axiom Raa : ∀A.(Not A → Bot) → A.
+axiom sort : Type.
+
+inductive Exists (A:Type) (P:A→CProp) : CProp ≝
+ Exists_intro: ∀w:A. P w → Exists A P.
+
+definition Exists_elim ≝
+ λA:Type.λP:A→CProp.λC:CProp.λc:Exists A P.λH:(Πx.P x → C).
+ match c with [ Exists_intro w p ⇒ H w p ].
+
+inductive Forall (A:Type) (P:A→CProp) : CProp ≝
+ Forall_intro: (∀n:A. P n) → Forall A P.
+
+definition Forall_elim ≝
+ λA:Type.λP:A→CProp.λn:A.λf:Forall A P.match f with [ Forall_intro g ⇒ g n ].
+
(* Dummy proposition *)
axiom unit : CProp.
interpretation "Not" 'not a = (Not a).
notation "✶" non associative with precedence 90 for @{'unit}.
interpretation "dummy prop" 'unit = unit.
-
+notation > "\exists list1 ident x sep , . term 19 Px" with precedence 20
+for ${ fold right @{$Px} rec acc @{'myexists (λ${ident x}.$acc)} }.
+notation < "hvbox(\exists ident i break . p)" with precedence 20
+for @{ 'myexists (\lambda ${ident i} : $ty. $p) }.
+interpretation "constructive ex" 'myexists \eta.x = (Exists sort x).
+notation > "\forall ident x.break term 19 Px" with precedence 20
+for @{ 'Forall (λ${ident x}.$Px) }.
+notation < "\forall ident x.break term 19 Px" with precedence 20
+for @{ 'Forall (λ${ident x}:$tx.$Px) }.
+interpretation "Forall" 'Forall \eta.Px = (Forall _ Px).
+
(* Variables *)
axiom A : CProp.
axiom B : CProp.
axiom M : CProp.
axiom N : CProp.
axiom O : CProp.
-axiom P : CProp.
-axiom Q : CProp.
-axiom R : CProp.
-axiom S : CProp.
-axiom T : CProp.
-axiom U : CProp.
-axiom V : CProp.
-axiom W : CProp.
-axiom X : CProp.
-axiom Y : CProp.
-axiom Z : CProp.
+axiom P : sort →CProp.
+axiom Q : sort →CProp.
+axiom R : sort →sort →CProp.
+axiom S : sort →sort →CProp.
+axiom f : sort → sort.
+axiom g : sort → sort.
+axiom h : sort → sort → sort.
(* Every formula user provided annotates its proof:
`A` becomes `(show A ?)` *)
-definition show : ∀A.A→A ≝ λA:CProp.λa:A.a.
+definition show : ΠA.A→A ≝ λA:CProp.λa:A.a.
(* When something does not fit, this daemon is used *)
-axiom cast: ∀A,B:CProp.B → A.
+axiom cast: ΠA,B:CProp.B → A.
(* begin a proof: draws the root *)
notation > "'prove' p" non associative with precedence 19
for @{ 'leaf_ko $a $t }.
interpretation "leaf KO" 'leaf_ko a t = (cast _ _ (show a t)).
+(* already proved lemma *)
+definition Lemma : ΠA.A→A ≝ λA:CProp.λa:A.a.
+
+notation < "\infrule \nbsp p mstyle color #ff0000 (\lem\emsp H)" non associative with precedence 19
+for @{ 'lemma_ko_1 $p $H }.
+interpretation "lemma_ko_1" 'lemma_ko_1 p H =
+ (show p (cast _ _ (Lemma _ H))).
+
+notation < "\infrule \nbsp mstyle color #ff0000 (p) mstyle color #ff0000 (\lem\emsp H)" non associative with precedence 19
+for @{ 'lemma_ko_2 $p $H }.
+interpretation "lemma_ko_2" 'lemma_ko_2 p H =
+ (cast _ _ (show p (cast _ _ (Lemma _ H)))).
+
+notation < "\infrule \nbsp p (\lem\emsp H)" non associative with precedence 19
+for @{ 'lemma_ok_1 $p $H }.
+interpretation "lemma_ok_1" 'lemma_ok_1 p H =
+ (show p (Lemma _ H)).
+interpretation "lemma_ok_11" 'lemma_ok_1 p H =
+ (show p (Lemma _ H _)).
