommGroupWithTop.zsmul_succ' Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [self : LinearOrderedAddCommGroupWithTop α] (n : ℕ) (a : α), LinearOrderedAddCommGroupWithTop.zsmul (Int.ofNat n.succ) a = LinearOrderedAddCommGroupWithTop.zsmul (Int.ofNat n) a + a - OrderedAddCommGroup.to_contravariantClass_left_le Mathlib.Algebra.Order.Group.Defs
∀ (α : Type u) [inst : OrderedAddCommGroup α], ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 - OrderedAddCommGroup.to_covariantClass_left_le Mathlib.Algebra.Order.Group.Defs
∀ (α : Type u) [inst : OrderedAddCommGroup α], CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 - LinearOrderedAddCommGroupWithTop.sub_eq_add_neg Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [self : LinearOrderedAddCommGroupWithTop α] (a b : α), a - b = a + -b - OrderedAddCommGroup.to_contravariantClass_right_le Mathlib.Algebra.Order.Group.Defs
∀ (α : Type u) [inst : OrderedAddCommGroup α], ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1 - OrderedAddCommGroup.add_lt_add_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [inst : Add α] [inst_1 : LT α] [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {b c : α}, b < c → ∀ (a : α), a + b < a + c - OrderedAddCommGroup.le_of_add_le_add_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [inst : Add α] [inst_1 : LE α] [inst_2 : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}, a + b ≤ a + c → b ≤ c - OrderedAddCommGroup.lt_of_add_lt_add_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [inst : Add α] [inst_1 : LT α] [inst_2 : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}, a + b < a + c → b < c - OrderedAddCommGroup.mk Mathlib.Algebra.Order.Group.Defs
{α : Type u} → [toAddCommGroup : AddCommGroup α] → [toPartialOrder : PartialOrder α] → (∀ (a b : α), a ≤ b → ∀ (c : α), c + a ≤ c + b) → OrderedAddCommGroup α - OrderedAddCommGroup.add_le_add_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [self : OrderedAddCommGroup α] (a b : α), a ≤ b → ∀ (c : α), c + a ≤ c + b - LinearOrderedAddCommGroup.add_lt_add_left Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : LinearOrderedAddCommGroup α] (a b : α), a < b → ∀ (c : α), c + a < c + b - LinearOrderedAddCommGroupWithTop.add_neg_cancel Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u_1} [self : LinearOrderedAddCommGroupWithTop α] (a : α), a ≠ ⊤ → a + -a = 0 - le_sub_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : AddCommGroup α] [inst_1 : LE α] [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}, a ≤ b - c ↔ c ≤ b - a - lt_sub_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : AddCommGroup α] [inst_1 : LT α] [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}, a < b - c ↔ c < b - a - sub_le_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : AddCommGroup α] [inst_1 : LE α] [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}, a - b ≤ c ↔ a - c ≤ b - sub_lt_comm Mathlib.Algebra.Order.Group.Defs
∀ {α : Type u} [inst : AddCommGroup α] [inst_1 : LT α] [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a b c : α}, a - b < c ↔ a - c < b - LinearOrderedAddCommGroupWithTop.mk Mathlib.Algebra.Order.Group.Defs
{α : Type u_1} → [toLinearOrderedAddCommMonoidWithTop : LinearOrderedAddCommMonoidWithTop α] → [toNeg : Neg α] → [toSub : Sub α] → autoParam (∀ (a b : α), a - b = a + -b) _auto✝ → (zsmul : ℤ → α → α) → autoParam (∀ (a : α), zsmul 0 a = 0) _auto✝¹ → autoParam (∀ (n : ℕ) (a : α), zsmul (Int.ofNat n.succ) a = zsmul (Int.ofNat n) a + a) _auto✝² → autoParam (∀ (n : ℕ) (a : α), zsmul (Int.negSucc n) a = -zsmul (↑n.succ) a) _auto✝³ → [toNontrivial : Nontrivial α] → -⊤ = ⊤ → (∀ (a : α), a ≠ ⊤ → a + -a = 0) → LinearOrderedAddCommGroupWithTop α - CanonicallyOrderedAddCommMonoid.le_self_add Mathlib.Algebra.Order.Monoid.Canonical.Defs
