--- /dev/null
+LambdaDelta.vo: LambdaDelta.v terms_defs.vo tlt_defs.vo contexts_defs.vo lift_defs.vo lift_gen.vo lift_props.vo lift_tlt.vo subst0_defs.vo subst0_gen.vo subst0_lift.vo subst0_subst0.vo subst0_confluence.vo subst0_tlt.vo subst1_defs.vo subst1_gen.vo subst1_lift.vo subst1_subst1.vo subst1_confluence.vo drop_defs.vo drop_props.vo csubst0_defs.vo csubst1_defs.vo pr0_defs.vo pr0_lift.vo pr0_gen.vo pr0_subst0.vo pr0_confluence.vo pr0_subst1.vo pr1_defs.vo pr1_confluence.vo pr2_defs.vo pr2_lift.vo pr2_gen.vo pr2_confluence.vo pr2_subst1.vo pr2_gen_context.vo pr3_defs.vo pr3_props.vo pr3_gen.vo pr3_confluence.vo pr3_subst1.vo pr3_gen_context.vo cpr0_defs.vo cpr0_props.vo pc1_defs.vo pc3_defs.vo pc3_props.vo pc3_gen.vo pc3_subst0.vo pc3_gen_context.vo ty0_defs.vo ty0_lift.vo ty0_props.vo ty0_subst0.vo ty0_gen_context.vo csub0_defs.vo ty0_sred.vo ty0_sred_props.vo
+ty0_sred_props.vo: ty0_sred_props.v lift_props.vo drop_props.vo pc3_props.vo pc3_gen.vo ty0_defs.vo ty0_props.vo ty0_sred.vo
+ty0_sred.vo: ty0_sred.v lift_gen.vo subst1_gen.vo csubst1_defs.vo pr0_lift.vo pr0_subst1.vo cpr0_defs.vo cpr0_props.vo pc3_props.vo pc3_gen.vo ty0_defs.vo ty0_lift.vo ty0_props.vo ty0_subst0.vo ty0_gen_context.vo csub0_defs.vo
+csub0_defs.vo: csub0_defs.v ty0_defs.vo
+ty0_gen_context.vo: ty0_gen_context.v lift_gen.vo lift_props.vo subst1_defs.vo subst1_lift.vo subst1_confluence.vo drop_props.vo csubst1_defs.vo pc3_gen.vo pc3_gen_context.vo ty0_defs.vo ty0_lift.vo
+ty0_subst0.vo: ty0_subst0.v drop_props.vo csubst0_defs.vo pc3_props.vo pc3_subst0.vo ty0_defs.vo ty0_lift.vo ty0_props.vo
+ty0_props.vo: ty0_props.v drop_props.vo pc3_props.vo ty0_defs.vo ty0_lift.vo
+ty0_lift.vo: ty0_lift.v lift_props.vo drop_props.vo pc3_props.vo ty0_defs.vo
+ty0_defs.vo: ty0_defs.v pc3_defs.vo
+pc3_gen_context.vo: pc3_gen_context.v subst1_confluence.vo csubst1_defs.vo pr3_gen_context.vo pc3_defs.vo pc3_props.vo
+pc3_subst0.vo: pc3_subst0.v subst0_subst0.vo csubst0_defs.vo pr0_subst0.vo pc3_defs.vo
+pc3_gen.vo: pc3_gen.v lift_gen.vo pr3_props.vo pr3_gen.vo pc3_defs.vo pc3_props.vo
+pc3_props.vo: pc3_props.v subst0_subst0.vo pr0_subst0.vo pr3_defs.vo pr3_props.vo pr3_confluence.vo cpr0_defs.vo cpr0_props.vo pc3_defs.vo
+pc3_defs.vo: pc3_defs.v pr2_defs.vo pr3_defs.vo pc1_defs.vo
+pc1_defs.vo: pc1_defs.v pr0_defs.vo
+cpr0_props.vo: cpr0_props.v pr0_subst0.vo pr3_defs.vo pr3_props.vo cpr0_defs.vo
+cpr0_defs.vo: cpr0_defs.v contexts_defs.vo pr0_defs.vo
+pr3_gen_context.vo: pr3_gen_context.v csubst1_defs.vo pr2_gen_context.vo pr3_defs.vo
+pr3_subst1.vo: pr3_subst1.v subst1_defs.vo pr2_subst1.vo pr3_defs.vo
+pr3_confluence.vo: pr3_confluence.v pr2_confluence.vo pr3_defs.vo
+pr3_gen.vo: pr3_gen.v pr2_gen.vo pr3_defs.vo pr3_props.vo
+pr3_props.vo: pr3_props.v subst0_subst0.vo pr0_subst0.vo pr2_lift.vo pr3_defs.vo
+pr3_defs.vo: pr3_defs.v pr1_defs.vo pr2_defs.vo
+pr2_gen_context.vo: pr2_gen_context.v drop_props.vo subst1_gen.vo subst1_subst1.vo subst1_confluence.vo csubst1_defs.vo pr0_gen.vo pr0_subst1.vo pr2_defs.vo pr2_subst1.vo
+pr2_subst1.vo: pr2_subst1.v subst1_defs.vo subst1_confluence.vo drop_props.vo pr0_subst1.vo pr2_defs.vo
+pr2_confluence.vo: pr2_confluence.v subst0_confluence.vo drop_props.vo pr0_subst0.vo pr0_confluence.vo pr2_defs.vo
+pr2_gen.vo: pr2_gen.v subst0_gen.vo drop_props.vo pr0_gen.vo pr2_defs.vo
+pr2_lift.vo: pr2_lift.v subst0_lift.vo drop_props.vo pr0_lift.vo pr2_defs.vo
+pr2_defs.vo: pr2_defs.v drop_defs.vo pr0_defs.vo
+pr1_confluence.vo: pr1_confluence.v pr0_confluence.vo pr1_defs.vo
+pr1_defs.vo: pr1_defs.v pr0_defs.vo
+pr0_subst1.vo: pr0_subst1.v subst1_defs.vo pr0_defs.vo pr0_subst0.vo
+pr0_confluence.vo: pr0_confluence.v tlt_defs.vo lift_gen.vo lift_tlt.vo subst0_gen.vo subst0_confluence.vo pr0_defs.vo pr0_lift.vo pr0_gen.vo pr0_subst0.vo
+pr0_subst0.vo: pr0_subst0.v subst0_gen.vo subst0_lift.vo subst0_subst0.vo subst0_confluence.vo pr0_defs.vo pr0_lift.vo
+pr0_gen.vo: pr0_gen.v lift_gen.vo lift_props.vo subst0_gen.vo pr0_defs.vo pr0_lift.vo
+pr0_lift.vo: pr0_lift.v lift_props.vo subst0_lift.vo pr0_defs.vo
+pr0_defs.vo: pr0_defs.v subst0_defs.vo
+csubst1_defs.vo: csubst1_defs.v subst1_defs.vo csubst0_defs.vo
+csubst0_defs.vo: csubst0_defs.v contexts_defs.vo subst0_defs.vo drop_defs.vo
+drop_props.vo: drop_props.v lift_gen.vo drop_defs.vo
+drop_defs.vo: drop_defs.v contexts_defs.vo lift_defs.vo
+subst1_confluence.vo: subst1_confluence.v lift_gen.vo subst0_gen.vo subst0_confluence.vo subst1_defs.vo subst1_gen.vo
+subst1_subst1.vo: subst1_subst1.v subst0_subst0.vo subst1_defs.vo
+subst1_lift.vo: subst1_lift.v lift_props.vo subst0_lift.vo subst1_defs.vo
+subst1_gen.vo: subst1_gen.v subst0_gen.vo subst1_defs.vo
+subst1_defs.vo: subst1_defs.v subst0_defs.vo
+subst0_tlt.vo: subst0_tlt.v tlt_defs.vo lift_tlt.vo subst0_defs.vo
+subst0_confluence.vo: subst0_confluence.v lift_gen.vo subst0_gen.vo subst0_defs.vo
+subst0_subst0.vo: subst0_subst0.v subst0_defs.vo subst0_gen.vo subst0_lift.vo
+subst0_lift.vo: subst0_lift.v lift_props.vo subst0_defs.vo
+subst0_gen.vo: subst0_gen.v lift_props.vo subst0_defs.vo
+subst0_defs.vo: subst0_defs.v lift_defs.vo
+lift_tlt.vo: lift_tlt.v tlt_defs.vo lift_defs.vo
+lift_props.vo: lift_props.v lift_defs.vo
+lift_gen.vo: lift_gen.v lift_defs.vo
+lift_defs.vo: lift_defs.v terms_defs.vo
+contexts_defs.vo: contexts_defs.v terms_defs.vo
+tlt_defs.vo: tlt_defs.v terms_defs.vo
+terms_defs.vo: terms_defs.v Base.vo
+Base.vo: Base.v base_tactics.vo base_hints.vo base_types.vo base_blt.vo base_rewrite.vo
+base_rewrite.vo: base_rewrite.v
+base_blt.vo: base_blt.v base_tactics.vo base_hints.vo
+base_types.vo: base_types.v base_tactics.vo base_hints.vo
+base_hints.vo: base_hints.v base_tactics.vo
+base_tactics.vo: base_tactics.v
--- /dev/null
+Require Export Arith.
+Require Export Wf_nat.
+Require Export base_tactics.
+Require Export base_hints.
+Require Export base_types.
+Require Export base_blt.
+Require Export base_rewrite.
--- /dev/null
+Require Export terms_defs.
+Require Export tlt_defs.
+Require Export contexts_defs.
+Require Export lift_defs.
+Require Export lift_gen.
+Require Export lift_props.
+Require Export lift_tlt.
+Require Export subst0_defs.
+Require Export subst0_gen.
+Require Export subst0_lift.
+Require Export subst0_subst0.
+Require Export subst0_confluence.
+Require Export subst0_tlt.
+Require Export subst1_defs.
+Require Export subst1_gen.
+Require Export subst1_lift.
+Require Export subst1_subst1.
+Require Export subst1_confluence.
+Require Export drop_defs.
+Require Export drop_props.
+Require Export csubst0_defs.
+Require Export csubst1_defs.
+Require Export pr0_defs.
+Require Export pr0_lift.
+Require Export pr0_gen.
+Require Export pr0_subst0.
+Require Export pr0_confluence.
+Require Export pr0_subst1.
+Require Export pr1_defs.
+Require Export pr1_confluence.
+Require Export pr2_defs.
+Require Export pr2_lift.
+Require Export pr2_gen.
+Require Export pr2_confluence.
+Require Export pr2_subst1.
+Require Export pr2_gen_context.
+Require Export pr3_defs.
+Require Export pr3_props.
+Require Export pr3_gen.
+Require Export pr3_confluence.
+Require Export pr3_subst1.
+Require Export pr3_gen_context.
+Require Export cpr0_defs.
+Require Export cpr0_props.
+Require Export pc1_defs.
+Require Export pc3_defs.
+Require Export pc3_props.
+Require Export pc3_gen.
+Require Export pc3_subst0.
+Require Export pc3_gen_context.
+Require Export ty0_defs.
+Require Export ty0_lift.
+Require Export ty0_props.
+Require Export ty0_subst0.
+Require Export ty0_gen_context.
+Require Export csub0_defs.
+Require Export ty0_sred.
+Require Export ty0_sred_props.
--- /dev/null
+# List of vernac files to compile
+base_tactics.v
+base_hints.v
+base_types.v
+base_blt.v
+base_rewrite.v
+Base.v
+terms_defs.v
+tlt_defs.v
+contexts_defs.v
+lift_defs.v
+lift_gen.v
+lift_props.v
+lift_tlt.v
+subst0_defs.v
+subst0_gen.v
+subst0_lift.v
+subst0_subst0.v
+subst0_confluence.v
+subst0_tlt.v
+subst1_defs.v
+subst1_gen.v
+subst1_lift.v
+subst1_subst1.v
+subst1_confluence.v
+drop_defs.v
+drop_props.v
+csubst0_defs.v
+csubst1_defs.v
+pr0_defs.v
+pr0_lift.v
+pr0_gen.v
+pr0_subst0.v
+pr0_confluence.v
+pr0_subst1.v
+pr1_defs.v
+pr1_confluence.v
+pr2_defs.v
+pr2_lift.v
+pr2_gen.v
+pr2_confluence.v
+pr2_subst1.v
+pr2_gen_context.v
+pr3_defs.v
+pr3_props.v
+pr3_gen.v
+pr3_confluence.v
+pr3_subst1.v
+pr3_gen_context.v
+cpr0_defs.v
+cpr0_props.v
+pc1_defs.v
+pc3_defs.v
+pc3_props.v
+pc3_gen.v
+pc3_subst0.v
+pc3_gen_context.v
+ty0_defs.v
+ty0_lift.v
+ty0_props.v
+ty0_subst0.v
+ty0_gen_context.v
+csub0_defs.v
+ty0_sred.v
+ty0_sred_props.v
+LambdaDelta.v
--- /dev/null
+##############################################################################
+## The Calculus of Inductive Constructions ##
+## ##
+## Projet Coq ##
+## ##
+## INRIA ENS-CNRS ##
+## Rocquencourt Lyon ##
+## ##
+## Coq V7 ##
+## ##
+## ##
+##############################################################################
+
+# WARNING
+#
+# This Makefile has been automagically generated by coq_makefile
+# Edit at your own risks !
+#
+# END OF WARNING
+
+#
+# This Makefile was generated by the command line :
+# coq_makefile -f Make -o Makefile
+#
+
+##########################
+# #
+# Variables definitions. #
+# #
+##########################
+
+CAMLP4LIB=`camlp4 -where`
+COQSRC=-I $(COQTOP)/kernel -I $(COQTOP)/lib \
+ -I $(COQTOP)/library -I $(COQTOP)/parsing -I $(COQTOP)/pretyping \
+ -I $(COQTOP)/proofs -I $(COQTOP)/syntax -I $(COQTOP)/tactics \
+ -I $(COQTOP)/toplevel -I $(CAMLP4LIB)
+ZFLAGS=$(OCAMLLIBS) $(COQSRC)
+OPT=
+COQFLAGS=-q $(OPT) $(COQLIBS)
+COQC=$(COQBIN)coqc
+GALLINA=gallina
+COQWEB=coqweb
+CAMLC=ocamlc -c
+CAMLOPTC=ocamlopt -c
+CAMLLINK=ocamlc
+CAMLOPTLINK=ocamlopt
+COQDEP=$(COQBIN)coqdep -c
+COQVO2XML=coq_vo2xml
+
+#########################
+# #
+# Libraries definition. #
+# #
+#########################
+
+OCAMLLIBS=-I .
+COQLIBS=-I .
+
+###################################
+# #
+# Definition of the "all" target. #
+# #
+###################################
+
+VFILES=base_tactics.v\
+ base_hints.v\
+ base_types.v\
+ base_blt.v\
+ base_rewrite.v\
+ Base.v\
+ terms_defs.v\
+ tlt_defs.v\
+ contexts_defs.v\
+ lift_defs.v\
+ lift_gen.v\
+ lift_props.v\
+ lift_tlt.v\
+ subst0_defs.v\
+ subst0_gen.v\
+ subst0_lift.v\
+ subst0_subst0.v\
+ subst0_confluence.v\
+ subst0_tlt.v\
+ subst1_defs.v\
+ subst1_gen.v\
+ subst1_lift.v\
+ subst1_subst1.v\
+ subst1_confluence.v\
+ drop_defs.v\
+ drop_props.v\
+ csubst0_defs.v\
+ csubst1_defs.v\
+ pr0_defs.v\
+ pr0_lift.v\
+ pr0_gen.v\
+ pr0_subst0.v\
+ pr0_confluence.v\
+ pr0_subst1.v\
+ pr1_defs.v\
+ pr1_confluence.v\
+ pr2_defs.v\
+ pr2_lift.v\
+ pr2_gen.v\
+ pr2_confluence.v\
+ pr2_subst1.v\
+ pr2_gen_context.v\
+ pr3_defs.v\
+ pr3_props.v\
+ pr3_gen.v\
+ pr3_confluence.v\
+ pr3_subst1.v\
+ pr3_gen_context.v\
+ cpr0_defs.v\
+ cpr0_props.v\
+ pc1_defs.v\
+ pc3_defs.v\
+ pc3_props.v\
+ pc3_gen.v\
+ pc3_subst0.v\
+ pc3_gen_context.v\
+ ty0_defs.v\
+ ty0_lift.v\
+ ty0_props.v\
+ ty0_subst0.v\
+ ty0_gen_context.v\
+ csub0_defs.v\
+ ty0_sred.v\
+ ty0_sred_props.v\
+ LambdaDelta.v
+VOFILES=$(VFILES:.v=.vo)
+VIFILES=$(VFILES:.v=.vi)
+GFILES=$(VFILES:.v=.g)
+HTMLFILES=$(VFILES:.v=.html)
+GHTMLFILES=$(VFILES:.v=.g.html)
+
+all: base_tactics.vo\
+ base_hints.vo\
+ base_types.vo\
+ base_blt.vo\
+ base_rewrite.vo\
+ Base.vo\
+ terms_defs.vo\
+ tlt_defs.vo\
+ contexts_defs.vo\
+ lift_defs.vo\
+ lift_gen.vo\
+ lift_props.vo\
+ lift_tlt.vo\
+ subst0_defs.vo\
+ subst0_gen.vo\
+ subst0_lift.vo\
+ subst0_subst0.vo\
+ subst0_confluence.vo\
+ subst0_tlt.vo\
+ subst1_defs.vo\
+ subst1_gen.vo\
+ subst1_lift.vo\
+ subst1_subst1.vo\
+ subst1_confluence.vo\
+ drop_defs.vo\
+ drop_props.vo\
+ csubst0_defs.vo\
+ csubst1_defs.vo\
+ pr0_defs.vo\
+ pr0_lift.vo\
+ pr0_gen.vo\
+ pr0_subst0.vo\
+ pr0_confluence.vo\
+ pr0_subst1.vo\
+ pr1_defs.vo\
+ pr1_confluence.vo\
+ pr2_defs.vo\
+ pr2_lift.vo\
+ pr2_gen.vo\
+ pr2_confluence.vo\
+ pr2_subst1.vo\
+ pr2_gen_context.vo\
+ pr3_defs.vo\
+ pr3_props.vo\
+ pr3_gen.vo\
+ pr3_confluence.vo\
+ pr3_subst1.vo\
+ pr3_gen_context.vo\
+ cpr0_defs.vo\
+ cpr0_props.vo\
+ pc1_defs.vo\
+ pc3_defs.vo\
+ pc3_props.vo\
+ pc3_gen.vo\
+ pc3_subst0.vo\
+ pc3_gen_context.vo\
+ ty0_defs.vo\
+ ty0_lift.vo\
+ ty0_props.vo\
+ ty0_subst0.vo\
+ ty0_gen_context.vo\
+ csub0_defs.vo\
+ ty0_sred.vo\
+ ty0_sred_props.vo\
+ LambdaDelta.vo
+
+spec: $(VIFILES)
+
+gallina: $(GFILES)
+
+html: $(HTMLFILES)
+
+gallinahtml: $(GHTMLFILES)
+
+all.ps: $(VFILES)
+ $(COQWEB) -ps -o $@ `$(COQDEP) -sort -suffix .v $(VFILES)`
+
+all-gal.ps: $(GFILES)
+ $(COQWEB) -ps -o $@ `$(COQDEP) -sort -suffix .g $(VFILES)`
+
+xml:: .xml_time_stamp
+.xml_time_stamp: base_tactics.vo\
+ base_hints.vo\
+ base_types.vo\
+ base_blt.vo\
+ base_rewrite.vo\
+ Base.vo\
+ terms_defs.vo\
+ tlt_defs.vo\
+ contexts_defs.vo\
+ lift_defs.vo\
+ lift_gen.vo\
+ lift_props.vo\
+ lift_tlt.vo\
+ subst0_defs.vo\
+ subst0_gen.vo\
+ subst0_lift.vo\
+ subst0_subst0.vo\
+ subst0_confluence.vo\
+ subst0_tlt.vo\
+ subst1_defs.vo\
+ subst1_gen.vo\
+ subst1_lift.vo\
+ subst1_subst1.vo\
+ subst1_confluence.vo\
+ drop_defs.vo\
+ drop_props.vo\
+ csubst0_defs.vo\
+ csubst1_defs.vo\
+ pr0_defs.vo\
+ pr0_lift.vo\
+ pr0_gen.vo\
+ pr0_subst0.vo\
+ pr0_confluence.vo\
+ pr0_subst1.vo\
+ pr1_defs.vo\
+ pr1_confluence.vo\
+ pr2_defs.vo\
+ pr2_lift.vo\
+ pr2_gen.vo\
+ pr2_confluence.vo\
+ pr2_subst1.vo\
+ pr2_gen_context.vo\
+ pr3_defs.vo\
+ pr3_props.vo\
+ pr3_gen.vo\
+ pr3_confluence.vo\
+ pr3_subst1.vo\
+ pr3_gen_context.vo\
+ cpr0_defs.vo\
+ cpr0_props.vo\
+ pc1_defs.vo\
+ pc3_defs.vo\
+ pc3_props.vo\
+ pc3_gen.vo\
+ pc3_subst0.vo\
+ pc3_gen_context.vo\
+ ty0_defs.vo\
+ ty0_lift.vo\
+ ty0_props.vo\
+ ty0_subst0.vo\
+ ty0_gen_context.vo\
+ csub0_defs.vo\
+ ty0_sred.vo\
+ ty0_sred_props.vo\
+ LambdaDelta.vo
+ $(COQVO2XML) $(COQFLAGS) $(?:%.o=%)
+ touch .xml_time_stamp
+
+####################
+# #
+# Special targets. #
+# #
+####################
+
+.PHONY: all opt byte archclean clean install depend xml
+
+.SUFFIXES: .v .vo .vi .g .html .tex .g.tex .g.html
+
+.v.vo:
+ $(COQC) $(COQDEBUG) $(COQFLAGS) $*
+
+.v.vi:
+ $(COQC) -i $(COQDEBUG) $(COQFLAGS) $*
+
+.v.g:
+ $(GALLINA) $<
+
+.v.tex:
+ $(COQWEB) $< -o $@
+
+.v.html:
+ $(COQWEB) -html $< -o $@
+
+.g.g.tex:
+ $(COQWEB) $< -o $@
+
+.g.g.html:
+ $(COQWEB) -html $< -o $@
+
+byte:
+ $(MAKE) all "OPT="
+
+opt:
+ $(MAKE) all "OPT=-opt"
+
+include .depend
+
+depend:
+ rm .depend
+ $(COQDEP) -i $(COQLIBS) *.v *.ml *.mli >.depend
+ $(COQDEP) $(COQLIBS) -suffix .html *.v >>.depend
+
+xml::
+
+install:
+ mkdir -p `$(COQC) -where`/user-contrib
+ cp -f *.vo `$(COQC) -where`/user-contrib
+
+Makefile: Make
+ mv -f Makefile Makefile.bak
+ $(COQBIN)coq_makefile -f Make -o Makefile
+
+clean:
+ rm -f *.cmo *.cmi *.cmx *.o *.vo *.vi *.g *~
+ rm -f all.ps all-gal.ps $(HTMLFILES) $(GHTMLFILES)
+
+archclean:
+ rm -f *.cmx *.o
+
+# WARNING
+#
+# This Makefile has been automagically generated by coq_makefile
+# Edit at your own risks !
+#
+# END OF WARNING
+
--- /dev/null
+
+ Contribution Bologna/LAMBDA-TYPES
+ ==================================
+
+This directory contains a formalization in Coq of "lambda-delta", a
+lambda-typed lambda-calculus with abbreviations.
+
+Author & Date: Ferruccio Guidi
+ Department of Computer Science, University of Bologna
+ March 2005
+E-mail : fguidi@cs.unibo.it
+WWW : http://www.cs.unibo.it/~fguidi
+
+Installation procedure:
+-----------------------
+
+ To get this contribution compiled, type
+
+ make
+
+ or
+
+ make opt
+
+ The main modules produced by the compilation are:
+
+ LambdaDelta provides the theory of the "lambda-delta" calculus
+ and its prerequisites
+
+ Base provides just the prerequisites (mainly some arithmetic
+ properties missing in the standard library of Coq)
+
+Description:
+------------
+
+ The present work, which is meant to be improved in the future, contains
+ a formalization of the "lambda-delta" calculus, defined in "item notation"
+ and with De Bruijn indices, and includes the proofs of some standard
+ properties of this calculus. In particular the user will find:
+
+ - Confluence of reduction
+ - Generation lemma
+ - Thinning lemma
+ - Substitution lemma
+ - Type Correctness
+ - Type Uniqueness
+ - Subject Reduction
+
+ Other properties to be added in the future versions of this contribution
+ include (but are not limited to):
+
+ - Strong Normalization
+ - Decidability of Type Inference and Type Checking
+
+Further information on this contribution:
+-----------------------------------------
+
+ The latest version of this development is maintained in the CVS repository
+ of the HELM project <helm.cs.unibo.it> and can be downloaded at:
+
+ www.cs.unibo.it/cgi-bin/viewcvs.cgi/helm/coq-contribs/LAMBDA-TYPES.tgz
--- /dev/null
+Require Arith.
+Require base_tactics.
+Require base_hints.
+
+(*#* #stop file *)
+
+ Fixpoint blt [m,n: nat] : bool := Cases n m of
+ | (0) m => false
+ | (S n) (0) => true
+ | (S n) (S m) => (blt m n)
+ end.
+
+ Section blt_props. (******************************************************)
+
+ Theorem lt_blt: (x,y:?) (lt y x) -> (blt y x) = true.
+ XElim x; [ Intros; Inversion H | XElim y; Simpl; XAuto ].
+ Qed.
+
+ Theorem le_bge: (x,y:?) (le x y) -> (blt y x) = false.
+ XElim x; [ XAuto | XElim y; Intros; [ Inversion H0 | Simpl; XAuto ] ].
+ Qed.
+
+ Theorem blt_lt: (x,y:?) (blt y x) = true -> (lt y x).
+ XElim x; [ Intros; Inversion H | XElim y; Simpl; XAuto ].
+ Qed.
+
+ Theorem bge_le: (x,y:?) (blt y x) = false -> (le x y).
+ XElim x; [ XAuto | XElim y; Intros; [ Inversion H0 | Simpl; XAuto ] ].
+ Qed.
+
+ End blt_props.
+
+ Hints Resolve lt_blt le_bge : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require Arith.
+Require base_tactics.
+
+(* eq ***********************************************************************)
+
+Hint eq : ltlc := Constructors eq.
+
+Hint f1N : ltlc := Resolve (f_equal nat).
+
+Hint f2NN : ltlc := Resolve (f_equal2 nat nat).
+
+Hints Resolve sym_equal : ltlc.
+
+Hints Resolve plus_sym plus_n_Sm plus_assoc_r simpl_plus_l : ltlc.
+
+Hints Resolve minus_n_O : ltlc.
+
+(* le ***********************************************************************)
+
+Hint le : ltlc := Constructors le.
+
+Hints Resolve le_O_n le_n_S le_S_n le_trans : ltlc.
+
+Hints Resolve lt_le_S : ltlc.
+
+Hints Resolve le_plus_plus le_plus_trans le_plus_l le_plus_r : ltlc.
+
+(* lt ***********************************************************************)
+
+Hints Resolve lt_trans : ltlc.
+
+Hints Resolve lt_le_trans le_lt_n_Sm : ltlc.
+
+Hints Resolve lt_reg_r lt_le_plus_plus le_lt_plus_plus : ltlc.
+
+(* not **********************************************************************)
+
+Hints Resolve sym_not_equal : ltlc.
+
+(* missing in the standard library ******************************************)
+
+ Theorem simpl_plus_r: (n,m,p:?) (plus m n) = (plus p n) -> m = p.
+ Intros.
+ Apply (simpl_plus_l n).
+ Rewrite plus_sym.
+ Rewrite H; XAuto.
+ Qed.
+
+ Theorem minus_plus_r: (m,n:?) (minus (plus m n) n) = m.
+ Intros.
+ Rewrite plus_sym.
+ Apply minus_plus.
+ Qed.
+
+ Theorem plus_permute_2_in_3: (x,y,z:?) (plus (plus x y) z) = (plus (plus x z) y).
+ Intros.
+ Rewrite plus_assoc_r.
+ Rewrite (plus_sym y z).
+ Rewrite <- plus_assoc_r; XAuto.
+ Qed.
+
+ Theorem plus_permute_2_in_3_assoc: (n,h,k:?) (plus (plus n h) k) = (plus n (plus k h)).
+ Intros.
+ Rewrite plus_permute_2_in_3; Rewrite plus_assoc_l; XAuto.
+ Qed.
+
+ Theorem plus_O: (x,y:?) (plus x y) = (0) -> x = (O) /\ y = (O).
+ XElim x; [ XAuto | Intros; Inversion H0 ].
+ Qed.
+
+ Theorem minus_Sx_SO: (x:?) (minus (S x) (1)) = x.
+ Intros; Simpl; Rewrite <- minus_n_O; XAuto.
+ Qed.
+
+ Theorem eq_nat_dec: (i,j:nat) ~i=j \/ i=j.
+ XElim i; XElim j; Intros; XAuto.
+ Elim (H n0); XAuto.
+ Qed.
+
+ Theorem neq_eq_e: (i,j:nat; P:Prop) (~i=j -> P) -> (i=j -> P) -> P.
+ Intros.
+ Pose (eq_nat_dec i j).
+ XElim o; XAuto.
+ Qed.
+
+ Theorem le_false: (m,n:?; P:Prop) (le m n) -> (le (S n) m) -> P.
+ XElim m.
+(* case 1 : m = 0 *)
+ Intros; Inversion H0.
+(* case 2 : m > 0 *)
+ XElim n0; Intros.
+(* case 2.1 : n = 0 *)
+ Inversion H0.
+(* case 2.2 : n > 0 *)
+ Simpl in H1.
+ Apply (H n0); XAuto.
+ Qed.
+
+ Theorem le_plus_minus_sym: (n,m:?) (le n m) -> m = (plus (minus m n) n).
+ Intros.
+ Rewrite plus_sym; Apply le_plus_minus; XAuto.
+ Qed.
+
+ Theorem le_minus_minus: (x,y:?) (le x y) -> (z:?) (le y z) ->
+ (le (minus y x) (minus z x)).
+ Intros.
+ EApply simpl_le_plus_l.
+ Rewrite le_plus_minus_r; [ Idtac | XAuto ].
+ Rewrite le_plus_minus_r; XEAuto.
+ Qed.
+
+ Theorem le_minus_plus: (z,x:?) (le z x) -> (y:?)
+ (minus (plus x y) z) = (plus (minus x z) y).
+ XElim z.
+(* case 1 : z = 0 *)
+ Intros x H; Inversion H; XAuto.
+(* case 2 : z > 0 *)
+ Intros z; XElim x; Intros.
+(* case 2.1 : x = 0 *)
+ Inversion H0.
+(* case 2.2 : x > 0 *)
+ Simpl; XAuto.
+ Qed.
+
+ Theorem le_minus: (x,z,y:?) (le (plus x y) z) -> (le x (minus z y)).
+ Intros.
+ Rewrite <- (minus_plus_r x y); XAuto.
+ Apply le_minus_minus; XAuto.
+ Qed.
+
+ Theorem le_trans_plus_r: (x,y,z:?) (le (plus x y) z) -> (le y z).
+ Intros.
+ EApply le_trans; [ EApply le_plus_r | Idtac ]; XEAuto.
+ Qed.
+
+ Theorem le_gen_S: (m,x:?) (le (S m) x) ->
+ (EX n | x = (S n) & (le m n)).
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ Theorem lt_x_plus_x_Sy: (x,y:?) (lt x (plus x (S y))).
+ Intros; Rewrite plus_sym; Simpl; XAuto.
+ Qed.
+
+ Theorem simpl_lt_plus_r: (p,n,m:?) (lt (plus n p) (plus m p)) -> (lt n m).
+ Intros.
+ EApply simpl_lt_plus_l.
+ Rewrite plus_sym in H; Rewrite (plus_sym m p) in H; Apply H.
+ Qed.
+
+ Theorem minus_x_Sy: (x,y:?) (lt y x) ->
+ (minus x y) = (S (minus x (S y))).
+ XElim x.
+(* case 1 : x = 0 *)
+ Intros; Inversion H.
+(* case 2 : x > 0 *)
+ XElim y; Intros; Simpl.
+(* case 2.1 : y = 0 *)
+ Rewrite <- minus_n_O; XAuto.
+(* case 2.2 : y > 0 *)
+ Cut (lt n0 n); XAuto.
+ Qed.
+
+ Theorem lt_plus_minus: (x,y:?) (lt x y) ->
+ y = (S (plus x (minus y (S x)))).
+ Intros.
+ Apply (le_plus_minus (S x) y); XAuto.
+ Qed.
+
+ Theorem lt_plus_minus_r: (x,y:?) (lt x y) ->
+ y = (S (plus (minus y (S x)) x)).
+ Intros.
+ Rewrite plus_sym; Apply lt_plus_minus; XAuto.
+ Qed.
+
+ Theorem minus_x_SO: (x:?) (lt (0) x) -> x = (S (minus x (1))).
+ Intros.
+ Rewrite <- minus_x_Sy; [ Rewrite <- minus_n_O; XEAuto | XEAuto ].
+ Qed.
+
+ Theorem lt_le_minus: (x,y:?) (lt x y) -> (le x (minus y (1))).
+ Intros; Apply le_minus; Rewrite plus_sym; Simpl; XAuto.
+ Qed.
+
+ Theorem lt_le_e: (n,d:?; P:Prop)
+ ((lt n d) -> P) -> ((le d n) -> P) -> P.
+ Intros.
+ Cut (le d n) \/ (lt n d); [ Intros H1; XElim H1; XAuto | Apply le_or_lt ].
+ Qed.
+
+ Theorem lt_eq_e: (x,y:?; P:Prop) ((lt x y) -> P) ->
+ (x = y -> P) -> (le x y) -> P.
+ Intros.
+ LApply (le_lt_or_eq x y); [ Clear H1; Intros H1 | XAuto ].
+ XElim H1; XAuto.
+ Qed.
+
+ Theorem lt_eq_gt_e: (x,y:?; P:Prop) ((lt x y) -> P) ->
+ (x = y -> P) -> ((lt y x) -> P) -> P.
+ Intros.
+ Apply (lt_le_e x y); [ XAuto | Intros ].
+ Apply (lt_eq_e y x); XAuto.
+ Qed.
+
+ Theorem lt_gen_S': (x,n:?) (lt x (S n)) ->
+ x = (0) \/ (EX m | x = (S m) & (lt m n)).
+ XElim x; XEAuto.
+ Qed.
+
+Hints Resolve le_lt_trans : ltlc.
+
+Hints Resolve simpl_plus_r minus_plus_r minus_x_Sy
+ plus_permute_2_in_3 plus_permute_2_in_3_assoc : ltlc.
+
+Hints Resolve le_minus_minus le_minus_plus le_minus le_trans_plus_r : ltlc.
+
+Hints Resolve lt_x_plus_x_Sy simpl_lt_plus_r lt_le_minus lt_plus_minus
+ lt_plus_minus_r : ltlc.
+
+ Theorem lt_neq: (x,y:?) (lt x y) -> ~x=y.
+ Unfold not; Intros; Rewrite H0 in H; Clear H0 x.
+ LApply (lt_n_n y); XAuto.
+ Qed.
+
+Hints Resolve lt_neq : ltlc.
+
+ Theorem arith0: (h2,d2,n:?) (le (plus d2 h2) n) ->
+ (h1:?) (le (plus d2 h1) (minus (plus n h1) h2)).
+ Intros.
+ Rewrite <- (minus_plus h2 (plus d2 h1)).
+ Apply le_minus_minus; [ XAuto | Idtac ].
+ Rewrite plus_assoc_l; Rewrite (plus_sym h2 d2); XAuto.
+ Qed.
+
+Hints Resolve arith0 : ltlc.
+
+ Tactic Definition EqFalse :=
+ Match Context With
+ [ H: ~?1=?1 |- ? ] ->
+ LApply H; [ Clear H; Intros H; Inversion H | XAuto ].
+
+ Tactic Definition PlusO :=
+ Match Context With
+ | [ H: (plus ?0 ?1) = (0) |- ? ] ->
+ LApply (plus_O ?0 ?1); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Tactic Definition SymEqual :=
+ Match Context With
+ | [ H: ?1 = ?2 |- ? ] ->
+ Cut ?2 = ?1; [ Clear H; Intros H | Apply sym_equal; XAuto ].
+
+ Tactic Definition LeLtGen :=
+ Match Context With
+ | [ H: (le (S ?1) ?2) |- ? ] ->
+ LApply (le_gen_S ?1 ?2); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (lt ?1 (S ?2)) |- ? ] ->
+ LApply (lt_gen_S' ?1 ?2); [ Clear H; Intros H | XAuto ];
+ XElim H; [ Intros | Intros H; XElim H; Intros ].
--- /dev/null
+(*#* #stop file *)
+
+Require Arith.
+
+ Tactic Definition Arith0 x :=
+ Replace (S x) with (plus (1) x); XAuto.
+
+ Tactic Definition Arith1 x :=
+ Replace x with (plus x (0)); [XAuto | Auto with arith].
+
+ Tactic Definition Arith1In H x :=
+ XReplaceIn H x '(plus x (0)).
+
+ Tactic Definition Arith2 x :=
+ Replace x with (plus (0) x); XAuto.
+
+ Tactic Definition Arith3 x :=
+ Replace (S x) with (S (plus (0) x)); XAuto.
+
+ Tactic Definition Arith3In H x :=
+ XReplaceIn H '(S x) '(S (plus (0) x)).
+
+ Tactic Definition Arith4 x y :=
+ Replace (S (plus x y)) with (plus (S x) y); XAuto.
+
+ Tactic Definition Arith4In H x y :=
+ XReplaceIn H '(S (plus x y)) '(plus (S x) y).
+
+ Tactic Definition Arith4c x y :=
+ Arith4 x y; Rewrite plus_sym.
+
+ Tactic Definition Arith5 x y :=
+ Replace (S (plus x y)) with (plus x (S y)); Auto with arith.
+
+ Tactic Definition Arith5In H x y :=
+ XReplaceIn H '(S (plus x y)) '(plus x (S y)); Auto with arith.
+
+ Tactic Definition Arith5' x y :=
+ Replace (plus x (S y)) with (S (plus x y)); Auto with arith.
+
+ Tactic Definition Arith5'In H x y :=
+ XReplaceIn H '(plus x (S y)) '(S (plus x y)); Auto with arith.
+
+ Tactic Definition Arith5'c x y :=
+ Arith5' x y; Rewrite plus_sym.
+
+ Tactic Definition Arith6In H x y :=
+ XReplaceIn H '(plus x (S y)) '(plus (1) (plus x y));
+ [ Idtac | Simpl; Auto with arith ].
+
+ Tactic Definition Arith7 x :=
+ Replace (S x) with (plus x (1));
+ [ Idtac | Rewrite plus_sym; Auto with arith ].
+
+ Tactic Definition Arith7In H x :=
+ XReplaceIn H '(S x) '(plus x (1)) ;
+ [ Idtac | Rewrite plus_sym; Auto with arith ].
+
+ Tactic Definition Arith7' x :=
+ Replace (plus x (1)) with (S x);
+ [ Idtac | Rewrite plus_sym; Auto with arith ].
+
+ Tactic Definition Arith8 x y :=
+ Replace x with (plus y (minus x y));
+ [ Idtac | Auto with arith ].
+
+ Tactic Definition Arith8' x y :=
+ Replace (plus y (minus x y)) with x;
+ [ Idtac | Auto with arith ].
+
+ Tactic Definition Arith9'In H x :=
+ XReplaceIn H '(S (plus x (0))) '(S x).
+
+ Tactic Definition Arith10 x :=
+ Replace x with (minus (S x) (1));
+ [ Idtac | Simpl; Rewrite <- minus_n_O; Auto with arith ].
--- /dev/null
+(*#* #stop file *)
+
+ Tactic Definition XAuto := Auto with ltlc.
+
+ Tactic Definition XEAuto := EAuto with ltlc.
+
+ Tactic Definition XDEAuto d := EAuto d with ltlc.
+
+ Tactic Definition XElimUsing e v :=
+ Try Intros until v; Elim v using e; Try Clear v.
+
+ Tactic Definition XElim v := Try Intros until v; Elim v; Try Clear v.
+
+ Tactic Definition XCase v := Try Intros until v; Case v; Try Clear v.
+
+ Tactic Definition XReplaceIn Z0 y1 y2 :=
+ Cut y1=y2; [ Intros Z; Rewrite Z in Z0; Clear Z | XAuto ].
+
+ Theorem insert_eq: (S:Set; x:S; P:S->Prop; G:Prop)
+ ((y:S) (P y) -> y = x -> G) -> (P x) -> G.
+ EAuto. Qed.
+
+ Tactic Definition InsertEq H y :=
+ Pattern 1 y in H; Match Context With [ _: (?1 y) |- ? ] ->
+ Apply insert_eq with x:=y P:=?1;
+ [ Clear H; Intros until 1 | Pattern y; Apply H ].
+
+ Theorem unintro : (A:Set; a:A; P:A->Prop) ((x:A) (P x)) -> (P a).
+ Auto.
+ Qed.
+
+ Tactic Definition UnIntro Last H :=
+ Move H after Last;
+ Match Context With [ y: ?1 |- ?2 ] ->
+ Apply (unintro ?1 y); Clear y.
+
+ Tactic Definition NonLinear :=
+ Match Context With
+ [ H: ?1 |- ? ] -> Cut ?1; [ Intros | XAuto ].
+
+ Tactic Definition XRewrite x :=
+ Match Context With
+ | [ H0: x = ? |- ? ] -> Try Rewrite H0
+ | [ H0: ? = x |- ? ] -> Try Rewrite <- H0
+ | _ -> Idtac.
--- /dev/null
+(* This file was generated by coqgen *)
+
+Require base_tactics.
