diff --git a/src/Lean/Elab/Coinductive.lean b/src/Lean/Elab/Coinductive.lean
new file mode 100644
index 000000000000..f5df85f6e95b
--- /dev/null
+++ b/src/Lean/Elab/Coinductive.lean
@@ -0,0 +1,253 @@
+/-
+Copyright (c) 2020 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: William Sørensen
+-/
+prelude
+import Lean.Elab.Command
+import Lean.Elab.DeclarationRange
+import Lean.Elab.Exception
+import Lean.Elab.Inductive
+import Lean.Util.Trace
+
+namespace Lean.Elab.Command
+
+open Lean Lean.Elab Lean.Elab.Term Lean.Parser Command
+
+builtin_initialize
+  registerTraceClass `Elab.Coinductive
+
+structure CoinductiveView.CtorView where
+  ref       : Syntax
+  modifiers : Modifiers
+  declName  : Name
+  binders   : TSyntaxArray ``Lean.Parser.Term.bracketedBinder
+  type?     : Option Term
+  deriving Inhabited
+
+structure CoinductiveView where
+  ref             : TSyntax ``Lean.Parser.Command.coinductive
+  declId          : TSyntax ``Parser.Command.declId
+  modifiers       : Modifiers
+  shortDeclName   : Name
+  declName        : Name
+  levelNames      : List Name
+  binders         : TSyntaxArray ``Lean.Parser.Term.bracketedBinder
+  type            : Term
+  ctors           : Array CoinductiveView.CtorView
+  deriving Inhabited
+
+namespace CoinductiveView
+
+/--
+  Create a CtorView from a Name, some Modifiers, and Syntax
+-/
+def CtorView.ofStx (declName : Name) (modifiers : Modifiers) (ref : Syntax) : CommandElabM CoinductiveView.CtorView := do
+  let mut ctorModifiers ← elabModifiers ref[2]
+  if let some leadingDocComment := ref[0].getOptional? then
+    if ctorModifiers.docString?.isSome then
+      logErrorAt leadingDocComment "duplicate doc string"
+    ctorModifiers := { ctorModifiers with docString? := TSyntax.getDocString ⟨leadingDocComment⟩ }
+  if ctorModifiers.isPrivate && modifiers.isPrivate then
+    throwError "invalid 'private' constructor in a 'private' inductive datatype"
+  if ctorModifiers.isProtected && modifiers.isPrivate then
+    throwError "invalid 'protected' constructor in a 'private' inductive datatype"
+
+  checkValidCtorModifier ctorModifiers
+  let ctorName := ref.getIdAt 3
+  let ctorName := declName ++ ctorName
+  let ctorName ← withRef ref[3] $ applyVisibility ctorModifiers.visibility ctorName
+  let (binders, type?) := expandOptDeclSig ref[4]
+  addDocString' ctorName ctorModifiers.docString?
+  addAuxDeclarationRanges ctorName ref ref[3]
+
+  return { ref, modifiers := ctorModifiers, declName := ctorName, binders := binders.getArgs.map (⟨·⟩), type? := type?.map (⟨·⟩) }
+
+/--
+  Create a CoinductiveView from Modifiers and Syntax
+-/
+def ofModifiersAndStx (modifiers : Modifiers) (decl : Syntax) : CommandElabM CoinductiveView := do
+  let (binders, type) := expandDeclSig decl[2]!