+interpretation "lemma_ok_111" 'lemma_ok_1 p H =
+ (show p (Lemma _ H _ _)).
+
+notation < "\infrule \nbsp mstyle color #ff0000 (p) (\lem\emsp H)" non associative with precedence 19
+for @{ 'lemma_ok_2 $p $H }.
+interpretation "lemma_ok_2" 'lemma_ok_2 p H =
+ (cast _ _ (show p (Lemma _ H))).
+interpretation "lemma_ok_22" 'lemma_ok_2 p H =
+ (cast _ _ (show p (Lemma _ H _))).
+interpretation "lemma_ok_22" 'lemma_ok_2 p H =
+ (cast _ _ (show p (Lemma _ H _ _))).
+
+notation > "'lem' term 90 p" non associative with precedence 19
+for @{ 'Lemma $p }.
+interpretation "lemma KO" 'Lemma p = (cast _ _ (Lemma _ p)).
+interpretation "lemma OK" 'Lemma p = (Lemma _ p).
+
+
(* discharging *)
notation < "[ a ] \sup mstyle color #ff0000 (H)" with precedence 19
for @{ 'discharge_ko_1 $a $H }.
(show ab (Or_intro_l _ _ a)).
notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\l))" with precedence 19
-for @{ 'Or_intro_l_ok_1 $a $ab }.
+for @{ 'Or_intro_l_ok_2 $a $ab }.
interpretation "Or_intro_l_ok_2" 'Or_intro_l_ok_2 a ab =
(cast _ _ (show ab (Or_intro_l _ _ a))).
(show ab (Or_intro_r _ _ a)).
notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\r))" with precedence 19
-for @{ 'Or_intro_r_ok_1 $a $ab }.
+for @{ 'Or_intro_r_ok_2 $a $ab }.
interpretation "Or_intro_r_ok_2" 'Or_intro_r_ok_2 a ab =
(cast _ _ (show ab (Or_intro_r _ _ a))).
interpretation "Bot_elim OK" 'Bot_elim a = (Bot_elim _ a).
(* ¬ introduction *)
-notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
+notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (\lnot\sub(\emsp\i)) \emsp ident H)" with precedence 19
for @{ 'Not_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
interpretation "Not_intro_ko_1" 'Not_intro_ko_1 ab \eta.b =
(show ab (cast _ _ (Not_intro _ (cast _ _ b)))).
-notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (\lnot\sub(\emsp\i)) \emsp ident H)" with precedence 19
for @{ 'Not_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
interpretation "Not_intro_ko_2" 'Not_intro_ko_2 ab \eta.b =
(cast _ _ (show ab (cast _ _ (Not_intro _ (cast _ _ b))))).
-notation < "\infrule hbox(\emsp b \emsp) ab (\lnot\sub\i \emsp ident H) " with precedence 19
+notation < "\infrule hbox(\emsp b \emsp) ab (\lnot\sub(\emsp\i) \emsp ident H) " with precedence 19
for @{ 'Not_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
interpretation "Not_intro_ok_1" 'Not_intro_ok_1 ab \eta.b =
(show ab (Not_intro _ b)).
-notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (\lnot\sub\i \emsp ident H) " with precedence 19
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (\lnot\sub(\emsp\i) \emsp ident H) " with precedence 19
for @{ 'Not_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
interpretation "Not_intro_ok_2" 'Not_intro_ok_2 ab \eta.b =
(cast _ _ (show ab (Not_intro _ b))).
interpretation "Not_intro OK" 'Not_intro a = (Not_intro _ a).
(* ¬ elimination *)
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (\lnot\sub\e) " with precedence 19
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (\lnot\sub(\emsp\e)) " with precedence 19
for @{ 'Not_elim_ko_1 $ab $a $b }.
interpretation "Not_elim_ko_1" 'Not_elim_ko_1 ab a b =
(show b (cast _ _ (Not_elim _ (cast _ _ ab) (cast _ _ a)))).
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (\lnot\sub\e) " with precedence 19
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (\lnot\sub(\emsp\e)) " with precedence 19
for @{ 'Not_elim_ko_2 $ab $a $b }.
interpretation "Not_elim_ko_2" 'Not_elim_ko_2 ab a b =
(cast _ _ (show b (cast _ _ (Not_elim _ (cast _ _ ab) (cast _ _ a))))).
