(**************************************************************************)
include "lambda/types.ma".
+include "lambda/lambda_notation.ma".
(* MATTER CONCERNING STRONG NORMALIZATION TO BE PUT ELSEWHERE *****************)
-(* from sn.ma *****************************************************************)
+(* arithmetics ****************************************************************)
+
+theorem arith1: ∀x,y. (S x) ≰ (S y) → x ≰ y.
+#x #y #HS @nmk (elim HS) -HS /3/
+qed.
+
+theorem arith2: ∀i,p,k. k ≤ i → i + p - (k + p) = i - k.
+#i #p #k #H @plus_to_minus
+>commutative_plus >(commutative_plus k) >associative_plus @eq_f /2/
+qed.
+
+theorem arith3: ∀x,y,z. x ≰ y → x + z ≰ y + z.
+#x #y #z #H @nmk (elim H) -H /3/
+qed.
+
+(* lists **********************************************************************)
(* all(P,l) holds when P holds for all members of l *)
-let rec all (P:T→Prop) l on l ≝ match l with
- [ nil ⇒ True
- | cons A D ⇒ P A ∧ all P D
+let rec all (A:Type[0]) (P:A→Prop) l on l ≝ match l with
+ [ nil ⇒ True
+ | cons hd tl ⇒ P hd ∧ all A P tl
].
+theorem all_append: ∀A,P,l2,l1. all A P l1 → all A P l2 → all A P (l1 @ l2).
+#A #P #l2 #l1 (elim l1) -l1 (normalize) // #hd #tl #IH1 #H (elim H) /3/
+qed.
+
(* all(?,P,l1,l2) holds when P holds for all paired members of l1 and l2 *)
let rec all2 (A:Type[0]) (P:A→A→Prop) l1 l2 on l1 ≝ match l1 with
[ nil ⇒ l2 = nil ?
]
].
+theorem length_append: ∀A. ∀(l2,l1:list A). |l1@l2| = |l1| + |l2|.
+#A #l2 #l1 (elim l1) -l1 (normalize) //
+qed.
+
+(* terms **********************************************************************)
+
(* Appl F l generalizes App applying F to a list of arguments
* The head of l is applied first
*)
| cons A D ⇒ Appl (App F A) D
].
+theorem appl_append: ∀N,l,M. Appl M (l @ [N]) = App (Appl M l) N.
+#N #l (elim l) -l // #hd #tl #IHl #M >IHl //
+qed.
+
+(* FG: not needed for now
+(* nautral terms *)
+inductive neutral: T → Prop ≝
+ | neutral_sort: ∀n.neutral (Sort n)
+ | neutral_rel: ∀i.neutral (Rel i)
+ | neutral_app: ∀M,N.neutral (App M N)
+.
+*)
+
+(* substitution ***************************************************************)
+
(* FG: do we need this?
definition lift0 ≝ λp,k,M . lift M p k. (**) (* remove definition *)
qed.
*)
-(* from rc.ma *****************************************************************)
-
-theorem arith1: ∀x,y. (S x) ≰ (S y) → x ≰ y.
-#x #y #HS @nmk (elim HS) -HS /3/
-qed.
-
-theorem arith2: ∀i,p,k. k ≤ i → i + p - (k + p) = i - k.
-#i #p #k #H @plus_to_minus
->commutative_plus >(commutative_plus k) >associative_plus @eq_f /2/
-qed.
-
-theorem arith3: ∀x,y,z. x ≰ y → x + z ≰ y + z.
-#x #y #z #H @nmk (elim H) -H /3/
-qed.
-
-theorem length_append: ∀A. ∀(l2,l1:list A). |l1@l2| = |l1| + |l2|.
-#A #l2 #l1 (elim l1) -l1 (normalize) //
-qed.
-
theorem lift_rel_lt: ∀i,p,k. (S i) ≤ k → lift (Rel i) k p = Rel i.
#i #p #k #Hik normalize >(le_to_leb_true … Hik) //
qed.
| cons A D ⇒ (lift (substc M[0≝A] D) 0 1)
].
-notation "M [ l ]" non associative with precedence 90 for @{'Substc $M $l}.
-
-interpretation "Substc" 'Substc M l = (substc M l).
+interpretation "Substc" 'Subst1 M l = (substc M l).
(* this is just to test that substitution works as expected
theorem test1: ∀A,B,C. (App (App (Rel 0) (Rel 1)) (Rel 2))[A::B::C::nil ?] =
theorem substc_sort: ∀n,l. (Sort n)[l] = Sort n.
//
qed.
-(* FG: not needed for now
-(* nautral terms *)
-inductive neutral: T → Prop ≝
- | neutral_sort: ∀n.neutral (Sort n)
- | neutral_rel: ∀i.neutral (Rel i)
- | neutral_app: ∀M,N.neutral (App M N)
-.
-*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE LAMBDA CALCULUS *******************************************)
+
+(* lifting, substitution *)
+
+notation "hvbox(M break [ l ])"
+ non associative with precedence 90
+ for @{'Subst1 $M $l}.
+
+(* evaluation, interpretation *)
+
+notation "hvbox(〚term 90 T〛)"
+ non associative with precedence 50
+ for @{'Eval $T}.
