import Lean import Lean.Meta.Tactic.Simp import Init.Data.List.Basic import Base.Utils import Base.Diverge.Base import Base.Diverge.ElabBase namespace Diverge /- Automating the generation of the encoding and the proofs so as to use nice syntactic sugar. -/ syntax (name := divergentDef) declModifiers "divergent" "def" declId ppIndent(optDeclSig) declVal : command open Lean Elab Term Meta Primitives Lean.Meta open Utils def normalize_let_bindings := true /- The following was copied from the `wfRecursion` function. -/ open WF in -- TODO: use those def UnitType := Expr.const ``PUnit [Level.succ .zero] def UnitValue := Expr.const ``PUnit.unit [Level.succ .zero] def mkProdType (x y : Expr) : MetaM Expr := mkAppM ``Prod #[x, y] def mkProd (x y : Expr) : MetaM Expr := mkAppM ``Prod.mk #[x, y] def mkInOutTy (x y z : Expr) : MetaM Expr := do mkAppM ``FixII.mk_in_out_ty #[x, y, z] -- Return the `a` in `Result a` def getResultTy (ty : Expr) : MetaM Expr := ty.withApp fun f args => do if ¬ f.isConstOf ``Result ∨ args.size ≠ 1 then throwError "Invalid argument to getResultTy: {ty}" else pure (args.get! 0) /- Deconstruct a sigma type. For instance, deconstructs `(a : Type) × List a` into `Type` and `λ a => List a`. -/ def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do ty.withApp fun f args => do if ¬ f.isConstOf ``Sigma ∨ args.size ≠ 2 then throwError "Invalid argument to getSigmaTypes: {ty}" else pure (args.get! 0, args.get! 1) /- Make a sigma type. `x` should be a variable, and `ty` and type which (might) uses `x` -/ def mkSigmaType (x : Expr) (sty : Expr) : MetaM Expr := do trace[Diverge.def.sigmas] "mkSigmaType: {x} {sty}" let alpha ← inferType x let beta ← mkLambdaFVars #[x] sty trace[Diverge.def.sigmas] "mkSigmaType: ({alpha}) ({beta})" mkAppOptM ``Sigma #[some alpha, some beta] /- Generate a Sigma type from a list of *variables* (all the expressions must be variables). Example: - xl = [(a:Type), (ls:List a), (i:Int)] Generates: `(a:Type) × (ls:List a) × (i:Int)` -/ def mkSigmasType (xl : List Expr) : MetaM Expr := match xl with | [] => do trace[Diverge.def.sigmas] "mkSigmasType: []" pure (Expr.const ``PUnit [Level.succ .zero]) | [x] => do trace[Diverge.def.sigmas] "mkSigmasType: [{x}]" let ty ← inferType x pure ty | x :: xl => do trace[Diverge.def.sigmas] "mkSigmasType: [{x}::{xl}]" let sty ← mkSigmasType xl mkSigmaType x sty /- Generate a product type from a list of *variables* (this is similar to `mkSigmas`). Example: - xl = [(ls:List a), (i:Int)] Generates: `List a × Int` -/ def mkProdsType (xl : List Expr) : MetaM Expr := match xl with | [] => do trace[Diverge.def.prods] "mkProdsType: []" pure (Expr.const ``PUnit [Level.succ .zero]) | [x] => do trace[Diverge.def.prods] "mkProdsType: [{x}]" let ty ← inferType x pure ty | x :: xl => do trace[Diverge.def.prods] "mkProdsType: [{x}::{xl}]" let ty ← inferType x let xl_ty ← mkProdsType xl mkAppM ``Prod #[ty, xl_ty] /- Split the input arguments between the types and the "regular" arguments. We do something simple: we treat an input argument as an input type iff it appears in the type of the following arguments. Note that what really matters is that we find the arguments which appear in the output type. Also, we stop at the first input that we treat as an input type. -/ def splitInputArgs (in_tys : Array Expr) (out_ty : Expr) : MetaM (Array Expr × Array Expr) := do -- Look for the first parameter which appears in the subsequent parameters let rec splitAux (in_tys : List Expr) : MetaM (HashSet FVarId × List Expr × List Expr) := match in_tys with | [] => do let fvars ← getFVarIds (← inferType out_ty) pure (fvars, [], []) | ty :: in_tys => do let (fvars, in_tys, in_args) ← splitAux in_tys -- Have we already found where to split between type variables/regular -- variables? if ¬ in_tys.isEmpty then -- The fvars set is now useless: no need to update it anymore pure (fvars, ty :: in_tys, in_args) else -- Check if ty appears in the set of free variables: let ty_id := ty.fvarId! if fvars.contains ty_id then -- We must split here. Note that we don't need to update the fvars -- set: it is not useful anymore pure (fvars, [ty], in_args) else -- We must split later: update the fvars set let fvars := fvars.insertMany (← getFVarIds (← inferType ty)) pure (fvars, [], ty :: in_args) let (_, in_tys, in_args) ← splitAux in_tys.data pure (Array.mk in_tys, Array.mk in_args) /- Apply a lambda expression to some arguments, simplifying the lambdas -/ def applyLambdaToArgs (e : Expr) (xs : Array Expr) : MetaM Expr := do lambdaTelescopeN e xs.size fun vars body => -- Create the substitution let s : HashMap FVarId Expr := HashMap.ofList (List.zip (vars.toList.map Expr.fvarId!) xs.toList) -- Substitute in the body pure (body.replace fun e => match e with | Expr.fvar fvarId => match s.find? fvarId with | none => e | some v => v | _ => none) /- Group a list of expressions into a dependent tuple. Example: xl = [`a : Type`, `ls : List a`] returns: `⟨ (a:Type), (ls: List a) ⟩` We need the type argument because as the elements in the tuple are "concrete", we can't in all generality figure out the type of the tuple. Example: `⟨ True, 3 ⟩ : (x : Bool) × (if x then Int else Unit)` -/ def mkSigmasVal (ty : Expr) (xl : List Expr) : MetaM Expr := match xl with | [] => do trace[Diverge.def.sigmas] "mkSigmasVal: []" pure (Expr.const ``PUnit.unit [Level.succ .zero]) | [x] => do trace[Diverge.def.sigmas] "mkSigmasVal: [{x}]" pure x | fst :: xl => do trace[Diverge.def.sigmas] "mkSigmasVal: [{fst}::{xl}]" -- Deconstruct the type let (alpha, beta) ← getSigmaTypes ty -- Compute the "second" field -- Specialize beta for fst let nty ← applyLambdaToArgs beta #[fst] -- Recursive call let snd ← mkSigmasVal nty xl -- Put everything together trace[Diverge.def.sigmas] "mkSigmasVal:\n{alpha}\n{beta}\n{fst}\n{snd}" mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd] /- Group a list of expressions into a (non-dependent) tuple -/ def mkProdsVal (xl : List Expr) : MetaM Expr := match xl with | [] => pure (Expr.const ``PUnit.unit [Level.succ .zero]) | [x] => do pure x | x :: xl => do let xl ← mkProdsVal xl mkAppM ``Prod.mk #[x, xl] def mkAnonymous (s : String) (i : Nat) : Name := .num (.str .anonymous s) i /- Given a list of values `[x0:ty0, ..., xn:ty1]`, where every `xi` might use the previous `xj` (j < i) and a value `out` which uses `x0`, ..., `xn`, generate the following expression: ``` fun x:((x0:ty0) × ... × (xn:tyn) => -- **Dependent** tuple match x with | (x0, ..., xn) => out ``` The `index` parameter is used for naming purposes: we use it to numerotate the bound variables that we introduce. We use this function to currify functions (the function bodies given to the fixed-point operator must be unary functions). Example: ======== - xl = `[a:Type, ls:List a, i:Int]` - out = `a` - index = 0 generates (getting rid of most of the syntactic sugar): ``` λ scrut0 => match scrut0 with | Sigma.mk x scrut1 => match scrut1 with | Sigma.mk ls i => a ``` -/ partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : MetaM Expr := match xl with | [] => do -- This would be unexpected throwError "mkSigmasMatch: empty list of input parameters" | [x] => do -- In the example given for the explanations: this is the inner match case trace[Diverge.def.sigmas] "mkSigmasMatch: [{x}]" mkLambdaFVars #[x] out | fst :: xl => do /- In the example given for the explanations: this is the outer match case Remark: for the naming purposes, we