- new component "s_transition" for the restored fqu and fquq

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / reduction / cir.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma
index d9509e17a509d1a5156124fbb3d67d653d396445..412297546be35b30893396504a2dcc35a00dc568 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma
@@ -12,67 +12,68 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/notreducible_2.ma".
+include "basic_2/notation/relations/prednotreducible_3.ma".
 include "basic_2/reduction/crr.ma".
 
-(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************)
+(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE REDUCTION ************************)
 
-definition cir: lenv → predicate term ≝ λL,T. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → ⊥.
+definition cir: relation3 genv lenv term ≝ λG,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⊥.
 
-interpretation "context-sensitive irreducibility (term)"
-   'NotReducible L T = (cir L T).
+interpretation "irreducibility for context-sensitive reduction (term)"
+   'PRedNotReducible G L T = (cir G L T).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cir_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ 𝐈⦃#i⦄ → ⊥.
-/3 width=3/ qed-.
+lemma cir_inv_delta: ∀G,L,K,V,i. ⬇[i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ ➡ 𝐈⦃#i⦄ → ⊥.
+/3 width=3 by crr_delta/ qed-.
 
-lemma cir_inv_ri2: ∀I,L,V,T. ri2 I → ⦃G, L⦄ ⊢ 𝐈⦃②{I}V.T⦄ → ⊥.
-/3 width=1/ qed-.
+lemma cir_inv_ri2: ∀I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃②{I}V.T⦄ → ⊥.
+/3 width=1 by crr_ri2/ qed-.
 
-lemma cir_inv_ib2: ∀a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
-                   ⦃G, L⦄ ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄.
-/4 width=1/ qed-.
+lemma cir_inv_ib2: ∀a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ →
+                   ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄.
+/4 width=1 by crr_ib2_sn, crr_ib2_dx, conj/ qed-.
 
-lemma cir_inv_bind: ∀a,I,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I.
+lemma cir_inv_bind: ∀a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ & ib2 a I.
 #a * [ elim a -a ]
-[ #L #V #T #H elim H -H /3 width=1/
-|*: #L #V #T #H elim (cir_inv_ib2 … H) -H /2 width=1/ /3 width=1/
-]
+#G #L #V #T #H [ elim H -H /3 width=1 by crr_ri2, or_introl/ ]
+elim (cir_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/
 qed-.
 
-lemma cir_inv_appl: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄.
-#L #V #T #HVT @and3_intro /3 width=1/
+lemma cir_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄.
+#G #L #V #T #HVT @and3_intro /3 width=1 by crr_appl_sn, crr_appl_dx/
 generalize in match HVT; -HVT elim T -T //
-* // #a * #U #T #_ #_ #H elim H -H /2 width=1/
+* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crr_beta, crr_theta/
 qed-.
 
-lemma cir_inv_flat: ∀I,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓕ{I}V.T⦄ →
-                    ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
-* #L #V #T #H
-[ elim (cir_inv_appl … H) -H /2 width=1/
-| elim (cir_inv_ri2 … H) -H /2 width=1/
+lemma cir_inv_flat: ∀I,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓕ{I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+* #G #L #V #T #H
+[ elim (cir_inv_appl … H) -H /2 width=1 by and4_intro/
+| elim (cir_inv_ri2 … H) -H //
 ]
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cir_sort: ∀L,k. ⦃G, L⦄ ⊢ 𝐈⦃⋆k⦄.
-/2 width=3 by crr_inv_sort/ qed.
+lemma cir_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐈⦃⋆s⦄.
+/2 width=4 by crr_inv_sort/ qed.
 
-lemma cir_gref: ∀L,p. ⦃G, L⦄ ⊢ 𝐈⦃§p⦄.
-/2 width=3 by crr_inv_gref/ qed.
+lemma cir_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐈⦃§p⦄.
+/2 width=4 by crr_inv_gref/ qed.
 
-lemma tir_atom: ∀I. ⋆ ⊢ 𝐈⦃⓪{I}⦄.
-/2 width=2 by trr_inv_atom/ qed.
+lemma tir_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ ➡ 𝐈⦃⓪{I}⦄.
+/2 width=3 by trr_inv_atom/ qed.
 
-lemma cir_ib2: ∀a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄.
-#a #I #L #V #T #HI #HV #HT #H
-elim (crr_inv_ib2 … HI H) -HI -H /2 width=1/
+lemma cir_ib2: ∀a,I,G,L,V,T.
+               ib2 a I → ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓑ{a,I}V.T⦄.
+#a #I #G #L #V #T #HI #HV #HT #H
+elim (crr_inv_ib2 … HI H) -HI -H /2 width=1 by/
 qed.
 
-lemma cir_appl: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃ⓐV.T⦄.
-#L #V #T #HV #HT #H1 #H2
-elim (crr_inv_appl … H2) -H2 /2 width=1/
+lemma cir_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃ⓐV.T⦄.
+#G #L #V #T #HV #HT #H1 #H2
+elim (crr_inv_appl … H2) -H2 /2 width=1 by/
 qed.