+
+  let binders := binders.getArgs.map (⟨·⟩)
+
+  let declId  := ⟨decl[1]⟩
+  let ⟨shortDeclName, declName, levelNames⟩ ← expandDeclId declId.raw modifiers
+
+  let ctors ← decl[4].getArgs.mapM $ CtorView.ofStx declName modifiers
+
+  addDeclarationRanges declName decl
+
+  return {
+    ref := ⟨decl⟩
+
+    declName
+    shortDeclName
+
+    levelNames
+    type := ⟨type⟩
+
+    binders
+    ctors
+
+    declId
+    modifiers
+  }
+
+def ofStx (stx : Syntax) : CommandElabM CoinductiveView := elabModifiers stx[0] >>= (ofModifiersAndStx · stx[1])
+
+def toBinderIds (c : CoinductiveView) : Array Ident := (c.binders.map getBracketedBinderIds).flatten.map mkIdent
+
+def toRelType (c : CoinductiveView) : CommandElabM Term :=
+  c.binders.reverse.foldlM (fun curr acc => `($acc:bracketedBinder → $curr)) c.type
+
+end CoinductiveView
+
+open Parser.Term in
+section
+
+/-- `bb`s are an alias for `bracketedBinder` and used in quotations -/
+private abbrev bb := bracketedBinder
+/-- Since `bb` is an alias for `bracketedBinder`, we can safely coerce syntax of these categories  -/
+instance : Coe (TSyntax ``bb) (TSyntax ``bracketedBinder) where coe x := ⟨x.raw⟩
+
+end
+
+-- TODO: Use elaboration over syntax manipulation
+/--
+  Extract the types and the bottom value like the following:
+
+  > a → ⋯ → bot
+  > ⟨#[a, ⋯], bot⟩
+-/
+private partial def typeToArgArr (type : Term) : Array Term × Term := Prod.map List.toArray id $ go type.raw
+  where go
+    | .node _ ``Parser.Term.arrow #[hd, _, tail] => Prod.map (⟨hd⟩ :: ·) id $ go tail
+    | rest => ⟨[], ⟨rest⟩⟩
+
+/--
+  Extract the args of a function
+
+  > bot a ⋯
+  > ⟨#[a, ⋯], bot⟩
+-/
+private def appsToArgArr (type : Term) : Array Term × Term := match type.raw with
+  | .node _ ``Parser.Term.app #[v, cont] => ⟨cont.getArgs.map (⟨·⟩), ⟨v⟩⟩
+  | rest => ⟨#[], ⟨rest⟩⟩
+
+/--
+  Given a list [a, b, c] split lets you split it into two arrays
+
+  > example : split 2 [1, 2, 3] = ⟨[1, 2], [3]⟩ := by simp [split]
+-/
+private def split : Nat → List α → (List α) × (List α)
+  | _, []  => ⟨[], []⟩
+  | 0, arr => ⟨[], arr⟩
+  | n+1, hd :: tl => Prod.map (hd :: ·) id $ split n tl
+
+/--
+  Coming in these have the form of  | name ... : ... → Nm       topBinders...         args...
+  But we want them to have the form | name ... : ... → Nm.Shape topBinders... RecName args...
+
+  To do this we simply replace the out type.
+-/
+private def handleCtor (names : Array Ident) (topView : CoinductiveView) (isTy : Ident) (view : CoinductiveView.CtorView) : CommandElabM CtorView := do
+  let nm := view.declName.replacePrefix topView.declName (topView.declName ++ `Shape)
+
+  let type? ← view.type?.mapM fun type => do
+    let ⟨args, out⟩ := typeToArgArr type
+    let ⟨arr, _⟩ := appsToArgArr out
+
+    let ⟨pre, post⟩ := (split topView.binders.size arr.toList).map (·.toArray) (·.toArray)
+
+    let out ← `($isTy $pre* $names* $post*)
+
+    args.reverse.foldlM (fun acc curr => `($curr → $acc)) out
+
+  return {
+    ref       := .missing
+    modifiers := view.modifiers
+    declName  := nm
+    binders   := .node .none `null (view.binders.map (·.raw))
+    type?     := type?
+  }
+
+/--
+  Coinductive predicates need a shape construct.
+  These are inductive datatypes that have the same constructors without any level of recursion.
+  These are then applied in the fixed point.
+-/
+private def generateShapes (vss : Array (CoinductiveView × Array Ident)) : CommandElabM Unit := do
+  -- It could be worth making this extract only the names that are required.