+Require base_hints.
+
+(*#* #stop file *)
+
+(* extensions for ex2 *)
+
+Syntactic Definition ex2_intro := ex_intro2.
+
+Theorem ex2_sym: (A:Set; P,Q:A->Prop)
+ (EX x | (P x) & (Q x)) -> (EX x | (Q x) & (P x)).
+ Intros; XElim H; XEAuto.
+ Qed.
+
+Hints Resolve ex2_sym : ltlc.
+
+(* or3 *)
+
+Inductive or3 [P0,P1,P2:Prop] : Prop :=
+ | or3_intro0 : P0 -> (or3 P0 P1 P2)
+ | or3_intro1 : P1 -> (or3 P0 P1 P2)
+ | or3_intro2 : P2 -> (or3 P0 P1 P2).
+
+Hint or3 : ltlc := Constructors or3.
+
+Grammar constr constr10 :=
+ | or3
+ [ "OR" constr($c0) "|" constr($c1) "|" constr($c2) ] ->
+ [ (or3 $c0 $c1 $c2) ].
+
+(* or4 *)
+
+Inductive or4 [P0,P1,P2,P3:Prop] : Prop :=
+ | or4_intro0 : P0 -> (or4 P0 P1 P2 P3)
+ | or4_intro1 : P1 -> (or4 P0 P1 P2 P3)
+ | or4_intro2 : P2 -> (or4 P0 P1 P2 P3)
+ | or4_intro3 : P3 -> (or4 P0 P1 P2 P3).
+
+Hint or4 : ltlc := Constructors or4.
+
+Grammar constr constr10 :=
+ | or4
+ [ "OR" constr($c0) "|" constr($c1) "|" constr($c2) "|" constr($c3) ] ->
+ [ (or4 $c0 $c1 $c2 $c3) ].
+
+(* ex2_2 *)
+
+Inductive ex2_2 [A0,A1:Set; P0,P1:A0->A1->Prop] : Prop :=
+ ex2_2_intro : (x0:A0; x1:A1)(P0 x0 x1)->(P1 x0 x1)->(ex2_2 A0 A1 P0 P1).
+
+Hint ex2_2 : ltlc := Constructors ex2_2.
+
+Syntactic Definition Ex2_2 := ex2_2 | 1.
+
+Grammar constr constr10 :=
+ | ex2_2implicit
+ [ "EX" ident($v0) ident($v1) "|" constr($c0) "&" constr($c1) ] ->
+ [ (ex2_2 ? ? [$v0;$v1]$c0 [$v0;$v1]$c1) ].
+
+(* ex3_2 *)
+
+Inductive ex3_2 [A0,A1:Set; P0,P1,P2:A0->A1->Prop] : Prop :=
+ ex3_2_intro : (x0:A0; x1:A1)(P0 x0 x1)->(P1 x0 x1)->(P2 x0 x1)->(ex3_2 A0 A1 P0 P1 P2).
+
+Hint ex3_2 : ltlc := Constructors ex3_2.
+
+Syntactic Definition Ex3_2 := ex3_2 | 1.
+
+Grammar constr constr10 :=
+ | ex3_2implicit
+ [ "EX" ident($v0) ident($v1) "|" constr($c0) "&" constr($c1) "&" constr($c2) ] ->
+ [ (ex3_2 ? ? [$v0;$v1]$c0 [$v0;$v1]$c1 [$v0;$v1]$c2) ].
+
+(* ex3_3 *)
+
+Inductive ex3_3 [A0,A1,A2:Set; P0,P1,P2:A0->A1->A2->Prop] : Prop :=
+ ex3_3_intro : (x0:A0; x1:A1; x2:A2)(P0 x0 x1 x2)->(P1 x0 x1 x2)->(P2 x0 x1 x2)->(ex3_3 A0 A1 A2 P0 P1 P2).
+
+Hint ex3_3 : ltlc := Constructors ex3_3.
+
+Syntactic Definition Ex3_3 := ex3_3 | 1.
+
+Grammar constr constr10 :=
+ | ex3_3implicit
+ [ "EX" ident($v0) ident($v1) ident($v2) "|" constr($c0) "&" constr($c1) "&" constr($c2) ] ->
+ [ (ex3_3 ? ? ? [$v0;$v1;$v2]$c0 [$v0;$v1;$v2]$c1 [$v0;$v1;$v2]$c2) ].
+
+(* ex4_3 *)
+
+Inductive ex4_3 [A0,A1,A2:Set; P0,P1,P2,P3:A0->A1->A2->Prop] : Prop :=
+ ex4_3_intro : (x0:A0; x1:A1; x2:A2)(P0 x0 x1 x2)->(P1 x0 x1 x2)->(P2 x0 x1 x2)->(P3 x0 x1 x2)->(ex4_3 A0 A1 A2 P0 P1 P2 P3).
+
+Hint ex4_3 : ltlc := Constructors ex4_3.
+
+Syntactic Definition Ex4_3 := ex4_3 | 1.
+
+Grammar constr constr10 :=
+ | ex4_3implicit
+ [ "EX" ident($v0) ident($v1) ident($v2) "|" constr($c0) "&" constr($c1) "&" constr($c2) "&" constr($c3) ] ->
+ [ (ex4_3 ? ? ? [$v0;$v1;$v2]$c0 [$v0;$v1;$v2]$c1 [$v0;$v1;$v2]$c2 [$v0;$v1;$v2]$c3) ].
+
+(* ex3_4 *)
+
+Inductive ex3_4 [A0,A1,A2,A3:Set; P0,P1,P2:A0->A1->A2->A3->Prop] : Prop :=
+ ex3_4_intro : (x0:A0; x1:A1; x2:A2; x3:A3)(P0 x0 x1 x2 x3)->(P1 x0 x1 x2 x3)->(P2 x0 x1 x2 x3)->(ex3_4 A0 A1 A2 A3 P0 P1 P2).
+
+Hint ex3_4 : ltlc := Constructors ex3_4.
+
+Syntactic Definition Ex3_4 := ex3_4 | 1.
+
+Grammar constr constr10 :=
+ | ex3_4implicit
+ [ "EX" ident($v0) ident($v1) ident($v2) ident($v3) "|" constr($c0) "&" constr($c1) "&" constr($c2) ] ->
+ [ (ex3_4 ? ? ? ? [$v0;$v1;$v2;$v3]$c0 [$v0;$v1;$v2;$v3]$c1 [$v0;$v1;$v2;$v3]$c2) ].
+
+(* ex4_4 *)
+
+Inductive ex4_4 [A0,A1,A2,A3:Set; P0,P1,P2,P3:A0->A1->A2->A3->Prop] : Prop :=
+ ex4_4_intro : (x0:A0; x1:A1; x2:A2; x3:A3)(P0 x0 x1 x2 x3)->(P1 x0 x1 x2 x3)->(P2 x0 x1 x2 x3)->(P3 x0 x1 x2 x3)->(ex4_4 A0 A1 A2 A3 P0 P1 P2 P3).
+
+Hint ex4_4 : ltlc := Constructors ex4_4.
+
+Syntactic Definition Ex4_4 := ex4_4 | 1.
+
+Grammar constr constr10 :=
+ | ex4_4implicit
+ [ "EX" ident($v0) ident($v1) ident($v2) ident($v3) "|" constr($c0) "&" constr($c1) "&" constr($c2) "&" constr($c3) ] ->
+ [ (ex4_4 ? ? ? ? [$v0;$v1;$v2;$v3]$c0 [$v0;$v1;$v2;$v3]$c1 [$v0;$v1;$v2;$v3]$c2 [$v0;$v1;$v2;$v3]$c3) ].
+
+(* ex4_5 *)
+
+Inductive ex4_5 [A0,A1,A2,A3,A4:Set; P0,P1,P2,P3:A0->A1->A2->A3->A4->Prop] : Prop :=
+ ex4_5_intro : (x0:A0; x1:A1; x2:A2; x3:A3; x4:A4)(P0 x0 x1 x2 x3 x4)->(P1 x0 x1 x2 x3 x4)->(P2 x0 x1 x2 x3 x4)->(P3 x0 x1 x2 x3 x4)->(ex4_5 A0 A1 A2 A3 A4 P0 P1 P2 P3).
+
+Hint ex4_5 : ltlc := Constructors ex4_5.
+
+Syntactic Definition Ex4_5 := ex4_5 | 1.
+
+Grammar constr constr10 :=
+ | ex4_5implicit
+ [ "EX" ident($v0) ident($v1) ident($v2) ident($v3) ident($v4) "|" constr($c0) "&" constr($c1) "&" constr($c2) "&" constr($c3) ] ->
+ [ (ex4_5 ? ? ? ? ? [$v0;$v1;$v2;$v3;$v4]$c0 [$v0;$v1;$v2;$v3;$v4]$c1 [$v0;$v1;$v2;$v3;$v4]$c2 [$v0;$v1;$v2;$v3;$v4]$c3) ].
+
+(* ex6_6 *)
+
+Inductive ex6_6 [A0,A1,A2,A3,A4,A5:Set; P0,P1,P2,P3,P4,P5:A0->A1->A2->A3->A4->A5->Prop] : Prop :=
+ ex6_6_intro : (x0:A0; x1:A1; x2:A2; x3:A3; x4:A4; x5:A5)(P0 x0 x1 x2 x3 x4 x5)->(P1 x0 x1 x2 x3 x4 x5)->(P2 x0 x1 x2 x3 x4 x5)->(P3 x0 x1 x2 x3 x4 x5)->(P4 x0 x1 x2 x3 x4 x5)->(P5 x0 x1 x2 x3 x4 x5)->(ex6_6 A0 A1 A2 A3 A4 A5 P0 P1 P2 P3 P4 P5).
+
+Hint ex6_6 : ltlc := Constructors ex6_6.
+
+Syntactic Definition Ex6_6 := ex6_6 | 1.
+
+Grammar constr constr10 :=
+ | ex6_6implicit
+ [ "EX" ident($v0) ident($v1) ident($v2) ident($v3) ident($v4) ident($v5) "|" constr($c0) "&" constr($c1) "&" constr($c2) "&" constr($c3) "&" constr($c4) "&" constr($c5) ] ->
+ [ (ex6_6 ? ? ? ? ? ? [$v0;$v1;$v2;$v3;$v4;$v5]$c0 [$v0;$v1;$v2;$v3;$v4;$v5]$c1 [$v0;$v1;$v2;$v3;$v4;$v5]$c2 [$v0;$v1;$v2;$v3;$v4;$v5]$c3 [$v0;$v1;$v2;$v3;$v4;$v5]$c4 [$v0;$v1;$v2;$v3;$v4;$v5]$c5) ].
+
--- /dev/null
+(*#* #stop file *)
+
+Require Export terms_defs.
+
+ Inductive Set C := CSort : nat -> C
+ | CTail : C -> K -> T -> C.
+
+ Hint f3CKT : ltlc := Resolve (f_equal3 C K T).
+
+ Definition r: K -> nat -> nat := [k;i] Cases k of
+ | (Bind _) => i
+ | (Flat _) => (S i)
+ end.
+
+ Fixpoint app [c:C] : nat -> T -> T := [j;t]Cases j c of
+ | (0) _ => t
+ | _ (CSort _) => t
+ | (S i) (CTail c k u) => (app c (r k i) (TTail k u t))
+ end.
+
+ Section r_props. (********************************************************)
+
+ Theorem r_S: (k:?; i:?) (r k (S i)) = (S (r k i)).
+ XElim k; XAuto.
+ Qed.
+
+ Theorem r_plus_sym: (k:?; i,j:?) (r k (plus i j)) = (plus i (r k j)).
+ XElim k; Intros; Simpl; XAuto.
+ Qed.
+
+ Theorem r_minus: (i,n:?) (lt n i) ->
+ (k:?) (minus (r k i) (S n)) = (r k (minus i (S n))).
+ XElim k; Intros; Simpl; XEAuto.
+ Qed.
+
+ Theorem r_dis: (k:?; P:Prop)
+ (((i:?) (r k i) = i) -> P) ->
+ (((i:?) (r k i) = (S i)) -> P) -> P.
+ XElim k; XAuto.
+ Qed.
+
+ End r_props.
+
+ Tactic Definition RRw :=
+ Repeat (Rewrite r_S Orelse Rewrite r_plus_sym).
+
+ Section r_arith. (********************************************************)
+
+ Theorem r_arith0: (k:?; i:?) (minus (r k (S i)) (1)) = (r k i).
+ Intros; RRw; Rewrite minus_Sx_SO; XAuto.
+ Qed.
+
+ Theorem r_arith1: (k:?; i,j:?) (minus (r k (S i)) (S j)) = (minus (r k i) j).
+ Intros; RRw; XAuto.
+ Qed.
+
+ End r_arith.
+
+ Section app_props. (******************************************************)
+
+ Theorem app_csort : (t:?; i,n:?) (app (CSort n) i t) = t.
+ XElim i; Intros; Simpl; XAuto.
+ Qed.
+
+ Theorem app_O : (c:?; t:?) (app c (0) t) = t.
+ XElim c; XAuto.
+ Qed.
+
+ End app_props.
+
+ Hints Resolve app_csort app_O : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require Export contexts_defs.
+Require Export pr0_defs.
+
+ Inductive cpr0 : C -> C -> Prop :=
+ | cpr0_refl : (c:?) (cpr0 c c)
+ | cpr0_cont : (c1,c2:?) (cpr0 c1 c2) -> (u1,u2:?) (pr0 u1 u2) ->
+ (k:?) (cpr0 (CTail c1 k u1) (CTail c2 k u2)).
+
+ Hint cpr0 : ltlc := Constructors cpr0.
--- /dev/null
+(*#* #stop file *)
+
+Require pr0_subst0.
+Require pr3_defs.
+Require pr3_props.
+Require cpr0_defs.
+
+ Section cpr0_drop. (******************************************************)
+
+ Theorem cpr0_drop : (c1,c2:?) (cpr0 c1 c2) -> (h:?; e1:?; u1:?; k:?)
+ (drop h (0) c1 (CTail e1 k u1)) ->
+ (EX e2 u2 | (drop h (0) c2 (CTail e2 k u2)) &
+ (cpr0 e1 e2) & (pr0 u1 u2)
+ ).
+ Intros until 1; XElim H.
+(* case 1 : cpr0_refl *)
+ XEAuto.
+(* case 2 : cpr0_cont *)
+ XElim h.
+(* case 2.1 : h = 0 *)
+ Intros; DropGenBase.
+ Inversion H2; Rewrite H6 in H1; Rewrite H4 in H; XEAuto.
+(* case 2.2 : h > 0 *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H0 n e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H0 (S n) e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+ Qed.
+
+ Theorem cpr0_drop_back : (c1,c2:?) (cpr0 c2 c1) -> (h:?; e1:?; u1:?; k:?)
+ (drop h (0) c1 (CTail e1 k u1)) ->
+ (EX e2 u2 | (drop h (0) c2 (CTail e2 k u2)) &
+ (cpr0 e2 e1) & (pr0 u2 u1)
+ ).
+ Intros until 1; XElim H.
+(* case 1 : cpr0_refl *)
+ XEAuto.
+(* case 2 : cpr0_cont *)
+ XElim h.
+(* case 2.1 : h = 0 *)
+ Intros; DropGenBase.
+ Inversion H2; Rewrite H6 in H1; Rewrite H4 in H; XEAuto.
+(* case 2.2 : h > 0 *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H0 n e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H0 (S n) e1 u0 k); [ Clear H0 H3; Intros H0 | XAuto ].
+ XElim H0; XEAuto.
+ Qed.
+
+ End cpr0_drop.
+
+ Tactic Definition Cpr0Drop :=
+ Match Context With
+ | [ _: (drop ?1 (0) ?2 (CTail ?3 ?4 ?5));
+ _: (cpr0 ?2 ?6) |- ? ] ->
+ LApply (cpr0_drop ?2 ?6); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ _: (drop ?1 (0) ?2 (CTail ?3 ?4 ?5));
+ _: (cpr0 ?6 ?2) |- ? ] ->
+ LApply (cpr0_drop_back ?2 ?6); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ _: (drop ?1 (0) (CTail ?2 ?7 ?8) (CTail ?3 ?4 ?5));
+ _: (cpr0 ?2 ?6) |- ? ] ->
+ LApply (cpr0_drop (CTail ?2 ?7 ?8) (CTail ?6 ?7 ?8)); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ _: (drop ?1 (0) (CTail ?2 ?7 ?8) (CTail ?3 ?4 ?5));
+ _: (cpr0 ?6 ?2) |- ? ] ->
+ LApply (cpr0_drop_back (CTail ?2 ?7 ?8) (CTail ?6 ?7 ?8)); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?3 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros.
+
+ Section cpr0_pr3. (*******************************************************)
+
+ Theorem cpr0_pr3_t : (c1,c2:?) (cpr0 c2 c1) -> (t1,t2:?) (pr3 c1 t1 t2) ->
+ (pr3 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : cpr0_refl *)
+ XAuto.
+(* case 2 : cpr0_cont *)
+ Pr3Context.
+ XElim H1; Intros.
+(* case 2.1 : pr3_r *)
+ XAuto.
+(* case 2.2 : pr3_u *)
+ EApply pr3_t; [ Idtac | XEAuto ]. Clear H2 H3 c1 c2 t1 t2 t4 u2.
+ Inversion_clear H1.
+(* case 2.2.1 : pr2_pr0 *)
+ XAuto.
+(* case 2.2.1 : pr2_delta *)
+ Cpr0Drop; Pr0Subst0.
+ EApply pr3_u; [ EApply pr2_delta; XEAuto | XAuto ].
+ Qed.
+
+ End cpr0_pr3.
--- /dev/null
+(*#* #stop file *)
+
+Require Export ty0_defs.
+
+ Inductive csub0 [g:G] : C -> C -> Prop :=
+(* structural rules *)
+ | csub0_sort : (n:?) (csub0 g (CSort n) (CSort n))
+ | csub0_tail : (c1,c2:?) (csub0 g c1 c2) -> (k,u:?)
+ (csub0 g (CTail c1 k u) (CTail c2 k u))
+(* axioms *)
+ | csub0_void : (c1,c2:?) (csub0 g c1 c2) -> (b:?) ~b=Void -> (u1,u2:?)
+ (csub0 g (CTail c1 (Bind Void) u1) (CTail c2 (Bind b) u2))
+ | csub0_abst : (c1,c2:?) (csub0 g c1 c2) -> (u,t:?) (ty0 g c2 u t) ->
+ (csub0 g (CTail c1 (Bind Abst) t) (CTail c2 (Bind Abbr) u)).
+
+ Hint csub0 : ltlc := Constructors csub0.
+
+ Section csub0_props. (****************************************************)
+
+ Theorem csub0_refl : (g:?; c:?) (csub0 g c c).
+ XElim c; XAuto.
+ Qed.
+
+ End csub0_props.
+
+ Hints Resolve csub0_refl : ltlc.
+
+ Section csub0_drop. (*****************************************************)
+
+ Theorem csub0_drop_abbr : (g:?; n:?; c1,c2:?) (csub0 g c1 c2) -> (d1,u:?)
+ (drop n (0) c1 (CTail d1 (Bind Abbr) u)) ->
+ (EX d2 | (csub0 g d1 d2) &
+ (drop n (0) c2 (CTail d2 (Bind Abbr) u))
+ ).
+ XElim n.
+(* case 1 : n = 0 *)
+ Intros; DropGenBase; Rewrite H0 in H; Inversion H; XEAuto.
+(* case 2 : n > 0 *)
+ Intros until 2; XElim H0.
+(* case 2.1 : csub0_sort *)
+ Intros; Inversion H0.
+(* case 2.2 : csub0_tail *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 u0); [ Clear H; Intros H | XAuto ].
+ XElim H; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H1 d1 u0); [ Clear H1; Intros H1 | XAuto ].
+ XElim H1; XEAuto.
+(* case 2.3 : csub0_void *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 u); [ Clear H; Intros H | XAuto ].
+ XElim H; XEAuto.
+(* case 2.4 : csub0_abst *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 u0); [ Clear H; Intros H | XAuto ].
+ XElim H; XEAuto.
+ Qed.
+
+ Theorem csub0_drop_abst : (g:?; n:?; c1,c2:?) (csub0 g c1 c2) -> (d1,t:?)
+ (drop n (0) c1 (CTail d1 (Bind Abst) t)) ->
+ (EX d2 | (csub0 g d1 d2) &
+ (drop n (0) c2 (CTail d2 (Bind Abst) t))
+
+ ) \/
+ (EX d2 u | (csub0 g d1 d2) &
+ (drop n (0) c2 (CTail d2 (Bind Abbr) u)) &
+ (ty0 g d2 u t)
+ ).
+ XElim n.
+(* case 1 : n = 0 *)
+ Intros; DropGenBase; Rewrite H0 in H; Inversion H; XEAuto.
+(* case 2 : n > 0 *)
+ Intros until 2; XElim H0.
+(* case 2.1 : csub0_sort *)
+ Intros; Inversion H0.
+(* case 2.2 : csub0_tail *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 t); [ Clear H; Intros H | XAuto ].
+ XElim H; Intros; XElim H; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H1 d1 t); [ Clear H1; Intros H1 | XAuto ].
+ XElim H1; Intros; XElim H1; XEAuto.
+(* case 2.3 : csub0_void *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 t); [ Clear H; Intros H | XAuto ].
+ XElim H; Intros; XElim H; XEAuto.
+(* case 2.4 : csub0_abst *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 t0); [ Clear H; Intros H | XAuto ].
+ XElim H; Intros; XElim H; XEAuto.
+ Qed.
+
+ End csub0_drop.
+
+ Tactic Definition CSub0Drop :=
+ Match Context With
+ | [ H1: (csub0 ?1 ?2 ?3);
+ H2: (drop ?4 (0) ?2 (CTail ?5 (Bind Abbr) ?6)) |- ? ] ->
+ LApply (csub0_drop_abbr ?1 ?4 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (csub0 ?1 ?2 ?3);
+ H2: (drop ?4 (0) ?2 (CTail ?5 (Bind Abst) ?6)) |- ? ] ->
+ LApply (csub0_drop_abst ?1 ?4 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros H1; XElim H1; Intros.
+
+ Section csub0_pc3. (*****************************************************)
+
+ Theorem csub0_pr2 : (g:?; c1:?; t1,t2:?) (pr2 c1 t1 t2) ->
+ (c2:?) (csub0 g c1 c2) -> (pr2 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr2_pr0 *)
+ XAuto.
+(* case 2 : pr2_delta *)
+ CSub0Drop; XEAuto.
+ Qed.
+
+ Theorem csub0_pc2 : (g:?; c1:?; t1,t2:?) (pc2 c1 t1 t2) ->
+ (c2:?) (csub0 g c1 c2) -> (pc2 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pc2_r *)
+ Apply pc2_r; EApply csub0_pr2; XEAuto.
+(* case 2 : pc2_x *)
+ Apply pc2_x; EApply csub0_pr2; XEAuto.
+ Qed.
+
+ Theorem csub0_pc3 : (g:?; c1:?; t1,t2:?) (pc3 c1 t1 t2) ->
+ (c2:?) (csub0 g c1 c2) -> (pc3 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pc3_r *)
+ XAuto.
+(* case 2 : pc3_u *)
+ EApply pc3_u; [ EApply csub0_pc2; XEAuto | XAuto ].
+ Qed.
+
+ End csub0_pc3.
+
+ Hints Resolve csub0_pc3 : ltlc.
+
+ Section csub0_ty0. (*****************************************************)
+
+ Theorem csub0_ty0 : (g:?; c1:?; t1,t2:?) (ty0 g c1 t1 t2) ->
+ (c2:?) (wf0 g c2) -> (csub0 g c1 c2) ->
+ (ty0 g c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ EApply ty0_conv; XEAuto.
+(* case 2 : ty0_sort *)
+ XEAuto.
+(* case 3 : ty0_abbr *)
+ CSub0Drop; EApply ty0_abbr; XEAuto.
+(* case 4 : ty0_abst *)
+ CSub0Drop; [ EApply ty0_abst | EApply ty0_abbr ]; XEAuto.
+(* case 5 : ty0_bind *)
+ EApply ty0_bind; XEAuto.
+(* case 6 : ty0_appl *)
+ EApply ty0_appl; XEAuto.
+(* case 7 : ty0_cast *)
+ EApply ty0_cast; XAuto.
+ Qed.
+
+ Theorem csub0_ty0_ld : (g:?; c:?; u,v:?) (ty0 g c u v) -> (t1,t2:?)
+ (ty0 g (CTail c (Bind Abst) v) t1 t2) ->
+ (ty0 g (CTail c (Bind Abbr) u) t1 t2).
+ Intros; EApply csub0_ty0; XEAuto.
+ Qed.
+
+ End csub0_ty0.
+
+ Hints Resolve csub0_ty0 csub0_ty0_ld : ltlc.
+
+ Tactic Definition CSub0Ty0 :=
+ Match Context With
+ [ _: (ty0 ?1 ?2 ?4 ?); _: (ty0 ?1 ?2 ?3 ?7); _: (pc3 ?2 ?4 ?7);
+ H: (ty0 ?1 (CTail ?2 (Bind Abst) ?4) ?5 ?6) |- ? ] ->
+ LApply (csub0_ty0_ld ?1 ?2 ?3 ?4); [ Intros H_x | EApply ty0_conv; XEAuto ];
+ LApply (H_x ?5 ?6); [ Clear H_x H; Intros | XAuto ].
--- /dev/null
+(*#* #stop file *)
+
+Require Export contexts_defs.
+Require Export subst0_defs.
+Require Export drop_defs.
+
+ Inductive csubst0 : nat -> T -> C -> C -> Prop :=
+ | csubst0_fst : (k:?; i:?; u,v,w:?) (subst0 (r k i) v u w) -> (c:?)
+ (csubst0 (S i) v (CTail c k u) (CTail c k w))
+ | csubst0_snd : (k:?; i:?; c1,c2:?; v:?) (csubst0 (r k i) v c1 c2) ->
+ (u:?) (csubst0 (S i) v (CTail c1 k u) (CTail c2 k u))
+ | csubst0_both : (k:?; i:?; u,v,w:?) (subst0 (r k i) v u w) ->
+ (c1,c2:?) (csubst0 (r k i) v c1 c2) ->
+ (csubst0 (S i) v (CTail c1 k u) (CTail c2 k w)).
+
+ Hint csubst0 : ltlc := Constructors csubst0.
+
+ Inductive fsubst0 [i:nat; v:T; c1:C; t1:T] : C -> T -> Prop :=
+ | fsubst0_t : (t2:?) (subst0 i v t1 t2) -> (fsubst0 i v c1 t1 c1 t2)
+ | fsubst0_c : (c2:?) (csubst0 i v c1 c2) -> (fsubst0 i v c1 t1 c2 t1)
+ | fsubst0_b : (t2:?) (subst0 i v t1 t2) ->
+ (c2:?) (csubst0 i v c1 c2) -> (fsubst0 i v c1 t1 c2 t2).
+
+ Hint fsubst0 : ltlc := Constructors fsubst0.
+
+ Section csubst0_gen_base. (***********************************************)
+
+ Theorem csubst0_gen_tail: (k:?; c1,x:?; u1,v:?; i:?)
+ (csubst0 (S i) v (CTail c1 k u1) x) -> (OR
+ (EX u2 | x = (CTail c1 k u2) &
+ (subst0 (r k i) v u1 u2)
+ ) |
+ (EX c2 | x = (CTail c2 k u1) &
+ (csubst0 (r k i) v c1 c2)
+ ) |
+ (EX u2 c2 | x = (CTail c2 k u2) &
+ (subst0 (r k i) v u1 u2) &
+ (csubst0 (r k i) v c1 c2)
+ )).
+ Intros until 1; InsertEq H '(S i); InsertEq H '(CTail c1 k u1).
+ XCase H; Clear x v y y0; Intros; Inversion H1.
+(* case 1: csubst0_fst *)
+ Inversion H0; Rewrite H3 in H; Rewrite H5 in H; Rewrite H6 in H; XEAuto.
+(* case 2: csubst0_snd *)
+ Inversion H0; Rewrite H3 in H; Rewrite H4 in H; Rewrite H5 in H; XEAuto.
+(* case 2: csubst0_both *)
+ Inversion H2; Rewrite H5 in H; Rewrite H6 in H; Rewrite H7 in H;
+ Rewrite H4 in H0; Rewrite H5 in H0; Rewrite H7 in H0; XEAuto.
+ Qed.
+
+ End csubst0_gen_base.
+
+ Tactic Definition CSubst0GenBase :=
+ Match Context With
+ | [ H: (csubst0 (S ?1) ?2 (CTail ?3 ?4 ?5) ?6) |- ? ] ->
+ LApply (csubst0_gen_tail ?4 ?3 ?6 ?5 ?2 ?1); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros H; XElim H; Intros.
+
+ Section csubst0_drop. (***************************************************)
+
+ Theorem csubst0_drop_ge : (i,n:?) (le i n) ->
+ (c1,c2:?; v:?) (csubst0 i v c1 c2) ->
+ (e:?) (drop n (0) c1 e) ->
+ (drop n (0) c2 e).
+ XElim i.
+(* case 1: i = 0 *)
+ Intros; Inversion H0.
+(* case 2: i > 0 *)
+ Intros i; XElim n.
+(* case 2.1: n = 0 *)
+ Intros; Inversion H0.
+(* case 2.2: n > 0 *)
+ Intros until 3; Clear H0; InsertEq H2 '(S i); XElim H0; Intros.
+ DropGenBase.
+(* case 2.2.1: csubst0_fst *)
+ XAuto.
+(* case 2.2.2: csubst0_snd *)
+ XReplaceIn H0 i0 i; DropGenBase; NewInduction k; XEAuto.
+(* case 2.2.3: csubst0_both *)
+ XReplaceIn H0 i0 i; XReplaceIn H2 i0 i.
+ DropGenBase; NewInduction k; XEAuto.
+ Qed.
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H0: (n:?) (lt n ?1) -> (c1,c2:?; v:?) (csubst0 ?1 v c1 c2) -> (e:C) (drop n (0) c1 e) -> ?;
+ H1: (csubst0 ?1 ?2 ?3 ?4); H2: (drop ?5 (0) ?3 ?6) |- ? ] ->
+ LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?3 ?4 ?2); [ Clear H0 H1; Intros H0 | XAuto ];
+ LApply (H0 ?6); [ Clear H0 H2; Intros H0 | XAuto ];
+ XElim H0; Intros H0; [ Idtac | XElim H0 | XElim H0 | XElim H0 ]; Intros
+ | [ H0: (r ? ?1) = (S ?1) -> (e:?) (drop (S ?2) (0) ?3 e) -> ?;
+ H1: (drop (S ?2) (0) ?3 ?4) |- ? ] ->
+ LApply H0; [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?4); [ Clear H0 H1; Intros H0 | XAuto ];
+ XElim H0; Intros H0; [ Idtac | XElim H0 | XElim H0 | XElim H0 ]; Intros.
+
+ Theorem csubst0_drop_lt : (i,n:?) (lt n i) ->
+ (c1,c2:?; v:?) (csubst0 i v c1 c2) ->
+ (e:?) (drop n (0) c1 e) -> (OR
+ (drop n (0) c2 e) |
+ (EX k e0 u w | e = (CTail e0 k u) &
+ (drop n (0) c2 (CTail e0 k w)) &
+ (subst0 (minus (r k i) (S n)) v u w)
+ ) |
+ (EX k e1 e2 u | e = (CTail e1 k u) &
+ (drop n (0) c2 (CTail e2 k u)) &
+ (csubst0 (minus (r k i) (S n)) v e1 e2)
+ ) |
+ (EX k e1 e2 u w | e = (CTail e1 k u) &
+ (drop n (0) c2 (CTail e2 k w)) &
+ (subst0 (minus (r k i) (S n)) v u w) &
+ (csubst0 (minus (r k i) (S n)) v e1 e2)
+ )).
+ XElim i.
+(* case 1: i = 0 *)
+ Intros; Inversion H.
+(* case 2: i > 0 *)
+ Intros i; XElim n.
+(* case 2.1: n = 0 *)
+ Intros H0; Clear H0; Intros until 1; InsertEq H0 '(S i); XElim H0;
+ Clear H c1 c2 v y; Intros; DropGenBase; XRewrite e;
+ Rewrite <- r_arith0 in H; Try Rewrite <- r_arith0 in H0; Replace i with i0; XEAuto.
+(* case 2.2: n > 0 *)
+ Intros until 3; Clear H0; InsertEq H2 '(S i); XElim H0; Clear c1 c2 v y;
+ Intros; DropGenBase.
+(* case 2.2.1: csubst0_fst *)
+ XEAuto.
+(* case 2.2.2: csubst0_snd *)
+ Replace i0 with i; XAuto; XReplaceIn H0 i0 i; XReplaceIn H2 i0 i; Clear H3 i0.
+ Apply (r_dis k); Intros; Rewrite (H3 i) in H0; Rewrite (H3 n) in H4.
+(* case 2.2.2.1: bind *)
+ IH; XRewrite e; Try Rewrite <- (H3 n) in H; Try Rewrite <- (H3 n) in H0;
+ Try Rewrite <- r_arith1 in H4; Try Rewrite <- r_arith1 in H5; XEAuto.
+(* case 2.2.2.2: flat *)
+ IH; XRewrite e; Try Rewrite <- (H3 n) in H2; Try Rewrite <- (H3 n) in H4; XEAuto.
+(* case 2.2.3: csubst0_both *)
+ Replace i0 with i; XAuto; XReplaceIn H0 i0 i; XReplaceIn H2 i0 i; XReplaceIn H3 i0 i; Clear H4 i0.
+ Apply (r_dis k); Intros; Rewrite (H4 i) in H2; Rewrite (H4 n) in H5.
+(* case 2.2.2.1: bind *)
+ IH; XRewrite e; Try Rewrite <- (H4 n) in H; Try Rewrite <- (H4 n) in H2;
+ Try Rewrite <- r_arith1 in H5; Try Rewrite <- r_arith1 in H6; XEAuto.
+(* case 2.2.3.2: flat *)
+ IH; XRewrite e; Try Rewrite <- (H4 n) in H3; Try Rewrite <- (H4 n) in H5; XEAuto.
+ Qed.
+
+ Theorem csubst0_drop_ge_back : (i,n:?) (le i n) ->
+ (c1,c2:?; v:?) (csubst0 i v c1 c2) ->
+ (e:?) (drop n (0) c2 e) ->
+ (drop n (0) c1 e).
+ XElim i.
+(* case 1 : i = 0 *)
+ Intros; Inversion H0.
+(* case 2 : i > 0 *)
+ Intros i; XElim n.
+(* case 2.1 : n = 0 *)
+ Intros; Inversion H0.
+(* case 2.2 : n > 0 *)
+ Intros until 3; Clear H0; InsertEq H2 '(S i); XElim H0; Intros;
+ DropGenBase.
+(* case 2.2.1 : csubst0_fst *)
+ XAuto.
+(* case 2.2.2 : csubst0_snd *)
+ XReplaceIn H0 i0 i; NewInduction k; XEAuto.
+(* case 2.2.3 : csubst0_both *)
+ XReplaceIn H0 i0 i; XReplaceIn H2 i0 i; NewInduction k; XEAuto.
+ Qed.
+
+ End csubst0_drop.
+
+ Tactic Definition CSubst0Drop :=
+ Match Context With
+ | [ H1: (lt ?2 ?1);
+ H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?4 ?6) |- ? ] ->
+ LApply (csubst0_drop_lt ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x H3; Intros H3 | XAuto ];
+ XElim H3;
+ [ Intros | Intros H3; XElim H3; Intros
+ | Intros H3; XElim H3; Intros | Intros H3; XElim H3; Intros ]
+ | [ H1: (le ?1 ?2);
+ H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?4 ?6) |- ? ] ->
+ LApply (csubst0_drop_ge ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x H3; Intros | XAuto ]
+ | [H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?1 (0) ?4 ?6) |- ? ] ->
+ LApply (csubst0_drop_ge ?1 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x H2; Intros H2 | XAuto ];
+ LApply (H2 ?6); [ Clear H2 H3; Intros | XAuto ]
+ | [H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?1 (0) ?5 ?6) |- ? ] ->
+ LApply (csubst0_drop_ge_back ?1 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x H3; Intros | XAuto ]
+ | [H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?5 ?6) |- ? ] ->
+ LApply (csubst0_drop_ge_back ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x H3; Intros | XAuto ].
+
--- /dev/null
+(*#* #stop file *)
+
+Require Export subst1_defs.
+Require Export csubst0_defs.
+
+ Inductive csubst1 [i:nat; v:T; c1:C] : C -> Prop :=
+ | csubst1_refl : (csubst1 i v c1 c1)
+ | csubst1_single : (c2:?) (csubst0 i v c1 c2) -> (csubst1 i v c1 c2).
+
+ Hint csubst1 : ltlc := Constructors csubst1.
+
+ Section csubst1_props. (**************************************************)
+
+ Theorem csubst1_tail: (k:?; i:?; v,u1,u2:?) (subst1 (r k i) v u1 u2) ->
+ (c1,c2:?) (csubst1 (r k i) v c1 c2) ->
+ (csubst1 (S i) v (CTail c1 k u1) (CTail c2 k u2)).
+ Intros until 1; XElim H; Clear u2.
+(* case 1: csubst1_refl *)
+ Intros until 1; XElim H; Clear c2; XAuto.
+(* case 2: csubst1_single *)
+ Intros until 2; XElim H0; Clear c2; XAuto.
+ Qed.
+
+ End csubst1_props.
+
+ Hints Resolve csubst1_tail : ltlc.
+
+ Section csubst1_gen_base. (***********************************************)
+
+ Theorem csubst1_gen_tail: (k:?; c1,x:?; u1,v:?; i:?)
+ (csubst1 (S i) v (CTail c1 k u1) x) ->
+ (EX u2 c2 | x = (CTail c2 k u2) &
+ (subst1 (r k i) v u1 u2) &
+ (csubst1 (r k i) v c1 c2)
+ ).
+ Intros; InsertEq H '(CTail c1 k u1); InsertEq H '(S i);
+ XElim H; Clear x; Intros.
+(* case 1: csubst1_refl *)
+ Rewrite H0; XEAuto.
+(* case 2: csubst1_single *)
+ Rewrite H0 in H; Rewrite H1 in H; Clear H0 H1 y y0.
+ CSubst0GenBase; Rewrite H; XEAuto.
+ Qed.
+
+ End csubst1_gen_base.
+
+ Tactic Definition CSubst1GenBase :=
+ Match Context With
+ | [ H: (csubst1 (S ?1) ?2 (CTail ?3 ?4 ?5) ?6) |- ? ] ->
+ LApply (csubst1_gen_tail ?4 ?3 ?6 ?5 ?2 ?1); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Section csubst1_drop. (***************************************************)
+
+ Theorem csubst1_drop_ge : (i,n:?) (le i n) ->
+ (c1,c2:?; v:?) (csubst1 i v c1 c2) ->
+ (e:?) (drop n (0) c1 e) ->
+ (drop n (0) c2 e).
+ Intros until 2; XElim H0; Intros;
+ Try CSubst0Drop; XAuto.
+ Qed.
+
+ Theorem csubst1_drop_lt : (i,n:?) (lt n i) ->
+ (c1,c2:?; v:?) (csubst1 i v c1 c2) ->
+ (e1:?) (drop n (0) c1 e1) ->
+ (EX e2 | (csubst1 (minus i n) v e1 e2) &
+ (drop n (0) c2 e2)
+ ).
+ Intros until 2; XElim H0; Intros;
+ Try (
+ CSubst0Drop; Try Rewrite H1; Try Rewrite minus_x_Sy;
+ Try Rewrite r_minus in H3; Try Rewrite r_minus in H4
+ ); XEAuto.
+ Qed.
+
+ Theorem csubst1_drop_ge_back : (i,n:?) (le i n) ->
+ (c1,c2:?; v:?) (csubst1 i v c1 c2) ->
+ (e:?) (drop n (0) c2 e) ->
+ (drop n (0) c1 e).
+ Intros until 2; XElim H0; Intros;
+ Try CSubst0Drop; XAuto.
+ Qed.
+
+ End csubst1_drop.
+
+ Tactic Definition CSubst1Drop :=
+ Match Context With
+ | [ H1: (lt ?2 ?1);
+ H2: (csubst1 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?4 ?6) |- ? ] ->
+ LApply (csubst1_drop_lt ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x H3; Intros H3 | XAuto ];
+ XElim H3; Intros
+ | [H2: (csubst1 ?1 ?3 ?4 ?5); H3: (drop ?1 (0) ?4 ?6) |- ? ] ->
+ LApply (csubst1_drop_ge ?1 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x H2; Intros H2 | XAuto ];
+ LApply (H2 ?6); [ Clear H2 H3; Intros | XAuto ]
+ | [ H2: (csubst1 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?4 ?6) |- ? ] ->
+ LApply (csubst1_drop_ge ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x H3; Intros | XAuto ].
--- /dev/null
+Name: lambda-delta
+Title: A formalization of a lambda-typed lambda-calculus with abbreviations
+Author: Ferruccio Guidi
+Email: fguidi@cs.unibo.it
+Homepage: http://www.cs.unibo.it/~fguidi
+Institution: Department of Computer Science, University of Bologna
+Address: Mura Anteo Zamboni 7, 40127 Bologna, ITALY
+Date: March 31, 2005
+Description:
+Url:
+Keywords: lambda-types, lambda-calculus, abbreviations
+Version: 7.3.1
+Require:
--- /dev/null
+(*#* #stop file *)
+
+Require Export contexts_defs.
+Require Export lift_defs.
+
+ Inductive drop : nat -> nat -> C -> C -> Prop :=
+ | drop_sort : (h,d,n:?) (drop h d (CSort n) (CSort n))
+ | drop_tail : (c,e:?) (drop (0) (0) c e) ->
+ (k:?; u:?) (drop (0) (0) (CTail c k u) (CTail e k u))
+ | drop_drop : (k:?; h:?; c,e:?) (drop (r k h) (0) c e) ->
+ (u:?) (drop (S h) (0) (CTail c k u) e)
+ | drop_skip : (k:?; h,d:?; c,e:?) (drop h (r k d) c e) -> (u:?)
