- ground_2: relocation with nstream is now based on two basic functions (push and...

- matex: scope management completed! stylesheet improved (also with colors)

diff --git a/matita/components/binaries/matex/engine.ml b/matita/components/binaries/matex/engine.ml
index b2117aa5e78648973ff53414b18b62612690f793..0cfb1108781b7ef04c36f4788b22b3e440cb244d 100644 (file)
--- a/matita/components/binaries/matex/engine.ml
+++ b/matita/components/binaries/matex/engine.ml
@@ -128,14 +128,14 @@ let mk_open st ris =
    if st.n = "" then ris else
    T.free (scope st) :: T.free st.n :: T.arg st.n :: T.Macro "OPEN" :: ris
 
-let mk_dec w s ris =
+let mk_dec kind w s ris =
    let w = if !G.no_types then [] else w in
-   T.Group w :: T.free s :: T.arg s :: T.Macro "DECL" :: ris
+   T.Group w :: T.free s :: T.arg s :: T.Macro kind :: ris
 
 let mk_inferred st c t ris =
    let u = typeof c t in
    let is_u = proc_term c u in
-   mk_dec is_u st.n ris
+   mk_dec "DECL" is_u st.n ris
 
 let rec proc_proof st ris c t = match t with
    | C.Appl []
@@ -149,7 +149,7 @@ let rec proc_proof st ris c t = match t with
       let s = alpha c s in
       let is_w = proc_term c w in
       let ris = mk_open st ris in
-      proc_proof (next st) (T.Macro "PRIM" :: mk_dec is_w s ris) (K.add_dec s w c) t
+      proc_proof (next st) (mk_dec "PRIM" is_w s ris) (K.add_dec s w c) t
    | C.Appl ts              ->
       let rts = X.rev_neg_filter (A.not_prop2 c) [] ts in
       let ris = T.Macro "STEP" :: mk_inferred st c t ris in
@@ -166,7 +166,8 @@ let rec proc_proof st ris c t = match t with
       let ris = mk_open st ris in
       if A.not_prop1 c w then
          let is_v = proc_term c v in
-         proc_proof (next st) (T.Group is_v :: T.Macro "BODY" :: mk_dec is_w s ris) (K.add_def s w v c) t
+         let ris = T.Group is_v :: T.Macro "BODY" :: mk_dec "DECL" is_w s ris in
+         proc_proof (next st) ris (K.add_def s w v c) t
       else
          let ris_v = proc_proof (push st s) ris c v in
          proc_proof (next st) ris_v (K.add_def s w v c) t

diff --git a/matita/components/binaries/matex/test/matex.sty b/matita/components/binaries/matex/test/matex.sty
index 505d68176b7ad466449aa3bf7502cf25f33afa75..0785eea9211c2717b31071a985a2e8a54907e457 100644 (file)
--- a/matita/components/binaries/matex/test/matex.sty
+++ b/matita/components/binaries/matex/test/matex.sty
@@ -1,10 +1,21 @@
 \NeedsTeXFormat{LaTeX2e}[1995/12/01]
-\ProvidesPackage{matex}[2015/12/21 MaTeX Package]
+\ProvidesPackage{matex}[2016/02/03 MaTeX Package]
+\RequirePackage{xcolor}
 \ExecuteOptions{}
 \ProcessOptions*
 
 \makeatletter
 
+\definecolor{ma@black}{HTML}{000000}
+\definecolor{ma@blue}{HTML}{00005F}
+\definecolor{ma@purple}{HTML}{3F005F}
+
+\newcommand*\ma@fwd{ma@black}
+\newcommand*\ma@open{ma@blue}
+\newcommand*\ma@exit{ma@blue}
+\newcommand*\ma@prim{ma@purple}
+\newcommand*\ma@qed{ma@blue}
+
 \newcommand*\setlabel[1]{\protected@edef\@currentlabel{#1}}
 
 \newcommand*\ObjLabel[1]{\label{obj:#1}}
@@ -12,7 +23,7 @@
 
 \newtheorem{prop}{Proposition}
 \newenvironment{proof}{\setlength\parindent{0pt}}{}
-\newenvironment{ma@step}{}{\vskip0pt}
+\newenvironment{ma@step}[1]{\color{#1}}{\\}
 
 \newcommand*\Object[3]{\begin{prop}[#1]\hfil\\\setlabel{#1}\ObjLabel{#2}#3\end{prop}}
 
@@ -37,14 +48,20 @@
 \newcommand*\ma@with{ with }
 \newcommand*\ma@comma{, }
 \newcommand*\ma@stop{.\end{ma@step}}
+\newcommand*\ma@head[4]{\def\ma@tmp{#4}%
+   \ifx\ma@tmp\empty\begin{ma@step}{#1}\textbf{#2}%
+   \else\begin{ma@step}{#3}\textbf{#4}%
+   \fi
+}
+\newcommand*\ma@tail{\ma@next\ma@with\ma@arg\ma@comma\ma@stop}
 
-\newcommand*\DECL[3]{\def\ma@tmp{#1}\begin{ma@step}\textbf{\ifx\ma@tmp\empty\_conlusion\else #1\fi} of type #3}
-\newcommand*\PRIM{\end{ma@step}}
-\newcommand*\BODY[1]{\\is #1\end{ma@step}}
-\newcommand*\STEP[1]{\\by #1\ma@next\ma@with\ma@arg\ma@comma\ma@stop}
-\newcommand*\DEST[1]{\\by cases on #1\ma@next\ma@with\ma@arg\ma@comma\ma@stop}
-\newcommand*\OPEN[3]{\begin{ma@step}\textbf{#1} is the following scope #3.\end{ma@step}}
-\newcommand*\EXIT[1]{\begin{ma@step}\textbf{end} of scope #1.\end{ma@step}}
+\newcommand*\EXIT[1]{\ma@head{}{}{\ma@exit}{end} of block #1\ma@stop}
+\newcommand*\OPEN[3]{\ma@head{}{}{\ma@open}{#1} is this block #3\ma@stop}
+\newcommand*\PRIM[3]{\ma@head{}{}{\ma@prim}{#1} will have type #3\ma@stop}
+\newcommand*\DECL[3]{\ma@head{\ma@qed}{\_QED}{\ma@fwd}{#1} has type #3\\}
+\newcommand*\BODY[1]{being #1\ma@stop}
+\newcommand*\STEP[1]{by #1\ma@tail}
+\newcommand*\DEST[1]{by cases on #1\ma@tail}
 
 \makeatother
 

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma
index ae7add375539397cb2ef84e0aab7fbadab3b4af8..79e028a9917db9e8cd9f5c86211192015daeca17 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_after.ma
@@ -19,26 +19,20 @@ include "ground_2/relocation/nstream_at.ma".
 (* RELOCATION N-STREAM ******************************************************)
 
 let corec compose: rtmap → rtmap → rtmap ≝ ?.
-#f1 * #b2 #f2 @(seq … (f1@❴b2❵)) @(compose ? f2) -compose -f2
-@(tln … (⫯b2) f1)
-qed.
+#f1 * #n2 #f2 @(seq … (f1@❴n2❵)) @(compose ? f2) -compose -f2
+@(tln … (⫯n2) f1)
+defined.
 
 interpretation "functional composition (nstream)"
    'compose f1 f2 = (compose f1 f2).
 
 coinductive after: relation3 rtmap rtmap rtmap ≝
-| after_zero: ∀f1,f2,f,b1,b2,b.
-              after f1 f2 f →
-              b1 = 0 → b2 = 0 → b = 0 →
-              after (b1@f1) (b2@f2) (b@f)
-| after_skip: ∀f1,f2,f,b1,b2,b,a2,a.
-              after f1 (a2@f2) (a@f) →
-              b1 = 0 → b2 = ⫯a2 → b = ⫯a →
-              after (b1@f1) (b2@f2) (b@f)
-| after_drop: ∀f1,f2,f,b1,b,a1,a.
-              after (a1@f1) f2 (a@f) →
-              b1 = ⫯a1 → b = ⫯a →
-              after (b1@f1) f2 (b@f)
+| after_refl: ∀f1,f2,f,g1,g2,g.
+              after f1 f2 f → g1 = ↑f1 → g2 = ↑f2 → g = ↑f → after g1 g2 g
+| after_push: ∀f1,f2,f,g1,g2,g.
+              after f1 f2 f → g1 = ↑f1 → g2 = ⫯f2 → g = ⫯f → after g1 g2 g
+| after_next: ∀f1,f2,f,g1,g.
+              after f1 f2 f → g1 = ⫯f1 → g = ⫯f → after g1 f2 g
 .
 
 interpretation "relational composition (nstream)"
@@ -46,70 +40,70 @@ interpretation "relational composition (nstream)"
 
 (* Basic properies on compose ***********************************************)
 
-lemma compose_unfold: ∀f1,f2,a2. f1∘(a2@f2) = f1@❴a2❵@tln … (⫯a2) f1∘f2.
-#f1 #f2 #a2 >(stream_expand … (f1∘(a2@f2))) normalize //
+lemma compose_unfold: ∀f1,f2,n2. f1∘(n2@f2) = f1@❴n2❵@tln … (⫯n2) f1∘f2.
+#f1 #f2 #n2 >(stream_expand … (f1∘(n2@f2))) normalize //
 qed.
 
-lemma compose_drop: ∀f1,f2,f,a1,a. (a1@f1)∘f2 = a@f → (⫯a1@f1)∘f2 = ⫯a@f.
-#f1 * #a2 #f2 #f #a1 #a >compose_unfold >compose_unfold
+lemma compose_next: ∀f1,f2,f. f1∘f2 = f → (⫯f1)∘f2 = ⫯f.
+* #n1 #f1 * #n2 #f2 #f >compose_unfold >compose_unfold
 #H destruct normalize //
 qed.
 
