3 Require Export terms_defs.
5 Definition wadd : (nat -> nat) -> nat -> (nat -> nat) :=
6 [f;w;n] Cases n of (0) => w | (S m) => (f m) end.
8 Fixpoint weight_map [f:nat->nat; t:T] : nat := Cases t of
11 | (TTail (Bind Abbr) u t) =>
12 (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f u))) t)))
13 | (TTail (Bind _) u t) =>
14 (S (plus (weight_map f u) (weight_map (wadd f (0)) t)))
15 | (TTail _ u t) => (S (plus (weight_map f u) (weight_map f t)))
18 Definition weight : T -> nat := (weight_map [_](0)).
20 Definition tlt : T -> T -> Prop := [t1,t2](lt (weight t1) (weight t2)).
22 Section wadd_props. (*****************************************************)
24 Theorem wadd_le: (f,g:?) ((n:?) (le (f n) (g n))) -> (v,w:?) (le v w) ->
25 (n:?) (le (wadd f v n) (wadd g w n)).
26 XElim n; Simpl; XAuto.
29 Theorem wadd_lt: (f,g:?) ((n:?) (le (f n) (g n))) -> (v,w:?) (lt v w) ->
30 (n:?) (le (wadd f v n) (wadd g w n)).
31 XElim n; Simpl; XAuto.
34 Theorem wadd_O: (n:?) (wadd [_](0) (0) n) = (0).
40 Hints Resolve wadd_le wadd_lt wadd_O : ltlc.
42 Section weight_props. (***************************************************)
44 Theorem weight_le : (t:?; f,g:?) ((n:?) (le (f n) (g n))) ->
45 (le (weight_map f t) (weight_map g t)).
46 XElim t; [ XAuto | Simpl; XAuto | Idtac ].
47 XElim k; Simpl; [ Idtac | XAuto ].
48 XElim b; Auto 7 with ltlc. (**)
51 Theorem weight_eq : (t:?; f,g:?) ((n:?) (f n) = (g n)) ->
52 (weight_map f t) = (weight_map g t).
53 Intros; Apply le_antisym; Apply weight_le;
54 Intros; Rewrite (H n); XAuto.
57 Hints Resolve weight_le weight_eq : ltlc.
59 Theorem weight_add_O : (t:?) (weight_map (wadd [_](0) (0)) t) = (weight_map [_](0) t).
63 Theorem weight_add_S : (t:?; m:?) (le (weight_map (wadd [_](0) (0)) t) (weight_map (wadd [_](0) (S m)) t)).
69 Hints Resolve weight_le weight_add_S : ltlc.
71 Section tlt_props. (******************************************************)
73 Theorem tlt_trans: (v,u,t:?) (tlt u v) -> (tlt v t) -> (tlt u t).
77 Theorem tlt_tail_sx: (k:?; u,t:?) (tlt u (TTail k u t)).
79 XElim k; Simpl; [ XElim b | Idtac ]; XAuto.
82 Theorem tlt_tail_dx: (k:?; u,t:?) (tlt t (TTail k u t)).
84 XElim k; Simpl; [ Idtac | XAuto ].
85 XElim b; Intros; Try Rewrite weight_add_O; [ Idtac | XAuto | XAuto ].
86 EApply lt_le_trans; [ Apply lt_n_Sn | Apply le_n_S ].
87 EApply le_trans; [ Rewrite <- (weight_add_O t); Apply weight_add_S | XAuto ].
92 Hints Resolve tlt_tail_sx tlt_tail_dx tlt_trans : ltlc.
94 Section tlt_wf. (*********************************************************)
96 Local Q: (T -> Prop) -> nat -> Prop :=
97 [P;n] (t:?) (weight t) = n -> (P t).
99 Remark q_ind: (P:T->Prop)((n:?) (Q P n)) -> (t:?) (P t).
101 Apply (H (weight t) t); XAuto.
104 Theorem tlt_wf_ind: (P:T->Prop)
105 ((t:?)((v:?)(tlt v t) -> (P v)) -> (P t)) ->
108 XElimUsing q_ind t; Intros.
109 Apply lt_wf_ind; Clear n; Intros.
110 Unfold Q in H0; Unfold Q; Intros.
111 Rewrite <- H1 in H0; Clear H1.