Theory Transitive_Closure
section ‹Reflexive and Transitive closure of a relation›
theory Transitive_Closure
imports Finite_Set
abbrevs "^*" = "⇧*" "⇧*⇧*"
and "^+" = "⇧+" "⇧+⇧+"
and "^=" = "⇧=" "⇧=⇧="
begin
ML_file ‹~~/src/Provers/trancl.ML›
text ‹
‹rtrancl› is reflexive/transitive closure,
‹trancl› is transitive closure,
‹reflcl› is reflexive closure.
These postfix operators have ∗‹maximum priority›, forcing their
operands to be atomic.
›
context notes [[inductive_internals]]
begin
inductive_set rtrancl :: "('a × 'a) set ⇒ ('a × 'a) set" ("(_⇧*)" [1000] 999)
for r :: "('a × 'a) set"
where
rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) ∈ r⇧*"
| rtrancl_into_rtrancl [Pure.intro]: "(a, b) ∈ r⇧* ⟹ (b, c) ∈ r ⟹ (a, c) ∈ r⇧*"
inductive_set trancl :: "('a × 'a) set ⇒ ('a × 'a) set" ("(_⇧+)" [1000] 999)
for r :: "('a × 'a) set"
where
r_into_trancl [intro, Pure.intro]: "(a, b) ∈ r ⟹ (a, b) ∈ r⇧+"
| trancl_into_trancl [Pure.intro]: "(a, b) ∈ r⇧+ ⟹ (b, c) ∈ r ⟹ (a, c) ∈ r⇧+"
notation
rtranclp ("(_⇧*⇧*)" [1000] 1000) and
tranclp ("(_⇧+⇧+)" [1000] 1000)
declare
rtrancl_def [nitpick_unfold del]
rtranclp_def [nitpick_unfold del]
trancl_def [nitpick_unfold del]
tranclp_def [nitpick_unfold del]
end
abbreviation reflcl :: "('a × 'a) set ⇒ ('a × 'a) set" ("(_⇧=)" [1000] 999)
where "r⇧= ≡ r ∪ Id"
abbreviation reflclp :: "('a ⇒ 'a ⇒ bool) ⇒ 'a ⇒ 'a ⇒ bool" ("(_⇧=⇧=)" [1000] 1000)
where "r⇧=⇧= ≡ sup r (=)"
notation (ASCII)
rtrancl ("(_^*)" [1000] 999) and
trancl ("(_^+)" [1000] 999) and
reflcl ("(_^=)" [1000] 999) and
rtranclp ("(_^**)" [1000] 1000) and
tranclp ("(_^++)" [1000] 1000) and
reflclp ("(_^==)" [1000] 1000)
subsection ‹Reflexive closure›
lemma reflcl_set_eq [pred_set_conv]: "(sup (λx y. (x, y) ∈ r) (=)) = (λx y. (x, y) ∈ r ∪ Id)"
by (auto simp: fun_eq_iff)
lemma refl_reflcl[simp]: "refl (r⇧=)"
by (simp add: refl_on_def)
lemma reflp_on_reflclp[simp]: "reflp_on A R⇧=⇧="
by (simp add: reflp_on_def)
lemma antisym_on_reflcl[simp]: "antisym_on A (r⇧=) ⟷ antisym_on A r"
by (simp add: antisym_on_def)
lemma antisymp_on_reflclp[simp]: "antisymp_on A R⇧=⇧= ⟷ antisymp_on A R"
by (rule antisym_on_reflcl[to_pred])
lemma trans_on_reflcl[simp]: "trans_on A r ⟹ trans_on A (r⇧=)"
by (auto intro: trans_onI dest: trans_onD)
lemma transp_on_reflclp[simp]: "transp_on A R ⟹ transp_on A R⇧=⇧="
by (rule trans_on_reflcl[to_pred])
lemma antisymp_on_reflclp_if_asymp_on:
assumes "asymp_on A R"
shows "antisymp_on A R⇧=⇧="
unfolding antisymp_on_reflclp
using antisymp_on_if_asymp_on[OF ‹asymp_on A R›] .
lemma antisym_on_reflcl_if_asym_on: "asym_on A R ⟹ antisym_on A (R⇧=)"
using antisymp_on_reflclp_if_asymp_on[to_set] .
lemma reflclp_idemp [simp]: "(P⇧=⇧=)⇧=⇧= = P⇧=⇧="
by blast
lemma reflclp_ident_if_reflp[simp]: "reflp R ⟹ R⇧=⇧= = R"
by (auto dest: reflpD)
text ‹The following are special cases of @{thm [source] reflclp_ident_if_reflp},
but they appear duplicated in multiple, independent theories, which causes name clashes.›
lemma (in preorder) reflclp_less_eq[simp]: "(≤)⇧=⇧= = (≤)"
using reflp_on_le by (simp only: reflclp_ident_if_reflp)
lemma (in preorder) reflclp_greater_eq[simp]: "(≥)⇧=⇧= = (≥)"
using reflp_on_ge by (simp only: reflclp_ident_if_reflp)
lemma order_reflclp_if_transp_and_asymp:
assumes "transp R" and "asymp R"
shows "class.order R⇧=⇧= R"
proof unfold_locales
show "⋀x y. R x y = (R⇧=⇧= x y ∧ ¬ R⇧=⇧= y x)"
using ‹asymp R› asympD by fastforce
next
show "⋀x. R⇧=⇧= x x"
by simp
next
show "⋀x y z. R⇧=⇧= x y ⟹ R⇧=⇧= y z ⟹ R⇧=⇧= x z"
using transp_on_reflclp[OF ‹transp R›, THEN transpD] .
next
show "⋀x y. R⇧=⇧= x y ⟹ R⇧=⇧= y x ⟹ x = y"
using antisymp_on_reflclp_if_asymp_on[OF ‹asymp R›, THEN antisympD] .
qed
subsection ‹Reflexive-transitive closure›
lemma r_into_rtrancl [intro]: "⋀p. p ∈ r ⟹ p ∈ r⇧*"
by (simp add: split_tupled_all rtrancl_refl [THEN rtrancl_into_rtrancl])
lemma r_into_rtranclp [intro]: "r x y ⟹ r⇧*⇧* x y"
by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
lemma rtranclp_mono: "r ≤ s ⟹ r⇧*⇧* ≤ s⇧*⇧*"
proof (rule predicate2I)
show "s⇧*⇧* x y" if "r ≤ s" "r⇧*⇧* x y" for x y
using ‹r⇧*⇧* x y› ‹r ≤ s›
by (induction rule: rtranclp.induct) (blast intro: rtranclp.rtrancl_into_rtrancl)+
qed
lemma mono_rtranclp[mono]: "(⋀a b. x a b ⟶ y a b) ⟹ x⇧*⇧* a b ⟶ y⇧*⇧* a b"
using rtranclp_mono[of x y] by auto
lemmas rtrancl_mono = rtranclp_mono [to_set]
theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
assumes a: "r⇧*⇧* a b"
and cases: "P a" "⋀y z. r⇧*⇧* a y ⟹ r y z ⟹ P y ⟹ P z"
shows "P b"
using a by (induct x≡a b) (rule cases)+
lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]
lemmas rtranclp_induct2 =
rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names refl step]
lemmas rtrancl_induct2 =
rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), consumes 1, case_names refl step]
lemma refl_rtrancl: "refl (r⇧*)"
unfolding refl_on_def by fast
text ‹Transitivity of transitive closure.›
lemma trans_rtrancl: "trans (r⇧*)"
proof (rule transI)
fix x y z
assume "(x, y) ∈ r⇧*"
assume "(y, z) ∈ r⇧*"
then show "(x, z) ∈ r⇧*"
proof induct
case base
show "(x, y) ∈ r⇧*" by fact
next
case (step u v)
from ‹(x, u) ∈ r⇧*› and ‹(u, v) ∈ r›
show "(x, v) ∈ r⇧*" ..
