证明coq函数终止的一些帮助 [英] Some help proving coq function terminates
问题描述
我知道这是一个常见问题:)我将继续阅读,但是我做了一些搜索,但我还没有完全理解如何衡量"问题.有效
I know this is a common issue :) I will keep reading up, but I've done some searching and thing I don't fully understand how "measure" works
我正在完成本杰明·皮尔斯(Benjamin Pierce)的有关依赖类型的认证编程的课堂练习.这是我的代码.
I'm working through Benjamin Pierce's class exercises for Certified Programming with Dependent Types. Here's my code.
Inductive type : Type :=
| Nat
| Bool
| Pair : type -> type -> type.
Inductive tbinop : type -> type -> type -> Set :=
| TPlus : tbinop Nat Nat Nat
| TTimes : tbinop Nat Nat Nat
| TEq : forall t, tbinop t t Bool
| TLt : tbinop Nat Nat Bool
| TPair : forall in1 in2, tbinop in1 in2 (Pair in1 in2).
Inductive texp : type -> Set :=
| TNConst : nat -> texp Nat
| TBConst : bool -> texp Bool
| TBinop : forall t1 t2 t, tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
Fixpoint typeDenote (t : type) : Type :=
match t with
| Nat => nat
| Bool => bool
| Pair l r => prod (typeDenote l) (typeDenote r)
end.
Fixpoint typeDepth (t: type): nat :=
match t with
| Nat => 1
| Bool => 1
| Pair A B => 1 + Nat.max (typeDepth A) (typeDepth B)
end.
Program Fixpoint tbinopDepth arg1 arg2 res (b: tbinop arg1 arg2 res)
{measure (Nat.max (typeDepth arg1) (typeDepth arg2))}
: nat :=
match b with
| TPlus => 1
| TTimes => 1
| TEq Nat => 1
| TEq Bool => 1
| TEq (Pair A B) => tbinopDepth (TPair A B)
| TLt => 1
| TPair A B => 1 + Nat.max (typeDepth A) (typeDepth B)
end.
Next Obligation.
simpl.
rewrite Nat.max_idempotent.
omega.
Qed.
Eval compute in tbinopDepth (TEq (Pair Nat Nat)). (* 2 *)
Eval compute in tbinopDepth (TEq Nat). (* 1 *)
Program Fixpoint tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
{measure (tbinopDepth b)} : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
match b with
(*| TPlus => plus*)
| TPlus => fun (a:typeDenote Nat) (b:typeDenote Nat) => plus a b : typeDenote Nat
| TTimes => mult
| TEq Nat => beq_nat
| TEq Bool => eqb
| TEq (Pair A B) => fun (a:typeDenote (Pair A B)) (b:typeDenote (Pair A B)) =>
match a, b with
| (x1, x2), (y1, y2) => eqb (tbinopDenote (TEq A) x1 y1) (tbinopDenote (TEq B) x2 y2)
end : typeDenote Bool
| TLt => leb
| TPair _ _ => fun a b => (a,b)
end.
但是,当我尝试编译它时,出现类型错误.注意:如果有重组的方法可以避免证明这一点,那当然是理想的!我对此表示欢迎.就是说,我想了解我的措施出了什么问题.
However, when I try to compile this I get a type error. Note: if there are ways to restructure this to avoid having to prove this, of course that is ideal! And I welcome any suggestions in that vein. That said, I'd like to understand where I'm going wrong with my measure.
我收到这样的错误:
The term "x1" has type
"(fix typeDenote (t : type) : Type :=
match t with
| Nat => nat
| Bool => bool
| Pair l r => (typeDenote l * typeDenote r)%type
end) A" while it is expected to have type
"tbinopDepth (TEq A) < tbinopDepth b".
这就是为什么我认为很清楚我不太了解度量与代码如何交互的原因,因为我认为度量将产生证明义务,而不改变我定义的函数的类型.
Which is why I think it's clear I'm not quite understanding how the measure interacts with the code, as I thought the measure would generate a proof obligation, not change the type of the function I'm defining.
我应该补充一点,之所以包含两个评估值,是因为如果我能达到证明目标,则"tbinopDepth(TEq A)<tbinopDepth b"
是真的,因为我们知道b是 TEq(Pair AB)
,所以 tbinopDepth(TEq A)
和 tbinopDepth(TEqB)
小于那个.但这不会进行类型检查...
I should add that the reason I included the two Evals is because if I can get to a proof goal, "tbinopDepth (TEq A) < tbinopDepth b"
is true, since we know b is TEq (Pair A B)
so it's probable that tbinopDepth (TEq A)
and tbinopDepth (TEq B)
are smaller than that. But it won't typecheck...
推荐答案
您可以通过分别定义相等运算符来解决此问题:
You can solve this issue by defining the equality operator separately:
Require Import Coq.Arith.Arith.
Set Implicit Arguments.
Inductive type : Type :=
| Nat
| Bool
| Pair : type -> type -> type.
Inductive tbinop : type -> type -> type -> Set :=
| TPlus : tbinop Nat Nat Nat
| TTimes : tbinop Nat Nat Nat
| TEq : forall t, tbinop t t Bool
| TLt : tbinop Nat Nat Bool
| TPair : forall in1 in2, tbinop in1 in2 (Pair in1 in2).
Inductive texp : type -> Set :=
| TNConst : nat -> texp Nat
| TBConst : bool -> texp Bool
| TBinop : forall t1 t2 t, tbinop t1 t2 t -> texp t1 -> texp t2 -> texp t.
Fixpoint typeDenote (t : type) : Type :=
match t with
| Nat => nat
| Bool => bool
| Pair l r => prod (typeDenote l) (typeDenote r)
end.
Fixpoint typeDepth (t: type): nat :=
match t with
| Nat => 1
| Bool => 1
| Pair A B => 1 + Nat.max (typeDepth A) (typeDepth B)
end.
Fixpoint eqb arg : typeDenote arg -> typeDenote arg -> bool :=
match arg return typeDenote arg -> typeDenote arg -> bool with
| Nat => Nat.eqb
| Bool => Bool.eqb
| Pair A B => fun '(x1, y1) '(x2, y2) => andb (eqb _ x1 x2) (eqb _ y1 y2)
end.
Fixpoint tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res) {struct arg1}
: typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
match b in tbinop arg1 arg2 res return typeDenote arg1 -> typeDenote arg2 -> typeDenote res with
| TPlus => Nat.add
| TTimes => Nat.mul
| TEq arg => eqb arg
| TLt => leb
| TPair _ _ => fun a b => (a,b)
end.
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