fact stringlengths 1 2.11k | type stringclasses 27 values | library stringclasses 888 values | imports listlengths 0 81 | filename stringlengths 9 106 | symbolic_name stringlengths 1 113 | docstring stringlengths 0 1.34k ⌀ |
|---|---|---|---|---|---|---|
NF_join_imp_reach:
assumes "y \<in> NF A" and "(x, y) \<in> A\<^sup>\<down>"
shows "(x, y) \<in> A\<^sup>*"
using assms by (auto simp: join_def) (metis NF_not_suc rtrancl_converseD) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | NF_join_imp_reach | CR r |
conversion_O_conversion[simp]:
"A\<^sup>\<leftrightarrow>\<^sup>* O A\<^sup>\<leftrightarrow>\<^sup>* = A\<^sup>\<leftrightarrow>\<^sup>*"
by (force simp: converse_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_O_conversion | null |
trans_O_iff: "trans A \<longleftrightarrow> A O A \<subseteq> A" unfolding trans_def by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | trans_O_iff | null |
refl_O_iff: "refl A \<longleftrightarrow> Id \<subseteq> A" unfolding refl_on_def by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | refl_O_iff | null |
relpow_Suc: "r ^^ Suc n = r O r ^^ n"
using relpow_add[of 1 n r] by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | relpow_Suc | null |
converse_power: fixes r :: "'a rel" shows "(r\<inverse>)^^n = (r^^n)\<inverse>"
proof (induct n)
case (Suc n)
show ?case unfolding relpow.simps(2)[of _ "r\<inverse>"] relpow_Suc[of _ r]
by (simp add: Suc converse_relcomp)
qed simp | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | converse_power | null |
conversion_mono: "A \<subseteq> B \<Longrightarrow> A\<^sup>\<leftrightarrow>\<^sup>* \<subseteq> B\<^sup>\<leftrightarrow>\<^sup>*"
by (auto simp: conversion_def intro!: rtrancl_mono) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_mono | null |
conversion_conversion_idemp[simp]: "(A\<^sup>\<leftrightarrow>\<^sup>*)\<^sup>\<leftrightarrow>\<^sup>* = A\<^sup>\<leftrightarrow>\<^sup>*"
by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_conversion_idemp | null |
lower_set_imp_not_SN_on:
assumes "s \<in> X" "\<forall>t \<in> X. \<exists>u \<in> X. (t,u) \<in> R" shows "\<not> SN_on R {s}"
by (meson SN_on_imp_on_minimal assms) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | lower_set_imp_not_SN_on | null |
SN_on_Image_rtrancl_iff[simp]: "SN_on R (R\<^sup>* `` X) \<longleftrightarrow> SN_on R X" (is "?l = ?r")
proof(intro iffI)
assume "?l" show "?r" by (rule SN_on_subset2[OF _ \<open>?l\<close>], auto)
qed (fact SN_on_Image_rtrancl) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_on_Image_rtrancl_iff | null |
O_mono1: "R \<subseteq> R' \<Longrightarrow> S O R \<subseteq> S O R'" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | O_mono1 | ?l |
O_mono2: "R \<subseteq> R' \<Longrightarrow> R O T \<subseteq> R' O T" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | O_mono2 | ?l |
