+++ /dev/null
-(*#* #stop file *)
-
-Require Arith.
-Require base_tactics.
-
-(* eq ***********************************************************************)
-
-Hint eq : ltlc := Constructors eq.
-
-Hint f1N : ltlc := Resolve (f_equal nat).
-
-Hint f2NN : ltlc := Resolve (f_equal2 nat nat).
-
-Hints Resolve sym_equal : ltlc.
-
-Hints Resolve plus_sym plus_n_Sm plus_assoc_r simpl_plus_l : ltlc.
-
-Hints Resolve minus_n_O : ltlc.
-
-(* le ***********************************************************************)
-
-Hint le : ltlc := Constructors le.
-
-Hints Resolve le_O_n le_n_S le_S_n le_trans : ltlc.
-
-Hints Resolve lt_le_S : ltlc.
-
-Hints Resolve le_plus_plus le_plus_trans le_plus_l le_plus_r : ltlc.
-
-(* lt ***********************************************************************)
-
-Hints Resolve lt_trans : ltlc.
-
-Hints Resolve lt_le_trans le_lt_n_Sm : ltlc.
-
-Hints Resolve lt_reg_r lt_le_plus_plus le_lt_plus_plus : ltlc.
-
-(* not **********************************************************************)
-
-Hints Resolve sym_not_equal : ltlc.
-
-(* missing in the standard library ******************************************)
-
- Theorem simpl_plus_r: (n,m,p:?) (plus m n) = (plus p n) -> m = p.
- Intros.
- Apply (simpl_plus_l n).
- Rewrite plus_sym.
- Rewrite H; XAuto.
- Qed.
-
- Theorem minus_plus_r: (m,n:?) (minus (plus m n) n) = m.
- Intros.
- Rewrite plus_sym.
- Apply minus_plus.
- Qed.
-
- Theorem plus_permute_2_in_3: (x,y,z:?) (plus (plus x y) z) = (plus (plus x z) y).
- Intros.
- Rewrite plus_assoc_r.
- Rewrite (plus_sym y z).
- Rewrite <- plus_assoc_r; XAuto.
- Qed.
-
- Theorem plus_permute_2_in_3_assoc: (n,h,k:?) (plus (plus n h) k) = (plus n (plus k h)).
- Intros.
- Rewrite plus_permute_2_in_3; Rewrite plus_assoc_l; XAuto.
- Qed.
-
- Theorem plus_O: (x,y:?) (plus x y) = (0) -> x = (O) /\ y = (O).
- XElim x; [ XAuto | Intros; Inversion H0 ].
- Qed.
-
- Theorem minus_Sx_SO: (x:?) (minus (S x) (1)) = x.
- Intros; Simpl; Rewrite <- minus_n_O; XAuto.
- Qed.
-
- Theorem eq_nat_dec: (i,j:nat) ~i=j \/ i=j.
- XElim i; XElim j; Intros; XAuto.
- Elim (H n0); XAuto.
- Qed.
-
- Theorem neq_eq_e: (i,j:nat; P:Prop) (~i=j -> P) -> (i=j -> P) -> P.
- Intros.
- Pose (eq_nat_dec i j).
- XElim o; XAuto.
- Qed.
-
- Theorem le_false: (m,n:?; P:Prop) (le m n) -> (le (S n) m) -> P.
- XElim m.
-(* case 1 : m = 0 *)
- Intros; Inversion H0.
-(* case 2 : m > 0 *)
- XElim n0; Intros.
-(* case 2.1 : n = 0 *)
- Inversion H0.
-(* case 2.2 : n > 0 *)
- Simpl in H1.
- Apply (H n0); XAuto.
- Qed.
-
- Theorem le_plus_minus_sym: (n,m:?) (le n m) -> m = (plus (minus m n) n).
- Intros.
- Rewrite plus_sym; Apply le_plus_minus; XAuto.
- Qed.
-
- Theorem le_minus_minus: (x,y:?) (le x y) -> (z:?) (le y z) ->
- (le (minus y x) (minus z x)).
- Intros.
- EApply simpl_le_plus_l.
- Rewrite le_plus_minus_r; [ Idtac | XAuto ].
- Rewrite le_plus_minus_r; XEAuto.
- Qed.
-
- Theorem le_minus_plus: (z,x:?) (le z x) -> (y:?)
- (minus (plus x y) z) = (plus (minus x z) y).
- XElim z.
-(* case 1 : z = 0 *)
- Intros x H; Inversion H; XAuto.
-(* case 2 : z > 0 *)
- Intros z; XElim x; Intros.
-(* case 2.1 : x = 0 *)
- Inversion H0.
-(* case 2.2 : x > 0 *)
- Simpl; XAuto.
- Qed.
-
- Theorem le_minus: (x,z,y:?) (le (plus x y) z) -> (le x (minus z y)).
- Intros.
- Rewrite <- (minus_plus_r x y); XAuto.
- Apply le_minus_minus; XAuto.
- Qed.
-
- Theorem le_trans_plus_r: (x,y,z:?) (le (plus x y) z) -> (le y z).
- Intros.
- EApply le_trans; [ EApply le_plus_r | Idtac ]; XEAuto.
- Qed.
-
- Theorem le_gen_S: (m,x:?) (le (S m) x) ->
- (EX n | x = (S n) & (le m n)).
- Intros; Inversion H; XEAuto.
- Qed.
-
- Theorem lt_x_plus_x_Sy: (x,y:?) (lt x (plus x (S y))).
