Creation du fichier Zhints.v repertoriant les thms de ZArith et definissant les thms interessants en hints

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@1775 85f007b7-540e-0410-9357-904b9bb8a0f7

4 files changed, 384 insertions, 3 deletions

diff --git a/.depend.coq b/.depend.coq
index 73cf87b8a3..bd6e04358c 100644
--- a/.depend.coq
+++ b/.depend.coq
@@ -1,10 +1,12 @@
 theories/ZArith/zarith_aux.vo: theories/ZArith/zarith_aux.v theories/Arith/Arith.vo theories/ZArith/fast_integer.vo
+theories/ZArith/positive_hints.vo: theories/ZArith/positive_hints.v
 theories/ZArith/fast_integer.vo: theories/ZArith/fast_integer.v theories/Arith/Le.vo theories/Arith/Lt.vo theories/Arith/Plus.vo theories/Arith/Mult.vo theories/Arith/Minus.vo
 theories/ZArith/auxiliary.vo: theories/ZArith/auxiliary.v theories/Arith/Arith.vo theories/ZArith/fast_integer.vo theories/ZArith/zarith_aux.vo theories/Logic/Decidable.vo theories/Arith/Peano_dec.vo theories/Arith/Compare_dec.vo
 theories/ZArith/Zsyntax.vo: theories/ZArith/Zsyntax.v theories/ZArith/fast_integer.vo theories/ZArith/zarith_aux.vo
 theories/ZArith/Zmisc.vo: theories/ZArith/Zmisc.v theories/ZArith/fast_integer.vo theories/ZArith/zarith_aux.vo theories/ZArith/auxiliary.vo theories/ZArith/Zsyntax.vo theories/Bool/Bool.vo
+theories/ZArith/Zhints.vo: theories/ZArith/Zhints.v theories/ZArith/zarith_aux.vo theories/ZArith/auxiliary.vo theories/ZArith/Zsyntax.vo theories/ZArith/Zmisc.vo theories/ZArith/Wf_Z.vo
 theories/ZArith/ZArith_dec.vo: theories/ZArith/ZArith_dec.v theories/Bool/Sumbool.vo theories/ZArith/fast_integer.vo theories/ZArith/zarith_aux.vo theories/ZArith/auxiliary.vo theories/ZArith/Zsyntax.vo
-theories/ZArith/ZArith.vo: theories/ZArith/ZArith.v theories/ZArith/fast_integer.vo theories/ZArith/zarith_aux.vo theories/ZArith/auxiliary.vo theories/ZArith/Zsyntax.vo theories/ZArith/ZArith_dec.vo theories/ZArith/Zmisc.vo theories/ZArith/Wf_Z.vo
+theories/ZArith/ZArith.vo: theories/ZArith/ZArith.v theories/ZArith/fast_integer.vo theories/ZArith/zarith_aux.vo theories/ZArith/auxiliary.vo theories/ZArith/Zsyntax.vo theories/ZArith/ZArith_dec.vo theories/ZArith/Zmisc.vo theories/ZArith/Wf_Z.vo theories/ZArith/Zhints.vo
 theories/ZArith/Wf_Z.vo: theories/ZArith/Wf_Z.v theories/ZArith/fast_integer.vo theories/ZArith/zarith_aux.vo theories/ZArith/auxiliary.vo theories/ZArith/Zsyntax.vo
 theories/Wellfounded/Wellfounded.vo: theories/Wellfounded/Wellfounded.v theories/Wellfounded/Disjoint_Union.vo theories/Wellfounded/Inclusion.vo theories/Wellfounded/Inverse_Image.vo theories/Wellfounded/Lexicographic_Exponentiation.vo theories/Wellfounded/Lexicographic_Product.vo theories/Wellfounded/Transitive_Closure.vo theories/Wellfounded/Union.vo theories/Wellfounded/Well_Ordering.vo
 theories/Wellfounded/Well_Ordering.vo: theories/Wellfounded/Well_Ordering.v theories/Logic/Eqdep.vo
@@ -214,7 +216,7 @@ contrib/omega/Omega.vo: contrib/omega/Omega.v theories/ZArith/ZArith.vo theories
 contrib/interface/Centaur.vo: contrib/interface/Centaur.v contrib/interface/paths.cmo contrib/interface/name_to_ast.cmo contrib/interface/xlate.cmo contrib/interface/vtp.cmo contrib/interface/translate.cmo contrib/interface/pbp.cmo contrib/interface/dad.cmo contrib/interface/showproof_ct.cmo contrib/interface/showproof.cmo contrib/interface/debug_tac.cmo contrib/interface/history.cmo contrib/interface/centaur.cmo
