X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_function.ma;h=d3d63233c37520c7d982a414ed767a51f6eaa64f;hb=88b32d4e8fe371d59e41cd272064c9d486ae7ec5;hp=8f0f472a9a48b5658595f26da4a598234afa09b4;hpb=ca41435a6021292ccba239aa173651c0be705b45;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/models/q_function.ma b/helm/software/matita/contribs/dama/dama/models/q_function.ma
index 8f0f472a9..d3d63233c 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -12,128 +12,300 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Q/q/q.ma".
-include "list/list.ma".
-include "cprop_connectives.ma". 
+include "nat_ordered_set.ma".
+include "models/q_bars.ma".
 
-notation "\rationals" non associative with precedence 99 for @{'q}.
-interpretation "Q" 'q = Q. 
+axiom le_le_eq: ∀x,y:Q. x ≤ y → y ≤ x → x = y.
 
-record q_f : Type ≝ {
- start : ℚ;
- bars: list (ℚ × ℚ) (* base, height *)
-}.
+lemma initial_shift_same_values:
+  ∀l1:q_f.∀init.init < start l1 →
+   same_values l1 
+     (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).  
+[apply hide; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
+intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
+cases (unpos (start l1-init) H1); intro input;
+simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
+cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input);
+simplify in ⊢ (? ? ? (? ? ? %));
+cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
+whd in ⊢ (% → ?); simplify in H3;
+[1: intro; cases H4; clear H4; rewrite > H3;
+    cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
+    [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
+    |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
+    |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
+        rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
+        symmetry; apply le_n_O_to_eq;
+        rewrite > (sum_bases_O (mk_q_f init (〈w,OQ〉::bars l1)) (\fst w1)); [apply le_n]   
+        clear H6 w2; simplify in H5:(? ? (? ? %));  
+        destruct H3; rewrite > q_d_x_x in H5; assumption;]
+|2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
+    cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
+    [1: cases (?:False); clear w2 H4 w1 H2 w H1; 
+        apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
+    |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
+    |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
+        apply (q_lt_trans ??? H3 H);]
+|3: intro; cases H4; clear H4;   
+    cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
+    [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+        simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+        cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
+        cut (\fst w2 = O); [2: clear H10;
+          symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O l1 (\fst w2)); [apply le_n]
+          apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x; 
+          apply q_eq_to_le; reflexivity;]
+        rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
+        cut (ⅆ[input,init] = Qpos w) as E; [2:
+          rewrite > H2; rewrite < H4; rewrite > q_d_sym; 
+          rewrite > q_d_noabs; [reflexivity] apply q_lt_to_le; assumption;]
+        cases (\fst w1) in H5 H6; intros;
+        [1: cases (?:False); clear H5; simplify in H6;
+            apply (q_lt_corefl ⅆ[input,init]);
+            rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
+            rewrite > q_plus_sym; assumption;
+        |2: cases n in H5 H6; [intros; reflexivity] intros;
+            cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
+            [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));] 
+            apply (q_le_S ??? (sum_bases_ge_OQ ? n1));[apply []|3:apply l]
+            simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus (Qpos w));
+            rewrite > q_elim_minus; apply q_le_minus_r; 
+            rewrite > q_elim_opp; rewrite < E in ⊢ (??%); assumption;]
+    |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
+        simplify in H5 H6 ⊢ %; 
+        cases (\fst w1) in H5 H6; [intros; reflexivity]
+        cases (bars l1);
+        [1: intros; simplify; elim n [reflexivity] simplify; assumption;
+        |2: simplify; intros; cases (?:False); clear H6;
+            apply (q_lt_le_incompat (input - init) (Qpos w) );
+            [1: rewrite > H2; do 2 rewrite > q_elim_minus;
+                apply q_lt_plus; rewrite > q_elim_minus;
+                rewrite < q_plus_assoc; rewrite < q_elim_minus;
+                rewrite > q_plus_minus;rewrite > q_plus_OQ; assumption;
+            |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
+                rewrite > q_d_sym
+                
+                ; apply (q_le_S ???? H5);apply sum_bases_ge_OQ;]]
+    |3: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+        simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
 
-axiom qp : ℚ → ℚ → ℚ.
+axiom nth_nil: ∀T,n.∀d:T. nth [] d n = d.
 