∀ {α : Type u_1} [self : CanonicallyOrderedAddCommMonoid α] (a b : α), a ≤ a + b - CanonicallyOrderedAddCommMonoid.exists_add_of_le Mathlib.Algebra.Order.Monoid.Canonical.Defs
∀ {α : Type u_1} [self : CanonicallyOrderedAddCommMonoid α] {a b : α}, a ≤ b → ∃ c, b = a + c - CanonicallyOrderedAddCommMonoid.mk Mathlib.Algebra.Order.Monoid.Canonical.Defs
{α : Type u_1} → [toOrderedAddCommMonoid : OrderedAddCommMonoid α] → [toOrderBot : OrderBot α] → (∀ {a b : α}, a ≤ b → ∃ c, b = a + c) → (∀ (a b : α), a ≤ a + b) → CanonicallyOrderedAddCommMonoid α - AddCommGroup.toDistribLattice Mathlib.Algebra.Order.Group.Lattice
(α : Type u_3) → [inst : Lattice α] → [inst_1 : AddCommGroup α] → [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] → DistribLattice α - AddCommGroup.toDistribLattice.eq_1 Mathlib.Algebra.Order.Group.Lattice
∀ (α : Type u_3) [inst : Lattice α] [inst_1 : AddCommGroup α] [inst_2 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1], AddCommGroup.toDistribLattice α = DistribLattice.mk ⋯ - Function.Injective.orderedAddCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → [inst : OrderedAddCommMonoid α] → {β : Type u_2} → [inst_1 : Zero β] → [inst_2 : Add β] → [inst_3 : SMul ℕ β] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x : β) (n : ℕ), f (n • x) = n • f x) → OrderedAddCommMonoid β - Function.Injective.orderedCancelAddCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → {β : Type u_1} → [inst : OrderedCancelAddCommMonoid α] → [inst_1 : Zero β] → [inst_2 : Add β] → [inst_3 : SMul ℕ β] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x : β) (n : ℕ), f (n • x) = n • f x) → OrderedCancelAddCommMonoid β - Function.Injective.orderedCancelAddCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} {β : Type u_1} [inst : OrderedCancelAddCommMonoid α] [inst_1 : Zero β] [inst_2 : Add β] [inst_3 : SMul ℕ β] (f : β → α) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ℕ), f (n • x) = n • f x), Function.Injective.orderedCancelAddCommMonoid f hf one mul npow = OrderedCancelAddCommMonoid.mk ⋯ - Function.Injective.orderedAddCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} [inst : OrderedAddCommMonoid α] {β : Type u_2} [inst_1 : Zero β] [inst_2 : Add β] [inst_3 : SMul ℕ β] (f : β → α) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ℕ), f (n • x) = n • f x), Function.Injective.orderedAddCommMonoid f hf one mul npow = OrderedAddCommMonoid.mk ⋯ - Function.Injective.linearOrderedAddCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → [inst : LinearOrderedAddCommMonoid α] → {β : Type u_2} → [inst_1 : Zero β] → [inst_2 : Add β] → [inst_3 : SMul ℕ β] → [inst_4 : Sup β] → [inst_5 : Inf β] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x : β) (n : ℕ), f (n • x) = n • f x) → (∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) → (∀ (x y : β), f (x ⊓ y) = min (f x) (f y)) → LinearOrderedAddCommMonoid β - Function.Injective.linearOrderedCancelAddCommMonoid Mathlib.Algebra.Order.Monoid.Basic
{α : Type u} → {β : Type u_1} → [inst : LinearOrderedCancelAddCommMonoid α] → [inst_1 : Zero β] → [inst_2 : Add β] → [inst_3 : SMul ℕ β] → [inst_4 : Sup β] → [inst_5 : Inf β] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x : β) (n : ℕ), f (n • x) = n • f x) → (∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) → (∀ (x y : β), f (x ⊓ y) = min (f x) (f y)) → LinearOrderedCancelAddCommMonoid β - Function.Injective.linearOrderedAddCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} [inst : LinearOrderedAddCommMonoid α] {β : Type u_2} [inst_1 : Zero β] [inst_2 : Add β] [inst_3 : SMul ℕ β] [inst_4 : Sup β] [inst_5 : Inf β] (f : β → α) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ℕ), f (n • x) = n • f x) (sup : ∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) (inf : ∀ (x y : β), f (x ⊓ y) = min (f x) (f y)), Function.Injective.linearOrderedAddCommMonoid f hf one mul npow sup inf = let __spread.0 := LinearOrder.lift f hf sup inf; LinearOrderedAddCommMonoid.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯ - Function.Injective.linearOrderedCancelAddCommMonoid.eq_1 Mathlib.Algebra.Order.Monoid.Basic