use the same convention as for the fields and parameters in `Sigma.casesOn` and `Sigma.mk` (looking at those definitions might help) We want to build the match expression: ``` λ scrut => match scrut with | Sigma.mk x ... -- the hole is given by a recursive call on the tail ``` -/ trace[Diverge.def.sigmas] "mkSigmasMatch: [{fst}::{xl}]" let alpha ← inferType fst let snd_ty ← mkSigmasType xl let beta ← mkLambdaFVars #[fst] snd_ty let snd ← mkSigmasMatch xl out (index + 1) let mk ← mkLambdaFVars #[fst] snd -- Introduce the "scrut" variable let scrut_ty ← mkSigmaType fst snd_ty withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do trace[Diverge.def.sigmas] "mkSigmasMatch: scrut: ({scrut}) : ({← inferType scrut})" -- TODO: make the computation of the motive more efficient let motive ← do let out_ty ← inferType out match out_ty with | .sort _ | .lit _ | .const .. => -- The type of the motive doesn't depend on the scrutinee mkLambdaFVars #[scrut] out_ty | _ => -- The type of the motive *may* depend on the scrutinee -- TODO: make this more efficient (we could change the output type of -- mkSigmasMatch mkSigmasMatch (fst :: xl) out_ty -- The final expression: putting everything together trace[Diverge.def.sigmas] "mkSigmasMatch:\n ({alpha})\n ({beta})\n ({motive})\n ({scrut})\n ({mk})" let sm ← mkAppOptM ``Sigma.casesOn #[some alpha, some beta, some motive, some scrut, some mk] -- Abstracting the "scrut" variable let sm ← mkLambdaFVars #[scrut] sm trace[Diverge.def.sigmas] "mkSigmasMatch: sm: {sm}" pure sm /- This is similar to `mkSigmasMatch`, but with non-dependent tuples Remark: factor out with `mkSigmasMatch`? This is extremely similar. -/ partial def mkProdsMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : MetaM Expr := match xl with | [] => do -- This would be unexpected throwError "mkProdsMatch: empty list of input parameters" | [x] => do -- In the example given for the explanations: this is the inner match case trace[Diverge.def.prods] "mkProdsMatch: [{x}]" mkLambdaFVars #[x] out | fst :: xl => do trace[Diverge.def.prods] "mkProdsMatch: [{fst}::{xl}]" let alpha ← inferType fst let beta ← mkProdsType xl let snd ← mkProdsMatch xl out (index + 1) let mk ← mkLambdaFVars #[fst] snd -- Introduce the "scrut" variable let scrut_ty ← mkProdType alpha beta withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do trace[Diverge.def.prods] "mkProdsMatch: scrut: ({scrut}) : ({← inferType scrut})" -- TODO: make the computation of the motive more efficient let motive ← do let out_ty ← inferType out mkLambdaFVars #[scrut] out_ty -- The final expression: putting everything together trace[Diverge.def.prods] "mkProdsMatch:\n ({alpha})\n ({beta})\n ({motive})\n ({scrut})\n ({mk})" let sm ← mkAppOptM ``Prod.casesOn #[some alpha, some beta, some motive, some scrut, some mk] -- Abstracting the "scrut" variable let sm ← mkLambdaFVars #[scrut] sm trace[Diverge.def.prods] "mkProdsMatch: sm: {sm}" pure sm /- Same as `mkSigmasMatch` but also accepts an empty list of inputs, in which case it generates the expression: ``` λ () => e ``` -/ def mkSigmasMatchOrUnit (xl : List Expr) (out : Expr) : MetaM Expr := if xl.isEmpty then do let scrut_ty := Expr.const ``PUnit [Level.succ .zero] withLocalDeclD (mkAnonymous "scrut" 0) scrut_ty fun scrut => do mkLambdaFVars #[scrut] out else mkSigmasMatch xl out /- Same as `mkProdsMatch` but also accepts an empty list of inputs, in which case it generates the expression: ``` λ () => e ``` -/ def mkProdsMatchOrUnit (xl : List Expr) (out : Expr) : MetaM Expr := if xl.isEmpty then do let scrut_ty := Expr.const ``PUnit [Level.succ .zero] withLocalDeclD (mkAnonymous "scrut" 0) scrut_ty fun scrut => do mkLambdaFVars #[scrut] out else mkProdsMatch xl out /- Small tests for list_nth: give a model of what `mkSigmasMatch` should generate -/ private def list_nth_out_ty_inner (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) := @Sigma.casesOn (List a) (fun (_ls : List a) => Int) (fun (_scrut1:@Sigma (List a) (fun (_ls : List a) => Int)) => Type) scrut1 (fun (_ls : List a) (_i : Int) => Primitives.Result a) private def list_nth_out_ty_outer (scrut0 : @Sigma (Type) (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))) := @Sigma.casesOn (Type) (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int)) (fun (_scrut0:@Sigma (Type) (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))) => Type) scrut0 (fun (a : Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) => list_nth_out_ty_inner a scrut1) /- -/ -- Return the expression: `Fin n` -- TODO: use more def mkFin (n : Nat) : Expr := mkAppN (.const ``Fin []) #[.lit (.natVal n)] -- Return the expression: `i : Fin n` def mkFinVal (n i : Nat) : MetaM Expr := do let n_lit : Expr := .lit (.natVal (n - 1)) let i_lit : Expr := .lit (.natVal i) mkAppOptM ``Fin.ofNat #[.some n_lit, .some i_lit] /- Information about the type of a function in a declaration group. In the comments about the fields, we take as example the `list_nth (α : Type) (ls : List α) (i : Int) : Result α` function. -/ structure TypeInfo where /- The total number of input arguments. For list_nth: 3 -/ total_num_args : ℕ /- The number of type parameters (they should be a prefix of the input arguments). For `list_nth`: 1 -/ num_params : ℕ /- The type of the dependent tuple grouping the type parameters. For `list_nth`: `Type` -/ params_ty : Expr /- The type of the tuple grouping the input values. This is a function taking as input a value of type `params_ty`. For `list_nth`: `λ a => List a × Int` -/ in_ty : Expr /- The output type, without the `Result`. This is a function taking as input a value of type `params_ty`. For `list_nth`: `λ a => a` -/ out_ty : Expr def mkInOutTyFromTypeInfo (info : TypeInfo) : MetaM Expr := do mkInOutTy info.params_ty info.in_ty info.out_ty instance : Inhabited TypeInfo := { default := { total_num_args := 0, num_params := 0, params_ty := UnitType, in_ty := UnitType, out_ty := UnitType } } instance : ToMessageData TypeInfo := ⟨ λ ⟨ total_num_args, num_params, params_ty, in_ty, out_ty ⟩ => f!"\{ total_num_args: {total_num_args}, num_params: {num_params}, params_ty: {params_ty}, in_ty: {in_ty}, out_ty: {out_ty} }}" ⟩ /- Generate and declare as individual definitions the bodies for the individual funcions: - replace the recursive calls with calls to the continutation `k` - make those bodies take one single dependent tuple as input We name the declarations: "[original_name].body". We return the new declarations. -/ def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr) (paramInOutTys : Array TypeInfo) (preDefs : Array PreDefinition) : MetaM (Array Expr) := do let grSize := preDefs.size /- Compute the map from name to (index, type info). Remark: the continuation has an indexed type; we use the index (a finite number of type `Fin`) to control which function we call at the recursive call site. -/ let nameToInfo : HashMap Name (Nat × TypeInfo) := let bl := preDefs.mapIdx fun i d => (d.declName, (i.val, paramInOutTys.get! i.val)) HashMap.ofList bl.toList trace[Diverge.def.genBody] "nameToId: {nameToInfo.toList}" -- Auxiliary function to explore the function bodies and replace the -- recursive calls let visit_e (i : Nat) (e : Expr) : MetaM Expr := do trace[Diverge.def.genBody.visit] "visiting expression (dept: {i}): {e}" let ne ← do match e with | .app .. => do e.withApp fun f args => do trace[Diverge.def.genBody.visit] "this is an app: {f} {args}" -- Check if this