(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/test/".
+set "baseuri" "cic:/matita/test/prova".
-include "nat/nat.ma".
+include "../legacy/coq.ma".
-theorem pippo: \forall (P,Q,R:nat \to Prop).
- \forall x,y. x=y \to P x \to Q x \to R x.
- intros.
- rewrite > P in Q.
-(*
-theorem pippo: \forall (P,Q,R:nat \to Prop).
- \forall x,y. x=y \to P x \to Q x \to R x.
- intros. rewrite > H in y.
-*)
\ No newline at end of file
+theorem pippo: \forall (A,B:Prop). A \land B \to A.
+ intros; decompose; assumption.
+qed.
+
+inline procedural "cic:/matita/test/prova/pippo.con".
+
+alias id "plus" = "cic:/Coq/Init/Peano/plus.con".
+alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
+alias id "eq" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1)".
+theorem plus_comm: \forall n:nat.\forall m:nat.eq nat (plus n m) (plus m n).
+ intros; alias id "plus_comm" = "cic:/Coq/Arith/Plus/plus_comm.con".
+apply plus_comm.
+qed.
+(*
+include "LAMBDA-TYPES/LambdaDelta-1/preamble.ma".
+alias id "ty3" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/defs/ty3.ind#xpointer(1/1)".
+alias id "pc3" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/defs/pc3.con".
+alias id "THead" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs/T.ind#xpointer(1/1/3)".
+alias id "T" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs/T.ind#xpointer(1/1)".
+alias id "G" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/G/defs/G.ind#xpointer(1/1)".
+alias id "Flat" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs/K.ind#xpointer(1/1/2)".
+alias id "Cast" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs/F.ind#xpointer(1/1/2)".
+alias id "C" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/C/defs/C.ind#xpointer(1/1)".
+theorem ty3_gen_cast:
+ (\forall g:G
+ .\forall c:C
+ .\forall t1:T
+ .\forall t2:T
+ .\forall x:T
+ .ty3 g c (THead (Flat Cast) t2 t1) x
+ \rarr pc3 c t2 x\land ty3 g c t1 t2)
+.
+(* tactics: 80 *)
+intros 6 (g c t1 t2 x H).
+apply insert_eq;(* 6 P P P C I I 3 0 *)
+[apply T(* dependent *)
+|apply (THead (Flat Cast) t2 t1)(* dependent *)
+|apply (\lambda t:T.ty3 g c t x)(* dependent *)
+|intros 2 (y H0).
+alias id "ty3_ind" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/defs/ty3_ind.con".
+elim H0 using ty3_ind names 0;(* 13 P C I I I I I I I C C C I 12 3 *)
+[intros 11 (c0 t0 t UNUSED UNUSED u t3 UNUSED H4 H5 H6).
+letin H7 \def (f_equal T T (\lambda e:T.e) u (THead (Flat Cast) t2 t1) H6).(* 6 C C C C C I *)
+rewrite > H7 in H4:(%) as (H8).
+cut (pc3 c0 t2 t3\land ty3 g c0 t1 t2) as H10;
+[id
+|apply H8.(* 1 I *)
+apply refl_equal(* 2 C C *)
+].
+elim H10 using and_ind names 0.(* 5 P P C I I 3 0 *)
+intros 2 (H11 H12).
+apply conj;(* 4 C C I I *)
+[alias id "pc3_t" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/props/pc3_t.con".
+apply pc3_t;(* 6 P C C I C I *)
+[apply t3(* dependent *)
+|apply H11(* assumption *)
+|apply H5(* assumption *)
+]
+|apply H12(* assumption *)
+]
+|intros 3 (c0 m H1).
+alias id "K" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs/K.ind#xpointer(1/1)".
+cut match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr True
+|TLRef (_:nat)\rArr False
+|THead (_:K) (_:T) (_:T)\rArr False] as H2;
+[id
+|rewrite < H1 in \vdash (%).
+apply I
+].
+clearbody H2.
+change in H2:(%) with match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr True
+|TLRef (_:nat)\rArr False
+|THead (_:K) (_:T) (_:T)\rArr False].
+elim H2 using False_ind names 0(* 2 C I 2 0 *)
+|intros 9 (n c0 d u UNUSED t UNUSED UNUSED H4).
+cut match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr True
+|THead (_:K) (_:T) (_:T)\rArr False] as H5;
+[id
+|rewrite < H4 in \vdash (%).
+apply I
+].
+clearbody H5.
+change in H5:(%) with match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr True
+|THead (_:K) (_:T) (_:T)\rArr False].
+elim H5 using False_ind names 0(* 2 C I 2 0 *)
+|intros 9 (n c0 d u UNUSED t UNUSED UNUSED H4).
+cut match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr True
+|THead (_:K) (_:T) (_:T)\rArr False] as H5;
+[id
+|rewrite < H4 in \vdash (%).
+apply I
+].
+clearbody H5.
+change in H5:(%) with match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr True
+|THead (_:K) (_:T) (_:T)\rArr False].
