X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Farith%2Fnat_lt_minus.ma;h=d7654a0bb43294fcfc57c6a4605e3b46959243e9;hb=19b0a814861157ba05f23877d5cd94059f52c2e8;hp=34a16e01f572fd379ba4ed35b8dece84a669e9ba;hpb=5e72e41f4f86814e56d4b00959ccc56c71042a4c;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma index 34a16e01f..d7654a0bb 100644 --- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma @@ -17,16 +17,6 @@ include "ground/arith/nat_lt_pred.ma". (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************) -(* Rewrites with nminus *****************************************************) - -(*** minus_pred_pred *) -lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m. -#m #n #Hm #Hn ->(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm ->(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn -// -qed-. - (* Constructions with nminus ************************************************) (*** monotonic_lt_minus_l *) @@ -35,3 +25,31 @@ lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o. lapply (nle_minus_sn_bi … o Hmn) -Hmn <(nminus_succ_sn … Hom) // qed. + +(*** monotonic_lt_minus_r *) +lemma nlt_minus_dx_bi (o) (m) (n): + m < o -> m < n → o-n < o-m. +#o #m #n #Ho #H +lapply (nle_minus_dx_bi … o H) -H #H +@(le_nlt_trans … H) -n +@nlt_i >(nminus_succ_sn … Ho) // +qed. + +(* Destructions with nminus *************************************************) + +(*** minus_pred_pred *) +lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m. +#m #n #Hm #Hn +>(nlt_des_gen … Hm) in ⊢ (??%?); -Hm +>(nlt_des_gen … Hn) in ⊢ (??%?); -Hn +// +qed-. + +lemma nlt_des_minus_dx (o) (m) (n): m < n - o → o < n. +#o @(nat_ind_succ … o) -o +[ #m #n