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub\e) " with precedence 19
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub(\emsp\e)) " with precedence 19
for @{ 'Not_elim_ok_1 $ab $a $b }.
interpretation "Not_elim_ok_1" 'Not_elim_ok_1 ab a b =
(show b (Not_elim _ ab a)).
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (\lnot\sub\e) " with precedence 19
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (\lnot\sub(\emsp\e)) " with precedence 19
for @{ 'Not_elim_ok_2 $ab $a $b }.
interpretation "Not_elim_ok_2" 'Not_elim_ok_2 ab a b =
(cast _ _ (show b (Not_elim _ ab a))).
notation > "¬_'e' term 90 ab term 90 a" with precedence 19
for @{ 'Not_elim (show $ab ?) (show $a ?) }.
interpretation "Not_elim KO" 'Not_elim ab a =
- (cast _ _ (Not_elim _ (cast _ _ ab) (cast _ _ a))).
+ (cast _ _ (Not_elim unit (cast _ _ ab) (cast _ _ a))).
interpretation "Not_elim OK" 'Not_elim ab a =
(Not_elim _ ab a).
interpretation "RAA KO" 'Raa p = (cast _ unit (Raa _ (cast _ (unit_to _) p))).
interpretation "RAA OK" 'Raa p = (Raa _ p).
-(*DOCBEGIN
-Templates for rules:
-
- apply rule (⇒_i […] (…)).
-
- apply rule (∧_i (…) (…));
- [
- |
- ]
-
- apply rule (∨_i_l (…)).
-
- apply rule (∨_i_r (…)).
-
- apply rule (¬_i […] (…)).
-
- apply rule (⊤_i).
-
- apply rule (⇒_e (…) (…));
- [
- |
- ]
-
- apply rule (∧_e_l (…)).
-
- apply rule (∧_e_r (…)).
-
- apply rule (∨_e (…) […] (…) […] (…));
- [
- |
- |
- ]
-
- apply rule (¬_e (…) (…));
- [
- |
- ]
-
- apply rule (⊥_e (…)).
-
- apply rule (prove (…)).
-
- apply rule (RAA […] (…)).
-
- apply rule (discharge […]).
-
-DOCEND*)
-
-
-
-
+(* ∃ introduction *)
+notation < "\infrule hbox(\emsp Pn \emsp) Px mstyle color #ff0000 (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ko_1 $Pn $Px }.
+interpretation "Exists_intro_ko_1" 'Exists_intro_ko_1 Pn Px =
+ (show Px (cast _ _ (Exists_intro _ _ _ (cast _ _ Pn)))).
+
+notation < "\infrule hbox(\emsp Pn \emsp) mstyle color #ff0000 (Px) mstyle color #ff0000 (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ko_2 $Pn $Px }.
+interpretation "Exists_intro_ko_2" 'Exists_intro_ko_2 Pn Px =
+ (cast _ _ (show Px (cast _ _ (Exists_intro _ _ _ (cast _ _ Pn))))).
+
+notation < "\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ok_1 $Pn $Px }.
+interpretation "Exists_intro_ok_1" 'Exists_intro_ok_1 Pn Px =
+ (show Px (Exists_intro _ _ _ Pn)).
+
+notation < "\infrule hbox(\emsp Pn \emsp) mstyle color #ff0000 (Px) (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ok_2 $Pn $Px }.
+interpretation "Exists_intro_ok_2" 'Exists_intro_ok_2 Pn Px =
+ (cast _ _ (show Px (Exists_intro _ _ _ Pn))).
+
+notation >"∃_'i' {term 90 t} term 90 Pt" non associative with precedence 19
+for @{'Exists_intro $t (λ_.?) (show $Pt ?)}.
+interpretation "Exists_intro KO" 'Exists_intro t P Pt =
+ (cast _ _ (Exists_intro sort P t (cast _ _ Pt))).
+interpretation "Exists_intro OK" 'Exists_intro t P Pt =
+ (Exists_intro sort P t Pt).
