(* Title: HOL/Proofs/Lambda/Commutation.thy
Author: Tobias Nipkow
Copyright 1995 TU Muenchen
*)
section \<open>Abstract commutation and confluence notions\<close>
theory Commutation
imports Main
begin
declare [[syntax_ambiguity_warning = false]]
subsection \<open>Basic definitions\<close>
definition
square :: "['a => 'a => bool, 'a => 'a => bool, 'a => 'a => bool, 'a => 'a => bool] => bool" where
"square R S T U =
(\<forall>x y. R x y --> (\<forall>z. S x z --> (\<exists>u. T y u \<and> U z u)))"
definition
commute :: "['a => 'a => bool, 'a => 'a => bool] => bool" where
"commute R S = square R S S R"
definition
diamond :: "('a => 'a => bool) => bool" where
"diamond R = commute R R"
definition
Church_Rosser :: "('a => 'a => bool) => bool" where
"Church_Rosser R =
(\<forall>x y. (sup R (R\<inverse>\<inverse>))\<^sup>*\<^sup>* x y \<longrightarrow> (\<exists>z. R\<^sup>*\<^sup>* x z \<and> R\<^sup>*\<^sup>* y z))"
abbreviation
confluent :: "('a => 'a => bool) => bool" where
"confluent R == diamond (R\<^sup>*\<^sup>*)"
subsection \<open>Basic lemmas\<close>
subsubsection \<open>\<open>square\<close>\<close>
lemma square_sym: "square R S T U ==> square S R U T"
apply (unfold square_def)
apply blast
done
lemma square_subset:
"[| square R S T U; T \ T' |] ==> square R S T' U"
apply (unfold square_def)
apply (blast dest: predicate2D)
done
lemma square_reflcl:
"[| square R S T (R\<^sup>=\<^sup>=); S \ T |] ==> square (R\<^sup>=\<^sup>=) S T (R\<^sup>=\<^sup>=)"
apply (unfold square_def)
apply (blast dest: predicate2D)
done
lemma square_rtrancl:
"square R S S T ==> square (R\<^sup>*\<^sup>*) S S (T\<^sup>*\<^sup>*)"
apply (unfold square_def)
apply (intro strip)
apply (erule rtranclp_induct)
apply blast
apply (blast intro: rtranclp.rtrancl_into_rtrancl)
done
lemma square_rtrancl_reflcl_commute:
"square R S (S\<^sup>*\<^sup>*) (R\<^sup>=\<^sup>=) ==> commute (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*)"
apply (unfold commute_def)
apply (fastforce dest: square_reflcl square_sym [THEN square_rtrancl])
done
subsubsection \<open>\<open>commute\<close>\<close>
lemma commute_sym: "commute R S ==> commute S R"
apply (unfold commute_def)
apply (blast intro: square_sym)
done
lemma commute_rtrancl: "commute R S ==> commute (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*)"
apply (unfold commute_def)
apply (blast intro: square_rtrancl square_sym)
done
lemma commute_Un:
"[| commute R T; commute S T |] ==> commute (sup R S) T"
apply (unfold commute_def square_def)
apply blast
done
subsubsection \<open>\<open>diamond\<close>, \<open>confluence\<close>, and \<open>union\<close>\<close>
lemma diamond_Un:
"[| diamond R; diamond S; commute R S |] ==> diamond (sup R S)"
apply (unfold diamond_def)
apply (blast intro: commute_Un commute_sym)
done
lemma diamond_confluent: "diamond R ==> confluent R"
apply (unfold diamond_def)
apply (erule commute_rtrancl)
done
lemma square_reflcl_confluent:
"square R R (R\<^sup>=\<^sup>=) (R\<^sup>=\<^sup>=) ==> confluent R"
apply (unfold diamond_def)
apply (fast intro: square_rtrancl_reflcl_commute elim: square_subset)
done
lemma confluent_Un:
"[| confluent R; confluent S; commute (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*) |] ==> confluent (sup R S)"
apply (rule rtranclp_sup_rtranclp [THEN subst])
apply (blast dest: diamond_Un intro: diamond_confluent)
done
lemma diamond_to_confluence:
"[| diamond R; T \ R; R \ T\<^sup>*\<^sup>* |] ==> confluent T"
apply (force intro: diamond_confluent
dest: rtranclp_subset [symmetric])
done
subsection \<open>Church-Rosser\<close>
lemma Church_Rosser_confluent: "Church_Rosser R = confluent R"
apply (unfold square_def commute_def diamond_def Church_Rosser_def)
apply (tactic \<open>safe_tac (put_claset HOL_cs \<^context>)\<close>)
apply (tactic \<open>
blast_tac (put_claset HOL_cs \<^context> addIs
[@{thm sup_ge2} RS @{thm rtranclp_mono} RS @{thm predicate2D} RS @{thm rtranclp_trans},
@{thm rtranclp_converseI}, @{thm conversepI},
@{thm sup_ge1} RS @{thm rtranclp_mono} RS @{thm predicate2D}]) 1\<close>)
apply (erule rtranclp_induct)
apply blast
apply (blast del: rtranclp.rtrancl_refl intro: rtranclp_trans)
done
subsection \<open>Newman's lemma\<close>
text \<open>Proof by Stefan Berghofer\<close>
theorem newman:
assumes wf: "wfP (R\\)"
and lc: "\a b c. R a b \ R a c \
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
shows "\b c. R\<^sup>*\<^sup>* a b \ R\<^sup>*\<^sup>* a c \
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
using wf
proof induct
case (less x b c)
have xc: "R\<^sup>*\<^sup>* x c" by fact
have xb: "R\<^sup>*\<^sup>* x b" by fact thus ?case
proof (rule converse_rtranclpE)
assume "x = b"
with xc have "R\<^sup>*\<^sup>* b c" by simp
thus ?thesis by iprover
next
fix y
assume xy: "R x y"
assume yb: "R\<^sup>*\<^sup>* y b"
from xc show ?thesis
proof (rule converse_rtranclpE)
assume "x = c"
with xb have "R\<^sup>*\<^sup>* c b" by simp
thus ?thesis by iprover
next
fix y'
assume y'c: "R\<^sup>*\<^sup>* y' c"
assume xy': "R x y'"
with xy have "\u. R\<^sup>*\<^sup>* y u \ R\<^sup>*\<^sup>* y' u" by (rule lc)
then obtain u where yu: "R\<^sup>*\<^sup>* y u" and y'u: "R\<^sup>*\<^sup>* y' u" by iprover
from xy have "R\\ y x" ..