+ (drop h (S d) (CTail c k (lift h (r k d) u)) (CTail e k u)).
+
+ Hint drop : ltlc := Constructors drop.
+
+ Hint discr : ltlc := Extern 4 (drop ? ? ? ?) Simpl.
+
+ Section drop_gen_base. (**************************************************)
+
+ Theorem drop_gen_sort : (n,h,d:?; x:?)
+ (drop h d (CSort n) x) -> x = (CSort n).
+ Intros until 1; InsertEq H '(CSort n); XElim H; Intros;
+ Try Inversion H1; XAuto.
+ Qed.
+
+ Theorem drop_gen_refl : (x,e:?) (drop (0) (0) x e) -> x = e.
+ Intros until 1; Repeat InsertEq H '(0); XElim H; Intros.
+(* case 1 : drop_sort *)
+ XAuto.
+(* case 2 : drop_tail *)
+ Rewrite H0; XAuto.
+(* case 3 : drop_drop *)
+ Inversion H2.
+(* case 4 : drop_skip *)
+ Inversion H1.
+ Qed.
+
+ Theorem drop_gen_drop : (k:?; c,x:?; u:?; h:?)
+ (drop (S h) (0) (CTail c k u) x) ->
+ (drop (r k h) (0) c x).
+ Intros until 1;
+ InsertEq H '(CTail c k u); InsertEq H '(0); InsertEq H '(S h);
+ XElim H; Intros.
+(* case 1 : drop_sort *)
+ Inversion H1.
+(* case 2 : drop_tail *)
+ Inversion H1.
+(* case 3 : drop_drop *)
+ Inversion H1; Inversion H3.
+ Rewrite <- H5; Rewrite <- H6; Rewrite <- H7; XAuto.
+(* case 4 : drop_skip *)
+ Inversion H2.
+ Qed.
+
+ Theorem drop_gen_skip_r : (c,x:?; u:?; h,d:?; k:?)
+ (drop h (S d) x (CTail c k u)) ->
+ (EX e | x = (CTail e k (lift h (r k d) u)) & (drop h (r k d) e c)).
+ Intros; Inversion_clear H; XEAuto.
+ Qed.
+
+ Theorem drop_gen_skip_l : (c,x:?; u:?; h,d:?; k:?)
+ (drop h (S d) (CTail c k u) x) ->
+ (EX e v | x = (CTail e k v) &
+ u = (lift h (r k d) v) &
+ (drop h (r k d) c e)
+ ).
+ Intros; Inversion_clear H; XEAuto.
+ Qed.
+
+ End drop_gen_base.
+
+ Hints Resolve drop_gen_refl : ltlc.
+
+ Tactic Definition DropGenBase :=
+ Match Context With
+ | [ H: (drop (0) (0) ?0 ?1) |- ? ] ->
+ LApply (drop_gen_refl ?0 ?1); [ Clear H; Intros | XAuto ]
+ | [ H: (drop ?0 ?1 (CSort ?2) ?3) |- ? ] ->
+ LApply (drop_gen_sort ?2 ?0 ?1 ?3); [ Clear H; Intros | XAuto ]
+ | [ H: (drop (S ?0) (0) (CTail ?1 ?2 ?3) ?4) |- ? ] ->
+ LApply (drop_gen_drop ?2 ?1 ?4 ?3 ?0); [ Clear H; Intros | XAuto ]
+ | [ H: (drop ?1 (S ?2) ?3 (CTail ?4 ?5 ?6)) |- ? ] ->
+ LApply (drop_gen_skip_r ?4 ?3 ?6 ?1 ?2 ?5); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (drop ?1 (S ?2) (CTail ?4 ?5 ?6) ?3) |- ? ] ->
+ LApply (drop_gen_skip_l ?4 ?3 ?6 ?1 ?2 ?5); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Section drop_props. (*****************************************************)
+
+ Theorem drop_skip_bind: (h,d:?; c,e:?) (drop h d c e) -> (b:?; u:?)
+ (drop h (S d) (CTail c (Bind b) (lift h d u)) (CTail e (Bind b) u)).
+ Intros; Pattern 2 d; Replace d with (r (Bind b) d); XAuto.
+ Qed.
+
+ Theorem drop_refl: (c:?) (drop (0) (0) c c).
+ XElim c; XAuto.
+ Qed.
+
+ Hints Resolve drop_refl : ltlc.
+
+ Theorem drop_S : (b:?; c,e:?; u:?; h:?)
+ (drop h (0) c (CTail e (Bind b) u)) ->
+ (drop (S h) (0) c e).
+ XElim c.
+(* case 1: CSort *)
+ Intros; DropGenBase; Inversion H.
+(* case 2: CTail *)
+ XElim h; Intros; DropGenBase.
+(* case 2.1: h = 0 *)
+ Inversion H0; XAuto.
+(* case 2.1: h > 0 *)
+ Apply drop_drop; RRw; XEAuto. (**) (* explicit constructor *)
+ Qed.
+
+ End drop_props.
+
+ Hints Resolve drop_skip_bind drop_refl drop_S : ltlc.
+
+ Tactic Definition DropS :=
+ Match Context With
+ [ _: (drop ?1 (0) ?2 (CTail ?3 (Bind ?4) ?5)) |- ? ] ->
+ LApply (drop_S ?4 ?2 ?3 ?5 ?1); [ Intros | XAuto ].
+
--- /dev/null
+Require lift_gen.
+Require drop_defs.
+
+(*#* #caption "main properties of drop" #clauses *)
+
+ Section confluence. (*****************************************************)
+
+(*#* #stop macro *)
+
+ Tactic Definition IH :=
+ Match Context With
+ [ H1: (drop ?1 ?2 c ?3); H2: (drop ?1 ?2 c ?4) |- ? ] ->
+ LApply (H ?4 ?2 ?1); [ Clear H H2; Intros H | XAuto ];
+ LApply (H ?3); [ Clear H H1; Intros | XAuto ].
+
+(*#* #start macro *)
+
+(*#* #caption "confluence, first case" *)
+(*#* #cap #alpha c in C, x1 in C1, x2 in C2, d in i *)
+
+ Theorem drop_mono : (c,x1:?; d,h:?) (drop h d c x1) ->
+ (x2:?) (drop h d c x2) -> x1 = x2.
+
+(*#* #stop proof *)
+
+ XElim c.
+(* case 1: CSort *)
+ Intros; Repeat DropGenBase; Rewrite H0; XAuto.
+(* case 2: CTail k *)
+ XElim d.
+(* case 2.1: d = 0 *)
+ XElim h; Intros; Repeat DropGenBase; Try Rewrite <- H0; XEAuto.
+(* case 2.2: d > 0 *)
+ Intros; Repeat DropGenBase; Rewrite H1; Rewrite H2; Rewrite H5 in H3;
+ LiftGen; IH; XAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "confluence, second case" *)
+(*#* #cap #alpha c in C1, c0 in E1, e in C2, e0 in E2, u in V1, v in V2, i in k, d in i *)
+
+ Theorem drop_conf_lt: (b:?; i:?; u:?; c0,c:?)
+ (drop i (0) c (CTail c0 (Bind b) u)) ->
+ (e:?; h,d:?) (drop h (S (plus i d)) c e) ->
+ (EX v e0 | u = (lift h d v) &
+ (drop i (0) e (CTail e0 (Bind b) v)) &
+ (drop h d c0 e0)
+ ).
+
+(*#* #stop proof *)
+
+ XElim i.
+(* case 1 : i = 0 *)
+ Intros until 1.
+ DropGenBase.
+ Rewrite H in H0; Clear H.
+ Inversion H0; XEAuto.
+(* case 2 : i > 0 *)
+ Intros i; XElim c.
+(* case 2.1 : CSort *)
+ Intros; Inversion H0.
+(* case 2.2 : CTail k *)
+ XElim k; Intros; Repeat DropGenBase; Rewrite H2; Clear H2 H3 e t.
+(* case 2.2.1 : Bind *)
+ LApply (H u c0 c); [ Clear H H0 H1; Intros H | XAuto ].
+ LApply (H x0 h d); [ Clear H H9; Intros H | XAuto ].
+ XElim H; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply H0; [ Clear H H0 H1; Intros H | XAuto ].
+ LApply (H x0 h d); [ Clear H H9; Intros H | XAuto ].
+ XElim H; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "confluence, third case" *)
+(*#* #cap #alpha c in C, a in C1, e in C2, i in k, d in i *)
+
+ Theorem drop_conf_ge: (i:?; a,c:?) (drop i (0) c a) ->
+ (e:?; h,d:?) (drop h d c e) -> (le (plus d h) i) ->
+ (drop (minus i h) (0) e a).
+
+(*#* #stop proof *)
+
+ XElim i.
+(* case 1 : i = 0 *)
+ Intros until 1.
+ DropGenBase; Rewrite H in H0; Clear H c.
+ Inversion H1; Rewrite H2; Simpl; Clear H1.
+ PlusO; Rewrite H in H0; Rewrite H1 in H0; Clear H H1 d h.
+ DropGenBase; Rewrite <- H; XAuto.
+(* case 2 : i > 0 *)
+ Intros i; XElim c.
+(* case 2.1 : CSort *)
+ Intros; Repeat DropGenBase; Rewrite H1; Rewrite H0; XAuto.
+(* case 2.2 : CTail k *)
+ XElim k; Intros; DropGenBase;
+ ( NewInduction d;
+ [ NewInduction h; DropGenBase;
+ [ Rewrite <- H2; Simpl; XAuto | Clear IHh ]
+ | DropGenBase; Rewrite H2; Clear IHd H2 H4 e t ] ).
+(* case 2.2.1 : Bind, d = 0, h > 0 *)
+ LApply (H a c); [ Clear H H0 H1; Intros H | XAuto ].
+ LApply (H e h (0)); XAuto.
+(* case 2.2.2 : Bind, d > 0 *)
+ LApply (H a c); [ Clear H H0 H1; Intros H | XAuto ].
+ LApply (H x0 h d); [ Clear H H4; Intros H | XAuto ].
+ LApply H; [ Clear H; Simpl in H3; Intros H | XAuto ].
+ Rewrite <- minus_Sn_m; XEAuto.
+(* case 2.2.3 : Flat, d = 0, h > 0 *)
+ LApply H0; [ Clear H H0 H1; Intros H | XAuto ].
+ LApply (H e (S h) (0)); XAuto.
+(* case 2.2.4 : Flat, d > 0 *)
+ LApply H0; [ Clear H H0 H1; Intros H | XAuto ].
+ LApply (H x0 h (S d)); [ Clear H H4; Intros H | XAuto ].
+ LApply H; [ Clear H; Simpl in H3; Intros H | XAuto ].
+ Rewrite <- minus_Sn_m in H; [ Idtac | XEAuto ].
+ Rewrite <- minus_Sn_m; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+ End confluence.
+
+ Section transitivity. (***************************************************)
+
+(*#* #caption "transitivity, first case" *)
+(*#* #cap #alpha c1 in C1, c2 in C2, e1 in D1, e2 in D2, d in i, i in k *)
+
+ Theorem drop_trans_le : (i,d:?) (le i d) ->
+ (c1,c2:?; h:?) (drop h d c1 c2) ->
+ (e2:?) (drop i (0) c2 e2) ->
+ (EX e1 | (drop i (0) c1 e1) & (drop h (minus d i) e1 e2)).
+
+(*#* #stop proof *)
+
+ XElim i.
+(* case 1 : i = 0 *)
+ Intros.
+ DropGenBase; Rewrite H1 in H0.
+ Rewrite <- minus_n_O; XEAuto.
+(* case 2 : i > 0 *)
+ Intros i IHi; XElim d.
+(* case 2.1 : d = 0 *)
+ Intros; Inversion H.
+(* case 2.2 : d > 0 *)
+ Intros d IHd; XElim c1.
+(* case 2.2.1 : CSort *)
+ Intros.
+ DropGenBase; Rewrite H0 in H1.
+ DropGenBase; Rewrite H1; XEAuto.
+(* case 2.2.2 : CTail k *)
+ Intros c1 IHc; XElim k; Intros;
+ DropGenBase; Rewrite H0 in H1; Rewrite H2; Clear IHd H0 H2 c2 t;
+ DropGenBase.
+(* case 2.2.2.1 : Bind *)
+ LApply (IHi d); [ Clear IHi; Intros IHi | XAuto ].
+ LApply (IHi c1 x0 h); [ Clear IHi H8; Intros IHi | XAuto ].
+ LApply (IHi e2); [ Clear IHi H0; Intros IHi | XAuto ].
+ XElim IHi; XEAuto.
+(* case 2.2.2.2 : Flat *)
+ LApply (IHc x0 h); [ Clear IHc H8; Intros IHc | XAuto ].
+ LApply (IHc e2); [ Clear IHc H0; Intros IHc | XAuto ].
+ XElim IHc; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "transitivity, second case" *)
+(*#* #cap #alpha c1 in C1, c2 in C, e2 in C2, d in i, i in k *)
+
+ Theorem drop_trans_ge : (i:?; c1,c2:?; d,h:?) (drop h d c1 c2) ->
+ (e2:?) (drop i (0) c2 e2) -> (le d i) ->
+ (drop (plus i h) (0) c1 e2).
+
+(*#* #stop proof *)
+
+ XElim i.
+(* case 1: i = 0 *)
+ Intros.
+ DropGenBase; Rewrite <- H0.
+ Inversion H1; Rewrite H2 in H; XAuto.
+(* case 2 : i > 0 *)
+ Intros i IHi; XElim c1; Simpl.
+(* case 2.1: CSort *)
+ Intros.
+ DropGenBase; Rewrite H in H0.
+ DropGenBase; Rewrite H0; XAuto.
+(* case 2.2: CTail *)
+ Intros c1 IHc; XElim d.
+(* case 2.2.1: d = 0 *)
+ XElim h; Intros.
+(* case 2.2.1.1: h = 0 *)
+ DropGenBase; Rewrite <- H in H0;
+ DropGenBase; Rewrite <- plus_n_O; XAuto.
+(* case 2.2.1.2: h > 0 *)
+ DropGenBase; Rewrite <- plus_n_Sm.
+ Apply drop_drop; RRw; XEAuto. (**) (* explicit constructor *)
+(* case 2.2.2: d > 0 *)
+ Intros d IHd; Intros.
+ DropGenBase; Rewrite H in IHd; Rewrite H in H0; Rewrite H2 in IHd; Rewrite H2; Clear IHd H H2 c2 t;
+ DropGenBase; Apply drop_drop; NewInduction k; Simpl; XEAuto. (**) (* explicit constructor *)
+ Qed.
+
+(*#* #start proof *)
+
+ End transitivity.
+
+(*#* #stop macro *)
+
+ Tactic Definition DropDis :=
+ Match Context With
+ [ H1: (drop ?1 ?2 ?3 ?4); H2: (drop ?1 ?2 ?3 ?5) |- ? ] ->
+ LApply (drop_mono ?3 ?5 ?2 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?4); [ Clear H_x H1; Intros H1; Rewrite H1 in H2 | XAuto ]
+ | [ H1: (drop ?0 (0) ?1 (CTail ?2 (Bind ?3) ?4));
+ H2: (drop ?5 (S (plus ?0 ?6)) ?1 ?7) |- ? ] ->
+ LApply (drop_conf_lt ?3 ?0 ?4 ?2 ?1); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?7 ?5 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ _: (drop ?0 (0) ?1 ?2); _: (drop ?5 (0) ?1 ?7);
+ _: (lt ?5 ?0) |- ? ] ->
+ LApply (drop_conf_ge ?0 ?2 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?7 ?5 (0)); [ Clear H_x; Intros H_x | XAuto ];
+ Simpl in H_x; LApply H_x; [ Clear H_x; Intros | XAuto ]
+ | [ _: (drop ?1 (0) ?2 (CTail ?3 (Bind ?) ?));
+ _: (drop (1) ?1 ?2 ?4) |- ? ] ->
+ LApply (drop_conf_ge (S ?1) ?3 ?2); [ Intros H_x | XEAuto ];
+ LApply (H_x ?4 (1) ?1); [ Clear H_x; Intros H_x | XAuto ];
+ LApply H_x; [ Clear H_x; Intros | Rewrite plus_sym; XAuto ]; (
+ Match Context With
+ [ H: (drop (minus (S ?1) (1)) (0) ?4 ?3) |- ? ] ->
+ Simpl in H; Rewrite <- minus_n_O in H )
+ | [ H0: (drop ?0 (0) ?1 ?2); H2: (lt ?6 ?0);
+ H1: (drop (1) ?6 ?1 ?7) |- ? ] ->
+ LApply (drop_conf_ge ?0 ?2 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?7 (1) ?6); [ Clear H_x; Intros H_x | XAuto ];
+ LApply H_x; [ Clear H_x; Intros | Rewrite plus_sym; XAuto ]
+ | [ H0: (drop ?0 (0) ?1 ?2);
+ H1: (drop ?5 ?6 ?1 ?7) |- ? ] ->
+ LApply (drop_conf_ge ?0 ?2 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?7 ?5 ?6); [ Clear H_x; Intros H_x | XAuto ];
+ LApply H_x; [ Clear H_x; Intros | XAuto ]
+ | [ H0: (lt ?1 ?2);
+ H1: (drop ?3 ?2 ?4 ?5); H2: (drop ?1 (0) ?5 ?6) |- ? ] ->
+ LApply (drop_trans_le ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5 ?3); [ Clear H_x H1; Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x H2; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ H0: (le ?1 ?2);
+ H1: (drop ?3 ?1 ?4 ?5); H2: (drop ?2 (0) ?5 ?6) |- ? ] ->
+ LApply (drop_trans_ge ?2 ?4 ?5 ?1 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
+ LApply H1; [ Clear H1; Intros | XAuto ].
+
+(*#* #start macro *)
+
+(*#* #single *)
--- /dev/null
+(*#* #stop file *)
+
+Require Export terms_defs.
+
+ Fixpoint bref_map [g:nat->nat; d:nat; t:T] : T := Cases t of
+ | (TSort n) => (TSort n)
+ | (TBRef n) =>
+ if (blt n d) then (TBRef n) else (TBRef (g n))
+ | (TTail k u t) =>
+ (TTail k (bref_map g d u) (bref_map g (s k d) t))
+ end.
+
+ Definition lift : nat -> nat -> T -> T :=
+ [h](bref_map [x](plus x h)).
+
+ Section lift_rw. (********************************************************)
+
+ Theorem lift_sort: (n:?; h,d:?) (lift h d (TSort n)) = (TSort n).
+ XAuto.
+ Qed.
+
+ Theorem lift_bref_lt: (n:?; h,d:?) (lt n d) ->
+ (lift h d (TBRef n)) = (TBRef n).
+ Intros; Unfold lift; Simpl.
+ Replace (blt n d) with true; XAuto.
+ Qed.
+
+ Theorem lift_bref_ge: (n:?; h,d:?) (le d n) ->
+ (lift h d (TBRef n)) = (TBRef (plus n h)).
+
+ Intros; Unfold lift; Simpl.
+ Replace (blt n d) with false; XAuto.
+ Qed.
+
+ Theorem lift_tail: (k:?; u,t:?; h,d:?)
+ (lift h d (TTail k u t)) =
+ (TTail k (lift h d u) (lift h (s k d) t)).
+ XAuto.
+ Qed.
+
+ Theorem lift_bind: (b:?; u,t:?; h,d:?)
+ (lift h d (TTail (Bind b) u t)) =
+ (TTail (Bind b) (lift h d u) (lift h (S d) t)).
+ XAuto.
+ Qed.
+
+ Theorem lift_flat: (f:?; u,t:?; h,d:?)
+ (lift h d (TTail (Flat f) u t)) =
+ (TTail (Flat f) (lift h d u) (lift h d t)).
+ XAuto.
+ Qed.
+
+ End lift_rw.
+
+ Hints Resolve lift_bref_lt lift_bind lift_flat : ltlc.
+
+ Tactic Definition LiftTailRw :=
+ Repeat (Rewrite lift_tail Orelse Rewrite lift_bind Orelse Rewrite lift_flat).
+
+ Tactic Definition LiftTailRwBack :=
+ Repeat (Rewrite <- lift_tail Orelse Rewrite <- lift_bind Orelse Rewrite <- lift_flat).
+
+ Section lift_gen. (*******************************************************)
+
+ Theorem lift_gen_sort: (h,d,n:?; t:?) (TSort n) = (lift h d t) ->
+ t = (TSort n).
+ XElim t; Intros.
+(* case 1 : TSort *)
+ XAuto.
+(* case 2 : TBRef n0 *)
+ Apply (lt_le_e n0 d); Intros.
+(* case 2.1 : n0 < d *)
+ Rewrite lift_bref_lt in H; [ Inversion H | XAuto ].
+(* case 2.2 : n0 >= d *)
+ Rewrite lift_bref_ge in H; [ Inversion H | XAuto ].
+(* case 3 : TTail k *)
+ Rewrite lift_tail in H1; Inversion H1.
+ Qed.
+
+ Theorem lift_gen_bref_lt: (h,d,n:?) (lt n d) ->
+ (t:?) (TBRef n) = (lift h d t) ->
+ t = (TBRef n).
+ XElim t; Intros.
+(* case 1 : TSort *)
+ XAuto.
+(* case 2 : TBRef n0 *)
+ Apply (lt_le_e n0 d); Intros.
+(* case 2.1 : n0 < d *)
+ Rewrite lift_bref_lt in H0; XAuto.
+(* case 2.2 : n0 >= d *)
+ Rewrite lift_bref_ge in H0; [ Inversion H0; Clear H0 | XAuto ].
+ Rewrite H3 in H; Clear H3 n.
+ EApply le_false; [ Apply H1 | XEAuto ].
+(* case 3 : TTail k *)
+ Rewrite lift_tail in H2; Inversion H2.
+ Qed.
+
+ Theorem lift_gen_bref_false: (h,d,n:?) (le d n) -> (lt n (plus d h)) ->
+ (t:?) (TBRef n) = (lift h d t) ->
+ (P:Prop) P.
+ XElim t; Intros.
+(* case 1 : TSort *)
+ Inversion H1.
+(* case 2 : TBRef n0 *)
+ Apply (lt_le_e n0 d); Intros.
+(* case 2.1 : n0 < d *)
+ Rewrite lift_bref_lt in H1; [ Inversion H1; Clear H1 | XAuto ].
+ Rewrite <- H4 in H2; Clear H4 n0.
+ EApply le_false; [ Apply H | XEAuto ].
+(* case 2.2 : n0 >= d *)
+ Rewrite lift_bref_ge in H1; [ Inversion H1; Clear H1 | XAuto ].
+ Rewrite H4 in H0; Clear H4.
+ EApply le_false; [ Apply H2 | XEAuto ].
+(* case 3 : TTail k *)
+ Rewrite lift_tail in H3; Inversion H3.
+ Qed.
+
+ Theorem lift_gen_bref_ge: (h,d,n:?) (le d n) ->
+ (t:?) (TBRef (plus n h)) = (lift h d t) ->
+ t = (TBRef n).
+ XElim t; Intros.
+(* case 1 : TSort *)
+ Inversion H0.
+(* case 2 : TBRef n0 *)
+ Apply (lt_le_e n0 d); Intros.
+(* case 2.1 : n0 < d *)
+ Rewrite lift_bref_lt in H0; [ Inversion H0; Clear H0 | XAuto ].
+ Rewrite <- H3 in H1; Clear H3 n0.
+ EApply le_false; [ Apply H | XEAuto ].
+(* case 2.2 : n0 >= d *)
+ Rewrite lift_bref_ge in H0; [ Inversion H0; XEAuto | XAuto ].
+(* case 3 : TTail k *)
+ Rewrite lift_tail in H2; Inversion H2.
+ Qed.
+
+(* NOTE: lift_gen_tail should be used instead of these two *) (**)
+ Theorem lift_gen_bind: (b:?; u,t,x:?; h,d:?)
+ (TTail (Bind b) u t) = (lift h d x) ->
+ (EX y z | x = (TTail (Bind b) y z) &
+ u = (lift h d y) &
+ t = (lift h (S d) z)
+ ).
+ XElim x; Intros.
+(* case 1 : TSort *)
+ Inversion H.
+(* case 2 : TBRef n *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1 : n < d *)
+ Rewrite lift_bref_lt in H; [ Inversion H | XAuto ].
+(* case 2.2 : n >= d *)
+ Rewrite lift_bref_ge in H; [ Inversion H | XAuto ].
+(* case 3 : TTail k *)
+ Rewrite lift_tail in H1; Inversion H1.
+ XEAuto.
+ Qed.
+
+ Theorem lift_gen_flat: (f:?; u,t,x:?; h,d:?)
+ (TTail (Flat f) u t) = (lift h d x) ->
+ (EX y z | x = (TTail (Flat f) y z) &
+ u = (lift h d y) &
+ t = (lift h d z)
+ ).
+ XElim x; Intros.
+(* case 1 : TSort *)
+ Inversion H.
+(* case 2 : TBRef n *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1 : n < d *)
+ Rewrite lift_bref_lt in H; [ Inversion H | XAuto ].
+(* case 2.2 : n >= d *)
+ Rewrite lift_bref_ge in H; [ Inversion H | XAuto ].
+(* case 3 : TTail k *)
+ Rewrite lift_tail in H1; Inversion H1.
+ XEAuto.
+ Qed.
+
+ End lift_gen.
+
+ Tactic Definition LiftGenBase :=
+ Match Context With
+ | [ H: (TSort ?0) = (lift ?1 ?2 ?3) |- ? ] ->
+ LApply (lift_gen_sort ?1 ?2 ?0 ?3); [ Clear H; Intros | XAuto ]
+ | [ H1: (le ?1 ?2); H2: (lt ?2 (plus ?1 ?3));
+ H3: (TBRef ?2) = (lift ?3 ?1 ?4) |- ? ] ->
+ Apply (lift_gen_bref_false ?3 ?1 ?2 H1 H2 ?4 H3); XAuto
+ | [ H: (TBRef ?1) = (lift (1) ?1 ?2) |- ? ] ->
+ LApply (lift_gen_bref_false (1) ?1 ?1); [ Intros H_x | XAuto ];
+ LApply H_x; [ Clear H_x; Intros H_x | Arith7' ?1; XAuto ];
+ LApply (H_x ?2); [ Clear H_x; Intros H_x | XAuto ];
+ Apply H_x
+ | [ H: (TBRef (plus ?0 ?1)) = (lift ?1 ?2 ?3) |- ? ] ->
+ LApply (lift_gen_bref_ge ?1 ?2 ?0); [ Intros H_x | XAuto ];
+ LApply (H_x ?3); [ Clear H_x H; Intros | XAuto ]
+ | [ H1: (TBRef ?0) = (lift ?1 ?2 ?3); H2: (lt ?0 ?4) |- ? ] ->
+ LApply (lift_gen_bref_lt ?1 ?2 ?0);
+ [ Intros H_x | Apply lt_le_trans with m:=?4; XEAuto ];
+ LApply (H_x ?3); [ Clear H_x H1; Intros | XAuto ]
+ | [ H: (TBRef ?0) = (lift ?1 ?2 ?3) |- ? ] ->
+ LApply (lift_gen_bref_lt ?1 ?2 ?0); [ Intros H_x | XEAuto ];
+ LApply (H_x ?3); [ Clear H_x H; Intros | XAuto ]
+ | [ H: (TTail (Bind ?0) ?1 ?2) = (lift ?3 ?4 ?5) |- ? ] ->
+ LApply (lift_gen_bind ?0 ?1 ?2 ?5 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (TTail (Flat ?0) ?1 ?2) = (lift ?3 ?4 ?5) |- ? ] ->
+ LApply (lift_gen_flat ?0 ?1 ?2 ?5 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Section lift_props. (*****************************************************)
+
+ Theorem lift_r: (t:?; d:?) (lift (0) d t) = t.
+ XElim t; Intros.
+(* case 1: TSort *)
+ XAuto.
+(* case 2: TBRef n *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1: n < d *)
+ Rewrite lift_bref_lt; XAuto.
+(* case 2.2: n >= d *)
+ Rewrite lift_bref_ge; XAuto.
+(* case 3: TTail *)
+ LiftTailRw; XAuto.
+ Qed.
+
+ Theorem lift_bref_gt : (d,n:?) (lt d n) ->
+ (lift (1) d (TBRef (pred n))) = (TBRef n).
+ Intros.
+ Rewrite lift_bref_ge.
+(* case 1: first branch *)
+ Rewrite <- plus_sym; Simpl; Rewrite <- (S_pred n d); XAuto.
+(* case 2: second branch *)
+ Apply le_S_n; Rewrite <- (S_pred n d); XAuto.
+ Qed.
+
+ End lift_props.
+
+ Hints Resolve lift_r lift_bref_gt : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require lift_defs.
+
+ Section lift_ini. (*******************************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H1: (lift ?1 ?2 t) = (lift ?1 ?2 ?3) |- ? ] ->
+ LApply (H ?3 ?1 ?2); [ Clear H H1; Intros | XAuto ]
+ | [ H1: (lift ?1 ?2 t0) = (lift ?1 ?2 ?3) |- ? ] ->
+ LApply (H0 ?3 ?1 ?2); [ Clear H0 H1; Intros | XAuto ].
+
+(*#* #start file *)
+
+(*#* #caption "main properties of lift" #clauses lift_props *)
+
+(*#* #caption "injectivity" *)
+(*#* #cap #alpha x in T1, t in T2, d in i *)
+
+ Theorem lift_inj : (x,t:?; h,d:?) (lift h d x) = (lift h d t) -> x = t.
+
+(*#* #stop file *)
+
+ XElim x.
+(* case 1 : TSort *)
+ Intros; Rewrite lift_sort in H; LiftGenBase; XAuto.
+(* case 2 : TBRef n *)
+ Intros; Apply (lt_le_e n d); Intros.
+(* case 2.1 : n < d *)
+ Rewrite lift_bref_lt in H; [ LiftGenBase; XAuto | XAuto ].
+(* case 2.2 : n >= d *)
+ Rewrite lift_bref_ge in H; [ LiftGenBase; XAuto | XAuto ].
+(* case 3 : TTail k *)
+ XElim k; Intros; [ Rewrite lift_bind in H1 | Rewrite lift_flat in H1 ];
+ LiftGenBase; Rewrite H1; IH; IH; XAuto.
+ Qed.
+
+ End lift_ini.
+
+ Section lift_gen_lift. (**************************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H_x: (lift ?0 ?1 t) = (lift ?2 (plus ?3 ?0) ?4) |- ? ] ->
+ LApply (H ?4 ?0 ?2 ?1 ?3); [ Clear H; Intros H | XAuto ];
+ LApply H; [ Clear H H_x; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H_x: (lift ?0 ?1 t0) = (lift ?2 (plus ?3 ?0) ?4) |- ? ] ->
+ LApply (H0 ?4 ?0 ?2 ?1 ?3); [ Clear H0; Intros H0 | XAuto ];
+ LApply H0; [ Clear H0 H_x; Intros H0 | XAuto ];
+ XElim H0; Intros.
+
+(*#* #start file *)
+
+(*#* #caption "generation lemma for lift" *)
+(*#* #cap #cap t1 #alpha t2 in T, x in T2, d1 in i1, d2 in i2 *)
+
+ Theorem lift_gen_lift : (t1,x:?; h1,h2,d1,d2:?) (le d1 d2) ->
+ (lift h1 d1 t1) = (lift h2 (plus d2 h1) x) ->
+ (EX t2 | x = (lift h1 d1 t2) &
+ t1 = (lift h2 d2 t2)
+ ).
+
+(*#* #stop file *)
+
+ XElim t1; Intros.
+(* case 1 : TSort *)
+ Rewrite lift_sort in H0.
+ LiftGenBase; Rewrite H0; Clear H0 x.
+ EApply ex2_intro; Rewrite lift_sort; XAuto.
+(* case 2 : TBRef n *)
+ Apply (lt_le_e n d1); Intros.
+(* case 2.1 : n < d1 *)
+ Rewrite lift_bref_lt in H0; [ Idtac | XAuto ].
+ LiftGenBase; Rewrite H0; Clear H0 x.
+ EApply ex2_intro; Rewrite lift_bref_lt; XEAuto.
+(* case 2.2 : n >= d1 *)
+ Rewrite lift_bref_ge in H0; [ Idtac | XAuto ].
+ Apply (lt_le_e n d2); Intros.
+(* case 2.2.1 : n < d2 *)
+ LiftGenBase; Rewrite H0; Clear H0 x.
+ EApply ex2_intro; [ Rewrite lift_bref_ge | Rewrite lift_bref_lt ]; XEAuto.
+(* case 2.2.2 : n >= d2 *)
+ Apply (lt_le_e n (plus d2 h2)); Intros.
+(* case 2.2.2.1 : n < d2 + h2 *)
+ EApply lift_gen_bref_false; [ Idtac | Idtac | Apply H0 ];
+ [ XAuto | Rewrite plus_permute_2_in_3; XAuto ].
+(* case 2.2.2.2 : n >= d2 + h2 *)
+ Rewrite (le_plus_minus_sym h2 (plus n h1)) in H0; [ Idtac | XEAuto ].
+ LiftGenBase; Rewrite H0; Clear H0 x.
+ EApply ex2_intro;
+ [ Rewrite le_minus_plus; [ Idtac | XEAuto ]
+ | Rewrite (le_plus_minus_sym h2 n); [ Idtac | XEAuto ] ];
+ Rewrite lift_bref_ge; XEAuto.
+(* case 3 : TTail k *)
+ NewInduction k.
+(* case 3.1 : Bind *)
+ Rewrite lift_bind in H2.
+ LiftGenBase; Rewrite H2; Clear H2 x.
+ IH; Rewrite H; Rewrite H2; Clear H H2 x0.
+ Arith4In H4 d2 h1; IH; Rewrite H; Rewrite H0; Clear H H0 x1 t t0.
+ EApply ex2_intro; Rewrite lift_bind; XAuto.
+(* case 3.2 : Flat *)
+ Rewrite lift_flat in H2.
+ LiftGenBase; Rewrite H2; Clear H2 x.
+ IH; Rewrite H; Rewrite H2; Clear H H2 x0.
+ IH; Rewrite H; Rewrite H0; Clear H H0 x1 t t0.
+ EApply ex2_intro; Rewrite lift_flat; XAuto.
+ Qed.
+
+ End lift_gen_lift.
+
+ Tactic Definition LiftGen :=
+ Match Context With
+ | [ H: (lift ?1 ?2 ?3) = (lift ?1 ?2 ?4) |- ? ] ->
+ LApply (lift_inj ?3 ?4 ?1 ?2); [ Clear H; Intros | XAuto ]
+ | [ H: (lift ?0 ?1 ?2) = (lift ?3 (plus ?4 ?0) ?5) |- ? ] ->
+ LApply (lift_gen_lift ?2 ?5 ?0 ?3 ?1 ?4); [ Intros H_x | XAuto ];
+ LApply H_x; [ Clear H H_x; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (lift (1) (0) ?1) = (lift (1) (S ?2) ?3) |- ? ] ->
+ LApply (lift_gen_lift ?1 ?3 (1) (1) (0) ?2); [ Intros H_x | XAuto ];
+ LApply H_x; [ Clear H_x H; Intros H | Arith7' ?2; XAuto ];
+ XElim H; Intros
+ | _ -> LiftGenBase.
--- /dev/null
+(*#* #stop file *)
+
+Require lift_defs.
+
+ Section lift_props. (*****************************************************)
+
+ Theorem lift_free: (t:?; h,k,d,e:?) (le e (plus d h)) -> (le d e) ->
+ (lift k e (lift h d t)) = (lift (plus k h) d t).
+ XElim t; Intros.
+(* case 1: TSort *)
+ Repeat Rewrite lift_sort; XAuto.
+(* case 2: TBRef n *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1: n < d *)
+ Repeat Rewrite lift_bref_lt; XEAuto.
+(* case 2.2: n >= d *)
+ Repeat Rewrite lift_bref_ge; XEAuto.
+(* case 3: TTail k *)
+ LiftTailRw; XAuto.
+ Qed.
+
+ Theorem lift_d : (t:?; h,k,d,e:?) (le e d) ->
+ (lift h (plus k d) (lift k e t)) = (lift k e (lift h d t)).
+ XElim t; Intros.
+(* case 1: TSort *)
+ Repeat Rewrite lift_sort; XAuto.
+(* case 2: TBRef n *)
+ Apply (lt_le_e n e); Intros.
+(* case 2.1: n < e *)
+ Cut (lt n d); Intros; Repeat Rewrite lift_bref_lt; XEAuto.
+(* case 2.2: n >= e *)
+ Rewrite lift_bref_ge; [ Idtac | XAuto ].
+ Rewrite plus_sym; Apply (lt_le_e n d); Intros.
+(* case 2.2.1: n < d *)
+ Do 2 (Rewrite lift_bref_lt; [ Idtac | XAuto ]).
+ Rewrite lift_bref_ge; XAuto.
+(* case 2.2.2: n >= d *)
+ Repeat Rewrite lift_bref_ge; XAuto.
+(* case 3: TTail k *)
+ LiftTailRw; SRw; XAuto.
+ Qed.
+
+ End lift_props.
--- /dev/null
+(*#* #stop file *)
+
+Require tlt_defs.
+Require lift_defs.
+
+ Hint discr : ltlc := Extern 4 (lt (weight_map (wadd ? ?) (lift (S ?) ? ?)) (wadd ? ? ?))
+ Simpl; Rewrite <- lift_weight_add_O.
+
+ Hint discr : ltlc := Extern 4 (lt (weight_map ? (lift (0) (0) ?)) ?)
+ Rewrite lift_r.
+
+ Section lift_tlt_props. (*************************************************)
+
+ Theorem lift_weight_map: (t:?; h,d:?; f:?)
+ ((m:?) (le d m) -> (f m)=(0)) ->
+ (weight_map f (lift h d t)) = (weight_map f t).
+ XElim t; Intros.
+(* case 1: TSort *)
+ XAuto.
+(* case 2: TBRef n *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1: n < d *)
+ Rewrite lift_bref_lt; XAuto.
+(* case 2.2: n >= d *)
+ Rewrite lift_bref_ge; [ Simpl | XAuto ].
+ Rewrite (H n); XAuto.
+(* case 3: TTail k *)
+ XElim k; Intros; LiftTailRw; Simpl.
+(* case 3.1: Bind *)
+ XElim b; [ Rewrite H; [ Idtac | XAuto ] | Idtac | Idtac ];
+ Rewrite H0; Intros; Try (LeLtGen; Rewrite H2; Simpl); XAuto.
+(* case 3.2: Flat *)
+ XAuto.
+ Qed.
+
+ Hints Resolve lift_weight_map : ltlc.
+
+ Theorem lift_weight : (t:?; h,d:?) (weight (lift h d t)) = (weight t).
+ Unfold weight; XAuto.
+ Qed.
+
+ Theorem lift_weight_add : (w:?; t:?; h,d:?; f,g:?)
+ ((m:?) (lt m d) -> (g m) = (f m)) ->
+ (g d) = w ->
+ ((m:?) (le d m) -> (g (S m)) = (f m)) ->
+ (weight_map f (lift h d t)) =
+ (weight_map g (lift (S h) d t)).
+ XElim t; Intros.
+(* case 1: TSort *)
+ XAuto.
+(* case 2: TBRef *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1: n < d *)
+ Repeat Rewrite lift_bref_lt; Simpl; XAuto.
+(* case 2.2: n >= d *)
+ Repeat Rewrite lift_bref_ge; Simpl; Try Rewrite <- plus_n_Sm; XAuto.
+(* case 3: TTail k *)
+ XElim k; Intros; LiftTailRw; Simpl.
+(* case 1 : bind b *)
+ XElim b; Simpl;
+ Apply (f_equal nat);
+ (Apply (f_equal2 nat nat); [ XAuto | Idtac ]);
+ ( Apply H0; Simpl; Intros; Try (LeLtGen; Rewrite H4; Simpl); XAuto).
+(* case 2 : Flat *)
+ XAuto.
+ Qed.
+
+ Theorem lift_weight_add_O: (w:?; t:?; h:?; f:?)
+ (weight_map f (lift h (0) t)) =
+ (weight_map (wadd f w) (lift (S h) (0) t)).
+ Intros.
+ EApply lift_weight_add; XAuto.
+ Intros; Inversion H.
+ Qed.
+
+ Theorem lift_tlt_dx: (k:?; u,t:?; h,d:?)
+ (tlt t (TTail k u (lift h d t))).
+ Unfold tlt; Intros.
+ Rewrite <- (lift_weight t h d).
+ Fold (tlt (lift h d t) (TTail k u (lift h d t))); XAuto.
+ Qed.
+
+ End lift_tlt_props.
+
+ Hints Resolve lift_tlt_dx : ltlc.
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+(*#* #stop file *)
+
+Require Export pr0_defs.
+
+ Inductive pc0 [t1,t2:T] : Prop :=
+ | pc0_r : (pr0 t1 t2) -> (pc0 t1 t2)
+ | pc0_x : (pr0 t2 t1) -> (pc0 t1 t2).
+
+ Hint pc0 : ltlc := Constructors pc0.
+
+ Inductive pc1 : T -> T -> Prop :=
+ | pc1_r : (t:?) (pc1 t t)
+ | pc1_u : (t2,t1:?) (pc0 t1 t2) -> (t3:?) (pc1 t2 t3) -> (pc1 t1 t3).
+
+ Hint pc1 : ltlc := Constructors pc1.
+
+ Section pc0_props. (******************************************************)
+
+ Theorem pc0_s : (t2,t1:?) (pc0 t1 t2) -> (pc0 t2 t1).
+ Intros.
+ Inversion H; XAuto.
+ Qed.
+
+ End pc0_props.
