notation ". r" with precedence 50 for @{'fi $r}.
interpretation "fi" 'fi r = (fi' ?? r).
-ndefinition and_morphism: unary_morphism1 CPROP (unary_morphism1_setoid1 CPROP CPROP).
- napply (mk_binary_morphism1 … And);
- #a; #a'; #b; #b'; #Ha; #Hb; @; *; #x; #y; @
- [ napply (. Ha^-1) | napply (. Hb^-1) | napply (. Ha) | napply (. Hb)] //.
+ndefinition and_morphism: binary_morphism1 CPROP CPROP CPROP.
+ napply mk_binary_morphism1
+ [ napply And
+ | #a; #a'; #b; #b'; *; #H1; #H2; *; #H3; #H4; napply mk_iff; *; #K1; #K2; napply conj
+ [ napply (H1 K1)
+ | napply (H3 K2)
+ | napply (H2 K1)
+ | napply (H4 K2)]##]
nqed.
-unification hint 0 ≔ A,B:CProp[0];
- T ≟ CPROP,
- MM ≟ mk_unary_morphism1 ??
- (λX.mk_unary_morphism1 ?? (And X) (prop11 ?? (fun11 ?? and_morphism X)))
- (prop11 ?? and_morphism)
-(*-------------------------------------------------------------*) ⊢
- fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ And A B.
-
-(*
-naxiom daemon: False.
-
-nlemma test: ∀A,A',B: CProp[0]. A=A' → (B ∨ A) = B → (B ∧ A) ∧ B.
- #A; #A'; #B; #H1; #H2; napply (. (#‡H1)‡H2^-1); nelim daemon.
+unification hint 0 ≔ A,B ⊢ fun21 … (mk_binary_morphism1 … And (prop21 … and_morphism)) A B ≡ And A B.
+
+(*nlemma test: ∀A,A',B: CProp[0]. A=A' → (B ∨ A) = B → (B ∧ A) ∧ B.
+ #A; #A'; #B; #H1; #H2;
+ napply (. ((#‡H1)‡H2^-1)); nnormalize;
+nqed.*)
+
+ndefinition or_morphism: binary_morphism1 CPROP CPROP CPROP.
+ napply mk_binary_morphism1
+ [ napply Or
+ | #a; #a'; #b; #b'; *; #H1; #H2; *; #H3; #H4; napply mk_iff; *; #H;
+ ##[##1,3: napply or_introl |##*: napply or_intror ]
+ ##[ napply (H1 H)
+ | napply (H2 H)
+ | napply (H3 H)
+ | napply (H4 H)]##]
nqed.
-CSC: ugly proof term
-ncheck test.
-*)
+unification hint 0 ≔ A,B ⊢ fun21 … (mk_binary_morphism1 … Or (prop21 … or_morphism)) A B ≡ Or A B.
-ndefinition or_morphism: unary_morphism1 CPROP (unary_morphism1_setoid1 CPROP CPROP).
- napply (mk_binary_morphism1 … Or);
- #a; #a'; #b; #b'; #Ha; #Hb; @; *; #x
- [ @1; napply (. Ha^-1) | @2; napply (. Hb^-1) | @1; napply (. Ha) | @2; napply (. Hb)] //.
+ndefinition if_morphism: binary_morphism1 CPROP CPROP CPROP.
+ napply mk_binary_morphism1
+ [ napply (λA,B. A → B)
+ | #a; #a'; #b; #b'; #H1; #H2; napply mk_iff; #H; #w
+ [ napply (if … H2); napply H; napply (fi … H1); nassumption
+ | napply (fi … H2); napply H; napply (if … H1); nassumption]##]
nqed.
-
-unification hint 0 ≔ A,B:CProp[0];
- T ≟ CPROP,
- MM ≟ mk_unary_morphism1 …
- (λX.mk_unary_morphism1 … (Or X) (prop11 … (fun11 ?? or_morphism X)))
- (prop11 … or_morphism)
-(*-------------------------------------------------------------*) ⊢
- fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ Or A B.