from this and yb yu have "\d. R\<^sup>*\<^sup>* b d \ R\<^sup>*\<^sup>* u d" by (rule less)
then obtain v where bv: "R\<^sup>*\<^sup>* b v" and uv: "R\<^sup>*\<^sup>* u v" by iprover
from xy' have "R\\ y' x" ..
moreover from y'u and uv have "R\<^sup>*\<^sup>* y' v" by (rule rtranclp_trans)
moreover note y'c
ultimately have "\d. R\<^sup>*\<^sup>* v d \ R\<^sup>*\<^sup>* c d" by (rule less)
then obtain w where vw: "R\<^sup>*\<^sup>* v w" and cw: "R\<^sup>*\<^sup>* c w" by iprover
from bv vw have "R\<^sup>*\<^sup>* b w" by (rule rtranclp_trans)
with cw show ?thesis by iprover
qed
qed
qed
text \<open>
Alternative version. Partly automated by Tobias
Nipkow. Takes 2 minutes (2002).
This is the maximal amount of automation possible using \<open>blast\<close>.
\<close>
theorem newman':
assumes wf: "wfP (R\\)"
and lc: "\a b c. R a b \ R a c \
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
shows "\b c. R\<^sup>*\<^sup>* a b \ R\<^sup>*\<^sup>* a c \
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
using wf
proof induct
case (less x b c)
note IH = \<open>\<And>y b c. \<lbrakk>R\<inverse>\<inverse> y x; R\<^sup>*\<^sup>* y b; R\<^sup>*\<^sup>* y c\<rbrakk>
\<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d\<close>
have xc: "R\<^sup>*\<^sup>* x c" by fact
have xb: "R\<^sup>*\<^sup>* x b" by fact
thus ?case
proof (rule converse_rtranclpE)
assume "x = b"
with xc have "R\<^sup>*\<^sup>* b c" by simp
thus ?thesis by iprover
next
fix y
assume xy: "R x y"
assume yb: "R\<^sup>*\<^sup>* y b"
from xc show ?thesis
proof (rule converse_rtranclpE)
assume "x = c"
with xb have "R\<^sup>*\<^sup>* c b" by simp
thus ?thesis by iprover
next
fix y'
assume y'c: "R\<^sup>*\<^sup>* y' c"
assume xy': "R x y'"
with xy obtain u where u: "R\<^sup>*\<^sup>* y u" "R\<^sup>*\<^sup>* y' u"
by (blast dest: lc)
from yb u y'c show ?thesis
by (blast del: rtranclp.rtrancl_refl
intro: rtranclp_trans
dest: IH [OF conversepI, OF xy] IH [OF conversepI, OF xy'])
qed
qed
qed
text \<open>
Using the coherent logic prover, the proof of the induction step
is completely automatic.
\<close>
lemma eq_imp_rtranclp: "x = y \ r\<^sup>*\<^sup>* x y"
by simp
theorem newman'':
assumes wf: "wfP (R\\)"
and lc: "\a b c. R a b \ R a c \
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
shows "\b c. R\<^sup>*\<^sup>* a b \ R\<^sup>*\<^sup>* a c \
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
using wf
proof induct
case (less x b c)
note IH = \<open>\<And>y b c. \<lbrakk>R\<inverse>\<inverse> y x; R\<^sup>*\<^sup>* y b; R\<^sup>*\<^sup>* y c\<rbrakk>
\<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d\<close>
show ?case
by (coherent
\<open>R\<^sup>*\<^sup>* x c\<close> \<open>R\<^sup>*\<^sup>* x b\<close>
refl [where 'a='a] sym
eq_imp_rtranclp
r_into_rtranclp [of R]
rtranclp_trans
lc IH [OF conversepI]
converse_rtranclpE)
qed
end
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