 contrib/interface/AddDad.vo: contrib/interface/AddDad.v
 contrib/fourier/Fourier_util.vo: contrib/fourier/Fourier_util.v theories/Reals/Rbase.vo
-contrib/fourier/Fourier.vo: contrib/fourier/Fourier.v contrib/ring/quote.cmo contrib/ring/ring.cmo contrib/fourier/fourier.cmo contrib/fourier/fourierR.cmo contrib/fourier/Fourier_util.vo
+contrib/fourier/Fourier.vo: contrib/fourier/Fourier.v contrib/ring/quote.cmo contrib/ring/ring.cmo contrib/fourier/fourier.cmo contrib/fourier/fourierR.cmo contrib/field/field.cmo contrib/fourier/Fourier_util.vo contrib/field/Field.vo theories/Reals/DiscrR.vo
 contrib/field/Field_Theory.vo: contrib/field/Field_Theory.v theories/Arith/Peano_dec.vo contrib/ring/Ring.vo contrib/field/Field_Compl.vo
 contrib/field/Field_Tactic.vo: contrib/field/Field_Tactic.v contrib/ring/Ring.vo contrib/field/Field_Compl.vo contrib/field/Field_Theory.vo
 contrib/field/Field_Compl.vo: contrib/field/Field_Compl.v
diff --git a/Makefile b/Makefile
index 81bac199a7..a1db4ccebd 100644
--- a/Makefile
+++ b/Makefile
@@ -367,7 +367,8 @@ BOOLVO=theories/Bool/Bool.vo  		theories/Bool/IfProp.vo \
 ZARITHVO=theories/ZArith/Wf_Z.vo        theories/ZArith/Zsyntax.vo \
 	 theories/ZArith/ZArith.vo      theories/ZArith/auxiliary.vo \
 	 theories/ZArith/ZArith_dec.vo  theories/ZArith/fast_integer.vo \
-	 theories/ZArith/Zmisc.vo       theories/ZArith/zarith_aux.vo
+	 theories/ZArith/Zmisc.vo       theories/ZArith/zarith_aux.vo \
+	 theories/ZArith/Zhints.vo
 
 LISTSVO=theories/Lists/List.vo      theories/Lists/PolyListSyntax.vo \
         theories/Lists/ListSet.vo   theories/Lists/Streams.vo \
diff --git a/theories/ZArith/ZArith.v b/theories/ZArith/ZArith.v
index b9b572137d..840d00a7fb 100644
--- a/theories/ZArith/ZArith.v
+++ b/theories/ZArith/ZArith.v
@@ -21,3 +21,5 @@ Require Export Wf_Z.
 Hints Resolve Zle_n Zplus_sym Zplus_assoc Zmult_sym Zmult_assoc
   Zero_left Zero_right Zmult_one Zplus_inverse_l Zplus_inverse_r 
   Zmult_plus_distr_l Zmult_plus_distr_r : zarith.
+
+Require Export Zhints.
diff --git a/theories/ZArith/Zhints.v b/theories/ZArith/Zhints.v
new file mode 100644
index 0000000000..6c8cc88fc0
--- /dev/null
+++ b/theories/ZArith/Zhints.v
@@ -0,0 +1,376 @@
+(***********************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
+(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
+(*   \VV/  *************************************************************)
+(*    //   *      This file is distributed under the terms of the      *)
+(*         *       GNU Lesser General Public License Version 2.1       *)
+(***********************************************************************)
+
+(*i $Id$ i*)
+
+(* This file centralizes the lemmas about Z, classifying them *)
+(* according to the way they can be used in automatic search  *)
+
+(* Lemmas which clearly leads to simplification during proof search are *)
+(* declared as Hints. A definite status (Hint or not) for the other lemmas *)
+(* remains to be given *)
+
+(* Structure of the file *)
+(* - simplification lemmas (only those are declared as Hints) *)
+(* - reversible lemmas relating operators *)
+(* - useful Bottom-up lemmas              *)
+(* - irreversible lemmas with meta-variables *)
+(* - unclear or too specific lemmas       *)
+(* - lemmas to be used as rewrite rules   *)
+
+(* Lemmas involving positive and compare are not taken into account *)
+
+Require zarith_aux.