rtrancl_O_shift: "(S O R)\<^sup>* O S = S O (R O S)\<^sup>*"
proof(intro equalityI subrelI)
fix x y
assume "(x,y) \<in> (S O R)\<^sup>* O S"
then obtain n where "(x,y) \<in> (S O R)^^n O S" by blast
then show "(x,y) \<in> S O (R O S)\<^sup>*"
proof(induct n arbitrary: y)
case IH: (Suc n)
then obtain z where xz: "(x,z) \<in> (S O R)^^n O S" and zy: "(z,y) \<in> R O S" by auto
from IH.hyps[OF xz] zy have "(x,y) \<in> S O (R O S)\<^sup>* O R O S" by auto
then show ?case by(fold trancl_unfold_right, auto)
qed auto
next
fix x y
assume "(x,y) \<in> S O (R O S)\<^sup>*"
then obtain n where "(x,y) \<in> S O (R O S)^^n" by blast
then show "(x,y) \<in> (S O R)\<^sup>* O S"
proof(induct n arbitrary: y)
case IH: (Suc n)
then obtain z where xz: "(x,z) \<in> S O (R O S)^^n" and zy: "(z,y) \<in> R O S" by auto
from IH.hyps[OF xz] zy have "(x,y) \<in> ((S O R)\<^sup>* O S O R) O S" by auto
from this[folded trancl_unfold_right]
show ?case by (rule rev_subsetD[OF _ O_mono2], auto simp: O_assoc)
qed auto
qed | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | rtrancl_O_shift | null |
O_rtrancl_O_O: "R O (S O R)\<^sup>* O S = (R O S)\<^sup>+"
by (unfold rtrancl_O_shift trancl_unfold_left, auto) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | O_rtrancl_O_O | null |
SN_on_subset_SN_terms:
assumes SN: "SN_on R X" shows "X \<subseteq> {x. SN_on R {x}}"
proof(intro subsetI, unfold mem_Collect_eq)
fix x assume x: "x \<in> X"
show "SN_on R {x}" by (rule SN_on_subset2[OF _ SN], insert x, auto)
qed | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_on_subset_SN_terms | null |
SN_on_Un2:
assumes "SN_on R X" and "SN_on R Y" shows "SN_on R (X \<union> Y)"
using assms by fast | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_on_Un2 | null |
SN_on_UN:
assumes "\<And>x. SN_on R (X x)" shows "SN_on R (\<Union>x. X x)"
using assms by fast | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_on_UN | null |
Image_subsetI: "R \<subseteq> R' \<Longrightarrow> R `` X \<subseteq> R' `` X" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | Image_subsetI | null |
SN_on_O_comm:
assumes SN: "SN_on ((R :: ('a\<times>'b) set) O (S :: ('b\<times>'a) set)) (S `` X)"
shows "SN_on (S O R) X"
proof
fix seq :: "nat \<Rightarrow> 'b" assume seq0: "seq 0 \<in> X" and chain: "chain (S O R) seq"
from SN have SN: "SN_on (R O S) ((R O S)\<^sup>* `` S `` X)" by simp
{ fix i a
assume ia: "(seq i,a) \<in> S" and aSi: "(a,seq (Suc i)) \<in> R"
have "seq i \<in> (S O R)\<^sup>* `` X"
proof (induct i)
case 0 from seq0 show ?case by auto
next
case (Suc i) with chain have "seq (Suc i) \<in> ((S O R)\<^sup>* O S O R) `` X" by blast
also have "... \<subseteq> (S O R)\<^sup>* `` X" by (fold trancl_unfold_right, auto)
finally show ?case.