qed
qed
lemmas rtrancl_trans = trans_rtrancl [THEN transD]
lemma rtranclp_trans:
assumes "r⇧*⇧* x y"
and "r⇧*⇧* y z"
shows "r⇧*⇧* x z"
using assms(2,1) by induct iprover+
lemma rtranclE [cases set: rtrancl]:
fixes a b :: 'a
assumes major: "(a, b) ∈ r⇧*"
obtains
(base) "a = b"
| (step) y where "(a, y) ∈ r⇧*" and "(y, b) ∈ r"
proof -
have "a = b ∨ (∃y. (a, y) ∈ r⇧* ∧ (y, b) ∈ r)"
by (rule major [THEN rtrancl_induct]) blast+
then show ?thesis
by (auto intro: base step)
qed
lemma rtrancl_Int_subset: "Id ⊆ s ⟹ (r⇧* ∩ s) O r ⊆ s ⟹ r⇧* ⊆ s"
by (fastforce elim: rtrancl_induct)
lemma converse_rtranclp_into_rtranclp: "r a b ⟹ r⇧*⇧* b c ⟹ r⇧*⇧* a c"
by (rule rtranclp_trans) iprover+
lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
text ‹┉ More \<^term>‹r⇧*› equations and inclusions.›
lemma rtranclp_idemp [simp]: "(r⇧*⇧*)⇧*⇧* = r⇧*⇧*"
proof -
have "r⇧*⇧*⇧*⇧* x y ⟹ r⇧*⇧* x y" for x y
by (induction rule: rtranclp_induct) (blast intro: rtranclp_trans)+
then show ?thesis
by (auto intro!: order_antisym)
qed
lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
lemma rtrancl_idemp_self_comp [simp]: "R⇧* O R⇧* = R⇧*"
by (force intro: rtrancl_trans)
lemma rtrancl_subset_rtrancl: "r ⊆ s⇧* ⟹ r⇧* ⊆ s⇧*"
by (drule rtrancl_mono, simp)
lemma rtranclp_subset: "R ≤ S ⟹ S ≤ R⇧*⇧* ⟹ S⇧*⇧* = R⇧*⇧*"
by (fastforce dest: rtranclp_mono)
lemmas rtrancl_subset = rtranclp_subset [to_set]
lemma rtranclp_sup_rtranclp: "(sup (R⇧*⇧*) (S⇧*⇧*))⇧*⇧* = (sup R S)⇧*⇧*"
by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
lemma rtranclp_reflclp [simp]: "(R⇧=⇧=)⇧*⇧* = R⇧*⇧*"
by (blast intro!: rtranclp_subset)
lemmas rtrancl_reflcl [simp] = rtranclp_reflclp [to_set]
lemma rtrancl_r_diff_Id: "(r - Id)⇧* = r⇧*"
by (rule rtrancl_subset [symmetric]) auto
lemma rtranclp_r_diff_Id: "(inf r (≠))⇧*⇧* = r⇧*⇧*"
by (rule rtranclp_subset [symmetric]) auto
theorem rtranclp_converseD:
assumes "(r¯¯)⇧*⇧* x y"
shows "r⇧*⇧* y x"
using assms by induct (iprover intro: rtranclp_trans dest!: conversepD)+
lemmas rtrancl_converseD = rtranclp_converseD [to_set]
theorem rtranclp_converseI:
assumes "r⇧*⇧* y x"
shows "(r¯¯)⇧*⇧* x y"
using assms by induct (iprover intro: rtranclp_trans conversepI)+
lemmas rtrancl_converseI = rtranclp_converseI [to_set]
lemma rtrancl_converse: "(r¯)⇧* = (r⇧*)¯"
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
lemma sym_rtrancl: "sym r ⟹ sym (r⇧*)"
by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
theorem converse_rtranclp_induct [consumes 1, case_names base step]:
assumes major: "r⇧*⇧* a b"
and cases: "P b" "⋀y z. r y z ⟹ r⇧*⇧* z b ⟹ P z ⟹ P y"
shows "P a"
using rtranclp_converseI [OF major]
by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
lemmas converse_rtranclp_induct2 =
converse_rtranclp_induct [of _ "(ax, ay)" "(bx, by)", split_rule, consumes 1, case_names refl step]
lemmas converse_rtrancl_induct2 =
converse_rtrancl_induct [of "(ax, ay)" "(bx, by)", split_format (complete),
consumes 1, case_names refl step]
lemma converse_rtranclpE [consumes 1, case_names base step]:
assumes major: "r⇧*⇧* x z"
and cases: "x = z ⟹ P" "⋀y. r x y ⟹ r⇧*⇧* y z ⟹ P"
shows P
proof -
have "x = z ∨ (∃y. r x y ∧ r⇧*⇧* y z)"
by (rule major [THEN converse_rtranclp_induct]) iprover+
then show ?thesis
by (auto intro: cases)
qed
lemmas converse_rtranclE = converse_rtranclpE [to_set]
lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]
lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
lemma r_comp_rtrancl_eq: "r O r⇧* = r⇧* O r"
by (blast elim: rtranclE converse_rtranclE
intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
lemma rtrancl_unfold: "r⇧* = Id ∪ r⇧* O r"
by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
lemma rtrancl_Un_separatorE:
"(a, b) ∈ (P ∪ Q)⇧* ⟹ ∀x y. (a, x) ∈ P⇧* ⟶ (x, y) ∈ Q ⟶ x = y ⟹ (a, b) ∈ P⇧*"
proof (induct rule: rtrancl.induct)
case rtrancl_refl
then show ?case by blast
next
case rtrancl_into_rtrancl
then show ?case by (blast intro: rtrancl_trans)
qed
lemma rtrancl_Un_separator_converseE:
"(a, b) ∈ (P ∪ Q)⇧* ⟹ ∀x y. (x, b) ∈ P⇧* ⟶ (y, x) ∈ Q ⟶ y = x ⟹ (a, b) ∈ P⇧*"
proof (induct rule: converse_rtrancl_induct)
case base
then show ?case by blast
next
case step
then show ?case by (blast intro: rtrancl_trans)
qed
lemma Image_closed_trancl:
assumes "r `` X ⊆ X"
shows "r⇧* `` X = X"
proof -
from assms have **: "{y. ∃x∈X. (x, y) ∈ r} ⊆ X"
by auto
have "x ∈ X" if 1: "(y, x) ∈ r⇧*" and 2: "y ∈ X" for x y
proof -
from 1 show "x ∈ X"
proof induct
case base
show ?case by (fact 2)
next
case step
with ** show ?case by auto
qed
qed
then show ?thesis by auto
qed
lemma rtranclp_ident_if_reflp_and_transp:
assumes "reflp R" and "transp R"
shows "R⇧*⇧* = R"
proof (intro ext iffI)
fix x y
show "R⇧*⇧* x y ⟹ R x y"
proof (induction y rule: rtranclp_induct)
case base
show ?case
using ‹reflp R›[THEN reflpD] .
next
case (step y z)
thus ?case
using ‹transp R›[THEN transpD, of x y z] by simp
qed
next
fix x y
show "R x y ⟹ R⇧*⇧* x y"
using r_into_rtranclp .
