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 x: sort.
+axiom y: sort.
+axiom z: sort.
+axiom w: 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 : ∀A.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 }.
interpretation "Imply_intro_ko_2" 'Imply_intro_ko_2 ab \eta.b =
(cast _ _ (show ab (cast _ _ (Imply_intro _ _ b)))).
-notation < "\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) " with precedence 19
+notation < "maction (\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) ) (\vdots)" with precedence 19
for @{ 'Imply_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
interpretation "Imply_intro_ok_1" 'Imply_intro_ok_1 ab \eta.b =
(show ab (Imply_intro _ _ b)).
interpretation "Imply_elim_ko_2" 'Imply_elim_ko_2 ab a b =
(cast _ _ (show b (cast _ _ (Imply_elim _ _ (cast _ _ ab) (cast _ _ a))))).
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) " with precedence 19
+notation < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) ) (\vdots)" with precedence 19
for @{ 'Imply_elim_ok_1 $ab $a $b }.
interpretation "Imply_elim_ok_1" 'Imply_elim_ok_1 ab a b =
(show b (Imply_elim _ _ ab a)).
interpretation "And_intro_ko_2" 'And_intro_ko_2 a b ab =
(cast _ _ (show ab (cast _ _ (And_intro _ _ a b)))).
-notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)" with precedence 19
+notation < "maction (\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)) (\vdots)" with precedence 19
for @{ 'And_intro_ok_1 $a $b $ab }.
interpretation "And_intro_ok_1" 'And_intro_ok_1 a b ab =
(show ab (And_intro _ _ a b)).
interpretation "And_elim_l_ko_2" 'And_elim_l_ko_2 ab a =
(cast _ _ (show a (cast _ _ (And_elim_l _ _ (cast _ _ ab))))).
-notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\l))" with precedence 19
+notation < "maction (\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\l))) (\vdots)" with precedence 19
for @{ 'And_elim_l_ok_1 $ab $a }.
interpretation "And_elim_l_ok_1" 'And_elim_l_ok_1 ab a =
(show a (And_elim_l _ _ ab)).
interpretation "And_elim_r_ko_2" 'And_elim_r_ko_2 ab a =
(cast _ _ (show a (cast _ _ (And_elim_r _ _ (cast _ _ ab))))).
-notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\r))" with precedence 19
+notation < "maction (\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\r))) (\vdots)" with precedence 19
for @{ 'And_elim_r_ok_1 $ab $a }.
interpretation "And_elim_r_ok_1" 'And_elim_r_ok_1 ab a =
(show a (And_elim_r _ _ ab)).
interpretation "Or_intro_l_ko_2" 'Or_intro_l_ko_2 a ab =
(cast _ _ (show ab (cast _ _ (Or_intro_l _ _ a)))).
-notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))" with precedence 19
+notation < "maction (\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))) (\vdots)" with precedence 19
for @{ 'Or_intro_l_ok_1 $a $ab }.
interpretation "Or_intro_l_ok_1" 'Or_intro_l_ok_1 a ab =
(show ab (Or_intro_l _ _ a)).
interpretation "Or_intro_r_ko_2" 'Or_intro_r_ko_2 a ab =
(cast _ _ (show ab (cast _ _ (Or_intro_r _ _ a)))).
-notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))" with precedence 19
+notation < "maction (\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))) (\vdots)" with precedence 19
for @{ 'Or_intro_r_ok_1 $a $ab }.
interpretation "Or_intro_r_ok_1" 'Or_intro_r_ok_1 a ab =
(show ab (Or_intro_r _ _ a)).
interpretation "Or_elim_ko_2" 'Or_elim_ko_2 ab \eta.ac \eta.bc c =
(cast _ _ (show c (cast _ _ (Or_elim _ _ _ (cast _ _ ab) (cast _ _ ac) (cast _ _ bc))))).
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (∨\sub\e \emsp ident Ha \emsp ident Hb)" with precedence 19
+notation < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (∨\sub\e \emsp ident Ha \emsp ident Hb)) (\vdots)" with precedence 19
for @{ 'Or_elim_ok_1 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
interpretation "Or_elim_ok_1" 'Or_elim_ok_1 ab \eta.ac \eta.bc c =
(show c (Or_elim _ _ _ ab ac bc)).
interpretation "Top_intro_ko_2" 'Top_intro_ko_2 =
(cast _ _ (show _ (cast _ _ Top_intro))).
-notation < "\infrule \nbsp ⊤ (⊤\sub\i)" with precedence 19
+notation < "maction (\infrule \nbsp ⊤ (⊤\sub\i)) (\vdots)" with precedence 19
for @{'Top_intro_ok_1}.
interpretation "Top_intro_ok_1" 'Top_intro_ok_1 = (show _ Top_intro).
-notation < "\infrule \nbsp ⊤ (⊤\sub\i)" with precedence 19
+notation < "maction (\infrule \nbsp ⊤ (⊤\sub\i)) (\vdots)" with precedence 19
for @{'Top_intro_ok_2 }.
interpretation "Top_intro_ok_2" 'Top_intro_ok_2 = (cast _ _ (show _ Top_intro)).
interpretation "Bot_elim_ko_2" 'Bot_elim_ko_2 a b =
(cast _ _ (show a (Bot_elim _ (cast _ _ b)))).