qed
with ia have "a \<in> ((S O R)\<^sup>* O S) `` X" by auto
then have a: "a \<in> ((R O S)\<^sup>*) `` S `` X" by (auto simp: rtrancl_O_shift)
with ia aSi have False
proof(induct "a" arbitrary: i rule: SN_on_induct[OF SN])
case 1 show ?case by (fact a)
next
case IH: (2 a)
from chain obtain b
where *: "(seq (Suc i), b) \<in> S" "(b, seq (Suc (Suc i))) \<in> R" by auto
with IH have ab: "(a,b) \<in> R O S" by auto
with \<open>a \<in> (R O S)\<^sup>* `` S `` X\<close> have "b \<in> ((R O S)\<^sup>* O R O S) `` S `` X" by auto
then have "b \<in> (R O S)\<^sup>* `` S `` X"
by (rule rev_subsetD, intro Image_subsetI, fold trancl_unfold_right, auto)
from IH.hyps[OF ab * this] IH.prems ab show False by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_on_O_comm | null |
SN_O_comm: "SN (R O S) \<longleftrightarrow> SN (S O R)"
by (intro iffI; rule SN_on_O_comm[OF SN_on_subset2], auto) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_O_comm | null |
chain_mono: assumes "R' \<subseteq> R" "chain R' seq" shows "chain R seq"
using assms by auto
context
fixes S R
assumes push: "S O R \<subseteq> R O S\<^sup>*"
begin | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | chain_mono | null |
rtrancl_O_push: "S\<^sup>* O R \<subseteq> R O S\<^sup>*"
proof-
{ fix n
have "\<And>s t. (s,t) \<in> S ^^ n O R \<Longrightarrow> (s,t) \<in> R O S\<^sup>*"
proof(induct n)
case (Suc n)
then obtain u where "(s,u) \<in> S" "(u,t) \<in> R O S\<^sup>*" unfolding relpow_Suc by blast
then have "(s,t) \<in> S O R O S\<^sup>*" by auto
also have "... \<subseteq> R O S\<^sup>* O S\<^sup>*" using push by blast
also have "... \<subseteq> R O S\<^sup>*" by auto
finally show ?case.
qed auto
}
thus ?thesis by blast
qed | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | rtrancl_O_push | null |
rtrancl_U_push: "(S \<union> R)\<^sup>* = R\<^sup>* O S\<^sup>*"
proof(intro equalityI subrelI)
fix x y
assume "(x,y) \<in> (S \<union> R)\<^sup>*"
also have "... \<subseteq> (S\<^sup>* O R)\<^sup>* O S\<^sup>*" by regexp
finally obtain z where xz: "(x,z) \<in> (S\<^sup>* O R)\<^sup>*" and zy: "(z,y) \<in> S\<^sup>*" by auto
from xz have "(x,z) \<in> R\<^sup>* O S\<^sup>*"
proof (induct rule: rtrancl_induct)
case (step z w)
then have "(x,w) \<in> R\<^sup>* O S\<^sup>* O S\<^sup>* O R" by auto
also have "... \<subseteq> R\<^sup>* O S\<^sup>* O R" by regexp
also have "... \<subseteq> R\<^sup>* O R O S\<^sup>*" using rtrancl_O_push by auto
also have "... \<subseteq> R\<^sup>* O S\<^sup>*" by regexp
finally show ?case.
qed auto
with zy show "(x,y) \<in> R\<^sup>* O S\<^sup>*" by auto
qed regexp | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | rtrancl_U_push | null |
SN_on_O_push:
assumes SN: "SN_on R X" shows "SN_on (R O S\<^sup>*) X"
proof
fix seq
have SN: "SN_on R (R\<^sup>* `` X)" using SN_on_Image_rtrancl[OF SN].
moreover assume "seq (0::nat) \<in> X"
then have "seq 0 \<in> R\<^sup>* `` X" by auto
ultimately show "chain (R O S\<^sup>*) seq \<Longrightarrow> False"
proof(induct "seq 0" arbitrary: seq rule: SN_on_induct)
case IH
then have 01: "(seq 0, seq 1) \<in> R O S\<^sup>*"
and 12: "(seq 1, seq 2) \<in> R O S\<^sup>*"
and 23: "(seq 2, seq 3) \<in> R O S\<^sup>*" by (auto simp: eval_nat_numeral)
then obtain s t
where s: "(seq 0, s) \<in> R" and s1: "(s, seq 1) \<in> S\<^sup>*"
and t: "(seq 1, t) \<in> R" and t2: "(t, seq 2) \<in> S\<^sup>*" by auto
from s1 t have "(s,t) \<in> S\<^sup>* O R" by auto
with rtrancl_O_push have st: "(s,t) \<in> R O S\<^sup>*" by auto
from t2 23 have "(t, seq 3) \<in> S\<^sup>* O R O S\<^sup>*" by auto
also from rtrancl_O_push have "... \<subseteq> R O S\<^sup>* O S\<^sup>*" by blast
finally have t3: "(t, seq 3) \<in> R O S\<^sup>*" by regexp
let ?seq = "\<lambda>i. case i of 0 \<Rightarrow> s | Suc 0 \<Rightarrow> t | i \<Rightarrow> seq (Suc i)"
show ?case
proof(rule IH)
from s show "(seq 0, ?seq 0) \<in> R" by auto
show "chain (R O S\<^sup>*) ?seq"
proof (intro allI)
fix i show "(?seq i, ?seq (Suc i)) \<in> R O S\<^sup>*"
proof (cases i)
case 0 with st show ?thesis by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_on_O_push | null |
SN_on_Image_push:
assumes SN: "SN_on R X" shows "SN_on R (S\<^sup>* `` X)"
proof-
{ fix n
have "SN_on R ((S^^n) `` X)"
proof(induct n)
case 0 from SN show ?case by auto
case (Suc n)
from SN_on_O_push[OF this] have "SN_on (R O S\<^sup>*) ((S ^^ n) `` X)".