+Require auxiliary.
+Require Zsyntax.
+Require Zmisc.
+Require Wf_Z.
+
+(**********************************************************************)
+(*                  Simplification lemmas                             *)
+(* No subgoal or smaller subgoals                                     *)
+
+Hints Resolve
+  (* A) Reversible simplification lemmas (no loss of information)       *)
+  (* Should clearly declared as hints                                   *)
+  
+  (* Lemmas ending by eq *)
+  Zeq_S (* :(n,m:Z)`n = m`->`(Zs n) = (Zs m)` *)
+  
+  (* Lemmas ending by Zgt *)
+  Zgt_n_S (* :(n,m:Z)`m > n`->`(Zs m) > (Zs n)` *)
+  Zgt_Sn_n (* :(n:Z)`(Zs n) > n` *)
+  POS_gt_ZERO (* :(p:positive)`(POS p) > 0` *)
+  Zgt_reg_l (* :(n,m,p:Z)`n > m`->`p+n > p+m` *)
+  Zgt_reg_r (* :(n,m,p:Z)`n > m`->`n+p > m+p` *)
+  
+  (* Lemmas ending by Zlt *)
+  Zlt_n_Sn (* :(n:Z)`n < (Zs n)` *)
+  Zlt_n_S (* :(n,m:Z)`n < m`->`(Zs n) < (Zs m)` *)
+  Zlt_pred_n_n (* :(n:Z)`(Zpred n) < n` *)
+  Zlt_reg_l (* :(n,m,p:Z)`n < m`->`p+n < p+m` *)
+  Zlt_reg_r (* :(n,m,p:Z)`n < m`->`n+p < m+p` *)
+  
+  (* Lemmas ending by Zle *)
+  ZERO_le_inj (* :(n:nat)`0 <= (inject_nat n)` *)
+  ZERO_le_POS (* :(p:positive)`0 <= (POS p)` *)
+  Zle_n (* :(n:Z)`n <= n` *)
+  Zle_n_Sn (* :(n:Z)`n <= (Zs n)` *)
+  Zle_n_S (* :(n,m:Z)`m <= n`->`(Zs m) <= (Zs n)` *)
+  Zle_pred_n (* :(n:Z)`(Zpred n) <= n` *)
+  Zle_min_l (* :(n,m:Z)`(Zmin n m) <= n` *)
+  Zle_min_r (* :(n,m:Z)`(Zmin n m) <= m` *)
+  Zle_reg_l (* :(n,m,p:Z)`n <= m`->`p+n <= p+m` *)
+  Zle_reg_r (* :(a,b,c:Z)`a <= b`->`a+c <= b+c` *)
+  
+  (* B) Irreversible simplification lemmas : Probably to be declared as *)
+  (* hints, when no other simplification is possible *)
+  
+  (* Lemmas ending by eq *)
+  Z_eq_mult (* :(x,y:Z)`y = 0`->`y*x = 0` *)
+  Zplus_simpl (* :(n,m,p,q:Z)`n = m`->`p = q`->`n+p = m+q` *)
+  
+  (* Lemmas ending by Zge *)
+  Zge_Zmult_pos_right (* :(a,b,c:Z)`a >= b`->`c >= 0`->`a*c >= b*c` *)
+  Zge_Zmult_pos_left (* :(a,b,c:Z)`a >= b`->`c >= 0`->`c*a >= c*b` *)
+  Zge_Zmult_pos_compat (* :
+    (a,b,c,d:Z)`a >= c`->`b >= d`->`c >= 0`->`d >= 0`->`a*b >= c*d` *)
+  
+  (* Lemmas ending by Zlt *)
+  Zgt_ZERO_mult (* :(a,b:Z)`a > 0`->`b > 0`->`a*b > 0` *)
+  Zlt_S (* :(n,m:Z)`n < m`->`n < (Zs m)` *)
+  
+  (* Lemmas ending by Zle *)
+  Zle_ZERO_mult (* :(x,y:Z)`0 <= x`->`0 <= y`->`0 <= x*y` *)
+  Zle_Zmult_pos_right (* :(a,b,c:Z)`a <= b`->`0 <= c`->`a*c <= b*c` *)
+  Zle_Zmult_pos_left (* :(a,b,c:Z)`a <= b`->`0 <= c`->`c*a <= c*b` *)
+  OMEGA2 (* :(x,y:Z)`0 <= x`->`0 <= y`->`0 <= x+y` *)
+  Zle_le_S (* :(x,y:Z)`x <= y`->`x <= (Zs y)` *)
+  Zle_plus_plus (* :(n,m,p,q:Z)`n <= m`->`p <= q`->`n+p <= m+q` *)
+
+: zarith.
