(* *)
(**************************************************************************)
-(*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 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 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) 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)).
-
-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 \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)).
-
-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 \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) }.
-interpretation "Forall" 'Forall \eta.Px = (Forall _ 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)).
-
-axiom A: CProp.
-axiom B: CProp.
-axiom C: CProp.
-axiom D: CProp.
-axiom E: CProp.
-
+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.
-notation > "[H]" with precedence 90
-for @{ assumpt ? (cast ? ? $H)}.
-notation > "⇒\sub\i [ident H] term 90 b" with precedence 19
-for @{ Imply_intro ?? (λ${ident H}.cast ? $b ?) }.
-notation > "⇒\sub\e term 90 ab term 90 a" with precedence 19
-for @{ Imply_elim ?? (cast ? $ab ?) (cast $a $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 (? ∧ 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 $ab ?) }. (* CSC: WRONG *)
-notation > "∧\sub\e\sup\r term 90 ab" with precedence 19
-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 ?) }.
-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 $ab ?) (λ${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 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.
- 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));
- apply (∧\sub\i E C);
- [ apply (⇒\sub\e (A⇒E) A);
- [ apply [C1];
- | apply (∧\sub\e\sup\l (A∧C));
- apply [K];
+
+ 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 (∧\sub\e\sup\r (A∧C));
- apply [K];
+ | apply rule (∧_e_r (A∧C)); apply rule (discharge [K]);
]
-| apply (∨\sub\i\sup\r B);
- apply [C2];
+| 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 ? ((∀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.
projects / helm.git / blobdiff