 (* Basic inversion lemmas on compose ****************************************)
 
-lemma compose_inv_unfold: ∀f1,f2,f,a2,a. f1∘(a2@f2) = a@f →
-                          f1@❴a2❵ = a ∧ tln … (⫯a2) f1∘f2 = f.
-#f1 #f2 #f #a2 #a >(stream_expand … (f1∘(a2@f2))) normalize
+lemma compose_inv_unfold: ∀f1,f2,f,n2,n. f1∘(n2@f2) = n@f →
+                          f1@❴n2❵ = n ∧ tln … (⫯n2) f1∘f2 = f.
+#f1 #f2 #f #n2 #n >(stream_expand … (f1∘(n2@f2))) normalize
 #H destruct /2 width=1 by conj/
 qed-.
 
-lemma compose_inv_O2: ∀f1,f2,f,a1,a. (a1@f1)∘(O@f2) = a@f →
-                      a = a1 ∧ f1∘f2 = f.
-#f1 #f2 #f #a1 #a >compose_unfold
+lemma compose_inv_O2: ∀f1,f2,f,n1,n. (n1@f1)∘(↑f2) = n@f →
+                      n = n1 ∧ f1∘f2 = f.
+#f1 #f2 #f #n1 #n >compose_unfold
 #H destruct /2 width=1 by conj/
 qed-.
 
-lemma compose_inv_S2: ∀f1,f2,f,a1,a2,a. (a1@f1)∘(⫯a2@f2) = a@f →
-                      a = ⫯(a1+f1@❴a2❵) ∧ f1∘(a2@f2) = f1@❴a2❵@f.
-#f1 #f2 #f #a1 #a2 #a >compose_unfold
+lemma compose_inv_S2: ∀f1,f2,f,n1,n2,n. (n1@f1)∘(⫯n2@f2) = n@f →
+                      n = ⫯(n1+f1@❴n2❵) ∧ f1∘(n2@f2) = f1@❴n2❵@f.
+#f1 #f2 #f #n1 #n2 #n >compose_unfold
 #H destruct /2 width=1 by conj/
 qed-.
 
-lemma compose_inv_S1: ∀f1,f2,f,a1,a2,a. (⫯a1@f1)∘(a2@f2) = a@f →
-                      a = ⫯((a1@f1)@❴a2❵) ∧ (a1@f1)∘(a2@f2) = (a1@f1)@❴a2❵@f.
-#f1 #f2 #f #a1 #a2 #a >compose_unfold
+lemma compose_inv_S1: ∀f1,f2,f,n1,n2,n. (⫯n1@f1)∘(n2@f2) = n@f →
+                      n = ⫯((n1@f1)@❴n2❵) ∧ (n1@f1)∘(n2@f2) = (n1@f1)@❴n2❵@f.
+#f1 #f2 #f #n1 #n2 #n >compose_unfold
 #H destruct /2 width=1 by conj/
 qed-.
 
 (* Basic properties on after ************************************************)
 
 lemma after_O2: ∀f1,f2,f. f1 ⊚ f2 ≡ f →
-                ∀b. b@f1 ⊚ O@f2 ≡ b@f.
-#f1 #f2 #f #Ht #b elim b -b /2 width=5 by after_drop, after_zero/
+                ∀n. n@f1 ⊚ ↑f2 ≡ n@f.
+#f1 #f2 #f #Ht #n elim n -n /2 width=7 by after_refl, after_next/
 qed.
 
-lemma after_S2: ∀f1,f2,f,b2,b. f1 ⊚ b2@f2 ≡ b@f →
-                ∀b1. b1@f1 ⊚ ⫯b2@f2 ≡ ⫯(b1+b)@f.
-#f1 #f2 #f #b2 #b #Ht #b1 elim b1 -b1 /2 width=5 by after_drop, after_skip/
+lemma after_S2: ∀f1,f2,f,n2,n. f1 ⊚ n2@f2 ≡ n@f →
+                ∀n1. n1@f1 ⊚ ⫯n2@f2 ≡ ⫯(n1+n)@f.
+#f1 #f2 #f #n2 #n #Ht #n1 elim n1 -n1 /2 width=7 by after_next, after_push/
 qed.
 
-lemma after_apply: ∀b2,f1,f2,f. (tln … (⫯b2) f1) ⊚ f2 ≡ f → f1 ⊚ b2@f2 ≡ f1@❴b2❵@f.
-#b2 elim b2 -b2
+lemma after_apply: ∀n2,f1,f2,f. (tln … (⫯n2) f1) ⊚ f2 ≡ f → f1 ⊚ n2@f2 ≡ f1@❴n2❵@f.
+#n2 elim n2 -n2
 [ * /2 width=1 by after_O2/
-| #b2 #IH * /3 width=1 by after_S2/
+| #n2 #IH * /3 width=1 by after_S2/
 ]
 qed-.
 
 let corec after_total_aux: ∀f1,f2,f. f1 ∘ f2 = f → f1 ⊚ f2 ≡ f ≝ ?.
-* #a1 #f1 * #a2 #f2 * #a #f cases a1 -a1
-[ cases a2 -a2
+* #n1 #f1 * #n2 #f2 * #n #f cases n1 -n1
+[ cases n2 -n2
   [ #H cases (compose_inv_O2 … H) -H
-    /3 width=1 by after_zero/
-  | #a2 #H cases (compose_inv_S2 … H) -H
-    /3 width=5 by after_skip, eq_f/
+    /3 width=7 by after_refl, eq_f2/
+  | #n2 #H cases (compose_inv_S2 … H) -H
+    /3 width=7 by after_push/
   ]
-| #a1 #H cases (compose_inv_S1 … H) -H
-  /3 width=5 by after_drop, eq_f/
+| #n1 #H cases (compose_inv_S1 … H) -H
+  /4 width=7 by after_next, next_rew_sn/
 ]
 qed-.
 
@@ -118,167 +112,179 @@ theorem after_total: ∀f2,f1. f1 ⊚ f2 ≡ f1 ∘ f2.
 
 (* Basic inversion lemmas on after ******************************************)
 
-fact after_inv_O1_aux: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1. f1 = 0@g1 →
-                       (∃∃g2,g. g1 ⊚ g2 ≡ g & f2 = 0@g2 & f = 0@g) ∨
-                       ∃∃g2,g,b2,b. g1 ⊚ b2@g2 ≡ b@g & f2 = ⫯b2@g2 & f = ⫯b@g.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #Ht #H1 #H2 #H3 #g1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
-| #b2 #b #a2 #a #Ht #H1 #H2 #H3 #g1 #H destruct /3 width=7 by ex3_4_intro, or_intror/
-| #b #a1 #a #_ #H1 #H3 #g1 #H destruct
+fact after_inv_OOx_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. g1 = ↑f1 → g2 = ↑f2 →
+                        ∃∃f. f1 ⊚ f2 ≡ f & g = ↑f.
+#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
+[ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
+  <(injective_push … Hx1) <(injective_push … Hx2) -x2 -x1
+  /2 width=3 by ex2_intro/
+| #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
+  elim (discr_next_push … Hx2)
+| #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
+  elim (discr_next_push … Hx1)
 ]
 qed-.
 
-fact after_inv_O1_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 →
-                        (∧∧ f1 ⊚ f2 ≡ f & b2 = 0 & b = 0) ∨
-                        ∃∃a2,a. f1 ⊚ a2@f2 ≡ a@f & b2 = ⫯a2 & b = ⫯a.
-#f1 #f2 #f #b1 #b2 #b #Ht #H elim (after_inv_O1_aux … Ht) -Ht [4: // |2: skip ] *
-[ #g2 #g #Hu #H1 #H2 destruct /3 width=1 by and3_intro, or_introl/
-| #g2 #g #a2 #a #Hu #H1 #H2 destruct /3 width=5 by ex3_2_intro, or_intror/
+lemma after_inv_OOx: ∀f1,f2,g. ↑f1 ⊚ ↑f2 ≡ g → ∃∃f. f1 ⊚ f2 ≡ f & g = ↑f.
+/2 width=5 by after_inv_OOx_aux/ qed-.
+
+fact after_inv_OSx_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. g1 = ↑f1 → g2 = ⫯f2 →
+                        ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
+[ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
+  elim (discr_push_next … Hx2)
+| #g2 #g #Hf #H1 #H2 #H3 #x1 #x2 #Hx1 #Hx2 destruct
+  <(injective_push … Hx1) <(injective_next … Hx2) -x2 -x1
+  /2 width=3 by ex2_intro/
+| #g #_ #H1 #_ #x1 #x2 #Hx1 #_ destruct
+  elim (discr_next_push … Hx1)
 ]
 qed-.
 
-lemma after_inv_O1: ∀g1,f2,f. 0@g1 ⊚ f2 ≡ f →
-                    (∃∃g2,g. g1 ⊚ g2 ≡ g & f2 = 0@g2 & f = 0@g) ∨
-                    ∃∃g2,g,b2,b. g1 ⊚ b2@g2 ≡ b@g & f2 = ⫯b2@g2 & f = ⫯b@g.
-/2 width=3 by after_inv_O1_aux/ qed-.
+lemma after_inv_OSx: ∀f1,f2,g. ↑f1 ⊚ ⫯f2 ≡ g → ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+/2 width=5 by after_inv_OSx_aux/ qed-.
 
-fact after_inv_zero_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 → b2 = 0 →
-                          f1 ⊚ f2 ≡ f ∧ b = 0.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 #H2 elim (after_inv_O1_aux2 … Ht H1) -Ht -H1 *
-[ /2 width=1 by conj/
-| #a1 #a2 #_ #H0 destruct
+fact after_inv_Sxx_aux: ∀g1,f2,g. g1 ⊚ f2 ≡ g → ∀f1. g1 = ⫯f1 →
+                        ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+#g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
+[ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
+  elim (discr_push_next … Hx1)
+| #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
+  elim (discr_push_next … Hx1)
+| #g #Hf #H1 #H #x1 #Hx1 destruct
+  <(injective_next … Hx1) -x1
+  /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma after_inv_zero: ∀g1,g2,f. 0@g1 ⊚ 0@g2 ≡ f →
-                      ∃∃g. g1 ⊚ g2 ≡ g & f = 0@g.
-#g1 #g2 #f #H elim (after_inv_O1 … H) -H *
-[ #x2 #g #Hu #H1 #H2 destruct /2 width=3 by ex2_intro/
-| #x2 #g #a2 #a #Hu #H destruct
-]
-qed-.
+lemma after_inv_Sxx: ∀f1,f2,g. ⫯f1 ⊚ f2 ≡ g → ∃∃f. f1 ⊚ f2 ≡ f & g = ⫯f.
+/2 width=5 by after_inv_Sxx_aux/ qed-.
 