is a recursive call if f.isConst then let name := f.constName! match nameToInfo.find? name with | none => pure e | some (id, type_info) => trace[Diverge.def.genBody.visit] "this is a recursive call" -- This is a recursive call: replace it -- Compute the index let i ← mkFinVal grSize id -- It can happen that there are no input values given to the -- recursive calls, and only type parameters. let num_args := args.size if num_args ≠ type_info.total_num_args ∧ num_args ≠ type_info.num_params then throwError "Invalid number of arguments for the recursive call: {e}" -- Split the arguments, and put them in two tuples (the first -- one is a dependent tuple) let (param_args, args) := args.toList.splitAt type_info.num_params trace[Diverge.def.genBody.visit] "param_args: {param_args}, args: {args}" let param_args ← mkSigmasVal type_info.params_ty param_args -- Check if there are input values if num_args = type_info.total_num_args then do trace[Diverge.def.genBody.visit] "Recursive call with input values" let args ← mkProdsVal args mkAppM' kk_var #[i, param_args, args] else do trace[Diverge.def.genBody.visit] "Recursive call without input values" mkAppM' kk_var #[i, param_args] else -- Not a recursive call: do nothing pure e | .const name _ => /- This might refer to the one of the top-level functions if we use it without arguments (if we give it to a higher-order function for instance) and there are actually no type parameters. -/ if (nameToInfo.find? name).isSome then do -- Checking the type information match nameToInfo.find? name with | none => pure e | some (id, type_info) => trace[Diverge.def.genBody.visit] "this is a recursive call" -- This is a recursive call: replace it -- Compute the index let i ← mkFinVal grSize id -- Check that there are no type parameters if type_info.num_params ≠ 0 then throwError "mkUnaryBodies: a recursive call was not eliminated" -- We might be in a degenerate case, if the function takes no arguments -- at all (i.e., the function is a constant). -- For instance: -- ``` -- divergent def infinite_loop : Result Unit := infinite_loop -- `` if type_info.total_num_args == 0 then do trace[Diverge.def.genBody.visit] "Degenerate case: total_num_args=0" mkAppM' kk_var #[i, UnitValue, UnitValue] else do -- Introduce the call to the continuation let param_args ← mkSigmasVal type_info.params_ty [] mkAppM' kk_var #[i, param_args] else pure e | _ => pure e trace[Diverge.def.genBody.visit] "done with expression (depth: {i}): {e}" pure ne -- Explore the bodies preDefs.mapM fun preDef => do -- Replace the recursive calls trace[Diverge.def.genBody] "About to replace recursive calls in {preDef.declName}" let body ← mapVisit visit_e preDef.value trace[Diverge.def.genBody] "Body after replacement of the recursive calls: {body}" -- Currify the function by grouping the arguments into dependent tuples -- (over which we match to retrieve the individual arguments). lambdaTelescope body fun args body => do -- Split the arguments between the type parameters and the "regular" inputs let (_, type_info) := nameToInfo.find! preDef.declName let (param_args, args) := args.toList.splitAt type_info.num_params let body ← mkProdsMatchOrUnit args body trace[Diverge.def.genBody] "Body after mkProdsMatchOrUnit: {body}" let body ← mkSigmasMatchOrUnit param_args body trace[Diverge.def.genBody] "Body after mkSigmasMatchOrUnit: {body}" -- Add the declaration let value ← mkLambdaFVars #[kk_var] body trace[Diverge.def.genBody] "Body after abstracting kk: {value}" let name := preDef.declName.append "body" let levelParams := grLvlParams let decl := Declaration.defnDecl { name := name levelParams := levelParams type := ← inferType value -- TODO: change the type value := value hints := ReducibilityHints.regular (getMaxHeight (← getEnv) value + 1) safety := .safe all := [name] } trace[Diverge.def.genBody] "About to add decl" addDecl decl trace[Diverge.def] "individual body of {preDef.declName}: {body}" -- Return the constant let body := Lean.mkConst name (levelParams.map .param) trace[Diverge.def] "individual body (after decl): {body}" pure body /- Generate a unique function body from the bodies of the mutually recursive group, and add it as a declaration in the context. We return the list of bodies (of type `FixI.Funs ...`) and the mutually recursive body. -/ def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name) (kk_var i_var : Expr) (param_ty in_ty out_ty : Expr) (paramInOutTys : Array TypeInfo) (bodies : Array Expr) : MetaM (Expr × Expr) := do -- Generate the body let grSize := bodies.size let finTypeExpr := mkFin grSize -- TODO: not very clean let paramInOutTyType ← do let info := paramInOutTys.get! 0 inferType (← mkInOutTyFromTypeInfo info) let rec mkFuns (paramInOutTys : List TypeInfo) (bl : List Expr) : MetaM Expr := match paramInOutTys, bl with | [], [] => mkAppOptM ``FixII.Funs.Nil #[finTypeExpr, param_ty, in_ty, out_ty] | info :: paramInOutTys, b :: bl => do let pty := info.params_ty let ity := info.in_ty let oty := info.out_ty -- Retrieving ity and oty - this is not very clean let paramInOutTysExpr ← mkListLit paramInOutTyType (← paramInOutTys.mapM mkInOutTyFromTypeInfo) let fl ← mkFuns paramInOutTys bl mkAppOptM ``FixII.Funs.Cons #[finTypeExpr, param_ty, in_ty, out_ty, pty, ity, oty, paramInOutTysExpr, b, fl] | _, _ => throwError "mkDeclareMutRecBody: `tys` and `bodies` don't have the same length" let bodyFuns ← mkFuns paramInOutTys.toList bodies.toList -- Wrap in `get_fun` let body ← mkAppM ``FixII.get_fun #[bodyFuns, i_var, kk_var] -- Add the index `i` and the continuation `k` as a variables let body ← mkLambdaFVars #[kk_var, i_var] body trace[Diverge.def] "mkDeclareMutRecBody: body: {body}" -- Add the declaration let name := grName.append "mut_rec_body" let levelParams := grLvlParams let decl := Declaration.defnDecl { name := name levelParams := levelParams type := ← inferType body value := body hints := ReducibilityHints.regular (getMaxHeight (← getEnv) body + 1) safety := .safe all := [name] } addDecl decl -- Return the bodies and the constant pure (bodyFuns, Lean.mkConst name (levelParams.map .param)) def isCasesExpr (e : Expr) : MetaM Bool := do let e := e.getAppFn if e.isConst then return isCasesOnRecursor (← getEnv) e.constName else return false structure MatchInfo where matcherName : Name matcherLevels : Array Level params : Array Expr motive : Expr scruts : Array Expr branchesNumParams : Array Nat branches : Array Expr instance : ToMessageData MatchInfo where -- This is not a very clean formatting, but we don't need more toMessageData := fun me => m!"