+  let coRecArgs   ← vss.mapM (fun ⟨v, _⟩ => do `(bb| ( $(mkIdent $ v.shortDeclName) : $(←v.toRelType))))
+  let names      := vss.map  (mkIdent ·.fst.shortDeclName)
+
+  elabInductiveViews $ ←vss.mapM fun ⟨topView, _⟩ => do
+    let shortDeclName := topView.shortDeclName ++ `Shape
+
+    let view := {
+      ref             := .missing
+      declId          := ←`(declId| $(mkIdent shortDeclName))
+      modifiers       := topView.modifiers
+      shortDeclName
+      declName        := topView.declName ++ `Shape
+      levelNames      := topView.levelNames
+      binders         := .node .none `null $ topView.binders.append coRecArgs
+      type?           := some topView.type
+      ctors           := ←topView.ctors.mapM $ handleCtor names topView $ mkIdent shortDeclName
+      derivingClasses := #[]
+      computedFields  := #[]
+    }
+
+    trace[Elab.Coinductive] s!"{repr topView.binders}"
+
+    return view
+/--
+  `Is` is a predicate on the relation used to generate the coinductive predicate.
+  It ensures it satisfies the declaration given.
+-/
+private def generateIs (vss : Array (CoinductiveView × Array Ident)) (rNameEntries : Array (Ident × Term)) : CommandElabM Unit := do
+  -- It could be worth making this extract only the names that are required.
+  let boundRNames ← rNameEntries.mapM fun ⟨i, v⟩ => do `(bb| ( $i : $v ) )
+
+  for ⟨idx, topView, argArr⟩ in vss.toList.enum do
+    let shortDeclName := topView.shortDeclName ++ `Shape
+
+    let boundNames := rNameEntries.map Prod.fst
+    let i := boundNames[idx]! -- OK since these come from the same source
+
+    let stx ← `(command|
+      abbrev $(mkIdent $ topView.shortDeclName ++ `Is) $(topView.binders)* $boundRNames* : Prop :=
+        ∀ { $argArr* }, $i $(topView.toBinderIds)* $argArr* → $(mkIdent shortDeclName) $(topView.toBinderIds)* $boundNames* $argArr*)
+
+    trace[Elab.Coinductive] "Generating Is check:"
+    trace[Elab.Coinductive] stx
+
+    elabCommand stx
+
+private def generateCoinductive (vss : Array (CoinductiveView × Array Ident)) (rNameEntries : Array (Ident × Term)) : CommandElabM Unit := do
+  for ⟨idx, view, argArr⟩ in vss.toList.enum do
+    let boundNames := rNameEntries.map Prod.fst
+    let i := boundNames[idx]!
+
+    let stx ← `(def $(mkIdent view.shortDeclName) $(view.binders)* : $(view.type) := fun $argArr* =>
+      ∃ $[$boundNames:ident]*, @$(mkIdent $ view.shortDeclName ++ `Is) $(view.toBinderIds)* $boundNames* ∧ $i $(view.toBinderIds)* $argArr*)
+
+    trace[Elab.Coinductive] "Generating co-inductive:"
+    trace[Elab.Coinductive] stx
+    elabCommand stx
+
+def elabCoinductiveViews (views : Array CoinductiveView) : CommandElabM Unit := do
+  let viewCheck ← views.mapM fun view => do
+    let ⟨tyArr, out⟩ := typeToArgArr view.type
+    let argArr := (← tyArr.mapM (fun _ => mkFreshBinderName)).map mkIdent
+
+    let .node _ ``Parser.Term.prop _ := out.raw | throwErrorAt out "Expected return type to be a Prop"
+    return Prod.mk view argArr
+
+  let rNameEntries ← viewCheck.mapM (fun ⟨v, _⟩ => return Prod.mk (mkIdent $ ←mkFreshBinderName) (←v.toRelType))
+
+  generateShapes viewCheck
+  generateIs viewCheck rNameEntries
+  generateCoinductive viewCheck rNameEntries
+
