X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Flib%2Fstar.ma;h=e8d881af0ad899b7279866481589f546e645f905;hb=1fd63df4c77f5c24024769432ea8492748b4ac79;hp=b0e3e6be601368d027e964864d73c371eb1bb8d3;hpb=4b1dee70d9f24b47ab1f93cf1e63862b7b71a645;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/star.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/star.ma index b0e3e6be6..e8d881af0 100644 --- a/matita/matita/contribs/lambdadelta/ground_2/lib/star.ma +++ b/matita/matita/contribs/lambdadelta/ground_2/lib/star.ma @@ -13,56 +13,18 @@ (**************************************************************************) include "basics/star1.ma". -include "ground_2/xoa/xoa_props.ma". +include "ground_2/lib/relations.ma". -(* PROPERTIES OF RELATIONS **************************************************) +(* TRANSITIVE CLOSURE *******************************************************) -definition relation5 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] -≝ λA,B,C,D,E.A→B→C→D→E→Prop. - -definition relation6 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] -≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop. - -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 left_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R. - ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2. - -definition right_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R. - ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2. - -definition pw_confluent2: ∀A. relation A → relation A → predicate A ≝ λA,R1,R2,a0. - ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 → - ∃∃a. R2 a1 a & R1 a2 a. - -definition confluent2: ∀A. relation (relation A) ≝ λA,R1,R2. - ∀a0. pw_confluent2 A R1 R2 a0. - -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) ≝ +definition CTC: ∀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. + ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → CTC … 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. + ∀L2,T1,T2. CTC … R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → CTC … R1 L1 T1 T2. lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 → ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 → @@ -134,66 +96,28 @@ lemma TC_transitive2: ∀A,R1,R2. ] 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. +lemma CTC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (CTC … 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. +lemma s_r_conf1_CTC1: ∀A,B,S,R. s_r_confluent1 A B S R → s_r_confluent1 A B (CTC … 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 → +lemma s_r_trans_CTC1: ∀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/ +/4 width=5 by s_r_conf1_CTC1, 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). +lemma s_r_trans_CTC2: ∀A,B,S,R. s_rs_transitive A B S R → s_r_transitive A B S (CTC … 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 → +lemma s_r_to_s_rs_trans: ∀A,B,S,R. s_r_transitive A B (CTC … 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) @@ -201,21 +125,31 @@ elim (TC_idem … (S L1) … T1 T2) 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. + s_r_transitive A B (CTC … 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. + s_rs_transitive A B (CTC … 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 ***********************************************) +(* Normal form and strong normalization *************************************) + +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-. + +(* 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 → @@ -256,14 +190,7 @@ lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B. ] 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. +(* Relations on unboxed triples *********************************************) definition tri_star: ∀A,B,C,R. tri_relation A B C ≝ λA,B,C,R. tri_RC A B C (tri_TC … R).