-notation < "\infrule b a (⊥\sub\e)" with precedence 19
+notation < "maction (\infrule b a (⊥\sub\e)) (\vdots)" with precedence 19
for @{'Bot_elim_ok_1 $a $b}.
interpretation "Bot_elim_ok_1" 'Bot_elim_ok_1 a b =
(show a (Bot_elim _ b)).
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 < "maction (\infrule hbox(\emsp b \emsp) ab (\lnot\sub(\emsp\i) \emsp ident H) ) (\vdots)" 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_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(\emsp\e)) " with precedence 19
+notation < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub(\emsp\e)) ) (\vdots)" with precedence 19
for @{ 'Not_elim_ok_1 $ab $a $b }.
interpretation "Not_elim_ok_1" 'Not_elim_ok_1 ab a b =
(show b (Not_elim _ ab a)).
interpretation "RAA_ko_2" 'RAA_ko_2 Px Pn =
(cast _ _ (show Pn (cast _ _ (Raa _ (cast _ _ Px))))).
-notation < "\infrule hbox(\emsp Px \emsp) Pn (\RAA \emsp ident x)" with precedence 19
+notation < "maction (\infrule hbox(\emsp Px \emsp) Pn (\RAA \emsp ident x)) (\vdots)" with precedence 19
for @{ 'RAA_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
interpretation "RAA_ok_1" 'RAA_ok_1 Px Pn =
(show Pn (Raa _ Px)).
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 […]).
+(* ∃ 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 < "maction (\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)) (\vdots)" 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 (λw.? w) (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 < "maction (\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (∃\sub\e \emsp ident n \emsp ident HPn)) (\vdots)" 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:sort → CProp.∀y:sort.unit → x y.
+definition ex_concl_dummy := ∀y:sort.unit → unit.
+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 (λx.? x) (show $ExPt ?) (λ${ident t}:sort.λ${ident H}.show $c ?)}.
+interpretation "Exists_elim KO" 'Exists_elim P ExPt c =
+ (cast _ _ (Exists_elim sort P _
+ (cast (Exists _ P) _ ExPt)
+ (cast ex_concl_dummy (ex_concl _) c))).
+interpretation "Exists_elim OK" 'Exists_elim P ExPt c =
+ (Exists_elim sort P _ ExPt c).
+
+(* ∀ introduction *)
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_1 (λ${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) $Pn }.
+interpretation "Forall_intro_ko_2" 'Forall_intro_ko_2 Px Pn =
+ (cast _ _ (show Pn (cast _ _ (Forall_intro _ _ (cast _ _ Px))))).
+
+notation < "maction (\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i \emsp ident x)) (\vdots)" with precedence 19
+for @{ 'Forall_intro_ok_1 (λ${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 < "maction (\infrule hbox(\emsp Px \emsp) Pn (∀\sub\e)) (\vdots)" 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).
-DOCEND*)
+(* already proved lemma *)
+definition hide_args : ∀A:Type.∀a:A.A := λA:Type.λa:A.a.
+notation < "t" non associative with precedence 90 for @{'hide_args $t}.
+interpretation "hide 0 args" 'hide_args t = (hide_args _ t).
+interpretation "hide 1 args" 'hide_args t = (hide_args _ t _).
+interpretation "hide 2 args" 'hide_args t = (hide_args _ t _ _).
+interpretation "hide 3 args" 'hide_args t = (hide_args _ t _ _ _).
+interpretation "hide 4 args" 'hide_args t = (hide_args _ t _ _ _ _).
+interpretation "hide 5 args" 'hide_args t = (hide_args _ t _ _ _ _ _).
+interpretation "hide 6 args" 'hide_args t = (hide_args _ t _ _ _ _ _ _).
+interpretation "hide 7 args" 'hide_args t = (hide_args _ t _ _ _ _ _ _ _).
+
+(* more args crashes the pattern matcher *)
+
+(* already proved lemma, 0 assumptions *)
+definition Lemma : ΠA.A→A ≝ λA:CProp.λa:A.a.
+
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma_ko_1 $p ($H : $_) }.
+interpretation "lemma_ko_1" 'lemma_ko_1 p H =
+ (show p (cast _ _ (Lemma _ (cast _ _ H)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+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 _ (cast _ _ H))))).
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma_ok_1 $p ($H : $_) }.
+interpretation "lemma_ok_1" 'lemma_ok_1 p H =
+ (show p (Lemma _ H)).
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+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))).
+notation > "'lem' 0 term 90 l" non associative with precedence 19
+for @{ 'Lemma (hide_args ? $l : ?) }.
+interpretation "lemma KO" 'Lemma l =
+ (cast _ _ (Lemma unit (cast unit _ l))).
+interpretation "lemma OK" 'Lemma l = (Lemma _ l).