- Intros; Rewrite plus_sym; Simpl; XAuto.
- Qed.
-
- Theorem simpl_lt_plus_r: (p,n,m:?) (lt (plus n p) (plus m p)) -> (lt n m).
- Intros.
- EApply simpl_lt_plus_l.
- Rewrite plus_sym in H; Rewrite (plus_sym m p) in H; Apply H.
- Qed.
-
- Theorem minus_x_Sy: (x,y:?) (lt y x) ->
- (minus x y) = (S (minus x (S y))).
- XElim x.
-(* case 1 : x = 0 *)
- Intros; Inversion H.
-(* case 2 : x > 0 *)
- XElim y; Intros; Simpl.
-(* case 2.1 : y = 0 *)
- Rewrite <- minus_n_O; XAuto.
-(* case 2.2 : y > 0 *)
- Cut (lt n0 n); XAuto.
- Qed.
-
- Theorem lt_plus_minus: (x,y:?) (lt x y) ->
- y = (S (plus x (minus y (S x)))).
- Intros.
- Apply (le_plus_minus (S x) y); XAuto.
- Qed.
-
- Theorem lt_plus_minus_r: (x,y:?) (lt x y) ->
- y = (S (plus (minus y (S x)) x)).
- Intros.
- Rewrite plus_sym; Apply lt_plus_minus; XAuto.
- Qed.
-
- Theorem minus_x_SO: (x:?) (lt (0) x) -> x = (S (minus x (1))).
- Intros.
- Rewrite <- minus_x_Sy; [ Rewrite <- minus_n_O; XEAuto | XEAuto ].
- Qed.
-
- Theorem lt_le_minus: (x,y:?) (lt x y) -> (le x (minus y (1))).
- Intros; Apply le_minus; Rewrite plus_sym; Simpl; XAuto.
- Qed.
-
- Theorem lt_le_e: (n,d:?; P:Prop)
- ((lt n d) -> P) -> ((le d n) -> P) -> P.
- Intros.
- Cut (le d n) \/ (lt n d); [ Intros H1; XElim H1; XAuto | Apply le_or_lt ].
- Qed.
-
- Theorem lt_eq_e: (x,y:?; P:Prop) ((lt x y) -> P) ->
- (x = y -> P) -> (le x y) -> P.
- Intros.
- LApply (le_lt_or_eq x y); [ Clear H1; Intros H1 | XAuto ].
- XElim H1; XAuto.
- Qed.
-
- Theorem lt_eq_gt_e: (x,y:?; P:Prop) ((lt x y) -> P) ->
- (x = y -> P) -> ((lt y x) -> P) -> P.
- Intros.
- Apply (lt_le_e x y); [ XAuto | Intros ].
- Apply (lt_eq_e y x); XAuto.
- Qed.
-
- Theorem lt_gen_S': (x,n:?) (lt x (S n)) ->
- x = (0) \/ (EX m | x = (S m) & (lt m n)).
- XElim x; XEAuto.
- Qed.
-
-Hints Resolve le_lt_trans : ltlc.
-
-Hints Resolve simpl_plus_r minus_plus_r minus_x_Sy
- plus_permute_2_in_3 plus_permute_2_in_3_assoc : ltlc.
-
-Hints Resolve le_minus_minus le_minus_plus le_minus le_trans_plus_r : ltlc.
-
-Hints Resolve lt_x_plus_x_Sy simpl_lt_plus_r lt_le_minus lt_plus_minus
- lt_plus_minus_r : ltlc.
-
- Theorem lt_neq: (x,y:?) (lt x y) -> ~x=y.
- Unfold not; Intros; Rewrite H0 in H; Clear H0 x.
- LApply (lt_n_n y); XAuto.
- Qed.
-
-Hints Resolve lt_neq : ltlc.
-
- Theorem arith0: (h2,d2,n:?) (le (plus d2 h2) n) ->
- (h1:?) (le (plus d2 h1) (minus (plus n h1) h2)).
- Intros.
- Rewrite <- (minus_plus h2 (plus d2 h1)).
- Apply le_minus_minus; [ XAuto | Idtac ].
- Rewrite plus_assoc_l; Rewrite (plus_sym h2 d2); XAuto.
- Qed.
-
-Hints Resolve arith0 : ltlc.
-
- Tactic Definition EqFalse :=
- Match Context With
- [ H: ~?1=?1 |- ? ] ->
- LApply H; [ Clear H; Intros H; Inversion H | XAuto ].
-
- Tactic Definition PlusO :=
- Match Context With
- | [ H: (plus ?0 ?1) = (0) |- ? ] ->
- LApply (plus_O ?0 ?1); [ Clear H; Intros H | XAuto ];
- XElim H; Intros.
-
- Tactic Definition SymEqual :=
- Match Context With
- | [ H: ?1 = ?2 |- ? ] ->
- Cut ?2 = ?1; [ Clear H; Intros H | Apply sym_equal; XAuto ].
-
- Tactic Definition LeLtGen :=
- Match Context With
- | [ H: (le (S ?1) ?2) |- ? ] ->
- LApply (le_gen_S ?1 ?2); [ Clear H; Intros H | XAuto ];
- XElim H; Intros
- | [ H: (lt ?1 (S ?2)) |- ? ] ->
- LApply (lt_gen_S' ?1 ?2); [ Clear H; Intros H | XAuto ];
- XElim H; [ Intros | Intros H; XElim H; Intros ].
projects / helm.git / blobdiff