qed
text ‹The following are special cases of @{thm [source] rtranclp_ident_if_reflp_and_transp},
but they appear duplicated in multiple, independent theories, which causes name clashes.›
lemma (in preorder) rtranclp_less_eq[simp]: "(≤)⇧*⇧* = (≤)"
using reflp_on_le transp_on_le by (simp only: rtranclp_ident_if_reflp_and_transp)
lemma (in preorder) rtranclp_greater_eq[simp]: "(≥)⇧*⇧* = (≥)"
using reflp_on_ge transp_on_ge by (simp only: rtranclp_ident_if_reflp_and_transp)
subsection ‹Transitive closure›
lemma totalp_on_tranclp: "totalp_on A R ⟹ totalp_on A (tranclp R)"
by (auto intro: totalp_onI dest: totalp_onD)
lemma total_on_trancl: "total_on A r ⟹ total_on A (trancl r)"
by (rule totalp_on_tranclp[to_set])
lemma trancl_mono:
assumes "p ∈ r⇧+" "r ⊆ s"
shows "p ∈ s⇧+"
proof -
have "⟦(a, b) ∈ r⇧+; r ⊆ s⟧ ⟹ (a, b) ∈ s⇧+" for a b
by (induction rule: trancl.induct) (iprover dest: subsetD)+
with assms show ?thesis
by (cases p) force
qed
lemma r_into_trancl': "⋀p. p ∈ r ⟹ p ∈ r⇧+"
by (simp only: split_tupled_all) (erule r_into_trancl)
text ‹┉ Conversions between ‹trancl› and ‹rtrancl›.›
lemma tranclp_into_rtranclp: "r⇧+⇧+ a b ⟹ r⇧*⇧* a b"
by (erule tranclp.induct) iprover+
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
lemma rtranclp_into_tranclp1:
assumes "r⇧*⇧* a b"
shows "r b c ⟹ r⇧+⇧+ a c"
using assms by (induct arbitrary: c) iprover+
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
lemma rtranclp_into_tranclp2:
assumes "r a b" "r⇧*⇧* b c" shows "r⇧+⇧+ a c"
using ‹r⇧*⇧* b c›
proof (cases rule: rtranclp.cases)
case rtrancl_refl
with assms show ?thesis
by iprover
next
case rtrancl_into_rtrancl
with assms show ?thesis
by (auto intro: rtranclp_trans [THEN rtranclp_into_tranclp1])
qed
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
text ‹Nice induction rule for ‹trancl››
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
assumes a: "r⇧+⇧+ a b"
and cases: "⋀y. r a y ⟹ P y" "⋀y z. r⇧+⇧+ a y ⟹ r y z ⟹ P y ⟹ P z"
shows "P b"
using a by (induct x≡a b) (iprover intro: cases)+
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
lemmas tranclp_induct2 =
tranclp_induct [of _ "(ax, ay)" "(bx, by)", split_rule, consumes 1, case_names base step]
lemmas trancl_induct2 =
trancl_induct [of "(ax, ay)" "(bx, by)", split_format (complete),
consumes 1, case_names base step]
lemma tranclp_trans_induct:
assumes major: "r⇧+⇧+ x y"
and cases: "⋀x y. r x y ⟹ P x y" "⋀x y z. r⇧+⇧+ x y ⟹ P x y ⟹ r⇧+⇧+ y z ⟹ P y z ⟹ P x z"
shows "P x y"
by (iprover intro: major [THEN tranclp_induct] cases)
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
lemma tranclE [cases set: trancl]:
assumes "(a, b) ∈ r⇧+"
obtains
(base) "(a, b) ∈ r"
| (step) c where "(a, c) ∈ r⇧+" and "(c, b) ∈ r"
using assms by cases simp_all
lemma trancl_Int_subset: "r ⊆ s ⟹ (r⇧+ ∩ s) O r ⊆ s ⟹ r⇧+ ⊆ s"
by (fastforce simp add: elim: trancl_induct)
lemma trancl_unfold: "r⇧+ = r ∪ r⇧+ O r"
by (auto intro: trancl_into_trancl elim: tranclE)
text ‹Transitivity of \<^term>‹r⇧+››
lemma trans_trancl [simp]: "trans (r⇧+)"
proof (rule transI)
fix x y z
assume "(x, y) ∈ r⇧+"
assume "(y, z) ∈ r⇧+"
then show "(x, z) ∈ r⇧+"
proof induct
case (base u)
from ‹(x, y) ∈ r⇧+› and ‹(y, u) ∈ r›
show "(x, u) ∈ r⇧+" ..
next
case (step u v)
from ‹(x, u) ∈ r⇧+› and ‹(u, v) ∈ r›
show "(x, v) ∈ r⇧+" ..
qed
qed
lemmas trancl_trans = trans_trancl [THEN transD]
lemma tranclp_trans:
assumes "r⇧+⇧+ x y"
and "r⇧+⇧+ y z"
shows "r⇧+⇧+ x z"
using assms(2,1) by induct iprover+
lemma trancl_id [simp]: "trans r ⟹ r⇧+ = r"
unfolding trans_def by (fastforce simp add: elim: trancl_induct)
lemma rtranclp_tranclp_tranclp:
assumes "r⇧*⇧* x y"
shows "⋀z. r⇧+⇧+ y z ⟹ r⇧+⇧+ x z"
using assms by induct (iprover intro: tranclp_trans)+
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
lemma tranclp_into_tranclp2: "r a b ⟹ r⇧+⇧+ b c ⟹ r⇧+⇧+ a c"
by (erule tranclp_trans [OF tranclp.r_into_trancl])
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
lemma tranclp_converseI:
assumes "(r⇧+⇧+)¯¯ x y" shows "(r¯¯)⇧+⇧+ x y"
using conversepD [OF assms]
proof (induction rule: tranclp_induct)
case (base y)
then show ?case
by (iprover intro: conversepI)
next
case (step y z)
then show ?case
by (iprover intro: conversepI tranclp_trans)
qed
lemmas trancl_converseI = tranclp_converseI [to_set]
lemma tranclp_converseD:
assumes "(r¯¯)⇧+⇧+ x y" shows "(r⇧+⇧+)¯¯ x y"
proof -
have "r⇧+⇧+ y x"
using assms
by (induction rule: tranclp_induct) (iprover dest: conversepD intro: tranclp_trans)+
then show ?thesis
by (rule conversepI)
qed
lemmas trancl_converseD = tranclp_converseD [to_set]
lemma tranclp_converse: "(r¯¯)⇧+⇧+ = (r⇧+⇧+)¯¯"
by (fastforce simp add: fun_eq_iff intro!: tranclp_converseI dest!: tranclp_converseD)
lemmas trancl_converse = tranclp_converse [to_set]
lemma sym_trancl: "sym r ⟹ sym (r⇧+)"
by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
lemma converse_tranclp_induct [consumes 1, case_names base step]:
assumes major: "r⇧+⇧+ a b"
and cases: "⋀y. r y b ⟹ P y" "⋀y z. r y z ⟹ r⇧+⇧+ z b ⟹ P z ⟹ P y"
shows "P a"
proof -
have "r¯¯⇧+⇧+ b a"
by (intro tranclp_converseI conversepI major)
then show ?thesis
by (induction rule: tranclp_induct) (blast intro: cases dest: tranclp_converseD)+
qed
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
lemma tranclpD: "R⇧+⇧+ x y ⟹ ∃z. R x z ∧ R⇧*⇧* z y"
proof (induction rule: converse_tranclp_induct)
case (step u v)
then show ?case
by (blast intro: rtranclp_trans)
qed auto
lemmas tranclD = tranclpD [to_set]
lemma converse_tranclpE:
assumes major: "tranclp r x z"
and base: "r x z ⟹ P"
and step: "⋀y. r x y ⟹ tranclp r y z ⟹ P"
shows P
proof -
from tranclpD [OF major] obtain y where "r x y" and "rtranclp r y z"
by iprover
from this(2) show P
proof (cases rule: rtranclp.cases)
case rtrancl_refl
with ‹r x y› base show P
by iprover
next
case rtrancl_into_rtrancl
then have "tranclp r y z"
by (iprover intro: rtranclp_into_tranclp1)
with ‹r x y› step show P
by iprover
qed
qed
lemmas converse_tranclE = converse_tranclpE [to_set]
lemma tranclD2: "(x, y) ∈ R⇧+ ⟹ ∃z. (x, z) ∈ R⇧* ∧ (z, y) ∈ R"
by (blast elim: tranclE intro: trancl_into_rtrancl)
lemma irrefl_tranclI: "r¯ ∩ r⇧* = {} ⟹ (x, x) ∉ r⇧+"
by (blast elim: tranclE dest: trancl_into_rtrancl)
lemma irrefl_trancl_rD: "∀x. (x, x) ∉ r⇧+ ⟹ (x, y) ∈ r ⟹ x ≠ y"