+
+(* ∃ elimination *)
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (mstyle color #ff0000 (∃\sub\e) \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ko_1 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ko_1" 'Exists_elim_ko_1 ExPx Pc c =
+ (show c (cast _ _ (Exists_elim _ _ _ (cast _ _ ExPx) (cast _ _ Pc)))).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) mstyle color #ff0000 (c) (mstyle color #ff0000 (∃\sub\e) \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ko_2 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ko_2" 'Exists_elim_ko_2 ExPx Pc c =
+ (cast _ _ (show c (cast _ _ (Exists_elim _ _ _ (cast _ _ ExPx) (cast _ _ Pc))))).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (∃\sub\e \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ok_1 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ok_1" 'Exists_elim_ok_1 ExPx Pc c =
+ (show c (Exists_elim _ _ _ ExPx Pc)).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) mstyle color #ff0000 (c) (∃\sub\e \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ok_2 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ok_2" 'Exists_elim_ok_2 ExPx Pc c =
+ (cast _ _ (show c (Exists_elim _ _ _ ExPx Pc))).
+
+definition ex_concl := λx:CProp.sort → unit → x.
+definition fake_pred := λx:sort.unit.
+
+notation >"∃_'e' term 90 ExPt {ident t} [ident H] term 90 c" non associative with precedence 19
+for @{'Exists_elim (λ_.?) (show $ExPt ?) (λ${ident t}:sort.λ${ident H}.show $c ?)}.
+interpretation "Exists_elim KO" 'Exists_elim P ExPt c =
+ (cast _ _ (Exists_elim sort P _
+ (cast (Exists _ P) _ ExPt) (cast (ex_concl unit) (ex_concl _) c))).
+interpretation "Exists_elim OK" 'Exists_elim P ExPt c =
+ (Exists_elim sort P _ ExPt c).
+
+(* ∀ introduction *)
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ko_1" 'Forall_intro_ko_1 Px Pn =
+ (show Pn (cast _ _ (Forall_intro _ _ (cast _ _ Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000(Pn) (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ko_2" 'Forall_intro_ko_2 Px Pn =
+ (cast _ _ (show Pn (cast _ _ (Forall_intro _ _ (cast _ _ Px))))).
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ok_1" 'Forall_intro_ok_1 Px Pn =
+ (show Pn (Forall_intro _ _ Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (∀\sub\i \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ok_2" 'Forall_intro_ok_2 Px Pn =
+ (cast _ _ (show Pn (Forall_intro _ _ Px))).
+
+notation > "∀_'i' {ident y} term 90 Px" non associative with precedence 19
+for @{ 'Forall_intro (λ_.?) (λ${ident y}.show $Px ?) }.
+interpretation "Forall_intro KO" 'Forall_intro P Px =
+ (cast _ _ (Forall_intro sort P (cast _ _ Px))).
+interpretation "Forall_intro OK" 'Forall_intro P Px =
+ (Forall_intro sort P Px).
+
+(* ∀ elimination *)
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\e))" with precedence 19
+for @{ 'Forall_elim_ko_1 $Px $Pn }.
+interpretation "Forall_elim_ko_1" 'Forall_elim_ko_1 Px Pn =
+ (show Pn (cast _ _ (Forall_elim _ _ _ (cast _ _ Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000(Pn) (mstyle color #ff0000 (∀\sub\e))" with precedence 19
+for @{ 'Forall_elim_ko_2 $Px $Pn }.
+interpretation "Forall_elim_ko_2" 'Forall_elim_ko_2 Px Pn =
+ (cast _ _ (show Pn (cast _ _ (Forall_elim _ _ _ (cast _ _ Px))))).
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\e)" with precedence 19
+for @{ 'Forall_elim_ok_1 $Px $Pn }.
+interpretation "Forall_elim_ok_1" 'Forall_elim_ok_1 Px Pn =
+ (show Pn (Forall_elim _ _ _ Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (∀\sub\e)" with precedence 19
+for @{ 'Forall_elim_ok_2 $Px $Pn }.
+interpretation "Forall_elim_ok_2" 'Forall_elim_ok_2 Px Pn =
+ (cast _ _ (show Pn (Forall_elim _ _ _ Px))).
+
+notation > "∀_'e' {term 90 t} term 90 Pn" non associative with precedence 19
+for @{ 'Forall_elim (λ_.?) $t (show $Pn ?) }.
+interpretation "Forall_elim KO" 'Forall_elim P t Px =
+ (cast _ unit (Forall_elim sort P t (cast _ _ Px))).
+interpretation "Forall_elim OK" 'Forall_elim P t Px =
+ (Forall_elim sort P t Px).
projects / helm.git / blobdiff