from SN_on_Image[OF this]
have "SN_on (R O S\<^sup>*) ((R O S\<^sup>*) `` (S ^^ n) `` X)".
then have "SN_on R ((R O S\<^sup>*) `` (S ^^ n) `` X)" by (rule SN_on_mono, auto)
from SN_on_subset2[OF Image_mono[OF push subset_refl] this]
have "SN_on R (R `` (S ^^ Suc n) `` X)" by (auto simp: relcomp_Image)
then show ?case by fast
qed
}
then show ?thesis by fast
qed | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | SN_on_Image_push | null |
not_SN_onI[intro]: "f 0 \<in> X \<Longrightarrow> chain R f \<Longrightarrow> \<not> SN_on R X"
by (unfold SN_on_def not_not, intro exI conjI) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | not_SN_onI | null |
shift_comp[simp]: "shift (f \<circ> seq) n = f \<circ> (shift seq n)" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | shift_comp | null |
Id_on_union: "Id_on (A \<union> B) = Id_on A \<union> Id_on B" unfolding Id_on_def by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | Id_on_union | null |
relpow_union_cases: "(a,d) \<in> (A \<union> B)^^n \<Longrightarrow> (a,d) \<in> B^^n \<or> (\<exists> b c k m. (a,b) \<in> B^^k \<and> (b,c) \<in> A \<and> (c,d) \<in> (A \<union> B)^^m \<and> n = Suc (k + m))"
proof (induct n arbitrary: a d)
case (Suc n a e)
let ?AB = "A \<union> B"
from Suc(2) obtain b where ab: "(a,b) \<in> ?AB" and be: "(b,e) \<in> ?AB^^n" by (rule relpow_Suc_E2)
from ab
show ?case
proof
assume "(a,b) \<in> A"
show ?thesis
proof (rule disjI2, intro exI conjI)
show "Suc n = Suc (0 + n)" by simp
show "(a,b) \<in> A" by fact
qed (insert be, auto)
next
assume ab: "(a,b) \<in> B"
from Suc(1)[OF be]
show ?thesis
proof
assume "(b,e) \<in> B ^^ n"
with ab show ?thesis
by (intro disjI1 relpow_Suc_I2)
next
assume "\<exists> c d k m. (b, c) \<in> B ^^ k \<and> (c, d) \<in> A \<and> (d, e) \<in> ?AB ^^ m \<and> n = Suc (k + m)"
then obtain c d k m where "(b, c) \<in> B ^^ k" and *: "(c, d) \<in> A" "(d, e) \<in> ?AB ^^ m" "n = Suc (k + m)" by blast
with ab have ac: "(a,c) \<in> B ^^ (Suc k)" by (intro relpow_Suc_I2)
show ?thesis
by (intro disjI2 exI conjI, rule ac, (rule *)+, simp add: *)
qed
qed | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | relpow_union_cases | null |
trans_refl_imp_rtrancl_id:
assumes "trans r" "refl r"
shows "r\<^sup>* = r"
proof
show "r\<^sup>* \<subseteq> r"
proof
fix x y
assume "(x,y) \<in> r\<^sup>*"
thus "(x,y) \<in> r"
by (induct, insert assms, unfold refl_on_def trans_def, blast+)
qed
qed regexp | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | trans_refl_imp_rtrancl_id | by (intro disjI2 exI conjI, rule ac, (rule +, simp add: |
trans_refl_imp_O_id:
assumes "trans r" "refl r"
shows "r O r = r"
proof(intro equalityI)
show "r O r \<subseteq> r" by(fact trans_O_subset[OF assms(1)])
have "r \<subseteq> r O Id" by auto
moreover have "Id \<subseteq> r" by(fact assms(2)[unfolded refl_O_iff])
ultimately show "r \<subseteq> r O r" by auto
qed | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | trans_refl_imp_O_id | null |
relcomp3_I:
assumes "(t, u) \<in> A" and "(s, t) \<in> B" and "(u, v) \<in> B"
