X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2A%2Flib%2Fstar.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2A%2Flib%2Fstar.ma;h=0000000000000000000000000000000000000000;hb=1fd63df4c77f5c24024769432ea8492748b4ac79;hp=0f193f0b1ffff0de8da31220d1846452a4e23b9f;hpb=277fc8ff21ce3dbd6893b1994c55cf5c06a98355;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma b/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma deleted file mode 100644 index 0f193f0b1..000000000 --- a/matita/matita/contribs/lambdadelta/ground_2A/lib/star.ma +++ /dev/null @@ -1,329 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basics/star1.ma". -include "ground_2A/xoa/xoa_props.ma". - -(* PROPERTIES OF RELATIONS **************************************************) - -definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥). - -definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R. - ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2. - -definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. - ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & R1 a2 a. - -definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. - ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & R1 a a2. - -definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R. - ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 → - ∃∃a,b. R a1 b1 a b & R a2 b2 a b. - -definition LTC: ∀A:Type[0]. ∀B. (A→relation B) → (A→relation B) ≝ - λA,B,R,a. TC … (R a). - -definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2. - ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2. - -definition s_r_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2. - ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2. - -definition s_rs_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2. - ∀L2,T1,T2. LTC … R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2. - -definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2. - ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2. - -lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 → - ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & TC … R1 a2 a. -#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 -[ #a1 #Ha01 #a2 #Ha02 - elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 - elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 - elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=5 by step, ex2_intro/ -] -qed. - -lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 → - ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 → - ∃∃a. TC … R2 a1 a & R1 a2 a. -#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 -[ #a2 #Ha02 #a1 #Ha01 - elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01 - elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0 - elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3 by step, ex2_intro/ -] -qed. - -lemma TC_confluent2: ∀A,R1,R2. - confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2). -#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1 -[ #a1 #Ha01 #a2 #Ha02 - elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 - elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 - elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=5 by step, ex2_intro/ -] -qed. - -lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 → - ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & TC … R1 a a2. -#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 -[ #a0 #Ha10 #a2 #Ha02 - elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 - elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02 - elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/ -] -qed. - -lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 → - ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 → - ∃∃a. TC … R2 a1 a & R1 a a2. -#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2 -[ #a2 #Ha02 #a1 #Ha10 - elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a2 #_ #Ha02 #IHa #a1 #Ha10 - elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0 - elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3 by step, ex2_intro/ -] -qed. - -lemma TC_transitive2: ∀A,R1,R2. - transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2). -#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 -[ #a0 #Ha10 #a2 #Ha02 - elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3 by inj, ex2_intro/ -| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 - elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02 - elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/ -] -qed. - -definition NF: ∀A. relation A → relation A → predicate A ≝ - λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1. - -definition NF_dec: ∀A. relation A → relation A → Prop ≝ - λA,R,S. ∀a1. NF A R S a1 ∨ - ∃∃a2. R … a1 a2 & (S a2 a1 → ⊥). - -inductive SN (A) (R,S:relation A): predicate A ≝ -| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1 -. - -lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a. -#A #R #S #a1 #Ha1 -@SN_intro #a2 #HRa12 #HSa12 -elim HSa12 -HSa12 /2 width=1 by/ -qed. - -lemma SN_to_NF: ∀A,R,S. NF_dec A R S → - ∀a1. SN A R S a1 → - ∃∃a2. star … R a1 a2 & NF A R S a2. -#A #R #S #HRS #a1 #H elim H -a1 -#a1 #_ #IHa1 elim (HRS a1) -HRS /2 width=3 by srefl, ex2_intro/ -* #a0 #Ha10 #Ha01 elim (IHa1 … Ha10 Ha01) -IHa1 -Ha01 /3 width=3 by star_compl, ex2_intro/ -qed-. - -definition NF_sn: ∀A. relation A → relation A → predicate A ≝ - λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1. - -inductive SN_sn (A) (R,S:relation A): predicate A ≝ -| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2 -. - -lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a. -#A #R #S #a2 #Ha2 -@SN_sn_intro #a1 #HRa12 #HSa12 -elim HSa12 -HSa12 /2 width=1 by/ -qed. - -lemma LTC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (LTC … R) S. -#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 /3 width=3 by inj/ -#T #T2 #_ #HT2 #IHT1 #L1 #HL12 -lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3 by step/ -qed-. - -lemma s_r_conf1_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R → s_r_confluent1 A B (LTC … S) R. -#A #B #S #R #HSR #L1 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3 by/ -qed-. - -lemma s_r_trans_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R → - s_r_transitive A B S R → s_rs_transitive A B S R. -#A #B #S #R #H1SR #H2SR #L2 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /2 width=3 by/ -#T1 #T #HT1 #_ #IHT2 #L1 #HL12 lapply (H2SR … HT1 … HL12) -H2SR -HT1 -/4 width=5 by s_r_conf1_LTC1, trans_TC/ -qed-. - -lemma s_r_trans_LTC2: ∀A,B,S,R. s_rs_transitive A B S R → s_r_transitive A B S (LTC … R). -#A #B #S #R #HSR #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /3 width=3 by inj/ -qed-. - -lemma s_r_to_s_rs_trans: ∀A,B,S,R. s_r_transitive A B (LTC … S) R → - s_rs_transitive A B S R. -#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1 -elim (TC_idem … (S L1) … T1 T2) -#_ #H @H @HSR // -qed-. - -lemma s_rs_to_s_r_trans: ∀A,B,S,R. s_rs_transitive A B S R → - s_r_transitive A B (LTC … S) R. -#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1 -elim (TC_idem … (S L1) … T1 T2) -#H #_ @H @HSR // -qed-. - -lemma s_rs_trans_TC1: ∀A,B,S,R. s_rs_transitive A B S R → - s_rs_transitive A B (LTC … S) R. -#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1 -elim (TC_idem … (S L1) … T1 T2) -elim (TC_idem … (S L2) … T1 T2) -#_ #H1 #H2 #_ @H2 @HSR /3 width=3 by/ -qed-. - -(* relations on unboxed pairs ***********************************************) - -lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R → - ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 → - ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b. -#A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2 -[ #a2 #b2 #H02 - elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/ -| #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2 - elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4 by ex2_2_intro, bi_step/ -] -qed. - -lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R → - bi_confluent A B (bi_TC … R). -#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1 -[ #a1 #b1 #H01 #a2 #b2 #H02 - elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/ -| #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02 - elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20 - elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7 by ex2_2_intro, bi_step/ -] -qed. - -lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B. - ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 → - R a1 b1 a2 b2 ∨ - ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2. -#A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4 by ex2_2_intro, or_intror/ -qed-. - -lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B. - ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 → - R a1 b1 a2 b2 ∨ - ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2. -#A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1 -[ /2 width=1 by or_introl/ -| #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4 by ex2_2_intro, or_intror/ (**) (* auto fails without #_ *) -] -qed-. - -(* relations on unboxed triples *********************************************) - -definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝ - λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨ - ∧∧ a1 = a2 & b1 = b2 & c1 = c2. - -lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R). -/3 width=1 by and3_intro, or_intror/ qed. - -definition tri_star: ∀A,B,C,R. tri_relation A B C ≝ - λA,B,C,R. tri_RC A B C (tri_TC … R). - -lemma tri_star_tri_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_star … R). -/2 width=1 by/ qed. - -lemma tri_TC_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2. - tri_TC A B C R a1 b1 c1 a2 b2 c2 → - tri_star A B C R a1 b1 c1 a2 b2 c2. -/2 width=1 by or_introl/ qed. - -lemma tri_R_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2. - R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2. -/3 width=1 by tri_TC_to_tri_star, tri_inj/ qed. - -lemma tri_star_strap1: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. - tri_star A B C R a1 b1 c1 a b c → - R a b c a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 * -[ /3 width=5 by tri_TC_to_tri_star, tri_step/ -| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/ -] -qed. - -lemma tri_star_strap2: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. R a1 b1 c1 a b c → - tri_star A B C R a b c a2 b2 c2 → - tri_star A B C R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H * -[ /3 width=5 by tri_TC_to_tri_star, tri_TC_strap/ -| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/ -] -qed. - -lemma tri_star_to_tri_TC_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. - tri_star A B C R a1 b1 c1 a b c → - tri_TC A B C R a b c a2 b2 c2 → - tri_TC A B C R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 * -[ /2 width=5 by tri_TC_transitive/ -| * #H1 #H2 #H3 destruct /2 width=1 by/ -] -qed. - -lemma tri_TC_to_tri_star_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. - tri_TC A B C R a1 b1 c1 a b c → - tri_star A B C R a b c a2 b2 c2 → - tri_TC A B C R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H * -[ /2 width=5 by tri_TC_transitive/ -| * #H1 #H2 #H3 destruct /2 width=1 by/ -] -qed. - -lemma tri_tansitive_tri_star: ∀A,B,C,R. tri_transitive A B C (tri_star … R). -#A #B #C #R #a1 #a #b1 #b #c1 #c #H #a2 #b2 #c2 * -[ /3 width=5 by tri_star_to_tri_TC_to_tri_TC, tri_TC_to_tri_star/ -| * #H1 #H2 #H3 destruct /2 width=1 by/ -] -qed. - -lemma tri_star_ind: ∀A,B,C,R,a1,b1,c1. ∀P:relation3 A B C. P a1 b1 c1 → - (∀a,a2,b,b2,c,c2. tri_star … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) → - ∀a2,b2,c2. tri_star … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2. -#A #B #C #R #a1 #b1 #c1 #P #H #IH #a2 #b2 #c2 * -[ #H12 elim H12 -a2 -b2 -c2 /3 width=6 by tri_TC_to_tri_star/ -| * #H1 #H2 #H3 destruct // -] -qed-. - -lemma tri_star_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 → - (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_star … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) → - ∀a1,b1,c1. tri_star … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1. -#A #B #C #R #a2 #b2 #c2 #P #H #IH #a1 #b1 #c1 * -[ #H12 @(tri_TC_ind_dx … a1 b1 c1 H12) -a1 -b1 -c1 /3 width=6 by tri_TC_to_tri_star/ -| * #H1 #H2 #H3 destruct // -] -qed-.