by (blast dest: r_into_trancl)
lemma trancl_subset_Sigma_aux: "(a, b) ∈ r⇧* ⟹ r ⊆ A × A ⟹ a = b ∨ a ∈ A"
by (induct rule: rtrancl_induct) auto
lemma trancl_subset_Sigma:
assumes "r ⊆ A × A" shows "r⇧+ ⊆ A × A"
proof (rule trancl_Int_subset [OF assms])
show "(r⇧+ ∩ A × A) O r ⊆ A × A"
using assms by auto
qed
lemma reflclp_tranclp [simp]: "(r⇧+⇧+)⇧=⇧= = r⇧*⇧*"
by (fast elim: rtranclp.cases tranclp_into_rtranclp dest: rtranclp_into_tranclp1)
lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
lemma trancl_reflcl [simp]: "(r⇧=)⇧+ = r⇧*"
proof -
have "(a, b) ∈ (r⇧=)⇧+ ⟹ (a, b) ∈ r⇧*" for a b
by (force dest: trancl_into_rtrancl)
moreover have "(a, b) ∈ (r⇧=)⇧+" if "(a, b) ∈ r⇧*" for a b
using that
proof (cases a b rule: rtranclE)
case step
show ?thesis
by (rule rtrancl_into_trancl1) (use step in auto)
qed auto
ultimately show ?thesis
by auto
qed
lemma rtrancl_trancl_reflcl [code]: "r⇧* = (r⇧+)⇧="
by simp
lemma trancl_empty [simp]: "{}⇧+ = {}"
by (auto elim: trancl_induct)
lemma rtrancl_empty [simp]: "{}⇧* = Id"
by (rule subst [OF reflcl_trancl]) simp
lemma rtranclpD: "R⇧*⇧* a b ⟹ a = b ∨ a ≠ b ∧ R⇧+⇧+ a b"
by (force simp: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
lemmas rtranclD = rtranclpD [to_set]
lemma rtrancl_eq_or_trancl: "(x,y) ∈ R⇧* ⟷ x = y ∨ x ≠ y ∧ (x, y) ∈ R⇧+"
by (fast elim: trancl_into_rtrancl dest: rtranclD)
lemma trancl_unfold_right: "r⇧+ = r⇧* O r"
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
lemma trancl_unfold_left: "r⇧+ = r O r⇧*"
by (auto dest: tranclD intro: rtrancl_into_trancl2)
lemma trancl_insert: "(insert (y, x) r)⇧+ = r⇧+ ∪ {(a, b). (a, y) ∈ r⇧* ∧ (x, b) ∈ r⇧*}"
proof -
have "⋀a b. (a, b) ∈ (insert (y, x) r)⇧+ ⟹
(a, b) ∈ r⇧+ ∪ {(a, b). (a, y) ∈ r⇧* ∧ (x, b) ∈ r⇧*}"
by (erule trancl_induct) (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)+
moreover have "r⇧+ ∪ {(a, b). (a, y) ∈ r⇧* ∧ (x, b) ∈ r⇧*} ⊆ (insert (y, x) r)⇧+"
by (blast intro: trancl_mono rtrancl_mono [THEN [2] rev_subsetD]
rtrancl_trancl_trancl rtrancl_into_trancl2)
ultimately show ?thesis
by auto
qed
lemma trancl_insert2:
"(insert (a, b) r)⇧+ = r⇧+ ∪ {(x, y). ((x, a) ∈ r⇧+ ∨ x = a) ∧ ((b, y) ∈ r⇧+ ∨ y = b)}"
by (auto simp: trancl_insert rtrancl_eq_or_trancl)
lemma rtrancl_insert: "(insert (a,b) r)⇧* = r⇧* ∪ {(x, y). (x, a) ∈ r⇧* ∧ (b, y) ∈ r⇧*}"
using trancl_insert[of a b r]
by (simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast
text ‹Simplifying nested closures›
lemma rtrancl_trancl_absorb[simp]: "(R⇧*)⇧+ = R⇧*"
by (simp add: trans_rtrancl)
lemma trancl_rtrancl_absorb[simp]: "(R⇧+)⇧* = R⇧*"
by (subst reflcl_trancl[symmetric]) simp
lemma rtrancl_reflcl_absorb[simp]: "(R⇧*)⇧= = R⇧*"
by auto
text ‹‹Domain› and ‹Range››
lemma Domain_rtrancl [simp]: "Domain (R⇧*) = UNIV"
by blast
lemma Range_rtrancl [simp]: "Range (R⇧*) = UNIV"
by blast
lemma rtrancl_Un_subset: "(R⇧* ∪ S⇧*) ⊆ (R ∪ S)⇧*"
by (rule rtrancl_Un_rtrancl [THEN subst]) fast
lemma in_rtrancl_UnI: "x ∈ R⇧* ∨ x ∈ S⇧* ⟹ x ∈ (R ∪ S)⇧*"
by (blast intro: subsetD [OF rtrancl_Un_subset])
lemma trancl_domain [simp]: "Domain (r⇧+) = Domain r"
by (unfold Domain_unfold) (blast dest: tranclD)
lemma trancl_range [simp]: "Range (r⇧+) = Range r"
unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
lemma Not_Domain_rtrancl:
assumes "x ∉ Domain R" shows "(x, y) ∈ R⇧* ⟷ x = y"
proof -
have "(x, y) ∈ R⇧* ⟹ x = y"
by (erule rtrancl_induct) (use assms in auto)
then show ?thesis
by auto
qed
lemma trancl_subset_Field2: "r⇧+ ⊆ Field r × Field r"
by (rule trancl_Int_subset) (auto simp: Field_def)
lemma finite_trancl[simp]: "finite (r⇧+) = finite r"
proof
show "finite (r⇧+) ⟹ finite r"
by (blast intro: r_into_trancl' finite_subset)
show "finite r ⟹ finite (r⇧+)"
by (auto simp: finite_Field trancl_subset_Field2 [THEN finite_subset])
qed
lemma finite_rtrancl_Image[simp]: assumes "finite R" "finite A" shows "finite (R⇧* `` A)"
proof (rule ccontr)
assume "infinite (R⇧* `` A)"
with assms show False
by(simp add: rtrancl_trancl_reflcl Un_Image del: reflcl_trancl)
qed
text ‹More about converse ‹rtrancl› and ‹trancl›, should
be merged with main body.›
lemma single_valued_confluent:
assumes "single_valued r" and xy: "(x, y) ∈ r⇧*" and xz: "(x, z) ∈ r⇧*"
shows "(y, z) ∈ r⇧* ∨ (z, y) ∈ r⇧*"
using xy
proof (induction rule: rtrancl_induct)
case base
show ?case
by (simp add: assms)
next
case (step y z)
with xz ‹single_valued r› show ?case
by (auto elim: converse_rtranclE dest: single_valuedD intro: rtrancl_trans)
qed
lemma r_r_into_trancl: "(a, b) ∈ R ⟹ (b, c) ∈ R ⟹ (a, c) ∈ R⇧+"
by (fast intro: trancl_trans)
lemma trancl_into_trancl: "(a, b) ∈ r⇧+ ⟹ (b, c) ∈ r ⟹ (a, c) ∈ r⇧+"
by (induct rule: trancl_induct) (fast intro: r_r_into_trancl trancl_trans)+
lemma tranclp_rtranclp_tranclp:
assumes "r⇧+⇧+ a b" "r⇧*⇧* b c" shows "r⇧+⇧+ a c"
proof -
obtain z where "r a z" "r⇧*⇧* z c"
using assms by (iprover dest: tranclpD rtranclp_trans)
then show ?thesis
by (blast dest: rtranclp_into_tranclp2)
qed
lemma rtranclp_conversep: "r¯¯⇧*⇧* = r⇧*⇧*¯¯"
by(auto simp add: fun_eq_iff intro: rtranclp_converseI rtranclp_converseD)
lemmas symp_rtranclp = sym_rtrancl[to_pred]
lemmas symp_conv_conversep_eq = sym_conv_converse_eq[to_pred]
lemmas rtranclp_tranclp_absorb [simp] = rtrancl_trancl_absorb[to_pred]
lemmas tranclp_rtranclp_absorb [simp] = trancl_rtrancl_absorb[to_pred]
lemmas rtranclp_reflclp_absorb [simp] = rtrancl_reflcl_absorb[to_pred]
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
lemmas transitive_closure_trans [trans] =
r_r_into_trancl trancl_trans rtrancl_trans
trancl.trancl_into_trancl trancl_into_trancl2
rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
rtrancl_trancl_trancl trancl_rtrancl_trancl
lemmas transitive_closurep_trans' [trans] =
tranclp_trans rtranclp_trans
tranclp.trancl_into_trancl tranclp_into_tranclp2
rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
declare trancl_into_rtrancl [elim]
lemma tranclp_ident_if_transp:
assumes "transp R"
shows "R⇧+⇧+ = R"
proof (intro ext iffI)
fix x y
show "R⇧+⇧+ x y ⟹ R x y"
proof (induction y rule: tranclp_induct)
case (base y)
thus ?case
by simp
next
case (step y z)
thus ?case
using ‹transp R›[THEN transpD, of x y z] by simp
qed
next
fix x y
show "R x y ⟹ R⇧+⇧+ x y"
using tranclp.r_into_trancl .