shows "(s, v) \<in> B O A O B"
using assms by blast | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | relcomp3_I | null |
relcomp3_transI:
assumes "trans B" and "(t, u) \<in> B O A O B" and "(s, t) \<in> B" and "(u, v) \<in> B"
shows "(s, v) \<in> B O A O B"
using assms by (auto simp: trans_def intro: relcomp3_I)
lemmas converse_inward = rtrancl_converse[symmetric] converse_Un converse_UNION converse_relcomp
converse_converse converse_Id | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | relcomp3_transI | null |
qc_SN_relto_iff:
assumes "r O s \<subseteq> s O (s \<union> r)\<^sup>*"
shows "SN (r\<^sup>* O s O r\<^sup>*) = SN s"
proof -
from converse_mono [THEN iffD2 , OF assms]
have *: "s\<inverse> O r\<inverse> \<subseteq> (s\<inverse> \<union> r\<inverse>)\<^sup>* O s\<inverse>" unfolding converse_inward .
have "(r\<^sup>* O s O r\<^sup>*)\<inverse> = (r\<inverse>)\<^sup>* O s\<inverse> O (r\<inverse>)\<^sup>*"
by (simp only: converse_relcomp O_assoc rtrancl_converse)
with qc_wf_relto_iff [OF *]
show ?thesis by (simp add: SN_iff_wf)
qed | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | qc_SN_relto_iff | null |
conversion_empty[simp]: "conversion {} = Id"
by (auto simp: conversion_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_empty | null |
symcl_idemp[simp]: "(r\<^sup>\<leftrightarrow>)\<^sup>\<leftrightarrow> = r\<^sup>\<leftrightarrow>" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | symcl_idemp | null |
SN_rel_on:: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set \<Rightarrow> bool" where
"SN_rel_on R S \<equiv> SN_on (relto R S)" | definition | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on | null |
SN_rel_on_alt:: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set \<Rightarrow> bool" where
"SN_rel_on_alt R S T = (\<forall>f. chain (R \<union> S) f \<and> f 0 \<in> T \<longrightarrow> \<not> (INFM j. (f j, f (Suc j)) \<in> R))" | definition | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_alt | null |
SN_rel:: "'a rel \<Rightarrow> 'a rel \<Rightarrow> bool" where
"SN_rel R S \<equiv> SN_rel_on R S UNIV" | abbreviation | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel | null |
SN_rel_alt:: "'a rel \<Rightarrow> 'a rel \<Rightarrow> bool" where
"SN_rel_alt R S \<equiv> SN_rel_on_alt R S UNIV" | abbreviation | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_alt | null |
relto_absorb[simp]: "relto R E O E\<^sup>* = relto R E" "E\<^sup>* O relto R E = relto R E"
using O_assoc and rtrancl_idemp_self_comp by (metis)+ | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_absorb | null |
steps_preserve_SN_on_relto:
assumes steps: "(a, b) \<in> (R \<union> S)^*"
and SN: "SN_on (relto R S) {a}"
shows "SN_on (relto R S) {b}"
proof -
let ?RS = "relto R S"
have "(R \<union> S)^* \<subseteq> S^* \<union> ?RS^*" by regexp
with steps have "(a,b) \<in> S^* \<or> (a,b) \<in> ?RS^*" by auto
thus ?thesis
proof
assume "(a,b) \<in> ?RS^*"
from steps_preserve_SN_on[OF this SN] show ?thesis .
