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[helm.git] / matita / contribs / library_auto / auto / nat / compare.ma

diff --git a/matita/contribs/library_auto/auto/nat/compare.ma b/matita/contribs/library_auto/auto/nat/compare.ma
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+(**************************************************************************)
+(*       ___                                                             *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/compare".
+
+include "datatypes/bool.ma".
+include "datatypes/compare.ma".
+include "auto/nat/orders.ma".
+
+let rec eqb n m \def 
+match n with 
+  [ O \Rightarrow 
+     match m with 
+     [ O \Rightarrow true
+          | (S q) \Rightarrow false] 
+  | (S p) \Rightarrow
+          match m with 
+     [ O \Rightarrow false
+          | (S q) \Rightarrow eqb p q]].
+          
+theorem eqb_to_Prop: \forall n,m:nat. 
+match (eqb n m) with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m].
+intros.
+apply (nat_elim2
+(\lambda n,m:nat.match (eqb n m) with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m]))
+[ intro.
+  elim n1;simplify;autobatch
+  (*[ simplify
+    reflexivity
+  | simplify.
+    apply not_eq_O_S
+  ]*)
+| intro.
+  simplify.
+  unfold Not.
+  intro.
+  apply (not_eq_O_S n1).
+  autobatch
+  (*apply sym_eq.
+  assumption*)
+| intros.
+  simplify.
+  generalize in match H.
+  elim ((eqb n1 m1));simplify
+  [ apply eq_f.    
+    apply H1
+  | unfold Not.
+    intro.
+    apply H1.
+    autobatch
+    (*apply inj_S.
+    assumption*)
+  ]
+]
+qed.
+
+theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
+(n=m \to (P true)) \to (n \neq m \to (P false)) \to (P (eqb n m)). 
+intros.
+cut 
+(match (eqb n m) with
+[ true  \Rightarrow n = m
+| false \Rightarrow n \neq m] \to (P (eqb n m)))
+[ apply Hcut.
+  (* qui autobatch non conclude il goal*)
+  apply eqb_to_Prop
+| elim (eqb n m)
+  [ (*qui autobatch non conclude il goal*)
+    apply ((H H2))
+  | (*qui autobatch non conclude il goal*)
+    apply ((H1 H2))
+  ]
+]
+qed.
+
+theorem eqb_n_n: \forall n. eqb n n = true.
+intro.
+elim n;simplify;autobatch.
+(*[ simplify.reflexivity
+| simplify.assumption.
+]*)
+qed.
+
+theorem eqb_true_to_eq: \forall n,m:nat.
+eqb n m = true \to n = m.
+intros.
+change with 
+match true with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m].
+rewrite < H.
+(*qui autobatch non conclude il goal*)
+apply eqb_to_Prop. 
+qed.
+
+theorem eqb_false_to_not_eq: \forall n,m:nat.
+eqb n m = false \to n \neq m.
+intros.
+change with 
+match false with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m].
+rewrite < H.
+(*qui autobatch non conclude il goal*)
+apply eqb_to_Prop. 
+qed.
+
+theorem eq_to_eqb_true: \forall n,m:nat.
+n = m \to eqb n m = true.
+intros.
+autobatch.
+(*apply (eqb_elim n m)
+[ intros. reflexivity
+| intros.apply False_ind.apply (H1 H)
+]*)
+qed.
+
+theorem not_eq_to_eqb_false: \forall n,m:nat.
+\lnot (n = m) \to eqb n m = false.
+intros.apply (eqb_elim n m);intros
+[ apply False_ind.
+  apply (H H1)
+| reflexivity
+]
+qed.
+
+let rec leb n m \def 
+match n with 
+    [ O \Rightarrow true
+    | (S p) \Rightarrow
+       match m with 
+        [ O \Rightarrow false
+       | (S q) \Rightarrow leb p q]].
+       
+theorem leb_to_Prop: \forall n,m:nat. 
+match (leb n m) with
+[ true  \Rightarrow n \leq m 
+| false \Rightarrow n \nleq m].
+intros.
+apply (nat_elim2
+(\lambda n,m:nat.match (leb n m) with
+[ true  \Rightarrow n \leq m 
+| false \Rightarrow n \nleq m]))
+[ simplify.
+  exact le_O_n
+| simplify.
+  exact not_le_Sn_O
+| intros 2.
+  simplify.
+  elim ((leb n1 m1));simplify
+  [ apply le_S_S.
+     (*qui autobatch non conclude il goal*)
+    apply H
+  | unfold Not.
+    intros.
+    apply H.