\n- matcherName: {me.matcherName}\n- params: {me.params}\n- motive: {me.motive}\n- scruts: {me.scruts}\n- branchesNumParams: {me.branchesNumParams}\n- branches: {me.branches}" /- Small helper: prove that an expression which doesn't use the continuation `kk` is valid, and return the proof. -/ def proveNoKExprIsValid (k_var : Expr) (e : Expr) : MetaM Expr := do trace[Diverge.def.valid] "proveNoKExprIsValid: {e}" let eIsValid ← mkAppM ``FixII.is_valid_p_same #[k_var, e] trace[Diverge.def.valid] "proveNoKExprIsValid: result:\n{eIsValid}:\n{← inferType eIsValid}" pure eIsValid mutual /- Prove that an expression is valid, and return the proof. More precisely, if `e` is an expression which potentially uses the continution `kk`, return an expression of type: ``` is_valid_p k (λ kk => e) ``` -/ partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do trace[Diverge.def.valid] "proveExprIsValid: {e}" -- Normalize to eliminate the lambdas - TODO: this is slightly dangerous. let e ← do if e.isLet ∧ normalize_let_bindings then do let updt_config (config : Lean.Meta.Config) := { config with transparency := .reducible } let e ← withConfig updt_config (whnf e) trace[Diverge.def.valid] "e (after normalization): {e}" pure e else pure e match e with | .const _ _ => throwError "Unimplemented" -- Shouldn't get there? | .bvar _ | .fvar _ | .lit _ | .mvar _ | .sort _ => throwError "Unreachable" | .lam .. => throwError "Unimplemented" -- TODO | .forallE .. => throwError "Unreachable" -- Shouldn't get there | .letE .. => do -- Remark: this branch is not taken if we normalize the expressions (above) -- Telescope all the let-bindings (remark: this also telescopes the lambdas) lambdaLetTelescope e fun xs body => do -- Note that we don't visit the bound values: there shouldn't be -- recursive calls, lambda expressions, etc. inside -- Prove that the body is valid let isValid ← proveExprIsValid k_var kk_var body -- Add the let-bindings around. -- Rem.: the let-binding should be *inside* the `is_valid_p`, not outside, -- but because it reduces in the end it doesn't matter. More precisely: -- `P (let x := v in y)` and `let x := v in P y` reduce to the same expression. mkLambdaFVars xs isValid (usedLetOnly := false) | .mdata _ b => proveExprIsValid k_var kk_var b | .proj _ _ _ => -- The projection shouldn't use the continuation proveNoKExprIsValid k_var e | .app .. => e.withApp fun f args => do proveAppIsValid k_var kk_var e f args partial def proveAppIsValid (k_var kk_var : Expr) (e : Expr) (f : Expr) (args : Array Expr): MetaM Expr := do trace[Diverge.def.valid] "proveAppIsValid: {e}\nDecomposed: {f} {args}" /- There are several cases: first, check if this is a match/if Check if the expression is a (dependent) if then else. We treat the if then else expressions differently from the other matches, and have dedicated theorems for them. -/ let isIte := e.isIte if isIte || e.isDIte then do e.withApp fun f args => do trace[Diverge.def.valid] "ite/dite: {f}:\n{args}" if args.size ≠ 5 then throwError "Wrong number of parameters for {f}: {args}" let cond := args.get! 1 let dec := args.get! 2 -- Prove that the branches are valid let br0 := args.get! 3 let br1 := args.get! 4 let proveBranchValid (br : Expr) : MetaM Expr := if isIte then proveExprIsValid k_var kk_var br else do -- There is a lambda lambdaOne br fun x br => do let brValid ← proveExprIsValid k_var kk_var br mkLambdaFVars #[x] brValid let br0Valid ← proveBranchValid br0 let br1Valid ← proveBranchValid br1 let const := if isIte then ``FixII.is_valid_p_ite else ``FixII.is_valid_p_dite let eIsValid ← mkAppOptM const #[none, none, none, none, none, some k_var, some cond, some dec, none, none, some br0Valid, some br1Valid] trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}" pure eIsValid /- Check if the expression is a match (this case is for when the elaborator introduces auxiliary definitions to hide the match behind syntactic sugar): -/ else if let some me := ← matchMatcherApp? e then do trace[Diverge.def.valid] "matcherApp: - params: {me.params} - motive: {me.motive} - discrs: {me.discrs} - altNumParams: {me.altNumParams} - alts: {me.alts} - remaining: {me.remaining}" -- matchMatcherApp does all the work for us: we simply need to gather -- the information and call the auxiliary helper `proveMatchIsValid` if me.remaining.size ≠ 0 then throwError "MatcherApp: non empty remaining array: {me.remaining}" let me : MatchInfo := { matcherName := me.matcherName matcherLevels := me.matcherLevels params := me.params motive := me.motive scruts := me.discrs branchesNumParams := me.altNumParams branches := me.alts } proveMatchIsValid k_var kk_var me /- Check if the expression is a raw match (this case is for when the expression is a direct call to the primitive `casesOn` function, without syntactic sugar). We have to check this case because functions like `mkSigmasMatch`, which we use to currify function bodies, introduce such raw matches. -/ else if ← isCasesExpr f then do trace[Diverge.def.valid] "rawMatch: {e}" /- Deconstruct the match, and call the auxiliary helper `proveMatchIsValid`. The casesOn definition is always of the following shape: - input parameters (implicit parameters) - motive (implicit), -- the motive gives the return type of the match - scrutinee (explicit) - branches (explicit). In particular, we notice that the scrutinee is the first *explicit* parameter - this is how we spot it. -/ let matcherName := f.constName! let matcherLevels := f.constLevels!.toArray -- Find the first explicit parameter: this is the scrutinee forallTelescope (← inferType f) fun xs _ => do let rec findFirstExplicit (i : Nat) : MetaM Nat := do if i ≥ xs.size then throwError "Unexpected: could not find an explicit parameter" else let x := xs.get! i let xFVarId := x.fvarId! let localDecl ← xFVarId.getDecl match localDecl.binderInfo with | .default => pure i | _ => findFirstExplicit (i + 1) let scrutIdx ← findFirstExplicit 0 -- Split the arguments let params := args.extract 0 (scrutIdx - 1) let motive := args.get! (scrutIdx - 1) let scrut := args.get! scrutIdx let branches := args.extract (scrutIdx + 1) args.size /- Compute the number of parameters for the branches: for this we use the type of the uninstantiated casesOn constant (we can't just destruct the lambdas in the branch expressions because the result of a match might be a lambda expression). -/ let branchesNumParams : Array Nat ← do let env ← getEnv let decl := env.constants.find! matcherName let ty := decl.type forallTelescope ty fun xs _ => do let xs := xs.extract (scrutIdx + 1) xs.size xs.mapM fun x => do let xty ← inferType x forallTelescope xty fun ys _ => do pure ys.size let me : MatchInfo := { matcherName, matcherLevels, params, motive, scruts := #[scrut], branchesNumParams, branches, } proveMatchIsValid k_var kk_var me -- Check if this is a monadic let-binding else if f.isConstOf ``Bind.bind then do trace[Diverge.def.valid] "bind:\n{args}" -- We simply need to prove that the subexpressions are valid, and call -- the appropriate lemma. let x := args.get! 4 let y := args.get! 5 -- Prove that the subexpressions are valid let xValid ← proveExprIsValid k_var kk_var x trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}" let yValid ← do -- This is a lambda expression lambdaOne y fun x y => do trace[Diverge.def.valid] "bind: y: {y}" let yValid ← proveExprIsValid k_var kk_var y trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}" trace[Diverge.def.valid] "bind: yValid: x: {x}" let yValid ← mkLambdaFVars #[x] yValid trace[Diverge.def.valid] "bind: yValid (forall): {yValid}: {← inferType yValid}" pure yValid -- Put everything together trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}" mkAppM ``FixII.is_valid_p_bind #[xValid, yValid] -- Check if this is a recursive call, i.e., a call to the continuation `kk` else if f.isFVarOf kk_var.fvarId! then do trace[Diverge.def.valid] "rec: args: \n{args}" if args.size ≠ 3 then throwError "Recursive call with invalid number of parameters: {args}" let i_arg := args.get! 0 let t_arg := args.get! 1 let x_arg := args.get! 2 let eIsValid ← mkAppM ``FixII.is_valid_p_rec #[k_var, i_arg, t_arg, x_arg] trace[Diverge.def.valid] "rec: result: \n{eIsValid}" pure eIsValid else do /- Remaining case: normal application. Check if the arguments use the continuation: - if no: this is simple - if yes: we have to lookup theorems in div spec database and continue -/ trace[Diverge.def.valid] "regular app: {e}, f: {f}, args: {args}" let argsFVars ← args.mapM getFVarIds let allArgsFVars := argsFVars.foldl (fun hs fvars => hs.insertMany fvars) HashSet.empty trace[Diverge.def.valid] "allArgsFVars: {allArgsFVars.toList.map mkFVar}" if ¬ allArgsFVars.contains kk_var.fvarId! then do -- Simple case trace[Diverge.def.valid] "kk doesn't appear in the arguments" proveNoKExprIsValid k_var e else do -- Lookup in the database for suitable theorems trace[Diverge.def.valid] "kk appears in the arguments" let thms ← divspecAttr.find? e trace[Diverge.def.valid] "Looked up theorems: {thms}" -- Try the theorems one by one proveAppIsValidApplyThms k_var kk_var e f args thms.toList partial def proveAppIsValidApplyThms (k_var kk_var : Expr) (e : Expr) (f : Expr) (args : Array Expr) (thms : List Name) : MetaM Expr := do match thms with | [] => throwError "Could not prove that the following expression is valid: {e}" | thName :: thms => -- Lookup the theorem itself let env ← getEnv let thDecl := env.constants.find! thName -- Introduce fresh meta-variables for the universes let ul : List (Name × Level) ← thDecl.levelParams.mapM (λ x => do pure (x, ← mkFreshLevelMVar)) let ulMap : HashMap Name Level := HashMap.ofList ul let thTy := thDecl.type.instantiateLevelParamsCore (λ x => ulMap.find! x) trace[Diverge.def.valid] "Trying with theorem {thName}: {thTy}" -- Introduce meta variables for the universally quantified variables let (mvars, _binders, thTyBody) ← forallMetaTelescope thTy let thTermToMatch := thTyBody trace[Diverge.def.valid] "thTermToMatch: {thTermToMatch}" -- Create the term: `is_valid_p k (λ kk => e)` let termToMatch ← mkLambdaFVars #[kk_var] e let termToMatch ← mkAppM ``FixII.is_valid_p #[k_var, termToMatch] trace[Diverge.def.valid] "termToMatch: {termToMatch}" -- Attempt to match trace[Diverge.def.valid] "Matching terms:\n\n{termToMatch}\n\n{thTermToMatch}" let ok ← isDefEq termToMatch thTermToMatch if ¬ ok then -- Failure: attempt with the other theorems proveAppIsValidApplyThms k_var kk_var e f args thms else do /- Success: continue with this theorem Instantiate the meta variables (some of them will not be instantiated: they are new subgoals) -/ let mvars ← mvars.mapM instantiateMVars let th ← mkAppOptM thName (Array.map some mvars) trace[Diverge.def.valid] "Instantiated theorem: {th}\n{← inferType th}" -- Filter the instantiated meta variables let mvars := mvars.filter (fun v => v.isMVar) let mvars := mvars.map (fun v => v.mvarId!) trace[Diverge.def.valid] "Remaining subgoals: {mvars}" for mvarId in mvars do -- Prove the subgoal (i.e., the precondition of the theorem) let mvarDecl ← mvarId.getDecl let declType ← instantiateMVars mvarDecl.type -- Reduce the subgoal before diving in, if necessary trace[Diverge.def.valid] "Subgoal: {declType}" -- Dive in the type forallTelescope declType fun forall_vars mvar_e => do trace[Diverge.def.valid] "forall_vars: {forall_vars}" -- `mvar_e` should have the shape `is_valid_p k (λ kk => ...)` -- We need to retrieve the new `k` variable, and dive into the -- `λ kk => ...` mvar_e.consumeMData.withApp fun is_valid args => do if is_valid.constName? ≠ ``FixII.is_valid_p ∨ args.size ≠ 7 then throwError "Invalid precondition: {mvar_e}" else do let k_var := args.get! 5 let e_lam := args.get! 6 trace[Diverge.def.valid] "k_var: {k_var}\ne_lam: {e_lam}" -- The outer lambda should be for the kk_var lambdaOne e_lam.consumeMData fun kk_var e => do -- Continue trace[Diverge.def.valid] "kk_var: {kk_var}\ne: {e}" -- We sometimes need to reduce the term - TODO: this is really dangerous let e ← do let updt_config config := { config with transparency := .reducible } withConfig updt_config (whnf e) trace[Diverge.def.valid] "e (after normalization): {e}" let e_valid ← proveExprIsValid k_var kk_var e trace[Diverge.def.valid] "e_valid (for e): {e_valid}" let e_valid ← mkLambdaFVars forall_vars e_valid trace[Diverge.def.valid] "e_valid (with foralls): {e_valid}" let _ ← inferType e_valid -- Sanity check -- Assign the meta variable mvarId.assign e_valid pure th -- Prove that a match expression is valid. partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Expr := do trace[Diverge.def.valid] "proveMatchIsValid: {me}" -- Prove the validity of the branch expressions let branchesValid:Array Expr ← me.branches.mapIdxM fun idx br => do /- Go inside the lambdas - note that we have to be careful: some of the binders might come from the match, and some of the binders might come from the fact that the expression in the match is a lambda expression: we use the branchesNumParams field for this reason. -/ let numParams := me.branchesNumParams.get! idx lambdaTelescopeN br numParams fun xs br => do -- Prove that the branch expression is valid let brValid ← proveExprIsValid k_var kk_var br -- Reconstruct the lambda expression mkLambdaFVars xs brValid trace[Diverge.def.valid] "branchesValid:\n{branchesValid}" /- Compute the motive, which has the following shape: ``` λ scrut => is_valid_p k (λ k => match scrut with ...) ^^^^^^^^^^^^^^^^^^^^ this is the original match expression, with the the difference that the scrutinee(s) is a variable ``` -/ let validMotive : Expr ← do -- The motive is a function of the scrutinees (i.e., a lambda expression): -- introduce binders for the scrutinees let declInfos := me.scruts.mapIdx fun idx scrut => let name : Name := mkAnonymous "scrut" idx let ty := λ (_ : Array Expr) => inferType scrut (name, ty) withLocalDeclsD declInfos fun scrutVars => do -- Create a match expression but where the scrutinees have been replaced -- by variables let params : Array (Option Expr) := me.params.map some let motive : Option Expr := some me.motive let scruts : Array (Option Expr) := scrutVars.map some let branches : Array (Option Expr) := me.branches.map some let args := params ++ [motive] ++ scruts ++ branches let matchE ← mkAppOptM me.matcherName args -- Wrap in the `is_valid_p` predicate let matchE ← mkLambdaFVars #[kk_var] matchE let validMotive ← mkAppM ``FixII.is_valid_p #[k_var, matchE] -- Abstract away the scrutinee variables mkLambdaFVars scrutVars validMotive trace[Diverge.def.valid] "valid motive: {validMotive}" -- Put together let valid ← do -- We let Lean infer the parameters let params : Array (Option Expr) := me.params.map (λ _ => none) let motive := some validMotive let scruts := me.scruts.map some let branches := branchesValid.map some let args := params ++ [motive] ++ scruts ++ branches mkAppOptM me.matcherName args trace[Diverge.def.valid] "proveMatchIsValid:\n{valid}:\n{← inferType valid}" pure valid end /- Prove that a single body (in the mutually recursive group) is valid. For instance, if we define the mutually recursive group [`is_even`, `is_odd`], we prove that `is_even.body` and `is_odd.body` are valid. -/ partial def proveSingleBodyIsValid (k_var : Expr) (preDef : PreDefinition) (bodyConst : Expr) : MetaM Expr := do trace[Diverge.def.valid] "proveSingleBodyIsValid: bodyConst: {bodyConst}" -- Lookup the definition (`bodyConst` is a const, we want to retrieve its -- definition to dive inside) let name := bodyConst.constName! let env ← getEnv let body := (env.constants.find! name).value! trace[Diverge.def.valid] "body: {body}" lambdaTelescope body fun xs body => do trace[Diverge.def.valid] "xs: {xs}" if xs.size ≠ 3 then throwError "Invalid number