∀ {α : Type u} {β : Type u_1} [inst : LinearOrderedCancelAddCommMonoid α] [inst_1 : Zero β] [inst_2 : Add β] [inst_3 : SMul ℕ β] [inst_4 : Sup β] [inst_5 : Inf β] (f : β → α) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (npow : ∀ (x : β) (n : ℕ), f (n • x) = n • f x) (hsup : ∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x ⊓ y) = min (f x) (f y)), Function.Injective.linearOrderedCancelAddCommMonoid f hf one mul npow hsup hinf = let __spread.0 := Function.Injective.linearOrderedAddCommMonoid f hf one mul npow hsup hinf; LinearOrderedCancelAddCommMonoid.mk ⋯ LinearOrderedAddCommMonoid.decidableLE LinearOrderedAddCommMonoid.decidableEq LinearOrderedAddCommMonoid.decidableLT ⋯ ⋯ ⋯ - CanonicallyOrderedAddCommMonoid.toAddCancelCommMonoid Mathlib.Algebra.Order.Sub.Canonical
(α : Type u_1) → [inst : CanonicallyOrderedAddCommMonoid α] → [inst_1 : Sub α] → [inst_2 : OrderedSub α] → [inst : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] → AddCancelCommMonoid α - AddLECancellable.tsub_add_tsub_comm Mathlib.Algebra.Order.Sub.Canonical
∀ {α : Type u_1} [inst : AddCommSemigroup α] [inst_1 : PartialOrder α] [inst_2 : ExistsAddOfLE α] [inst_3 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [inst_4 : Sub α] [inst_5 : OrderedSub α] {a b c d : α}, AddLECancellable b → AddLECancellable d → b ≤ a → d ≤ c → a - b + (c - d) = a + c - (b + d) - tsub_add_tsub_comm Mathlib.Algebra.Order.Sub.Canonical
∀ {α : Type u_1} [inst : AddCommSemigroup α] [inst_1 : PartialOrder α] [inst_2 : ExistsAddOfLE α] [inst_3 : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [inst_4 : Sub α] [inst_5 : OrderedSub α] {a b c d : α} [inst_6 : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1], b ≤ a → d ≤ c → a - b + (c - d) = a + c - (b + d) - CanonicallyOrderedCommSemiring.npow_succ Mathlib.Algebra.Order.Ring.Canonical
∀ {α : Type u_2} [self : CanonicallyOrderedCommSemiring α] (n : ℕ) (x : α), CanonicallyOrderedCommSemiring.npow (n + 1) x = CanonicallyOrderedCommSemiring.npow n x * x - CanonicallyOrderedCommSemiring.natCast_succ Mathlib.Algebra.Order.Ring.Canonical
∀ {α : Type u_2} [self : CanonicallyOrderedCommSemiring α] (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1 - CanonicallyOrderedCommSemiring.left_distrib Mathlib.Algebra.Order.Ring.Canonical
∀ {α : Type u_2} [self : CanonicallyOrderedCommSemiring α] (a b c : α), a * (b + c) = a * b + a * c - CanonicallyOrderedCommSemiring.right_distrib Mathlib.Algebra.Order.Ring.Canonical
∀ {α : Type u_2} [self : CanonicallyOrderedCommSemiring α] (a b c : α), (a + b) * c = a * c + b * c - CanonicallyOrderedCommSemiring.mk Mathlib.Algebra.Order.Ring.Canonical
{α : Type u_2} → [toCanonicallyOrderedAddCommMonoid : CanonicallyOrderedAddCommMonoid α] → [toMul : Mul α] → (∀ (a b c : α), a * (b + c) = a * b + a * c) → (∀ (a b c : α), (a + b) * c = a * c + b * c) → (∀ (a : α), 0 * a = 0) → (∀ (a : α), a * 0 = 0) → (∀ (a b c : α), a * b * c = a * (b * c)) → [toOne : One α] → (∀ (a : α), 1 * a = a) → (∀ (a : α), a * 1 = a) → [toNatCast : NatCast α] → autoParam (NatCast.natCast 0 = 0) _auto✝ → autoParam (∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1) _auto✝¹ → (npow : ℕ → α → α) → autoParam (∀ (x : α), npow 0 x = 1) _auto✝² → autoParam (∀ (n : ℕ) (x : α), npow (n + 1) x = npow n x * x) _auto✝³ → (∀ (a b : α), a * b = b * a) → (∀ {a b : α}, a * b = 0 → a = 0 ∨ b = 0) → CanonicallyOrderedCommSemiring α - Commute.inv_add_inv Mathlib.Algebra.Field.Basic
∀ {α : Type u_1} [inst : DivisionSemiring α] {a b : α}, Commute a b → a ≠ 0 → b ≠ 0 → a⁻¹ + b⁻¹ = (a + b) / (a * b) - Commute.one_div_add_one_div Mathlib.Algebra.Field.Basic