+  
+(**********************************************************************)
+(*         Reversible lemmas relating operators                       *)
+(* Probably to be declared as hints but need to define precedences    *)
+
+(* A) Conversion between comparisons/predicates and arithmetic operators
+
+(* Lemmas ending by eq *)
+Zegal_left: (x,y:Z)`x = y`->`x+(-y) = 0`
+Zabs_eq: (x:Z)`0 <= x`->`|x| = x`
+Zeven_div2: (x:Z)(Zeven x)->`x = 2*(Zdiv2 x)`
+Zodd_div2: (x:Z)`x >= 0`->(Zodd x)->`x = 2*(Zdiv2 x)+1`
+
+(* Lemmas ending by Zgt *)
+Zgt_left_rev: (x,y:Z)`x+(-y) > 0`->`x > y`
+Zgt_left_gt: (x,y:Z)`x > y`->`x+(-y) > 0`
+
+(* Lemmas ending by Zlt *)
+Zlt_left_rev: (x,y:Z)`0 < y+(-x)`->`x < y`
+Zlt_left_lt: (x,y:Z)`x < y`->`0 < y+(-x)`
+Zlt_O_minus_lt: (n,m:Z)`0 < n-m`->`m < n`
+
+(* Lemmas ending by Zle *)
+Zle_left: (x,y:Z)`x <= y`->`0 <= y+(-x)`
+Zle_left_rev: (x,y:Z)`0 <= y+(-x)`->`x <= y`
+Zlt_left: (x,y:Z)`x < y`->`0 <= y+(-1)+(-x)`
+Zge_left: (x,y:Z)`x >= y`->`0 <= x+(-y)`
+Zgt_left: (x,y:Z)`x > y`->`0 <= x+(-1)+(-y)`
+
+(* B) Conversion between nat comparisons and Z comparisons *)
+
+(* Lemmas ending by eq *)
+inj_eq: (x,y:nat)x=y->`(inject_nat x) = (inject_nat y)`
+
+(* Lemmas ending by Zge *)
+inj_ge: (x,y:nat)(ge x y)->`(inject_nat x) >= (inject_nat y)`
+
+(* Lemmas ending by Zgt *)
+inj_gt: (x,y:nat)(gt x y)->`(inject_nat x) > (inject_nat y)`
+
+(* Lemmas ending by Zlt *)
+inj_lt: (x,y:nat)(lt x y)->`(inject_nat x) < (inject_nat y)`
+
+(* Lemmas ending by Zle *)
+inj_le: (x,y:nat)(le x y)->`(inject_nat x) <= (inject_nat y)`
+
+(* C) Conversion between comparisons *)
+
+(* Lemmas ending by Zge *)
+not_Zlt: (x,y:Z)~`x < y`->`x >= y`
+Zle_ge: (m,n:Z)`m <= n`->`n >= m`
+
+(* Lemmas ending by Zgt *)
+Zle_gt_S: (n,p:Z)`n <= p`->`(Zs p) > n`
+not_Zle: (x,y:Z)~`x <= y`->`x > y`
+Zlt_gt: (m,n:Z)`m < n`->`n > m`
+Zle_S_gt: (n,m:Z)`(Zs n) <= m`->`m > n`
+
+(* Lemmas ending by Zlt *)
+not_Zge: (x,y:Z)~`x >= y`->`x < y`
+Zgt_lt: (m,n:Z)`m > n`->`n < m`
+Zle_lt_n_Sm: (n,m:Z)`n <= m`->`n < (Zs m)`
+
+(* Lemmas ending by Zle *)
+Zlt_ZERO_pred_le_ZERO: (x:Z)`0 < x`->`0 <= (Zpred x)`
+not_Zgt: (x,y:Z)~`x > y`->`x <= y`
+Zgt_le_S: (n,p:Z)`p > n`->`(Zs n) <= p`
+Zgt_S_le: (n,p:Z)`(Zs p) > n`->`n <= p`
+Zge_le: (m,n:Z)`m >= n`->`n <= m`
+Zlt_le_S: (n,p:Z)`n < p`->`(Zs n) <= p`
+Zlt_n_Sm_le: (n,m:Z)`n < (Zs m)`->`n <= m`
+Zlt_le_weak: (n,m:Z)`n < m`->`n <= m`