of lambdas: {xs} (expected 3)" let kk_var := xs.get! 0 let t_var := xs.get! 1 let x_var := xs.get! 2 -- State the type of the theorem to prove trace[Diverge.def.valid] "bodyConst: {bodyConst} : {← inferType bodyConst}" let bodyApp ← mkAppOptM' bodyConst #[.some kk_var, .some t_var, .some x_var] trace[Diverge.def.valid] "bodyApp: {bodyApp}" let bodyApp ← mkLambdaFVars #[kk_var] bodyApp trace[Diverge.def.valid] "bodyApp: {bodyApp}" let thmTy ← mkAppM ``FixII.is_valid_p #[k_var, bodyApp] trace[Diverge.def.valid] "thmTy: {thmTy}" -- Prove that the body is valid trace[Diverge.def.valid] "body: {body}" let proof ← proveExprIsValid k_var kk_var body let proof ← mkLambdaFVars #[k_var, t_var, x_var] proof trace[Diverge.def.valid] "proveSingleBodyIsValid: proof:\n{proof}:\n{← inferType proof}" -- The target type (we don't have to do this: this is simply a sanity check, -- and this allows a nicer debugging output) let thmTy ← do let ty ← mkAppM ``FixII.is_valid_p #[k_var, bodyApp] mkForallFVars #[k_var, t_var, x_var] ty trace[Diverge.def.valid] "proveSingleBodyIsValid: thmTy\n{thmTy}:\n{← inferType thmTy}" -- Save the theorem let name := preDef.declName ++ "body_is_valid" let decl := Declaration.thmDecl { name levelParams := preDef.levelParams type := thmTy value := proof all := [name] } addDecl decl trace[Diverge.def.valid] "proveSingleBodyIsValid: added thm: {name}" -- Return the theorem pure (Expr.const name (preDef.levelParams.map .param)) /- Prove that the list of bodies are valid. For instance, if we define the mutually recursive group [`is_even`, `is_odd`], we prove that `Funs.Cons is_even.body (Funs.Cons is_odd.body Funs.Nil)` is valid. -/ partial def proveFunsBodyIsValid (paramInOutTys: Expr) (bodyFuns : Expr) (k_var : Expr) (bodiesValid : Array Expr) : MetaM Expr := do -- Create the big "and" expression, which groups the validity proof of the individual bodies let rec mkValidConj (i : Nat) : MetaM Expr := do if i = bodiesValid.size then -- We reached the end mkAppM ``FixII.Funs.is_valid_p_Nil #[k_var] else do -- We haven't reached the end: introduce a conjunction let valid := bodiesValid.get! i let valid ← mkAppM' valid #[k_var] mkAppM ``And.intro #[valid, ← mkValidConj (i + 1)] let andExpr ← mkValidConj 0 -- Wrap in the `is_valid_p_is_valid_p` theorem, and abstract the continuation let isValid ← mkAppM ``FixII.Funs.is_valid_p_is_valid_p #[paramInOutTys, k_var, bodyFuns, andExpr] mkLambdaFVars #[k_var] isValid /- Prove that the mut rec body (i.e., the unary body which groups the bodies of all the functions in the mutually recursive group and on which we will apply the fixed-point operator) is valid. We save the proof in the theorem "[GROUP_NAME]."mut_rec_body_is_valid", which we return. TODO: maybe this function should introduce k_var itself -/ def proveMutRecIsValid (grName : Name) (grLvlParams : List Name) (paramInOutTys : Expr) (bodyFuns mutRecBodyConst : Expr) (k_var : Expr) (preDefs : Array PreDefinition) (bodies : Array Expr) : MetaM Expr := do -- First prove that the individual bodies are valid let bodiesValid ← bodies.mapIdxM fun idx body => do let preDef := preDefs.get! idx trace[Diverge.def.valid] "## Proving that the body {body} is valid" proveSingleBodyIsValid k_var preDef body -- Then prove that the mut rec body is valid trace[Diverge.def.valid] "## Proving that the 'Funs' body is valid" let isValid ← proveFunsBodyIsValid paramInOutTys bodyFuns k_var bodiesValid trace[Diverge.def.valid] "Generated the term: {isValid}" -- Save the theorem let thmTy ← mkAppM ``FixII.is_valid #[mutRecBodyConst] let name := grName ++ "mut_rec_body_is_valid" let decl := Declaration.thmDecl { name levelParams := grLvlParams type := thmTy value := isValid all := [name] } addDecl decl trace[Diverge.def.valid] "proveFunsBodyIsValid: added thm: {name}:\n{thmTy}" -- Return the theorem pure (Expr.const name (grLvlParams.map .param)) /- Generate the final definions by using the mutual body and the fixed point operator. For instance: ``` def is_even (i : Int) : Result Bool := mut_rec_body 0 i def is_odd (i : Int) : Result Bool := mut_rec_body 1 i ``` -/ def mkDeclareFixDefs (mutRecBody : Expr) (paramInOutTys : Array TypeInfo) (preDefs : Array PreDefinition) : TermElabM (Array Name) := do let grSize := preDefs.size let defs ← preDefs.mapIdxM fun idx preDef => do lambdaTelescope preDef.value fun xs _ => do -- Retrieve the parameters info let type_info := paramInOutTys.get! idx.val -- Create the index let idx ← mkFinVal grSize idx.val -- Group the inputs into two tuples let (params_args, input_args) := xs.toList.splitAt type_info.num_params let params ← mkSigmasVal type_info.params_ty params_args let input ← mkProdsVal input_args -- Apply the fixed point let fixedBody ← mkAppM ``FixII.fix #[mutRecBody, idx, params, input] let fixedBody ← mkLambdaFVars xs fixedBody -- Create the declaration let name := preDef.declName let decl := Declaration.defnDecl { name := name levelParams := preDef.levelParams type := preDef.type value := fixedBody hints := ReducibilityHints.regular (getMaxHeight (← getEnv) fixedBody + 1) safety := .safe all := [name] } addDecl decl pure name pure defs -- Prove the equations that we will use as unfolding theorems partial def proveUnfoldingThms (isValidThm : Expr) (paramInOutTys : Array TypeInfo) (preDefs : Array PreDefinition) (decls : Array Name) : MetaM Unit := do let grSize := preDefs.size let proveIdx (i : Nat) : MetaM Unit := do let preDef := preDefs.get! i let defName := decls.get! i -- Retrieve the arguments lambdaTelescope preDef.value fun xs body => do trace[Diverge.def.unfold] "proveUnfoldingThms: xs: {xs}" trace[Diverge.def.unfold] "proveUnfoldingThms: body: {body}" -- The theorem statement let thmTy ← do -- The equation: the declaration gives the lhs, the pre-def gives the rhs let lhs ← mkAppOptM defName (xs.map some) let rhs := body let eq ← mkAppM ``Eq #[lhs, rhs] mkForallFVars xs eq trace[Diverge.def.unfold] "proveUnfoldingThms: thm statement: {thmTy}" -- The proof -- Use the fixed-point equation let proof ← mkAppM ``FixII.is_valid_fix_fixed_eq #[isValidThm] -- Add the index let idx ← mkFinVal grSize i let proof ← mkAppM ``congr_fun #[proof, idx] -- Add the input arguments let type_info := paramInOutTys.get! i let (params, args) := xs.toList.splitAt type_info.num_params let params ← mkSigmasVal type_info.params_ty params let args ← mkProdsVal args let proof ← mkAppM ``congr_fun #[proof, params] let proof ← mkAppM ``congr_fun #[proof, args] -- Abstract all the arguments away let proof ← mkLambdaFVars xs proof trace[Diverge.def.unfold] "proveUnfoldingThms: proof: {proof}:\n{← inferType proof}" -- Declare the theorem let name := preDef.declName ++ "unfold" let decl := Declaration.thmDecl { name levelParams := preDef.levelParams type := thmTy value := proof all := [name] } addDecl decl -- Add the unfolding theorem to the equation compiler eqnsAttribute.add preDef.declName #[name] trace[Diverge.def.unfold] "proveUnfoldingThms: added thm: {name}:\n{thmTy}" let rec prove (i : Nat) : MetaM Unit := do if i = preDefs.size then pure () else do proveIdx i prove (i + 1) -- prove 0 def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value) trace[Diverge.def] ("divRecursion: defs:\n" ++ msg) -- Apply all the "attribute" functions (for instance, the function which -- registers the theorem in the simp database if there is the `simp` attribute, -- etc.) for preDef in preDefs do applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation -- Retrieve the name of the first definition, that we will use as the namespace -- for the definitions common to the group let def0 := preDefs[0]! let grName := def0.declName trace[Diverge.def] "group name: {grName}" /- # Compute the input/output types of the continuation `k`. -/ let grLvlParams := def0.levelParams trace[Diverge.def] "def0 universe levels: {def0.levelParams}" /- We first compute the tuples: (type parameters × input type × output type) - type parameters: this is a sigma type - input type: λ params_type => product type - output type: λ params_type => output type For instance, on the function: `list_nth (α : Type) (ls : List α) (i : Int) : Result α`: we generate: `(Type, λ α => List α × i, λ α => Result α)` -/ let paramInOutTys : Array TypeInfo ← preDefs.mapM (fun preDef => do -- Check the universe parameters - TODO: I'm not sure what the best thing -- to do is. In practice, all the type parameters should be in Type 0, so -- we shouldn't have universe issues. if preDef.levelParams ≠ grLvlParams then throwError "Non-uniform polymorphism in the universes" forallTelescope preDef.type (fun in_tys out_ty => do let total_num_args := in_tys.size let (params, in_tys) ← splitInputArgs in_tys out_ty trace[Diverge.def] "Decomposed arguments: {preDef.declName}: {params}, {in_tys}, {out_ty}" let num_params := params.size let params_ty ← mkSigmasType params.data let in_ty ← mkSigmasMatchOrUnit params.data (← mkProdsType in_tys.data) -- Retrieve the type in the "Result" let out_ty ← getResultTy out_ty let out_ty ← mkSigmasMatchOrUnit params.data out_ty trace[Diverge.def] "inOutTy: {preDef.declName}: {params_ty}, {in_tys}, {out_ty}" pure ⟨ total_num_args, num_params, params_ty, in_ty, out_ty ⟩)) trace[Diverge.def] "paramInOutTys: {paramInOutTys}" -- Turn the list of input types/input args/output type tuples into expressions let paramInOutTysExpr ← liftM (paramInOutTys.mapM mkInOutTyFromTypeInfo) let paramInOutTysExpr ← mkListLit (← inferType (paramInOutTysExpr.get! 0)) paramInOutTysExpr.toList trace[Diverge.def] "paramInOutTys: {paramInOutTys}" /- From the list of pairs of input/output types, actually compute the type of the continuation `k`. We first introduce the index `i : Fin n` where `n` is the number of functions in the group. -/ let i_var_ty := mkFin preDefs.size withLocalDeclD (mkAnonymous "i" 0) i_var_ty fun i_var => do let param_in_out_ty ← mkAppM ``List.get #[paramInOutTysExpr, i_var] trace[Diverge.def] "param_in_out_ty := {param_in_out_ty} : {← inferType param_in_out_ty}" -- Add an auxiliary definition for `param_in_out_ty` (this is a potentially big term) let param_in_out_ty ← do let value ← mkLambdaFVars #[i_var] param_in_out_ty let name := grName.append "param_in_out_ty" let levelParams := grLvlParams let decl := Declaration.defnDecl { name := name levelParams := levelParams type := ← inferType value value := value hints := .abbrev safety := .safe all := [name] } addDecl decl -- Return the constant let param_in_out_ty := Lean.mkConst name (levelParams.map .param) mkAppM' param_in_out_ty #[i_var] trace[Diverge.def] "param_in_out_ty (after decl) := {param_in_out_ty} : {← inferType param_in_out_ty}" -- Decompose between: param_ty, in_ty, out_ty let param_ty ← mkAppM ``Sigma.fst #[param_in_out_ty] let in_out_ty ← mkAppM ``Sigma.snd #[param_in_out_ty] let in_ty ← mkAppM ``Prod.fst #[in_out_ty] let out_ty ← mkAppM ``Prod.snd #[in_out_ty] trace[Diverge.def] "param_ty: {param_ty}" trace[Diverge.def] "in_ty: {in_ty}" trace[Diverge.def] "out_ty: {out_ty}" withLocalDeclD (mkAnonymous "t" 1) param_ty fun param => do let in_ty ← mkAppM' in_ty #[param] let out_ty ← mkAppM' out_ty #[param] trace[Diverge.def] "in_ty: {in_ty}" trace[Diverge.def] "out_ty: {out_ty}" -- Introduce the continuation `k` let param_ty ← mkLambdaFVars #[i_var] param_ty let in_ty ← mkLambdaFVars #[i_var, param] in_ty let out_ty ← mkLambdaFVars #[i_var, param] out_ty let kk_var_ty ← mkAppM ``FixII.kk_ty #[i_var_ty, param_ty, in_ty, out_ty] trace[Diverge.def] "kk_var_ty: {kk_var_ty}" withLocalDeclD (mkAnonymous "kk" 2) kk_var_ty fun kk_var => do trace[Diverge.def] "kk_var: {kk_var}" -- Replace the recursive calls in all the function bodies by calls to the -- continuation `k` and and generate for those bodies declarations trace[Diverge.def] "# Generating the unary bodies" let bodies ← mkDeclareUnaryBodies grLvlParams kk_var paramInOutTys preDefs trace[Diverge.def] "Unary bodies (after decl): {bodies}" -- Generate the mutually recursive body trace[Diverge.def] "# Generating the mut rec body" let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var param_ty in_ty out_ty paramInOutTys bodies trace[Diverge.def] "mut rec body (after decl): {mutRecBody}" -- Prove that the mut rec body satisfies the validity criteria required by -- our fixed-point let k_var_ty ← mkAppM ``FixII.k_ty #[i_var_ty, param_ty, in_ty, out_ty] withLocalDeclD (mkAnonymous "k" 3) k_var_ty fun k_var => do trace[Diverge.def] "# Proving that the mut rec body is valid" let isValidThm ← proveMutRecIsValid grName grLvlParams paramInOutTysExpr bodyFuns mutRecBody k_var preDefs bodies -- Generate the final definitions trace[Diverge.def] "# Generating the final definitions" let decls ← mkDeclareFixDefs mutRecBody paramInOutTys preDefs -- Prove the unfolding theorems trace[Diverge.def] "# Proving the unfolding theorems" proveUnfoldingThms isValidThm paramInOutTys preDefs decls -- Generating code addAndCompilePartialRec preDefs -- The following function is copy&pasted from Lean.Elab.PreDefinition.Main -- This is the only part where we make actual changes and hook into the equation compiler. -- (I've removed all the well-founded stuff to make it easier to read though.) open private ensureNoUnassignedMVarsAtPreDef betaReduceLetRecApps partitionPreDefs addAndCompilePartial addAsAxioms from Lean.Elab.PreDefinition.Main def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLCtx {} {} do for preDef in preDefs do trace[Diverge.elab] "{preDef.declName} : {preDef.type} :=\n{preDef.value}" let preDefs ← preDefs.mapM ensureNoUnassignedMVarsAtPreDef let preDefs ← betaReduceLetRecApps preDefs let cliques := partitionPreDefs preDefs let mut hasErrors := false for preDefs in cliques do trace[Diverge.elab] "{preDefs.map (·.declName)}" try withRef (preDefs[0]!.ref) do divRecursion preDefs catch ex => -- If it failed, we add the functions as partial functions hasErrors := true logException ex let s ← saveState try if preDefs.all fun preDef => preDef.kind == DefKind.def || preDefs.all fun preDef => preDef.kind == DefKind.abbrev then -- try to add as partial definition try addAndCompilePartial preDefs (useSorry := true) catch _ => -- Compilation failed try again just as axiom s.restore addAsAxioms preDefs else return () catch _ => s.restore -- The following two functions are copy-pasted from Lean.Elab.MutualDef open private elabHeaders levelMVarToParamHeaders getAllUserLevelNames withFunLocalDecls elabFunValues instantiateMVarsAtHeader instantiateMVarsAtLetRecToLift checkLetRecsToLiftTypes withUsed from Lean.Elab.MutualDef def Term.elabMutualDef (vars : Array Expr) (views : Array DefView) : TermElabM Unit := do let scopeLevelNames ← getLevelNames let headers ← elabHeaders views let headers ← levelMVarToParamHeaders views headers let allUserLevelNames := getAllUserLevelNames headers withFunLocalDecls headers fun funFVars => do for view in views, funFVar in funFVars