-
-(* XXX always applied, generates hard unif problems
-ndefinition if_morphism: unary_morphism1 CPROP (unary_morphism1_setoid1 CPROP CPROP).
- napply (mk_binary_morphism1 … (λA,B:CProp[0]. A → B));
- #a; #a'; #b; #b'; #Ha; #Hb; @; #H; #x
- [ napply (. Hb^-1); napply H; napply (. Ha) | napply (. Hb); napply H; napply (. Ha^-1)]
- //.
-nqed.
-
-unification hint 0 ≔ A,B:CProp[0];
- T ≟ CPROP,
- R ≟ mk_unary_morphism1 …
- (λX:CProp[0].mk_unary_morphism1 …
- (λY:CProp[0]. X → Y) (prop11 … (if_morphism X)))
- (prop11 … if_morphism)
-(*----------------------------------------------------------------------*) ⊢
- fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R A) B ≡ A → B.
-*)
-
-(* not as morphism *)
-nlemma Not_morphism : CProp[0] ⇒_1 CProp[0].
-@(λx:CProp[0].¬ x); #a b; *; #; @; /3/; nqed.
-
-unification hint 0 ≔ P : CProp[0];
- A ≟ CPROP,
- B ≟ CPROP,
- M ≟ mk_unary_morphism1 ?? (λP.¬ P) (prop11 ?? Not_morphism)
-(*------------------------*)⊢
- fun11 A B M P ≡ ¬ P.
-
-(* Ex setoid support *)
-
-(* The caml, as some patches for it
-ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid on _s: setoid to setoid1.
-*)
-
-(* simple case where the whole predicate can be rewritten *)
-nlemma Ex_morphism : ∀S:setoid.((setoid1_of_setoid S) ⇒_1 CProp[0]) ⇒_1 CProp[0].
-#S; @(λP: (setoid1_of_setoid S) ⇒_1 CProp[0].Ex S P);
-#P Q E; @; *; #x Px; @x; ncases (E x x #); /2/; nqed.
-
-unification hint 0 ≔ S : setoid, P : (setoid1_of_setoid S) ⇒_1 CPROP;
- A ≟ unary_morphism1_setoid1 (setoid1_of_setoid S) CPROP,
- B ≟ CPROP,
- M ≟ mk_unary_morphism1 ??
- (λP: (setoid1_of_setoid S) ⇒_1 CPROP.Ex (carr S) (fun11 ?? P))
- (prop11 ?? (Ex_morphism S))
-(*------------------------*)⊢
- fun11 A B M P ≡ Ex (carr S) (fun11 (setoid1_of_setoid S) CPROP P).
-
-nlemma Ex_morphism_eta : ∀S:setoid.((setoid1_of_setoid S) ⇒_1 CProp[0]) ⇒_1 CProp[0].
-#S; @(λP: (setoid1_of_setoid S) ⇒_1 CProp[0].Ex S (λx.P x));
-#P Q E; @; *; #x Px; @x; ncases (E x x #); /2/; nqed.
-
-unification hint 0 ≔ S : setoid, P : (setoid1_of_setoid S) ⇒_1 CPROP;
- A ≟ unary_morphism1_setoid1 (setoid1_of_setoid S) CPROP,
- B ≟ CPROP,
- M ≟ mk_unary_morphism1 ??
- (λP: (setoid1_of_setoid S) ⇒_1 CPROP.Ex (carr S) (λx.fun11 ?? P x))
- (prop11 ?? (Ex_morphism_eta S))
-(*------------------------*)⊢
- fun11 A B M P ≡ Ex (carr S) (λx.fun11 (setoid1_of_setoid S) CPROP P x).
-
-nlemma Ex_setoid : ∀S:setoid.((setoid1_of_setoid S) ⇒_1 CPROP) → setoid.
-#T P; @ (Ex T (λx:T.P x)); @; ##[ #H1 H2; napply True |##*: //; ##] nqed.