-interpretation "Q plus" 'plus x y = (qp x y).
+lemma key:
+  ∀init,input,l1,w1,w2,w.
+  Qpos w = start l1 - init →   
+  init < start l1 → 
+  start l1 < input →
+  sum_bases (〈w,OQ〉::bars l1) w1 ≤ ⅆ[input,init] →
+    ⅆ[input,init] < sum_bases (bars l1) w1 + (start l1-init) →
+  sum_bases (bars l1) w2 ≤ ⅆ[input,start l1] →
+    ⅆ[input,start l1] < sum_bases (bars l1) (S w2) →
+    \snd (nth (bars l1) ▭ w2) = \snd (nth (〈w,OQ〉::bars l1) ▭ w1).
+intros 4 (init input l); cases l (st l);
+change in match (start (mk_q_f st l)) with st;
+change in match (bars (mk_q_f st l)) with l;
+elim l;
+[1: rewrite > nth_nil; cases w1 in H4;
+    [1: rewrite > q_d_sym; rewrite > q_d_noabs; [2:
+          apply (q_le_trans ? st); apply q_lt_to_le; assumption]
+        do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc;
+        intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
+        simplify in Y; cases (?:False); 
+        apply (q_lt_corefl st); apply (q_lt_trans ??? H2);
+        apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
+        apply q_eq_to_le; reflexivity;
+    |2: intros; simplify; rewrite > nth_nil; reflexivity;]       
+|2: FACTORIZE w1>0    
+    
+     (* interesting case: init < start < input *)
+        intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+        simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+        elim (\fst w2) in H9 H10;
+        [1: elim (\fst w1) in H5 H6;
+            [1: cases (?:False); clear H5 H8 H7; 
+                apply (q_lt_antisym input (start l1)); [2: assumption]
+                rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6; 
+                  [2: apply q_lt_to_le; assumption]
+                rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6; 
+                rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init)); 
+                do 2 rewrite < q_elim_minus; assumption;
+            |2: 
+                
+        cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
+        cases (\fst w1) in H5 H6; intros; [1: 
+          cases (?:False); clear H5 H9 H10; 
+          apply (q_lt_antisym input (start l1)); [2: assumption]
+          rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6; 
+            [2: apply q_lt_to_le; assumption]
+          rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6; 
+          rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init)); 
+          do 2 rewrite < q_elim_minus; assumption;]
+       apply eq_f;
+            cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n));[2:
+              apply (q_le_lt_trans ??? H9);
+              apply (q_lt_trans ??? ? H6);
+              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+              do 2 rewrite > q_elim_minus; rewrite > (q_plus_sym ? (Qopp init));
+              apply q_lt_plus; rewrite > q_plus_sym;
+              rewrite > q_elim_minus; rewrite < q_plus_assoc;
+              rewrite < q_elim_minus; rewrite > q_plus_minus;
+              rewrite > q_plus_OQ; apply q_lt_opp_opp; assumption]
+            clear H9 H6;
+            cut (ⅆ[input,init] - Qpos w = ⅆ[input,start l1]);[2:
+              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+              rewrite > H2; rewrite > (q_elim_minus (start ?));
+              rewrite > q_minus_distrib; rewrite > q_elim_opp;
+              do 2 rewrite > q_elim_minus;
+              do 2 rewrite < q_plus_assoc;
+              rewrite > (q_plus_sym ? init);
+              rewrite > (q_plus_assoc ? init);
+              rewrite > (q_plus_sym ? init);
+              rewrite < (q_elim_minus init); rewrite > q_plus_minus;
+              rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
+              rewrite < q_elim_minus; reflexivity;]
+            cut (sum_bases (bars l1) n < sum_bases (bars l1) (S (\fst w2)));[2:
+              apply (q_le_lt_trans ???? H10); rewrite < Hcut1;
+              rewrite > q_elim_minus; apply q_le_minus_r; rewrite > q_elim_opp;
+              assumption;] clear Hcut1 H5 H10;
+            generalize in match Hcut;generalize in match Hcut2;clear Hcut Hcut2; 
+            apply (nat_elim2 ???? n (\fst w2)); 
+            [3: intros (x y); apply eq_f; apply H5; clear H5;
+                [1: clear H7; apply sum_bases_lt_canc; assumption;
+                |2: clear H6; ]
+            |2: intros; cases (?:False); clear H6;
+                cases n1 in H5; intro;
+                [1: apply (q_lt_corefl ? H5);
+                |2: cases (bars l1) in H5; intro;
+                    [1: simplify in H5; 
+                        apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+                        apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+                        apply q_le_OQ_Qpos;
+                    |2: simplify in H5:(??%);
+                        lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+                        apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+            |1: intro; cases n1 [intros; reflexivity] intros; cases (?:False);
+                elim n2 in H5 H6;
+            
+            
+             elim (bars l1) 0; 
+                [1: intro; elim n1; [reflexivity] cases (?:False);
+                 
+            
+                intros; clear H5;
+                elim n1 in H6; [reflexivity] cases (?:False);
+                [1: apply (q_lt_corefl ? H5);
+                |2: cases (bars l1) in H5; intro;
+                    [1: simplify in H5; 
+                        apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+                        apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+                        apply q_le_OQ_Qpos;
+                    |2: simplify in H5:(??%);
+                        lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+                        apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+qed.
 
