(* Title: HOL/Lambda/Commutation.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1995 TU Muenchen
*)
header {* Abstract commutation and confluence notions *}
theory Commutation imports Main begin
subsection {* Basic definitions *}
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^--1))^** x y --> (\<exists>z. R^** x z \<and> R^** y z))"
abbreviation
confluent :: "('a => 'a => bool) => bool" where
"confluent R == diamond (R^**)"
subsection {* Basic lemmas *}
subsubsection {* @{text "square"} *}
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 \<le> T' |] ==> square R S T' U"
apply (unfold square_def)
apply (blast dest: predicate2D)
done
lemma square_reflcl:
"[| square R S T (R^==); S \<le> T |] ==> square (R^==) S T (R^==)"
apply (unfold square_def)
apply (blast dest: predicate2D)
done
lemma square_rtrancl:
"square R S S T ==> square (R^**) S S (T^**)"
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^**) (R^==) ==> commute (R^**) (S^**)"
apply (unfold commute_def)
apply (fastsimp dest: square_reflcl square_sym [THEN square_rtrancl])
done
subsubsection {* @{text "commute"} *}
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^**) (S^**)"
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 {* @{text "diamond"}, @{text "confluence"}, and @{text "union"} *}
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^==) (R^==) ==> 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^**) (S^**) |] ==> 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 \<le> R; R \<le> T^** |] ==> confluent T"
apply (force intro: diamond_confluent
dest: rtranclp_subset [symmetric])
done
subsection {* Church-Rosser *}
lemma Church_Rosser_confluent: "Church_Rosser R = confluent R"
apply (unfold square_def commute_def diamond_def Church_Rosser_def)
apply (tactic {* safe_tac HOL_cs *})
apply (tactic {*
blast_tac (HOL_cs 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 *})
apply (erule rtranclp_induct)
apply blast
apply (blast del: rtranclp.rtrancl_refl intro: rtranclp_trans)
done
subsection {* Newman's lemma *}
text {* Proof by Stefan Berghofer *}
theorem newman:
assumes wf: "wfP (R\<inverse>\<inverse>)"
and lc: "\<And>a b c. R a b \<Longrightarrow> R a c \<Longrightarrow>
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
shows "\<And>b c. R\<^sup>*\<^sup>* a b \<Longrightarrow> R\<^sup>*\<^sup>* a c \<Longrightarrow>
\<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 "\<exists>u. R\<^sup>*\<^sup>* y u \<and> 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\<inverse>\<inverse> y x" ..
from this and yb yu have "\<exists>d. R\<^sup>*\<^sup>* b d \<and> 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\<inverse>\<inverse> y' x" ..
moreover from y'u and uv have "R\<^sup>*\<^sup>* y' v" by (rule rtranclp_trans)
moreover note y'c
ultimately have "\<exists>d. R\<^sup>*\<^sup>* v d \<and> 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 {*
Alternative version. Partly automated by Tobias
Nipkow. Takes 2 minutes (2002).
This is the maximal amount of automation possible using @{text blast}.
*}
theorem newman':
assumes wf: "wfP (R\<inverse>\<inverse>)"
and lc: "\<And>a b c. R a b \<Longrightarrow> R a c \<Longrightarrow>
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
shows "\<And>b c. R\<^sup>*\<^sup>* a b \<Longrightarrow> R\<^sup>*\<^sup>* a c \<Longrightarrow>
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
using wf
proof induct
case (less x b c)
note IH = `\<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`
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 {*
Using the coherent logic prover, the proof of the induction step
is completely automatic.
*}
lemma eq_imp_rtranclp: "x = y \<Longrightarrow> r\<^sup>*\<^sup>* x y"
by simp
theorem newman'':
assumes wf: "wfP (R\<inverse>\<inverse>)"
and lc: "\<And>a b c. R a b \<Longrightarrow> R a c \<Longrightarrow>
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
shows "\<And>b c. R\<^sup>*\<^sup>* a b \<Longrightarrow> R\<^sup>*\<^sup>* a c \<Longrightarrow>
\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
using wf
proof induct
case (less x b c)
note IH = `\<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`
show ?case
by (coherent
`R\<^sup>*\<^sup>* x c` `R\<^sup>*\<^sup>* x b`
refl [where 'a='a] sym
eq_imp_rtranclp
r_into_rtranclp [of R]
rtranclp_trans
lc IH [OF conversepI]
converse_rtranclpE)
qed
end