+
+ Hints Resolve pc0_s : ltlc.
+
+ Section pc1_props. (******************************************************)
+
+ Theorem pc1_pr0_r : (t1,t2:?) (pr0 t1 t2) -> (pc1 t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pc1_pr0_x : (t1,t2:?) (pr0 t2 t1) -> (pc1 t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pc1_pc0 : (t1,t2:?) (pc0 t1 t2) -> (pc1 t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pc1_t : (t2,t1:?) (pc1 t1 t2) ->
+ (t3:?) (pc1 t2 t3) -> (pc1 t1 t3).
+ Intros t2 t1 H; XElim H; XEAuto.
+ Qed.
+
+ Hints Resolve pc1_t : ltlc.
+
+ Theorem pc1_s : (t2,t1:?) (pc1 t1 t2) -> (pc1 t2 t1).
+ Intros; XElim H; XEAuto.
+ Qed.
+
+ Theorem pc1_tail_1: (u1,u2:?) (pc1 u1 u2) ->
+ (t:?; k:?) (pc1 (TTail k u1 t) (TTail k u2 t)).
+ Intros; XElim H; Intros.
+(* case 1: pc1_r *)
+ XAuto.
+(* case 2: pc1_u *)
+ XElim H; Intros; XEAuto.
+ Qed.
+
+ Theorem pc1_tail_2: (t1,t2:?) (pc1 t1 t2) ->
+ (u:?; k:?) (pc1 (TTail k u t1) (TTail k u t2)).
+ Intros; XElim H; Intros.
+(* case 1: pc1_r *)
+ XAuto.
+(* case 2: pc1_u *)
+ XElim H; Intros; XEAuto.
+ Qed.
+
+ Theorem pc1_tail: (u1,u2:?) (pc1 u1 u2) -> (t1,t2:?) (pc1 t1 t2) ->
+ (k:?) (pc1 (TTail k u1 t1) (TTail k u2 t2)).
+ Intros; EApply pc1_t; [ EApply pc1_tail_1 | EApply pc1_tail_2 ]; XAuto.
+ Qed.
+
+ End pc1_props.
+
+ Hints Resolve pc1_pr0_r pc1_pr0_x pc1_pc0 pc1_t pc1_s
+ pc1_tail_1 pc1_tail_2 pc1_tail : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require Export pr2_defs.
+Require Export pr3_defs.
+Require Export pc1_defs.
+
+ Inductive pc2 [c:C; t1,t2:T] : Prop :=
+ | pc2_r : (pr2 c t1 t2) -> (pc2 c t1 t2)
+ | pc2_x : (pr2 c t2 t1) -> (pc2 c t1 t2).
+
+ Hint pc2 : ltlc := Constructors pc2.
+
+ Inductive pc3 [c:C] : T -> T -> Prop :=
+ | pc3_r : (t:?) (pc3 c t t)
+ | pc3_u : (t2,t1:?) (pc2 c t1 t2) ->
+ (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
+
+ Hint pc3 : ltlc := Constructors pc3.
+
+ Section pc2_props. (******************************************************)
+
+ Theorem pc2_s : (c,t2,t1:?) (pc2 c t1 t2) -> (pc2 c t2 t1).
+ Intros.
+ Inversion H; XAuto.
+ Qed.
+
+ Theorem pc2_shift : (h:?; c,e:?) (drop h (0) c e) ->
+ (t1,t2:?) (pc2 c t1 t2) ->
+ (pc2 e (app c h t1) (app c h t2)).
+ Intros until 2; XElim H0; Intros.
+(* case 1 : pc2_r *)
+ XAuto.
+(* case 2 : pc2_x *)
+ XEAuto.
+ Qed.
+
+ End pc2_props.
+
+ Hints Resolve pc2_s pc2_shift : ltlc.
+
+ Section pc3_props. (******************************************************)
+
+ Theorem pc3_pr2_r : (c,t1,t2:?) (pr2 c t1 t2) -> (pc3 c t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pc3_pr2_x : (c,t1,t2:?) (pr2 c t2 t1) -> (pc3 c t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pc3_pc2 : (c,t1,t2:?) (pc2 c t1 t2) -> (pc3 c t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pc3_t : (t2,c,t1:?) (pc3 c t1 t2) ->
+ (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
+ Intros t2 c t1 H; XElim H; XEAuto.
+ Qed.
+
+ Hints Resolve pc3_t : ltlc.
+
+ Theorem pc3_s : (c,t2,t1:?) (pc3 c t1 t2) -> (pc3 c t2 t1).
+ Intros; XElim H; [ XAuto | XEAuto ].
+ Qed.
+
+ Hints Resolve pc3_s : ltlc.
+
+ Theorem pc3_pr3_r : (c:?; t1,t2) (pr3 c t1 t2) -> (pc3 c t1 t2).
+ Intros; XElim H; XEAuto.
+ Qed.
+
+ Theorem pc3_pr3_x : (c:?; t1,t2) (pr3 c t2 t1) -> (pc3 c t1 t2).
+ Intros; XElim H; XEAuto.
+ Qed.
+
+ Hints Resolve pc3_pr3_r pc3_pr3_x : ltlc.
+
+ Theorem pc3_pr3_t : (c:?; t1,t0:?) (pr3 c t1 t0) ->
+ (t2:?) (pr3 c t2 t0) -> (pc3 c t1 t2).
+ Intros; Apply (pc3_t t0); XAuto.
+ Qed.
+
+ Theorem pc3_thin_dx : (c:? ;t1,t2:?) (pc3 c t1 t2) ->
+ (u:?; f:?) (pc3 c (TTail (Flat f) u t1)
+ (TTail (Flat f) u t2)).
+ Intros; XElim H; [XAuto | Intros ].
+ EApply pc3_u; [ Inversion H | Apply H1 ]; XAuto.
+ Qed.
+
+ Theorem pc3_tail_1 : (c:?; u1,u2:?) (pc3 c u1 u2) ->
+ (k:?; t:?) (pc3 c (TTail k u1 t) (TTail k u2 t)).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pc3_r *)
+ XAuto.
+(* case 2 : pc3_u *)
+ EApply pc3_u; [ Inversion H | Apply H1 ]; XAuto.
+ Qed.
+
+ Theorem pc3_tail_2 : (c:?; u,t1,t2:?; k:?) (pc3 (CTail c k u) t1 t2) ->
+ (pc3 c (TTail k u t1) (TTail k u t2)).
+ Intros.
+ XElim H; [ Idtac | Intros; Inversion H ]; XEAuto.
+ Qed.
+
+ Theorem pc3_tail_12 : (c:?; u1,u2:?) (pc3 c u1 u2) ->
+ (k:?; t1,t2:?) (pc3 (CTail c k u2) t1 t2) ->
+ (pc3 c (TTail k u1 t1) (TTail k u2 t2)).
+ Intros.
+ EApply pc3_t; [ Apply pc3_tail_1 | Apply pc3_tail_2 ]; XAuto.
+ Qed.
+
+ Theorem pc3_tail_21 : (c:?; u1,u2:?) (pc3 c u1 u2) ->
+ (k:?; t1,t2:?) (pc3 (CTail c k u1) t1 t2) ->
+ (pc3 c (TTail k u1 t1) (TTail k u2 t2)).
+ Intros.
+ EApply pc3_t; [ Apply pc3_tail_2 | Apply pc3_tail_1 ]; XAuto.
+ Qed.
+
+ Theorem pc3_pr3_u : (c:?; t2,t1:?) (pr2 c t1 t2) ->
+ (t3:?) (pc3 c t2 t3) -> (pc3 c t1 t3).
+ XEAuto.
+ Qed.
+
+ Theorem pc3_pr3_u2 : (c:?; t0,t1:?) (pr2 c t0 t1) ->
+ (t2:?) (pc3 c t0 t2) -> (pc3 c t1 t2).
+ Intros; Apply (pc3_t t0); XAuto.
+ Qed.
+
+ Theorem pc3_shift : (h:?; c,e:?) (drop h (0) c e) ->
+ (t1,t2:?) (pc3 c t1 t2) ->
+ (pc3 e (app c h t1) (app c h t2)).
+ Intros until 2; XElim H0; Clear t1 t2; Intros.
+(* case 1 : pc3_r *)
+ XAuto.
+(* case 2 : pc3_u *)
+ XEAuto.
+ Qed.
+
+ Theorem pc3_pc1: (t1,t2:?) (pc1 t1 t2) -> (c:?) (pc3 c t1 t2).
+ Intros; XElim H; Intros.
+(* case 1: pc1_r *)
+ XAuto.
+(* case 2 : pc1_u *)
+ XElim H; XEAuto.
+ Qed.
+
+ End pc3_props.
+
+ Hints Resolve pc3_pr2_r pc3_pr2_x pc3_pc2 pc3_pr3_r pc3_pr3_x
+ pc3_t pc3_s pc3_pr3_t pc3_thin_dx pc3_tail_1 pc3_tail_2
+ pc3_tail_12 pc3_tail_21 pc3_pr3_u pc3_shift pc3_pc1 : ltlc.
+
+ Tactic Definition Pc3T :=
+ Match Context With
+ | [ _: (pr3 ?1 ?2 (TTail ?3 ?4 ?5)); _: (pc3 ?1 ?6 ?4) |- ? ] ->
+ LApply (pc3_t (TTail ?3 ?4 ?5) ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x (TTail ?3 ?6 ?5)); [ Clear H_x; Intros | Apply pc3_s; XAuto ]
+ | [ _: (pc3 ?1 ?2 ?3); _: (pr3 ?1 ?3 ?4) |- ? ] ->
+ LApply (pc3_t ?3 ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?4); [ Clear H_x; Intros | XAuto ].
--- /dev/null
+(*#* #stop file *)
+
+Require lift_gen.
+Require pr3_props.
+Require pr3_gen.
+Require pc3_defs.
+Require pc3_props.
+
+ Section pc3_gen. (********************************************************)
+
+ Theorem pc3_gen_abst : (c:?; u1,u2,t1,t2:?)
+ (pc3 c (TTail (Bind Abst) u1 t1)
+ (TTail (Bind Abst) u2 t2)
+ ) ->
+ (pc3 c u1 u2) /\
+ (b:?; u:?) (pc3 (CTail c (Bind b) u) t1 t2).
+ Intros.
+ Pc3Confluence; Pr3Gen; Pr3Gen; Rewrite H0 in H; Clear H0 x.
+ Inversion H; Rewrite H5 in H1; Rewrite H6 in H2.
+ Split; XEAuto.
+ Qed.
+
+ Theorem pc3_gen_lift: (c:?; t1,t2:?; h,d:?)
+ (pc3 c (lift h d t1) (lift h d t2)) ->
+ (e:?) (drop h d c e) ->
+ (pc3 e t1 t2).
+ Intros.
+ Pc3Confluence; Pr3Gen; Pr3Gen; Rewrite H1 in H; Clear H1 x.
+ LiftGen; Rewrite H in H2; XEAuto.
+ Qed.
+
+ Theorem pc3_gen_not_abst : (b:?) ~b=Abst -> (c:?; t1,t2,u1,u2:?)
+ (pc3 c (TTail (Bind b) u1 t1)
+ (TTail (Bind Abst) u2 t2)
+ ) ->
+ (pc3 (CTail c (Bind b) u1) t1
+ (lift (1) (0) (TTail (Bind Abst) u2 t2))
+ )
+ .
+ Intros b; XElim b; Intros;
+ Try EqFalse; Pc3Confluence; Pr3Gen; Pr3Gen;
+ Try (Rewrite H1 in H0; Inversion H0);
+ Rewrite H1 in H4; Pr3Context;
+ EApply pc3_pr3_t; XEAuto.
+ Qed.
+
+ Theorem pc3_gen_lift_abst : (c:?; t,t2,u2:?; h,d:?)
+ (pc3 c (lift h d t)
+ (TTail (Bind Abst) u2 t2)
+ ) ->
+ (e:?) (drop h d c e) ->
+ (EX u1 t1 | (pr3 e t (TTail (Bind Abst) u1 t1)) &
+ (pr3 c u2 (lift h d u1)) &
+ (b:B; u:T)(pr3 (CTail c (Bind b) u) t2 (lift h (S d) t1))
+ ).
+ Intros.
+ Pc3Confluence; Pr3Gen; Pr3Gen; Rewrite H1 in H; Clear H1 x.
+ LiftGenBase; Rewrite H in H4; Rewrite H1 in H2; Rewrite H5 in H3; XEAuto.
+ Qed.
+
+ End pc3_gen.
+
+ Tactic Definition Pc3Gen :=
+ Match Context With
+ | [ _: (pc3 ?1 (lift ?2 ?3 ?4) (lift ?2 ?3 ?5));
+ _: (drop ?2 ?3 ?1 ?6) |- ? ] ->
+ LApply (pc3_gen_lift ?1 ?4 ?5 ?2 ?3); [ Intros H_x | XAuto ];
+ LApply (H_x ?6); [ Clear H_x; Intros | XAuto ]
+ | [ H: (pc3 ?1 (TTail (Bind Abst) ?2 ?3) (TTail (Bind Abst) ?4 ?5)) |- ? ] ->
+ LApply (pc3_gen_abst ?1 ?2 ?4 ?3 ?5);[ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (pc3 ?1 (TTail (Bind ?2) ?3 ?4) (TTail (Bind Abst) ?5 ?6));
+ _: ~ ?2 = Abst |- ? ] ->
+ LApply (pc3_gen_not_abst ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?1 ?4 ?6 ?3 ?5); [ Clear H H_x; Intros | XAuto ]
+ | [ _: (pc3 ?1 (lift ?2 ?3 ?4) (TTail (Bind Abst) ?5 ?6));
+ _: (drop ?2 ?3 ?1 ?7) |- ? ] ->
+ LApply (pc3_gen_lift_abst ?1 ?4 ?6 ?5 ?2 ?3); [ Intros H_x | XAuto ];
+ LApply (H_x ?7); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros.
--- /dev/null
+(*#* #stop file *)
+
+Require subst1_confluence.
+Require csubst1_defs.
+Require pr3_gen_context.
+Require pc3_defs.
+Require pc3_props.
+
+ Section pc3_gen_context. (************************************************)
+
+ Theorem pc3_gen_cabbr: (c:?; t1,t2:?) (pc3 c t1 t2) -> (e:?; u:?; d:?)
+ (drop d (0) c (CTail e (Bind Abbr) u)) ->
+ (a0:?) (csubst1 d u c a0) ->
+ (a:?) (drop (1) d a0 a) ->
+ (x1:?) (subst1 d u t1 (lift (1) d x1)) ->
+ (x2:?) (subst1 d u t2 (lift (1) d x2)) ->
+ (pc3 a x1 x2).
+ Intros; Pc3Confluence; Repeat Pr3GenContext.
+ Subst1Confluence; Rewrite H in H3; Clear H x3; XEAuto.
+ Qed.
+
+ End pc3_gen_context.
--- /dev/null
+(*#* #stop file *)
+
+Require subst0_subst0.
+Require pr0_subst0.
+Require pr3_defs.
+Require pr3_props.
+Require pr3_confluence.
+Require cpr0_defs.
+Require cpr0_props.
+Require pc3_defs.
+
+ Section pc3_confluence. (*************************************************)
+
+ Theorem pc3_confluence : (c:?; t1,t2:?) (pc3 c t1 t2) ->
+ (EX t0 | (pr3 c t1 t0) & (pr3 c t2 t0)).
+ Intros; XElim H; Intros.
+(* case 1 : pc3_r *)
+ XEAuto.
+(* case 2 : pc3_u *)
+ Clear H0; XElim H1; Intros.
+ Inversion_clear H; [ XEAuto | Pr3Confluence; XEAuto ].
+ Qed.
+
+ End pc3_confluence.
+
+ Tactic Definition Pc3Confluence :=
+ Match Context With
+ [ H: (pc3 ?1 ?2 ?3) |- ? ] ->
+ LApply (pc3_confluence ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Section pc3_context. (****************************************************)
+
+ Theorem pc3_pr0_pr2_t : (u1,u2:?) (pr0 u2 u1) ->
+ (c:?; t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros.
+ Inversion H0; Clear H0; [ XAuto | NewInduction i ].
+(* case 1 : pr2_delta i = 0 *)
+ DropGenBase; Inversion H0; Clear H0 H3 H4 c k.
+ Rewrite H5 in H; Clear H5 u2.
+ Pr0Subst0; XEAuto.
+(* case 2 : pr2_delta i > 0 *)
+ NewInduction k; DropGenBase; XEAuto.
+ Qed.
+
+ Theorem pc3_pr2_pr2_t : (c:?; u1,u2:?) (pr2 c u2 u1) ->
+ (t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros until 1; Inversion H; Clear H; Intros.
+(* case 1 : pr2_pr0 *)
+ EApply pc3_pr0_pr2_t; [ Apply H0 | XAuto ].
+(* case 2 : pr2_delta *)
+ Inversion H; [ XAuto | NewInduction i0 ].
+(* case 2.1 : i0 = 0 *)
+ DropGenBase; Inversion H2; Clear H2.
+ Rewrite <- H5; Rewrite H6 in H; Rewrite <- H7 in H3; Clear H5 H6 H7 d0 k u0.
+ Subst0Subst0; Arith9'In H4 i.
+ EApply pc3_pr3_u.
+ EApply pr2_delta; XEAuto.
+ Apply pc3_pr2_x; EApply pr2_delta; [ Idtac | XEAuto ]; XEAuto.
+(* case 2.2 : i0 > 0 *)
+ Clear IHi0; NewInduction k; DropGenBase; XEAuto.
+ Qed.
+
+ Theorem pc3_pr2_pr3_t : (c:?; u2,t1,t2:?; k:?)
+ (pr3 (CTail c k u2) t1 t2) ->
+ (u1:?) (pr2 c u2 u1) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply pc3_t.
+ EApply pc3_pr2_pr2_t; [ Apply H2 | Apply H ].
+ XAuto.
+ Qed.
+
+ Theorem pc3_pr3_pc3_t : (c:?; u1,u2:?) (pr3 c u2 u1) ->
+ (t1,t2:?; k:?) (pc3 (CTail c k u2) t1 t2) ->
+ (pc3 (CTail c k u1) t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ Apply H1; Pc3Confluence.
+ EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_pr2_pr3_t; XEAuto.
+ Qed.
+
+ End pc3_context.
+
+ Tactic Definition Pc3Context :=
+ Match Context With
+ | [ H1: (pr0 ?3 ?2); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr0_pr2_t ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr0 ?3 ?2); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+ LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr2 ?1 ?3 ?2); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr2_pr2_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr2 ?1 ?3 ?2); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+ LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr3 ?1 ?3 ?2); H2: (pc3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pc3_pr3_pc3_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | _ -> Pr3Context.
+
+ Section pc3_lift. (*******************************************************)
+
+ Theorem pc3_lift : (c,e:?; h,d:?) (drop h d c e) ->
+ (t1,t2:?) (pc3 e t1 t2) ->
+ (pc3 c (lift h d t1) (lift h d t2)).
+
+ Intros.
+ Pc3Confluence.
+ EApply pc3_pr3_t; (EApply pr3_lift; [ XEAuto | Apply H0 Orelse Apply H1 ]).
+ Qed.
+
+ End pc3_lift.
+
+ Hints Resolve pc3_lift : ltlc.
+
+ Section pc3_cpr0. (*******************************************************)
+
+ Remark pc3_cpr0_t_aux : (c1,c2:?) (cpr0 c1 c2) ->
+ (k:?; u,t1,t2:?) (pr3 (CTail c1 k u) t1 t2) ->
+ (pc3 (CTail c2 k u) t1 t2).
+ Intros; XElim H0; Intros.
+(* case 1.1 : pr3_r *)
+ XAuto.
+(* case 1.2 : pr3_u *)
+ EApply pc3_t; [ Idtac | XEAuto ]. Clear H2 t1 t2.
+ Inversion_clear H0.
+(* case 1.2.1 : pr2_pr0 *)
+ XAuto.
+(* case 1.2.2 : pr2_delta *)
+ Cpr0Drop; Pr0Subst0.
+ EApply pc3_pr3_u; [ EApply pr2_delta; XEAuto | XAuto ].
+ Qed.
+
+ Theorem pc3_cpr0_t : (c1,c2:?) (cpr0 c1 c2) ->
+ (t1,t2:?) (pr3 c1 t1 t2) ->
+ (pc3 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : cpr0_refl *)
+ XAuto.
+(* case 2 : cpr0_cont *)
+ Pc3Context; Pc3Confluence.
+ EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_cpr0_t_aux; XEAuto.
+ Qed.
+
+ Theorem pc3_cpr0 : (c1,c2:?) (cpr0 c1 c2) -> (t1,t2:?) (pc3 c1 t1 t2) ->
+ (pc3 c2 t1 t2).
+ Intros; Pc3Confluence.
+ EApply pc3_t; [ Idtac | Apply pc3_s ]; EApply pc3_cpr0_t; XEAuto.
+ Qed.
+
+ End pc3_cpr0.
+
+ Hints Resolve pc3_cpr0 : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require subst0_subst0.
+Require csubst0_defs.
+Require pr0_subst0.
+Require pc3_defs.
+
+ Section pc3_fsubst0. (****************************************************)
+
+ Theorem pc3_pr2_fsubst0: (c1:?; t1,t:?) (pr2 c1 t1 t) ->
+ (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+ (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+ (pc3 c2 t2 t).
+ Intros until 1; XElim H.
+(* case 1: pr2_pr0 *)
+ Intros until 2; XElim H0; Intros.
+(* case 1.1: fsubst0_t *)
+ Pr0Subst0; [ XAuto | Apply (pc3_pr3_u c1 x); XEAuto ].
+(* case 1.2: fsubst0_c *)
+ XAuto.
+(* case 1.3: fsubst0_b *)
+ Pr0Subst0; CSubst0Drop; [ XAuto | Apply (pc3_pr3_u c0 x); XEAuto ].
+(* case 2 : pr2_delta *)
+ Intros until 3; XElim H1; Intros.
+(* case 2.1: fsubst0_t. *)
+ Apply (pc3_t t1); [ Apply pc3_s; XEAuto | XEAuto ].
+(* case 2.2: fsubst0_c. *)
+ Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.2.1: i < i0, none *)
+ XEAuto.
+(* case 2.2.2: i < i0, csubst0_fst *)
+ Inversion H; Rewrite <- H7 in H4; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Clear H H7 H8 H9 c2 t2 x0 x1 x2.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H6; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.3: i < i0, csubst0_snd *)
+ Inversion H; Rewrite <- H7 in H5; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H4; Clear H H7 H8 H9 c2 t2 x0 x1 x3.
+ Apply pc3_pr2_r; XEAuto.
+(* case 2.2.4: i < i0, csubst0_both *)
+ Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H4; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.5: i >= i0 *)
+ XEAuto.
+(* case 2.3: fsubst0_b *)
+ Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.3.1 : i < i0, none *)
+ CSubst0Drop; Apply pc3_pr3_u2 with t0 := t1; XEAuto.
+(* case 2.3.2 : i < i0, csubst0_fst *)
+ Inversion H; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H6; Clear H H8 H9 H10 c2 t2 x0 x1 x2.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u2 c0 t1); [ Idtac | Apply (pc3_pr3_u c0 x) ]; XEAuto.
+(* case 2.3.3: i < i0, csubst0_snd *)
+ Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+ CSubst0Drop; Apply (pc3_pr3_u2 c0 t1); [ Idtac | Apply pc3_pr2_r ]; XEAuto.
+(* case 2.3.4: i < i0, csubst0_both *)
+ Inversion H; Rewrite <- H9 in H7; Rewrite <- H10 in H5; Rewrite <- H10 in H6; Rewrite <- H10 in H7; Rewrite <- H11 in H6; Clear H H9 H10 H11 c2 t2 x0 x1 x3.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H8; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u2 c0 t1); [ Idtac | Apply (pc3_pr3_u c0 x) ]; XEAuto.
+(* case 2.3.5: i >= i0 *)
+ CSubst0Drop; Apply (pc3_pr3_u2 c0 t1); XEAuto.
+ Qed.
+
+ Theorem pc3_pr2_fsubst0_back: (c1:?; t,t1:?) (pr2 c1 t t1) ->
+ (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+ (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+ (pc3 c2 t t2).
+ Intros until 1; XElim H.
+(* case 1: pr2_pr0 *)
+ Intros until 2; XElim H0; Intros.
+(* case 1.1: fsubst0_t. *)
+ Apply (pc3_pr3_u c1 t1); XEAuto.
+(* case 1.2: fsubst0_c. *)
+ XAuto.
+(* case 1.3: fsubst0_c. *)
+ CSubst0Drop; Apply (pc3_pr3_u c0 t1); XEAuto.
+(* case 2: pr2_delta *)
+ Intros until 3; XElim H1; Intros.
+(* case 2.1: fsubst0_t. *)
+ Apply (pc3_t t1); XEAuto.
+(* case 2.2: fsubst0_c. *)
+ Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.2.1: i < i0, none *)
+ XEAuto.
+(* case 2.2.2: i < i0, csubst0_bind *)
+ Inversion H; Rewrite <- H7 in H4; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Clear H H7 H8 H9 c2 t2 x0 x1 x2.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H6; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.3: i < i0, csubst0_snd *)
+ Inversion H; Rewrite <- H7 in H5; Rewrite <- H8 in H4; Rewrite <- H8 in H5; Rewrite <- H9 in H4; Clear H H7 H8 H9 c2 t2 x0 x1 x3.
+ Apply pc3_pr2_r; XEAuto.
+(* case 2.2.4: i < i0, csubst0_both *)
+ Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H4; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u c0 x); XEAuto.
+(* case 2.2.5: i >= i0 *)
+ XEAuto.
+(* case 2.3: fsubst0_b *)
+ Apply (lt_le_e i i0); Intros; CSubst0Drop.
+(* case 2.3.1 : i < i0, none *)
+ CSubst0Drop; Apply pc3_pr3_u with t2 := t1; XEAuto.
+(* case 2.3.2 : i < i0, csubst0_fst *)
+ Inversion H; Rewrite <- H8 in H5; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H6; Clear H H8 H9 H10 c2 t2 x0 x1 x2.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H7; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u c0 x); [ Idtac | Apply (pc3_pr3_u2 c0 t1) ]; XEAuto.
+(* case 2.3.3: i < i0, csubst0_snd *)
+ Inversion H; Rewrite <- H8 in H6; Rewrite <- H9 in H5; Rewrite <- H9 in H6; Rewrite <- H10 in H5; Clear H H8 H9 H10 c2 t2 x0 x1 x3.
+ CSubst0Drop; Apply (pc3_pr3_u c0 t1); [ Idtac | Apply pc3_pr2_r ]; XEAuto.
+(* case 2.3.4: i < i0, csubst0_both *)
+ Inversion H; Rewrite <- H9 in H7; Rewrite <- H10 in H5; Rewrite <- H10 in H6; Rewrite <- H10 in H7; Rewrite <- H11 in H6; Clear H H9 H10 H11 c2 t2 x0 x1 x3.
+ Subst0Subst0; Rewrite <- lt_plus_minus_r in H8; [ CSubst0Drop | XAuto ].
+ Apply (pc3_pr3_u c0 x); [ Idtac | Apply (pc3_pr3_u2 c0 t1) ]; XEAuto.
+(* case 2.3.5: i >= i0 *)
+ CSubst0Drop; Apply (pc3_pr3_u c0 t1); XEAuto.
+ Qed.
+
+ Theorem pc3_pc2_fsubst0: (c1:?; t1,t:?) (pc2 c1 t1 t) ->
+ (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+ (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+ (pc3 c2 t2 t).
+ Intros until 1; XElim H; Intros.
+(* case 1: pc2_r *)
+ EApply pc3_pr2_fsubst0; XEAuto.
+(* case 2: pc2_x *)
+ Apply pc3_s; EApply pc3_pr2_fsubst0_back; XEAuto.
+ Qed.
+
+ Theorem pc3_fsubst0: (c1:?; t1,t:?) (pc3 c1 t1 t) ->
+ (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+ (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+ (pc3 c2 t2 t).
+ Intros until 1; XElim H.
+(* case 1: pc3_r *)
+ Intros until 1; XElim H; Intros; Try CSubst0Drop; XEAuto.
+(* case 2: pc3_u *)
+ Intros until 4; XElim H2; Intros;
+ (Apply (pc3_t t2); [ EApply pc3_pc2_fsubst0; XEAuto | XEAuto ]).
+ Qed.
+
+ End pc3_fsubst0.
+
+ Hints Resolve pc3_fsubst0.
--- /dev/null
+(*#* #stop file *)
+
+Require tlt_defs.
+Require lift_gen.
+Require lift_tlt.
+Require subst0_gen.
+Require subst0_confluence.
+Require pr0_defs.
+Require pr0_lift.
+Require pr0_gen.
+Require pr0_subst0.
+
+ Section pr0_confluence. (*************************************************)
+
+ Tactic Definition SSubstInv :=
+ Match Context With
+ | [ H0: (TTail ? ? ?) = (TTail ? ? ?) |- ? ] ->
+ Inversion H0; Clear H0
+ | [ H0: (pr0 (TTail (Bind ?) ? ?) ?) |- ? ] ->
+ Inversion H0; Clear H0
+ | _ -> EqFalse Orelse LiftGen Orelse Pr0Gen.
+
+ Tactic Definition SSubstBack :=
+ Match Context With
+ | [ H0: Abst = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1
+ | [ H0: Abbr = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1
+ | [ H0: (? ?) = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1
+ | [ H0: (? ? ? ?) = ?1; H1:? |- ? ] ->
+ Rewrite <- H0 in H1 Orelse Rewrite <- H0 Orelse Clear H0 ?1.
+
+ Tactic Definition SSubst :=
+ Match Context With
+ [ H0: ?1 = ?; H1:? |- ? ] ->
+ Rewrite H0 in H1 Orelse Rewrite H0 Orelse Clear H0 ?1.
+
+ Tactic Definition XSubst :=
+ Repeat (SSubstInv Orelse SSubstBack Orelse SSubst).
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H0: (pr0 ?1 ?2); H1: (pr0 ?1 ?3) |- ? ] ->
+ LApply (H ?1); [ Intros H_x | XEAuto ];
+ LApply (H_x ?2); [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x ?3); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Clear H0 H1; Intros.
+
+(* case pr0_cong pr0_gamma pr0_refl *****************************************)
+
+ Remark pr0_cong_gamma_refl: (b:?) ~ b = Abst ->
+ (u0,u3:?) (pr0 u0 u3) ->
+ (t4,t5:?) (pr0 t4 t5) ->
+ (u2,v2,x:?) (pr0 u2 x) -> (pr0 v2 x) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 (TTail (Bind b) u0 t4)) t) &
+ (pr0 (TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) v2) t5)) t)).
+ Intros.
+ Apply ex2_intro with x:=(TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) x) t5)); XAuto.
+ Qed.
+
+(* case pr0_cong pr0_gamma pr0_cong *****************************************)
+
+ Remark pr0_cong_gamma_cong: (b:?) ~ b = Abst ->
+ (u2,v2,x:?) (pr0 u2 x) -> (pr0 v2 x) ->
+ (t2,t5,x0:?) (pr0 t2 x0) -> (pr0 t5 x0) ->
+ (u5,u3,x1:?) (pr0 u5 x1) -> (pr0 u3 x1) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 (TTail (Bind b) u5 t2)) t) &
+ (pr0 (TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) v2) t5)) t)).
+ Intros.
+ Apply ex2_intro with x:=(TTail (Bind b) x1 (TTail (Flat Appl) (lift (1) (0) x) x0)); XAuto.
+ Qed.
+
+(* case pr0_cong pr0_gamma pr0_delta *****************************************)
+
+ Remark pr0_cong_gamma_delta: ~ Abbr = Abst ->
+ (u5,t2,w:?) (subst0 (0) u5 t2 w) ->
+ (u2,v2,x:?) (pr0 u2 x) -> (pr0 v2 x) ->
+ (t5,x0:?) (pr0 t2 x0) -> (pr0 t5 x0) ->
+ (u3,x1:?) (pr0 u5 x1) -> (pr0 u3 x1) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 (TTail (Bind Abbr) u5 w)) t) &
+ (pr0 (TTail (Bind Abbr) u3 (TTail (Flat Appl) (lift (1) (0) v2) t5)) t)).
+ Intros; Pr0Subst0.
+(* case 1: x0 is a lift *)
+ Apply ex2_intro with x:=(TTail (Bind Abbr) x1 (TTail (Flat Appl) (lift (1) (0) x) x0)); XAuto.
+(* case 2: x0 is not a lift *)
+ Apply ex2_intro with x:=(TTail (Bind Abbr) x1 (TTail (Flat Appl) (lift (1) (0) x) x2)); XEAuto.
+ Qed.
+
+(* case pr0_cong pr0_gamma pr0_zeta *****************************************)
+
+ Remark pr0_cong_gamma_zeta: (b:?) ~ b = Abst ->
+ (u0,u3:?) (pr0 u0 u3) ->
+ (u2,v2,x0:?) (pr0 u2 x0) -> (pr0 v2 x0) ->
+ (x,t3,x1:?) (pr0 x x1) -> (pr0 t3 x1) ->
+ (EX t:T | (pr0 (TTail (Flat Appl) u2 t3) t) &
+ (pr0 (TTail (Bind b) u3 (TTail (Flat Appl) (lift (1) (0) v2) (lift (1) (0) x))) t)).
+ Intros; LiftTailRwBack; XEAuto.
+ Qed.
+
+(* case pr0_cong pr0_delta **************************************************)
+
+ Remark pr0_cong_delta: (u3,t5,w:?) (subst0 (0) u3 t5 w) ->
+ (u2,x:?) (pr0 u2 x) -> (pr0 u3 x) ->
+ (t3,x0:?) (pr0 t3 x0) -> (pr0 t5 x0) ->
+ (EX t:T | (pr0 (TTail (Bind Abbr) u2 t3) t) &
+ (pr0 (TTail (Bind Abbr) u3 w) t)).
+ Intros; Pr0Subst0; XEAuto.
+ Qed.
+
+(* case pr0_gamma pr0_gamma *************************************************)
+
+ Remark pr0_gamma_gamma: (b:?) ~ b = Abst ->
+ (v1,v2,x0:?) (pr0 v1 x0) -> (pr0 v2 x0) ->
+ (u1,u2,x1:?) (pr0 u1 x1) -> (pr0 u2 x1) ->
+ (t1,t2,x2:?) (pr0 t1 x2) -> (pr0 t2 x2) ->
+ (EX t:T | (pr0 (TTail (Bind b) u1 (TTail (Flat Appl) (lift (1) (0) v1) t1)) t) &
+ (pr0 (TTail (Bind b) u2 (TTail (Flat Appl) (lift (1) (0) v2) t2)) t)).
+ Intros.
+ Apply ex2_intro with x:=(TTail (Bind b) x1 (TTail (Flat Appl) (lift (1) (0) x0) x2)); XAuto.
+ Qed.
+
+(* case pr0_delta pr0_delta *************************************************)
+
+ Remark pr0_delta_delta: (u2,t3,w:?) (subst0 (0) u2 t3 w) ->
+ (u3,t5,w0:?) (subst0 (0) u3 t5 w0) ->
+ (x:?) (pr0 u2 x) -> (pr0 u3 x) ->
+ (x0:?) (pr0 t3 x0) -> (pr0 t5 x0) ->
+ (EX t:T | (pr0 (TTail (Bind Abbr) u2 w) t) &
+ (pr0 (TTail (Bind Abbr) u3 w0) t)).
+ Intros; Pr0Subst0; Pr0Subst0; Try Subst0Confluence; XSubst; XEAuto.
+ Qed.
+
+(* case pr0_delta pr0_epsilon ***********************************************)
+
+ Remark pr0_delta_epsilon: (u2,t3,w:?) (subst0 (0) u2 t3 w) ->
+ (t4:?) (pr0 (lift (1) (0) t4) t3) ->
+ (t2:?) (EX t:T | (pr0 (TTail (Bind Abbr) u2 w) t) & (pr0 t2 t)).
+ Intros; Pr0Gen; XSubst; Subst0Gen.
+ Qed.
+
+(* main *********************************************************************)
+
+ Hints Resolve pr0_cong_gamma_refl pr0_cong_gamma_cong : ltlc.
+ Hints Resolve pr0_cong_gamma_delta pr0_cong_gamma_zeta : ltlc.
+ Hints Resolve pr0_cong_delta : ltlc.
+ Hints Resolve pr0_gamma_gamma : ltlc.
+ Hints Resolve pr0_delta_delta pr0_delta_epsilon : ltlc.
+
+(*#* #start proof *)
+
+(*#* #caption "confluence with itself: Church-Rosser property" *)
+(*#* #cap #cap t0, t1, t2, t *)
+
+ Theorem pr0_confluence : (t0,t1:?) (pr0 t0 t1) -> (t2:?) (pr0 t0 t2) ->
+ (EX t | (pr0 t1 t) & (pr0 t2 t)).
+
+(*#* #stop file *)
+
+ XElimUsing tlt_wf_ind t0; Intros.
+ Inversion H0; Inversion H1; Clear H0 H1;
+ XSubst; Repeat IH; XEAuto.
+ Qed.
+
+ End pr0_confluence.
+
+ Tactic Definition Pr0Confluence :=
+ Match Context With
+ [ H1: (pr0 ?1 ?2); H2: (pr0 ?1 ?3) |-? ] ->
+ LApply (pr0_confluence ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros.
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+(*#* #stop file *)
+
+Require Export subst0_defs.
+
+ Inductive pr0 : T -> T -> Prop :=
+(* structural rules *)
+ | pr0_refl : (t:?) (pr0 t t)
+ | pr0_thin : (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (k:?) (pr0 (TTail k u1 t1) (TTail k u2 t2))
+(* axiom rules *)
+ | pr0_beta : (k,v1,v2:?) (pr0 v1 v2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (pr0 (TTail (Flat Appl) v1 (TTail (Bind Abst) k t1))
+ (TTail (Bind Abbr) v2 t2))
+ | pr0_gamma : (b:?) ~b=Abst -> (v1,v2:?) (pr0 v1 v2) ->
+ (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (pr0 (TTail (Flat Appl) v1 (TTail (Bind b) u1 t1))
+ (TTail (Bind b) u2 (TTail (Flat Appl) (lift (1) (0) v2) t2)))
+ | pr0_delta : (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (w:?) (subst0 (0) u2 t2 w) ->
+ (pr0 (TTail (Bind Abbr) u1 t1) (TTail (Bind Abbr) u2 w))
+ | pr0_zeta : (b:?) ~b=Abst -> (t1,t2:?) (pr0 t1 t2) ->
+ (u:?) (pr0 (TTail (Bind b) u (lift (1) (0) t1)) t2)
+ | pr0_eps : (t1,t2:?) (pr0 t1 t2) ->
+ (u:?) (pr0 (TTail (Flat Cast) u t1) t2).
+
+ Hint pr0 : ltlc := Constructors pr0.
+
+ Section pr0_gen. (********************************************************)
+
+ Theorem pr0_gen_sort : (x:?; n:?) (pr0 (TSort n) x) -> x = (TSort n).
+ Intros; Inversion H; XAuto.
+ Qed.
+
+ Theorem pr0_gen_bref : (x:?; n:?) (pr0 (TBRef n) x) -> x = (TBRef n).
+ Intros; Inversion H; XAuto.
+ Qed.
+
+ Theorem pr0_gen_abst : (u1,t1,x:?) (pr0 (TTail (Bind Abst) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
+ (pr0 u1 u2) & (pr0 t1 t2)
+ ).
+
+ Intros; Inversion H; Clear H.
+(* case 1 : pr0_refl *)
+ XEAuto.
+(* case 2 : pr0_cont *)
+ XEAuto.
+(* case 3 : pr0_zeta *)
+ XElim H4; XAuto.
+ Qed.
+
+ Theorem pr0_gen_appl : (u1,t1,x:?) (pr0 (TTail (Flat Appl) u1 t1) x) -> (OR
+ (EX u2 t2 | x = (TTail (Flat Appl) u2 t2) &
+ (pr0 u1 u2) & (pr0 t1 t2)
+ ) |
+ (EX y1 z1 u2 t2 | t1 = (TTail (Bind Abst) y1 z1) &
+ x = (TTail (Bind Abbr) u2 t2) &
+ (pr0 u1 u2) & (pr0 z1 t2)
+ ) |
+ (EX b y1 z1 u2 v2 t2 |
+ ~b=Abst &
+ t1 = (TTail (Bind b) y1 z1) &
+ x = (TTail (Bind b) v2 (TTail (Flat Appl) (lift (1) (0) u2) t2)) &
+ (pr0 u1 u2) & (pr0 y1 v2) & (pr0 z1 t2))
+ ).
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ Theorem pr0_gen_cast : (u1,t1,x:?) (pr0 (TTail (Flat Cast) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Flat Cast) u2 t2) &
+ (pr0 u1 u2) & (pr0 t1 t2)
+ ) \/
+ (pr0 t1 x).
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ End pr0_gen.
+
+ Hints Resolve pr0_gen_sort pr0_gen_bref : ltlc.
+
+ Tactic Definition Pr0GenBase :=
+ Match Context With
+ | [ H: (pr0 (TTail (Bind Abst) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_abst ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (pr0 (TTail (Flat Appl) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_appl ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros H; XElim H; Intros
+ | [ H: (pr0 (TTail (Flat Cast) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_cast ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; [ Intros H; XElim H; Intros | Intros ].
--- /dev/null
+(*#* #stop file *)
+
+Require lift_gen.
+Require lift_props.
+Require subst0_gen.
+Require pr0_defs.
+Require pr0_lift.
+
+ Section pr0_gen_abbr. (***************************************************)
+
+ Theorem pr0_gen_abbr : (u1,t1,x:?) (pr0 (TTail (Bind Abbr) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abbr) u2 t2) &
+ (pr0 u1 u2) &
+ (pr0 t1 t2) \/
+ (EX y | (pr0 t1 y) & (subst0 (0) u2 y t2))
+ ) \/
+ (pr0 t1 (lift (1) (0) x)).