+
+
+(* already proved lemma, 1 assumption *)
+definition Lemma1 : ΠA,B. (A ⇒ B) → A → B ≝
+ λA,B:CProp.λf:A⇒B.λa:A.
+ Imply_elim A B f a.
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ko_1 $a $p ($H : $_) }.
+interpretation "lemma1_ko_1" 'lemma1_ko_1 a p H =
+ (show p (cast _ _ (Lemma1 _ _ (cast _ _ H) (cast _ _ a)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ko_2 $a $p ($H : $_) }.
+interpretation "lemma1_ko_2" 'lemma1_ko_2 a p H =
+ (cast _ _ (show p (cast _ _
+ (Lemma1 _ _ (cast _ _ H) (cast _ _ a))))).
+
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ok_1 $a $p ($H : $_) }.
+interpretation "lemma1_ok_1" 'lemma1_ok_1 a p H =
+ (show p (Lemma1 _ _ H a)).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ok_2 $a $p ($H : $_) }.
+interpretation "lemma1_ok_2" 'lemma1_ok_2 a p H =
+ (cast _ _ (show p (Lemma1 _ _ H a))).
+
+
+notation > "'lem' 1 term 90 l term 90 p" non associative with precedence 19
+for @{ 'Lemma1 (hide_args ? $l : ?) (show $p ?) }.
+interpretation "lemma 1 KO" 'Lemma1 l p =
+ (cast _ _ (Lemma1 unit unit (cast (Imply unit unit) _ l) (cast unit _ p))).
+interpretation "lemma 1 OK" 'Lemma1 l p = (Lemma1 _ _ l p).
+
+(* already proved lemma, 2 assumptions *)
+definition Lemma2 : ΠA,B,C. (A ⇒ B ⇒ C) → A → B → C ≝
+ λA,B,C:CProp.λf:A⇒B⇒C.λa:A.λb:B.
+ Imply_elim B C (Imply_elim A (B⇒C) f a) b.
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ko_1 $a $b $p ($H : $_) }.
+interpretation "lemma2_ko_1" 'lemma2_ko_1 a b p H =
+ (show p (cast _ _ (Lemma2 _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ko_2 $a $b $p ($H : $_) }.
+interpretation "lemma2_ko_2" 'lemma2_ko_2 a b p H =
+ (cast _ _ (show p (cast _ _
+ (Lemma2 _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b))))).
+
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ok_1 $a $b $p ($H : $_) }.
+interpretation "lemma2_ok_1" 'lemma2_ok_1 a b p H =
+ (show p (Lemma2 _ _ _ H a b)).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ok_2 $a $b $p ($H : $_) }.
+interpretation "lemma2_ok_2" 'lemma2_ok_2 a b p H =
+ (cast _ _ (show p (Lemma2 _ _ _ H a b))).
+
+notation > "'lem' 2 term 90 l term 90 p term 90 q" non associative with precedence 19
+for @{ 'Lemma2 (hide_args ? $l : ?) (show $p ?) (show $q ?) }.
+interpretation "lemma 2 KO" 'Lemma2 l p q =
+ (cast _ _ (Lemma2 unit unit unit (cast (Imply unit (Imply unit unit)) _ l) (cast unit _ p) (cast unit _ q))).
+interpretation "lemma 2 OK" 'Lemma2 l p q = (Lemma2 _ _ _ l p q).
+
+(* already proved lemma, 3 assumptions *)
+definition Lemma3 : ΠA,B,C,D. (A ⇒ B ⇒ C ⇒ D) → A → B → C → D ≝
+ λA,B,C,D:CProp.λf:A⇒B⇒C⇒D.λa:A.λb:B.λc:C.
+ Imply_elim C D (Imply_elim B (C⇒D) (Imply_elim A (B⇒C⇒D) f a) b) c.
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ko_1 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ko_1" 'lemma3_ko_1 a b c p H =
+ (show p (cast _ _
+ (Lemma3 _ _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b) (cast _ _ c)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ko_2 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ko_2" 'lemma3_ko_2 a b c p H =
+ (cast _ _ (show p (cast _ _
+ (Lemma3 _ _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b) (cast _ _ c))))).
+
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ok_1 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ok_1" 'lemma3_ok_1 a b c p H =
+ (show p (Lemma3 _ _ _ _ H a b c)).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ok_2 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ok_2" 'lemma3_ok_2 a b c p H =
+ (cast _ _ (show p (Lemma3 _ _ _ _ H a b c))).
+
+notation > "'lem' 3 term 90 l term 90 p term 90 q term 90 r" non associative with precedence 19
+for @{ 'Lemma3 (hide_args ? $l : ?) (show $p ?) (show $q ?) (show $r ?) }.
+interpretation "lemma 3 KO" 'Lemma3 l p q r =
+ (cast _ _ (Lemma3 unit unit unit unit (cast (Imply unit (Imply unit (Imply unit unit))) _ l) (cast unit _ p) (cast unit _ q) (cast unit _ r))).
+interpretation "lemma 3 OK" 'Lemma3 l p q r = (Lemma3 _ _ _ _ l p q r).
projects / helm.git / blobdiff