qed
text ‹The following are special cases of @{thm [source] tranclp_ident_if_transp},
but they appear duplicated in multiple, independent theories, which causes name clashes.›
lemma (in preorder) tranclp_less[simp]: "(<)⇧+⇧+ = (<)"
using transp_on_less by (simp only: tranclp_ident_if_transp)
lemma (in preorder) tranclp_less_eq[simp]: "(≤)⇧+⇧+ = (≤)"
using transp_on_le by (simp only: tranclp_ident_if_transp)
lemma (in preorder) tranclp_greater[simp]: "(>)⇧+⇧+ = (>)"
using transp_on_greater by (simp only: tranclp_ident_if_transp)
lemma (in preorder) tranclp_greater_eq[simp]: "(≥)⇧+⇧+ = (≥)"
using transp_on_ge by (simp only: tranclp_ident_if_transp)
subsection ‹Symmetric closure›
definition symclp :: "('a ⇒ 'a ⇒ bool) ⇒ 'a ⇒ 'a ⇒ bool"
where "symclp r x y ⟷ r x y ∨ r y x"
lemma symclpI [simp, intro?]:
shows symclpI1: "r x y ⟹ symclp r x y"
and symclpI2: "r y x ⟹ symclp r x y"
by(simp_all add: symclp_def)
lemma symclpE [consumes 1, cases pred]:
assumes "symclp r x y"
obtains (base) "r x y" | (sym) "r y x"
using assms by(auto simp add: symclp_def)
lemma symclp_pointfree: "symclp r = sup r r¯¯"
by(auto simp add: symclp_def fun_eq_iff)
lemma symclp_greater: "r ≤ symclp r"
by(simp add: symclp_pointfree)
lemma symclp_conversep [simp]: "symclp r¯¯ = symclp r"
by(simp add: symclp_pointfree sup.commute)
lemma symp_on_symclp [simp]: "symp_on A (symclp R)"
by(auto simp add: symp_on_def elim: symclpE intro: symclpI)
lemma symp_symclp_eq: "symp r ⟹ symclp r = r"
by(simp add: symclp_pointfree symp_conv_conversep_eq)
lemma symp_rtranclp_symclp [simp]: "symp (symclp r)⇧*⇧*"
by(simp add: symp_rtranclp)
lemma rtranclp_symclp_sym [sym]: "(symclp r)⇧*⇧* x y ⟹ (symclp r)⇧*⇧* y x"
by(rule sympD[OF symp_rtranclp_symclp])
lemma symclp_idem [simp]: "symclp (symclp r) = symclp r"
by(simp add: symclp_pointfree sup_commute converse_join)
lemma reflp_on_rtranclp [simp]: "reflp_on A R⇧*⇧*"
by (simp add: reflp_on_def)
subsection ‹The power operation on relations›
text ‹‹R ^^ n = R O … O R›, the n-fold composition of ‹R››
overloading
relpow ≡ "compow :: nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
relpowp ≡ "compow :: nat ⇒ ('a ⇒ 'a ⇒ bool) ⇒ ('a ⇒ 'a ⇒ bool)"
begin
primrec relpow :: "nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
where
"relpow 0 R = Id"
| "relpow (Suc n) R = (R ^^ n) O R"
primrec relpowp :: "nat ⇒ ('a ⇒ 'a ⇒ bool) ⇒ ('a ⇒ 'a ⇒ bool)"
where
"relpowp 0 R = HOL.eq"
| "relpowp (Suc n) R = (R ^^ n) OO R"
end
lemma relpowp_relpow_eq [pred_set_conv]:
"(λx y. (x, y) ∈ R) ^^ n = (λx y. (x, y) ∈ R ^^ n)" for R :: "'a rel"
by (induct n) (simp_all add: relcompp_relcomp_eq)
text ‹For code generation:›
definition relpow :: "nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
where relpow_code_def [code_abbrev]: "relpow = compow"
definition relpowp :: "nat ⇒ ('a ⇒ 'a ⇒ bool) ⇒ ('a ⇒ 'a ⇒ bool)"
where relpowp_code_def [code_abbrev]: "relpowp = compow"
lemma [code]:
"relpow (Suc n) R = (relpow n R) O R"
"relpow 0 R = Id"
by (simp_all add: relpow_code_def)
lemma [code]:
"relpowp (Suc n) R = (R ^^ n) OO R"
"relpowp 0 R = HOL.eq"
by (simp_all add: relpowp_code_def)
hide_const (open) relpow
hide_const (open) relpowp
lemma relpow_1 [simp]: "R ^^ 1 = R"
for R :: "('a × 'a) set"
by simp
lemma relpowp_1 [simp]: "P ^^ 1 = P"
for P :: "'a ⇒ 'a ⇒ bool"
by (fact relpow_1 [to_pred])
lemma relpow_0_I: "(x, x) ∈ R ^^ 0"
by simp
lemma relpowp_0_I: "(P ^^ 0) x x"
by (fact relpow_0_I [to_pred])
lemma relpow_Suc_I: "(x, y) ∈ R ^^ n ⟹ (y, z) ∈ R ⟹ (x, z) ∈ R ^^ Suc n"
by auto
lemma relpowp_Suc_I: "(P ^^ n) x y ⟹ P y z ⟹ (P ^^ Suc n) x z"
by (fact relpow_Suc_I [to_pred])
lemma relpow_Suc_I2: "(x, y) ∈ R ⟹ (y, z) ∈ R ^^ n ⟹ (x, z) ∈ R ^^ Suc n"
by (induct n arbitrary: z) (simp, fastforce)
lemma relpowp_Suc_I2: "P x y ⟹ (P ^^ n) y z ⟹ (P ^^ Suc n) x z"