+ Intros.
+ Inversion H; Clear H; XDEAuto 6.
+ Qed.
+
+ End pr0_gen_abbr.
+
+ Section pr0_gen_void. (***************************************************)
+
+ Theorem pr0_gen_void : (u1,t1,x:?) (pr0 (TTail (Bind Void) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Void) u2 t2) &
+ (pr0 u1 u2) & (pr0 t1 t2)
+ ) \/
+ (pr0 t1 (lift (1) (0) x)).
+ Intros.
+ Inversion H; Clear H; XEAuto.
+ Qed.
+
+ End pr0_gen_void.
+
+ Section pr0_gen_lift. (***************************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H: (_:?; _:?) ?0 = (lift ? ? ?) -> ?;
+ H0: ?0 = (lift ? ?2 ?3) |- ? ] ->
+ LApply (H ?3 ?2); [ Clear H H0; Intros H_x | XAuto ];
+ XElim H_x; Intro; Intros H_x; Intro;
+ Try Rewrite H_x; Try Rewrite H_x in H3; Clear H_x.
+
+(*#* #start file *)
+
+(*#* #caption "generation lemma for lift" *)
+(*#* #cap #alpha t1 in U1, t2 in U2, x in T, d in i *)
+
+ Theorem pr0_gen_lift : (t1,x:?; h,d:?) (pr0 (lift h d t1) x) ->
+ (EX t2 | x = (lift h d t2) & (pr0 t1 t2)).
+
+(*#* #stop file *)
+
+ Intros until 1; InsertEq H '(lift h d t1);
+ UnIntro H d; UnIntro H t1; XElim H; Clear y x; Intros;
+ Rename x into t3; Rename x0 into d.
+(* case 1 : pr0_r *)
+ XEAuto.
+(* case 2 : pr0_c *)
+ NewInduction k; LiftGen; Rewrite H3; Clear H3 t0;
+ IH; IH; XEAuto.
+(* case 3 : pr0_beta *)
+ LiftGen; Rewrite H3; Clear H3 t0.
+ LiftGen; Rewrite H3; Clear H3 H5 x1 k.
+ IH; IH; XEAuto.
+(* case 4 : pr0_gamma *)
+ LiftGen; Rewrite H6; Clear H6 t0.
+ LiftGen; Rewrite H6; Clear H6 x1.
+ IH; IH; IH.
+ Rewrite <- lift_d; [ Simpl | XAuto ].
+ Rewrite <- lift_flat; XEAuto.
+(* case 5 : pr0_delta *)
+ LiftGen; Rewrite H4; Clear H4 t0.
+ IH; IH; Arith3In H3 d; Subst0Gen.
+ Rewrite H3; XEAuto.
+(* case 6 : pr0_zeta *)
+ LiftGen; Rewrite H2; Clear H2 t0.
+ Arith7In H4 d; LiftGen; Rewrite H2; Clear H2 x1.
+ IH; XEAuto.
+(* case 7 : pr0_zeta *)
+ LiftGen; Rewrite H1; Clear H1 t0.
+ IH; XEAuto.
+ Qed.
+
+ End pr0_gen_lift.
+
+ Tactic Definition Pr0Gen :=
+ Match Context With
+ | [ H: (pr0 (TTail (Bind Abbr) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_abbr ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H;
+ [ Intros H; XElim H; Do 4 Intro; Intros H_x;
+ XElim H_x; [ Intros | Intros H_x; XElim H_x; Intros ]
+ | Intros ]
+ | [ H: (pr0 (TTail (Bind Void) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_void ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; [ Intros H; XElim H; Intros | Intros ]
+ | [ H: (pr0 (lift ?0 ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_lift ?2 ?3 ?0 ?1); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | _ -> Pr0GenBase.
--- /dev/null
+Require lift_props.
+Require subst0_lift.
+Require pr0_defs.
+
+(*#* #caption "main properties of predicate \\texttt{pr0}" *)
+(*#* #clauses pr0_props *)
+
+ Section pr0_lift. (*******************************************************)
+
+(*#* #caption "conguence with lift" *)
+(*#* #cap #cap t1,t2 #alpha d in i *)
+
+ Theorem pr0_lift: (t1,t2:?) (pr0 t1 t2) ->
+ (h,d:?) (pr0 (lift h d t1) (lift h d t2)).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1: pr0_refl *)
+ XAuto.
+(* case 2: pr0_cong *)
+ LiftTailRw; XAuto.
+(* case 3 : pr0_beta *)
+ LiftTailRw; XAuto.
+(* case 4: pr0_gamma *)
+ LiftTailRw; Simpl; Arith0 d; Rewrite lift_d; XAuto.
+(* case 5: pr0_delta *)
+ LiftTailRw; Simpl.
+ EApply pr0_delta; [ XAuto | Apply H2 | Idtac ].
+ LetTac d' := (S d); Arith10 d; Unfold d'; XAuto.
+(* case 6: pr0_zeta *)
+ LiftTailRw; Simpl; Arith0 d; Rewrite lift_d; XAuto.
+(* case 7: pr0_epsilon *)
+ LiftTailRw; XAuto.
+ Qed.
+
+ End pr0_lift.
+
+ Hints Resolve pr0_lift : ltlc.
--- /dev/null
+Require subst0_gen.
+Require subst0_lift.
+Require subst0_subst0.
+Require subst0_confluence.
+Require pr0_defs.
+Require pr0_lift.
+
+(*#* #stop file *)
+
+ Section pr0_subst0. (*****************************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H1: (u1:?) (pr0 u1 ?1) -> ?; H2: (pr0 ?2 ?1) |- ? ] ->
+ LApply (H1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (u1:?) (pr0 ?1 u1) -> ?; H2: (pr0 ?1 ?2) |- ? ] ->
+ LApply (H1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (v1,w1:?; i:?) (subst0 i v1 ?1 w1) -> (v2:T) (pr0 v1 v2) -> ?;
+ H2: (subst0 ?2 ?3 ?1 ?4); H3: (pr0 ?3 ?5) |- ? ] ->
+ LApply (H1 ?3 ?4 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; [ Intros | Intros H1; XElim H1; Intros ].
+
+ Theorem pr0_subst0_back: (u2,t1,t2:?; i:?) (subst0 i u2 t1 t2) ->
+ (u1:?) (pr0 u1 u2) ->
+ (EX t | (subst0 i u1 t1 t) & (pr0 t t2)).
+ Intros until 1; XElim H; Intros;
+ Repeat IH; XEAuto.
+ Qed.
+
+ Theorem pr0_subst0_fwd: (u2,t1,t2:?; i:?) (subst0 i u2 t1 t2) ->
+ (u1:?) (pr0 u2 u1) ->
+ (EX t | (subst0 i u1 t1 t) & (pr0 t2 t)).
+ Intros until 1; XElim H; Intros;
+ Repeat IH; XEAuto.
+ Qed.
+
+ Hints Resolve pr0_subst0_fwd : ltlc.
+
+(*#* #start file *)
+
+(*#* #caption "confluence with substitution" *)
+(*#* #cap #cap t1,t2,v1,v2 #alpha w1 in U1, w2 in U2 *)
+
+ Theorem pr0_subst0: (t1,t2:?) (pr0 t1 t2) ->
+ (v1,w1:?; i:?) (subst0 i v1 t1 w1) ->
+ (v2:?) (pr0 v1 v2) ->
+ (pr0 w1 t2) \/
+ (EX w2 | (pr0 w1 w2) & (subst0 i v2 t2 w2)).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Clear t1 t2; Intros.
+(* case 1: pr0_refl *)
+ XEAuto.
+(* case 2: pr0_cong *)
+ Subst0Gen; Rewrite H3; Repeat IH; XEAuto.
+(* case 3: pr0_beta *)
+ Repeat Subst0Gen; Rewrite H3; Try Rewrite H5; Try Rewrite H6;
+ Repeat IH; XEAuto.
+(* case 4: pr0_gamma *)
+ Repeat Subst0Gen; Rewrite H6; Try Rewrite H8; Try Rewrite H9;
+ Repeat IH; XDEAuto 7.
+(* case 5: pr0_delta *)
+ Subst0Gen; Rewrite H4; Repeat IH;
+ [ XEAuto | Idtac | XEAuto | Idtac | XEAuto | Idtac | Idtac | Idtac ].
+ Subst0Subst0; Arith9'In H9 i; XEAuto.
+ Subst0Confluence; XEAuto.
+ Subst0Subst0; Arith9'In H10 i; XEAuto.
+ Subst0Confluence; XEAuto.
+ Subst0Subst0; Arith9'In H11 i; Subst0Confluence; XDEAuto 6.
+(* case 6: pr0_zeta *)
+ Repeat Subst0Gen; Rewrite H2; Try Rewrite H4; Try Rewrite H5;
+ Try (Simpl in H5; Rewrite <- minus_n_O in H5);
+ Try (Simpl in H6; Rewrite <- minus_n_O in H6);
+ Try IH; XEAuto.
+(* case 7: pr0_epsilon *)
+ Subst0Gen; Rewrite H1; Try IH; XEAuto.
+ Qed.
+
+ End pr0_subst0.
+
+ Tactic Definition Pr0Subst0 :=
+ Match Context With
+ | [ H1: (pr0 ?1 ?2); H2: (subst0 ?3 ?4 ?1 ?5);
+ H3: (pr0 ?4 ?6) |- ? ] ->
+ LApply (pr0_subst0 ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4 ?5 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ LApply (H1 ?6); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; [ Intros | Intros H1; XElim H1; Intros ]
+ | [ H1: (pr0 ?1 ?2); H2: (subst0 ?3 ?4 ?1 ?5) |- ? ] ->
+ LApply (pr0_subst0 ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4 ?5 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; [ Intros | Intros H1; XElim H1; Intros ]
+ | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (pr0 ?4 ?1) |- ? ] ->
+ LApply (pr0_subst0_back ?1 ?2 ?3 ?0); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (pr0 ?1 ?4) |- ? ] ->
+ LApply (pr0_subst0_fwd ?1 ?2 ?3 ?0); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros.
--- /dev/null
+(*#* #stop file *)
+
+Require subst1_defs.
+Require pr0_defs.
+Require pr0_subst0.
+
+ Section pr0_subst1_props. (***********************************************)
+
+ Theorem pr0_delta1: (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (w:?) (subst1 (0) u2 t2 w) ->
+ (pr0 (TTail (Bind Abbr) u1 t1) (TTail (Bind Abbr) u2 w)).
+ Intros until 3; XElim H1; Clear w; XEAuto.
+ Qed.
+
+ Theorem pr0_subst1_back: (u2,t1,t2:?; i:?) (subst1 i u2 t1 t2) ->
+ (u1:?) (pr0 u1 u2) ->
+ (EX t | (subst1 i u1 t1 t) & (pr0 t t2)).
+ Intros until 1; XElim H; Intros;
+ Try Pr0Subst0; XEAuto.
+ Qed.
+
+ Theorem pr0_subst1_fwd: (u2,t1,t2:?; i:?) (subst1 i u2 t1 t2) ->
+ (u1:?) (pr0 u2 u1) ->
+ (EX t | (subst1 i u1 t1 t) & (pr0 t2 t)).
+ Intros until 1; XElim H; Intros;
+ Try Pr0Subst0; XEAuto.
+ Qed.
+
+ Theorem pr0_subst1: (t1,t2:?) (pr0 t1 t2) ->
+ (v1,w1:?; i:?) (subst1 i v1 t1 w1) ->
+ (v2:?) (pr0 v1 v2) ->
+ (EX w2 | (pr0 w1 w2) & (subst1 i v2 t2 w2)).
+ Intros until 2; XElim H0; Intros;
+ Try Pr0Subst0; XEAuto.
+ Qed.
+
+ End pr0_subst1_props.
+
+ Hints Resolve pr0_delta1 : ltlc.
+
+ Tactic Definition Pr0Subst1 :=
+ Match Context With
+ | [ H1: (pr0 ?1 ?2); H2: (subst1 ?3 ?4 ?1 ?5);
+ H3: (pr0 ?4 ?6) |- ? ] ->
+ LApply (pr0_subst1 ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4 ?5 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ LApply (H1 ?6); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (pr0 ?1 ?2); H2: (subst1 ?3 ?4 ?1 ?5) |- ? ] ->
+ LApply (pr0_subst1 ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4 ?5 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (pr0 ?4 ?1) |- ? ] ->
+ LApply (pr0_subst1_back ?1 ?2 ?3 ?0); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (pr0 ?1 ?4) |- ? ] ->
+ LApply (pr0_subst1_fwd ?1 ?2 ?3 ?0); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | _ -> Pr0Subst0.
+
--- /dev/null
+Require pr0_confluence.
+Require pr1_defs.
+
+(*#* #caption "main properties of predicate \\texttt{pr1}" *)
+(*#* #clauses pr1_props *)
+
+ Section pr1_confluence. (*************************************************)
+
+(*#* #caption "confluence with single step reduction: strip lemma" *)
+(*#* #cap #cap t0, t1, t2, t *)
+
+ Theorem pr1_strip : (t0,t1:?) (pr1 t0 t1) -> (t2:?) (pr0 t0 t2) ->
+ (EX t | (pr1 t1 t) & (pr1 t2 t)).
+
+(*#* #stop proof *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr1_r *)
+ XEAuto.
+(* case 2 : pr1_u *)
+ Pr0Confluence.
+ LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
+ XElim H1; Intros; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "confluence with itself: Church-Rosser property" *)
+(*#* #cap #cap t0, t1, t2, t *)
+
+ Theorem pr1_confluence : (t0,t1:?) (pr1 t0 t1) -> (t2:?) (pr1 t0 t2) ->
+ (EX t | (pr1 t1 t) & (pr1 t2 t)).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr1_r *)
+ XEAuto.
+(* case 2 : pr1_u *)
+ LApply (pr1_strip t3 t5); [ Clear H2; Intros H2 | XAuto ].
+ LApply (H2 t2); [ Clear H H2; Intros H | XAuto ].
+ XElim H; Intros.
+ LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
+ XElim H1; Intros; XEAuto.
+ Qed.
+
+ End pr1_confluence.
+
+ Tactic Definition Pr1Confluence :=
+ Match Context With
+ | [ H1: (pr1 ?1 ?2); H2: (pr0 ?1 ?3) |-? ] ->
+ LApply (pr1_strip ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (pr1 ?1 ?2); H2: (pr1 ?1 ?3) |-? ] ->
+ LApply (pr1_confluence ?1 ?2); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?3); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | _ -> Pr0Confluence.
+
+(*#* #start file *)
+
+(*#* #single *)
+
--- /dev/null
+(*#* #stop file *)
+
+Require Export pr0_defs.
+
+ Inductive pr1 : T -> T -> Prop :=
+ | pr1_r : (t:?) (pr1 t t)
+ | pr1_u : (t2,t1:?) (pr0 t1 t2) -> (t3:?) (pr1 t2 t3) -> (pr1 t1 t3).
+
+ Hint pr1 : ltlc := Constructors pr1.
+
+ Section pr1_props. (******************************************************)
+
+ Theorem pr1_pr0 : (t1,t2:?) (pr0 t1 t2) -> (pr1 t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pr1_t : (t2,t1:?) (pr1 t1 t2) ->
+ (t3:?) (pr1 t2 t3) -> (pr1 t1 t3).
+ Intros t2 t1 H; XElim H; XEAuto.
+ Qed.
+
+ End pr1_props.
+
+ Hints Resolve pr1_pr0 pr1_t : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require subst0_confluence.
+Require drop_props.
+Require pr0_subst0.
+Require pr0_confluence.
+Require pr2_defs.
+
+ Section pr2_confluence. (*************************************************)
+
+(* case 1.1 : pr2_pr0 pr2_pr0 ***********************************************)
+
+ Remark pr2_pr0_pr0: (c:?; t0,t1,t2:?)
+ (pr0 t0 t1) -> (pr0 t0 t2) ->
+ (EX t:T | (pr2 c t1 t) & (pr2 c t2 t)).
+ Intros; Pr0Confluence; XEAuto.
+ Qed.
+
+(* case 1.2 : pr2_pr0 pr2_delta *********************************************)
+
+ Remark pr2_pr0_delta: (c,d:?; t0,t1,t2,u:?; i:?)
+ (pr0 t0 t1) ->
+ (drop i (0) c (CTail d (Bind Abbr) u)) ->
+ (subst0 i u t0 t2) ->
+ (EX t | (pr2 c t1 t) & (pr2 c t2 t)).
+ Intros; Pr0Subst0; XEAuto.
+ Qed.
+
+(* case 2.2 : pr2_delta pr2_delta *******************************************)
+
+ Remark pr2_delta_delta: (c,d,d0:?; t0,t1,t2,u,u0:?; i,i0:?)
+ (drop i (0) c (CTail d (Bind Abbr) u)) ->
+ (subst0 i u t0 t1) ->
+ (drop i0 (0) c (CTail d0 (Bind Abbr) u0)) ->
+ (subst0 i0 u0 t0 t2) ->
+ (EX t:T | (pr2 c t1 t) & (pr2 c t2 t)).
+ Intros.
+ Apply (neq_eq_e i i0); Intros.
+(* case 1 : i != i0 *)
+ Subst0Confluence; XEAuto.
+(* case 2 : i = i0 *)
+ Rewrite H3 in H; Rewrite H3 in H0; Clear H3 i.
+ DropDis; Inversion H1; Rewrite H5 in H0; Clear H1 H4 H5 d u.
+ Subst0Confluence; [ Rewrite H1; XEAuto | XEAuto | XEAuto | XEAuto ].
+ Qed.
+
+(* main *********************************************************************)
+
+ Hints Resolve pr2_pr0_pr0 pr2_pr0_delta pr2_delta_delta : ltlc.
+
+(*#* #start file *)
+
+(*#* #caption "confluence with itself: Church-Rosser property" *)
+(*#* #cap #cap c, t0, t1, t2, t *)
+
+ Theorem pr2_confluence : (c,t0,t1:?) (pr2 c t0 t1) ->
+ (t2:?) (pr2 c t0 t2) ->
+ (EX t | (pr2 c t1 t) & (pr2 c t2 t)).
+
+(*#* #stop file *)
+
+ Intros; Inversion H; Inversion H0; XEAuto.
+ Qed.
+
+ End pr2_confluence.
+
+ Tactic Definition Pr2Confluence :=
+ Match Context With
+ | [ H1: (pr2 ?1 ?2 ?3); H2: (pr2 ?1 ?2 ?4) |-? ] ->
+ LApply (pr2_confluence ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | _ -> Pr0Confluence.
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+(*#* #stop file *)
+
+Require Export drop_defs.
+Require Export pr0_defs.
+
+ Inductive pr2 [c:C; t1,t2:T] : Prop :=
+(* structural rules *)
+ | pr2_pr0 : (pr0 t1 t2) -> (pr2 c t1 t2)
+(* axiom rules *)
+ | pr2_delta : (d:?; u:?; i:?)
+ (drop i (0) c (CTail d (Bind Abbr) u)) ->
+ (subst0 i u t1 t2) -> (pr2 c t1 t2).
+
+ Hint pr2 : ltlc := Constructors pr2.
+
+ Section pr2_gen_base. (***************************************************)
+
+ Theorem pr2_gen_sort : (c:?; x:?; n:?) (pr2 c (TSort n) x) ->
+ x = (TSort n).
+ Intros; Inversion H;
+ Try Subst0GenBase; XEAuto.
+ Qed.
+
+ Theorem pr2_gen_bref : (c:?; x:?; n:?) (pr2 c (TBRef n) x) ->
+ x = (TBRef n) \/
+ (EX d u | (drop n (0) c (CTail d (Bind Abbr) u)) &
+ x = (lift (S n) (0) u)
+ ).
+ Intros; Inversion H;
+ Try Subst0GenBase; Try Rewrite <- H1 in H0; XEAuto.
+ Qed.
+
+ Theorem pr2_gen_abst : (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Bind Abst) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
+ (pr2 c u1 u2) & (b:?; u:?)
+ (pr2 (CTail c (Bind b) u) t1 t2)
+ ).
+ Intros; Inversion H;
+ Try Pr0GenBase; Try Subst0GenBase; XDEAuto 6.
+ Qed.
+
+ Theorem pr2_gen_appl : (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Flat Appl) u1 t1) x) -> (OR
+ (EX u2 t2 | x = (TTail (Flat Appl) u2 t2) &
+ (pr2 c u1 u2) & (pr2 c t1 t2)
+ ) |
+ (EX y1 z1 u2 t2 | t1 = (TTail (Bind Abst) y1 z1) &
+ x = (TTail (Bind Abbr) u2 t2) &
+ (pr0 u1 u2) & (pr0 z1 t2)
+ ) |
+ (EX b y1 z1 u2 v2 t2 |
+ ~b=Abst &
+ t1 = (TTail (Bind b) y1 z1) &
+ x = (TTail (Bind b) v2 (TTail (Flat Appl) (lift (1) (0) u2) t2)) &
+ (pr0 u1 u2) & (pr0 y1 v2) & (pr0 z1 t2))
+ ).
+ Intros; Inversion H;
+ Try Pr0GenBase; Try Subst0GenBase; XEAuto.
+ Qed.
+
+ Theorem pr2_gen_cast : (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Flat Cast) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Flat Cast) u2 t2) &
+ (pr2 c u1 u2) & (pr2 c t1 t2)
+ ) \/
+ (pr0 t1 x).
+ Intros; Inversion H;
+ Try Pr0GenBase; Try Subst0GenBase; XEAuto.
+ Qed.
+
+ End pr2_gen_base.
+
+ Tactic Definition Pr2GenBase :=
+ Match Context With
+ | [ H: (pr2 ?1 (TTail (Bind Abst) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr2_gen_abst ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Section pr2_props. (******************************************************)
+
+ Theorem pr2_thin_dx : (c:?; t1,t2:?) (pr2 c t1 t2) ->
+ (u:?; f:?) (pr2 c (TTail (Flat f) u t1)
+ (TTail (Flat f) u t2)).
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ Theorem pr2_tail_1 : (c:?; u1,u2:?) (pr2 c u1 u2) ->
+ (k:?; t:?) (pr2 c (TTail k u1 t) (TTail k u2 t)).
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ Theorem pr2_tail_2 : (c:?; u,t1,t2:?; k:?) (pr2 (CTail c k u) t1 t2) ->
+ (pr2 c (TTail k u t1) (TTail k u t2)).
+ XElim k; Intros;
+ (Inversion H; [ XAuto | Clear H ];
+ (NewInduction i; DropGenBase; [ Inversion H; XEAuto | XEAuto ])).
+ Qed.
+
+ Hints Resolve pr2_tail_2 : ltlc.
+
+ Theorem pr2_shift : (i:?; c,e:?) (drop i (0) c e) ->
+ (t1,t2:?) (pr2 c t1 t2) ->
+ (pr2 e (app c i t1) (app c i t2)).
+ XElim i.
+(* case 1 : i = 0 *)
+ Intros.
+ DropGenBase; Rewrite H in H0.
+ Repeat Rewrite app_O; XAuto.
+(* case 2 : i > 0 *)
+ XElim c.
+(* case 2.1 : CSort *)
+ Intros; DropGenBase; Rewrite H0; XAuto.
+(* case 2.2 : CTail *)
+ XElim k; Intros; Simpl; DropGenBase; XAuto.
+ Qed.
+
+ End pr2_props.
+
+ Hints Resolve pr2_thin_dx pr2_tail_1 pr2_tail_2 pr2_shift : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require subst0_gen.
+Require drop_props.
+Require pr0_gen.
+Require pr2_defs.
+
+ Section pr2_gen. (********************************************************)
+
+ Theorem pr2_gen_abbr : (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Bind Abbr) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abbr) u2 t2) &
+ (pr2 c u1 u2) &
+ ((b:?; u:?) (pr2 (CTail c (Bind b) u) t1 t2)) \/
+ (EX y | (pr0 t1 y) & (subst0 (0) u2 y t2))
+ ) \/
+ (pr0 t1 (lift (1) (0) x)).
+ Intros; Inversion H;
+ Try Pr0Gen; Try Subst0Gen; XDEAuto 8.
+ Qed.
+
+ Theorem pr2_gen_void : (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Bind Void) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Void) u2 t2) &
+ (pr2 c u1 u2) & (b:?; u:?)
+ (pr2 (CTail c (Bind b) u) t1 t2)
+ ) \/
+ (pr0 t1 (lift (1) (0) x)).
+ Intros; Inversion H;
+ Try Pr0Gen; Try Subst0Gen; XDEAuto 7.
+ Qed.
+
+(*#* #start file *)
+
+(*#* #caption "generation lemma for lift" *)
+(*#* #cap #cap c #alpha e in D, t1 in U1, t2 in U2, x in T, d in i *)
+
+ Theorem pr2_gen_lift : (c:?; t1,x:?; h,d:?) (pr2 c (lift h d t1) x) ->
+ (e:?) (drop h d c e) ->
+ (EX t2 | x = (lift h d t2) & (pr2 e t1 t2)).
+
+(*#* #stop file *)
+
+ Intros.
+ Inversion H; Clear H.
+(* case 1 : pr2_pr0 *)
+ Pr0Gen; XEAuto.
+(* case 2 : pr2_delta *)
+ Apply (lt_le_e i d); Intros.
+(* case 2.1 : i < d *)
+ Rewrite (lt_plus_minus i d) in H0; [ Idtac | XAuto ].
+ Rewrite (lt_plus_minus i d) in H2; [ Idtac | XAuto ].
+ DropDis; Rewrite H0 in H2; Clear H0 u.
+ Subst0Gen; Rewrite <- lt_plus_minus in H0; XEAuto.
+(* case 2.2 : i >= d *)
+ Apply (lt_le_e i (plus d h)); Intros.
+(* case 2.2.1 : i < d + h *)
+ EApply subst0_gen_lift_false; [ Apply H | Apply H3 | XEAuto ].
+(* case 2.2.2 : i >= d + h *)
+ DropDis; Subst0Gen; XEAuto.
+ Qed.
+
+ End pr2_gen.
+
+ Tactic Definition Pr2Gen :=
+ Match Context With
+ | [ H: (pr2 ?1 (TTail (Bind Abbr) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr2_gen_abbr ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H;
+ [ Intros H; XElim H; Do 4 Intro; Intros H_x; XElim H_x;
+ [ Intros | Intros H_x; XElim H_x; Intros ]
+ | Intros ]
+ | [ H: (pr2 ?1 (TTail (Bind Void) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr2_gen_void ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H; [ Intros H; XElim H; Intros | Intros ]
+ | [ H0: (pr2 ?1 (lift ?2 ?3 ?4) ?5);
+ H1: (drop ?2 ?3 ?1 ?6) |- ? ] ->
+ LApply (pr2_gen_lift ?1 ?4 ?5 ?2 ?3); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?6); [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | _ -> Pr2GenBase.
--- /dev/null
+(*#* #stop file *)
+
+Require drop_props.
+Require subst1_gen.
+Require subst1_subst1.
+Require subst1_confluence.
+Require csubst1_defs.
+Require pr0_gen.
+Require pr0_subst1.
+Require pr2_defs.
+Require pr2_subst1.
+
+ Section pr2_gen_context. (************************************************)
+
+ Theorem pr2_gen_cabbr: (c:?; t1,t2:?) (pr2 c t1 t2) -> (e:?; u:?; d:?)
+ (drop d (0) c (CTail e (Bind Abbr) u)) ->
+ (a0:?) (csubst1 d u c a0) ->
+ (a:?) (drop (1) d a0 a) ->
+ (x1:?) (subst1 d u t1 (lift (1) d x1)) ->
+ (EX x2 | (subst1 d u t2 (lift (1) d x2)) &
+ (pr2 a x1 x2)
+ ).
+ Intros until 1; XElim H; Intros.
+(* case 1: pr2_pr0 *)
+ Pr0Subst1; Pr0Gen; Rewrite H in H3; Clear H x; XEAuto.
+(* case 2: pr2_delta *)
+ Apply (lt_eq_gt_e i d0); Intros.
+(* case 2.1: i < d0 *)
+ Subst1Confluence; CSubst1Drop.
+ Rewrite minus_x_Sy in H; [ Idtac | XAuto ].
+ CSubst1GenBase; Rewrite H in H6; Clear H x0.
+ Rewrite (lt_plus_minus i d0) in H3; [ Idtac | XAuto ].
+ DropDis; Rewrite H in H7; Clear H x2.
+ Subst1Subst1; Pattern 2 d0 in H; Rewrite (lt_plus_minus i d0) in H; [ Idtac | XAuto ].
+ Subst1Gen; Rewrite H in H9; Simpl in H9; Clear H x2.
+ Rewrite <- lt_plus_minus in H9; [ Idtac | XAuto ].
+ Rewrite <- lt_plus_minus_r in H9; XEAuto.
+(* case 2.2: i = d0 *)
+ Rewrite H5 in H; Rewrite H5 in H0; Clear H5 i.
+ DropDis; Inversion H; Rewrite <- H7 in H0; Rewrite <- H7 in H1; Rewrite <- H7; Clear H H6 H7 e u.
+ Subst1Confluence; Subst1Gen; Rewrite H0 in H; Clear H0 x; XEAuto.
+(* case 2.3: i > d0 *)
+ Subst1Confluence; Subst1Gen; Rewrite H4 in H0; Clear H1 H4 x.
+ CSubst1Drop; DropDis; XEAuto.
+ Qed.
+
+ End pr2_gen_context.
+
+ Tactic Definition Pr2GenContext :=
+ Match Context With
+ | [ H0: (pr2 ?1 ?2 ?3); H1: (drop ?4 (0) ?1 (CTail ?5 (Bind Abbr) ?6));
+ H2: (csubst1 ?4 ?6 ?1 ?7); H3: (drop (1) ?4 ?7 ?8);
+ H4: (subst1 ?4 ?6 ?2 (lift (1) ?4 ?9)) |- ? ] ->
+ LApply (pr2_gen_cabbr ?1 ?2 ?3); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?5 ?6 ?4); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?8); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?9); [ Clear H0 H4; Intros H0 | XAuto ];
+ XElim H0; Intros.
--- /dev/null
+Require subst0_lift.
+Require drop_props.
+Require pr0_lift.
+Require pr2_defs.
+
+(*#* #caption "main properties of predicate \\texttt{pr2}" *)
+(*#* #clauses pr2_props *)
+
+ Section pr2_lift. (*******************************************************)
+
+(*#* #caption "conguence with lift" *)
+(*#* #cap #cap c, t1, t2 #alpha e in D, d in i *)
+
+ Theorem pr2_lift : (c,e:?; h,d:?) (drop h d c e) ->
+ (t1,t2:?) (pr2 e t1 t2) ->
+ (pr2 c (lift h d t1) (lift h d t2)).
+
+(*#* #stop file *)
+
+ Intros until 2; XElim H0; Intros.
+(* case 1 : pr2_pr0 *)
+ XAuto.
+(* case 2 : pr2_delta *)
+ Apply (lt_le_e i d); Intros; DropDis.
+(* case 2.1 : i < d *)
+ Rewrite minus_x_Sy in H0; [ Idtac | XAuto ].
+ DropGenBase; Rewrite H0 in H; Simpl in H; XEAuto.
+(* case 2.2 : i >= d *)
+ XEAuto.
+ Qed.
+
+ End pr2_lift.
+
+ Hints Resolve pr2_lift : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require subst1_defs.
+Require subst1_confluence.
+Require drop_props.
+Require pr0_subst1.
+Require pr2_defs.
+
+ Section pr2_subst1_props. (***********************************************)
+
+ Theorem pr2_delta1: (c,d:?; u:?; i:?) (drop i (0) c (CTail d (Bind Abbr) u)) ->
+ (t1,t2:?) (subst1 i u t1 t2) -> (pr2 c t1 t2).
+ Intros; XElim H0; Clear t2; XEAuto.
+ Qed.
+
+ Hints Resolve pr2_delta1 : ltlc.
+
+ Theorem pr2_subst1: (c,e:?; v:?; i:?) (drop i (0) c (CTail e (Bind Abbr) v)) ->
+ (t1,t2:?) (pr2 c t1 t2) ->
+ (w1:?) (subst1 i v t1 w1) ->
+ (EX w2 | (pr2 c w1 w2) & (subst1 i v t2 w2)).
+ Intros until 2; XElim H0; Intros.
+(* case 1: pr2_pr0 *)
+ Pr0Subst1; XEAuto.
+(* case 2: pr2_delta *)
+ Apply (neq_eq_e i i0); Intros.
+(* case 2.1: i <> i0 *)
+ Subst1Confluence; XEAuto.
+(* case 2.2: i = i0 *)
+ Rewrite <- H3 in H0; Rewrite <- H3 in H1; Clear H3 i0.
+ DropDis; Inversion H0; Rewrite H5 in H2; Clear H0 H4 H5 e v.
+ Subst1Confluence; XEAuto.
+ Qed.
+
+ End pr2_subst1_props.
+
+ Hints Resolve pr2_delta1 : ltlc.
+
+ Tactic Definition Pr2Subst1 :=
+ Match Context With
+ | [ H0: (drop ?1 (0) ?2 (CTail ?3 (Bind Abbr) ?4));
+ H1: (pr2 ?2 ?5 ?6); H3: (subst1 ?1 ?4 ?5 ?7) |- ? ] ->
+ LApply (pr2_subst1 ?2 ?3 ?4 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?5 ?6); [ Clear H_x H1; Intros H1 | XAuto ];
+ LApply (H1 ?7); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros.
--- /dev/null
+Require pr2_confluence.
+Require pr3_defs.
+
+ Section pr3_confluence. (*************************************************)
+
+(*#* #caption "confluence with single step reduction: strip lemma" *)
+(*#* #cap #cap c, t0, t1, t2, t *)
+
+ Theorem pr3_strip : (c:?; t0,t1:?) (pr3 c t0 t1) -> (t2:?) (pr2 c t0 t2) ->
+ (EX t | (pr3 c t1 t) & (pr3 c t2 t)).
+
+(*#* #stop proof *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XEAuto.
+(* case 2 : pr3_u *)
+ Pr2Confluence.
+ LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
+ XElim H1; Intros; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "confluence with itself: Church-Rosser property" *)
+(*#* #cap #cap c, t0, t1, t2, t *)
+
+ Theorem pr3_confluence : (c:?; t0,t1:?) (pr3 c t0 t1) -> (t2:?) (pr3 c t0 t2) ->
+ (EX t | (pr3 c t1 t) & (pr3 c t2 t)).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XEAuto.
+(* case 2 : pr3_u *)
+ LApply (pr3_strip c t3 t5); [ Clear H2; Intros H2 | XAuto ].
+ LApply (H2 t2); [ Clear H H2; Intros H | XAuto ].
+ XElim H; Intros.
+ LApply (H1 x); [ Clear H1 H2; Intros H1 | XAuto ].
+ XElim H1; Intros; XEAuto.
+ Qed.
+
+ End pr3_confluence.
+
+ Tactic Definition Pr3Confluence :=
+ Match Context With
+ | [ H1: (pr3 ?1 ?2 ?3); H2: (pr2 ?1 ?2 ?4) |-? ] ->
+ LApply (pr3_strip ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (pr3 ?1 ?2 ?3); H2: (pr3 ?1 ?2 ?4) |-? ] ->
+ LApply (pr3_confluence ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?4); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | _ -> Pr2Confluence.
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+(*#* #stop file *)
+
+Require Export pr1_defs.
+Require Export pr2_defs.
+
+ Inductive pr3 [c:C] : T -> T -> Prop :=
+ | pr3_r : (t:?) (pr3 c t t)
+ | pr3_u : (t2,t1:?) (pr2 c t1 t2) ->
+ (t3:?) (pr3 c t2 t3) -> (pr3 c t1 t3).
+
+ Hint pr3: ltlc := Constructors pr3.
+
+ Section pr3_props. (******************************************************)
+
+ Theorem pr3_pr2 : (c,t1,t2:?) (pr2 c t1 t2) -> (pr3 c t1 t2).
+ XEAuto.
+ Qed.
+
+ Theorem pr3_t : (t2,t1,c:?) (pr3 c t1 t2) ->
+ (t3:?) (pr3 c t2 t3) -> (pr3 c t1 t3).
+ Intros until 1; XElim H; XEAuto.
+ Qed.
+
+ Theorem pr3_tail_1 : (c:?; u1,u2:?) (pr3 c u1 u2) ->
+ (k:?; t:?) (pr3 c (TTail k u1 t) (TTail k u2 t)).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply pr3_u; [ Apply pr2_tail_1; Apply H | XAuto ].
+ Qed.
+
+ Theorem pr3_tail_2 : (c:?; u,t1,t2:?; k:?) (pr3 (CTail c k u) t1 t2) ->
+ (pr3 c (TTail k u t1) (TTail k u t2)).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply pr3_u; [ Apply pr2_tail_2; Apply H | XAuto ].
+ Qed.
+
+ Hints Resolve pr3_tail_1 pr3_tail_2 : ltlc.
+
+ Theorem pr3_tail_21 : (c:?; u1,u2:?) (pr3 c u1 u2) ->
+ (k:?; t1,t2:?) (pr3 (CTail c k u1) t1 t2) ->
+ (pr3 c (TTail k u1 t1) (TTail k u2 t2)).
+ Intros.
+ EApply pr3_t; [ Apply pr3_tail_2 | Apply pr3_tail_1 ]; XAuto.
+ Qed.
+
+ Theorem pr3_tail_12 : (c:?; u1,u2:?) (pr3 c u1 u2) ->
+ (k:?; t1,t2:?) (pr3 (CTail c k u2) t1 t2) ->
+ (pr3 c (TTail k u1 t1) (TTail k u2 t2)).
+ Intros.
+ EApply pr3_t; [ Apply pr3_tail_1 | Apply pr3_tail_2 ]; XAuto.
+ Qed.
+
+ Theorem pr3_shift : (h:?; c,e:?) (drop h (0) c e) ->
+ (t1,t2:?) (pr3 c t1 t2) ->
+ (pr3 e (app c h t1) (app c h t2)).
+ Intros until 2; XElim H0; Clear t1 t2; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ XEAuto.
+ Qed.
+
+ Theorem pr3_pr1: (t1,t2:?) (pr1 t1 t2) -> (c:?) (pr3 c t1 t2).
+ Intros until 1; XElim H; XEAuto.
+ Qed.
+
+ End pr3_props.
+
+ Hints Resolve pr3_pr2 pr3_t pr3_pr1
+ pr3_tail_12 pr3_tail_21 pr3_shift : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require pr2_gen.
+Require pr3_defs.
+Require pr3_props.
+
+ Section pr3_gen_void. (***************************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ [ H: (u,t:T) (TTail (Bind Void) ?1 ?2) = (TTail (Bind Void) u t) -> ? |- ? ] ->
+ LApply (H ?1 ?2); [ Clear H; Intros H | XAuto ];
+ XElim H1; Intros H1; XElim H1; Intros.
+
+ Theorem pr3_gen_void : (c:?; u1,t1,x:?) (pr3 c (TTail (Bind Void) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Void) u2 t2) &
+ (pr3 c u1 u2) & (b:?; u:?)
+ (pr3 (CTail c (Bind b) u) t1 t2)
+ ) \/
+ (EX u | (pr3 c u1 u) &
+ (pr3 (CTail c (Bind Void) u) t1 (lift (1) (0) x))
+ ).
+ Intros until 1; InsertEq H '(TTail (Bind Void) u1 t1);
+ UnIntro t1 H; UnIntro u1 H; XElim H; Intros.
+(* case 1 : pr3_r *)
+ Rewrite H; XEAuto.
+(* case 2 : pr3_u *)
+ Rewrite H2 in H; Clear H2 t0; Pr2Gen.
+(* case 2.1 : short step: compatibility *)
+ Rewrite H in H0; Rewrite H in H1; Clear H t2; IH.
+(* case 2.1.1 : long step: compatibility *)
+ Rewrite H; Rewrite H in H0; XEAuto.
+(* case 2.1.2 : long step: zeta *)
+ XEAuto.
+(* case 2.2 : short step: zeta *)
+ Clear H1; Right.
+ EApply ex2_intro; [ XAuto | Idtac ].
+ EApply pr3_u; [ Idtac | EApply pr3_lift ].
+ XEAuto. XAuto. XAuto.
+ Qed.
+
+ End pr3_gen_void.
+
+ Section pr3_gen_abbr. (***************************************************)
+
+ Tactic Definition IH :=
+ LApply (H1 x0 x1); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1;
+ [ Intros H1; XElim H1;
+ Do 4 Intro; Intros H_x; XElim H_x;
+ [ Intros | Intros H_x; XElim H_x; Intros ]
+ | Intros H1; XElim H1; Intros ].
+
+ Theorem pr3_gen_abbr : (c:?; u1,t1,x:?) (pr3 c (TTail (Bind Abbr) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abbr) u2 t2) &
+ (pr3 c u1 u2) &
+ ((b:?; u:?) (pr3 (CTail c (Bind b) u) t1 t2)) \/
+ (EX u3 t3 y | (pr3 c (TTail (Bind Abbr) u3 t3) x) &
+ (pr3 c u1 u3) &
+ (b:?; u:?) (pr3 (CTail c (Bind b) u) t1 y) &
+ (subst0 (0) u3 y t3)
+ )
+ ) \/
+ (EX u | (pr3 c u1 u) &
+ (pr3 (CTail c (Bind Abbr) u) t1 (lift (1) (0) x))
+ ).
+ Intros until 1; InsertEq H '(TTail (Bind Abbr) u1 t1);
+ UnIntro H t1; UnIntro H u1; XElim H; Clear y x; Intros;
+ Rename x into u1; Rename x0 into t4.
+(* case 1 : pr3_r *)
+ Rewrite H; XEAuto.
+(* case 2 : pr3_u *)
+ Rewrite H2 in H; Clear H2 t1; Pr2Gen.
+(* case 2.1 : short step: compatibility *)
+ Rewrite H in H0; Rewrite H in H1; Clear H t2; IH.