do addLocalVarInfo view.declId funFVar -- Add fake use site to prevent "unused variable" warning (if the -- function is actually not recursive, Lean would print this warning). -- Remark: we could detect this case and encode the function without -- using the fixed-point. In practice it shouldn't happen however: -- we define non-recursive functions with the `divergent` keyword -- only for testing purposes. addTermInfo' view.declId funFVar let values ← try let values ← elabFunValues headers Term.synthesizeSyntheticMVarsNoPostponing values.mapM (instantiateMVars ·) catch ex => logException ex headers.mapM fun header => mkSorry header.type (synthetic := true) let headers ← headers.mapM instantiateMVarsAtHeader let letRecsToLift ← getLetRecsToLift let letRecsToLift ← letRecsToLift.mapM instantiateMVarsAtLetRecToLift checkLetRecsToLiftTypes funFVars letRecsToLift withUsed vars headers values letRecsToLift fun vars => do let preDefs ← MutualClosure.main vars headers funFVars values letRecsToLift for preDef in preDefs do trace[Diverge.elab] "{preDef.declName} : {preDef.type} :=\n{preDef.value}" let preDefs ← withLevelNames allUserLevelNames <| levelMVarToParamPreDecls preDefs let preDefs ← instantiateMVarsAtPreDecls preDefs let preDefs ← fixLevelParams preDefs scopeLevelNames allUserLevelNames for preDef in preDefs do trace[Diverge.elab] "after eraseAuxDiscr, {preDef.declName} : {preDef.type} :=\n{preDef.value}" checkForHiddenUnivLevels allUserLevelNames preDefs addPreDefinitions preDefs open Command in def Command.elabMutualDef (ds : Array Syntax) : CommandElabM Unit := do let views ← ds.mapM fun d => do let `($mods:declModifiers divergent def $id:declId $sig:optDeclSig $val:declVal) := d | throwUnsupportedSyntax let modifiers ← elabModifiers mods let (binders, type) := expandOptDeclSig sig let deriving? := none pure { ref := d, kind := DefKind.def, modifiers, declId := id, binders, type? := type, value := val, deriving? } runTermElabM fun vars => Term.elabMutualDef vars views -- Special command so that we don't fall back to the built-in mutual when we produce an error. local syntax "_divergent" Parser.Command.mutual : command elab_rules : command | `(_divergent mutual $decls* end) => Command.elabMutualDef decls macro_rules | `(mutual $decls* end) => do unless !decls.isEmpty && decls.all (·.1.getKind == ``divergentDef) do Macro.throwUnsupported `(command| _divergent mutual $decls* end) open private setDeclIdName from Lean.Elab.Declaration elab_rules : command | `($mods:declModifiers divergent%$tk def $id:declId $sig:optDeclSig $val:declVal) => do let (name, _) := expandDeclIdCore id if (`_root_).isPrefixOf name then throwUnsupportedSyntax let view := extractMacroScopes name let .str ns shortName := view.name | throwUnsupportedSyntax let shortName' := { view with name := shortName }.review let cmd ← `(mutual $mods:declModifiers divergent%$tk def $(⟨setDeclIdName id shortName'⟩):declId $sig:optDeclSig $val:declVal end) if ns matches .anonymous then Command.elabCommand cmd else Command.elabCommand <| ← `(namespace $(mkIdentFrom id ns) $cmd end $(mkIdentFrom id ns)) namespace Tests /- Some examples of partial functions -/ --set_option trace.Diverge.def true --set_option trace.Diverge.def.genBody true --set_option trace.Diverge.def.valid true --set_option trace.Diverge.def.genBody.visit true divergent def list_nth {a: Type u} (ls : List a) (i : Int) : Result a := match ls with | [] => .fail .panic | x :: ls => do if i = 0 then pure x else pure (← list_nth ls (i - 1)) --set_option trace.Diverge false #check list_nth.unfold example {a: Type} (ls : List a) : ∀ (i : Int), 0 ≤ i → i < ls.length → ∃ x, list_nth ls i = .ok x := by induction ls . intro i hpos h; simp at h; linarith . rename_i hd tl ih intro i hpos h -- We can directly use `rw [list_nth]` rw [list_nth]; simp split <;> try simp [*] . tauto . -- We don't have to do this if we use scalar_tac have hneq : 0 < i := by cases i <;> rename_i a _ <;> simp_all; cases a <;> simp_all simp at h have ⟨ x, ih ⟩ := ih (i - 1) (by linarith) (by linarith) simp [ih] tauto -- Return a continuation divergent def list_nth_with_back {a: Type} (ls : List a) (i : Int) : Result (a × (a → Result (List a))) := match ls with | [] => .fail .panic | x :: ls => if i = 0 then return (x, (λ ret => return (ret :: ls))) else do let (x, back) ← list_nth_with_back ls (i - 1) return (x, (λ ret => do let ls ← back ret return (x :: ls))) #check list_nth_with_back.unfold mutual divergent def is_even (i : Int) : Result Bool := if i = 0 then return true else return (← is_odd (i - 1)) divergent def is_odd (i : Int) : Result Bool := if i = 0 then return false else return (← is_even (i - 1)) end #check is_even.unfold #check is_odd.unfold mutual divergent def foo (i : Int) : Result Nat := if i > 10 then return (← foo (i / 10)) + (← bar i) else bar 10 divergent def bar (i : Int) : Result Nat := if i > 20 then foo (i / 20) else .ok 42 end #check foo.unfold #check bar.unfold -- Testing dependent branching and let-bindings divergent def isNonZero (i : Int) : Result Bool := if _h:i = 0 then return false else let b := true return b #check isNonZero.unfold -- Testing let-bindings divergent def iInBounds {a : Type} (ls : List a) (i : Int) : Result Bool := let i0 := ls.length if i < i0 then Result.ok True else Result.ok False #check iInBounds.unfold divergent def isCons {a : Type} (ls : List a) : Result Bool := let ls1 := ls match ls1 with | [] => Result.ok False | _ :: _ => Result.ok True #check isCons.unfold -- Testing what happens when we use concrete arguments in dependent tuples divergent def test1 (_ : Option Bool) (_ : Unit) : Result Unit := test1 Option.none () #check test1.unfold -- Testing a degenerate case divergent def infinite_loop : Result Unit := do let _ ← infinite_loop Result.ok () #check infinite_loop.unfold -- Testing a degenerate case divergent def infinite_loop1 : Result Unit := infinite_loop1 #check infinite_loop1.unfold /- Tests with higher-order functions -/ section HigherOrder open Ex8 inductive Tree (a : Type u) := | leaf (x : a) | node (tl : List (Tree a)) divergent def id {a : Type u} (t : Tree a) : Result (Tree a) := match t with | .leaf x => .ok (.leaf x) | .node tl => do let tl ← map id tl .ok (.node tl) #check id.unfold divergent def id1 {a : Type u} (t : Tree a) : Result (Tree a) := match t with | .leaf x => .ok (.leaf x) | .node tl => do let tl ← map (fun x => id1 x) tl .ok (.node tl) #check id1.unfold divergent def id2 {a : Type u} (t : Tree a) : Result (Tree a) := match t with | .leaf x => .ok (.leaf x) | .node tl => do let tl ← map (fun x => do let _ ← id2 x; id2 x) tl .ok (.node tl) #check id2.unfold divergent def incr (t : Tree Nat) : Result (Tree Nat) := match t with | .leaf x => .ok (.leaf (x + 1)) | .node tl => do let tl ← map incr tl .ok (.node tl) -- We handle this by inlining the let-binding divergent def id3 (t : Tree Nat) : Result (Tree Nat) := match t with | .leaf x => .ok (.leaf (x + 1)) | .node tl => do let f := id3 let tl ← map f tl .ok (.node tl) #check id3.unfold /- -- This is not handled yet: we can only do it if we introduce "general" -- relations for the input types and output types (result_rel should -- be parameterized by something). divergent def id4 (t : Tree Nat) : Result (Tree Nat) := match t with | .leaf x => .ok (.leaf (x + 1)) | .node tl => do let f ← .ok id4 let tl ← map f tl .ok (.node tl) #check id4.unfold -/ end HigherOrder end Tests end Diverge