+Zle_refl: (n,m:Z)`n = m`->`n <= m`
+
+(* D) Irreversible simplification involving several comparaisons, *)
+(*    useful with clear precedences *)
+
+(* Lemmas ending by Zlt *)
+Zlt_le_reg :(a,b,c,d:Z)`a < b`->`c <= d`->`a+c < b+d`
+Zle_lt_reg : (a,b,c,d:Z)`a <= b`->`c < d`->`a+c < b+d`
+
+(* D) What is decreasing here ? *)
+
+(* Lemmas ending by eq *)
+Zplus_minus: (n,m,p:Z)`n = m+p`->`p = n-m`
+
+(* Lemmas ending by Zgt *)
+Zgt_pred: (n,p:Z)`p > (Zs n)`->`(Zpred p) > n`
+
+(* Lemmas ending by Zlt *)
+Zlt_pred: (n,p:Z)`(Zs n) < p`->`n < (Zpred p)`
+*)
+
+(**********************************************************************)
+(*                 Useful Bottom-up lemmas                            *)
+
+(* A) Bottom-up simplification: should be used 
+
+(* Lemmas ending by eq *)
+Zeq_add_S: (n,m:Z)`(Zs n) = (Zs m)`->`n = m`
+Zsimpl_plus_l: (n,m,p:Z)`n+m = n+p`->`m = p`
+Zplus_unit_left: (n,m:Z)`n+0 = m`->`n = m`
+Zplus_unit_right: (n,m:Z)`n = m+0`->`n = m`
+
+(* Lemmas ending by Zgt *)
+Zsimpl_gt_plus_l: (n,m,p:Z)`p+n > p+m`->`n > m`
+Zsimpl_gt_plus_r: (n,m,p:Z)`n+p > m+p`->`n > m`
+Zgt_S_n: (n,p:Z)`(Zs p) > (Zs n)`->`p > n`
+
+(* Lemmas ending by Zlt *)
+Zsimpl_lt_plus_l: (n,m,p:Z)`p+n < p+m`->`n < m`
+Zsimpl_lt_plus_r: (n,m,p:Z)`n+p < m+p`->`n < m`
+Zlt_S_n: (n,m:Z)`(Zs n) < (Zs m)`->`n < m`
+
+(* Lemmas ending by Zle *)
+Zsimpl_le_plus_l: (p,n,m:Z)`p+n <= p+m`->`n <= m`
+Zsimpl_le_plus_r: (p,n,m:Z)`n+p <= m+p`->`n <= m`
+Zle_S_n: (n,m:Z)`(Zs m) <= (Zs n)`->`m <= n`
+
+(* B) Bottom-up irreversible (syntactic) simplification *)
+
+(* Lemmas ending by Zle *)
+Zle_trans_S: (n,m:Z)`(Zs n) <= m`->`n <= m`
+
+(* C) Other unclearly simplifying lemmas *)
+
+(* Lemmas ending by Zeq *)
+Zmult_eq: (x,y:Z)`x <> 0`->`y*x = 0`->`y = 0`
+
+(* Lemmas ending by Zgt *)
+Zmult_gt: (x,y:Z)`x > 0`->`x*y > 0`->`y > 0`
+
+(* Lemmas ending by Zlt *)
+pZmult_lt: (x,y:Z)`x > 0`->`0 < y*x`->`0 < y`
+
+(* Lemmas ending by Zle *)
+Zmult_le: (x,y:Z)`x > 0`->`0 <= y*x`->`0 <= y`
+OMEGA1: (x,y:Z)`x = y`->`0 <= x`->`0 <= y`
+*)
+
+(**********************************************************************)
+(*           Irreversible lemmas with meta-variables                  *)
+(* To be used by EAuto                                                
+
+Hints Immediate
+(* Lemmas ending by eq *)
+Zle_antisym: (n,m:Z)`n <= m`->`m <= n`->`n = m`
+
+(* Lemmas ending by Zge *)