diff --git a/src/Lean/Elab/Declaration.lean b/src/Lean/Elab/Declaration.lean
index 720c1c899d2e..56f1a569a5fc 100644
--- a/src/Lean/Elab/Declaration.lean
+++ b/src/Lean/Elab/Declaration.lean
@@ -8,6 +8,7 @@ import Lean.Util.CollectLevelParams
 import Lean.Elab.DeclUtil
 import Lean.Elab.DefView
 import Lean.Elab.Inductive
+import Lean.Elab.Coinductive
 import Lean.Elab.Structure
 import Lean.Elab.MutualDef
 import Lean.Elab.DeclarationRange
@@ -47,6 +48,7 @@ private def isNamedDef (stx : Syntax) : Bool :=
     k == ``Lean.Parser.Command.opaque ||
     k == ``Lean.Parser.Command.axiom ||
     k == ``Lean.Parser.Command.inductive ||
+    k == ``Lean.Parser.Command.coinductive ||
     k == ``Lean.Parser.Command.classInductive ||
     k == ``Lean.Parser.Command.structure
 
@@ -183,6 +185,10 @@ def elabInductive (modifiers : Modifiers) (stx : Syntax) : CommandElabM Unit :=
   let v ← inductiveSyntaxToView modifiers stx
   elabInductiveViews #[v]
 
+def elabCoinductive (modifiers : Modifiers) (stx : Syntax) : CommandElabM Unit := do
+  let v ← CoinductiveView.ofModifiersAndStx modifiers stx
+  elabCoinductiveViews #[v]
+
 def elabClassInductive (modifiers : Modifiers) (stx : Syntax) : CommandElabM Unit := do
   let modifiers := modifiers.addAttr { name := `class }
   let v ← classInductiveSyntaxToView modifiers stx
@@ -216,6 +222,9 @@ def elabDeclaration : CommandElab := fun stx => do
     else if declKind == ``Lean.Parser.Command.«inductive» then
       let modifiers ← elabModifiers stx[0]
       elabInductive modifiers decl
+    else if declKind == ``Lean.Parser.Command.«coinductive» then
+      let modifiers ← elabModifiers stx[0]
+      elabCoinductive modifiers decl
     else if declKind == ``Lean.Parser.Command.classInductive then
       let modifiers ← elabModifiers stx[0]
       elabClassInductive modifiers decl
diff --git a/src/Lean/Parser/Command.lean b/src/Lean/Parser/Command.lean
index e457dc8703d7..03fe329617a6 100644
--- a/src/Lean/Parser/Command.lean
+++ b/src/Lean/Parser/Command.lean
@@ -198,6 +198,9 @@ For more information about [inductive types](https://lean-lang.org/theorem_provi
 def «inductive»      := leading_parser
   "inductive " >> recover declId skipUntilWsOrDelim >> ppIndent optDeclSig >> optional (symbol " :=" <|> " where") >>
   many ctor >> optional (ppDedent ppLine >> computedFields) >> optDeriving
+def «coinductive»      := leading_parser
+  "coinductive " >> recover declId skipUntilWsOrDelim >> ppIndent declSig >> optional (symbol " :=" <|> " where") >>
+  many ctor >> optional (ppDedent ppLine >> computedFields) -- >> optDeriving --TODO: Handle deriving
 def classInductive   := leading_parser
   atomic (group (symbol "class " >> "inductive ")) >>
   recover declId skipUntilWsOrDelim >> ppIndent optDeclSig >>
@@ -236,7 +239,7 @@ def «structure»          := leading_parser
 @[builtin_command_parser] def declaration := leading_parser
   declModifiers false >>
   («abbrev» <|> definition <|> «theorem» <|> «opaque» <|> «instance» <|> «axiom» <|> «example» <|>
-   «inductive» <|> classInductive <|> «structure»)
+   «coinductive» <|> «inductive» <|> classInductive <|> «structure»)
 @[builtin_command_parser] def «deriving»     := leading_parser