+elim H5 using False_ind names 0(* 2 C I 2 0 *)
+|intros 14 (c0 u t UNUSED UNUSED b t0 t3 UNUSED UNUSED t4 UNUSED UNUSED H7).
+alias id "F" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs/F.ind#xpointer(1/1)".
+alias id "B" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/T/defs/B.ind#xpointer(1/1)".
+cut match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr False
+|THead (k:K) (_:T) (_:T)\rArr
+ match k in K return \lambda _:K.Prop with
+ [Bind (_:B)\rArr True|Flat (_:F)\rArr False]] as H8;
+[id
+|rewrite < H7 in \vdash (%).
+apply I
+].
+clearbody H8.
+change in H8:(%) with match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr False
+|THead (k:K) (_:T) (_:T)\rArr
+ match k in K return \lambda _:K.Prop with
+ [Bind (_:B)\rArr True|Flat (_:F)\rArr False]].
+elim H8 using False_ind names 0(* 2 C I 2 0 *)
+|intros 10 (c0 w u UNUSED UNUSED v t UNUSED UNUSED H5).
+cut match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr False
+|THead (k:K) (_:T) (_:T)\rArr
+ match k in K return \lambda _:K.Prop with
+ [Bind (_:B)\rArr False
+ |Flat (f:F)\rArr
+ match f in F return \lambda _:F.Prop with
+ [Appl\rArr True|Cast\rArr False]]] as H6;
+[id
+|rewrite < H5 in \vdash (%).
+apply I
+].
+clearbody H6.
+change in H6:(%) with match THead (Flat Cast) t2 t1 in T return \lambda _:T.Prop with
+[TSort (_:nat)\rArr False
+|TLRef (_:nat)\rArr False
+|THead (k:K) (_:T) (_:T)\rArr
+ match k in K return \lambda _:K.Prop with
+ [Bind (_:B)\rArr False
+ |Flat (f:F)\rArr
+ match f in F return \lambda _:F.Prop with
+ [Appl\rArr True|Cast\rArr False]]].
+elim H6 using False_ind names 0(* 2 C I 2 0 *)
+|intros 9 (c0 t0 t3 H1 H2 t4 UNUSED UNUSED H5).
+letin H6 \def (f_equal T T
+ (\lambda e:T
+ .match e in T return \lambda _:T.T with
+ [TSort (_:nat)\rArr t3
+ |TLRef (_:nat)\rArr t3
+ |THead (_:K) (t:T) (_:T)\rArr t]) (THead (Flat Cast) t3 t0)
+ (THead (Flat Cast) t2 t1) H5).(* 6 C C C C C I *)
+letin H7 \def (f_equal T T
+ (\lambda e:T
+ .match e in T return \lambda _:T.T with
+ [TSort (_:nat)\rArr t0
+ |TLRef (_:nat)\rArr t0
+ |THead (_:K) (_:T) (t:T)\rArr t]) (THead (Flat Cast) t3 t0)
+ (THead (Flat Cast) t2 t1) H5).(* 6 C C C C C I *)
+cut (t3=t2\rarr pc3 c0 t2 t3\land ty3 g c0 t1 t2) as DEFINED;
+[id
+|intros 1 (H8).
+rewrite > H8 in H2:(%) as (UNUSED).
+rewrite > H8 in H1:(%) as (H12).
+rewrite > H8 in \vdash (%).
+clearbody H7.
+change in H7:(%) with (match THead (Flat Cast) t3 t0 in T return \lambda _:T.T with
+ [TSort (_:nat)\rArr t0
+ |TLRef (_:nat)\rArr t0
+ |THead (_:K) (_:T) (t:T)\rArr t]
+ =match THead (Flat Cast) t2 t1 in T return \lambda _:T.T with
+ [TSort (_:nat)\rArr t0
+ |TLRef (_:nat)\rArr t0
+ |THead (_:K) (_:T) (t:T)\rArr t]).
+rewrite > H7 in H12:(%) as (H14).
+apply conj;(* 4 C C I I *)
+[alias id "pc3_refl" = "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/pc3/props/pc3_refl.con".
+apply pc3_refl(* 2 C C *)
+|apply H14(* assumption *)
+]
+].
+apply DEFINED.(* 1 I *)
+apply H6(* assumption *)
+]
+|apply H(* assumption *)
+].
+qed.
+*)
+(*
+include "nat/orders.ma".
+
+theorem le_inv:
+ \forall P:nat \to Prop
+ .\forall p2
+ .\forall p1
+ .p2 <= p1 \to
+ (p1 = p2 \to P p2) \to
+ (\forall n1
+ .p2 <= n1 \to
+ (p1 = n1 \to P n1) \to
+ p1 = S n1 \to P (S n1)) \to
+ P p1.
+ intros 4; elim H names 0; clear H p1; intros;
+ [ apply H; reflexivity
+ | apply H3; clear H3; intros;
+ [ apply H | apply H1; clear H1; intros; subst;
+ [ apply H2; apply H3 | ]
+ | reflexivity
+ ]
+*)
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