-axiom qm : ℚ → ℚ → ℚ.
-
-interpretation "Q minus" 'minus x y = (qm x y).
-
-axiom qlt : ℚ → ℚ → CProp.
-
-interpretation "Q less than" 'lt x y = (qlt x y).
-
-inductive q_comparison (a,b:ℚ) : CProp ≝
- | q_eq : a = b → q_comparison a b 
- | q_lt : a < b → q_comparison a b
- | q_gt : b < a → q_comparison a b.
-
-axiom q_cmp:∀a,b:ℚ.q_comparison a b.
-
-definition qle ≝ λa,b:ℚ.a = b ∨ a < b.
-
-interpretation "Q less or equal than" 'le x y = (qle x y).
-
-notation "'nth'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp l \nbsp d \nbsp i" with precedence 71 
-for @{'nth_appl $l $d $i}.
-interpretation "list nth" 'nth = (cic:/matita/list/list/nth.con _).
-interpretation "list nth" 'nth_appl l d i = (cic:/matita/list/list/nth.con _ l d i).
-
-notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-let rec make_list (A:Type) (def:nat→A) (n:nat) on n ≝
-  match n with
-  [ O ⇒ []
-  | S m ⇒ def m :: make_list A def m].
-
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-interpretation "'mk_list'" 'mk_list = (make_list _).   
-notation < "'mk_list' \nbsp f \nbsp n" 
-with precedence 71 for @{'mk_list_appl $f $n}.
-interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
-
-definition q0 : ℚ × ℚ ≝ 〈OQ,OQ〉.
-notation < "0 \sub \rationals" with precedence 90 for @{'q0}.
-interpretation "q0" 'q0 = q0.
-
-notation < "[ \rationals \sup 2]" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list (Prod Q Q)).
-notation < "[ \rationals \sup 2] \sup 2" with precedence 90 for @{'lq22}.
-interpretation "lq22" 'lq22 = (Prod (list (Prod Q Q)) (list (Prod Q Q))).
+            
+        
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
+definition rebase_spec ≝ 
+ ∀l1,l2:q_f.∃p:q_f × q_f.
+   And4
+    (*len (bars (\fst p)) = len (bars (\snd p))*)
+    (start (\fst p) = start (\snd p))
+    (same_bases (\fst p) (\snd p))
+    (same_values l1 (\fst p)) 
+    (same_values l2 (\snd p)).
 
+definition rebase_spec_simpl ≝ 
+ λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
+   And3
+    (same_bases (mk_q_f start (\fst p)) (mk_q_f start (\snd p)))
+    (same_values (mk_q_f start l1) (mk_q_f start (\fst p))) 
+    (same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
 
-notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = length.
-notation < "'len' \nbsp l" with precedence 70 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
+(* a local letin makes russell fail *)
+definition cb0h : list bar → list bar ≝ 
+  λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l).
 