+Zge_trans: (n,m,p:Z)`n >= m`->`m >= p`->`n >= p`
+
+(* Lemmas ending by Zgt *)
+Zgt_trans: (n,m,p:Z)`n > m`->`m > p`->`n > p`
+Zgt_trans_S: (n,m,p:Z)`(Zs n) > m`->`m > p`->`n > p`
+Zle_gt_trans: (n,m,p:Z)`m <= n`->`m > p`->`n > p`
+Zgt_le_trans: (n,m,p:Z)`n > m`->`p <= m`->`n > p`
+
+(* Lemmas ending by Zlt *)
+Zlt_trans: (n,m,p:Z)`n < m`->`m < p`->`n < p`
+Zlt_le_trans: (n,m,p:Z)`n < m`->`m <= p`->`n < p`
+Zle_lt_trans: (n,m,p:Z)`n <= m`->`m < p`->`n < p`
+
+(* Lemmas ending by Zle *)
+Zle_trans: (n,m,p:Z)`n <= m`->`m <= p`->`n <= p`
+*)
+
+(**********************************************************************)
+(*               Unclear or too specific lemmas                       *)
+(* Not to be used ??                                                  *)
+
+(* A) Irreversible and too specific (not enough regular) 
+
+(* Lemmas ending by Zle *)
+Zle_mult: (x,y:Z)`x > 0`->`0 <= y`->`0 <= y*x`
+Zle_mult_approx: (x,y,z:Z)`x > 0`->`z > 0`->`0 <= y`->`0 <= y*x+z`
+OMEGA6: (x,y,z:Z)`0 <= x`->`y = 0`->`0 <= x+y*z`
+OMEGA7: (x,y,z,t:Z)`z > 0`->`t > 0`->`0 <= x`->`0 <= y`->`0 <= x*z+y*t`
+
+
+(* B) Expansion and too specific ? *)
+
+(* Lemmas ending by Zge *)
+Zge_mult_simpl: (a,b,c:Z)`c > 0`->`a*c >= b*c`->`a >= b`
+
+(* Lemmas ending by Zgt *)
+Zgt_mult_simpl: (a,b,c:Z)`c > 0`->`a*c > b*c`->`a > b`
+Zgt_square_simpl: (x,y:Z)`x >= 0`->`y >= 0`->`x*x > y*y`->`x > y`
+
+(* Lemmas ending by Zle *)
+Zle_mult_simpl: (a,b,c:Z)`c > 0`->`a*c <= b*c`->`a <= b`
+Zmult_le_approx: (x,y,z:Z)`x > 0`->`x > z`->`0 <= y*x+z`->`0 <= y`
+
+(* C) Reversible but too specific ? *)
+
+(* Lemmas ending by Zlt *)
+Zlt_minus: (n,m:Z)`0 < m`->`n-m < n`
+*)
+
+(**********************************************************************)
+(*               Lemmas to be used as rewrite rules                   *)
+(* but can also be used as hints                                      
+
+(* Left-to-right simplification lemmas (a symbol disappears) *)
+
+Zcompare_n_S: (n,m:Z)(Zcompare (Zs n) (Zs m))=(Zcompare n m)
+Zmin_n_n: (n:Z)`(Zmin n n) = n`
+Zmult_1_n: (n:Z)`1*n = n`
+Zmult_n_1: (n:Z)`n*1 = n`
+Zminus_plus: (n,m:Z)`n+m-n = m`
+Zle_plus_minus: (n,m:Z)`n+(m-n) = m`
+Zopp_Zopp: (x:Z)`(-(-x)) = x`
+Zero_left: (x:Z)`0+x = x`
+Zero_right: (x:Z)`x+0 = x`
+Zplus_inverse_r: (x:Z)`x+(-x) = 0`
+Zplus_inverse_l: (x:Z)`(-x)+x = 0`
+Zopp_intro: (x,y:Z)`(-x) = (-y)`->`x = y`
+Zmult_one: (x:Z)`1*x = x`
