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正如標題所暗示的，我的問題的關注證明在勒柯克算術表達式的惰性評估的正確性和整體性。我想證明的定理總共有三種：懶惰計算的正確性與總體（勒柯克）
計算中只給出規範 表達式作爲結果
定理Only_canonical_results： （FORALL X Y：AEXP，比較x和y - >規範Y）。
正確性：計算關係 保留表達式的外延
定理correct_wrt_semantics： （forall的x和y：AEXP，小樣X Y - > I N（外延X）（外延Y））。
每個輸入導致了一些結果。定理Comp_is_total：（forall x：Aexp， （Sigma Aexp（fun y => prod（Comp x y）（Canonical y））））。
必要的定義見下面的代碼。當談到Coq時，我應該說清楚我是一個新手;更有經驗的用戶可能會馬上注意到。在大多數情況下，或者甚至所有我編寫的背景材料都可以在標準庫中找到，這是絕對的情況。但是，如果我確切地知道從標準庫導入什麼以證明理想結果，我很可能不會在這裏困擾你。這就是爲什麼我向你提供我迄今爲止的材料，希望有一些善良的人可以幫助我。謝謝！
(* Sigma types *)
Inductive Sigma (A:Set)(B:A -> Set) :Set :=
Spair: forall a:A, forall b : B a,Sigma A B.
Definition E (A:Set)(B:A -> Set)
(C: Sigma A B -> Set)
(c: Sigma A B)
(d: (forall x:A, forall y:B x,
C (Spair A B x y))): C c :=
match c as c0 return (C c0) with
| Spair a b => d a b
end.
Definition project1 (A:Set)(B:A -> Set)(c: Sigma A B):=
E A B (fun z => A) c (fun x y => x).
(* Binary sum type *)
Inductive sum' (A B:Set):Set :=
inl': A -> sum' A B | inr': B -> sum' A B.
Print sum'_rect.
Definition D (A B : Set)(C: sum' A B -> Set)
(c: sum' A B)
(d: (forall x:A, C (inl' A B x)))
(e: (forall y:B, C (inr' A B y))): C c :=
match c as c0 return C c0 with
| inl' x => d x
| inr' y => e y
end.
(* Three useful finite sets *)
Inductive N_0: Set :=.
Definition R_0
(C:N_0 -> Set)
(c: N_0): C c :=
match c as c0 return (C c0) with
end.
Inductive N_1: Set := zero_1:N_1.
Definition R_1
(C:N_1 -> Set)
(c: N_1)
(d_zero: C zero_1): C c :=
match c as c0 return (C c0) with
| zero_1 => d_zero
end.
Inductive N_2: Set := zero_2:N_2 | one_2:N_2.
Definition R_2
(C:N_2 -> Set)
(c: N_2)
(d_zero: C zero_2)
(d_one: C one_2): C c :=
match c as c0 return (C c0) with
| zero_2 => d_zero
| one_2 => d_one
end.
(* Natural numbers *)
Inductive N:Set :=
zero: N | succ : N -> N.
Print N.
Print N_rect.
Definition R
(C:N -> Set)
(d: C zero)
(e: (forall x:N, C x -> C (succ x))):
(forall n:N, C n) :=
fix F (n: N): C n :=
match n as n0 return (C n0) with
| zero => d
| succ n0 => e n0 (F n0)
end.
(* Boolean to truth-value converter *)
Definition Tr (c:N_2) : Set :=
match c as c0 with
| zero_2 => N_0
| one_2 => N_1
end.
(* Identity type *)
Inductive I (A: Set)(x: A) : A -> Set :=
r : I A x x.
Print I_rect.
Theorem J
(A:Set)
(C: (forall x y:A,
forall z: I A x y, Set))
(d: (forall x:A, C x x (r A x)))
(a:A)(b:A)(c:I A a b): C a b c.
induction c.
apply d.
Defined.
(* functions are extensional wrt
identity types *)
Theorem I_I_extensionality (A B: Set)(f: A -> B):
(forall x y:A, I A x y -> I B (f x) (f y)).
Proof.
intros x y P.
induction P.
apply r.
Defined.
(* addition *)
Definition add (m n:N) : N
:= R (fun z=> N) m (fun x y => succ y) n.
(* multiplication *)
Definition mul (m n:N) : N
:= R (fun z=> N) zero (fun x y => add y m) n.