-fact after_inv_skip_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b1 = 0 → ∀a2. b2 = ⫯a2 →
-                          ∃∃a. f1 ⊚ a2@f2 ≡ a@f & b = ⫯a.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 #a2 #H2 elim (after_inv_O1_aux2 … Ht H1) -Ht -H1 *
-[ #_ #H0 destruct
-| #x2 #x #H #H0 #H1 destruct /2 width=3 by ex2_intro/
-]
-qed-.
+(* Advanced inversion lemmas on after ***************************************)
 
-lemma after_inv_skip: ∀g1,g2,f,b2. 0@g1 ⊚ ⫯b2@g2 ≡ f →
-                      ∃∃g,b. g1 ⊚ b2@g2 ≡ b@g & f = ⫯b@g.
-#g1 #g2 * #b #f #b2 #Ht elim (after_inv_skip_aux2 … Ht) [2,4: // |3: skip ] -Ht
-#a #Ht #H destruct /2 width=4 by ex2_2_intro/
+fact after_inv_OOO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+                        ∀f1,f2,f. g1 = ↑f1 → g2 = ↑f2 → g = ↑f → f1 ⊚ f2 ≡ f.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct >(injective_push … Hx) -f //
 qed-.
 
-fact after_inv_S1_aux: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1,b1. f1 = ⫯b1@g1 →
-                       ∃∃g,b. b1@g1 ⊚ f2 ≡ b@g & f = ⫯b@g.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #_ #H1 #H2 #H3 #g1 #a1 #H destruct
-| #b2 #b #a2 #a #_ #H1 #H2 #H3 #g1 #a1 #H destruct
-| #b #a1 #a #Ht #H1 #H3 #g1 #x1 #H destruct /2 width=4 by ex2_2_intro/
-]
+fact after_inv_OOS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+                        ∀f1,f2,f. g1 = ↑f1 → g2 = ↑f2 → g = ⫯f → ⊥.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct elim (discr_next_push … Hx)
 qed-.
 
-fact after_inv_S1_aux2: ∀f1,f2,f,b1,b. b1@f1 ⊚ f2 ≡ b@f → ∀a1. b1 = ⫯a1 →
-                        â\88\83â\88\83a. a1@f1 â\8a\9a f2 â\89¡ a@f & b = ⫯a.
-#f1 #f2 #f #b1 #b #Ht #a #H elim (after_inv_S1_aux … Ht) -Ht [4: // |2,3: skip ]
-#g #x #Hu #H0 destruct /2 width=3 by ex2_intro/ 
+fact after_inv_OSS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+                        â\88\80f1,f2,f. g1 = â\86\91f1 â\86\92 g2 = ⫯f2 â\86\92 g = ⫯f â\86\92 f1 â\8a\9a f2 â\89¡ f.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OSx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct >(injective_next … Hx) -f //
 qed-.
 
-lemma after_inv_S1: ∀g1,f2,f,b1. ⫯b1@g1 ⊚ f2 ≡ f →
-                    ∃∃g,b. b1@g1 ⊚ f2 ≡ b@g & f = ⫯b@g.
-/2 width=3 by after_inv_S1_aux/ qed-.
-
-fact after_inv_drop_aux2: ∀f1,f2,f,a1,a. a1@f1 ⊚ f2 ≡ a@f → ∀b1,b. a1 = ⫯b1 → a = ⫯b →
-                          b1@f1 ⊚ f2 ≡ b@f.
-#f1 #f2 #f #a1 #a #Ht #b1 #b #H1 #H elim (after_inv_S1_aux2 … Ht … H1) -a1
-#x #Ht #Hx destruct //
+fact after_inv_OSO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+                        ∀f1,f2,f. g1 = ↑f1 → g2 = ⫯f2 → g = ↑f → ⊥.
+#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H elim (after_inv_OSx_aux … Hg … H1 H2) -g1 -g2
+#x #Hf #Hx destruct elim (discr_push_next … Hx)
 qed-.
 
-lemma after_inv_drop: ∀f1,f2,f,b1,b. ⫯b1@f1 ⊚ f2 ≡ ⫯b@f → b1@f1 ⊚ f2 ≡ b@f.
-/2 width=5 by after_inv_drop_aux2/ qed-.
+fact after_inv_SxS_aux: ∀g1,f2,g. g1 ⊚ f2 ≡ g →
+                        ∀f1,f. g1 = ⫯f1 → g = ⫯f → f1 ⊚ f2 ≡ f.
+#g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_Sxx_aux … Hg … H1) -g1
+#x #Hf #Hx destruct >(injective_next … Hx) -f //
+qed-.
 
-fact after_inv_O3_aux1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g. f = 0@g →
-                        ∃∃g1,g2. g1 ⊚ g2 ≡ g & f1 = 0@g1 & f2 = 0@g2.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #Ht #H1 #H2 #H3 #g #H destruct /2 width=5 by ex3_2_intro/
-| #b2 #b #a2 #a #_ #H1 #H2 #H3 #g #H destruct
-| #b #a1 #a #_ #H1 #H3 #g #H destruct
-]
+fact after_inv_SxO_aux: ∀g1,f2,g. g1 ⊚ f2 ≡ g →
+                        ∀f1,f. g1 = ⫯f1 → g = ↑f → ⊥.
+#g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_Sxx_aux … Hg … H1) -g1
+#x #Hf #Hx destruct elim (discr_push_next … Hx)
 qed-.
 
-fact after_inv_O3_aux2: ∀f1,f2,f,b1,b2,b. b1@f1 ⊚ b2@f2 ≡ b@f → b = 0 →
-                        ∧∧ f1 ⊚ f2 ≡ f & b1 = 0 & b2 = 0.
-#f1 #f2 #f #b1 #b2 #b #Ht #H1 elim (after_inv_O3_aux1 … Ht) [2: // |3: skip ] -b
-#g1 #g2 #Ht #H1 #H2 destruct /2 width=1 by and3_intro/
+fact after_inv_OxO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+                        ∀f1,f. g1 = ↑f1 → g = ↑f →
+                        ∃∃f2. f1 ⊚ f2 ≡ f & g2 = ↑f2.
+#g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
+[ elim (after_inv_OSO_aux … Hg … H1 … H) -g1 -g -f1 -f //
+| lapply (after_inv_OOO_aux … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
+]
 qed-.
 
-lemma after_inv_O3: ∀f1,f2,g. f1 ⊚ f2 ≡ 0@g →
-                    ∃∃g1,g2. g1 ⊚ g2 ≡ g & f1 = 0@g1 & f2 = 0@g2.
-/2 width=3 by after_inv_O3_aux1/ qed-.
+lemma after_inv_OxO: ∀f1,g2,f. ↑f1 ⊚ g2 ≡ ↑f →
+                     ∃∃f2. f1 ⊚ f2 ≡ f & g2 = ↑f2.
+/2 width=5 by after_inv_OxO_aux/ qed-.
 
-fact after_inv_S3_aux1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g,b. f = ⫯b@g →
-                        (∃∃g1,g2,b2. g1 ⊚ b2@g2 ≡ b@g & f1 = 0@g1 & f2 = ⫯b2@g2) ∨
-                        ∃∃g1,b1. b1@g1 ⊚ f2 ≡ b@g & f1 = ⫯b1@g1.
-#f1 #f2 #f * -f1 -f2 -f #f1 #f2 #f #b1
-[ #b2 #b #_ #H1 #H2 #H3 #g #a #H destruct
-| #b2 #b #a2 #a #HT #H1 #H2 #H3 #g #x #H destruct /3 width=6 by ex3_3_intro, or_introl/
-| #b #a1 #a #HT #H1 #H3 #g #x #H destruct /3 width=4 by ex2_2_intro, or_intror/
+fact after_inv_OxS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
+                        ∀f1,f. g1 = ↑f1 → g = ⫯f →
+                        ∃∃f2. f1 ⊚ f2 ≡ f & g2 = ⫯f2.
+#g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
+[ lapply (after_inv_OSS_aux … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
+| elim (after_inv_OOS_aux … Hg … H1 … H) -g1 -g -f1 -f // 
 ]
 qed-.
 
-fact after_inv_S3_aux2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f → ∀b. a = ⫯b →
-                        (∃∃b2. f1 ⊚ b2@f2 ≡ b@f & a1 = 0 & a2 = ⫯b2) ∨
-                        ∃∃b1. b1@f1 ⊚ a2@f2 ≡ b@f & a1 = ⫯b1.
-#f1 #f2 #f #a1 #a2 #a #Ht #b #H elim (after_inv_S3_aux1 … Ht) [3: // |4,5: skip ] -a *
-[ #g1 #g2 #b2 #Ht #H1 #H2 destruct /3 width=3 by ex3_intro, or_introl/
-| #g1 #b1 #Ht #H1 destruct /3 width=3 by ex2_intro, or_intror/
+fact after_inv_xxO_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f. g = ↑f →
+                        ∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ↑f2.
+* * [2: #m1 ] #g1 #g2 #g #Hg #f #H
+[ elim (after_inv_SxO_aux … Hg … H) -g2 -g -f //
+| elim (after_inv_OxO_aux … Hg … H) -g /2 width=5 by ex3_2_intro/
 ]
 qed-.
 
-lemma after_inv_S3: ∀f1,f2,g,b. f1 ⊚ f2 ≡ ⫯b@g →
-                    (∃∃g1,g2,b2. g1 ⊚ b2@g2 ≡ b@g & f1 = 0@g1 & f2 = ⫯b2@g2) ∨
-                    ∃∃g1,b1. b1@g1 ⊚ f2 ≡ b@g & f1 = ⫯b1@g1.
-/2 width=3 by after_inv_S3_aux1/ qed-.
-
-(* Advanced inversion lemmas on after ***************************************)
+lemma after_inv_xxO: ∀g1,g2,f. g1 ⊚ g2 ≡ ↑f →
+                     ∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ↑f2.
+/2 width=3 by after_inv_xxO_aux/ qed-.
 