+Zero_mult_left: (x:Z)`0*x = 0`
+Zero_mult_right: (x:Z)`x*0 = 0`
+Zmult_Zopp_Zopp: (x,y:Z)`(-x)*(-y) = x*y`
+
+(* Right-to-left simplification lemmas (a symbol disappears) *)
+
+Zpred_Sn: (m:Z)`m = (Zpred (Zs m))`
+Zs_pred: (n:Z)`n = (Zs (Zpred n))`
+Zplus_n_O: (n:Z)`n = n+0`
+Zmult_n_O: (n:Z)`0 = n*0`
+Zminus_n_O: (n:Z)`n = n-0`
+Zminus_n_n: (n:Z)`0 = n-n`
+Zred_factor6: (x:Z)`x = x+0`
+Zred_factor0: (x:Z)`x = x*1`
+
+(* Unclear orientation (no symbol disappears) *)
+
+Zplus_n_Sm: (n,m:Z)`(Zs (n+m)) = n+(Zs m)`
+Zmult_n_Sm: (n,m:Z)`n*m+n = n*(Zs m)`
+Zmin_SS: (n,m:Z)`(Zs (Zmin n m)) = (Zmin (Zs n) (Zs m))`
+Zplus_assoc_l: (n,m,p:Z)`n+(m+p) = n+m+p`
+Zplus_assoc_r: (n,m,p:Z)`n+m+p = n+(m+p)`
+Zplus_permute: (n,m,p:Z)`n+(m+p) = m+(n+p)`
+Zplus_Snm_nSm: (n,m:Z)`(Zs n)+m = n+(Zs m)`
+Zminus_plus_simpl: (n,m,p:Z)`n-m = p+n-(p+m)`
+Zminus_Sn_m: (n,m:Z)`(Zs (n-m)) = (Zs n)-m`
+Zmult_plus_distr_l: (n,m,p:Z)`(n+m)*p = n*p+m*p`
+Zmult_minus_distr: (n,m,p:Z)`(n-m)*p = n*p-m*p`
+Zmult_assoc_r: (n,m,p:Z)`n*m*p = n*(m*p)`
+Zmult_assoc_l: (n,m,p:Z)`n*(m*p) = n*m*p`
+Zmult_permute: (n,m,p:Z)`n*(m*p) = m*(n*p)`
+Zmult_Sm_n: (n,m:Z)`n*m+m = (Zs n)*m`
+Zmult_Zplus_distr: (x,y,z:Z)`x*(y+z) = x*y+x*z`
+Zmult_plus_distr: (n,m,p:Z)`(n+m)*p = n*p+m*p`
+Zopp_Zplus: (x,y:Z)`(-(x+y)) = (-x)+(-y)`
+Zplus_sym: (x,y:Z)`x+y = y+x`
+Zplus_assoc: (x,y,z:Z)`x+(y+z) = x+y+z`
+Zmult_sym: (x,y:Z)`x*y = y*x`
+Zmult_assoc: (x,y,z:Z)`x*(y*z) = x*y*z`
+Zopp_Zmult: (x,y:Z)`(-x)*y = (-(x*y))`
+Zplus_S_n: (x,y:Z)`(Zs x)+y = (Zs (x+y))`
+Zopp_one: (x:Z)`(-x) = x*(-1)`
+Zopp_Zmult_r: (x,y:Z)`(-(x*y)) = x*(-y)`
+Zmult_Zopp_left: (x,y:Z)`(-x)*y = x*(-y)`
+Zopp_Zmult_l: (x,y:Z)`(-(x*y)) = (-x)*y`
+Zred_factor1: (x:Z)`x+x = x*2`
+Zred_factor2: (x,y:Z)`x+x*y = x*(1+y)`
+Zred_factor3: (x,y:Z)`x*y+x = x*(1+y)`
+Zred_factor4: (x,y,z:Z)`x*y+x*z = x*(y+z)`
+Zminus_Zplus_compatible: (x,y,n:Z)`x+n-(y+n) = x-y`
+Zmin_plus: (x,y,n:Z)`(Zmin (x+n) (y+n)) = (Zmin x y)+n`
+
+(* nat <-> Z *)
+inj_S: (y:nat)`(inject_nat (S y)) = (Zs (inject_nat y))`
+inj_plus: (x,y:nat)`(inject_nat (plus x y)) = (inject_nat x)+(inject_nat y)`
+inj_mult: (x,y:nat)`(inject_nat (mult x y)) = (inject_nat x)*(inject_nat y)`
+inj_minus1:
+ (x,y:nat)(le y x)->`(inject_nat (minus x y)) = (inject_nat x)-(inject_nat y)`
+inj_minus2: (x,y:nat)(gt y x)->`(inject_nat (minus x y)) = 0`
+
+(* Too specific ? *)
+Zred_factor5: (x,y:Z)`x*0+y = y`
+*)