by (fact relpow_Suc_I2 [to_pred])
lemma relpow_0_E: "(x, y) ∈ R ^^ 0 ⟹ (x = y ⟹ P) ⟹ P"
by simp
lemma relpowp_0_E: "(P ^^ 0) x y ⟹ (x = y ⟹ Q) ⟹ Q"
by (fact relpow_0_E [to_pred])
lemma relpow_Suc_E: "(x, z) ∈ R ^^ Suc n ⟹ (⋀y. (x, y) ∈ R ^^ n ⟹ (y, z) ∈ R ⟹ P) ⟹ P"
by auto
lemma relpowp_Suc_E: "(P ^^ Suc n) x z ⟹ (⋀y. (P ^^ n) x y ⟹ P y z ⟹ Q) ⟹ Q"
by (fact relpow_Suc_E [to_pred])
lemma relpow_E:
"(x, z) ∈ R ^^ n ⟹
(n = 0 ⟹ x = z ⟹ P) ⟹
(⋀y m. n = Suc m ⟹ (x, y) ∈ R ^^ m ⟹ (y, z) ∈ R ⟹ P) ⟹ P"
by (cases n) auto
lemma relpowp_E:
"(P ^^ n) x z ⟹
(n = 0 ⟹ x = z ⟹ Q) ⟹
(⋀y m. n = Suc m ⟹ (P ^^ m) x y ⟹ P y z ⟹ Q) ⟹ Q"
by (fact relpow_E [to_pred])
lemma relpow_Suc_D2: "(x, z) ∈ R ^^ Suc n ⟹ (∃y. (x, y) ∈ R ∧ (y, z) ∈ R ^^ n)"
by (induct n arbitrary: x z)
(blast intro: relpow_0_I relpow_Suc_I elim: relpow_0_E relpow_Suc_E)+
lemma relpowp_Suc_D2: "(P ^^ Suc n) x z ⟹ ∃y. P x y ∧ (P ^^ n) y z"
by (fact relpow_Suc_D2 [to_pred])
lemma relpow_Suc_E2: "(x, z) ∈ R ^^ Suc n ⟹ (⋀y. (x, y) ∈ R ⟹ (y, z) ∈ R ^^ n ⟹ P) ⟹ P"
by (blast dest: relpow_Suc_D2)
lemma relpowp_Suc_E2: "(P ^^ Suc n) x z ⟹ (⋀y. P x y ⟹ (P ^^ n) y z ⟹ Q) ⟹ Q"
by (fact relpow_Suc_E2 [to_pred])
lemma relpow_Suc_D2': "∀x y z. (x, y) ∈ R ^^ n ∧ (y, z) ∈ R ⟶ (∃w. (x, w) ∈ R ∧ (w, z) ∈ R ^^ n)"
by (induct n) (simp_all, blast)
lemma relpowp_Suc_D2': "∀x y z. (P ^^ n) x y ∧ P y z ⟶ (∃w. P x w ∧ (P ^^ n) w z)"
by (fact relpow_Suc_D2' [to_pred])
lemma relpow_E2:
assumes "(x, z) ∈ R ^^ n" "n = 0 ⟹ x = z ⟹ P"
"⋀y m. n = Suc m ⟹ (x, y) ∈ R ⟹ (y, z) ∈ R ^^ m ⟹ P"
shows "P"
proof (cases n)
case 0
with assms show ?thesis
by simp
next
case (Suc m)
with assms relpow_Suc_D2' [of m R] show ?thesis
by force
qed
lemma relpowp_E2:
"(P ^^ n) x z ⟹
(n = 0 ⟹ x = z ⟹ Q) ⟹
(⋀y m. n = Suc m ⟹ P x y ⟹ (P ^^ m) y z ⟹ Q) ⟹ Q"
by (fact relpow_E2 [to_pred])
lemma relpowp_trans[trans]: "(R ^^ i) x y ⟹ (R ^^ j) y z ⟹ (R ^^ (i + j)) x z"
proof (induction i arbitrary: x)
case 0
thus ?case by simp
next
case (Suc i)
obtain x' where "R x x'" and "(R ^^ i) x' y"
using ‹(R ^^ Suc i) x y›[THEN relpowp_Suc_D2] by auto
show "(R ^^ (Suc i + j)) x z"
unfolding add_Suc
proof (rule relpowp_Suc_I2)
show "R x x'"
using ‹R x x'› .
next
show "(R ^^ (i + j)) x' z"
using Suc.IH[OF ‹(R ^^ i) x' y› ‹(R ^^ j) y z›] .
qed
qed
lemma relpow_trans[trans]: "(x, y) ∈ R ^^ i ⟹ (y, z) ∈ R ^^ j ⟹ (x, z) ∈ R ^^ (i + j)"
using relpowp_trans[to_set] .
lemma relpowp_left_unique:
fixes R :: "'a ⇒ 'a ⇒ bool" and n :: nat and x y z :: 'a
assumes lunique: "⋀x y z. R x z ⟹ R y z ⟹ x = y"
shows "(R ^^ n) x z ⟹ (R ^^ n) y z ⟹ x = y"
proof (induction n arbitrary: x y z)
case 0
thus ?case
by simp
next
case (Suc n')
then obtain x' y' :: 'a where
"(R ^^ n') x x'" and "R x' z" and
"(R ^^ n') y y'" and "R y' z"
by auto
have "x' = y'"
using lunique[OF ‹R x' z› ‹R y' z›] .
show "x = y"
proof (rule Suc.IH)
show "(R ^^ n') x x'"
using ‹(R ^^ n') x x'› .
next
show "(R ^^ n') y x'"
using ‹(R ^^ n') y y'›
unfolding ‹x' = y'› .
qed
qed
lemma relpow_left_unique:
fixes R :: "('a × 'a) set" and n :: nat and x y z :: 'a
shows "(⋀x y z. (x, z) ∈ R ⟹ (y, z) ∈ R ⟹ x = y) ⟹
(x, z) ∈ R ^^ n ⟹ (y, z) ∈ R ^^ n ⟹ x = y"
using relpowp_left_unique[to_set] .
lemma relpowp_right_unique:
fixes R :: "'a ⇒ 'a ⇒ bool" and n :: nat and x y z :: 'a
assumes runique: "⋀x y z. R x y ⟹ R x z ⟹ y = z"
shows "(R ^^ n) x y ⟹ (R ^^ n) x z ⟹ y = z"
proof (induction n arbitrary: x y z)
case 0
thus ?case
by simp
next
case (Suc n')
then obtain x' :: 'a where
"(R ^^ n') x x'" and "R x' y" and "R x' z"
by auto
thus "y = z"
using runique by simp
qed
lemma relpow_right_unique:
fixes R :: "('a × 'a) set" and n :: nat and x y z :: 'a
shows "(⋀x y z. (x, y) ∈ R ⟹ (x, z) ∈ R ⟹ y = z) ⟹
(x, y) ∈ (R ^^ n) ⟹ (x, z) ∈ (R ^^ n) ⟹ y = z"
using relpowp_right_unique[to_set] .