(* Axioms of Peano verified *)
Theorem P1a: (forall x: N, I N (add x zero) x).
intro x.
(* force use of definitional equality
by applying reflexivity *)
apply r.
Defined.
Theorem P1b: (forall x y: N,
I N (add x (succ y)) (succ (add x y))).
intros.
apply r.
Defined.
Theorem P2a: (forall x: N, I N (mul x zero) zero).
intros.
apply r.
Defined.
Theorem P2b: (forall x y: N,
I N (mul x (succ y)) (add (mul x y) x)).
intros.
apply r.
Defined.
Definition pd (n: N): N :=
R (fun _=> N) zero (fun x y=> x) n.
(* alternatively
Definition pd (x: N): N :=
match x as x0 with
| zero => zero
| succ n0 => n0
end.
*)
Theorem P3: (forall x y:N,
I N (succ x) (succ y) -> I N x y).
intros x y p.
apply (I_I_extensionality N N pd (succ x) (succ y)).
apply p.
Defined.
Definition not (A:Set): Set:= (A -> N_0).
Definition isnonzero (n: N): N_2:=
R (fun _ => N_2) zero_2 (fun x y => one_2) n.
Theorem P4 : (forall x:N,
not (I N (succ x) zero)).
intro x.
intro p.
apply (J N (fun x y z =>
Tr (isnonzero x) -> Tr (isnonzero y))
(fun x => (fun t => t)) (succ x) zero)
.
apply p.
simpl.
apply zero_1.
Defined.
Theorem P5 (P:N -> Set):
P zero -> (forall x:N, P x -> P (succ x))
-> (forall x:N, P x).
intros base step n.
apply R.
apply base.
apply step.
Defined.
(* I(A,-,-) is an equivalence relation *)
Lemma Ireflexive (A:Set): (forall x:A, I A x x).
intro x.
apply r.
Defined.
Lemma Isymmetric (A:Set): (forall x y:A, I A x y -> I A y x).
intros x y P.
induction P.
apply r.
Defined.
Lemma Itransitive (A:Set):
(forall x y z:A, I A x y -> I A y z -> I A x z).
intros x y z P Q.
induction P.
assumption.
Defined.
Definition or (A B : Set):= sum' A B.
(* arithmetical expressions *)
Inductive Aexp :Set :=
zer: Aexp
| suc: Aexp -> Aexp
| pls: Aexp -> Aexp -> Aexp.
(* denotation of an expression *)
Definition denotation: Aexp->N:=
fix F (a: Aexp): N :=
match a as a0 with
| zer => zero
| suc a1 => succ (F a1)
| pls a1 a2 => add (F a1) (F a2)
end.
(* predicate for distinguishing
canonical expressions *)
Definition Canonical (x:Aexp):Set :=
or (I Aexp x zer)
(Sigma Aexp (fun y =>
I Aexp x (suc y))).
(* the computation relation is
an inductively defined relation *)
Inductive Comp : Aexp -> Aexp -> Set
:=
refrule: forall a: Aexp,
forall p: Canonical a, Comp a a
| zerrule: forall a b c:Aexp,
forall p: Comp b zer,
forall q: Comp a c,
Comp (pls a b) c
| sucrule: forall a b c:Aexp,
forall p: Comp b (suc c),
Comp (pls a b) (suc (pls a c)).
(* Computations only give canonical
expressions as results *)
Theorem Only_canonical_results:
(forall x y: Aexp, Comp x y -> Canonical y).
admit.
Defined.
(* Here is where help is needed *)
(* Correctness: the computation relation
preserves denotation of expressions *)
Theorem correct_wrt_semantics:
(forall x y: Aexp, Comp x y ->
I N (denotation x) (denotation y)).
admit.
(* Here is where help is need*)
Defined.
(* every input leads to some result *)
Theorem Comp_is_total: (forall x:Aexp,
(Sigma Aexp (fun y =>
prod (Comp x y) (Canonical y)))).
admit.
(* Proof required *)
Defined.
看來你知道'_rect'消除器Coq爲你的歸納數據類型自動定義，但是你重新實現了它們。這是爲了教育目的嗎？ – Ptival
@Ptival正確！我正在學習Coq，所以我認爲這將是一個很好的練習，可以自己寫出定義，而不是簡單地導入它們。 – user111731