-fact after_inv_O2_aux2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f → a2 = 0 →
-                         a1 = a ∧ f1 ⊚ f2 ≡ f.
-#f1 #f2 #f #a1 #a2 elim a1 -a1
-[ #a #H #H2 elim (after_inv_zero_aux2 … H … H2) -a2 /2 width=1 by conj/
-| #a1 #IH #a #H #H2 elim (after_inv_S1_aux2 … H) -H [3: // |2: skip ]
-  #b #H #H1 elim (IH … H) // -a2
-  #H2 destruct /2 width=1 by conj/
+fact after_inv_xxS_aux: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f. g = ⫯f →
+                        (∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ⫯f2) ∨
+                        ∃∃f1. f1 ⊚ g2 ≡ f & g1 = ⫯f1.
+* * [2: #m1 ] #g1 #g2 #g #Hg #f #H
+[ /4 width=5 by after_inv_SxS_aux, or_intror, ex2_intro/
+| elim (after_inv_OxS_aux … Hg … H) -g
+  /3 width=5 by or_introl, ex3_2_intro/
 ]
 qed-.
 
-lemma after_inv_O2: ∀f1,g2,f. f1 ⊚ 0@g2 ≡ f →
-                    ∃∃g1,g,a. f1 = a@g1 & f = a@g & g1 ⊚ g2 ≡ g.
-* #a1 #f1 #f2 * #a #f #H elim (after_inv_O2_aux2 … H) -H //
-/2 width=6 by ex3_3_intro/
+lemma after_inv_xxS: ∀g1,g2,f. g1 ⊚ g2 ≡ ⫯f →
+                     (∃∃f1,f2. f1 ⊚ f2 ≡ f & g1 = ↑f1 & g2 = ⫯f2) ∨
+                     ∃∃f1. f1 ⊚ g2 ≡ f & g1 = ⫯f1.
+/2 width=3 by after_inv_xxS_aux/ qed-.
+
+fact after_inv_xOx_aux: ∀f1,g2,f,n1,n. n1@f1 ⊚ g2 ≡ n@f → ∀f2. g2 = ↑f2 →
+                        f1 ⊚ f2 ≡ f ∧ n1 = n.
+#f1 #g2 #f #n1 elim n1 -n1
+[ #n #Hf #f2 #H2 elim (after_inv_OOx_aux … Hf … H2) -g2 [3: // |2: skip ]
+  #g #Hf #H elim (push_inv_seq_sn … H) -H destruct /2 width=1 by conj/
+| #n1 #IH #n #Hf #f2 #H2 elim (after_inv_Sxx_aux … Hf) -Hf [3: // |2: skip ]
+  #g1 #Hg #H1 elim (next_inv_seq_sn … H1) -H1
+  #x #Hx #H destruct elim (IH … Hg) [2: // |3: skip ] -IH -Hg
+  #H destruct /2 width=1 by conj/
+]
 qed-.
 
-lemma after_inv_const: ∀a,f1,b2,g2,f. a@f1 ⊚ b2@g2 ≡ a@f → b2 = 0.
-#a elim a -a
-[ #f1 #b2 #g2 #f #H elim (after_inv_O3 … H) -H
-  #g1 #x2 #_ #_ #H destruct //
-| #a #IH #f1 #b2 #g2 #f #H elim (after_inv_S1 … H) -H
-  #x #b #Hx #H destruct >(IH … Hx) -f1 -g2 -x -b2 -b //
+lemma after_inv_xOx: ∀f1,f2,f,n1,n. n1@f1 ⊚ ↑f2 ≡ n@f →
+                     f1 ⊚ f2 ≡ f ∧ n1 = n.
+/2 width=3 by after_inv_xOx_aux/ qed-.
+
+fact after_inv_xSx_aux: ∀f1,g2,f,n1,n. n1@f1 ⊚ g2 ≡ n@f → ∀f2. g2 = ⫯f2 →
+                        ∃∃m. f1 ⊚ f2 ≡ m@f & n = ⫯(n1+m).
+#f1 #g2 #f #n1 elim n1 -n1
+[ #n #Hf #f2 #H2 elim (after_inv_OSx_aux … Hf … H2) -g2 [3: // |2: skip ]
+  #g #Hf #H elim (next_inv_seq_sn … H) -H
+  #x #Hx #Hg destruct /2 width=3 by ex2_intro/
+| #n1 #IH #n #Hf #f2 #H2 elim (after_inv_Sxx_aux … Hf) -Hf [3: // |2: skip ]
+  #g #Hg #H elim (next_inv_seq_sn … H) -H
+  #x #Hx #H destruct elim (IH … Hg) -IH -Hg [3: // |2: skip ]
+  #m #Hf #Hm destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma after_inv_S2: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ ⫯a2@f2 ≡ a@f → ∀b. a = ⫯(a1+b) →
-                    f1 ⊚ a2@f2 ≡ b@f.
-#f1 #f2 #f #a1 elim a1 -a1
-[ #a2 #a #Ht #b #Hb
-  elim (after_inv_skip_aux2 … Ht) -Ht [3,4: // |2: skip ]
-  #c #Ht #Hc destruct //
-| #a1 #IH #a2 #a #Ht #b #Hb
-  lapply (after_inv_drop_aux2 … Ht … Hb) -a [ // | skip ]
-  /2 width=3 by/
+lemma after_inv_xSx: ∀f1,f2,f,n1,n. n1@f1 ⊚ ⫯f2 ≡ n@f →
+                     ∃∃m. f1 ⊚ f2 ≡ m@f & n = ⫯(n1+m).
+/2 width=3 by after_inv_xSx_aux/ qed-.
+
+lemma after_inv_const: ∀f1,f2,f,n2,n. n@f1 ⊚ n2@f2 ≡ n@f → f1 ⊚ f2 ≡ f ∧ n2 = 0.
+#f1 #f2 #f #n2 #n elim n -n
+[ #H elim (after_inv_OxO … H) -H
+  #g2 #Hf #H elim (push_inv_seq_sn … H) -H /2 width=1 by conj/
+| #n #IH #H lapply (after_inv_SxS_aux … H ????) -H /2 width=5 by/
 ]
 qed-.
 
@@ -287,16 +293,16 @@ qed-.
 lemma after_at_fwd: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f →
                     ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
 #f #i1 #i #H elim H -f -i1 -i
-[ #f #f2 #f1 #H elim (after_inv_O3 … H) -H
-  /2 width=3 by at_zero, ex2_intro/
-| #f #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_O3 … H) -H
-  #g2 #g1 #Hu #H1 #H2 destruct elim (IH … Hu) -f
+[ #f #f2 #f1 #H elim (after_inv_xxO … H) -H
+  /2 width=3 by at_refl, ex2_intro/
+| #f #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_xxO … H) -H
+  #g2 #g1 #Hg #H1 #H2 destruct elim (IH … Hg) -f
   /3 width=3 by at_S1, ex2_intro/
-| #f #b #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_S3 … H) -H *
-  [ #g2 #g1 #b2 #Hu #H1 #H2 destruct elim (IH … Hu) -f -b
-    /3 width=3 by at_S1, at_lift, ex2_intro/
-  | #g1 #b1 #Hu #H destruct elim (IH … Hu) -f -b
-    /3 width=3 by at_lift, ex2_intro/
+| #f #i1 #i #_ #IH #f2 #f1 #H elim (after_inv_xxS … H) -H *
+  [ #g2 #g1 #Hg #H2 #H1 destruct elim (IH … Hg) -f
+    /3 width=3 by at_S1, at_next, ex2_intro/
+  | #g1 #Hg #H destruct elim (IH … Hg) -f
+    /3 width=3 by at_next, ex2_intro/
   ]
 ]
 qed-.
@@ -304,23 +310,12 @@ qed-.
 lemma after_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≡ i2 → ∀f2,f. f2 ⊚ f1 ≡ f →
                      ∃∃i. @⦃i2, f2⦄ ≡ i & @⦃i1, f⦄ ≡ i.
 #f1 #i1 #i2 #H elim H -f1 -i1 -i2
-[ #f1 #f2 #f #H elim (after_inv_O2 … H) -H /2 width=3 by ex2_intro/
-| #f1 #i1 #i2 #_ #IH * #b2 elim b2 -b2
-  [ #f2 #f #H elim (after_inv_zero … H) -H
-    #g #Hu #H destruct elim (IH … Hu) -f1
-    /3 width=3 by at_S1, at_skip, ex2_intro/
-  | -IH #b2 #IH #f2 #f #H elim (after_inv_S1 … H) -H
-    #g #b #Hu #H destruct elim (IH … Hu) -f1
-    /3 width=3 by at_lift, ex2_intro/
-  ]
-| #f1 #b1 #i1 #i2 #_ #IH * #b2 elim b2 -b2
-  [ #f2 #f #H elim (after_inv_skip … H) -H
-    #g #a #Hu #H destruct elim (IH … Hu) -f1 -b1
-    /3 width=3 by at_S1, at_lift, ex2_intro/
-  | -IH #b2 #IH #f2 #f #H elim (after_inv_S1 … H) -H
-    #g #b #Hu #H destruct elim (IH … Hu) -f1 -b1
-    /3 width=3 by at_lift, ex2_intro/
-  ]
+[ #f1 * #n2 #f2 * #n #f #H elim (after_inv_xOx … H) -H /2 width=3 by ex2_intro/
+| #f1 #i1 #i2 #_ #IH * #n2 #f2 * #n #f #H elim (after_inv_xOx … H) -H
+  #Hf #H destruct elim (IH … Hf) -f1 /3 width=3 by at_S1, ex2_intro/
+| #f1 #i1 #i2 #_ #IH * #n2 #f2 * #n #f #H elim (after_inv_xSx … H) -H
+  #m #Hf #Hm destruct elim (IH … Hf) -f1
+  /4 width=3 by at_plus2, at_S1, at_next, ex2_intro/
 ]
 qed-.
 