+    autobatch
+    (*apply le_S_S_to_le.
+    assumption*)
+  ]
+]
+qed.
+
+theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop. 
+(n \leq m \to (P true)) \to (n \nleq m \to (P false)) \to
+P (leb n m).
+intros.
+cut 
+(match (leb n m) with
+[ true  \Rightarrow n \leq m
+| false \Rightarrow n \nleq m] \to (P (leb n m)))
+[ apply Hcut.
+  (*qui autobatch non conclude il goal*)
+  apply leb_to_Prop
+| elim (leb n m)
+  [ (*qui autobatch non conclude il goal*)
+    apply ((H H2))
+  | (*qui autobatch non conclude il goal*)
+    apply ((H1 H2))
+  ]
+]
+qed.
+
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow 
+    match m with 
+      [ O \Rightarrow EQ
+      | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow 
+    match m with 
+      [ O \Rightarrow GT
+      | (S q) \Rightarrow nat_compare p q]].
+(**********)
+theorem nat_compare_n_n: \forall n:nat. nat_compare n n = EQ.
+intro.elim n
+[ autobatch
+  (*simplify.
+  reflexivity*)
+| simplify.
+  assumption
+]
+qed.
+
+theorem nat_compare_S_S: \forall n,m:nat. 
+nat_compare n m = nat_compare (S n) (S m).
+intros.autobatch.
+(*simplify.reflexivity.*)
+qed.
+
+theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
+intro.
+elim n;autobatch.
+(*[ apply False_ind.
+  exact (not_le_Sn_O O H)
+| apply eq_f.
+  apply pred_Sn
+]*)
+qed.
+
+theorem nat_compare_pred_pred: 
+\forall n,m:nat.lt O n \to lt O m \to 
+eq compare (nat_compare n m) (nat_compare (pred n) (pred m)).
+intros.
+apply (lt_O_n_elim n H).
+apply (lt_O_n_elim m H1).
+intros.
+autobatch.
+(*simplify.reflexivity.*)
+qed.
+
+theorem nat_compare_to_Prop: \forall n,m:nat. 
+match (nat_compare n m) with
+  [ LT \Rightarrow n < m
+  | EQ \Rightarrow n=m
+  | GT \Rightarrow m < n ].
+intros.
+apply (nat_elim2 (\lambda n,m.match (nat_compare n m) with
+  [ LT \Rightarrow n < m
+  | EQ \Rightarrow n=m
+  | GT \Rightarrow m < n ]))
+[ intro.
+  elim n1;simplify;autobatch
+  (*[ reflexivity
+  | unfold lt.
+    apply le_S_S.
+    apply le_O_n
+  ]*)
+| intro.
+  simplify.
+  autobatch
+  (*unfold lt.
+  apply le_S_S. 
+  apply le_O_n*)
+| intros 2.
+  simplify.
+  elim ((nat_compare n1 m1));simplify
+  [ unfold lt.
+    apply le_S_S.
+    (*qui autobatch non chiude il goal*)
+    apply H
+  | apply eq_f.
+    (*qui autobatch non chiude il goal*)
+    apply H
+  | unfold lt.
+    apply le_S_S.
+    (*qui autobatch non chiude il goal*)
+    apply H
+  ]
+]
+qed.
+
+theorem nat_compare_n_m_m_n: \forall n,m:nat. 
+nat_compare n m = compare_invert (nat_compare m n).
+intros. 
+apply (nat_elim2 (\lambda n,m. nat_compare n m = compare_invert (nat_compare m n)));intros
+[ elim n1;autobatch(*;simplify;reflexivity*)
+| elim n1;autobatch(*;simplify;reflexivity*)  
+| autobatch
+  (*simplify.elim H.reflexivity*)
+]
+qed.
+     
+theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
+(n < m \to P LT) \to (n=m \to P EQ) \to (m < n \to P GT) \to 
+(P (nat_compare n m)).
+intros.
+cut (match (nat_compare n m) with
+[ LT \Rightarrow n < m
+| EQ \Rightarrow n=m
+| GT \Rightarrow m < n] \to
+(P (nat_compare n m)))
+[ apply Hcut.
+  (*autobatch non chiude il goal*)
+  apply nat_compare_to_Prop
+| elim ((nat_compare n m))
+  [ (*autobatch non chiude il goal*)
+    apply ((H H3))
+  | (*autobatch non chiude il goal*)
+    apply ((H1 H3))
+  | (*autobatch non chiude il goal*)
+    apply ((H2 H3))
+  ]
+]
+qed.