   "deriving " >> "instance " >> derivingClasses >> " for " >> sepBy1 (recover ident skip) ", "
 @[builtin_command_parser] def noncomputableSection := leading_parser
diff --git a/tests/lean/run/coindBisim.lean b/tests/lean/run/coindBisim.lean
new file mode 100644
index 000000000000..e1f7ebdf1b34
--- /dev/null
+++ b/tests/lean/run/coindBisim.lean
@@ -0,0 +1,87 @@
+structure FSM where
+  S : Type
+  d : S → Nat → S
+  A : S → Prop
+
+-- Example of a coinductive predicate defined over FSMs
+
+coinductive Bisim (fsm : FSM) : fsm.S → fsm.S → Prop :=
+  | step {s t : fsm.S} :
+    (fsm.A s ↔ fsm.A t)
+    → (∀ c, Bisim fsm (fsm.d s c) (fsm.d t c))
+    → Bisim fsm s t
+
+/--
+info: inductive Bisim.Shape : (fsm : FSM) → ((fsm : FSM) → fsm.S → fsm.S → Prop) → fsm.S → fsm.S → Prop
+number of parameters: 4
+constructors:
+Bisim.Shape.step : ∀ {fsm : FSM} {Bisim : (fsm : FSM) → fsm.S → fsm.S → Prop} {s t : fsm.S},
+  (fsm.A s ↔ fsm.A t) → (∀ (c : Nat), Bisim fsm (fsm.d s c) (fsm.d t c)) → Bisim.Shape fsm Bisim s t
+-/
+#guard_msgs in
+#print Bisim.Shape
+
+/--
+info: @[reducible] def Bisim.Is : FSM → ((fsm : FSM) → fsm.S → fsm.S → Prop) → Prop :=
+fun fsm x => ∀ {x_1 x_2 : fsm.S}, x fsm x_1 x_2 → Bisim.Shape fsm x x_1 x_2
+-/
+#guard_msgs in
+#print Bisim.Is
+
+/--
+info: def Bisim : (fsm : FSM) → fsm.S → fsm.S → Prop :=
+fun fsm x x_1 => ∃ x_2, Bisim.Is fsm x_2 ∧ x_2 fsm x x_1
+-/
+#guard_msgs in
+#print Bisim
+
+/-- info: 'Bisim' does not depend on any axioms -/
+#guard_msgs in
+#print axioms Bisim
+
+theorem bisim_refl : Bisim f a a := by
+  exists fun _ a b => a = b
+  simp only [and_true]
+  intro s t seqt
+  constructor <;> simp_all
+
+theorem bisim_symm (h : Bisim f a b): Bisim f b a := by
+  rcases h with ⟨rel, relIsBisim, rab⟩
+  exists fun f a b => rel f b a
+  simp_all
+  intro a b holds
+  specialize relIsBisim holds
+  rcases relIsBisim with ⟨imp, z⟩
+  constructor <;> simp_all only [implies_true, and_self]
+
+theorem Bisim.unfold {f} : Bisim.Is f Bisim := by
+  rintro s t ⟨R, h_is, h_Rst⟩
+  constructor
+  · exact (h_is h_Rst).1
+  · intro c; exact ⟨R, h_is, (h_is h_Rst).2 c⟩
+
+theorem bisim_trans (h_ab : Bisim f a b) (h_bc : Bisim f b c) :
+    Bisim f a c := by
+  exists (fun f s t => ∃ u, Bisim f s u ∧ Bisim f u t)
+  constructor
+  intro s t h_Rst
+  · rcases h_Rst with ⟨u, h_su, h_ut⟩
+    have ⟨h_su₁, h_su₂⟩ := h_su.unfold
+    have ⟨h_ut₁, h_ut₂⟩ := h_ut.unfold
+    refine ⟨?_, ?_⟩
+    · rw [h_su₁, h_ut₁]
+    · intro c; exact ⟨_, h_su₂ c, h_ut₂ c⟩
+  · exact ⟨b, h_ab, h_bc⟩
+
+/-- info: 'bisim_refl' depends on axioms: [propext, Quot.sound] -/
+#guard_msgs in
+#print axioms bisim_refl
+
+/-- info: 'bisim_symm' depends on axioms: [propext, Quot.sound] -/
+#guard_msgs in
+#print axioms bisim_symm
+
+/-- info: 'bisim_trans' depends on axioms: [propext] -/
+#guard_msgs in
+#print axioms bisim_trans
+
diff --git a/tests/lean/run/coindTypes.lean b/tests/lean/run/coindTypes.lean
new file mode 100644
index 000000000000..e2b44ffe43b2
--- /dev/null
+++ b/tests/lean/run/coindTypes.lean
@@ -0,0 +1,7 @@
+-- Coinductive currently does only work on Prop
+
+/-- error: Expected return type to be a Prop -/
+#guard_msgs in
+coinductive S (α : Type) : Type :=
+  | cons (hd : α) (tl : S α)
+