@@ -339,99 +334,104 @@ qed-.
 lemma after_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
                      ∀f2. f2 ⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
 #f #i1 #i #H elim H -f -i1 -i
-[ #f #f1 #i2 #Ht1 #f2 #H elim (after_inv_O3 … H) -H
+[ #f #f1 #i2 #Ht1 #f2 #H elim (after_inv_xxO … H) -H
   #g2 #g1 #_ #H1 #H2 destruct >(at_inv_OOx … Ht1) -f -g1 -i2 //
-| #f #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_O3 … H) -H
+| #f #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_xxO … H) -H
   #g2 #g1 #Hu #H1 #H2 destruct elim (at_inv_SOx … Ht1) -Ht1
-  /3 width=3 by at_skip/
-| #f #b #i1 #i #_ #IH #f1 #i2 #Ht1 #f2 #H elim (after_inv_S3 … H) -H *
-  [ #g2 #g1 #a1 #Hu #H1 #H2 destruct elim (at_inv_xSx … Ht1) -Ht1
-    /3 width=3 by at_skip/
-  | #g2 #a2 #Hu #H destruct /3 width=3 by at_lift/
+  /3 width=3 by at_push/
+| #f #i1 #i #_ #IH #f1 #i2 #Hf1 #f2 #H elim (after_inv_xxS … H) -H *
+  [ #g2 #g1 #Hg #H2 #H1 destruct elim (at_inv_xSx … Hf1) -Hf1
+    /3 width=3 by at_push/
+  | #g2 #Hg #H destruct /3 width=3 by at_next/
   ]
 ]
 qed-.
 
 (* Advanced forward lemmas on after *****************************************)
 
-lemma after_fwd_hd: ∀f1,f2,f,a2,a. f1 ⊚ a2@f2 ≡ a@f → a = f1@❴a2❵.
-#f1 #f2 #f #a2 #a #Ht lapply (after_fwd_at … 0 … Ht) -Ht [4: // | // |2,3: skip ]
+lemma after_fwd_hd: ∀f1,f2,f,n2,n. f1 ⊚ n2@f2 ≡ n@f → n = f1@❴n2❵.
+#f1 #f2 #f #n2 #n #H lapply (after_fwd_at … 0 … H) -H [1,4: // |2,3: skip ]
 /3 width=2 by at_inv_O1, sym_eq/
 qed-.
 
-lemma after_fwd_tl: ∀f,f2,a2,f1,a1,a. a1@f1 ⊚ a2@f2 ≡ a@f →
-                    tln … a2 f1 ⊚ f2 ≡ f.
-#f #f2 #a2 elim a2 -a2
-[ #f1 #a1 #a #Ht elim (after_inv_O2_aux2 … Ht) -Ht //
-| #a2 #IH * #b1 #f1 #a1 #a #Ht
-  lapply (after_fwd_hd … Ht) #Ha
-  lapply (after_inv_S2 … Ht … Ha) -a
-  /2 width=3 by/
+lemma after_fwd_tl: ∀f,f2,n2,f1,n1,n. n1@f1 ⊚ n2@f2 ≡ n@f →
+                    tln … n2 f1 ⊚ f2 ≡ f.
+#f #f2 #n2 elim n2 -n2
+[ #f1 #n1 #n #H elim (after_inv_xOx … H) -H //
+| #n2 #IH * #m1 #f1 #n1 #n #H elim (after_inv_xSx_aux … H ??) -H [3: // |2: skip ]
+  #m #Hm #H destruct /2 width=3 by/
 ]
 qed-.
 
 lemma after_inv_apply: ∀f1,f2,f,a1,a2,a. a1@f1 ⊚ a2@f2 ≡ a@f →
                        a = (a1@f1)@❴a2❵ ∧ tln … a2 f1 ⊚ f2 ≡ f.
-/3 width=3 by  after_fwd_tl, after_fwd_hd, conj/ qed-.
+/3 width=3 by after_fwd_tl, after_fwd_hd, conj/ qed-.
 
 (* Main properties on after *************************************************)
 
-let corec after_trans1: ∀f1,f2,f0. f1 ⊚ f2 ≡ f0 →
-                        ∀f3,f4. f0 ⊚ f3 ≡ f4 →
+let corec after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≡ f4 →
+                        ∀f1,f2. f1 ⊚ f2 ≡ f0 →
                         ∀f. f2 ⊚ f3 ≡ f → f1 ⊚ f ≡ f4 ≝ ?.
-#f1 #f2 #f0 * -f1 -f2 -f0 #f1 #f2 #f0 #b1 [1,2: #b2 ] #b0
-[ #Ht0 #H1 #H2 #H0 * #b3 #f3 * #b4 #f4 #Ht4 * #b #f #Ht
-  cases (after_inv_O1_aux2 … Ht4 H0) -Ht4 -H0 *
-  [ #Ht4 #H3 #H4 cases (after_inv_zero_aux2 … Ht H2 H3) -Ht -H2 -H3
-    #Ht #H /3 width=6 by after_zero/
-  | #a0 #a4 #Ht4 #H3 #H4 cases (after_inv_skip_aux2 … Ht H2 … H3) -Ht -H2 -H3
-    #a #Ht3 #H /3 width=6 by after_skip/
+#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
+[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (after_inv_xxO_aux … Hg0 … H0) -g0
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (after_inv_OOx_aux … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by after_refl/
+| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (after_inv_xxO_aux … Hg0 … H0) -g0
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (after_inv_OSx_aux … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by after_push/
+| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (after_inv_xxS_aux … Hg0 … H0) -g0 *
+  [ #f1 #f2 #Hf0 #H1 #H2
+    cases (after_inv_Sxx_aux … Hg … H2) -g2
+    #f #Hf #H /3 width=7 by after_push/
+  | #f1 #Hf0 #H1 /3 width=6 by after_next/
   ]
-| #a2 #a0 #Ht0 #H1 #H2 #H0 #f3 * #b4 #f4 #Ht4 cases (after_inv_S1_aux2 … Ht4 … H0) -Ht4 -H0
-  #a4 #Ht4 #H4 * #b #f #H cases (after_inv_S1_aux2 … H … H2) -H -H2
-  #a #Ht3 #H /3 width=6 by after_skip/
-| #a1 #a0 #Ht0 #H1 #H0 #f3 * #b4 #f4 #Ht4 cases (after_inv_S1_aux2 … Ht4 … H0) -Ht4 -H0
-  #a4 #Ht4 #H4 * #b #f #Ht /3 width=6 by after_drop/
 ]
 qed-.
 
 let corec after_trans2: ∀f1,f0,f4. f1 ⊚ f0 ≡ f4 →
                         ∀f2, f3. f2 ⊚ f3 ≡ f0 →
                         ∀f. f1 ⊚ f2 ≡ f → f ⊚ f3 ≡ f4 ≝ ?.
-#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #b1 [1,2: #b0 ] #b4
-[ #Ht4 #H1 #H0 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
-  cases (after_inv_O3_aux2 … Ht0 H0) -b0
-  #Ht0 #H2 #H3 cases (after_inv_zero_aux2 … Ht H1 H2) -b1 -b2
-  #Ht #H /3 width=6 by after_zero/
-| #a0 #a4 #Ht4 #H1 #H0 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
-  cases (after_inv_S3_aux2 … Ht0 … H0) -b0 *
-  [ #a3 #Ht0 #H2 #H3 cases (after_inv_zero_aux2 … Ht H1 H2) -b1 -b2
-    #Ht #H /3 width=6 by after_skip/
-  | #a2 #Ht0 #H2 cases (after_inv_skip_aux2 … Ht H1 … H2) -b1 -b2
-    #a #Ht #H /3 width=6 by after_drop/
+#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
+[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+  cases (after_inv_xxO_aux … Hg0 … H0) -g0
+  #f2 #f3 #Hf0 #H2 #H3
+  cases (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+  #f #Hf #H /3 width=7 by after_refl/
+| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+  cases (after_inv_xxS_aux … Hg0 … H0) -g0 *
+  [ #f2 #f3 #Hf0 #H2 #H3
+    cases (after_inv_OOx_aux … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=7 by after_push/
+  | #f2 #Hf0 #H2
+    cases (after_inv_OSx_aux … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=6 by after_next/
   ]
-| #a1 #a4 #Ht4 #H1 #H4 * #b2 #f2 * #b3 #f3 #Ht0 * #b #f #Ht
-  cases (after_inv_S1_aux2 … Ht … H1) -b1
-  #a #Ht #H /3 width=6 by after_drop/
+| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
+  cases (after_inv_Sxx_aux … Hg … H1) -g1
+  #f #Hg #H /3 width=6 by after_next/
 ]
 qed-.
 
+(* Main inversion lemmas on after *******************************************)
+
 let corec after_mono: ∀f1,f2,x. f1 ⊚ f2 ≡ x → ∀y. f1 ⊚ f2 ≡ y → x ≐ y ≝ ?.
-* #a1 #f1 * #a2 #f2 * #c #x #Hx * #d #y #Hy
-cases (after_inv_apply … Hx) -Hx #Hc #Hx
-cases (after_inv_apply … Hy) -Hy #Hd #Hy
+* #n1 #f1 * #n2 #f2 * #n #x #Hx * #m #y #Hy
+cases (after_inv_apply … Hx) -Hx #Hn #Hx
+cases (after_inv_apply … Hy) -Hy #Hm #Hy
 /3 width=4 by eq_seq/
 qed-.
 
 let corec after_inj: ∀f1,x,f. f1 ⊚ x ≡ f → ∀y. f1 ⊚ y ≡ f → x ≐ y ≝ ?.
-* #a1 #f1 * #c #x * #a #f #Hx * #d #y #Hy
-cases (after_inv_apply … Hx) -Hx #Hc #Hx
-cases (after_inv_apply … Hy) -Hy #Hd
-cases (apply_inj_aux … Hc Hd) //
-#Hy -a -d /3 width=4 by eq_seq/
+* #n1 #f1 * #n2 #x * #n #f #Hx * #m2 #y #Hy
+cases (after_inv_apply … Hx) -Hx #Hn2 #Hx
+cases (after_inv_apply … Hy) -Hy #Hm2
+cases (apply_inj_aux … Hn2 Hm2) -n -m2 /3 width=4 by eq_seq/
 qed-.
 