∀ {α : Type u_1} [inst : DivisionSemiring α] {a b : α}, Commute a b → a ≠ 0 → b ≠ 0 → 1 / a + 1 / b = (a + b) / (a * b) - Commute.div_add_div Mathlib.Algebra.Field.Basic
∀ {α : Type u_1} [inst : DivisionSemiring α] {a b c d : α}, Commute b c → Commute b d → b ≠ 0 → d ≠ 0 → a / b + c / d = (a * d + b * c) / (b * d) - Function.Injective.orderedAddCommGroup Mathlib.Algebra.Order.Group.InjSurj
{α : Type u_1} → [inst : OrderedAddCommGroup α] → {β : Type u_3} → [inst_1 : Zero β] → [inst_2 : Add β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (x : β) (n : ℕ), f (n • x) = n • f x) → (∀ (x : β) (n : ℤ), f (n • x) = n • f x) → OrderedAddCommGroup β - Function.Injective.linearOrderedAddCommGroup Mathlib.Algebra.Order.Group.InjSurj
{α : Type u_1} → [inst : LinearOrderedAddCommGroup α] → {β : Type u_3} → [inst_1 : Zero β] → [inst_2 : Add β] → [inst_3 : Neg β] → [inst_4 : Sub β] → [inst_5 : SMul ℕ β] → [inst_6 : SMul ℤ β] → [inst_7 : Sup β] → [inst_8 : Inf β] → (f : β → α) → Function.Injective f → f 0 = 0 → (∀ (x y : β), f (x + y) = f x + f y) → (∀ (x : β), f (-x) = -f x) → (∀ (x y : β), f (x - y) = f x - f y) → (∀ (x : β) (n : ℕ), f (n • x) = n • f x) → (∀ (x : β) (n : ℤ), f (n • x) = n • f x) → (∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) → (∀ (x y : β), f (x ⊓ y) = min (f x) (f y)) → LinearOrderedAddCommGroup β - Function.Injective.orderedAddCommGroup.eq_1 Mathlib.Algebra.Order.Group.InjSurj
∀ {α : Type u_1} [inst : OrderedAddCommGroup α] {β : Type u_3} [inst_1 : Zero β] [inst_2 : Add β] [inst_3 : Neg β] [inst_4 : Sub β] [inst_5 : SMul ℕ β] [inst_6 : SMul ℤ β] (f : β → α) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (inv : ∀ (x : β), f (-x) = -f x) (div : ∀ (x y : β), f (x - y) = f x - f y) (npow : ∀ (x : β) (n : ℕ), f (n • x) = n • f x) (zpow : ∀ (x : β) (n : ℤ), f (n • x) = n • f x), Function.Injective.orderedAddCommGroup f hf one mul inv div npow zpow = let __spread.0 := Function.Injective.orderedAddCommMonoid f hf one mul npow; OrderedAddCommGroup.mk ⋯ - Function.Injective.linearOrderedAddCommGroup.eq_1 Mathlib.Algebra.Order.Group.InjSurj
∀ {α : Type u_1} [inst : LinearOrderedAddCommGroup α] {β : Type u_3} [inst_1 : Zero β] [inst_2 : Add β] [inst_3 : Neg β] [inst_4 : Sub β] [inst_5 : SMul ℕ β] [inst_6 : SMul ℤ β] [inst_7 : Sup β] [inst_8 : Inf β] (f : β → α) (hf : Function.Injective f) (one : f 0 = 0) (mul : ∀ (x y : β), f (x + y) = f x + f y) (inv : ∀ (x : β), f (-x) = -f x) (div : ∀ (x y : β), f (x - y) = f x - f y) (npow : ∀ (x : β) (n : ℕ), f (n • x) = n • f x) (zpow : ∀ (x : β) (n : ℤ), f (n • x) = n • f x) (sup : ∀ (x y : β), f (x ⊔ y) = max (f x) (f y)) (inf : ∀ (x y : β), f (x ⊓ y) = min (f x) (f y)), Function.Injective.linearOrderedAddCommGroup f hf one mul inv div npow zpow sup inf = let __spread.0 := LinearOrder.lift f hf sup inf; LinearOrderedAddCommGroup.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯
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About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
🔍_ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.The pattern can also be non-linear, as in
🔍Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
🔍(?a -> ?b) -> List ?a -> List ?b
🔍List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all→
and∀
) has the given shape.As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍|- _ < _ → tsum _ < tsum _
will findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin
and tsum
, have "two"
as a substring of the
lemma name, include a product and a power somewhere in the type,
and have a hypothesis of the form _ < _
(if
there were any such lemmas). Metavariables (?a
) are
assigned independently in each filter.
The #lucky
button will directly send you to the
documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision c44d42e