(* *)
(**************************************************************************)
-definition cast ≝ λA:CProp.λa:A.a.
+(*definition cast ≝ λA,B:CProp.λa:A.a.*)
+axiom cast: ∀A,B:CProp.B → A.
+
+(*notation < "\infrule (t\atop ⋮) (b \ALPOSTODI a) (? \ERROR)" with precedence 19
+for @{ 'caste $a $b $t }.
+interpretation "cast" 'caste a b t = (cast a b t).*)
+notation < "\infrule (t\atop ⋮) mstyle color #ff0000 (b) (? \ERROR)" with precedence 19
+for @{ 'caste $a $b $t }.
+interpretation "cast" 'caste a b t = (cast a b t).
notation < "\infrule (t\atop ⋮) a ?" with precedence 19 for @{ 'cast $a $t }.
-interpretation "cast" 'cast a t = (cast a t).
+interpretation "cast" 'cast a t = (cast a a t).
definition assumpt ≝ λA:CProp.λa:A.a.
+notation < "[ a ] \sup mstyle color #ff0000 (H)" with precedence 19 for @{ 'asse $a $H }.
+interpretation "assumption" 'asse a H = (cast _ _ (assumpt a (cast _ _ H))).
+
notation < "[ a ] \sup H" with precedence 19 for @{ 'ass $a $H }.
-interpretation "assumption" 'ass a H = (cast _ (assumpt a H)).
+interpretation "assumption" 'ass a H = (cast a a (assumpt a (cast a a H))).
inductive Imply (A,B:CProp) : CProp ≝
Imply_intro: (A → B) → Imply A B.
notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
interpretation "Imply" 'Imply a b = (Imply a b).
-notation < "\infrule hbox(\emsp b \emsp) ab (⇒\sub\i) " with precedence 19 for @{ 'Imply_intro $ab (λ${ident H}:$p.$b) }.
-interpretation "Imply_intro" 'Imply_intro ab \eta.b = (cast ab (Imply_intro _ _ b)).
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000(ab) (⇒\sub\i \emsp ident H \ERROR) " with precedence 19
+for @{ 'Imply_introe $xxx $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro" 'Imply_introe xxx ab \eta.b = (cast xxx ab (Imply_intro _ _ b)).
+
+notation < "\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) " with precedence 19
+for @{ 'Imply_intro $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro" 'Imply_intro ab \eta.b = (cast ab ab (Imply_intro _ _ b)).
definition Imply_elim ≝ λA,B.λf:Imply A B.λa:A.match f with [ Imply_intro g ⇒ g a].
notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) " with precedence 19 for @{ 'Imply_elim $ab $a $b }.
-interpretation "Imply_elim" 'Imply_elim ab a b = (cast b (Imply_elim _ _ ab a)).
+interpretation "Imply_elim" 'Imply_elim ab a b = (cast _ b (Imply_elim _ _ ab a)).
inductive And (A,B:CProp) : CProp ≝
And_intro: A → B → And A B.
interpretation "constructive and" 'and x y = (And x y).
notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)" with precedence 19 for @{ 'And_intro $a $b $ab }.
-interpretation "And_intro" 'And_intro a b ab = (cast ab (And_intro _ _ a b)).
+interpretation "And_intro" 'And_intro a b ab = (cast _ ab (And_intro _ _ a b)).
definition And_elim_l ≝
λA,B.λc:A∧B.match c with [ And_intro a b ⇒ a ].
notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub\e\sup\l)" with precedence 19 for @{ 'And_elim_l $ab $a }.
-interpretation "And_elim_l" 'And_elim_l ab a = (cast a (And_elim_l _ _ ab)).
+interpretation "And_elim_l" 'And_elim_l ab a = (cast _ a (And_elim_l _ _ ab)).
definition And_elim_r ≝
λA,B.λc:A∧B.match c with [ And_intro a b ⇒ b ].
notation < "\infrule hbox(\emsp ab \emsp) b (∧\sub\e\sup\r)" with precedence 19 for @{ 'And_elim_r $ab $b }.
-interpretation "And_elim_r" 'And_elim_r ab b = (cast b (And_elim_r _ _ ab)).
+interpretation "And_elim_r" 'And_elim_r ab b = (cast _ b (And_elim_r _ _ ab)).
inductive Or (A,B:CProp) : CProp ≝
| Or_intro_l: A → Or A B
interpretation "constructive or" 'or x y = (Or x y).
notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub\i\sup\l)" with precedence 19 for @{ 'Or_intro_l $a $ab }.
-interpretation "Or_intro_l" 'Or_intro_l a ab = (cast ab (Or_intro_l _ _ a)).
+interpretation "Or_intro_l" 'Or_intro_l a ab = (cast _ ab (Or_intro_l _ _ a)).
notation < "\infrule hbox(\emsp b \emsp) ab (∨\sub\i\sup\l)" with precedence 19 for @{ 'Or_intro_r $b $ab }.