+(* case 2.1.1 : long step: compatibility *)
+ Rewrite H; Rewrite H in H0; Clear H t3.
+ Left; EApply ex3_2_intro; XEAuto.
+(* case 2.1.2 : long step: delta *)
+ Rewrite H; Rewrite H in H0; Rewrite H in H4; Clear H t3.
+ Left; EApply ex3_2_intro;
+ [ XEAuto | XEAuto
+ | Right; EApply ex4_3_intro;
+ [ EApply pr3_t; [ XAuto | Apply H4 ] | XEAuto | Idtac | Apply H7 ] ].
+ XEAuto.
+(* case 2.1.3 : long step: zeta *)
+ XEAuto.
+(* case 2.2 : short step: delta *)
+ Rewrite H in H0; Rewrite H in H1; Clear H t2; IH.
+(* case 2.2.1 : long step: compatibility *)
+ Left; EApply ex3_2_intro;
+ [ XEAuto | XEAuto
+ | Right; EApply ex4_3_intro; [ Rewrite H | Idtac | Idtac | Apply H4 ] ].
+ XAuto. XAuto. XAuto.
+(* case 2.2.2 : long step: delta *)
+ Left; EApply ex3_2_intro;
+ [ XEAuto | XEAuto
+ | Right; EApply ex4_3_intro;
+ [ EApply pr3_t; [ Apply pr3_tail_12 | Apply H5 ]
+ | Idtac | Idtac | Apply H4 ] ].
+ XAuto. EApply pr3_t; [ Apply H7 | XEAuto ]. XAuto. XAuto.
+(* case 2.2.3 : long step: zeta *)
+ Right; Apply ex2_intro with x := x0; [ XAuto | Idtac ].
+ Apply pr3_u with t2 := x; [ XAuto | Idtac ].
+ Apply pr3_u with t2 := x1; [ XEAuto | Idtac ].
+ Pr3Context; XAuto.
+(* case 2.3 : short step: zeta *)
+ Clear H1; Right.
+ EApply ex2_intro; [ XAuto | Idtac ].
+ EApply pr3_u; [ Idtac | EApply pr3_lift ].
+ XEAuto. XAuto. XAuto.
+ Qed.
+
+ End pr3_gen_abbr.
+
+ Section pr3_gen_abst. (***************************************************)
+
+ Theorem pr3_gen_abst : (c:?; u1,t1,x:?)
+ (pr3 c (TTail (Bind Abst) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
+ (pr3 c u1 u2) & (b:?; u:?)
+ (pr3 (CTail c (Bind b) u) t1 t2)
+ ).
+ Intros until 1; InsertEq H '(TTail (Bind Abst) u1 t1);
+ UnIntro H t1; UnIntro H u1; XElim H; Clear y x; Intros;
+ Rename x into u1; Rename x0 into t4.
+(* case 1 : pr3_r *)
+ Rewrite H; XEAuto.
+(* case 2 : pr3_u *)
+ Rewrite H2 in H; Clear H2 t1.
+ Pr2GenBase.
+ LApply (H1 x0 x1); [ Clear H H1; Intros | XAuto ].
+ XElim H; XEAuto.
+ Qed.
+
+ End pr3_gen_abst.
+
+ Section pr3_gen_lift. (***************************************************)
+
+(*#* #start file *)
+
+(*#* #caption "generation lemma for lift" *)
+(*#* #cap #cap c #alpha e in D, t1 in U1, t2 in U2, x in T, d in i *)
+
+ Theorem pr3_gen_lift : (c:?; t1,x:?; h,d:?) (pr3 c (lift h d t1) x) ->
+ (e:?) (drop h d c e) ->
+ (EX t2 | x = (lift h d t2) & (pr3 e t1 t2)).
+
+(*#* #stop file *)
+
+ Intros until 1; InsertEq H '(lift h d t1);
+ UnIntro H t1; XElim H; Clear y x; Intros; Rename x into t4.
+(* case 1 : pr3_r *)
+ XEAuto.
+(* case 2 : pr3_u *)
+ Rewrite H2 in H; Pr2Gen.
+ LApply (H1 x); [ Clear H1; Intros H1 | XAuto ].
+ LApply (H1 e); [ Clear H1; Intros H1 | XAuto ].
+ XElim H1; XEAuto.
+ Qed.
+
+ End pr3_gen_lift.
+
+ Tactic Definition Pr3Gen :=
+ Match Context With
+ | [ H: (pr3 ?1 (TTail (Bind Abst) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr3_gen_abst ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (pr3 ?1 (TTail (Bind Abbr) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr3_gen_abbr ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H;
+ [ Intros H; XElim H;
+ Do 4 Intro; Intros H_x; XElim H_x;
+ [ Intros | Intros H_x; XElim H_x; Intros ]
+ | Intros H; XElim H; Intros ]
+ | [ H: (pr3 ?1 (TTail (Bind Void) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr3_gen_void ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros H; XElim H; Intros
+ | [ H0: (pr3 ?1 (lift ?2 ?3 ?4) ?5);
+ H1: (drop ?2 ?3 ?1 ?6) |- ? ] ->
+ LApply (pr3_gen_lift ?1 ?4 ?5 ?2 ?3); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?6); [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | _ -> Pr2Gen.
--- /dev/null
+(*#* #stop file *)
+
+Require csubst1_defs.
+Require pr2_gen_context.
+Require pr3_defs.
+
+ Section pr3_gen_context. (************************************************)
+
+ Theorem pr3_gen_cabbr: (c:?; t1,t2:?) (pr3 c t1 t2) -> (e:?; u:?; d:?)
+ (drop d (0) c (CTail e (Bind Abbr) u)) ->
+ (a0:?) (csubst1 d u c a0) ->
+ (a:?) (drop (1) d a0 a) ->
+ (x1:?) (subst1 d u t1 (lift (1) d x1)) ->
+ (EX x2 | (subst1 d u t2 (lift (1) d x2)) &
+ (pr3 a x1 x2)
+ ).
+ Intros until 1; XElim H; Intros.
+(* case 1: pr3_r *)
+ XEAuto.
+(* case 1: pr3_r *)
+ Pr2GenContext.
+ LApply (H1 e u d); [ Clear H1; Intros H1 | XAuto ].
+ LApply (H1 a0); [ Clear H1; Intros H1 | XAuto ].
+ LApply (H1 a); [ Clear H1; Intros H1 | XAuto ].
+ LApply (H1 x); [ Clear H1; Intros H1 | XAuto ].
+ XElim H1; XEAuto.
+ Qed.
+
+ End pr3_gen_context.
+
+ Tactic Definition Pr3GenContext :=
+ Match Context With
+ | [ H0: (pr3 ?1 ?2 ?3); H1: (drop ?4 (0) ?1 (CTail ?5 (Bind Abbr) ?6));
+ H2: (csubst1 ?4 ?6 ?1 ?7); H3: (drop (1) ?4 ?7 ?8);
+ H4: (subst1 ?4 ?6 ?2 (lift (1) ?4 ?9)) |- ? ] ->
+ LApply (pr3_gen_cabbr ?1 ?2 ?3); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?5 ?6 ?4); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?8); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?9); [ Clear H0 H4; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | _ -> Pr2GenContext.
--- /dev/null
+Require subst0_subst0.
+Require pr0_subst0.
+Require pr2_lift.
+Require pr3_defs.
+
+(*#* #caption "main properties of predicate \\texttt{pr3}" *)
+(*#* #clauses *)
+
+(*#* #stop file *)
+
+ Section pr3_context. (****************************************************)
+
+ Theorem pr3_pr0_pr2_t : (u1,u2:?) (pr0 u1 u2) ->
+ (c:?; t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+ (pr3 (CTail c k u1) t1 t2).
+ Intros.
+ Inversion H0; Clear H0; XAuto.
+ NewInduction i.
+(* case 1 : pr2_delta i = 0 *)
+ DropGenBase; Inversion H0; Clear H0 H3 H4 c k.
+ Rewrite H5 in H; Clear H5 u2.
+ Pr0Subst0; XEAuto.
+(* case 2 : pr2_delta i > 0 *)
+ NewInduction k; DropGenBase; XEAuto.
+ Qed.
+
+ Theorem pr3_pr2_pr2_t : (c:?; u1,u2:?) (pr2 c u1 u2) ->
+ (t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
+ (pr3 (CTail c k u1) t1 t2).
+ Intros until 1; Inversion H; Clear H; Intros.
+(* case 1 : pr2_pr0 *)
+ EApply pr3_pr0_pr2_t; [ Apply H0 | XAuto ].
+(* case 2 : pr2_delta *)
+ Inversion H; [ XAuto | NewInduction i0 ].
+(* case 2.1 : i0 = 0 *)
+ DropGenBase; Inversion H2; Clear H2.
+ Rewrite <- H5; Rewrite H6 in H; Rewrite <- H7 in H3; Clear H5 H6 H7 d0 k u0.
+ Subst0Subst0; Arith9'In H4 i; XDEAuto 7.
+(* case 2.2 : i0 > 0 *)
+ Clear IHi0; NewInduction k; DropGenBase; XEAuto.
+ Qed.
+
+ Theorem pr3_pr2_pr3_t : (c:?; u2,t1,t2:?; k:?)
+ (pr3 (CTail c k u2) t1 t2) ->
+ (u1:?) (pr2 c u1 u2) ->
+ (pr3 (CTail c k u1) t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply pr3_t.
+ EApply pr3_pr2_pr2_t; [ Apply H2 | Apply H ].
+ XAuto.
+ Qed.
+
+(*#* #start file *)
+
+(*#* #caption "reduction inside context items" *)
+(*#* #cap #cap t1, t2 #alpha c in E, u1 in V1, u2 in V2, k in z *)
+
+ Theorem pr3_pr3_pr3_t : (c:?; u1,u2:?) (pr3 c u1 u2) ->
+ (t1,t2:?; k:?) (pr3 (CTail c k u2) t1 t2) ->
+ (pr3 (CTail c k u1) t1 t2).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply pr3_pr2_pr3_t; [ Apply H1; XAuto | XAuto ].
+ Qed.
+
+ End pr3_context.
+
+ Tactic Definition Pr3Context :=
+ Match Context With
+ | [ H1: (pr0 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pr3_pr0_pr2_t ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr0 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+ LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr2 ?1 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pr3_pr2_pr2_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr2 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
+ LApply (H2 ?2); [ Clear H1 H2; Intros | XAuto ]
+ | [ H1: (pr3 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
+ LApply (pr3_pr3_pr3_t ?1 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6 ?4); [ Clear H1 H2; Intros | XAuto ].
+
+ Section pr3_lift. (*******************************************************)
+
+(*#* #start file *)
+
+(*#* #caption "conguence with lift" *)
+(*#* #cap #cap c, t1, t2 #alpha e in D, d in i *)
+
+ Theorem pr3_lift : (c,e:?; h,d:?) (drop h d c e) ->
+ (t1,t2:?) (pr3 e t1 t2) ->
+ (pr3 c (lift h d t1) (lift h d t2)).
+
+(*#* #stop file *)
+
+ Intros until 2; XElim H0; Intros; XEAuto.
+ Qed.
+
+ End pr3_lift.
+
+ Hints Resolve pr3_lift : ltlc.
+
--- /dev/null
+(*#* #stop file *)
+
+Require subst1_defs.
+Require pr2_subst1.
+Require pr3_defs.
+
+ Section pr3_subst1_props. (***********************************************)
+
+ Theorem pr3_subst1: (c,e:?; v:?; i:?) (drop i (0) c (CTail e (Bind Abbr) v)) ->
+ (t1,t2:?) (pr3 c t1 t2) ->
+ (w1:?) (subst1 i v t1 w1) ->
+ (EX w2 | (pr3 c w1 w2) & (subst1 i v t2 w2)).
+ Intros until 2; XElim H0; Clear t1 t2; Intros.
+(* case 1: pr3_refl *)
+ XEAuto.
+(* case 2: pr3_single *)
+ Pr2Subst1.
+ LApply (H2 x); [ Clear H2; Intros H2 | XAuto ].
+ XElim H2; XEAuto.
+ Qed.
+
+ End pr3_subst1_props.
+
+ Tactic Definition Pr3Subst1 :=
+ Match Context With
+ | [ H0: (drop ?1 (0) ?2 (CTail ?3 (Bind Abbr) ?4));
+ H1: (pr3 ?2 ?5 ?6); H3: (subst1 ?1 ?4 ?5 ?7) |- ? ] ->
+ LApply (pr3_subst1 ?2 ?3 ?4 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?5 ?6); [ Clear H_x H1; Intros H1 | XAuto ];
+ LApply (H1 ?7); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | _ -> Pr2Subst1.
+
--- /dev/null
+(*#* #stop file *)
+
+Require lift_gen.
+Require subst0_gen.
+Require subst0_defs.
+
+ Section subst0_confluence. (**********************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H1: (t2,u2:?; i2:?) (subst0 i2 u2 ?1 t2) -> ~?2=i2 -> ?;
+ H2: (subst0 ?3 ?4 ?1 ?5); H3: ~?2=?3 |- ? ] ->
+ LApply (H1 ?5 ?4 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply H1; [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (t2,u2:?; i2:?) (subst0 i2 u2 ?1 t2) -> ~(s k ?2)=i2 -> ?;
+ H2: (subst0 (s k ?3) ?4 ?1 ?5); H3: ~?2=?3 |- ? ] ->
+ LApply (H1 ?5 ?4 (s k ?3)); [ Clear H1; Intros H1 | XAuto ];
+ LApply H1; [ Clear H1; Intros H1 | Unfold not in H3; Unfold not; XEAuto ];
+ XElim H1; Intros
+ | [ H1: (t2:T) (subst0 ?1 ?2 ?3 t2) -> ?; H2: (subst0 ?1 ?2 ?3 ?4) |- ? ] ->
+ LApply (H1 ?4); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros H1; [ Try Rewrite H1 | XElim H1; Intros | Idtac | Idtac ].
+
+ Theorem subst0_confluence_neq : (t0,t1,u1:?; i1:?) (subst0 i1 u1 t0 t1) ->
+ (t2,u2:?; i2:?) (subst0 i2 u2 t0 t2) ->
+ ~i1=i2 ->
+ (EX t | (subst0 i2 u2 t1 t) &
+ (subst0 i1 u1 t2 t)).
+
+ Intros until 1; XElim H; Intros;
+ Subst0GenBase; Try Rewrite H in H0; Try Rewrite H1; Try Rewrite H3;
+ Try EqFalse; Repeat IH; XEAuto.
+ Qed.
+
+ Theorem subst0_confluence_eq : (t0,t1,u:?; i:?) (subst0 i u t0 t1) ->
+ (t2:?) (subst0 i u t0 t2) -> (OR
+ t1 = t2 |
+ (EX t | (subst0 i u t1 t) & (subst0 i u t2 t)) |
+ (subst0 i u t1 t2) |
+ (subst0 i u t2 t1)).
+ Intros until 1; XElim H; Intros;
+ Subst0GenBase; Try Rewrite H1; Try Rewrite H3;
+ Repeat IH; XEAuto.
+ Qed.
+
+ End subst0_confluence.
+
+ Tactic Definition Subst0Confluence :=
+ Match Context With
+ | [ H0: (subst0 ?1 ?2 ?3 ?4);
+ H1: (subst0 ?1 ?2 ?3 ?5) |- ? ] ->
+ LApply (subst0_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; [ Intros | Intros H0; XElim H0; Intros | Intros | Intros ]
+ | [ H0: (subst0 ?1 ?2 ?3 ?4);
+ H1: (subst0 ?5 ?6 ?3 ?7) |- ? ] ->
+ LApply (subst0_confluence_neq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7 ?6 ?5); [ Clear H0 H1; Intros H0 | XAuto ];
+ LApply H0; [ Clear H0; Intros H0 | Simpl; XAuto ];
+ XElim H0; Intros.
+
+ Section subst0_confluence_lift. (*****************************************)
+
+ Theorem subst0_confluence_lift: (t0,t1,u:?; i:?) (subst0 i u t0 (lift (1) i t1)) ->
+ (t2:?) (subst0 i u t0 (lift (1) i t2)) ->
+ t1 = t2.
+ Intros; Subst0Confluence;
+ Try Subst0Gen; SymEqual; LiftGen; XEAuto.
+ Qed.
+
+ End subst0_confluence_lift.
+
+ Tactic Definition Subst0ConfluenceLift :=
+ Match Context With
+ | [ H0: (subst0 ?1 ?2 ?3 (lift (1) ?1 ?4));
+ H1: (subst0 ?1 ?2 ?3 (lift (1) ?1 ?5)) |- ? ] ->
+ LApply (subst0_confluence_lift ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?5); [ Clear H0; Intros | XAuto ]
+ | _ -> Subst0Confluence.
--- /dev/null
+(*#* #stop file *)
+
+Require Export lift_defs.
+
+ Inductive subst0 : nat -> T -> T -> T -> Prop :=
+ | subst0_bref : (v:?; i:?) (subst0 i v (TBRef i) (lift (S i) (0) v))
+ | subst0_fst : (v,w,u:?; i:?) (subst0 i v u w) ->
+ (t:?; k:?) (subst0 i v (TTail k u t) (TTail k w t))
+ | subst0_snd : (k:?; v,w,t:?; i:?) (subst0 (s k i) v t w) -> (u:?)
+ (subst0 i v (TTail k u t) (TTail k u w))
+ | subst0_both : (v,u1,u2:?; i:?) (subst0 i v u1 u2) ->
+ (k:?; t1,t2:?) (subst0 (s k i) v t1 t2) ->
+ (subst0 i v (TTail k u1 t1) (TTail k u2 t2)).
+
+ Hint subst0 : ltlc := Constructors subst0.
+
+ Section subst0_gen_base. (************************************************)
+
+ Theorem subst0_gen_sort : (v,x:?; i,n:?) (subst0 i v (TSort n) x) ->
+ (P:Prop) P.
+ Intros; Inversion H.
+ Qed.
+
+ Theorem subst0_gen_bref : (v,x:?; i,n:?) (subst0 i v (TBRef n) x) ->
+ n = i /\ x = (lift (S n) (0) v).
+ Intros; Inversion H; XAuto.
+ Qed.
+
+ Theorem subst0_gen_tail : (k:?; v,u1,t1,x:?; i:?)
+ (subst0 i v (TTail k u1 t1) x) -> (OR
+ (EX u2 | x = (TTail k u2 t1) &
+ (subst0 i v u1 u2)) |
+ (EX t2 | x = (TTail k u1 t2) &
+ (subst0 (s k i) v t1 t2)) |
+ (EX u2 t2 | x = (TTail k u2 t2) &
+ (subst0 i v u1 u2) &
+ (subst0 (s k i) v t1 t2))
+ ).
+
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ End subst0_gen_base.
+
+ Tactic Definition Subst0GenBase :=
+ Match Context With
+ | [ H: (subst0 ?1 ?2 (TSort ?3) ?4) |- ? ] ->
+ Apply (subst0_gen_sort ?2 ?4 ?1 ?3); Apply H
+ | [ H: (subst0 ?1 ?2 (TBRef ?3) ?4) |- ? ] ->
+ LApply (subst0_gen_bref ?2 ?4 ?1 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (subst0 ?1 ?2 (TTail ?3 ?4 ?5) ?6) |- ? ] ->
+ LApply (subst0_gen_tail ?3 ?2 ?4 ?5 ?6 ?1); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros H; XElim H; Intros.
+
--- /dev/null
+(*#* #stop file *)
+
+Require lift_props.
+Require subst0_defs.
+
+ Section subst0_gen_lift_lt. (*********************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ [ H1: (x:T; i,h,d:nat) (subst0 i (lift h d ?1) (lift h (S (plus i d)) ?2) x) -> ?;
+ H2: (subst0 ?3 (lift ?4 ?5 ?1) (lift ?4 (S (plus ?3 ?5)) ?2) ?6) |- ? ] ->
+ LApply (H1 ?6 ?3 ?4 ?5); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros.
+
+ Theorem subst0_gen_lift_lt : (u,t1,x:?; i,h,d:?) (subst0 i (lift h d u) (lift h (S (plus i d)) t1) x) ->
+ (EX t2 | x = (lift h (S (plus i d)) t2) & (subst0 i u t1 t2)).
+ XElim t1; Intros.
+(* case 1: TSort *)
+ Rewrite lift_sort in H; Subst0GenBase.
+(* case 2: TBRef n *)
+ Apply (lt_le_e n (S (plus i d))); Intros.
+(* case 2.1: n < 1 + i + d *)
+ Rewrite lift_bref_lt in H; [ Idtac | XAuto ].
+ Subst0GenBase; Rewrite H1; Rewrite H.
+ Rewrite <- lift_d; Simpl; XEAuto.
+(* case 2.2: n >= 1 + i + d *)
+ Rewrite lift_bref_ge in H; [ Idtac | XAuto ].
+ Subst0GenBase; Rewrite <- H in H0.
+ EApply le_false; [ Idtac | Apply H0 ]; XAuto.
+(* case 3: TTail k *)
+ Rewrite lift_tail in H1; Subst0GenBase; Rewrite H1; Clear H1 x.
+(* case 3.1: subst0_fst *)
+ IH; Rewrite H; Rewrite <- lift_tail; XEAuto.
+(* case 3.2: subst0_snd *)
+ SRwIn H2; IH; Rewrite H0; SRwBack; Rewrite <- lift_tail; XEAuto.
+(* case 3.2: subst0_snd *)
+ SRwIn H3; Repeat IH; Rewrite H; Rewrite H0; SRwBack;
+ Rewrite <- lift_tail; XEAuto.
+ Qed.
+
+ End subst0_gen_lift_lt.
+
+ Section subst0_gen_lift_false. (******************************************)
+
+ Theorem subst0_gen_lift_false : (t,u,x:?; h,d,i:?)
+ (le d i) -> (lt i (plus d h)) ->
+ (subst0 i u (lift h d t) x) ->
+ (P:Prop) P.
+ XElim t; Intros.
+(* case 1: TSort *)
+ Rewrite lift_sort in H1; Subst0GenBase.
+(* case 2: TBRef n *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1: n < d *)
+ Rewrite lift_bref_lt in H1; [ Idtac | XAuto ].
+ Subst0GenBase; Rewrite H1 in H2.
+ EApply le_false; [ Apply H | XAuto ].
+(* case 2.2: n >= d *)
+ Rewrite lift_bref_ge in H1; [ Idtac | XAuto ].
+ Subst0GenBase; Rewrite <- H1 in H0.
+ EApply le_false; [ Apply H2 | XEAuto ].
+(* case 3: TTail k *)
+ Rewrite lift_tail in H3; Subst0GenBase.
+(* case 3.1: subst0_fst *)
+ EApply H; XEAuto.
+(* case 3.2: subst0_snd *)
+ EApply H0; [ Idtac | Idtac | XEAuto ]; [ Idtac | SRwBack ]; XAuto.
+(* case 3.3: subst0_both *)
+ EApply H; XEAuto.
+ Qed.
+
+ End subst0_gen_lift_false.
+
+ Section subst0_gen_lift_ge. (*********************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ [ H1: (x:?; i,h,d:?) (subst0 i ?1 (lift h d ?2) x) -> ?;
+ H2: (subst0 ?3 ?1 (lift ?4 ?5 ?2) ?6) |- ? ] ->
+ LApply (H1 ?6 ?3 ?4 ?5); [ Clear H1 H2; Intros H1 | XAuto ];
+ LApply H1; [ Clear H1; Intros H1 | SRwBack; XAuto ];
+ XElim H1; Intros.
+
+ Theorem subst0_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst0 i u (lift h d t1) x) ->
+ (le (plus d h) i) ->
+ (EX t2 | x = (lift h d t2) & (subst0 (minus i h) u t1 t2)).
+ XElim t1; Intros.
+(* case 1: TSort *)
+ Rewrite lift_sort in H; Subst0GenBase.
+(* case 2: TBRef n *)
+ Apply (lt_le_e n d); Intros.
+(* case 2.1: n < d *)
+ Rewrite lift_bref_lt in H; [ Idtac | XAuto ].
+ Subst0GenBase; Rewrite H in H1.
+ EApply le_false; [ Apply H0 | XAuto ].
+(* case 2.2: n >= d *)
+ Rewrite lift_bref_ge in H; [ Idtac | XAuto ].
+ Subst0GenBase; Rewrite <- H; Rewrite H2.
+ Rewrite minus_plus_r.
+ EApply ex2_intro; [ Idtac | XAuto ].
+ Rewrite lift_free; [ Idtac | XEAuto (**) | XAuto ].
+ Rewrite plus_sym; Rewrite plus_n_Sm; XAuto.
+(* case 3: TTail k *)
+ Rewrite lift_tail in H1; Subst0GenBase; Rewrite H1; Clear H1 x;
+ Repeat IH; Try Rewrite H; Try Rewrite H0;
+ Rewrite <- lift_tail; Try Rewrite <- s_minus in H1; XEAuto.
+ Qed.
+
+ End subst0_gen_lift_ge.
+
+ Tactic Definition Subst0Gen :=
+ Match Context With
+ | [ H: (subst0 ?0 (lift ?1 ?2 ?3) (lift ?1 (S (plus ?0 ?2)) ?4) ?5) |- ? ] ->
+ LApply (subst0_gen_lift_lt ?3 ?4 ?5 ?0 ?1 ?2); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (subst0 ?0 ?1 (lift (1) ?0 ?2) ?3) |- ? ] ->
+ LApply (subst0_gen_lift_false ?2 ?1 ?3 (1) ?0 ?0); [ Intros H_x | XAuto ];
+ LApply H_x; [ Clear H_x; Intros H_x | Rewrite plus_sym; XAuto ];
+ LApply H_x; [ Clear H H_x; Intros H | XAuto ];
+ Apply H
+ | [ _: (le ?1 ?2); _: (lt ?2 (plus ?1 ?3));
+ _: (subst0 ?2 ?4 (lift ?3 ?1 ?5) ?6) |- ? ] ->
+ Apply (subst0_gen_lift_false ?5 ?4 ?6 ?3 ?1 ?2); XAuto
+ | [ _: (subst0 ?1 ?2 (lift (S ?1) (0) ?3) ?4) |- ? ] ->
+ Apply (subst0_gen_lift_false ?3 ?2 ?4 (S ?1) (0) ?1); XAuto
+ | [ H: (subst0 ?0 ?1 (lift ?2 ?3 ?4) ?5) |- ? ] ->
+ LApply (subst0_gen_lift_ge ?1 ?4 ?5 ?0 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ LApply H; [ Clear H; Intros H | Simpl; XAuto ];
+ XElim H; Intros
+ | _ -> Subst0GenBase.
--- /dev/null
+(*#* #stop file *)
+
+Require lift_props.
+Require subst0_defs.
+
+ Section subst0_lift. (****************************************************)
+
+ Theorem subst0_lift_lt : (t1,t2,u:?; i:?) (subst0 i u t1 t2) ->
+ (d:?) (lt i d) -> (h:?)
+ (subst0 i (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)).
+ Intros until 1; XElim H; Intros.
+(* case 1: subst0_bref *)
+ Rewrite lift_bref_lt; [ Idtac | XAuto ].
+ LetTac w := (minus d (S i0)).
+ Arith8 d '(S i0); Rewrite lift_d; XAuto.
+(* case 2: subst0_fst *)
+ LiftTailRw; XAuto.
+(* case 3: subst0_snd *)
+ SRwBackIn H0; LiftTailRw; Rewrite <- (minus_s_s k); XAuto.
+(* case 4: subst0_both *)
+ SRwBackIn H2; LiftTailRw.
+ Apply subst0_both; [ Idtac | Rewrite <- (minus_s_s k) ]; XAuto.
+ Qed.
+
+ Theorem subst0_lift_ge : (t1,t2,u:?; i,h:?) (subst0 i u t1 t2) ->
+ (d:?) (le d i) ->
+ (subst0 (plus i h) u (lift h d t1) (lift h d t2)).
+ Intros until 1; XElim H; Intros.
+(* case 1: subst0_bref *)
+ Rewrite lift_bref_ge; [ Idtac | XAuto ].
+ Rewrite lift_free; [ Idtac | Simpl; XAuto | XAuto ].
+ Arith5'c h i0; XAuto.
+(* case 2: subst0_fst *)
+ LiftTailRw; XAuto.
+(* case 3: subst0_snd *)
+ SRwBackIn H0; LiftTailRw; XAuto.
+(* case 4: subst0_snd *)
+ SRwBackIn H2; LiftTailRw; XAuto.
+ Qed.
+
+ Theorem subst0_lift_ge_S : (t1,t2,u:?; i:?) (subst0 i u t1 t2) ->
+ (d:?) (le d i) -> (b:?)
+ (subst0 (s (Bind b) i) u (lift (1) d t1) (lift (1) d t2)).
+ Intros; Simpl; Arith7 i; Apply subst0_lift_ge; XAuto.
+ Qed.
+
+ End subst0_lift.
+
+ Hints Resolve subst0_lift_lt subst0_lift_ge subst0_lift_ge_S : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require subst0_defs.
+Require subst0_gen.
+Require subst0_lift.
+
+ Section subst0_subst0. (**************************************************)
+
+ Tactic Definition IH :=
+ Match Context With
+ | [ H1: (u1,u:?; i:?) (subst0 i u u1 ?1) -> ?;
+ H2: (subst0 ?2 ?3 ?4 ?1) |- ? ] ->
+ LApply (H1 ?4 ?3 ?2); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (u1,u:?; i:?) (subst0 i u ?1 u1) -> ?;
+ H2: (subst0 ?2 ?3 ?1 ?4) |- ? ] ->
+ LApply (H1 ?4 ?3 ?2); [ Clear H1; Intros H1 | XAuto ];
+ XElim H1; Intros.
+
+ Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+ (u1,u:?; i:?) (subst0 i u u1 u2) ->
+ (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
+ Intros until 1; XElim H; Intros.
+(* case 1 : subst0_bref *)
+ Arith5 i0 i; XEAuto.
+(* case 2 : subst0_fst *)
+ IH; XEAuto.
+(* case 3 : subst0_snd *)
+ IH; SRwBackIn H2; XEAuto.
+(* case 4 : subst0_both *)
+ Repeat IH; SRwBackIn H4; XEAuto.
+ Qed.
+
+ Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+ (u1,u:?; i:?) (subst0 i u u2 u1) ->
+ (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
+ Intros until 1; XElim H; Intros.
+(* case 1 : subst0_bref *)
+ Arith5 i0 i; XEAuto.
+(* case 2 : subst0_fst *)
+ IH; XEAuto.
+(* case 3 : subst0_snd *)
+ IH; SRwBackIn H2; XEAuto.
+(* case 4 : subst0_both *)
+ Repeat IH; SRwBackIn H4; XEAuto.
+ Qed.
+
+ Theorem subst0_trans: (t2,t1,v:?; i:?) (subst0 i v t1 t2) ->
+ (t3:?) (subst0 i v t2 t3) ->
+ (subst0 i v t1 t3).
+ Intros until 1; XElim H; Intros;
+ Subst0Gen; Try Rewrite H1; Try Rewrite H3; XAuto.
+ Qed.
+
+ End subst0_subst0.
+
+ Hints Resolve subst0_trans : ltlc.
+
+ Tactic Definition Subst0Subst0 :=
+ Match Context With
+ | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?6 ?1) |- ? ] ->
+ LApply (subst0_subst0 ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
+ LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ H1: (subst0 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?1 ?6) |- ? ] ->
+ LApply (subst0_subst0_back ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
+ LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros.
--- /dev/null
+(*#* #stop file *)
+
+Require tlt_defs.
+Require lift_tlt.
+Require subst0_defs.
+
+ Section subst0_tlt_props. (***********************************************)
+
+ Theorem subst0_weight_le : (u,t,z:?; d:?) (subst0 d u t z) ->
+ (f,g:?) ((m:?) (le (f m) (g m))) ->
+ (lt (weight_map f (lift (S d) (0) u)) (g d)) ->
+ (le (weight_map f z) (weight_map g t)).
+ Intros until 1; XElim H.
+(* case 1: subst0_bref *)
+ Intros; XAuto.
+(* case 2: subst0_fst *)
+ XElim k; [ XElim b | Idtac ]; Simpl; [ Auto 7 with ltlc (**) | XAuto | XAuto | XAuto ].
+(* case 3: subst0_snd *)
+ XElim k; [ XElim b | Idtac ]; Simpl; [ Auto 7 with ltlc (**) | XAuto | XAuto | XAuto ].
+(* case 4: subst0_both *)
+ XElim k; [ XElim b | Idtac ]; Simpl; [ Auto 7 with ltlc (**) | XAuto | XAuto | XAuto ].
+ Qed.
+
+ Theorem subst0_weight_lt : (u,t,z:?; d:?) (subst0 d u t z) -> (f,g:?)
+ ((m:?) (le (f m) (g m))) ->
+ (lt (weight_map f (lift (S d) (0) u)) (g d)) ->
+ (lt (weight_map f z) (weight_map g t)).
+ Intros until 1; XElim H.
+(* case 1: subst0_bref *)
+ Intros; XAuto.
+(* case 2: subst0_fst *)
+ XElim k; [ XElim b | Idtac ]; Simpl; Intros;
+ Apply lt_n_S; (Apply lt_le_plus_plus; [ XAuto | Idtac ]); (**)
+ [ Auto 6 with ltlc (**) | XAuto | XAuto | XAuto ].
+(* case 3: subst0_snd *)
+ XElim k; [ XElim b | Idtac ]; Simpl;
+ [ Auto 8 with ltlc | Auto 6 with ltlc | Auto 6 with ltlc | XAuto ]. (**)
+(* case 3: subst0_both *)
+ XElim k; [ XElim b | Idtac ]; Simpl;
+ Intros; Apply lt_n_S; [ Apply lt_le_plus_plus | Apply lt_plus_plus | Apply lt_plus_plus | Apply lt_plus_plus ]; XAuto.
+ EApply subst0_weight_le; [ XEAuto | XAuto | XAuto ].
+ Qed.
+
+ Theorem subst0_tlt_tail: (u,t,z:?) (subst0 (0) u t z) ->
+ (tlt (TTail (Bind Abbr) u z)
+ (TTail (Bind Abbr) u t)
+ ).
+ Unfold tlt weight; Intros; Simpl.
+ Apply lt_n_S; Apply le_lt_plus_plus; [ XAuto | Idtac ].
+ EApply subst0_weight_lt; [ XEAuto | XAuto | XAuto ].
+ Qed.
+
+ Theorem subst0_tlt: (u,t,z:?) (subst0 (0) u t z) ->
+ (tlt z (TTail (Bind Abbr) u t)).
+ Intros.
+ EApply tlt_trans; [ Idtac | Apply subst0_tlt_tail; XEAuto].
+ XAuto.
+ Qed.
+
+ End subst0_tlt_props.
+
+ Hints Resolve subst0_tlt : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require lift_gen.
+Require subst0_gen.
+Require subst0_confluence.
+Require subst1_defs.
+Require subst1_gen.
+
+ Section subst1_confluence. (**********************************************)
+
+ Theorem subst1_confluence_neq: (t0,t1,u1:?; i1:?) (subst1 i1 u1 t0 t1) ->
+ (t2,u2:?; i2:?) (subst1 i2 u2 t0 t2) ->
+ ~i1=i2 ->
+ (EX t | (subst1 i2 u2 t1 t) &
+ (subst1 i1 u1 t2 t)
+ ).
+ Intros until 1; XElim H; Clear t1; Intros.
+(* case 1; subst1_refl *)
+ XEAuto.
+(* case 2; subst1_single *)
+ XElim H0; Intros; Try Subst0Confluence; XEAuto.
+ Qed.
+
+ Theorem subst1_confluence_eq : (t0,t1,u:?; i:?) (subst1 i u t0 t1) ->
+ (t2:?) (subst1 i u t0 t2) ->
+ (EX t | (subst1 i u t1 t) &
+ (subst1 i u t2 t)
+ ).
+ Intros until 1; XElim H; Intros.
+(* case 1; subst1_refl *)
+ XEAuto.
+(* case 2; subst1_single *)
+ XElim H0; Intros;
+ Try Subst0Confluence; Try Rewrite H0; XEAuto.
+ Qed.
+
+ Theorem subst1_confluence_lift: (t0,t1,u:?; i:?) (subst1 i u t0 (lift (1) i t1)) ->
+ (t2:?) (subst1 i u t0 (lift (1) i t2)) ->
+ t1 = t2.
+ Intros until 1; InsertEq H '(lift (1) i t1); XElim H; Clear y; Intros.
+(* case 1: subst1_refl *)
+ Rewrite H in H0; Clear H t0.
+ Subst1Gen; SymEqual; LiftGen; XEAuto.
+(* case 2: subst1_single *)
+ Rewrite H0 in H; Clear H0 t2.
+ InsertEq H1 '(lift (1) i t3); XElim H0; Clear y; Intros.
+(* case 2.1: subst1_refl *)
+ Rewrite H0 in H; Clear H0 t0; Subst0Gen.
+(* case 2.2: subst1_single *)
+ Rewrite H1 in H0; Clear H1 t2; Subst0ConfluenceLift; XAuto.
+ Qed.
+
+ End subst1_confluence.
+
+ Tactic Definition Subst1Confluence :=
+ Match Context With
+ | [ H0: (subst1 ?1 ?2 ?3 (lift (1) ?1 ?4));
+ H1: (subst1 ?1 ?2 ?3 (lift (1) ?1 ?5)) |- ? ] ->
+ LApply (subst1_confluence_lift ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?5); [ Clear H0; Intros | XAuto ]
+ | [ H0: (subst1 ?1 ?2 ?3 ?4);
+ H1: (subst1 ?1 ?2 ?3 ?5) |- ? ] ->
+ LApply (subst1_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | [ H0: (subst0 ?1 ?2 ?3 ?4);
+ H1: (subst1 ?1 ?2 ?3 ?5) |- ? ] ->
+ LApply (subst1_confluence_eq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | [ H0: (subst1 ?1 ?2 ?3 ?4);
+ H1: (subst1 ?5 ?6 ?3 ?7) |- ? ] ->
+ LApply (subst1_confluence_neq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7 ?6 ?5); [ Clear H0 H1; Intros H0 | XAuto ];
+ LApply H0; [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | [ H0: (subst0 ?1 ?2 ?3 ?4);
+ H1: (subst1 ?5 ?6 ?3 ?7) |- ? ] ->
+ LApply (subst1_confluence_neq ?3 ?4 ?2 ?1); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7 ?6 ?5); [ Clear H0 H1; Intros H0 | XAuto ];
+ LApply H0; [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros.
--- /dev/null
+(*#* #stop file *)
+
+Require Export subst0_defs.
+
+ Inductive subst1 [i:nat; v:T; t1:T] : T -> Prop :=
+ | subst1_refl : (subst1 i v t1 t1)
+ | subst1_single : (t2:?) (subst0 i v t1 t2) -> (subst1 i v t1 t2).
+
+ Hint subst1 : ltlc := Constructors subst1.
+
+ Section subst1_props. (***************************************************)
+
+ Theorem subst1_tail: (v,u1,u2:?; i:?) (subst1 i v u1 u2) ->
+ (k:?; t1,t2:?) (subst1 (s k i) v t1 t2) ->
+ (subst1 i v (TTail k u1 t1) (TTail k u2 t2)).
+ Intros until 1; XElim H; Clear u2.
+(* case 1: csubst1_refl *)
+ Intros until 1; XElim H; Clear t2; XAuto.
+(* case 2: csubst1_single *)
+ Intros until 2; XElim H0; Clear t3; XAuto.
+ Qed.
+
+ End subst1_props.
+
+ Hints Resolve subst1_tail : ltlc.
+
+ Section subst1_gen_base. (************************************************)
+
+ Theorem subst1_gen_sort : (v,x:?; i,n:?) (subst1 i v (TSort n) x) ->
+ x = (TSort n).
+ Intros; XElim H; Clear x; Intros;
+ Try Subst0GenBase; XAuto.
+ Qed.
+
+ Theorem subst1_gen_bref : (v,x:?; i,n:?) (subst1 i v (TBRef n) x) ->
+ x = (TBRef n) \/
+ n = i /\ x = (lift (S n) (0) v).
+ Intros; XElim H; Clear x; Intros;
+ Try Subst0GenBase; XAuto.
+ Qed.
+
+ Theorem subst1_gen_tail : (k:?; v,u1,t1,x:?; i:?)
+ (subst1 i v (TTail k u1 t1) x) ->
+ (EX u2 t2 | x = (TTail k u2 t2) &
+ (subst1 i v u1 u2) &
+ (subst1 (s k i) v t1 t2)
+ ).
+ Intros; XElim H; Clear x; Intros;
+ Try Subst0GenBase; XEAuto.
+ Qed.
+
+ End subst1_gen_base.
+
+ Tactic Definition Subst1GenBase :=
+ Match Context With
+ | [ H: (subst1 ?1 ?2 (TTail ?3 ?4 ?5) ?6) |- ? ] ->
+ LApply (subst1_gen_tail ?3 ?2 ?4 ?5 ?6 ?1); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
--- /dev/null
+(*#* #stop file *)
+
+Require subst0_gen.
+Require subst1_defs.
+
+ Section subst1_gen_lift. (************************************************)
+
+ Theorem subst1_gen_lift_lt : (u,t1,x:?; i,h,d:?) (subst1 i (lift h d u) (lift h (S (plus i d)) t1) x) ->
+ (EX t2 | x = (lift h (S (plus i d)) t2) & (subst1 i u t1 t2)).
+ Intros; XElim H; Clear x; Intros;
+ Try Subst0Gen; XEAuto.
+ Qed.
+
+ Theorem subst1_gen_lift_eq : (t,u,x:?; h,d,i:?)
+ (le d i) -> (lt i (plus d h)) ->
+ (subst1 i u (lift h d t) x) ->
+ x = (lift h d t).
+ Intros; XElim H1; Clear x; Intros;
+ Try Subst0Gen; XAuto.
+ Qed.
