(* *)
(**************************************************************************)
-definition cast ≝ λA:CProp.λa:A.a.
-
-notation < "[ a ] \sup H" with precedence 19 for @{ 'ass $a $H }.
-interpretation "assumption" 'ass a H = (cast 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)).
-
-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)).
-
-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)).
-
-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)).
-
-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)).
-
-inductive Or (A,B:CProp) : CProp ≝
- | Or_intro_l: A → Or A B
- | Or_intro_r: B → 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)).
-
-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)).
-
-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)).
-
-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)).
-
-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) $c }.
-interpretation "Exists_elim" 'Exists_elim ExPx Pc c = (cast c (Exists_elim _ _ _ ExPx Pc)).
-
-inductive Forall (A:Type) (P:A→CProp) : CProp ≝
- Forall_intro: (∀n:A. P n) → Forall A P.
-
-notation "\forall ident x:A.break term 19 Px" with precedence 20 for @{ 'Forall (λ${ident x}:$A.$Px) }.
-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)).
-
-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)).
+include "didactic/support/natural_deduction.ma".
+
+lemma RAA_to_EM : A ∨ ¬ A.
+
+ apply rule (prove (A ∨ ¬ A));
+
+ apply rule (RAA [H] ⊥);
+ apply rule (¬#e (¬A) A);
+ [ apply rule (¬#i [H1] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_l A);
+ apply rule (discharge [H1]);
+ ]
+ | apply rule (RAA [H2] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_r (¬A));
+ apply rule (discharge [H2]);
+ ]
+ ]
+qed.
-axiom A: CProp.
-axiom B: CProp.
-axiom C: CProp.
-axiom D: CProp.
-axiom E: CProp.
+lemma RA_to_EM1 : A ∨ ¬ A.
+
+ apply rule (prove (A ∨ ¬ A));
+
+ apply rule (RAA [H] ⊥);
+ apply rule (¬#e (¬¬A) (¬A));
+ [ apply rule (¬#i [H2] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_r (¬A));
+ apply rule (discharge [H2]);
+ ]
+ | apply rule (¬#i [H1] ⊥);
+ apply rule (¬#e (¬(A∨¬A)) (A∨¬A));
+ [ apply rule (discharge [H]);
+ | apply rule (∨#i_l A);
+ apply rule (discharge [H1]);
+ ]
+ ]
+qed.
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 rule (prove ((A⇒E)∨B⇒A∧C⇒E∧C∨B));
+
+ apply rule (⇒#i [H] (A∧C⇒E∧C∨B));
+ apply rule (⇒#i [K] (E∧C∨B));
+ apply rule (∨#e ((A⇒E)∨B) [C1] (E∧C∨B) [C2] (E∧C∨B));
+[ apply rule (discharge [H]);
+| apply rule (∨#i_l (E∧C));
+ apply rule (∧#i E C);
+ [ apply rule (⇒#e (A⇒E) A);
+ [ apply rule (discharge [C1]);
+ | apply rule (∧#e_l (A∧C)); apply rule (discharge [K]);
]
- | apply cast; apply (And_elim_r A C);
- apply cast; assumption
+ | apply rule (∧#e_r (A∧C)); apply rule (discharge [K]);
]
-| apply cast; apply Or_intro_r;
- apply cast; assumption
+| apply rule (∨#i_r B); apply rule (discharge [C2]);
]
qed.
-axiom N: Type.
-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
- ]
- ]
-qed.
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projects / helm.git / blobdiff