lemma relpow_add: "R ^^ (m + n) = R^^m O R^^n"
by (induct n) auto
lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
by (fact relpow_add [to_pred])
lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
by (induct n) (simp_all add: O_assoc [symmetric])
lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
by (fact relpow_commute [to_pred])
lemma relpow_empty: "0 < n ⟹ ({} :: ('a × 'a) set) ^^ n = {}"
by (cases n) auto
lemma relpowp_bot: "0 < n ⟹ (⊥ :: 'a ⇒ 'a ⇒ bool) ^^ n = ⊥"
by (fact relpow_empty [to_pred])
lemma rtrancl_imp_UN_relpow:
assumes "p ∈ R⇧*"
shows "p ∈ (⋃n. R ^^ n)"
proof (cases p)
case (Pair x y)
with assms have "(x, y) ∈ R⇧*" by simp
then have "(x, y) ∈ (⋃n. R ^^ n)"
proof induct
case base
show ?case by (blast intro: relpow_0_I)
next
case step
then show ?case by (blast intro: relpow_Suc_I)
qed
with Pair show ?thesis by simp
qed
lemma rtranclp_imp_Sup_relpowp:
assumes "(P⇧*⇧*) x y"
shows "(⨆n. P ^^ n) x y"
using assms and rtrancl_imp_UN_relpow [of "(x, y)", to_pred] by simp
lemma relpow_imp_rtrancl:
assumes "p ∈ R ^^ n"
shows "p ∈ R⇧*"
proof (cases p)
case (Pair x y)
with assms have "(x, y) ∈ R ^^ n" by simp
then have "(x, y) ∈ R⇧*"
proof (induct n arbitrary: x y)
case 0
then show ?case by simp
next
case Suc
then show ?case
by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
qed
with Pair show ?thesis by simp
qed
lemma relpowp_imp_rtranclp: "(P ^^ n) x y ⟹ (P⇧*⇧*) x y"
using relpow_imp_rtrancl [of "(x, y)", to_pred] by simp
lemma rtrancl_is_UN_relpow: "R⇧* = (⋃n. R ^^ n)"
by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
lemma rtranclp_is_Sup_relpowp: "P⇧*⇧* = (⨆n. P ^^ n)"
using rtrancl_is_UN_relpow [to_pred, of P] by auto
lemma rtrancl_power: "p ∈ R⇧* ⟷ (∃n. p ∈ R ^^ n)"
by (simp add: rtrancl_is_UN_relpow)
lemma rtranclp_power: "(P⇧*⇧*) x y ⟷ (∃n. (P ^^ n) x y)"
by (simp add: rtranclp_is_Sup_relpowp)
lemma trancl_power: "p ∈ R⇧+ ⟷ (∃n > 0. p ∈ R ^^ n)"
proof -
have "(a, b) ∈ R⇧+ ⟷ (∃n>0. (a, b) ∈ R ^^ n)" for a b
proof safe
show "(a, b) ∈ R⇧+ ⟹ ∃n>0. (a, b) ∈ R ^^ n"
by (fastforce simp: rtrancl_is_UN_relpow relcomp_unfold dest: tranclD2)
show "(a, b) ∈ R⇧+" if "n > 0" "(a, b) ∈ R ^^ n" for n
proof (cases n)
case (Suc m)
with that show ?thesis
by (auto simp: dest: relpow_imp_rtrancl rtrancl_into_trancl1)
qed (use that in auto)
qed
then show ?thesis
by (cases p) auto
qed
lemma tranclp_power: "(P⇧+⇧+) x y ⟷ (∃n > 0. (P ^^ n) x y)"
using trancl_power [to_pred, of P "(x, y)"] by simp
lemma rtrancl_imp_relpow: "p ∈ R⇧* ⟹ ∃n. p ∈ R ^^ n"
by (auto dest: rtrancl_imp_UN_relpow)
lemma rtranclp_imp_relpowp: "(P⇧*⇧*) x y ⟹ ∃n. (P ^^ n) x y"
by (auto dest: rtranclp_imp_Sup_relpowp)
text ‹By Sternagel/Thiemann:›
lemma relpow_fun_conv: "(a, b) ∈ R ^^ n ⟷ (∃f. f 0 = a ∧ f n = b ∧ (∀i<n. (f i, f (Suc i)) ∈ R))"
proof (induct n arbitrary: b)
case 0
show ?case by auto
next
case (Suc n)
show ?case
proof -
have "(∃y. (∃f. f 0 = a ∧ f n = y ∧ (∀i<n. (f i,f(Suc i)) ∈ R)) ∧ (y,b) ∈ R) ⟷
(∃f. f 0 = a ∧ f(Suc n) = b ∧ (∀i<Suc n. (f i, f (Suc i)) ∈ R))"
(is "?l ⟷ ?r")
proof
assume ?l
then obtain c f
where 1: "f 0 = a" "f n = c" "⋀i. i < n ⟹ (f i, f (Suc i)) ∈ R" "(c,b) ∈ R"
by auto
let ?g = "λ m. if m = Suc n then b else f m"
show ?r by (rule exI[of _ ?g]) (simp add: 1)
next
assume ?r
then obtain f where 1: "f 0 = a" "b = f (Suc n)" "⋀i. i < Suc n ⟹ (f i, f (Suc i)) ∈ R"
by auto
show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], auto simp add: 1)
qed
then show ?thesis by (simp add: relcomp_unfold Suc)
qed
qed
lemma relpowp_fun_conv: "(P ^^ n) x y ⟷ (∃f. f 0 = x ∧ f n = y ∧ (∀i<n. P (f i) (f (Suc i))))"
by (fact relpow_fun_conv [to_pred])
lemma relpow_finite_bounded1:
fixes R :: "('a × 'a) set"
assumes "finite R" and "k > 0"
shows "R^^k ⊆ (⋃n∈{n. 0 < n ∧ n ≤ card R}. R^^n)"
(is "_ ⊆ ?r")
proof -
have "(a, b) ∈ R^^(Suc k) ⟹ ∃n. 0 < n ∧ n ≤ card R ∧ (a, b) ∈ R^^n" for a b k
proof (induct k arbitrary: b)
case 0
then have "R ≠ {}" by auto
with card_0_eq[OF ‹finite R›] have "card R ≥ Suc 0" by auto
then show ?case using 0 by force
next
case (Suc k)
then obtain a' where "(a, a') ∈ R^^(Suc k)" and "(a', b) ∈ R"
by auto
from Suc(1)[OF ‹(a, a') ∈ R^^(Suc k)›] obtain n where "n ≤ card R" and "(a, a') ∈ R ^^ n"
by auto
have "(a, b) ∈ R^^(Suc n)"
using ‹(a, a') ∈ R^^n› and ‹(a', b)∈ R› by auto
from ‹n ≤ card R› consider "n < card R" | "n = card R" by force
then show ?case
proof cases
case 1
then show ?thesis
using ‹(a, b) ∈ R^^(Suc n)› Suc_leI[OF ‹n < card R›] by blast
next
case 2
from ‹(a, b) ∈ R ^^ (Suc n)› [unfolded relpow_fun_conv]
obtain f where "f 0 = a" and "f (Suc n) = b"
and steps: "⋀i. i ≤ n ⟹ (f i, f (Suc i)) ∈ R" by auto
let ?p = "λi. (f i, f(Suc i))"
let ?N = "{i. i ≤ n}"
have "?p ` ?N ⊆ R"
using steps by auto
from card_mono[OF assms(1) this] have "card (?p ` ?N) ≤ card R" .
also have "… < card ?N"
using ‹n = card R› by simp
finally have "¬ inj_on ?p ?N"
by (rule pigeonhole)
then obtain i j where i: "i ≤ n" and j: "j ≤ n" and ij: "i ≠ j" and pij: "?p i = ?p j"
by (auto simp: inj_on_def)
let ?i = "min i j"
let ?j = "max i j"
have i: "?i ≤ n" and j: "?j ≤ n" and pij: "?p ?i = ?p ?j" and ij: "?i < ?j"
using i j ij pij unfolding min_def max_def by auto
from i j pij ij obtain i j where i: "i ≤ n" and j: "j ≤ n" and ij: "i < j"
and pij: "?p i = ?p j"
by blast
let ?g = "λl. if l ≤ i then f l else f (l + (j - i))"
let ?n = "Suc (n - (j - i))"
have abl: "(a, b) ∈ R ^^ ?n"
unfolding relpow_fun_conv
proof (rule exI[of _ ?g], intro conjI impI allI)
show "?g ?n = b"
using ‹f(Suc n) = b› j ij by auto
next
fix k
assume "k < ?n"
show "(?g k, ?g (Suc k)) ∈ R"
proof (cases "k < i")
case True
with i have "k ≤ n"
by auto
from steps[OF this] show ?thesis
using True by simp
next
case False
then have "i ≤ k" by auto
show ?thesis
proof (cases "k = i")
case True
then show ?thesis
using ij pij steps[OF i] by simp
next
case False
with ‹i ≤ k› have "i < k" by auto
then have small: "k + (j - i) ≤ n"
using ‹k<?n› by arith
show ?thesis
using steps[OF small] ‹i<k› by auto
qed
qed
qed (simp add: ‹f 0 = a›)
moreover have "?n ≤ n"
using i j ij by arith
ultimately show ?thesis
using ‹n = card R› by blast
qed
qed
then show ?thesis
using gr0_implies_Suc[OF ‹k > 0›] by auto
qed
lemma relpow_finite_bounded:
fixes R :: "('a × 'a) set"
assumes "finite R"