+
+ Theorem subst1_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst1 i u (lift h d t1) x) ->
+ (le (plus d h) i) ->
+ (EX t2 | x = (lift h d t2) & (subst1 (minus i h) u t1 t2)).
+ Intros; XElim H; Clear x; Intros;
+ Try Subst0Gen; XEAuto.
+ Qed.
+
+ End subst1_gen_lift.
+
+ Tactic Definition Subst1Gen :=
+ Match Context With
+ | [ H: (subst1 ?0 (lift ?1 ?2 ?3) (lift ?1 (S (plus ?0 ?2)) ?4) ?5) |- ? ] ->
+ LApply (subst1_gen_lift_lt ?3 ?4 ?5 ?0 ?1 ?2); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (subst1 ?0 ?1 (lift (1) ?0 ?2) ?3) |- ? ] ->
+ LApply (subst1_gen_lift_eq ?2 ?1 ?3 (1) ?0 ?0); [ Intros H_x | XAuto ];
+ LApply H_x; [ Clear H_x; Intros H_x | Rewrite plus_sym; XAuto ];
+ LApply H_x; [ Clear H H_x; Intros | XAuto ]
+ | [ H0: (subst1 ?0 ?1 (lift (1) ?4 ?2) ?3); H1: (lt ?4 ?0) |- ? ] ->
+ LApply (subst1_gen_lift_ge ?1 ?2 ?3 ?0 (1) ?4); [ Clear H0; Intros H0 | XAuto ];
+ LApply H0; [ Clear H0; Intros H0 | Rewrite plus_sym; XAuto ];
+ XElim H0; Intros
+ | _ -> Subst1GenBase.
--- /dev/null
+(*#* #stop file *)
+
+Require lift_props.
+Require subst0_lift.
+Require subst1_defs.
+
+ Section subst1_lift. (****************************************************)
+
+ Theorem subst1_lift_lt : (t1,t2,u:?; i:?) (subst1 i u t1 t2) ->
+ (d:?) (lt i d) -> (h:?)
+ (subst1 i (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)).
+ Intros until 1; XElim H; Clear t2; XAuto.
+ Qed.
+
+ Theorem subst1_lift_ge : (t1,t2,u:?; i,h:?) (subst1 i u t1 t2) ->
+ (d:?) (le d i) ->
+ (subst1 (plus i h) u (lift h d t1) (lift h d t2)).
+ Intros until 1; XElim H; Clear t2; XAuto.
+ Qed.
+
+ End subst1_lift.
+
+ Hints Resolve subst1_lift_lt subst1_lift_ge : ltlc.
--- /dev/null
+(*#* #stop file *)
+
+Require subst0_subst0.
+Require subst1_defs.
+
+ Section subst1_subst1. (**************************************************)
+
+ Theorem subst1_subst1: (t1,t2,u2:?; j:?) (subst1 j u2 t1 t2) ->
+ (u1,u:?; i:?) (subst1 i u u1 u2) ->
+ (EX t | (subst1 j u1 t1 t) & (subst1 (S (plus i j)) u t t2)).
+ Intros until 1; XElim H; Clear t2.
+(* case 1: subst1_refl on first premise *)
+ XEAuto.
+(* case 2: subst1_single on first premise *)
+ Intros until 2; InsertEq H0 u2; XElim H0; Clear y; Intros.
+(* case 2.1: subst1_refl on second premise *)
+ Rewrite H0; Clear H0 u1; XEAuto.
+(* case 2.2: subst1_single on second premise *)
+ Rewrite H1 in H0; Clear H1 t0; Subst0Subst0; XEAuto.
+ Qed.
+
+ Theorem subst1_subst1_back: (t1,t2,u2:?; j:?) (subst1 j u2 t1 t2) ->
+ (u1,u:?; i:?) (subst1 i u u2 u1) ->
+ (EX t | (subst1 j u1 t1 t) & (subst1 (S (plus i j)) u t2 t)).
+ Intros until 1; XElim H; Clear t2.
+(* case 1: subst1_refl on first premise *)
+ XEAuto.
+(* case 2: subst1_single on first premise *)
+ Intros until 2; XElim H0; Clear u1; Intros;
+ Try Subst0Subst0; XEAuto.
+ Qed.
+
+ Theorem subst1_trans: (t2,t1,v:?; i:?) (subst1 i v t1 t2) ->
+ (t3:?) (subst1 i v t2 t3) ->
+ (subst1 i v t1 t3).
+ Intros until 1; XElim H; Clear t2.
+(* case 1: subst1_refl on first premise *)
+ XEAuto.
+(* case 2: subst1_single on first premise *)
+ Intros until 2; XElim H0; Clear t3; XEAuto.
+ Qed.
+
+ End subst1_subst1.
+
+ Hints Resolve subst1_trans : ltlc.
+
+ Tactic Definition Subst1Subst1 :=
+ Match Context With
+ | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (subst1 ?4 ?5 ?6 ?1) |- ? ] ->
+ LApply (subst1_subst1 ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
+ LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (subst0 ?4 ?5 ?6 ?1) |- ? ] ->
+ LApply (subst1_subst1 ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
+ LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros
+ | [ H1: (subst1 ?0 ?1 ?2 ?3); H2: (subst1 ?4 ?5 ?1 ?6) |- ? ] ->
+ LApply (subst1_subst1_back ?2 ?3 ?1 ?0); [ Intros H_x | XAuto ];
+ LApply (H_x ?6 ?5 ?4); [ Clear H_x; Intros H_x | XAuto ];
+ XElim H_x; Intros.
--- /dev/null
+(*#* #stop file *)
+
+Require Export Base.
+
+ Inductive Set B := Abbr : B
+ | Abst : B
+ | Void : B.
+
+ Inductive Set F := Appl : F
+ | Cast : F.
+
+ Inductive Set K := Bind : B -> K
+ | Flat : F -> K.
+
+ Inductive Set T := TSort : nat -> T
+ | TBRef : nat -> T
+ | TTail : K -> T -> T -> T.
+
+ Hint f3KTT : ltlc := Resolve (f_equal3 K T T).
+
+ Definition s: K -> nat -> nat := [k;i] Cases k of
+ | (Bind _) => (S i)
+ | (Flat _) => i
+ end.
+
+ Section s_props. (********************************************************)
+
+ Theorem s_S: (k:?; i:?) (s k (S i)) = (S (s k i)).
+ XElim k; XAuto.
+ Qed.
+
+ Theorem s_plus: (k:?; i,j:?) (s k (plus i j)) = (plus (s k i) j).
+ XElim k; XAuto.
+ Qed.
+
+ Theorem s_plus_sym: (k:?; i,j:?) (s k (plus i j)) = (plus i (s k j)).
+ XElim k; [ Intros; Simpl; Rewrite plus_n_Sm | Idtac ]; XAuto.
+ Qed.
+
+ Theorem s_minus: (k:?; i,j:?) (le j i) ->
+ (s k (minus i j)) = (minus (s k i) j).
+ XElim k; [ Intros; Unfold s; Cbv Iota | XAuto ].
+ Rewrite minus_Sn_m; XAuto.
+ Qed.
+
+ Theorem minus_s_s: (k:?; i,j:?) (minus (s k i) (s k j)) = (minus i j).
+ XElim k; XAuto.
+ Qed.
+
+ Theorem s_le: (k:?; i,j:?) (le i j) -> (le (s k i) (s k j)).
+ XElim k; Simpl; XAuto.
+ Qed.
+
+ Theorem s_lt: (k:?; i,j:?) (lt i j) -> (lt (s k i) (s k j)).
+ XElim k; Simpl; XAuto.
+ Qed.
+
+ Theorem s_inj: (k:?; i,j:?) (s k i) = (s k j) -> i = j.
+ XElim k; XEAuto.
+ Qed.
+
+ End s_props.
+
+ Hints Resolve s_le s_lt s_inj.
+
+ Tactic Definition SRw :=
+ Repeat (Rewrite s_S Orelse Rewrite s_plus_sym).
+
+ Tactic Definition SRwIn H :=
+ Repeat (Rewrite s_S in H Orelse Rewrite s_plus in H).
+
+ Tactic Definition SRwBack :=
+ Repeat (Rewrite <- s_S Orelse Rewrite <- s_plus Orelse Rewrite <- s_plus_sym).
+
+ Tactic Definition SRwBackIn H :=
+ Repeat (Rewrite <- s_S in H Orelse Rewrite <- s_plus in H Orelse Rewrite <- s_plus_sym in H).
+
+ Hint discr : ltlc := Extern 4 (le ? (plus (s ? ?) ?)) SRwBack.
--- /dev/null
+(*#* #stop file *)
+
+Require Export terms_defs.
+
+ Definition wadd : (nat -> nat) -> nat -> (nat -> nat) :=
+ [f;w;n] Cases n of (0) => w | (S m) => (f m) end.
+
+ Fixpoint weight_map [f:nat->nat; t:T] : nat := Cases t of
+ | (TSort n) => (0)
+ | (TBRef n) => (f n)
+ | (TTail (Bind Abbr) u t) =>
+ (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f u))) t)))
+ | (TTail (Bind _) u t) =>
+ (S (plus (weight_map f u) (weight_map (wadd f (0)) t)))
+ | (TTail _ u t) => (S (plus (weight_map f u) (weight_map f t)))
+ end.
+
+ Definition weight : T -> nat := (weight_map [_](0)).
+
+ Definition tlt : T -> T -> Prop := [t1,t2](lt (weight t1) (weight t2)).
+
+ Section wadd_props. (*****************************************************)
+
+ Theorem wadd_le: (f,g:?) ((n:?) (le (f n) (g n))) -> (v,w:?) (le v w) ->
+ (n:?) (le (wadd f v n) (wadd g w n)).
+ XElim n; Simpl; XAuto.
+ Qed.
+
+ Theorem wadd_lt: (f,g:?) ((n:?) (le (f n) (g n))) -> (v,w:?) (lt v w) ->
+ (n:?) (le (wadd f v n) (wadd g w n)).
+ XElim n; Simpl; XAuto.
+ Qed.
+
+ Theorem wadd_O: (n:?) (wadd [_](0) (0) n) = (0).
+ XElim n; XAuto.
+ Qed.
+
+ End wadd_props.
+
+ Hints Resolve wadd_le wadd_lt wadd_O : ltlc.
+
+ Section weight_props. (***************************************************)
+
+ Theorem weight_le : (t:?; f,g:?) ((n:?) (le (f n) (g n))) ->
+ (le (weight_map f t) (weight_map g t)).
+ XElim t; [ XAuto | Simpl; XAuto | Idtac ].
+ XElim k; Simpl; [ Idtac | XAuto ].
+ XElim b; Auto 7 with ltlc. (**)
+ Qed.
+
+ Theorem weight_eq : (t:?; f,g:?) ((n:?) (f n) = (g n)) ->
+ (weight_map f t) = (weight_map g t).
+ Intros; Apply le_antisym; Apply weight_le;
+ Intros; Rewrite (H n); XAuto.
+ Qed.
+
+ Hints Resolve weight_le weight_eq : ltlc.
+
+ Theorem weight_add_O : (t:?) (weight_map (wadd [_](0) (0)) t) = (weight_map [_](0) t).
+ XAuto.
+ Qed.
+
+ Theorem weight_add_S : (t:?; m:?) (le (weight_map (wadd [_](0) (0)) t) (weight_map (wadd [_](0) (S m)) t)).
+ XAuto.
+ Qed.
+
+ End weight_props.
+
+ Hints Resolve weight_le weight_add_S : ltlc.
+
+ Section tlt_props. (******************************************************)
+
+ Theorem tlt_trans: (v,u,t:?) (tlt u v) -> (tlt v t) -> (tlt u t).
+ Unfold tlt; XEAuto.
+ Qed.
+
+ Theorem tlt_tail_sx: (k:?; u,t:?) (tlt u (TTail k u t)).
+ Unfold tlt weight.
+ XElim k; Simpl; [ XElim b | Idtac ]; XAuto.
+ Qed.
+
+ Theorem tlt_tail_dx: (k:?; u,t:?) (tlt t (TTail k u t)).
+ Unfold tlt weight.
+ XElim k; Simpl; [ Idtac | XAuto ].
+ XElim b; Intros; Try Rewrite weight_add_O; [ Idtac | XAuto | XAuto ].
+ EApply lt_le_trans; [ Apply lt_n_Sn | Apply le_n_S ].
+ EApply le_trans; [ Rewrite <- (weight_add_O t); Apply weight_add_S | XAuto ].
+ Qed.
+
+ End tlt_props.
+
+ Hints Resolve tlt_tail_sx tlt_tail_dx tlt_trans : ltlc.
+
+ Section tlt_wf. (*********************************************************)
+
+ Local Q: (T -> Prop) -> nat -> Prop :=
+ [P;n] (t:?) (weight t) = n -> (P t).
+
+ Remark q_ind: (P:T->Prop)((n:?) (Q P n)) -> (t:?) (P t).
+ Unfold Q; Intros.
+ Apply (H (weight t) t); XAuto.
+ Qed.
+
+ Theorem tlt_wf_ind: (P:T->Prop)
+ ((t:?)((v:?)(tlt v t) -> (P v)) -> (P t)) ->
+ (t:?)(P t).
+ Unfold tlt; Intros.
+ XElimUsing q_ind t; Intros.
+ Apply lt_wf_ind; Clear n; Intros.
+ Unfold Q in H0; Unfold Q; Intros.
+ Rewrite <- H1 in H0; Clear H1.
+ Apply H; XEAuto.
+ Qed.
+
+ End tlt_wf.
--- /dev/null
+Require Export pc3_defs.
+
+(*#* #stop record *)
+
+ Record G : Set := {
+ f : nat -> nat;
+ f_inc : (n:?) (lt n (f n))
+ }.
+
+(*#* #start record *)
+
+(*#* #caption "typing",
+ "well formed context sort", "well formed context binder",
+ "conversion rule", "typed sort", "typed reference to abbreviation",
+ "typed reference to abstraction", "typed binder", "typed application",
+ "typed cast"
+*)
+ Inductive wf0 [g:G] : C -> Prop :=
+ | wf0_sort : (m:?) (wf0 g (CSort m))
+ | wf0_bind : (c:?; u,t:?) (ty0 g c u t) ->
+ (b:?) (wf0 g (CTail c (Bind b) u))
+ with ty0 [g:G] : C -> T -> T -> Prop :=
+(* structural rules *)
+ | ty0_conv : (c:?; t2,t:?) (ty0 g c t2 t) ->
+ (u,t1:?) (ty0 g c u t1) -> (pc3 c t1 t2) ->
+ (ty0 g c u t2)
+(* axiom rules *)
+ | ty0_sort : (c:?) (wf0 g c) ->
+ (m:?) (ty0 g c (TSort m) (TSort (f g m)))
+ | ty0_abbr : (c:?) (wf0 g c) ->
+ (n:?; d:?; u:?) (drop n (0) c (CTail d (Bind Abbr) u)) ->
+ (t:?) (ty0 g d u t) ->
+ (ty0 g c (TBRef n) (lift (S n) (0) t))
+ | ty0_abst : (c:?) (wf0 g c) ->
+ (n:?; d:?; u:?) (drop n (0) c (CTail d (Bind Abst) u)) ->
+ (t:?) (ty0 g d u t) ->
+ (ty0 g c (TBRef n) (lift (S n) (0) u))
+ | ty0_bind : (c:?; u,t:?) (ty0 g c u t) ->
+ (b:?; t1,t2:?) (ty0 g (CTail c (Bind b) u) t1 t2) ->
+ (t0:?) (ty0 g (CTail c (Bind b) u) t2 t0) ->
+ (ty0 g c (TTail (Bind b) u t1) (TTail (Bind b) u t2))
+ | ty0_appl : (c:?; w,u:?) (ty0 g c w u) ->
+ (v,t:?) (ty0 g c v (TTail (Bind Abst) u t)) ->
+ (ty0 g c (TTail (Flat Appl) w v)
+ (TTail (Flat Appl) w (TTail (Bind Abst) u t))
+ )
+ | ty0_cast : (c:?; t1,t2:?) (ty0 g c t1 t2) ->
+ (t0:?) (ty0 g c t2 t0) ->
+ (ty0 g c (TTail (Flat Cast) t2 t1) t2).
+
+ Hint wf0 : ltlc := Constructors wf0.
+
+ Hint ty0 : ltlc := Constructors ty0.
+
+(*#* #caption "generation lemma of typing" #clauses ty0_gen_base *)
+
+ Section ty0_gen_base. (***************************************************)
+
+(*#* #caption "generation lemma for sort" *)
+(*#* #cap #cap c #alpha x in T, n in h *)
+
+ Theorem ty0_gen_sort: (g:?; c:?; x:?; n:?)
+ (ty0 g c (TSort n) x) ->
+ (pc3 c (TSort (f g n)) x).
+
+(*#* #stop proof *)
+
+ Intros until 1; InsertEq H '(TSort n); XElim H; Intros.
+(* case 1 : ty0_conv *)
+ XEAuto.
+(* case 2 : ty0_sort *)
+ Inversion H0; XAuto.
+(* case 3 : ty0_abbr *)
+ Inversion H3.
+(* case 4 : ty0_abst *)
+ Inversion H3.
+(* case 5 : ty0_bind *)
+ Inversion H5.
+(* case 6 : ty0_appl *)
+ Inversion H3.
+(* case 7 : ty0_cast *)
+ Inversion H3.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "generation lemma for bound reference" *)
+(*#* #cap #cap c, e #alpha x in T, t in U, u in V, n in i *)
+
+ Theorem ty0_gen_bref: (g:?; c:?; x:?; n:?) (ty0 g c (TBRef n) x) ->
+ (EX e u t | (pc3 c (lift (S n) (0) t) x) &
+ (drop n (0) c (CTail e (Bind Abbr) u)) &
+ (ty0 g e u t)
+ ) \/
+ (EX e u t | (pc3 c (lift (S n) (0) u) x) &
+ (drop n (0) c (CTail e (Bind Abst) u)) &
+ (ty0 g e u t)
+ ).
+
+(*#* #stop proof *)
+
+ Intros until 1; InsertEq H '(TBRef n); XElim H; Intros.
+(* case 1 : ty0_conv *)
+ LApply H2; [ Clear H2; Intros H2 | XAuto ].
+ XElim H2; Intros; XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+ Inversion H0.
+(* case 3 : ty0_abbr *)
+ Inversion H3 ; Rewrite H5 in H0; XEAuto.
+(* case 4 : ty0_abst *)
+ Inversion H3; Rewrite H5 in H0; XEAuto.
+(* case 5 : ty0_bind *)
+ Inversion H5.
+(* case 6 : ty0_appl *)
+ Inversion H3.
+(* case 7 : ty0_cast *)
+ Inversion H3.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "generation lemma for binder" *)
+(*#* #cap #cap c #alpha x in T, t1 in U1, t2 in U2, u in V, t in U, t0 in U3 *)
+
+ Theorem ty0_gen_bind: (g:?; b:?; c:?; u,t1,x:?) (ty0 g c (TTail (Bind b) u t1) x) ->
+ (EX t2 t t0 | (pc3 c (TTail (Bind b) u t2) x) &
+ (ty0 g c u t) &
+ (ty0 g (CTail c (Bind b) u) t1 t2) &
+ (ty0 g (CTail c (Bind b) u) t2 t0)
+ ).
+
+(*#* #stop proof *)
+
+ Intros until 1; InsertEq H '(TTail (Bind b) u t1); XElim H; Intros.
+(* case 1 : ty0_conv *)
+ LApply H2; [ Clear H2; Intros H2 | XAuto ].
+ XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+ Inversion H0.
+(* case 3 : ty0_abbr *)
+ Inversion H3.
+(* case 4 : ty0_abst *)
+ Inversion H3.
+(* case 5 : ty0_bind *)
+ Inversion H5.
+ Rewrite H7 in H1; Rewrite H7 in H3.
+ Rewrite H8 in H; Rewrite H8 in H1; Rewrite H8 in H3.
+ Rewrite H9 in H1; XEAuto.
+(* case 6 : ty0_appl *)
+ Inversion H3.
+(* case 7 : ty0_cast *)
+ Inversion H3.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "generation lemma for application" *)
+(*#* #cap #cap c #alpha x in T, v in U1, w in V1, u in V2, t in U2 *)
+
+ Theorem ty0_gen_appl: (g:?; c:?; w,v,x:?) (ty0 g c (TTail (Flat Appl) w v) x) ->
+ (EX u t | (pc3 c (TTail (Flat Appl) w (TTail (Bind Abst) u t)) x) &
+ (ty0 g c v (TTail (Bind Abst) u t)) &
+ (ty0 g c w u)
+ ).
+
+(*#* #stop proof *)
+
+ Intros until 1; InsertEq H '(TTail (Flat Appl) w v); XElim H; Intros.
+(* case 1 : ty0_conv *)
+ LApply H2; [ Clear H2; Intros H2 | XAuto ].
+ XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+ Inversion H0.
+(* case 3 : ty0_abbr *)
+ Inversion H3.
+(* case 4 : ty0_abst *)
+ Inversion H3.
+(* case 5 : ty0_bind *)
+ Inversion H5.
+(* case 6 : ty0_appl *)
+ Inversion H3; Rewrite H5 in H; Rewrite H6 in H1; XEAuto.
+(* case 7 : ty0_cast *)
+ Inversion H3.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "generation lemma for cast" *)
+(*#* #cap #cap c #alpha x in T, t2 in V, t1 in U *)
+
+ Theorem ty0_gen_cast: (g:?; c:?; t1,t2,x:?)
+ (ty0 g c (TTail (Flat Cast) t2 t1) x) ->
+ (pc3 c t2 x) /\ (ty0 g c t1 t2).
+
+(*#* #stop proof *)
+
+ Intros until 1; InsertEq H '(TTail (Flat Cast) t2 t1); XElim H; Intros.
+(* case 1 : ty0_conv *)
+ LApply H2; [ Clear H2; Intros H2 | XAuto ].
+ XElim H2; XEAuto.
+(* case 2 : ty0_sort *)
+ Inversion H0.
+(* case 3 : ty0_abbr *)
+ Inversion H3.
+(* case 4 : ty0_abst *)
+ Inversion H3.
+(* case 5 : ty0_bind *)
+ Inversion H5.
+(* case 6 : ty0_appl *)
+ Inversion H3.
+(* case 7 : ty0_cast *)
+ Inversion H3; Rewrite H5 in H; Rewrite H5 in H1; Rewrite H6 in H; XAuto.
+ Qed.
+
+ End ty0_gen_base.
+
+ Tactic Definition Ty0GenBase :=
+ Match Context With
+ | [ H: (ty0 ?1 ?2 (TSort ?3) ?4) |- ? ] ->
+ LApply (ty0_gen_sort ?1 ?2 ?4 ?3); [ Clear H; Intros | XAuto ]
+ | [ H: (ty0 ?1 ?2 (TBRef ?3) ?4) |- ? ] ->
+ LApply (ty0_gen_bref ?1 ?2 ?4 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros H; XElim H; Intros
+ | [ H: (ty0 ?1 ?2 (TTail (Bind ?3) ?4 ?5) ?6) |- ? ] ->
+ LApply (ty0_gen_bind ?1 ?3 ?2 ?4 ?5 ?6); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (ty0 ?1 ?2 (TTail (Flat Appl) ?3 ?4) ?5) |- ? ] ->
+ LApply (ty0_gen_appl ?1 ?2 ?3 ?4 ?5); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (ty0 ?1 ?2 (TTail (Flat Cast) ?3 ?4) ?5) |- ? ] ->
+ LApply (ty0_gen_cast ?1 ?2 ?4 ?3 ?5); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+ Section wf0_props. (******************************************************)
+
+ Theorem wf0_ty0 : (g:?; c:?; u,t:?) (ty0 g c u t) -> (wf0 g c).
+ Intros; XElim H; XAuto.
+ Qed.
+
+ Hints Resolve wf0_ty0 : ltlc.
+
+ Theorem wf0_drop_O : (c,e:?; h:?) (drop h (0) c e) ->
+ (g:?) (wf0 g c) -> (wf0 g e).
+ XElim c.
+(* case 1 : CSort *)
+ Intros; DropGenBase; Rewrite H; XAuto.
+(* case 2 : CTail k *)
+ Intros c IHc; XElim k; (
+ XElim h; Intros; DropGenBase;
+ [ Rewrite H in H0; XAuto | Inversion H1; XEAuto ] ).
+ Qed.
+
+ End wf0_props.
+
+ Hints Resolve wf0_ty0 wf0_drop_O : ltlc.
+
+ Tactic Definition Wf0Ty0 :=
+ Match Context With
+ [ _: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
+ LApply (wf0_ty0 ?1 ?2 ?3 ?4); [ Intros H_x | XAuto ];
+ Inversion_clear H_x.
+
+ Tactic Definition Wf0DropO :=
+ Match Context With
+ | [ _: (drop ?1 (0) ?2 ?3); _: (wf0 ?4 ?2) |- ? ] ->
+ LApply (wf0_drop_O ?2 ?3 ?1); [ Intros H_x | XAuto ];
+ LApply (H_x ?4); [ Clear H_x; Intros | XAuto ].
+
+ Section wf0_facilities. (*************************************************)
+
+ Theorem wf0_drop_wf0 : (g:?; c:?) (wf0 g c) ->
+ (b:?; e:?; u:?; h:?)
+ (drop h (0) c (CTail e (Bind b) u)) -> (wf0 g e).
+ Intros.
+ Wf0DropO; Inversion H1; XEAuto.
+ Qed.
+
+ Theorem ty0_drop_wf0 : (g:?; c:?; t1,t2:?) (ty0 g c t1 t2) ->
+ (b:?; e:?; u:?; h:?)
+ (drop h (0) c (CTail e (Bind b) u)) -> (wf0 g e).
+ Intros.
+ EApply wf0_drop_wf0; [ Idtac | EApply H0 ]; XEAuto.
+ Qed.
+
+ End wf0_facilities.
+
+ Hints Resolve wf0_drop_wf0 ty0_drop_wf0 : ltlc.
+
+ Tactic Definition DropWf0 :=
+ Match Context With
+ | [ _: (ty0 ?1 ?2 ?3 ?4);
+ _: (drop ?5 (0) ?2 (CTail ?6 (Bind ?7) ?8)) |- ? ] ->
+ LApply (ty0_drop_wf0 ?1 ?2 ?3 ?4); [ Intros H_x | XAuto ];
+ LApply (H_x ?7 ?6 ?8 ?5); [ Clear H_x; Intros | XAuto ]
+ | [ _: (wf0 ?1 ?2);
+ _: (drop ?5 (0) ?2 (CTail ?6 (Bind ?7) ?8)) |- ? ] ->
+ LApply (wf0_drop_wf0 ?1 ?2); [ Intros H_x | XAuto ];
+ LApply (H_x ?7 ?6 ?8 ?5); [ Clear H_x; Intros | XAuto ].
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+(*#* #stop file *)
+
+Require lift_gen.
+Require lift_props.
+Require subst1_defs.
+Require subst1_lift.
+Require subst1_confluence.
+Require drop_props.
+Require csubst1_defs.
+Require pc3_gen.
+Require pc3_gen_context.
+Require ty0_defs.
+Require ty0_lift.
+
+(* NOTE: these break the recursion between ty0_sred_cpr0_pr0 and ty0_gen_lift *)
+
+ Section ty0_gen_cabbr. (**************************************************)
+
+ Tactic Definition IH d a0 a :=
+ Match Context With
+ [ H: (e:?; u:?; d:?) ? -> (a0:?) ? -> (a:?) ? -> ? -> ? |- ? ] ->
+ LApply (H e u0 d); [ Clear H; Intros H | XAuto ];
+ LApply (H a0); [ Clear H; Intros H | XAuto ];
+ LApply (H a); [ Clear H; Intros H | XEAuto ];
+ LApply H; [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+(* NOTE: This can be generalized removing the last three premises *)
+ Theorem ty0_gen_cabbr: (g:?; c:?; t1,t2:?) (ty0 g c t1 t2) ->
+ (e:?; u:?; d:?) (drop d (0) c (CTail e (Bind Abbr) u)) ->
+ (a0:?) (csubst1 d u c a0) ->
+ (a:?) (wf0 g a) -> (drop (1) d a0 a) ->
+ (EX y1 y2 | (subst1 d u t1 (lift (1) d y1)) &
+ (subst1 d u t2 (lift (1) d y2)) &
+ (ty0 g a y1 y2)
+ ).
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ Repeat IH d a0 a; EApply ex3_2_intro;
+ [ XEAuto | XEAuto | EApply ty0_conv; Try EApply pc3_gen_cabbr; XEAuto ].
+(* case 2 : ty0_sort *)
+ EApply ex3_2_intro; Try Rewrite lift_sort; XAuto.
+(* case 3 : ty0_abbr *)
+ Apply (lt_eq_gt_e n d0); Intros; Clear c t1 t2.
+(* case 3.1 : n < d0 *)
+ Clear H1; DropDis; Rewrite minus_x_Sy in H1; [ DropGenBase | XAuto ].
+ CSubst1Drop; Rewrite minus_x_Sy in H0; [ Idtac | XAuto ].
+ CSubst1GenBase; Rewrite H0 in H8; Clear H0 x; Simpl in H9.
+ Rewrite (lt_plus_minus n d0) in H6; [ Idtac | XAuto ].
+ DropDis; Rewrite H0 in H9; Clear H0 x0.
+ IH '(minus d0 (S n)) x1 x3.
+ Subst1Confluence; Rewrite H0 in H11; Clear H0 x0.
+ Pattern 3 d0; Rewrite (le_plus_minus_sym (S n) d0); [ Idtac | XAuto ].
+ Pattern 4 d0; Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
+ EApply ex3_2_intro;
+ [ Rewrite lift_bref_lt | Rewrite lift_d | EApply ty0_abbr ]; XEAuto.
+(* case 3.2 : n = d0 *)
+ Rewrite H7; Rewrite H7 in H0; Clear H2 H7 n.
+ DropDis; Inversion H0; Rewrite H8 in H4; Clear H0 H7 H8 e u0.
+ CSubst1Drop; DropDis.
+ EApply ex3_2_intro;
+ [ EApply subst1_single; Rewrite lift_free; Simpl; XEAuto
+ | Rewrite lift_free; Simpl; XEAuto
+ | XEAuto ].
+(* case 3.3 : n > d0 *)
+ Clear H2 H3 e; CSubst1Drop; DropDis.
+ Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
+ Arith4c '(0) '(minus n (1)).
+ EApply ex3_2_intro;
+ [ Rewrite lift_bref_ge
+ | Rewrite lift_free; Simpl
+ | Pattern 2 n; Rewrite (minus_x_SO n)
+ ]; XEAuto.
+(* case 4 : ty0_abst *)
+ Apply (lt_eq_gt_e n d0); Intros; Clear c t1 t2.
+(* case 4.1 : n < d0 *)
+ Clear H1; DropDis; Rewrite minus_x_Sy in H1; [ DropGenBase | XAuto ].
+ CSubst1Drop; Rewrite minus_x_Sy in H0; [ Idtac | XAuto ].
+ CSubst1GenBase; Rewrite H0 in H8; Clear H0 x; Simpl in H9.
+ Rewrite (lt_plus_minus n d0) in H6; [ Idtac | XAuto ].
+ DropDis; Rewrite H0 in H9; Clear H0 x0.
+ IH '(minus d0 (S n)) x1 x3.
+ Subst1Confluence; Rewrite H0 in H11; Clear H0 x0.
+ Pattern 3 d0; Rewrite (le_plus_minus_sym (S n) d0); [ Idtac | XAuto ].
+ Pattern 4 d0; Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
+ EApply ex3_2_intro;
+ [ Rewrite lift_bref_lt | Rewrite lift_d | EApply ty0_abst ]; XEAuto.
+(* case 4.2 : n = d0 *)
+ Rewrite H7; Rewrite H7 in H0; DropDis; Inversion H0.
+(* case 4.3 : n > d0 *)
+ Clear H2 H3 e; CSubst1Drop; DropDis.
+ Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
+ Arith4c '(0) '(minus n (1)).
+ EApply ex3_2_intro;
+ [ Rewrite lift_bref_ge
+ | Rewrite lift_free; Simpl
+ | Pattern 2 n; Rewrite (minus_x_SO n)
+ ]; XEAuto.
+(* case 5 : ty0_bind *)
+ IH d a0 a; Clear H H1 H3 c t1 t2.
+ IH '(S d) '(CTail a0 (Bind b) (lift (1) d x0)) '(CTail a (Bind b) x0).
+ IH '(S d) '(CTail a0 (Bind b) (lift (1) d x0)) '(CTail a (Bind b) x0).
+ Subst1Confluence; Rewrite H4 in H11; Clear H4 x5.
+ EApply ex3_2_intro; Try Rewrite lift_bind; XEAuto.
+(* case 6 : ty0_appl *)
+ Repeat IH d a0 a; Clear H H1 c t1 t2.
+ Subst1GenBase; SymEqual; LiftGenBase; Rewrite H in H8; Rewrite H11 in H1; Rewrite H12 in H7; Clear H H11 H12 x1 x4 x5.
+ Subst1Confluence; Rewrite H in H8; Clear H x6.
+ EApply ex3_2_intro; Try Rewrite lift_flat;
+ [ Idtac | EApply subst1_tail; [ Idtac | Rewrite lift_bind ] | Idtac ]; XEAuto.
+(* case 7 : ty0_cast *)
+ Rename u into u0; Repeat IH d a0 a; Clear H H1 c t1 t2.
+ Subst1Confluence; Rewrite H in H10; Clear H x3.
+ EApply ex3_2_intro; [ Rewrite lift_flat | Idtac | Idtac ]; XEAuto.
+ Qed.
+
+ End ty0_gen_cabbr.
+
+ Section ty0_gen_cvoid. (**************************************************)
+
+ Tactic Definition IH d a :=
+ Match Context With
+ [ H: (e:?; u:?; d:?) ? -> (a:?) ? -> ? -> ? |- ? ] ->
+ LApply (H e u0 d); [ Clear H; Intros H | XAuto ];
+ LApply (H a); [ Clear H; Intros H | XEAuto ];
+ LApply H; [ Clear H; Intros H | XAuto ];
+ XElim H; Intros.
+
+(* NOTE: This can be generalized removing the last two premises *)
+ Theorem ty0_gen_cvoid: (g:?; c:?; t1,t2:?) (ty0 g c t1 t2) ->
+ (e:?; u:?; d:?) (drop d (0) c (CTail e (Bind Void) u)) ->
+ (a:?) (wf0 g a) -> (drop (1) d c a) ->
+ (EX y1 y2 | t1 = (lift (1) d y1) &
+ t2 = (lift (1) d y2) &
+ (ty0 g a y1 y2)
+ ).
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ Repeat IH d a; Rewrite H0 in H3; Rewrite H7 in H3; Pc3Gen; XEAuto.
+(* case 2 : ty0_sort *)
+ EApply ex3_2_intro; Try Rewrite lift_sort; XEAuto.
+(* case 3 : ty0_abbr *)
+ Apply (lt_eq_gt_e n d0); Intros.
+(* case 3.1 : n < d0 *)
+ DropDis; Rewrite minus_x_Sy in H7; [ DropGenBase | XAuto ].
+ Rewrite (lt_plus_minus n d0) in H5; [ Idtac | XAuto ].
+ DropDis; Rewrite H0 in H2; Clear H0 H1 u.
+ IH '(minus d0 (S n)) x1; Rewrite H1; Clear H1 t.
+ LiftGen; Rewrite <- H0 in H2; Clear H0 x2.
+ Rewrite <- lift_d; [ Idtac | XAuto ].
+ Rewrite <- le_plus_minus; [ Idtac | XAuto ].
+ EApply ex3_2_intro; [ Rewrite lift_bref_lt | Idtac | EApply ty0_abbr ]; XEAuto.
+(* case 3.2 : n = d0 *)
+ Rewrite H6 in H0; DropDis; Inversion H0.
+(* case 3.3 : n > d0 *)
+ Clear H2 H3 c e t1 t2 u0; DropDis.
+ Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
+ Arith4c '(0) '(minus n (1)).
+ EApply ex3_2_intro;
+ [ Rewrite lift_bref_ge
+ | Rewrite lift_free; Simpl
+ | Pattern 2 n; Rewrite (minus_x_SO n)
+ ]; XEAuto.
+(* case 4 : ty0_abst *)
+ Apply (lt_eq_gt_e n d0); Intros.
+(* case 4.1 : n < d0 *)
+ DropDis; Rewrite minus_x_Sy in H7; [ DropGenBase | XAuto ].
+ Rewrite (lt_plus_minus n d0) in H5; [ Idtac | XAuto ].
+ DropDis; Rewrite H0; Rewrite H0 in H2; Clear H0 H1 u.
+ IH '(minus d0 (S n)) x1; Clear H1 t.
+ LiftGen; Rewrite <- H0 in H2; Clear H0 x2.
+ Rewrite <- lift_d; [ Idtac | XAuto ].
+ Rewrite <- le_plus_minus; [ Idtac | XAuto ].
+ EApply ex3_2_intro; [ Rewrite lift_bref_lt | Idtac | EApply ty0_abst ]; XEAuto.
+(* case 4.2 : n = d0 *)
+ Rewrite H6 in H0; DropDis; Inversion H0.
+(* case 4.3 : n > d0 *)
+ Clear H2 H3 c e t1 t2 u0; DropDis.
+ Pattern 1 n; Rewrite (lt_plus_minus (0) n); [ Idtac | XEAuto ].
+ Arith4c '(0) '(minus n (1)).
+ EApply ex3_2_intro;
+ [ Rewrite lift_bref_ge
+ | Rewrite lift_free; [ Simpl | Simpl | Idtac ]
+ | Pattern 2 n; Rewrite (minus_x_SO n)
+ ]; XEAuto.
+(* case 5 : ty0_bind *)
+ IH d a; Rewrite H0; Rewrite H0 in H2; Rewrite H0 in H4; Clear H H0 H1 H3 H8 u t.
+ IH '(S d) '(CTail a (Bind b) x0); Rewrite H; Rewrite H in H2; Clear H H0 t3 t4.
+ IH '(S d) '(CTail a (Bind b) x0); Rewrite H; Clear H t0.
+ LiftGen; Rewrite <- H in H2; Clear H x5.
+ LiftTailRwBack; XEAuto.
+(* case 6 : ty0_appl *)
+ IH d a; Rewrite H2; Clear H H1 H2 v.
+ LiftGenBase; Rewrite H in H7; Rewrite H1; Rewrite H1 in H0; Rewrite H2; Clear H H1 H2 u t x1.
+ IH d a; Rewrite H; Clear H w.
+ LiftGen; Rewrite <- H in H1; Clear H x4.
+ LiftTailRwBack; XEAuto.
+(* case 7 : ty0_cast *)
+ Rename u into u0.
+ IH d a; Rewrite H2 in H0; Rewrite H2; Clear H H1 H2 H6 t3 t4.
+ IH d a; Rewrite H; Clear H t0.
+ LiftGen; Rewrite <- H in H1; Clear H x3.
+ LiftTailRwBack; XEAuto.
+ Qed.
+
+ End ty0_gen_cvoid.
+
+ Tactic Definition Ty0GenContext :=
+ Match Context With
+ | [ H: (ty0 ?1 (CTail ?2 (Bind Abbr) ?3) ?4 ?5) |- ? ] ->
+ LApply (ty0_gen_cabbr ?1 (CTail ?2 (Bind Abbr) ?3) ?4 ?5); [ Clear H; Intros H | XAuto ];
+ LApply (H ?2 ?3 (0)); [ Clear H; Intros H | XAuto ];
+ LApply (H (CTail ?2 (Bind Abbr) ?3)); [ Clear H; Intros H | XAuto ];
+ LApply (H ?2); [ Clear H; Intros H | XAuto ];
+ LApply H; [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (ty0 ?1 (CTail ?2 (Bind Void) ?3) ?4 ?5) |- ? ] ->
+ LApply (ty0_gen_cvoid ?1 (CTail ?2 (Bind Void) ?3) ?4 ?5); [ Clear H; Intros H | XAuto ];
+ LApply (H ?2 ?3 (0)); [ Clear H; Intros H | XAuto ];
+ LApply (H ?2); [ Clear H; Intros H | XAuto ];
+ LApply H; [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | _ -> Ty0GenBase.
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+Require lift_props.
+Require drop_props.
+Require pc3_props.
+Require ty0_defs.
+
+(*#* #caption "main properties of typing" #clauses ty0_props *)
+
+ Section ty0_lift. (*******************************************************)
+
+(*#* #caption "lift preserves types" *)
+(*#* #cap #cap t1, t2 #alpha c in C1, e in C2, d in i *)
+
+ Theorem ty0_lift : (g:?; e:?; t1,t2:?) (ty0 g e t1 t2) ->
+ (c:?) (wf0 g c) -> (d,h:?) (drop h d c e) ->
+ (ty0 g c (lift h d t1) (lift h d t2)).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ XEAuto.
+(* case 2 : ty0_sort *)
+ Repeat Rewrite lift_sort; XAuto.
+(* case 3 : ty0_abbr *)
+ Apply (lt_le_e n d0); Intros; DropDis.
+(* case 3.1 : n < d0 *)
+ Rewrite minus_x_Sy in H4; [ Idtac | XAuto ].
+ DropGenBase; Rewrite H4 in H0; Clear H4 x.
+ Rewrite lift_bref_lt; [ Idtac | XAuto ].
+ Arith8 d0 '(S n); Rewrite lift_d; [ Arith8' d0 '(S n) | XAuto ].
+ EApply ty0_abbr; XEAuto.
+(* case 3.2 : n >= d0 *)
+ Rewrite lift_bref_ge; [ Idtac | XAuto ].
+ Arith7' n; Rewrite lift_free; [ Idtac | Simpl; XAuto | XAuto ].
+ Rewrite (plus_sym h (S n)); Simpl; XEAuto.
+(* case 4: ty0_abst *)
+ Apply (lt_le_e n d0); Intros; DropDis.
+(* case 4.1 : n < d0 *)
+ Rewrite minus_x_Sy in H4; [ Idtac | XAuto ].