next
assume Ssteps: "(a,b) \<in> S^*"
show ?thesis
proof
fix f
assume "f 0 \<in> {b}" and "chain ?RS f"
hence f0: "f 0 = b" and steps: "\<And>i. (f i, f (Suc i)) \<in> ?RS" by auto
let ?g = "\<lambda> i. if i = 0 then a else f i"
have "\<not> SN_on ?RS {a}" unfolding SN_on_def not_not
proof (rule exI[of _ ?g], intro conjI allI)
fix i
show "(?g i, ?g (Suc i)) \<in> ?RS"
proof (cases i)
case (Suc j)
show ?thesis using steps[of i] unfolding Suc by simp
next
case 0
from steps[of 0, unfolded f0] Ssteps have steps: "(a,f (Suc 0)) \<in> S^* O ?RS" by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | steps_preserve_SN_on_relto | null |
step_preserves_SN_on_relto: assumes st: "(s,t) \<in> R \<union> E"
and SN: "SN_on (relto R E) {s}"
shows "SN_on (relto R E) {t}"
by (rule steps_preserve_SN_on_relto[OF _ SN], insert st, auto) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | step_preserves_SN_on_relto | null |
SN_rel_on_imp_SN_rel_on_alt: "SN_rel_on R S T \<Longrightarrow> SN_rel_on_alt R S T"
proof (unfold SN_rel_on_def)
assume SN: "SN_on (relto R S) T"
show ?thesis
proof (unfold SN_rel_on_alt_def, intro allI impI)
fix f
assume steps: "chain (R \<union> S) f \<and> f 0 \<in> T"
with SN have SN: "SN_on (relto R S) {f 0}"
and steps: "\<And> i. (f i, f (Suc i)) \<in> R \<union> S" unfolding SN_defs by auto
obtain r where r: "\<And> j. r j \<equiv> (f j, f (Suc j)) \<in> R" by auto
show "\<not> (INFM j. (f j, f (Suc j)) \<in> R)"
proof (rule ccontr)
assume "\<not> ?thesis"
hence ih: "infinitely_many r" unfolding infinitely_many_def r by blast
obtain r_index where "r_index = infinitely_many.index r" by simp
with infinitely_many.index_p[OF ih] infinitely_many.index_ordered[OF ih] infinitely_many.index_not_p_between[OF ih]
have r_index: "\<And> i. r (r_index i) \<and> r_index i < r_index (Suc i) \<and> (\<forall> j. r_index i < j \<and> j < r_index (Suc i) \<longrightarrow> \<not> r j)" by auto
obtain g where g: "\<And> i. g i \<equiv> f (r_index i)" ..
