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[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_transition / rpx.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma
index f45abc6d72143506b98e7f040f58bcc866a10559..70351ad4f0bd7d1eeb8a5f2df770790fdf8dd497 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx.ma
@@ -27,111 +27,111 @@ interpretation
 
 (* Basic properties ***********************************************************)
 
-lemma rpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
+lemma rpx_atom: ∀h,I,G. ⦃G,⋆⦄ ⊢ ⬈[h,⓪{I}] ⋆.
 /2 width=1 by rex_atom/ qed.
 
 lemma rpx_sort: ∀h,I1,I2,G,L1,L2,s.
-                ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2}.
+                ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2}.
 /2 width=1 by rex_sort/ qed.
 
 lemma rpx_pair: ∀h,I,G,L1,L2,V1,V2.
-                ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2.
+                ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 → ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2.
 /2 width=1 by rex_pair/ qed.
 
 lemma rpx_lref: ∀h,I1,I2,G,L1,L2,i.
-                ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2}.
+                ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2}.
 /2 width=1 by rex_lref/ qed.
 
 lemma rpx_gref: ∀h,I1,I2,G,L1,L2,l.
-                ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2}.
+                ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2}.
 /2 width=1 by rex_gref/ qed.
 
 lemma rpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
-                        ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} →
-                        ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 →
-                        ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}.
+                        ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I1} →
+                        ∀I2. ⦃G,L1⦄ ⊢ I ⬈[h] I2 →
+                        ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I2}.
 /2 width=2 by rex_bind_repl_dx/ qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
+lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G,⋆⦄ ⊢ ⬈[h,T] Y2 → Y2 = ⋆.
 /2 width=3 by rex_inv_atom_sn/ qed-.
 
-lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
+lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G,Y1⦄ ⊢ ⬈[h,T] ⋆ → Y1 = ⋆.
 /2 width=3 by rex_inv_atom_dx/ qed-.
 
-lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
+lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] Y2 →
                     ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
-                     | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
+                     | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 &
                                       Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
 /2 width=1 by rex_inv_sort/ qed-.
 
-lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 →
+lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] Y2 →
                     ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
-                     | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
+                     | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 &
                                       Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
 /2 width=1 by rex_inv_lref/ qed-.
 
-lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
+lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] Y2 →
                     ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
-                     | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
+                     | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 &
                                       Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
 /2 width=1 by rex_inv_gref/ qed-.
 
-lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
-                    ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 →
+                    ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V.
 /2 width=2 by rex_inv_bind/ qed-.
 
-lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
-                    ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 →
+                    ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
 /2 width=2 by rex_inv_flat/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 →
-                            ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] Y2 →
+                            ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y2 = L2.ⓘ{I2}.
 /2 width=2 by rex_inv_sort_bind_sn/ qed-.
 
-lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} →
-                            ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2} →
+                            ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y1 = L1.ⓘ{I1}.
 /2 width=2 by rex_inv_sort_bind_dx/ qed-.
 
-lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
-                            ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] Y2 →
+                            ∃∃L2,V2. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
                                      Y2 = L2.ⓑ{I}V2.
 /2 width=1 by rex_inv_zero_pair_sn/ qed-.
 
-lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 →
-                            ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G,Y1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2 →
+                            ∃∃L1,V1. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
                                      Y1 = L1.ⓑ{I}V1.
 /2 width=1 by rex_inv_zero_pair_dx/ qed-.
 
-lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] Y2 →
-                            ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] Y2 →
+                            ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y2 = L2.ⓘ{I2}.
 /2 width=2 by rex_inv_lref_bind_sn/ qed-.
 
-lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} →
-                            ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2} →
+                            ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y1 = L1.ⓘ{I1}.
 /2 width=2 by rex_inv_lref_bind_dx/ qed-.
 
-lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 →
-                            ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] Y2 →
+                            ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y2 = L2.ⓘ{I2}.
 /2 width=2 by rex_inv_gref_bind_sn/ qed-.
 
-lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} →
-                            ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2} →
+                            ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y1 = L1.ⓘ{I1}.
 /2 width=2 by rex_inv_gref_bind_dx/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
 lemma rpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
-                       ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
+                       ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,V] L2.
 /2 width=3 by rex_fwd_pair_sn/ qed-.
 
 lemma rpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
-                       ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+                       ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V.
 /2 width=2 by rex_fwd_bind_dx/ qed-.
 
 lemma rpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
-                       ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+                       ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
 /2 width=3 by rex_fwd_flat_dx/ qed-.