-interpretation "Or_intro_l" 'Or_intro_r b ab = (cast ab (Or_intro_r _ _ b)).
+interpretation "Or_intro_l" 'Or_intro_r b ab = (cast _ ab (Or_intro_r _ _ b)).
definition Or_elim ≝
λA,B,C:CProp.λc:A∨B.λfa: A → C.λfb: B → C.
match c with [ Or_intro_l a ⇒ fa a | Or_intro_r b ⇒ fb b].
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (∨\sub\e)" with precedence 19 for @{ 'Or_elim $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
-interpretation "Or_elim" 'Or_elim ab ac bc c = (cast c (Or_elim _ _ _ ab ac 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
+for @{ 'Or_elim $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
+interpretation "Or_elim" 'Or_elim ab \eta.ac \eta.bc c = (cast _ c (Or_elim _ _ _ ab ac bc)).
inductive Exists (A:Type) (P:A→CProp) : CProp ≝
Exists_intro: ∀w:A. P w → Exists A P.
interpretation "constructive ex" 'exists \eta.x = (Exists _ x).
-notation < "\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)" with precedence 19 for @{ 'Exists_intro $Pn $Px }.
-interpretation "Exists_intro" 'Exists_intro Pn Px = (cast Px (Exists_intro _ _ _ Pn)).
+notation < "\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro $Pn $Px }.
+interpretation "Exists_intro" 'Exists_intro Pn Px = (cast _ Px (Exists_intro _ _ _ Pn)).
definition Exists_elim ≝
λA:Type.λP:A→CProp.λC:CProp.λc:∃x:A.P x.λH:(∀x.P x → C).
match c with [ Exists_intro w p ⇒ H w p ].
-notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (∃\sub\e)" with precedence 19 for @{ 'Exists_elim $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
-interpretation "Exists_elim" 'Exists_elim ExPx Pc c = (cast c (Exists_elim _ _ _ ExPx 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 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim" 'Exists_elim ExPx Pc c = (cast _ c (Exists_elim _ _ _ ExPx Pc)).
inductive Forall (A:Type) (P:A→CProp) : CProp ≝
Forall_intro: (∀n:A. P n) → Forall A P.
-notation "\forall ident x:A.break term 19 Px" with precedence 20 for @{ 'Forall (λ${ident x}:$A.$Px) }.
+notation "\forall ident x:A.break term 19 Px" with precedence 20
+for @{ 'Forall (λ${ident x}:$A.$Px) }.
interpretation "Forall" 'Forall \eta.Px = (Forall _ Px).
-notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i)" with precedence 19 for @{ 'Forall_intro (λ${ident x}:$tx.$Px) $Pn }.
-interpretation "Forall_intro" 'Forall_intro Px Pn = (cast Pn (Forall_intro _ _ Px)).
+notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i \emsp ident x)" with precedence 19
+for @{ 'Forall_intro (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro" 'Forall_intro Px Pn = (cast _ Pn (Forall_intro _ _ Px)).
definition Forall_elim ≝
λA:Type.λP:A→CProp.λn:A.λf:∀x:A.P x.match f with [ Forall_intro g ⇒ g n ].
notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i)" with precedence 19 for @{ 'Forall_elim $Px $Pn }.
-interpretation "Forall_elim" 'Forall_elim Px Pn = (cast Pn (Forall_elim _ _ _ Px)).
+interpretation "Forall_elim" 'Forall_elim Px Pn = (cast _ Pn (Forall_elim _ _ _ Px)).
axiom A: CProp.
axiom B: CProp.
notation > "[H]" with precedence 90
-for @{ assumpt ? $H}.
+for @{ assumpt ? (cast ? ? $H)}.
notation > "⇒\sub\i [ident H] term 90 b" with precedence 19
-for @{ Imply_intro ?? (λ${ident H}.cast $b ?) }.
+for @{ Imply_intro ?? (λ${ident H}.cast ? $b ?) }.
notation > "⇒\sub\e term 90 ab term 90 a" with precedence 19
-for @{ Imply_elim ?? (cast $ab ?) (cast $a ?) }.
+for @{ Imply_elim ?? (cast ? $ab ?) (cast $a $a ?) }.
notation > "∧\sub\i term 90 a term 90 b" with precedence 19
-for @{ And_intro ?? (cast $a ?) (cast $b ?) }.
+for @{ And_intro ?? (cast ? $a ?) (cast ? $b ?) }.
+(*notation > "∧\sub\e\sup\l term 90 ab" with precedence 19
+for @{ And_elim_l ?? (cast (? ∧ False) $ab ?) }.
+notation > "∧\sub\e\sup\l term 90 a ∧ term 90 b" with precedence 19
+for @{ And_elim_l ?? (cast (? ∧ $b) ($a ∧ $b) ?) }. *)
notation > "∧\sub\e\sup\l term 90 ab" with precedence 19
-for @{ And_elim_l ?? (cast $ab ?) }.