-
-unification hint 0 ≔ T : setoid,P ;
- S ≟ (Ex_setoid T P)
-(*---------------------------*) ⊢
- Ex (carr T) (λx:carr T.fun11 ?? P x) ≡ carr S.
-
-(* couts how many Ex we are traversing *)
-ninductive counter : Type[0] ≝
- | End : counter
- | Next : (Prop → Prop) → (* dummy arg please the notation mechanism *)
- counter → counter.
-
-(* to rewrite terms (live in setoid) *)
-nlet rec mk_P (S, T : setoid) (n : counter) on n ≝
- match n with [ End ⇒ T → CProp[0] | Next _ m ⇒ S → (mk_P S T m) ].
-
-nlet rec mk_F (S, T : setoid) (n : counter) on n ≝
- match n with [ End ⇒ T | Next _ m ⇒ S → (mk_F S T m) ].
-
-nlet rec mk_E (S, T : setoid) (n : counter) on n : ∀f,g : mk_F S T n. CProp[0] ≝
- match n with
- [ End ⇒ λf,g:T. f = g
- | Next q m ⇒ λf,g: mk_F S T (Next q m). ∀x:S.mk_E S T m (f x) (g x) ].
-
-nlet rec mk_H (S, T : setoid) (n : counter) on n :
-∀P1,P2: mk_P S T n.∀f,g : mk_F S T n. CProp[1] ≝
- match n with
- [ End ⇒ λP1,P2:mk_P S T End.λf,g:T. f = g → P1 f =_1 P2 g
- | Next q m ⇒ λP1,P2: mk_P S T (Next q m).λf,g: mk_F S T (Next q m).
- ∀x:S.mk_H S T m (P1 x) (P2 x) (f x) (g x) ].
-
-nlet rec mk_Ex (S, T : setoid) (n : counter) on n :
-∀P: mk_P S T n.∀f : mk_F S T n. CProp[0] ≝
- match n with
- [ End ⇒ λP:mk_P S T End.λf:T. P f
- | Next q m ⇒ λP: mk_P S T (Next q m).λf: mk_F S T (Next q m).
- ∃x:S.mk_Ex S T m (P x) (f x) ].
-
-nlemma Sig_generic : ∀S,T.∀n:counter.∀P,f,g.
- mk_E S T n f g → mk_H S T n P P f g → mk_Ex S T n P f =_1 mk_Ex S T n P g.
-#S T n; nelim n; nnormalize;
-##[ #P f g E H; /2/;
-##| #q m IH P f g E H; @; *; #x Px; @x; ncases (IH … (E x) (H x)); /3/; ##]
-nqed.
-
-(* to rewrite propositions (live in setoid1) *)
-nlet rec mk_P1 (S : setoid) (T : setoid1) (n : counter) on n ≝
- match n with [ End ⇒ T → CProp[0] | Next _ m ⇒ S → (mk_P1 S T m) ].
-
-nlet rec mk_F1 (S : setoid) (T : setoid1) (n : counter) on n ≝
- match n with [ End ⇒ T | Next _ m ⇒ S → (mk_F1 S T m) ].
-
-nlet rec mk_E1 (S : setoid) (T : setoid1) (n : counter) on n : ∀f,g : mk_F1 S T n. CProp[1] ≝
- match n with
- [ End ⇒ λf,g:T. f =_1 g
- | Next q m ⇒ λf,g: mk_F1 S T (Next q m). ∀x:S.mk_E1 S T m (f x) (g x) ].
-
-nlet rec mk_H1 (S : setoid) (T : setoid1) (n : counter) on n :
-∀P1,P2: mk_P1 S T n.∀f,g : mk_F1 S T n. CProp[1] ≝
- match n with
- [ End ⇒ λP1,P2:mk_P1 S T End.λf,g:T. f = g → P1 f =_1 P2 g
- | Next q m ⇒ λP1,P2: mk_P1 S T (Next q m).λf,g: mk_F1 S T (Next q m).
- ∀x:S.mk_H1 S T m (P1 x) (P2 x) (f x) (g x) ].