{
fix i
let ?ri = "r_index i"
let ?rsi = "r_index (Suc i)"
from r_index have isi: "?ri < ?rsi" by auto
obtain ri rsi where ri: "ri = ?ri" and rsi: "rsi = ?rsi" by auto
with r_index[of i] steps have inter: "\<And> j. ri < j \<and> j < rsi \<Longrightarrow> (f j, f (Suc j)) \<in> S" unfolding r by auto
from ri isi rsi have risi: "ri < rsi" by simp
{
fix n
assume "Suc n \<le> rsi - ri"
hence "(f (Suc ri), f (Suc (n + ri))) \<in> S^*" | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_imp_SN_rel_on_alt | null |
SN_rel_on_alt_imp_SN_rel_on: "SN_rel_on_alt R S T \<Longrightarrow> SN_rel_on R S T"
proof (unfold SN_rel_on_def)
assume SN: "SN_rel_on_alt R S T"
show "SN_on (relto R S) T"
proof
fix f
assume start: "f 0 \<in> T" and "chain (relto R S) f"
hence steps: "\<And> i. (f i, f (Suc i)) \<in> S^* O R O S^*" by auto
let ?prop = "\<lambda> i ai bi. (f i, bi) \<in> S^* \<and> (bi, ai) \<in> R \<and> (ai, f (Suc (i))) \<in> S^*"
{
fix i
from steps obtain bi ai where "?prop i ai bi" by blast
hence "\<exists> ai bi. ?prop i ai bi" by blast
}
hence "\<forall> i. \<exists> bi ai. ?prop i ai bi" by blast
from choice[OF this] obtain b where "\<forall> i. \<exists> ai. ?prop i ai (b i)" by blast
from choice[OF this] obtain a where steps: "\<And> i. ?prop i (a i) (b i)" by blast
from steps[of 0] have fa0: "(f 0, a 0) \<in> S^* O R" by auto
let ?prop = "\<lambda> i li. (b i, a i) \<in> R \<and> (\<forall> j < length li. ((a i # li) ! j, (a i # li) ! Suc j) \<in> S) \<and> last (a i # li) = b (Suc i)"
{
fix i
from steps[of i] steps[of "Suc i"] have "(a i, f (Suc i)) \<in> S^*" and "(f (Suc i), b (Suc i)) \<in> S^*" by auto
from rtrancl_trans[OF this] steps[of i] have R: "(b i, a i) \<in> R" and S: "(a i, b (Suc i)) \<in> S^*" by blast+
from S[unfolded rtrancl_list_conv] obtain li where "last (a i # li) = b (Suc i) \<and> (\<forall> j < length li. ((a i # li) ! j, (a i # li) ! Suc j) \<in> S)" ..
with R have "?prop i li" by blast
hence "\<exists> li. ?prop i li" ..
}
hence "\<forall> i. \<exists> li. ?prop i li" ..
from choice[OF this] obtain l where steps: "\<And> i. ?prop i (l i)" by auto
let ?p = "\<lambda> i. ?prop i (l i)" | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_alt_imp_SN_rel_on | null |
SN_rel_on_conv: "SN_rel_on = SN_rel_on_alt"
by (intro ext) (blast intro: SN_rel_on_imp_SN_rel_on_alt SN_rel_on_alt_imp_SN_rel_on)
lemmas SN_rel_defs = SN_rel_on_def SN_rel_on_alt_def | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_conv | null |
SN_rel_on_alt_r_empty: "SN_rel_on_alt {} S T"
unfolding SN_rel_defs by auto | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_alt_r_empty | null |
SN_rel_on_alt_s_empty: "SN_rel_on_alt R {} = SN_on R"
by (intro ext, unfold SN_rel_defs SN_defs, auto) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_alt_s_empty | null |
SN_rel_on_mono':
assumes R: "R \<subseteq> R'" and S: "S \<subseteq> R' \<union> S'" and SN: "SN_rel_on R' S' T"
shows "SN_rel_on R S T"
proof -
note conv = SN_rel_on_conv SN_rel_on_alt_def INFM_nat_le
show ?thesis unfolding conv
proof(intro allI impI)
fix f
assume "chain (R \<union> S) f \<and> f 0 \<in> T"
with R S have "chain (R' \<union> S') f \<and> f 0 \<in> T" by auto
from SN[unfolded conv, rule_format, OF this]
show "\<not> (\<forall> m. \<exists> n \<ge> m. (f n, f (Suc n)) \<in> R)" using R by auto