+for @{ And_elim_l ?? (cast $ab $ab ?) }. (* CSC: WRONG *)
notation > "∧\sub\e\sup\r term 90 ab" with precedence 19
-for @{ And_elim_r ?? (cast $ab ?) }.
+for @{ And_elim_r ?? (cast $ab $ab ?) }. (* CSC: WRONG *)
notation > "∨\sub\i\sup\l term 90 a" with precedence 19
-for @{ Or_intro_l ?? (cast $a ?) }.
+for @{ Or_intro_l ?? (cast ? $a ?) }.
notation > "∨\sub\i\sup\r term 90 a" with precedence 19
-for @{ Or_intro_r ?? (cast $a ?) }.
+for @{ Or_intro_r ?? (cast ? $a ?) }.
notation > "∨\sub\e term 90 ab [ident Ha] term 90 c1 [ident Hb] term 90 c2" with precedence 19
-for @{ Or_elim ??? (cast $ab ?) (λ${ident Ha}.cast $c1 ?) (λ${ident Hb}.cast $c2 ?) }.
-
+for @{ Or_elim ??? (cast $ab $ab ?) (λ${ident Ha}.cast ? $c1 ?) (λ${ident Hb}.cast ? $c2 ?) }.
+notation > "∀\sub\i [ident z] term 90 a" with precedence 19
+for @{ Forall_intro ?? (λ${ident z}.cast ? $a ?) }.
+notation > "∀\sub\e term 90 ab" with precedence 19
+for @{ Forall_elim ?? ? (cast $ab $ab ?) }. (* CSC: WRONG *)
+notation > "∃\sub\e term 90 enpn [ident z] [ident pz] term 90 c" with precedence 19
+for @{ Exists_elim ??? (cast $enpn $enpn ?) (λ${ident z}.λ${ident pz}.cast ? $c ?) }.
+notation > "∃\sub\i term 90 n term 90 pn" with precedence 19
+for @{ Exists_intro ? (λ_.?) $n (cast ? $pn ?) }.
lemma ex1 : (A ⇒ E) ∨ B ⇒ A ∧ C ⇒ (E ∧ C) ∨ B.
- apply cast;
+ apply (cast ? ((A⇒E)∨B⇒A∧C⇒E∧C∨B));
+ (*NICE: TRY THIS ERROR!
+ apply (⇒\sub\i [H] (A∧C⇒E∧E∧C∨B));
+ apply (⇒\sub\i [K] (E∧E∧C∨B));
+ OR DO THE RIGHT THING *)
apply (⇒\sub\i [H] (A∧C⇒E∧C∨B));
apply (⇒\sub\i [K] (E∧C∨B));
+
apply (∨\sub\e ((A⇒E)∨B) [C1] (E∧C∨B) [C2] (E∧C∨B));
[ apply [H];
| apply (∨\sub\i\sup\l (E∧C));
axiom R: N → N → CProp.
lemma ex2: (∀a:N.∀b:N.R a b ⇒ R b a) ⇒ ∀z:N.(∃x.R x z) ⇒ ∃y. R z y.
- apply cast; apply Imply_intro; intro;
- apply cast; apply Forall_intro; intro z;
- apply cast; apply Imply_intro; intro;
- apply cast; apply (Exists_elim N (λy.R y z)); try intros (n);
- [ apply cast; assumption
- | apply cast; apply (Exists_intro ? ? n);
- apply cast; apply (Imply_elim (R n z) (R z n));
- [ apply cast; apply (Forall_elim N (λb:N.R n b ⇒ R b n) z);
- apply cast; apply (Forall_elim N (λa:N.∀b:N.R a b ⇒ R b a) n);
- apply cast; assumption
- | apply cast; assumption
+ apply (cast ? ((∀a:N.∀b:N.R a b ⇒ R b a) ⇒ ∀z:N.(∃x.R x z) ⇒ ∃y. R z y));
+ apply (⇒\sub\i [H] (∀z:N.(∃x:N.R x z)⇒∃y:N.R z y));
+ apply (∀\sub\i [z] ((∃x:N.R x z)⇒∃y:N.R z y));
+ apply (⇒\sub\i [H2] (∃y:N.R z y));
+ apply (∃\sub\e (∃x:N.R x z) [n] [H3] (∃y:N.R z y));
+ [ apply [H2]
+ | apply (∃\sub\i n (R z n));
+ apply (⇒\sub\e (R n z ⇒ R z n) (R n z));
+ [ apply (∀\sub\e (∀b:N.R n b ⇒ R b n));
+ apply (∀\sub\e (∀a:N.∀b:N.R a b ⇒ R b a));
+ apply [H]
+ | apply [H3]
]
]
-qed.
\ No newline at end of file
+qed.
projects / helm.git / blobdiff