shows "R^^k ⊆ (⋃n∈{n. n ≤ card R}. R^^n)"
proof (cases k)
case (Suc k')
then show ?thesis
using relpow_finite_bounded1[OF assms, of k] by auto
qed force
lemma rtrancl_finite_eq_relpow: "finite R ⟹ R⇧* = (⋃n∈{n. n ≤ card R}. R^^n)"
by (fastforce simp: rtrancl_power dest: relpow_finite_bounded)
lemma trancl_finite_eq_relpow:
assumes "finite R" shows "R⇧+ = (⋃n∈{n. 0 < n ∧ n ≤ card R}. R^^n)"
proof -
have "⋀a b n. ⟦0 < n; (a, b) ∈ R ^^ n⟧ ⟹ ∃x>0. x ≤ card R ∧ (a, b) ∈ R ^^ x"
using assms by (auto dest: relpow_finite_bounded1)
then show ?thesis
by (auto simp: trancl_power)
qed
lemma finite_relcomp[simp,intro]:
assumes "finite R" and "finite S"
shows "finite (R O S)"
proof-
have "R O S = (⋃(x, y)∈R. ⋃(u, v)∈S. if u = y then {(x, v)} else {})"
by (force simp: split_def image_constant_conv split: if_splits)
then show ?thesis
using assms by clarsimp
qed
lemma finite_relpow [simp, intro]:
fixes R :: "('a × 'a) set"
assumes "finite R"
shows "n > 0 ⟹ finite (R^^n)"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
then show ?case by (cases n) (use assms in simp_all)
qed
lemma single_valued_relpow:
fixes R :: "('a × 'a) set"
shows "single_valued R ⟹ single_valued (R ^^ n)"
proof (induct n arbitrary: R)
case 0
then show ?case by simp
next
case (Suc n)
show ?case
by (rule single_valuedI)
(use Suc in ‹fast dest: single_valuedD elim: relpow_Suc_E›)
qed
subsection ‹Bounded transitive closure›
definition ntrancl :: "nat ⇒ ('a × 'a) set ⇒ ('a × 'a) set"
where "ntrancl n R = (⋃i∈{i. 0 < i ∧ i ≤ Suc n}. R ^^ i)"
lemma ntrancl_Zero [simp, code]: "ntrancl 0 R = R"
proof
show "R ⊆ ntrancl 0 R"
unfolding ntrancl_def by fastforce
have "0 < i ∧ i ≤ Suc 0 ⟷ i = 1" for i
by auto
then show "ntrancl 0 R ≤ R"
unfolding ntrancl_def by auto
qed
lemma ntrancl_Suc [simp]: "ntrancl (Suc n) R = ntrancl n R O (Id ∪ R)"
proof
have "(a, b) ∈ ntrancl n R O (Id ∪ R)" if "(a, b) ∈ ntrancl (Suc n) R" for a b
proof -
from that obtain i where "0 < i" "i ≤ Suc (Suc n)" "(a, b) ∈ R ^^ i"
unfolding ntrancl_def by auto
show ?thesis
proof (cases "i = 1")
case True
with ‹(a, b) ∈ R ^^ i› show ?thesis
by (auto simp: ntrancl_def)
next
case False
with ‹0 < i› obtain j where j: "i = Suc j" "0 < j"
by (cases i) auto
with ‹(a, b) ∈ R ^^ i› obtain c where c1: "(a, c) ∈ R ^^ j" and c2: "(c, b) ∈ R"
by auto
from c1 j ‹i ≤ Suc (Suc n)› have "(a, c) ∈ ntrancl n R"
by (fastforce simp: ntrancl_def)
with c2 show ?thesis by fastforce
qed
qed
then show "ntrancl (Suc n) R ⊆ ntrancl n R O (Id ∪ R)"
by auto
show "ntrancl n R O (Id ∪ R) ⊆ ntrancl (Suc n) R"
by (fastforce simp: ntrancl_def)
qed
lemma [code]: "ntrancl (Suc n) r = (let r' = ntrancl n r in r' ∪ r' O r)"
by (auto simp: Let_def)
lemma finite_trancl_ntranl: "finite R ⟹ trancl R = ntrancl (card R - 1) R"
by (cases "card R") (auto simp: trancl_finite_eq_relpow relpow_empty ntrancl_def)
subsection ‹Acyclic relations›
definition acyclic :: "('a × 'a) set ⇒ bool"
where "acyclic r ⟷ (∀x. (x,x) ∉ r⇧+)"
abbreviation acyclicP :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
where "acyclicP r ≡ acyclic {(x, y). r x y}"
lemma acyclic_irrefl [code]: "acyclic r ⟷ irrefl (r⇧+)"
by (simp add: acyclic_def irrefl_def)
lemma acyclicI: "∀x. (x, x) ∉ r⇧+ ⟹ acyclic r"
by (simp add: acyclic_def)
lemma (in preorder) acyclicI_order:
assumes *: "⋀a b. (a, b) ∈ r ⟹ f b < f a"
shows "acyclic r"
proof -
have "f b < f a" if "(a, b) ∈ r⇧+" for a b
using that by induct (auto intro: * less_trans)
then show ?thesis
by (auto intro!: acyclicI)
qed
lemma acyclic_insert [iff]: "acyclic (insert (y, x) r) ⟷ acyclic r ∧ (x, y) ∉ r⇧*"
by (simp add: acyclic_def trancl_insert) (blast intro: rtrancl_trans)
lemma acyclic_converse [iff]: "acyclic (r¯) ⟷ acyclic r"
by (simp add: acyclic_def trancl_converse)
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
lemma acyclic_impl_antisym_rtrancl: "acyclic r ⟹ antisym (r⇧*)"
by (simp add: acyclic_def antisym_def)
(blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
lemma acyclic_subset: "acyclic s ⟹ r ⊆ s ⟹ acyclic r"
unfolding acyclic_def by (blast intro: trancl_mono)
subsection ‹Setup of transitivity reasoner›
ML ‹
structure Trancl_Tac = Trancl_Tac
(
val r_into_trancl = @{thm trancl.r_into_trancl};
val trancl_trans = @{thm trancl_trans};
val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
val r_into_rtrancl = @{thm r_into_rtrancl};
val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
val rtrancl_trans = @{thm rtrancl_trans};
fun decomp \<^Const_>‹Trueprop for t› =
let
fun dec \<^Const_>‹Set.member _ for \<^Const_>‹Pair _ _ for a b› rel› =
let
fun decr \<^Const_>‹rtrancl _ for r› = (r,"r*")
| decr \<^Const_>‹trancl _ for r› = (r,"r+")
| decr r = (r,"r");
val (rel,r) = decr (Envir.beta_eta_contract rel);
in SOME (a,b,rel,r) end
| dec _ = NONE
in dec t end
| decomp _ = NONE;
);
structure Tranclp_Tac = Trancl_Tac
(
val r_into_trancl = @{thm tranclp.r_into_trancl};
val trancl_trans = @{thm tranclp_trans};
val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
val r_into_rtrancl = @{thm r_into_rtranclp};
val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
val rtrancl_trans = @{thm rtranclp_trans};
fun decomp \<^Const_>‹Trueprop for t› =
let
fun dec (rel $ a $ b) =
let
fun decr \<^Const_>‹rtranclp _ for r› = (r,"r*")
| decr \<^Const_>‹tranclp _ for r› = (r,"r+")
| decr r = (r,"r");
val (rel,r) = decr rel;
in SOME (a, b, rel, r) end
| dec _ = NONE
in dec t end
| decomp _ = NONE;
);
›
setup ‹
map_theory_simpset (fn ctxt => ctxt
addSolver (mk_solver "Trancl" Trancl_Tac.trancl_tac)
addSolver (mk_solver "Rtrancl" Trancl_Tac.rtrancl_tac)
addSolver (mk_solver "Tranclp" Tranclp_Tac.trancl_tac)
addSolver (mk_solver "Rtranclp" Tranclp_Tac.rtrancl_tac))
›
lemma transp_rtranclp [simp]: "transp R⇧*⇧*"
by(auto simp add: transp_def)
text ‹Optional methods.›
method_setup trancl =
‹Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac)›
‹simple transitivity reasoner›
method_setup rtrancl =
‹Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac)›
‹simple transitivity reasoner›
method_setup tranclp =
‹Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac)›
‹simple transitivity reasoner (predicate version)›
method_setup rtranclp =
‹Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac)›
‹simple transitivity reasoner (predicate version)›
end