-(* Main inversion lemmas on after *******************************************)
-
 theorem after_inv_total: ∀f1,f2,f. f1 ⊚ f2 ≡ f → f1 ∘ f2 ≐ f.
 /2 width=4 by after_mono/ qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_at.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_at.ma
index a40ba30ad1de6ccd87ff640e80803494c1835012..8fd1da8a77ea290539cc8fec796fcb230c900f5b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_at.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_at.ma
@@ -14,25 +14,25 @@
 
 include "ground_2/notation/functions/apply_2.ma".
 include "ground_2/notation/relations/rat_3.ma".
-include "ground_2/relocation/nstream.ma".
+include "ground_2/relocation/nstream_lift.ma".
 
 (* RELOCATION N-STREAM ******************************************************)
 
 let rec apply (i: nat) on i: rtmap → nat ≝ ?.
-* #b #f cases i -i
-[ @b
+* #n #f cases i -i
+[ @n
 | #i lapply (apply i f) -apply -i -f
-  #i @(⫯(b+i))
+  #i @(⫯(n+i))
 ]
-qed.
+defined.
 
 interpretation "functional application (nstream)"
    'Apply f i = (apply i f).
 
 inductive at: rtmap → relation nat ≝
-| at_zero: ∀f. at (0 @ f) 0 0
-| at_skip: ∀f,i1,i2. at f i1 i2 → at (0 @ f) (⫯i1) (⫯i2)
-| at_lift: ∀f,b,i1,i2. at (b @ f) i1 i2 → at (⫯b @ f) i1 (⫯i2)
+| at_refl: ∀f. at (↑f) 0 0
+| at_push: ∀f,i1,i2. at f i1 i2 → at (↑f) (⫯i1) (⫯i2)
+| at_next: ∀f,i1,i2. at f i1 i2 → at (⫯f) i1 (⫯i2)
 .
 
 interpretation "relational application (nstream)"
@@ -40,122 +40,123 @@ interpretation "relational application (nstream)"
 
 (* Basic properties on apply ************************************************)
 
-lemma apply_S1: ∀f,a,i. (⫯a@f)@❴i❵ = ⫯((a@f)@❴i❵).
-#a #f * //
+lemma apply_S1: ∀f,n,i. (⫯n@f)@❴i❵ = ⫯((n@f)@❴i❵).
+#n #f * //
 qed.
 
 (* Basic inversion lemmas on at *********************************************)
 
-fact at_inv_xOx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. f = 0@g →
-                     (i1 = 0 ∧ i2 = 0) ∨
-                     ∃∃j1,j2. @⦃j1, g⦄ ≡ j2 & i1 = ⫯j1 & i2 = ⫯j2.
-#f #i1 #i2 * -f -i1 -i2
-[ /3 width=1 by or_introl, conj/
-| #f #i1 #i2 #Hi #g #H destruct /3 width=5 by ex3_2_intro, or_intror/
-| #f #b #i1 #i2 #_ #g #H destruct
+fact at_inv_OOx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. i1 = 0 → f = ↑g → i2 = 0.
+#f #i1 #i2 * -f -i1 -i2 //
+[ #f #i1 #i2 #_ #g #H destruct
+| #f #i1 #i2 #_ #g #_ #H elim (discr_next_push … H)
 ]
 qed-.
 
-lemma at_inv_xOx: ∀f,i1,i2. @⦃i1, 0@f⦄ ≡ i2 →
-                  (i1 = 0 ∧ i2 = 0) ∨
-                  ∃∃j1,j2. @⦃j1, f⦄ ≡ j2 & i1 = ⫯j1 & i2 = ⫯j2.
-/2 width=3 by at_inv_xOx_aux/ qed-.
+lemma at_inv_OOx: ∀f,i2. @⦃0, ↑f⦄ ≡ i2 → i2 = 0.
+/2 width=6 by at_inv_OOx_aux/ qed-.
 
-lemma at_inv_OOx: ∀f,i. @⦃0, 0 @ f⦄ ≡ i → i = 0.
-#f #i #H elim (at_inv_xOx … H) -H * //
-#j1 #j2 #_ #H destruct
+fact at_inv_SOx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g,j1. i1 = ⫯j1 → f = ↑g →
+                     ∃∃j2. @⦃j1, g⦄ ≡ j2 & i2 = ⫯j2.
+#f #i1 #i2 * -f -i1 -i2
+[ #f #g #j1 #H destruct
+| #f #i1 #i2 #Hi #g #j1 #H #Hf <(injective_push … Hf) -g destruct /2 width=3 by ex2_intro/
+| #f #i1 #i2 #_ #g #j1 #_ #H elim (discr_next_push … H)
+]
 qed-.
 
-lemma at_inv_xOO: ∀f,i. @⦃i, 0@f⦄ ≡ 0 → i = 0.
-#f #i #H elim (at_inv_xOx … H) -H * //
-#j1 #j2 #_ #_ #H destruct
+lemma at_inv_SOx: ∀f,i1,i2. @⦃⫯i1, ↑f⦄ ≡ i2 →
+                  ∃∃j2. @⦃i1, f⦄ ≡ j2 & i2 = ⫯j2.
+/2 width=5 by at_inv_SOx_aux/ qed-.
+
+fact at_inv_xSx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. f = ⫯g →
+                     ∃∃j2. @⦃i1, g⦄ ≡ j2 & i2 = ⫯j2.
+#f #i1 #i2 * -f -i1 -i2
+[ #f #g #H elim (discr_push_next … H)
+| #f #i1 #i2 #_ #g #H elim (discr_push_next … H)
+| #f #i1 #i2 #Hi #g #H <(injective_next … H) -g /2 width=3 by ex2_intro/
+]
 qed-.
 
-lemma at_inv_SOx: ∀f,i1,i2. @⦃⫯i1, 0@f⦄ ≡ i2 →
+lemma at_inv_xSx: ∀f,i1,i2. @⦃i1, ⫯f⦄ ≡ i2 →
                   ∃∃j2. @⦃i1, f⦄ ≡ j2 & i2 = ⫯j2.
-#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
-[ #H destruct
-| #j1 #j2 #Hj #H1 #H2 destruct /2 width=3 by ex2_intro/
-]
+/2 width=3 by at_inv_xSx_aux/ qed-.
+
+(* Advanced inversion lemmas on at ******************************************)
+
+lemma at_inv_OOS: ∀f,i2. @⦃0, ↑f⦄ ≡ ⫯i2 → ⊥.
+#f #i2 #H lapply (at_inv_OOx … H) -H
+#H destruct
 qed-.
 
-lemma at_inv_xOS: ∀f,i1,i2. @⦃i1, 0@f⦄ ≡ ⫯i2 →
-                  ∃∃j1. @⦃j1, f⦄ ≡ i2 & i1 = ⫯j1.
-#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
-[ #_ #H destruct
-| #j1 #j2 #Hj #H1 #H2 destruct /2 width=3 by ex2_intro/
-]
+lemma at_inv_SOS: ∀f,i1,i2. @⦃⫯i1, ↑f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
+#f #i1 #i2 #H elim (at_inv_SOx … H) -H
+#j2 #H2 #H destruct //
 qed-.
 
-lemma at_inv_SOS: ∀f,i1,i2. @⦃⫯i1, 0@f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
-#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
-[ #H destruct
-| #j1 #j2 #Hj #H1 #H2 destruct //
-]
+lemma at_inv_SOO: ∀f,i1. @⦃⫯i1, ↑f⦄ ≡ 0 → ⊥.
+#f #i1 #H elim (at_inv_SOx … H) -H
+#j2 #_ #H destruct
 qed-.
 
-lemma at_inv_OOS: ∀f,i. @⦃0, 0@f⦄ ≡ ⫯i → ⊥.
-#f #i #H elim (at_inv_xOx … H) -H *
-[ #_ #H destruct
-| #j1 #j2 #_ #H destruct
-]
+lemma at_inv_xSS: ∀f,i1,i2. @⦃i1, ⫯f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
+#f #i1 #i2 #H elim (at_inv_xSx … H) -H
+#j2 #H #H2 destruct //
 qed-.
 
-lemma at_inv_SOO: ∀f,i. @⦃⫯i, 0@f⦄ ≡ 0 → ⊥.
-#f #i #H elim (at_inv_xOx … H) -H *
-[ #H destruct
-| #j1 #j2 #_ #_ #H destruct
-]
+lemma at_inv_xSO: ∀f,i1. @⦃i1, ⫯f⦄ ≡ 0 → ⊥.
+#f #i1 #H elim (at_inv_xSx … H) -H
+#j2 #_ #H destruct
 qed-.
 
-fact at_inv_xSx_aux: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g,a. f = ⫯a @ g →
-                     ∃∃j2. @⦃i1, a@g⦄ ≡ j2 & i2 = ⫯j2.
-#f #i1 #i2 * -f -i1 -i2
-[ #f #g #a #H destruct
-| #f #i1 #i2 #_ #g #a #H destruct
-| #f #b #i1 #i2 #Hi #g #a #H destruct /2 width=3 by ex2_intro/
+lemma at_inv_xOx: ∀f,i1,i2. @⦃i1, ↑f⦄ ≡ i2 →
+                  (i1 = 0 ∧ i2 = 0) ∨
+                  ∃∃j1,j2. @⦃j1, f⦄ ≡ j2 & i1 = ⫯j1 & i2 = ⫯j2.
+#f * [2: #i1 ] #i2 #H
+[ elim (at_inv_SOx … H) -H
+  #j2 #H2 #H destruct /3 width=5 by or_intror, ex3_2_intro/
+| >(at_inv_OOx … H) -i2 /3 width=1 by conj, or_introl/
 ]
 qed-.
 
-lemma at_inv_xSx: ∀f,b,i1,i2. @⦃i1, ⫯b@f⦄ ≡ i2 →
-                  ∃∃j2. @⦃i1, b@f⦄ ≡ j2 & i2 = ⫯j2.
-/2 width=3 by at_inv_xSx_aux/ qed-.
-
-lemma at_inv_xSS: ∀f,b,i1,i2. @⦃i1, ⫯b @ f⦄ ≡ ⫯i2 → @⦃i1, b@f⦄ ≡ i2.
-#f #b #i1 #i2 #H elim (at_inv_xSx … H) -H
-#j2 #Hj #H destruct //
+lemma at_inv_xOO: ∀f,i. @⦃i, ↑f⦄ ≡ 0 → i = 0.
+#f #i #H elim (at_inv_xOx … H) -H * //
+#j1 #j2 #_ #_ #H destruct
 qed-.
 