+
+notation "hvbox(〚term 90 T〛 break _ [term 90 E])"
+ non associative with precedence 50
+ for @{'Eval1 $T $E}.
+
+notation "hvbox(〚term 90 T〛 break _ [term 90 E1 break , term 90 E2])"
+ non associative with precedence 50
+ for @{'Eval2 $T $E1 $E2}.
#M #C1 #C2 #H1 #H12 (elim (H12 M)) -H12 /2/
qed.
-(* NOTE: hd_repl and tl_repl are proved essentially by the same script *)
+(* NOTE: hd_repl and tl_repl are proved essentially by the same script *)
theorem hd_repl: ∀C1,C2. C1 ≈ C2 → ∀l1,l2. l1 ≈ l2 → hd ? l1 C1 ≈ hd ? l2 C2.
#C1 #C2 #QC #l1 (elim l1) -l1 [ #l2 #Q >Q // ]
#hd1 #tl1 #_ #l2 (elim l2) -l2 [ #Q elim Q ]
definition dep_mem ≝ λB,C,M. ∀N. N ∈ B → App M N ∈ C.
-axiom dep_cr1: ∀B,C. CR1 (dep_mem B C).
+theorem dep_cr1: ∀B,C. CR1 (dep_mem B C).
+#B #C #M #Hdep (lapply (Hdep (Sort 0) ?)) /2 by SAT0_sort/ /3/ (**) (* adiacent auto *)
+qed.
-axiom dep_sat0: ∀B,C. SAT0 (dep_mem B C).
+theorem dep_sat0: ∀B,C. SAT0 (dep_mem B C).
+/5/ qed.
-axiom dep_sat1: ∀B,C. SAT1 (dep_mem B C).
+theorem dep_sat1: ∀B,C. SAT1 (dep_mem B C).
+/5/ qed.
-axiom dep_sat2: ∀B,C. SAT2 (dep_mem B C).
+(* NOTE: @sat2 is not needed if append_cons is enabled *)
+theorem dep_sat2: ∀B,C. SAT2 (dep_mem B C).
+#B #C #N #L #M #l #HN #HL #HM #K #HK <appl_append @sat2 /2/
+qed.
definition depRC: RC → RC → RC ≝ λB,C. mk_RC (dep_mem B C) ….
/2/ qed.
include "lambda/ext.ma".
-(* saturation conditions on an explicit subset ********************************)
+(* STRONGLY NORMALIZING TERMS *************************************************)
+
+(* SN(t) holds when t is strongly normalizing *)
+(* FG: we axiomatize it for now because we dont have reduction yet *)
+axiom SN: T → Prop.
-definition SAT0 ≝ λP. ∀l,n. all P l → P (Appl (Sort n) l).
+(* lists of strongly normalizing terms *)
+definition SNl ≝ all ? SN.
-definition SAT1 ≝ λP. ∀l,i. all P l → P (Appl (Rel i) l).
+(* saturation conditions ******************************************************)
-definition SAT2 ≝ λ(P:?→Prop). ∀F,A,B,l. P B → P A →
- P (Appl F[0:=A] l) → P (Appl (Lambda B F) (A::l)).
+definition CR1 ≝ λ(P:?→Prop). ∀M. P M → SN M.
+
+definition SAT0 ≝ λ(P:?→Prop). ∀n,l. SNl l → P (Appl (Sort n) l).
+
+definition SAT1 ≝ λ(P:?->Prop). ∀i,l. SNl l → P (Appl (Rel i) l).
+
+definition SAT2 ≝ λ(P:?→Prop). ∀N,L,M,l. SN N → SN L →
+ P (Appl M[0:=L] l) → P (Appl (Lambda N M) (L::l)).
theorem SAT0_sort: ∀P,n. SAT0 P → P (Sort n).
-#P #n #H @(H (nil ?) …) //
+#P #n #H @(H n (nil ?) …) //
qed.
theorem SAT1_rel: ∀P,i. SAT1 P → P (Rel i).
-#P #i #H @(H (nil ?) …) //
+#P #i #H @(H i (nil ?) …) //
qed.
-(* STRONGLY NORMALIZING TERMS *************************************************)
-
-(* SN(t) holds when t is strongly normalizing *)
-(* FG: we axiomatize it for now because we dont have reduction yet *)
-axiom SN: T → Prop.
-
-definition CR1 ≝ λ(P:?→Prop). ∀M. P M → SN M.
+(* axiomatization *************************************************************)
axiom sn_sort: SAT0 SN.
axiom sn_rel: SAT1 SN.
-axiom sn_lambda: ∀B,F. SN B → SN F → SN (Lambda B F).
-
axiom sn_beta: SAT2 SN.
-(* FG: do we need this?
-axiom sn_lift: ∀t,k,p. SN t → SN (lift t p k).
-*)
+axiom sn_lambda: ∀N,M. SN N → SN M → SN (Lambda N M).
+
+axiom sn_prod: ∀N,M. SN N → SN M → SN (Prod N M).
+
+axiom sn_dummy: ∀M. SN M → SN (D M).
+
+axiom sn_inv_app_1: ∀M,N. SN (App M N) → SN M.
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