-
-nlet rec mk_Ex1 (S : setoid) (T : setoid1) (n : counter) on n :
-∀P: mk_P1 S T n.∀f : mk_F1 S T n. CProp[0] ≝
- match n with
- [ End ⇒ λP:mk_P1 S T End.λf:T. P f
- | Next q m ⇒ λP: mk_P1 S T (Next q m).λf: mk_F1 S T (Next q m).
- ∃x:S.mk_Ex1 S T m (P x) (f x) ].
-
-nlemma Sig_generic1 : ∀S,T.∀n:counter.∀P,f,g.
- mk_E1 S T n f g → mk_H1 S T n P P f g → mk_Ex1 S T n P f =_1 mk_Ex1 S T n P g.
-#S T n; nelim n; nnormalize;
-##[ #P f g E H; /2/;
-##| #q m IH P f g E H; @; *; #x Px; @x; ncases (IH … (E x) (H x)); /3/; ##]
-nqed.
-
-(* notation "∑x1,...,xn. E / H ; P" were:
- - x1...xn are bound in E and P, H is bound in P
- - H is an identifier that will have the type of E in P
- - P is the proof that the two existentially quantified predicates are equal*)
-notation > "∑ list1 ident x sep , . term 56 E / ident nE ; term 19 H" with precedence 20
-for @{ 'Sig_gen
- ${ fold right @{ 'End } rec acc @{ ('Next (λ${ident x}.${ident x}) $acc) } }
- ${ fold right @{ $E } rec acc @{ λ${ident x}.$acc } }
- ${ fold right @{ λ${ident nE}.$H } rec acc @{ λ${ident x}.$acc } }
-}.
-
-interpretation "next" 'Next x y = (Next x y).
-interpretation "end" 'End = End.
-interpretation "sig_gen" 'Sig_gen n E H = (Sig_generic ?? n ??? E H).
-interpretation "sig_gen1" 'Sig_gen n E H = (Sig_generic1 ?? n ??? E H).
-
-(*
-nlemma test0 : ∀S:setoid. ∀P: (setoid1_of_setoid S) ⇒_1 CPROP.∀f,g:S → S.
- (∀x:S.f x = g x) → (Ex S (λw.P (f w))) =_1 (Ex S (λw.P (g w))).
-#S P f g E; napply (∑w. E w / H ; ┼_1H); nqed.
-
-nlemma test : ∀S:setoid. ∀P: (setoid1_of_setoid S) ⇒_1 CPROP.∀f,g:S → S.
- (∀x:S.f x = g x) → (Ex S (λw.P (f w)∧ True)) =_1 (Ex S (λw.P (g w)∧ True)).
-#S P f g E; napply (∑w. E w / H ; (┼_1H)╪_1#); nqed.
-
-nlemma test_bound : ∀S:setoid. ∀e,f: (setoid1_of_setoid S) ⇒_1 CPROP. e = f →
- (Ex S (λw.e w ∧ True)) =_1 (Ex S (λw.f w ∧ True)).
-#S f g E; napply (.=_1 ∑x. E x x # / H ; (H ╪_1 #)); //; nqed.
-
-nlemma test2 : ∀S:setoid. ∀ee: (setoid1_of_setoid S) ⇒_1 (setoid1_of_setoid S) ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y →
- (True ∧ (Ex S (λw.ee x w ∧ True))) =_1 (True ∧ (Ex S (λw.ee y w ∧ True))).
-#S m x y E; napply (.=_1 #╪_1(∑w. E / E ; ((E ╪_1 #) ╪_1 #))). //; nqed.
-
-nlemma test3 : ∀S:setoid. ∀ee: (setoid1_of_setoid S) ⇒_1 (setoid1_of_setoid S) ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y →
- ((Ex S (λw.ee x w ∧ True) ∨ True)) =_1 ((Ex S (λw.ee y w ∧ True) ∨ True)).
-#S m x y E; napply (.=_1 (∑w. E / E ; ((E ╪_1 #) ╪_1 #)) ╪_1 #). //; nqed.
-*)
-
\ No newline at end of file
projects / helm.git / blobdiff