 definition eject ≝
-  λP.λp:∃x:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).P x.match p with [ex_introT p _ ⇒ p].
-coercion cic:/matita/dama/models/q_function/eject.con.
-definition inject ≝
-  λP.λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).λh:P p. ex_introT ? P p h.
-(*coercion inject with 0 1 nocomposites.*)
-coercion cic:/matita/dama/models/q_function/inject.con 0 1 nocomposites.
-
-definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l q0 i),OQ〉) (length ? l)).
-
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
-definition rebase: 
-  q_f → q_f → 
-    ∃p:q_f × q_f.∀i.
-     \fst (nth (bars (\fst p)) q0 i) = 
-     \fst (nth (bars (\snd p)) q0 i).
-intros (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
-letin spec ≝ (λl1,l2:list (ℚ × ℚ).λm:nat.λz:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).True);
+  λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
+coercion eject.
+definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
+coercion inject with 0 1 nocomposites.
+        
+definition rebase: rebase_spec.
+intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
+letin spec ≝ (
+  λs.λl1,l2.λm.λz.len l1 + len l2 < m → rebase_spec_simpl s l1 l2 z);
+alias symbol "pi1" (instance 34) = "exT \fst".
+alias symbol "pi1" (instance 21) = "exT \fst".
 letin aux ≝ ( 
-let rec aux (l1,l2:list (ℚ × ℚ)) (n:nat) on n : (list (ℚ × ℚ)) × (list (ℚ × ℚ)) ≝
+let rec aux (l1,l2:list bar) (n:nat) on n : (list bar) × (list bar) ≝
 match n with
-[ O ⇒ 〈[],[]〉
-| S m ⇒
+[ O ⇒ 〈 nil ? , nil ? 〉
+| S m ⇒ 
   match l1 with
   [ nil ⇒ 〈cb0h l2, l2〉
   | cons he1 tl1 ⇒
      match l2 with
      [ nil ⇒ 〈l1, cb0h l1〉
      | cons he2 tl2 ⇒  
-         let base1 ≝ (\fst he1) in
-         let base2 ≝ (\fst he2) in
+         let base1 ≝ Qpos (\fst he1) in
+         let base2 ≝ Qpos (\fst he2) in
          let height1 ≝ (\snd he1) in
          let height2 ≝ (\snd he2) in
          match q_cmp base1 base2 with
-         [ q_eq _ ⇒
+         [ q_eq _ ⇒ 
              let rc ≝ aux tl1 tl2 m in 
-             〈he1 :: \fst rc,he2 :: \snd rc〉
-         | q_lt _ ⇒ 
+             〈he1 :: \fst rc,he2 :: \snd rc〉 
+         | q_lt Hp ⇒
              let rest ≝ base2 - base1 in
-             let rc ≝ aux tl1 (〈rest,height2〉 :: tl2) m in
-             〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉
-         | q_gt _ ⇒ 
+             let rc ≝ aux tl1 (〈\fst (unpos rest ?),height2〉 :: tl2) m in
+             〈〈\fst he1,height1〉 :: \fst rc,〈\fst he1,height2〉 :: \snd rc〉 
+         | q_gt Hp ⇒ 
              let rest ≝ base1 - base2 in
-             let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
-             〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉
+             let rc ≝ aux (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
+             〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
 ]]]]
-in aux); : ∀l1,l2,m.∃z.spec l1 l2 m z); 
-  
-cases (q_cmp s1 s2);
-[1: apply (mk_q_f s1);
-|2: apply (mk_q_f s1); cases l2;
-    [1: letin l2' ≝ (
-[1: (* offset: the smallest one *)
-    cases 
+in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
+[9: clearbody aux; unfold spec in aux; clear spec;
+    cases (q_cmp s1 s2);
+    [1: cases (aux l1 l2 (S (len l1 + len l2)));
+        cases (H1 s1 (le_n ?)); clear H1;
+        exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] split;
+        [1,2: assumption;
+        |3: intro; apply (H3 input);
+        |4: intro; rewrite > H in H4; 
+            rewrite > (H4 input); reflexivity;]
+    |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
+          apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+          assumption]
+        cases (aux l1 l2' (S (len l1 + len l2')));
+        cases (H1 s1 (le_n ?)); clear H1 aux;
+        exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s1 (\snd w)〉] split;
+        [1: reflexivity
+        |2: assumption;
+        |3: assumption;
+        |4: intro; rewrite < (H4 input); clear H3 H4 H2 w;
+            cases (value (mk_q_f s1 l2') input);
+            cases (q_cmp input (start (mk_q_f s1 l2'))) in H1;
+            whd in ⊢ (% → ?);
+            [1: intros; cases H2; clear H2; whd in ⊢ (??? %);
+                cases (value (mk_q_f s2 l2) input);
+                cases (q_cmp input (start (mk_q_f s2 l2))) in H2;
+                whd in ⊢ (% → ?);
+                [1: intros; cases H6; clear H6; change with (w1 = w);
+                          
+            (* TODO *) ]]    
+|1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+      assumption;        
+|3:(* TODO *)
+|4:(* TODO *)
+|5:(* TODO *)
+|6:(* TODO *)
+|7:(* TODO *)
+|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
+qed.