definition cast ≝ λA:CProp.λa:A.a.
+notation < "\infrule (t\atop ⋮) a ?" with precedence 19 for @{ 'cast $a $t }.
+interpretation "cast" 'cast a t = (cast a t).
+
+definition assumpt ≝ λA:CProp.λa:A.a.
+
notation < "[ a ] \sup H" with precedence 19 for @{ 'ass $a $H }.
-interpretation "assumption" 'ass a H = (cast a H).
+interpretation "assumption" 'ass a H = (cast _ (assumpt 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) }.
+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 (Imply_intro _ _ b)).
definition Imply_elim ≝ λA,B.λf:Imply A B.λa:A.match f with [ Imply_intro g ⇒ g a].
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 }.
+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 }.
+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) $Pn }.
+notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i \emsp ident x)" with precedence 19
+for @{ 'Forall_intro (λ${ident x}:$tx.$Px) $Pn }.
interpretation "Forall_intro" 'Forall_intro Px Pn = (cast Pn (Forall_intro _ _ Px)).
definition Forall_elim ≝
axiom D: CProp.
axiom E: CProp.
+
+notation > "[H]" with precedence 90
+for @{ assumpt ? $H}.
+notation > "⇒\sub\i [ident H] term 90 b" with precedence 19
+for @{ Imply_intro ?? (λ${ident H}.cast $b ?) }.
+notation > "⇒\sub\e term 90 ab term 90 a" with precedence 19
+for @{ Imply_elim ?? (cast $ab ?) (cast $a ?) }.
+notation > "∧\sub\i term 90 a term 90 b" with precedence 19
+for @{ And_intro ?? (cast $a ?) (cast $b ?) }.
+notation > "∧\sub\e\sup\l term 90 ab" with precedence 19
+for @{ And_elim_l ?? (cast $ab ?) }.
+notation > "∧\sub\e\sup\r term 90 ab" with precedence 19
+for @{ And_elim_r ?? (cast $ab ?) }.
+notation > "∨\sub\i\sup\l term 90 a" with precedence 19
+for @{ Or_intro_l ?? (cast $a ?) }.
+notation > "∨\sub\i\sup\r term 90 a" with precedence 19
+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 $c2 ?) }.
+notation > "∀\sub\i [ident z] term 90 a" with precedence 19
+for @{ Forall_intro ?? (λ${ident z}.cast $a ?) }.
+notation > "∀\sub\e term 90 ab" with precedence 19
+for @{ Forall_elim ?? ? (cast $ab ?) }.
+notation > "∃\sub\e term 90 enpn [ident z] [ident pz] term 90 c" with precedence 19
+for @{ Exists_elim ??? (cast $enpn ?) (λ${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.
-repeat (apply cast; constructor 1; intro);
-apply cast; apply (Or_elim (A ⇒ E) B (E∧C∨B)); try intro;
-[ apply cast; assumption
-| apply cast; apply Or_intro_l;
- apply cast; constructor 1;
- [ apply cast; apply (Imply_elim A E);
- [ apply cast; assumption
- | apply cast; apply (And_elim_l A C);
- apply cast; assumption
+ apply cast;
+ 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));
+ apply (∧\sub\i E C);
+ [ apply (⇒\sub\e (A⇒E) A);
+ [ apply [C1];
+ | apply (∧\sub\e\sup\l (A∧C));
+ apply [K];
]
- | apply cast; apply (And_elim_r A C);
- apply cast; assumption
+ | apply (∧\sub\e\sup\r (A∧C));
+ apply [K];
]
-| apply cast; apply Or_intro_r;
- apply cast; assumption
+| apply (∨\sub\i\sup\r B);
+ apply [C2];
]
qed.
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;
+ 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
projects / helm.git / blobdiff