-lemma at_inv_xSO: ∀f,b,i. @⦃i, ⫯b@f⦄ ≡ 0 → ⊥.
-#f #b #i #H elim (at_inv_xSx … H) -H
-#j2 #_ #H destruct
+lemma at_inv_xOS: ∀f,i1,i2. @⦃i1, ↑f⦄ ≡ ⫯i2 →
+                  ∃∃j1. @⦃j1, f⦄ ≡ i2 & i1 = ⫯j1.
+#f #i1 #i2 #H elim (at_inv_xOx … H) -H *
+[ #_ #H destruct
+| #j1 #j2 #Hj #H1 #H2 destruct /2 width=3 by ex2_intro/
+]
 qed-.
 
 (* alternative definition ***************************************************)
 
-lemma at_O1: ∀b,f. @⦃0, b@f⦄ ≡ b.
-#b elim b -b /2 width=1 by at_lift/
+lemma at_O1: ∀i2,f. @⦃0, i2@f⦄ ≡ i2.
+#i2 elim i2 -i2 /2 width=1 by at_refl, at_next/
 qed.
 
-lemma at_S1: ∀b,f,i1,i2. @⦃i1, f⦄ ≡ i2 → @⦃⫯i1, b@f⦄ ≡ ⫯(b+i2).
-#b elim b -b /3 width=1 by at_skip, at_lift/
+lemma at_S1: ∀n,f,i1,i2. @⦃i1, f⦄ ≡ i2 → @⦃⫯i1, n@f⦄ ≡ ⫯(n+i2).
+#n elim n -n /3 width=1 by at_push, at_next/
 qed.
 
-lemma at_inv_O1: ∀f,b,i2. @⦃0, b@f⦄ ≡ i2 → i2 = b.
-#f #b elim b -b /2 width=2 by at_inv_OOx/
-#b #IH #i2 #H elim (at_inv_xSx … H) -H
+lemma at_inv_O1: ∀f,n,i2. @⦃0, n@f⦄ ≡ i2 → i2 = n.
+#f #n elim n -n /2 width=2 by at_inv_OOx/
+#n #IH #i2 <next_rew #H elim (at_inv_xSx … H) -H
 #j2 #Hj #H destruct /3 width=1 by eq_f/
 qed-.
 
-lemma at_inv_S1: ∀f,b,j1,i2. @⦃⫯j1, b@f⦄ ≡ i2 → ∃∃j2. @⦃j1, f⦄ ≡ j2 & i2 =⫯(b+j2).
-#f #b elim b -b /2 width=1 by at_inv_SOx/
-#b #IH #j1 #i2 #H elim (at_inv_xSx … H) -H
+lemma at_inv_S1: ∀f,n,j1,i2. @⦃⫯j1, n@f⦄ ≡ i2 → ∃∃j2. @⦃j1, f⦄ ≡ j2 & i2 =⫯(n+j2).
+#f #n elim n -n /2 width=1 by at_inv_SOx/
+#n #IH #j1 #i2 <next_rew #H elim (at_inv_xSx … H) -H
 #j2 #Hj #H destruct elim (IH … Hj) -IH -Hj
 #i2 #Hi #H destruct /2 width=3 by ex2_intro/
 qed-.
 
-lemma at_total: ∀i,f. @⦃i, f⦄ ≡ f@❴i❵.
-#i elim i -i
+lemma at_total: ∀i1,f. @⦃i1, f⦄ ≡ f@❴i1❵.
+#i1 elim i1 -i1
 [ * // | #i #IH * /3 width=1 by at_S1/ ]
 qed.
 
@@ -165,8 +166,8 @@ lemma at_increasing: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → i1 ≤ i2.
 #f #i1 #i2 #H elim H -f -i1 -i2 /2 width=1 by le_S_S, le_S/
 qed-.
 
-lemma at_increasing_plus: ∀f,b,i1,i2. @⦃i1, b@f⦄ ≡ i2 → i1 + b ≤ i2.
-#f #b *
+lemma at_increasing_plus: ∀f,n,i1,i2. @⦃i1, n@f⦄ ≡ i2 → i1 + n ≤ i2.
+#f #n *
 [ #i2 #H >(at_inv_O1 … H) -i2 //
 | #i1 #i2 #H elim (at_inv_S1 … H) -H
   #j1 #Ht #H destruct
@@ -174,41 +175,49 @@ lemma at_increasing_plus: ∀f,b,i1,i2. @⦃i1, b@f⦄ ≡ i2 → i1 + b ≤ i2.
 ]
 qed-.
 
-lemma at_increasing_strict: ∀f,b,i1,i2. @⦃i1, ⫯b @ f⦄ ≡ i2 →
-                            i1 < i2 ∧ @⦃i1, b@f⦄ ≡ ⫰i2.
-#f #b #i1 #i2 #H elim (at_inv_xSx … H) -H
+lemma at_increasing_strict: ∀f,i1,i2. @⦃i1, ⫯f⦄ ≡ i2 →
+                            i1 < i2 ∧ @⦃i1, f⦄ ≡ ⫰i2.
+#f #i1 #i2 #H elim (at_inv_xSx … H) -H
 #j2 #Hj #H destruct /4 width=2 by conj, at_increasing, le_S_S/
 qed-.
 
-lemma at_fwd_id: ∀f,b,i. @⦃i, b@f⦄ ≡ i → b = 0.
-#f #b *
-[ #H <(at_inv_O1 … H) -f -b //
+lemma at_fwd_id: ∀f,n,i. @⦃i, n@f⦄ ≡ i → n = 0.
+#f #n *
+[ #H <(at_inv_O1 … H) -f -n //
 | #i #H elim (at_inv_S1 … H) -H
   #j #H #H0 destruct lapply (at_increasing … H) -H
   #H lapply (eq_minus_O … H) -H //
 ]
+qed-.
+
+(* Basic properties on at ***************************************************)
+
+lemma at_plus2: ∀f,i1,i,n,m. @⦃i1, n@f⦄ ≡ i → @⦃i1, (m+n)@f⦄ ≡ m+i.
+#f #i1 #i #n #m #H elim m -m /2 width=1 by at_next/
 qed.
 
-(* Main properties on at ****************************************************)
+(* Advanced properties on at ************************************************)
 
 lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2, f⦄ ≡ i2 → @⦃i1, f⦄ ≡ i1.
 #i1 #i2 #H @(le_elim … H) -i1 -i2 [ #i2 | #i1 #i2 #IH ]
-* #b #f #H lapply (at_fwd_id … H)
+* #n #f #H lapply (at_fwd_id … H)
 #H0 destruct /4 width=1 by at_S1, at_inv_SOS/
 qed-.
 
+(* Main properties on at ****************************************************)
+
 let corec at_ext: ∀f1,f2. (∀i,i1,i2. @⦃i, f1⦄ ≡ i1 → @⦃i, f2⦄ ≡ i2 → i1 = i2) → f1 ≐ f2 ≝ ?.
-* #b1 #f1 * #b2 #f2 #Hi lapply (Hi 0 b1 b2 ? ?) //
+* #n1 #f1 * #n2 #f2 #Hi lapply (Hi 0 n1 n2 ? ?) //
 #H lapply (at_ext f1 f2 ?) /2 width=1 by eq_seq/ -at_ext
-#j #j1 #j2 #H1 #H2 @(injective_plus_r … b2) /4 width=5 by at_S1, injective_S/ (**) (* full auto fails *)
+#j #j1 #j2 #H1 #H2 @(injective_plus_r … n2) /4 width=5 by at_S1, injective_S/ (**) (* full auto fails *)
 qed-.
 
 theorem at_monotonic: ∀i1,i2. i1 < i2 → ∀f1,f2. f1 ≐ f2 → ∀j1,j2. @⦃i1, f1⦄ ≡ j1 → @⦃i2, f2⦄ ≡ j2 → j1 < j2.
 #i1 #i2 #H @(lt_elim … H) -i1 -i2
-[ #i2 * #b1 #f1 * #b2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
+[ #i2 * #n1 #f1 * #n2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
   #H #Ht #j1 #j2 #H1 #H2 destruct
   >(at_inv_O1 … H1) elim (at_inv_S1 … H2) -H2 -j1 //
-| #i1 #i2 #IH * #b1 #f1 * #b2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
+| #i1 #i2 #IH * #n1 #f1 * #n2 #f2 #H elim (eq_stream_inv_seq ????? H) -H
   #H #Ht #j1 #j2 #H1 #H2 destruct
   elim (at_inv_S1 … H2) elim (at_inv_S1 … H1) -H1 -H2
   #x1 #Hx1 #H1 #x2 #Hx2 #H2 destruct /4 width=5 by lt_S_S, monotonic_lt_plus_r/
@@ -218,13 +227,13 @@ qed-.
 theorem at_inv_monotonic: ∀f1,i1,j1. @⦃i1, f1⦄ ≡ j1 → ∀f2,i2,j2. @⦃i2, f2⦄ ≡ j2 → f1 ≐ f2 → j2 < j1 → i2 < i1.
 #f1 #i1 #j1 #H elim H -f1 -i1 -j1
 [ #f1 #f2 #i2 #j2 #_ #_ #H elim (lt_le_false … H) //
-| #f1 #i1 #j1 #_ #IH * #b2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
+| #f1 #i1 #j1 #_ #IH * #n2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
   #H0 #Ht destruct elim (at_inv_xOx … H) -H *
   [ #H1 #H2 destruct //
   | #x2 #y2 #Hxy #H1 #H2 destruct /4 width=5 by lt_S_S_to_lt, lt_S_S/
   ]
-| #f1 #b1 #i1 #j1 #_ #IH * #b2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
-  #H0 #Ht destruct elim (at_inv_xSx … H) -H
+| * #n1 #f1 #i1 #j1 #_ #IH * #n2 #f2 #i2 #j2 #H #Ht #Hj elim (eq_stream_inv_seq ????? Ht) -Ht
+  #H0 #Ht destruct <next_rew in H; #H elim (at_inv_xSx … H) -H
   #y2 #Hy #H destruct /3 width=5 by eq_seq, lt_S_S_to_lt/
 ]
 qed-.
@@ -273,7 +282,7 @@ qed-.
 