+ DropGenBase; Rewrite H4 in H0; Clear H4 x.
+ Rewrite lift_bref_lt; [ Idtac | XAuto ].
+ Arith8 d0 '(S n); Rewrite lift_d; [ Arith8' d0 '(S n) | XAuto ].
+ EApply ty0_abst; XEAuto.
+(* case 4.2 : n >= d0 *)
+ Rewrite lift_bref_ge; [ Idtac | XAuto ].
+ Arith7' n; Rewrite lift_free; [ Idtac | Simpl; XAuto | XAuto ].
+ Rewrite (plus_sym h (S n)); Simpl; XEAuto.
+(* case 5: ty0_bind *)
+ LiftTailRw; Simpl; EApply ty0_bind; XEAuto.
+(* case 6: ty0_appl *)
+ LiftTailRw; Simpl; EApply ty0_appl; [ Idtac | Rewrite <- lift_bind ]; XEAuto.
+(* case 7: ty0_cast *)
+ LiftTailRw; XEAuto.
+ Qed.
+
+ End ty0_lift.
+
+ Hints Resolve ty0_lift : ltlc.
+
+ Tactic Definition Ty0Lift b u :=
+ Match Context With
+ [ H: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
+ LApply (ty0_lift ?1 ?2 ?3 ?4); [ Intros H_x | XAuto ];
+ LApply (H_x (CTail ?2 (Bind b) u)); [ Clear H_x; Intros H_x | XEAuto ];
+ LApply (H_x (0) (1)); [ Clear H_x; Intros | XAuto ].
--- /dev/null
+Require drop_props.
+Require pc3_props.
+Require ty0_defs.
+Require ty0_lift.
+
+ Section ty0_correct. (****************************************************)
+
+(*#* #caption "correctness of types" *)
+(*#* #cap #cap c, t1, t2 #alpha t in T3 *)
+
+ Theorem ty0_correct : (g:?; c:?; t1,t2:?)
+ (ty0 g c t1 t2) -> (EX t | (ty0 g c t2 t)).
+
+(*#* #stop file *)
+
+ Intros; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ XEAuto.
+(* case 2 : ty0_sort *)
+ XEAuto.
+(* case 3 : ty0_abbr *)
+ XElim H2; XEAuto.
+(* case 4 : ty0_abst *)
+ XEAuto.
+(* case 5 : ty0_bind *)
+ XElim H4; XEAuto.
+(* case 6 : ty0_appl *)
+ XElim H0; XElim H2; Intros.
+ Ty0GenBase; XEAuto.
+(* case 7 : ty0_cast *)
+ XAuto.
+ Qed.
+
+ End ty0_correct.
+
+ Tactic Definition Ty0Correct :=
+ Match Context With
+ [ _: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
+ LApply (ty0_correct ?1 ?2 ?3 ?4); [ Intros H_x | XAuto ];
+ XElim H_x; Intros.
+
+(*#* #start file *)
+
+ Section ty0_shift. (******************************************************)
+
+(*#* #caption "shift lemma for types" *)
+(*#* #cap #cap t1, t2 #alpha c in C1, e in C2 *)
+
+ Theorem ty0_shift : (h:?; c,e:?) (drop h (0) c e) ->
+ (g:?; t1,t2:?) (ty0 g c t1 t2) -> (wf0 g e) ->
+ (ty0 g e (app c h t1) (app c h t2)).
+
+(*#* #stop file *)
+
+ XElim h.
+(* case 1 : h = 0 *)
+ Intros; DropGenBase; Rewrite <- H.
+ Repeat Rewrite app_O; XAuto.
+(* case 2 : h > 0 *)
+ Intros h IHh; XElim c.
+(* case 2.1 : CSort *)
+ Intros; DropGenBase; Rewrite H.
+ Simpl; XAuto.
+(* case 2.2 : CTail k *)
+ Intros c IHc; Clear IHc; XElim k; Intros; Wf0Ty0.
+ DropGenBase; Move H0 after H2; Ty0Correct.
+ Simpl; Apply IHh; [ Idtac | EApply ty0_bind | Idtac ]; XEAuto.
+ Qed.
+
+ End ty0_shift.
+
+ Hints Resolve ty0_shift : ltlc.
+
+(*#* #start file *)
+
+ Section ty0_unique. (*****************************************************)
+
+(*#* #caption "uniqueness of types" *)
+(*#* #cap #cap c, t1, t2 #alpha u in T *)
+
+ Theorem ty0_unique : (g:?; c:?; u,t1:?) (ty0 g c u t1) ->
+ (t2:?) (ty0 g c u t2) -> (pc3 c t1 t2).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ XEAuto.
+(* case 2 : ty0_sort *)
+ Ty0GenBase; XAuto.
+(* case 3 : ty0_abbr *)
+ Ty0GenBase; DropDis; Inversion H4.
+ Rewrite H7 in H2; Rewrite H8 in H2; XEAuto.
+(* case 4 : ty0_abst *)
+ Ty0GenBase; DropDis; Inversion H4.
+ Rewrite H7 in H2; Rewrite H8 in H2; XEAuto.
+(* case 5 : ty0_bind *)
+ Ty0GenBase; XEAuto.
+(* case 6 : ty0_appl *)
+ Ty0GenBase; EApply pc3_t; [ Idtac | EApply H3 ]; XEAuto.
+(* case 7 : ty0_cast *)
+ Ty0GenBase; XEAuto.
+ Qed.
+
+ End ty0_unique.
+
+ Hints Resolve ty0_unique : ltlc.
--- /dev/null
+Require lift_gen.
+Require subst1_gen.
+Require csubst1_defs.
+Require pr0_lift.
+Require pr0_subst1.
+Require cpr0_defs.
+Require cpr0_props.
+Require pc3_props.
+Require pc3_gen.
+Require ty0_defs.
+Require ty0_lift.
+Require ty0_props.
+Require ty0_subst0.
+Require ty0_gen_context.
+Require csub0_defs.
+
+(*#* #caption "subject reduction" #clauses *)
+
+(*#* #stop file *)
+
+ Section ty0_sred_cpr0_pr0. (**********************************************)
+
+ Tactic Definition IH H c2 t2 :=
+ LApply (H c2); [ Intros H_x | XEAuto ];
+ LApply H_x; [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x t2); [ Clear H_x; Intros | XEAuto ].
+
+ Tactic Definition IH0 :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
+ IH H1 ?5 ?3.
+
+ Tactic Definition IH0c :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
+ IH H1 ?5 ?3; Clear H1.
+
+ Tactic Definition IH0B :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?6) ?7) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 (CTail ?2 (Bind ?6) ?7) ?3 ?4) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?6) ?7) ?3.
+
+ Tactic Definition IH0Bc :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?6) ?7) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 (CTail ?2 (Bind ?6) ?7) ?3 ?4) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?6) ?7) ?3; Clear H1.
+
+ Tactic Definition IH1 :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 ?5 ?6.
+
+ Tactic Definition IH1c :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 ?5 ?6; Clear H1.
+
+ Tactic Definition IH1Bc :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?7) ?8) ?6; Clear H1.
+
+ Tactic Definition IH1BLc :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 (lift ?10 ?11 ?3) t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?7) ?8) '(lift ?10 ?11 ?6); Clear H1.
+
+ Tactic Definition IH1T :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 (TTail ?7 ?8 ?3) t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 ?5 '(TTail ?7 ?8 ?6).
+
+ Tactic Definition IH1T2c :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 (TTail ?7 ?8 ?3) t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6); H4: (pr0 ?8 ?9) |- ? ] ->
+ IH H1 ?5 '(TTail ?7 ?9 ?6); Clear H1.
+
+ Tactic Definition IH3B :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6); H4: (pr0 ?8 ?9) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?7) ?9) ?6.
+
+(*#* #start file *)
+
+(*#* #caption "base case" *)
+(*#* #cap #cap c1, c2 #alpha t1 in T, t2 in T1, t in T2 *)
+
+ Theorem ty0_sred_cpr0_pr0 : (g:?; c1:?; t1,t:?) (ty0 g c1 t1 t) ->
+ (c2:?) (wf0 g c2) -> (cpr0 c1 c2) ->
+ (t2:?) (pr0 t1 t2) -> (ty0 g c2 t2 t).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ IH1c; IH0c; EApply ty0_conv; XEAuto.
+(* case 2 : ty0_sort *)
+ Inversion H2; XAuto.
+(* case 3 : ty0_abbr *)
+ Inversion H5; Cpr0Drop; IH1c; XEAuto.
+(* case 4 : ty0_abst *)
+ Intros; Inversion H5; Cpr0Drop; IH0; IH1.
+ EApply ty0_conv;
+ [ EApply ty0_lift; [ Idtac | XAuto | XEAuto ]
+ | EApply ty0_abst
+ | EApply pc3_lift ]; XEAuto.
+(* case 5 : ty0_bind *)
+ Intros; Inversion H7; Clear H7.
+(* case 5.1 : pr0_refl *)
+ IH0c; IH0Bc; IH0Bc.
+ EApply ty0_bind; XEAuto.
+(* case 5.2 : pr0_cont *)
+ IH0; IH0B; Ty0Correct; IH3B; Ty0Correct.
+ EApply ty0_conv; [ EApply ty0_bind | EApply ty0_bind | Idtac ]; XEAuto.
+(* case 5.3 : pr0_delta *)
+ Rewrite <- H8 in H1; Rewrite <- H8 in H2;
+ Rewrite <- H8 in H3; Rewrite <- H8 in H4; Clear H8 b.
+ IH0; IH0B; Ty0Correct; IH3B; Ty0Correct.
+ EApply ty0_conv; [ EApply ty0_bind | EApply ty0_bind | Idtac ]; XEAuto.
+(* case 5.4 : pr0_zeta *)
+ Rewrite <- H11 in H1; Rewrite <- H11 in H2; Clear H8 H9 H10 H11 b0 t2 t7 u0.
+ IH0; IH1BLc; Move H3 after H8; IH0Bc; Ty0Correct; Move H8 after H4; Clear H H0 H1 H3 H6 c c1 t t1;
+ NewInduction b.
+(* case 5.4.1 : Abbr *)
+ Ty0GenContext; Subst1Gen; LiftGen; Rewrite H in H1; Clear H x0.
+ EApply ty0_conv;
+ [ EApply ty0_bind; XEAuto | XEAuto
+ | EApply pc3_pr3_x;
+ EApply (pr3_t (TTail (Bind Abbr) u (lift (1) (0) x1))); XEAuto ].
+(* case 5.4.2 : Abst *)
+ EqFalse.
+(* case 5.4.3 : Void *)
+ Ty0GenContext; Rewrite H0; Rewrite H0 in H2; Clear H0 t3.
+ LiftGen; Rewrite <- H in H1; Clear H x0.
+ EApply ty0_conv; [ EApply ty0_bind; XEAuto | XEAuto | XAuto ].
+(* case 6 : ty0_appl *)
+ Intros; Inversion H5; Clear H5.
+(* case 6.1 : pr0_refl *)
+ IH0c; IH0c; EApply ty0_appl; XEAuto.
+(* case 6.2 : pr0_cont *)
+ Clear H6 H7 H8 H9 c1 k t t1 t2 t3 u1.
+ IH0; Ty0Correct; Ty0GenBase; IH1c; IH0; IH1c.
+ EApply ty0_conv;
+ [ EApply ty0_appl; [ XEAuto | EApply ty0_bind; XEAuto ]
+ | EApply ty0_appl; XEAuto
+ | XEAuto ].
+(* case 6.3 : pr0_beta *)
+ Rewrite <- H7 in H1; Rewrite <- H7 in H2; Clear H6 H7 H9 c1 t t1 t2 v v1.
+ IH1T; IH0c; Ty0Correct; Ty0GenBase; IH0; IH1c.
+ Move H5 after H13; Ty0GenBase; Pc3Gen; Repeat CSub0Ty0.
+ EApply ty0_conv;
+ [ Apply ty0_appl; [ Idtac | EApply ty0_bind ]
+ | EApply ty0_bind
+ | Apply (pc3_t (TTail (Bind Abbr) v2 t0))
+ ]; XEAuto.
+(* case 6.4 : pr0_delta *)
+ Rewrite <- H7 in H1; Rewrite <- H7 in H2; Clear H6 H7 H11 c1 t t1 t2 v v1.
+ IH1T2c; Clear H1; Ty0Correct; NonLinear; Ty0GenBase; IH1; IH0c.
+ Move H5 after H1; Ty0GenBase; Pc3Gen; Rewrite lift_bind in H0.
+ Move H1 after H0; Ty0Lift b u2; Rewrite lift_bind in H17.
+ Ty0GenBase.
+ EApply ty0_conv;
+ [ Apply ty0_appl; [ Idtac | EApply ty0_bind ]; XEAuto
+ | EApply ty0_bind;
+ [ Idtac
+ | EApply ty0_appl; [ EApply ty0_lift | EApply ty0_conv ]
+ | EApply ty0_appl; [ EApply ty0_lift | EApply ty0_bind ]
+ ]; XEAuto
+ | Idtac ].
+ Rewrite <- lift_bind; Apply pc3_pc1;
+ Apply (pc1_u (TTail (Flat Appl) v2 (TTail (Bind b) u2 (lift (1) (0) (TTail (Bind Abst) u t0))))); XAuto.
+(* case 7 : ty0_cast *)
+ Intros; Inversion H5; Clear H5.
+(* case 7.1 : pr0_refl *)
+ IH0c; IH0c; EApply ty0_cast; XEAuto.
+(* case 7.2 : pr0_cont *)
+ Clear H6 H7 H8 H9 c1 k u1 t t1 t4 t5.
+ IH0; IH1c; IH1c.
+ EApply ty0_conv;
+ [ XEAuto
+ | EApply ty0_cast; [ EApply ty0_conv; XEAuto | XEAuto ]
+ | XAuto ].
+(* case 7.3 : pr0_eps *)
+ XAuto.
+ Qed.
+
+ End ty0_sred_cpr0_pr0.
+
+ Section ty0_sred_pr3. (**********************************************)
+
+ Theorem ty0_sred_pr1 : (c:?; t1,t2:?) (pr1 t1 t2) ->
+ (g:?; t:?) (ty0 g c t1 t) ->
+ (ty0 g c t2 t).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr1_r *)
+ XAuto.
+(* case 2 : pr1_u *)
+ EApply H1; EApply ty0_sred_cpr0_pr0; XEAuto.
+ Qed.
+
+ Theorem ty0_sred_pr2 : (c:?; t1,t2:?) (pr2 c t1 t2) ->
+ (g:?; t:?) (ty0 g c t1 t) ->
+ (ty0 g c t2 t).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr2_pr0 *)
+ EApply ty0_sred_cpr0_pr0; XEAuto.
+(* case 2 : pr2_u *)
+ XEAuto.
+ Qed.
+
+(*#* #start file *)
+
+(*#* #caption "general case" *)
+(*#* #cap #cap c #alpha t1 in T, t2 in T1, t in T2 *)
+
+ Theorem ty0_sred_pr3 : (c:?; t1,t2:?) (pr3 c t1 t2) ->
+ (g:?; t:?) (ty0 g c t1 t) ->
+ (ty0 g c t2 t).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply H1; EApply ty0_sred_pr2; XEAuto.
+ Qed.
+
+ End ty0_sred_pr3.
+
+ Tactic Definition Ty0SRed :=
+ Match Context With
+ | [ H1: (pr3 ?1 ?2 ?3); H2: (ty0 ?4 ?1 ?2 ?5) |- ? ] ->
+ LApply (ty0_sred_pr3 ?1 ?2 ?3); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5); [ Clear H2 H_x; Intros | XAuto ].
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+Require lift_props.
+Require drop_props.
+Require pc3_props.
+Require pc3_gen.
+Require ty0_defs.
+Require ty0_props.
+Require ty0_sred.
+
+(*#* #caption "corollaries of subject reduction" #clauses *)
+
+(*#* #stop file *)
+
+ Section ty0_gen. (********************************************************)
+
+ Tactic Definition IH e :=
+ Match Context With
+ [ H0: (t:?; d:?) ?1 = (lift ?2 d t) -> ?; H1: ?1 = (lift ?2 ?3 ?4) |- ? ] ->
+ LApply (H0 ?4 ?3); [ Clear H0 H1; Intros H0 | XAuto ];
+ LApply (H0 e); [ Clear H0; Intros H0 | XEAuto ];
+ LApply H0; [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros.
+
+(*#* #start file *)
+
+(*#* #caption "generation lemma for lift" *)
+(*#* #cap #cap t2 #alpha c in C1, e in C2, t1 in T, x in T1, d in i *)
+
+ Theorem ty0_gen_lift : (g:?; c:?; t1,x:?; h,d:?)
+ (ty0 g c (lift h d t1) x) ->
+ (e:?) (wf0 g e) -> (drop h d c e) ->
+ (EX t2 | (pc3 c (lift h d t2) x) & (ty0 g e t1 t2)).
+
+(*#* #stop file *)
+
+ Intros until 1; InsertEq H '(lift h d t1);
+ UnIntro H d; UnIntro H t1; XElim H; Intros;
+ Rename x0 into t3; Rename x1 into d0.
+(* case 1 : ty0_conv *)
+ IH e; XEAuto.
+(* case 2 : ty0_sort *)
+ LiftGenBase; Rewrite H0; Clear H0 t.
+ EApply ex2_intro; [ Rewrite lift_sort; XAuto | XAuto ].
+(* case 3 : ty0_abbr *)
+ Apply (lt_le_e n d0); Intros.
+(* case 3.1 : n < d0 *)
+ LiftGenBase; DropS; Rewrite H3; Clear H3 t3.
+ Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
+ Rewrite (lt_plus_minus n d0) in H5; [ DropDis; IH x1 | XAuto ].
+ EApply ex2_intro;
+ [ Rewrite lift_d; [ EApply pc3_lift; XEAuto | XEAuto ]
+ | EApply ty0_abbr; XEAuto ].
+(* case 3.2 : n >= d0 *)
+ Apply (lt_le_e n (plus d0 h)); Intros.
+(* case 3.2.1 : n < d0 + h *)
+ LiftGenBase.
+(* case 3.2.2 : n >= d0 + h *)
+ Rewrite (le_plus_minus_sym h n) in H3; [ Idtac | XEAuto ].
+ LiftGenBase; DropDis; Rewrite H3; Clear H3 t3.
+ EApply ex2_intro; [ Idtac | EApply ty0_abbr; XEAuto ].
+ Rewrite lift_free; [ Idtac | XEAuto | XAuto ].
+ Rewrite <- plus_n_Sm; Rewrite <- le_plus_minus; XEAuto.
+(* case 4 : ty0_abst *)
+ Apply (lt_le_e n d0); Intros.
+(* case 4.1 : n < d0 *)
+ LiftGenBase; Rewrite H3; Clear H3 t3.
+ Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
+ Rewrite (lt_plus_minus n d0) in H5; [ DropDis; Rewrite H0; IH x1 | XAuto ].
+ EApply ex2_intro; [ Rewrite lift_d | EApply ty0_abst ]; XEAuto.
+(* case 4.2 : n >= d0 *)
+ Apply (lt_le_e n (plus d0 h)); Intros.
+(* case 4.2.1 : n < d0 + h *)
+ LiftGenBase.
+(* case 4.2.2 : n >= d0 + h *)
+ Rewrite (le_plus_minus_sym h n) in H3; [ Idtac | XEAuto ].
+ LiftGenBase; DropDis; Rewrite H3; Clear H3 t3.
+ EApply ex2_intro; [ Idtac | EApply ty0_abst; XEAuto ].
+ Rewrite lift_free; [ Idtac | XEAuto | XAuto ].
+ Rewrite <- plus_n_Sm; Rewrite <- le_plus_minus; XEAuto.
+(* case 5 : ty0_bind *)
+ LiftGenBase; Rewrite H5; Rewrite H8; Rewrite H8 in H2; Clear H5 t3.
+ Move H0 after H2; IH e; IH '(CTail e (Bind b) x0); Ty0Correct.
+ EApply ex2_intro; [ Rewrite lift_bind; XEAuto | XEAuto ].
+(* case 6 : ty0_appl *)
+ LiftGenBase; Rewrite H3; Rewrite H6; Clear H3 c t3 x y.
+ IH e; IH e; Pc3Gen; Pc3T; Pc3Gen; Pc3T.
+ Move H3 after H12; Ty0Correct; Ty0SRed; Ty0GenBase; Wf0Ty0.
+ EApply ex2_intro;
+ [ Rewrite lift_flat; Apply pc3_thin_dx;
+ Rewrite lift_bind; Apply pc3_tail_21; [ EApply pc3_pr3_x | Idtac ]
+ | EApply ty0_appl;
+ [ EApply ty0_conv
+ | EApply ty0_conv; [ EApply ty0_bind | Idtac | Idtac ] ]
+ ]; XEAuto.
+(* case 7 : ty0_cast *)
+ LiftGenBase; Rewrite H3; Rewrite H6; Rewrite H6 in H0.
+ IH e; IH e; Pc3Gen; XEAuto.
+ Qed.
+
+ End ty0_gen.
+
+ Tactic Definition Ty0Gen :=
+ Match Context With
+ | [ H0: (ty0 ?1 ?2 (lift ?3 ?4 ?5) ?6);
+ H1: (drop ?3 ?4 ?2 ?7) |- ? ] ->
+ LApply (ty0_gen_lift ?1 ?2 ?5 ?6 ?3 ?4); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7); [ Clear H0; Intros H0 | XEAuto ];
+ LApply H0; [ Clear H0 H1; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | [ H0: (ty0 ?1 ?2 (lift ?3 ?4 ?5) ?6);
+ _ : (wf0 ?1 ?7) |- ? ] ->
+ LApply (ty0_gen_lift ?1 ?2 ?5 ?6 ?3 ?4); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7); [ Clear H0; Intros H0 | XAuto ];
+ LApply H0; [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | _ -> Ty0GenContext.
+
+ Section ty0_sred_props. (*************************************************)
+
+(*#* #start file *)
+
+(*#* #caption "drop preserves well-formedness" *)
+(*#* #cap #alpha c in C1, e in C2, d in i *)
+
+ Theorem wf0_drop : (c,e:?; d,h:?) (drop h d c e) ->
+ (g:?) (wf0 g c) -> (wf0 g e).
+
+(*#* #stop proof *)
+
+ XElim c.
+(* case 1 : CSort *)
+ Intros; DropGenBase; Rewrite H; XAuto.
+(* case 2 : CTail k *)
+ Intros c IHc; XElim k; (
+ XElim d;
+ [ XEAuto
+ | Intros d IHd; Intros;
+ DropGenBase; Rewrite H; Rewrite H1 in H0; Clear IHd H H1 e t;
+ Inversion H0; Clear H3 H4 b0 u ]).
+(* case 2.1 : Bind, d > 0 *)
+ Ty0Gen; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "type reduction" *)
+(*#* #cap #cap c, t1, t2 #alpha u in T *)
+
+ Theorem ty0_tred : (g:?; c:?; u,t1:?) (ty0 g c u t1) ->
+ (t2:?) (pr3 c t1 t2) -> (ty0 g c u t2).
+
+(*#* #stop proof *)
+
+ Intros; Ty0Correct; Ty0SRed; EApply ty0_conv; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "subject conversion" *)
+(*#* #cap #cap c, u1, u2, t1, t2 *)
+
+ Theorem ty0_sconv : (g:?; c:?; u1,t1:?) (ty0 g c u1 t1) ->
+ (u2,t2:?) (ty0 g c u2 t2) ->
+ (pc3 c u1 u2) -> (pc3 c t1 t2).
+
+(*#* #stop file *)
+
+ Intros; Pc3Confluence; Repeat Ty0SRed; XEAuto.
+ Qed.
+
+
+ End ty0_sred_props.
+
+(*#* #start file *)
+
+(*#* #single *)
--- /dev/null
+Require drop_props.
+Require csubst0_defs.
+Require pc3_props.
+Require pc3_subst0.
+Require ty0_defs.
+Require ty0_lift.
+Require ty0_props.
+
+ Section ty0_fsubst0. (****************************************************)
+
+(*#* #stop macro *)
+
+ Tactic Definition IH H0 v1 v2 v3 v4 v5 :=
+ LApply (H0 v1 v2 v3 v4); [ Intros H_x | XEAuto ];
+ LApply H_x; [ Clear H_x; Intros H_x | XEAuto ];
+ LApply (H_x v5); [ Clear H_x; Intros | XEAuto ].
+
+ Tactic Definition IHT :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 ?1 ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) ?1 (CTail e (Bind Abbr) u0)) -> ?;
+ _: (subst0 ?4 ?5 ?2 ?6);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H ?4 ?5 ?1 ?6 ?9.
+
+ Tactic Definition IHTb1 :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 (CTail ?1 (Bind ?11) ?10) ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) (CTail ?1 (Bind ?11) ?10) (CTail e (Bind Abbr) u0)) -> ?;
+ _: (subst0 ?4 ?5 ?10 ?6);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H '(S ?4) ?5 '(CTail ?1 (Bind ?11) ?6) ?2 ?9.
+
+ Tactic Definition IHTb2 :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 (CTail ?1 (Bind ?11) ?10) ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) (CTail ?1 (Bind ?11) ?10) (CTail e (Bind Abbr) u0)) -> ?;
+ _: (subst0 (s (Bind ?11) ?4) ?5 ?2 ?6);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H '(S ?4) ?5 '(CTail ?1 (Bind ?11) ?10) ?6 ?9.
+
+ Tactic Definition IHC :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 ?1 ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) ?1 (CTail e (Bind Abbr) u0)) -> ?;
+ _: (csubst0 ?4 ?5 ?1 ?6);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H ?4 ?5 ?6 ?2 ?9.
+
+ Tactic Definition IHCb :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 (CTail ?1 (Bind ?11) ?10) ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) (CTail ?1 (Bind ?11) ?10) (CTail e (Bind Abbr) u0)) -> ?;
+ _: (csubst0 ?4 ?5 ?1 ?6);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H '(S ?4) ?5 '(CTail ?6 (Bind ?11) ?10) ?2 ?9.
+
+ Tactic Definition IHTTb :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 (CTail ?1 (Bind ?11) ?10) ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) (CTail ?1 (Bind ?11) ?10) (CTail e (Bind Abbr) u0)) -> ?;
+ _: (subst0 ?4 ?5 ?10 ?6);
+ _: (subst0 (s (Bind ?11) ?4) ?5 ?2 ?7);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H '(S ?4) ?5 '(CTail ?1 (Bind ?11) ?6) ?7 ?9.
+
+ Tactic Definition IHCT :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 ?1 ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) ?1 (CTail e (Bind Abbr) u0)) -> ?;
+ _: (csubst0 ?4 ?5 ?1 ?6);
+ _: (subst0 ?4 ?5 ?2 ?7);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H ?4 ?5 ?6 ?7 ?9.
+
+ Tactic Definition IHCTb1 :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 (CTail ?1 (Bind ?11) ?10) ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) (CTail ?1 (Bind ?11) ?10) (CTail e (Bind Abbr) u0)) -> ?;
+ _: (csubst0 ?4 ?5 ?1 ?6);
+ _: (subst0 ?4 ?5 ?10 ?7);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H '(S ?4) ?5 '(CTail ?6 (Bind ?11) ?7) ?2 ?9.
+
+ Tactic Definition IHCTb2 :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 (CTail ?1 (Bind ?11) ?10) ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) (CTail ?1 (Bind ?11) ?10) (CTail e (Bind Abbr) u0)) -> ?;
+ _: (csubst0 ?4 ?5 ?1 ?6);
+ _: (subst0 (s (Bind ?11) ?4) ?5 ?2 ?7);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H '(S ?4) ?5 '(CTail ?6 (Bind ?11) ?10) ?7 ?9.
+
+ Tactic Definition IHCTTb :=
+ Match Context With
+ [ H: (i:nat; u0:T; c2:C; t2:T) (fsubst0 i u0 (CTail ?1 (Bind ?11) ?10) ?2 c2 t2) ->
+ (wf0 ?3 c2) ->
+ (e:C) (drop i (0) (CTail ?1 (Bind ?11) ?10) (CTail e (Bind Abbr) u0)) -> ?;
+ _: (csubst0 ?4 ?5 ?1 ?6);
+ _: (subst0 ?4 ?5 ?10 ?7);
+ _: (subst0 (s (Bind ?11) ?4) ?5 ?2 ?8);
+ _: (drop ?4 (0) ?1 (CTail ?9 (Bind Abbr) ?5)) |- ? ] ->
+ IH H '(S ?4) ?5 '(CTail ?6 (Bind ?11) ?7) ?8 ?9.
+
+(*#* #start macro *)
+
+(*#* #caption "substitution preserves types" *)
+(*#* #cap #cap c1, c2, e, t1, t2, t #alpha u in V *)
+
+(* NOTE: This breaks the mutual recursion between ty0_subst0 and ty0_csubst0 *)
+ Theorem ty0_fsubst0: (g:?; c1:?; t1,t:?) (ty0 g c1 t1 t) ->
+ (i:?; u,c2,t2:?) (fsubst0 i u c1 t1 c2 t2) ->
+ (wf0 g c2) ->
+ (e:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+ (ty0 g c2 t2 t).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H.
+(* case 1: ty0_conv *)
+ Intros until 6; XElim H4; Intros.
+(* case 1.1: fsubst0_t *)
+ IHT; EApply ty0_conv; XEAuto.
+(* case 1.2: fsubst0_c *)
+ IHC; EApply ty0_conv; Try EApply pc3_fsubst0; XEAuto.
+(* case 1.3: fsubst0_b *)
+ IHCT; IHCT; EApply ty0_conv; Try EApply pc3_fsubst0; XEAuto.
+(* case 2: ty0_sort *)
+ Intros until 2; XElim H0; Intros.
+(* case 2.1: fsubst0_t *)
+ Subst0GenBase.
+(* case 2.2: fsubst0_c *)
+ XAuto.
+(* case 2.3: fsubst0_b *)
+ Subst0GenBase.
+(* case 3: ty0_abbr *)
+ Intros until 5; XElim H3; Intros; Clear c1 c2 t t1 t2.
+(* case 3.1: fsubst0_t *)
+ Subst0GenBase; Rewrite H6; Rewrite <- H3 in H5; Clear H3 H6 i t3.
+ DropDis; Inversion H5; Rewrite <- H6 in H0; Rewrite H7 in H1; XEAuto.
+(* case 3.2: fsubst0_c *)
+ Apply (lt_le_e n i); Intros; CSubst0Drop.
+(* case 3.2.1: n < i, none *)
+ EApply ty0_abbr; XEAuto.
+(* case 3.2.2: n < i, csubst0_fst *)
+ Inversion H0; CSubst0Drop.
+ Rewrite <- H10 in H7; Rewrite <- H11 in H7; Rewrite <- H11 in H8; Rewrite <- H12 in H8;
+ Clear H0 H10 H11 H12 x0 x1 x2.
+ DropDis; Rewrite minus_x_Sy in H0; [ DropGenBase | XAuto ].
+ IHT; EApply ty0_abbr; XEAuto.
+(* case 3.2.3: n < i, csubst0_snd *)
+ Inversion H0; CSubst0Drop.
+ Rewrite <- H10 in H8; Rewrite <- H11 in H7; Rewrite <- H11 in H8; Rewrite <- H12 in H7;
+ Clear H0 H10 H11 H12 x0 x1 x3.
+ DropDis; Rewrite minus_x_Sy in H0; [ DropGenBase; CSubst0Drop | XAuto ].
+ IHC; EApply ty0_abbr; XEAuto.
+(* case 3.2.4: n < i, csubst0_both *)
+ Inversion H0; CSubst0Drop.
+ Rewrite <- H11 in H9; Rewrite <- H12 in H7; Rewrite <- H12 in H8; Rewrite <- H12 in H9; Rewrite <- H13 in H8;
+ Clear H0 H11 H12 H13 x0 x1 x3.
+ DropDis; Rewrite minus_x_Sy in H0; [ DropGenBase; CSubst0Drop | XAuto ].
+ IHCT; EApply ty0_abbr; XEAuto.
+(* case 3.2.5: n >= i *)
+ EApply ty0_abbr; XEAuto.
+(* case 3.3: fsubst0_b *)
+ Subst0GenBase; Rewrite H7; Rewrite <- H3 in H4; Rewrite <- H3 in H6; Clear H3 H7 i t3.
+ DropDis; Inversion H6; Rewrite <- H7 in H0; Rewrite H8 in H1.
+ CSubst0Drop; XEAuto.
+(* case 4: ty0_abst *)
+ Intros until 5; XElim H3; Intros; Clear c1 c2 t t1 t2.
+(* case 4.1: fsubst0_t *)
+ Subst0GenBase; Rewrite H3 in H0; DropDis; Inversion H0.
+(* case 4.2: fsubst0_c *)
+ Apply (lt_le_e n i); Intros; CSubst0Drop.
+(* case 4.2.1: n < i, none *)
+ EApply ty0_abst; XEAuto.
+(* case 4.2.2: n < i, csubst0_fst *)
+ Inversion H0; CSubst0Drop.
+ Rewrite <- H10 in H7; Rewrite <- H11 in H7; Rewrite <- H11 in H8; Rewrite <- H12 in H8; Rewrite <- H12;
+ Clear H0 H10 H11 H12 x0 x1 x2.
+ DropDis; Rewrite minus_x_Sy in H0; [ DropGenBase | XAuto ].
+ IHT; EApply ty0_conv;
+ [ EApply ty0_lift | EApply ty0_abst | EApply pc3_lift ]; XEAuto.
+(* case 4.2.3: n < i, csubst0_snd *)
+ Inversion H0; CSubst0Drop.
+ Rewrite <- H10 in H8; Rewrite <- H11 in H7; Rewrite <- H11 in H8; Rewrite <- H12 in H7; Rewrite <- H12;
+ Clear H0 H10 H11 H12 x0 x1 x3.
+ DropDis; Rewrite minus_x_Sy in H0; [ DropGenBase; CSubst0Drop | XAuto ].
+ IHC; EApply ty0_abst; XEAuto.
+(* case 4.2.4: n < i, csubst0_both *)
+ Inversion H0; CSubst0Drop.
+ Rewrite <- H11 in H9; Rewrite <- H12 in H7; Rewrite <- H12 in H8; Rewrite <- H12 in H9; Rewrite <- H13 in H8; Rewrite <- H13;
+ Clear H0 H11 H12 H13 x0 x1 x3.
+ DropDis; Rewrite minus_x_Sy in H0; [ DropGenBase; CSubst0Drop | XAuto ].
+ IHCT; IHC; EApply ty0_conv;
+ [ EApply ty0_lift | EApply ty0_abst
+ | EApply pc3_lift; Try EApply pc3_fsubst0; Try Apply H0
+ ]; XEAuto.
+(* case 4.2.4: n >= i *)
+ EApply ty0_abst; XEAuto.
+(* case 4.3: fsubst0_b *)
+ Subst0GenBase; Rewrite H3 in H0; DropDis; Inversion H0.
+(* case 5: ty0_bind *)
+ Intros until 7; XElim H5; Intros; Clear H4.
+(* case 5.1: fsubst0_t *)
+ Subst0GenBase; Rewrite H4; Clear H4 t6.
+(* case 5.1.1: subst0 on left argument *)
+ Ty0Correct; IHT; IHTb1; Ty0Correct.
+ EApply ty0_conv;
+ [ EApply ty0_bind | EApply ty0_bind | EApply pc3_fsubst0 ]; XEAuto.
+(* case 5.1.2: subst0 on right argument *)
+ IHTb2; Ty0Correct; EApply ty0_bind; XEAuto.
+(* case 5.1.3: subst0 on both arguments *)
+ Ty0Correct; IHT; IHTb1; IHTTb; Ty0Correct.
+ EApply ty0_conv;
+ [ EApply ty0_bind | EApply ty0_bind | EApply pc3_fsubst0 ]; XEAuto.
+(* case 5.2: fsubst0_c *)
+ IHC; IHCb; Ty0Correct; EApply ty0_bind; XEAuto.
+(* case 5.3: fsubst0_b *)
+ Subst0GenBase; Rewrite H4; Clear H4 t6.
+(* case 5.3.1: subst0 on left argument *)
+ IHC; IHCb; Ty0Correct; Ty0Correct; IHCT; IHCTb1; Ty0Correct.
+ EApply ty0_conv;
+ [ EApply ty0_bind | EApply ty0_bind
+ | EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]; XEAuto.
+(* case 5.3.2: subst0 on right argument *)
+ IHC; IHCTb2; Ty0Correct; EApply ty0_bind; XEAuto.
+(* case 5.3.3: subst0 on both arguments *)
+ IHC; IHCb; Ty0Correct; Ty0Correct; IHCT; IHCTTb; Ty0Correct.
+ EApply ty0_conv;
+ [ EApply ty0_bind | EApply ty0_bind
+ | EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]; XEAuto.
+(* case 6: ty0_appl *)
+ Intros until 5; XElim H3; Intros.
+(* case 6.1: fsubst0_t *)
+ Subst0GenBase; Rewrite H3; Clear H3 c1 c2 t t1 t2 t3.
+(* case 6.1.1: subst0 on left argument *)
+ Ty0Correct; Ty0GenBase; IHT; Ty0Correct.
+ EApply ty0_conv;
+ [ EApply ty0_appl | EApply ty0_appl | EApply pc3_fsubst0 ]; XEAuto.
+(* case 6.1.2: subst0 on right argument *)
+ IHT; EApply ty0_appl; XEAuto.
+(* case 6.1.3: subst0 on both arguments *)
+ Ty0Correct; Ty0GenBase; Move H after H10; Ty0Correct; IHT; Clear H2; IHT.
+ EApply ty0_conv;
+ [ EApply ty0_appl | EApply ty0_appl | EApply pc3_fsubst0 ]; XEAuto.
+(* case 6.2: fsubst0_c *)
+ IHC; Clear H2; IHC; EApply ty0_appl; XEAuto.
+(* case 6.3: fsubst0_b *)
+ Subst0GenBase; Rewrite H3; Clear H3 c1 c2 t t1 t2 t3.
+(* case 6.3.1: subst0 on left argument *)
+ IHC; Ty0Correct; Ty0GenBase; Clear H2; IHC; IHCT.
+ EApply ty0_conv;
+ [ EApply ty0_appl | EApply ty0_appl
+ | EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]; XEAuto.
+(* case 6.3.2: subst0 on right argument *)
+ IHCT; Clear H2; IHC; EApply ty0_appl; XEAuto.
+(* case 6.3.3: subst0 on both arguments *)
+ IHC; Ty0Correct; Ty0GenBase; IHCT; Clear H2; IHC; Ty0Correct; IHCT.
+ EApply ty0_conv;
+ [ EApply ty0_appl | EApply ty0_appl
+ | EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]; XEAuto.
+(* case 7: ty0_cast *)
+ Clear c1 t t1; Intros until 5; XElim H3; Intros; Clear c2 t3.
+(* case 7.1: fsubst0_t *)
+ Subst0GenBase; Rewrite H3; Clear H3 t4.
+(* case 7.1.1: subst0 on left argument *)
+ IHT; EApply ty0_conv;
+ [ Idtac
+ | EApply ty0_cast;
+ [ EApply ty0_conv; [ Idtac | Idtac | Apply pc3_s; EApply pc3_fsubst0 ]
+ | Idtac ]
+ | EApply pc3_fsubst0 ]; XEAuto.
+(* case 7.1.2: subst0 on right argument *)
+ IHT; EApply ty0_cast; XEAuto.
+(* case 7.1.3: subst0 on both arguments *)
+ IHT; Clear H2; IHT.
+ EApply ty0_conv;
+ [ Idtac
+ | EApply ty0_cast;
+ [ EApply ty0_conv; [ Idtac | Idtac | Apply pc3_s; EApply pc3_fsubst0 ]
+ | Idtac ]
+ | EApply pc3_fsubst0 ]; XEAuto.
+(* case 7.2: fsubst0_c *)
+ IHC; Clear H2; IHC; EApply ty0_cast; XEAuto.
+(* case 6.3: fsubst0_b *)
+ Subst0GenBase; Rewrite H3; Clear H3 t4.
+(* case 7.3.1: subst0 on left argument *)
+ IHC; IHCT; Clear H2; IHC.
+ EApply ty0_conv;
+ [ Idtac
+ | EApply ty0_cast;
+ [ EApply ty0_conv; [ Idtac | Idtac | Apply pc3_s; EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]
+ | Idtac ]
+ | EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]; XEAuto.
+(* case 7.3.2: subst0 on right argument *)
+ IHCT; IHC; EApply ty0_cast; XEAuto.
+(* case 7.3.3: subst0 on both arguments *)
+ IHC; IHCT; Clear H2; IHCT.
+ EApply ty0_conv;
+ [ Idtac
+ | EApply ty0_cast;
+ [ EApply ty0_conv; [ Idtac | Idtac | Apply pc3_s; EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]
+ | Idtac ]
+ | EApply pc3_fsubst0; [ Idtac | Idtac | XEAuto ] ]; XEAuto.
+ Qed.
+
+ Theorem ty0_csubst0: (g:?; c1:?; t1,t2:?) (ty0 g c1 t1 t2) ->
+ (e:?; u:?; i:?) (drop i (0) c1 (CTail e (Bind Abbr) u)) ->
+ (c2:?) (wf0 g c2) -> (csubst0 i u c1 c2) ->
+ (ty0 g c2 t1 t2).
+ Intros; EApply ty0_fsubst0; XEAuto.
+ Qed.
+
+ Theorem ty0_subst0: (g:?; c:?; t1,t:?) (ty0 g c t1 t) ->
+ (e:?; u:?; i:?) (drop i (0) c (CTail e (Bind Abbr) u)) ->
+ (t2:?) (subst0 i u t1 t2) -> (ty0 g c t2 t).
+ Intros; EApply ty0_fsubst0; XEAuto.
+ Qed.
+
+ End ty0_fsubst0.
+
+ Hints Resolve ty0_subst0 : ltlc.
+
projects / helm.git / commitdiff