 (* Note: see also: trace_at/is_at_dec *)
 lemma is_at_dec: ∀f,i2. Decidable (∃i1. @⦃i1, f⦄ ≡ i2).
-#f #i2 @(is_at_dec_le ? ? (⫯i2)) /2 width=4 by lt_le_false/
+#f #i2 @(is_at_dec_le ?? (⫯i2)) /2 width=4 by lt_le_false/
 qed-.
 
 (* Advanced properties on apply *********************************************)

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_id.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_id.ma
index 5e3d1e3b222184f1c49bed19fd952b4b83dab37e..a24fd8a3ba23d80873def3d8b09534f2a0dc8344 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_id.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_id.ma
@@ -14,7 +14,6 @@
 
 include "ground_2/notation/functions/identity_0.ma".
 include "ground_2/notation/relations/isidentity_1.ma".
-include "ground_2/relocation/nstream_lift.ma".
 include "ground_2/relocation/nstream_after.ma".
 
 (* RELOCATION N-STREAM ******************************************************)
@@ -46,37 +45,37 @@ qed.
 
 (* Basic inversion lemmas on isid *******************************************)
 
-lemma isid_inv_seq: ∀f,a.  𝐈⦃a@f⦄ → 𝐈⦃f⦄ ∧ a = 0.
-#f #a normalize >id_unfold in ⊢ (???%→?);
+lemma isid_inv_seq: ∀f,n.  𝐈⦃n@f⦄ → 𝐈⦃f⦄ ∧ n = 0.
+#f #n normalize >id_unfold in ⊢ (???%→?);
 #H elim (eq_stream_inv_seq ????? H) -H /2 width=1 by conj/
 qed-.
 
 lemma isid_inv_push: ∀f. 𝐈⦃↑f⦄ → 𝐈⦃f⦄.
-* #a #f #H elim (isid_inv_seq … H) -H //
+* #n #f #H elim (isid_inv_seq … H) -H //
 qed-.
 
 lemma isid_inv_next: ∀f. 𝐈⦃⫯f⦄ → ⊥.
-* #a #f #H elim (isid_inv_seq … H) -H
+* #n #f #H elim (isid_inv_seq … H) -H
 #_ #H destruct
 qed-.
 
 (* inversion lemmas on at ***************************************************)
 
 let corec id_inv_at: ∀f. (∀i. @⦃i, f⦄ ≡ i) → f ≐ 𝐈𝐝 ≝ ?.
-* #a #f #Ht lapply (Ht 0)
+* #n #f #Ht lapply (Ht 0)
 #H lapply (at_inv_O1 … H) -H
 #H0 >id_unfold @eq_seq
-[ cases H0 -a //
+[ cases H0 -n //
 | @id_inv_at -id_inv_at
-  #i lapply (Ht (⫯i)) -Ht cases H0 -a
+  #i lapply (Ht (⫯i)) -Ht cases H0 -n
   #H elim (at_inv_SOx … H) -H //
 ]
 qed-.
 
 lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i, f⦄ ≡ i.
 #i elim i -i
-[ * #a #f #H elim (isid_inv_seq … H) -H //
-| #i #IH * #a #f #H elim (isid_inv_seq … H) -H
+[ * #n #f #H elim (isid_inv_seq … H) -H //
+| #i #IH * #n #f #H elim (isid_inv_seq … H) -H
   /3 width=1 by at_S1/
 ]
 qed-.
@@ -117,18 +116,15 @@ qed.
 (* Inversion lemmas on after ************************************************)
 
 let corec isid_after_sn: ∀f1,f2. 𝐈⦃f1⦄ → f1 ⊚ f2 ≡ f2 ≝ ?.
-* #a1 #f1 * * [ | #a2 ] #f2 #H cases (isid_inv_seq … H) -H
-#Ht1 #H1
-[ @(after_zero … H1) -H1 /2 width=1 by/
-| @(after_skip … H1) -H1 /2 width=5 by/
-]
+* #n1 #f1 * * [ | #n2 ] #f2 #H cases (isid_inv_seq … H) -H
+/3 width=7 by after_push, after_refl/
 qed-.
 
 let corec isid_after_dx: ∀f2,f1. 𝐈⦃f2⦄ → f1 ⊚ f2 ≡ f1 ≝ ?.
-* #a2 #f2 * *
+* #n2 #f2 * *
 [ #f1 #H cases (isid_inv_seq … H) -H
-  #Ht2 #H2 @(after_zero … H2) -H2 /2 width=1 by/
-| #a1 #f1 #H @(after_drop … a1 a1) /2 width=5 by/
+  /3 width=7 by after_refl/
+| #n1 #f1 #H @after_next [4,5: // |1,2: skip ] /2 width=1 by/
 ]
 qed-.
 

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_lift.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_lift.ma
index e98d8bee1f98515be0b45362e7b3f5d5f9330038..4e2f20bf68dc738a0ba30784e4363673a18fc046 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_lift.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/nstream_lift.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "ground_2/notation/functions/lift_1.ma".
-include "ground_2/relocation/nstream_at.ma".
+include "ground_2/relocation/nstream.ma".
 
 (* RELOCATION N-STREAM ******************************************************)
 
@@ -22,23 +22,23 @@ definition push: rtmap → rtmap ≝ λf. 0@f.
 interpretation "push (nstream)" 'Lift f = (push f).
 
 definition next: rtmap → rtmap.
-* #a #f @(⫯a@f)
+* #n #f @(⫯n@f)
 qed.
 
 interpretation "next (nstream)" 'Successor f = (next f).
 
-(* Basic properties on push *************************************************)
+(* Basic properties *********************************************************)
 
-lemma push_at_O: ∀f. @⦃0, ↑f⦄ ≡ 0.
+lemma push_rew: ∀f. ↑f = 0@f.
 // qed.
 
-lemma push_at_S: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → @⦃⫯i1, ↑f⦄ ≡ ⫯i2.
-/2 width=1 by at_S1/ qed.
+lemma next_rew: ∀f,n. ⫯(n@f) = (⫯n)@f.
+// qed.
 
-(* Basic inversion lemmas on push *******************************************)
+lemma next_rew_sn: ∀f,n1,n2. n1 = ⫯n2 → n1@f = ⫯(n2@f).
+// qed.
 
-lemma push_inv_at_S: ∀f,i1,i2. @⦃⫯i1, ↑f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
-/2 width=1 by at_inv_SOS/ qed-.
+(* Basic inversion lemmas ***************************************************)
 
 lemma injective_push: injective ? ? push.
 #f1 #f2 normalize #H destruct //
@@ -52,18 +52,22 @@ lemma discr_next_push: ∀f1,f2. ⫯f1 = ↑f2 → ⊥.
 * #n1 #f1 #f2 normalize #H destruct
 qed-.
 
-(* Basic properties on next *************************************************)
+lemma injective_next: injective ? ? next.
+* #n1 #f1 * #n2 #f2 normalize #H destruct //
+qed-.
 
-lemma next_at: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → @⦃i1, ⫯f⦄ ≡ ⫯i2.
-* /2 width=1 by at_lift/
-qed.
+lemma push_inv_seq_sn: ∀f,g,n. n@g = ↑f → n = 0 ∧ g = f.
+#f #g #n >push_rew #H destruct /2 width=1 by conj/
+qed-.
 
-(* Basic inversion lemmas on next *******************************************)
+lemma push_inv_seq_dx: ∀f,g,n. ↑f = n@g → n = 0 ∧ g = f.
+#f #g #n >push_rew #H destruct /2 width=1 by conj/
+qed-.
 
-lemma next_inv_at: ∀f,i1,i2. @⦃i1, ⫯f⦄ ≡ ⫯i2 → @⦃i1, f⦄ ≡ i2.
-* /2 width=1 by at_inv_xSS/
+lemma next_inv_seq_sn: ∀f,g,n. n@g = ⫯f → ∃∃m. n = ⫯m & f = m@g.
+* #m #f #g #n >next_rew #H destruct /2 width=3 by ex2_intro/
 qed-.
 
-lemma injective_next: injective ? ? next.
-* #a1 #f1 * #a2 #f2 normalize #H destruct //
+lemma next_inv_seq_dx: ∀f,g,n. ⫯f = n@g → ∃∃m. n = ⫯m & f = m@g.
+* #m #f #g #n >next_rew #H destruct /2 width=3 by ex2_intro/
 qed-.

diff --git a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl
index 50896c56c0db2f22e7d4a609f078b4482142fce6..39f75c7b6496ac29e489550350268036fdcda27e 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2_src.tbl
@@ -12,7 +12,7 @@ table {
    class "green"
    [ { "multiple relocation" * } {
         [ { "" * } {
-             [ "nstream" "nstream_at ( ?@❴?❵ ) ( @⦃?,?⦄ ≡ ? )" "nstream_lift ( ↑? ) ( ⫯? )" "nstream_after ( ? ∘ ? ) ( ? ⊚ ? ≡ ? )" "nstream_id ( 𝐈𝐝 ) ( 𝐈⦃?⦄ )" * ]
+             [ "nstream" "nstream_lift ( ↑? ) ( ⫯? )" "nstream_at ( ?@❴?❵ ) ( @⦃?,?⦄ ≡ ? )" "nstream_after ( ? ∘ ? ) ( ? ⊚ ? ≡ ? )" "nstream_id ( 𝐈𝐝 ) ( 𝐈⦃?⦄ )" * ]
              [ "trace ( ∥?∥ )" "trace_at ( @⦃?,?⦄ ≡ ? )" "trace_after ( ? ⊚ ? ≡ ? )" "trace_isid ( 𝐈⦃?⦄ )" "trace_isun ( 𝐔⦃?⦄ )"
                "trace_sle ( ? ⊆ ? )" "trace_sor ( ? ⋓ ? ≡ ? )" "trace_snot ( ∁ ? )" * ]
              [ "mr2" "mr2_at ( @⦃?,?⦄ ≡ ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? ▭ ? ≡ ? )" * ]