qed
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_mono' | null |
relto_mono:
assumes "R \<subseteq> R'" and "S \<subseteq> S'"
shows "relto R S \<subseteq> relto R' S'"
using assms rtrancl_mono by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_mono | null |
SN_rel_on_mono:
assumes R: "R \<subseteq> R'" and S: "S \<subseteq> S'"
and SN: "SN_rel_on R' S' T"
shows "SN_rel_on R S T"
using SN
unfolding SN_rel_on_def using SN_on_mono[OF _ relto_mono[OF R S]] by blast
lemmas SN_rel_on_alt_mono = SN_rel_on_mono[unfolded SN_rel_on_conv] | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_mono | null |
SN_rel_on_imp_SN_on:
assumes "SN_rel_on R S T" shows "SN_on R T"
proof
fix f
assume "chain R f"
and f0: "f 0 \<in> T"
hence "\<And>i. (f i, f (Suc i)) \<in> relto R S" by blast
thus False using assms f0 unfolding SN_rel_on_def SN_defs by blast
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_imp_SN_on | null |
relto_Id: "relto R (S \<union> Id) = relto R S" by simp | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_Id | null |
SN_rel_on_Id:
shows "SN_rel_on R (S \<union> Id) T = SN_rel_on R S T"
unfolding SN_rel_on_def by (simp only: relto_Id) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_Id | null |
SN_rel_on_empty[simp]: "SN_rel_on R {} T = SN_on R T"
unfolding SN_rel_on_def by auto | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_empty | null |
SN_rel_on_ideriv: "SN_rel_on R S T = (\<not> (\<exists> as. ideriv R S as \<and> as 0 \<in> T))" (is "?L = ?R")
proof
assume ?L
show ?R
proof
assume "\<exists> as. ideriv R S as \<and> as 0 \<in> T"
then obtain as where id: "ideriv R S as" and T: "as 0 \<in> T" by auto
note id = id[unfolded ideriv_def]
from \<open>?L\<close>[unfolded SN_rel_on_conv SN_rel_on_alt_def, THEN spec[of _ as]]
id T obtain i where i: "\<And> j. j \<ge> i \<Longrightarrow> (as j, as (Suc j)) \<notin> R" by auto
with id[unfolded INFM_nat, THEN conjunct2, THEN spec[of _ "Suc i"]] show False by auto
qed
next
assume ?R
show ?L
unfolding SN_rel_on_conv SN_rel_on_alt_def
proof(intro allI impI)
fix as
assume "chain (R \<union> S) as \<and> as 0 \<in> T"
with \<open>?R\<close>[unfolded ideriv_def] have "\<not> (INFM i. (as i, as (Suc i)) \<in> R)" by auto
from this[unfolded INFM_nat] obtain i where i: "\<And>j. i < j \<Longrightarrow> (as j, as (Suc j)) \<notin> R" by auto
show "\<not> (INFM j. (as j, as (Suc j)) \<in> R)" unfolding INFM_nat using i by blast
qed
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_on_ideriv | null |
SN_rel_to_SN_rel_alt: "SN_rel R S \<Longrightarrow> SN_rel_alt R S"
proof (unfold SN_rel_on_def)
assume SN: "SN (relto R S)"
show ?thesis
proof (unfold SN_rel_on_alt_def, intro allI impI)
fix f
presume steps: "chain (R \<union> S) f"
obtain r where r: "\<And>j. r j \<equiv> (f j, f (Suc j)) \<in> R" by auto
show "\<not> (INFM j. (f j, f (Suc j)) \<in> R)"
proof (rule ccontr)
assume "\<not> ?thesis"
hence ih: "infinitely_many r" unfolding infinitely_many_def r by blast
obtain r_index where "r_index = infinitely_many.index r" by simp
with infinitely_many.index_p[OF ih] infinitely_many.index_ordered[OF ih] infinitely_many.index_not_p_between[OF ih]
have r_index: "\<And> i. r (r_index i) \<and> r_index i < r_index (Suc i) \<and> (\<forall> j. r_index i < j \<and> j < r_index (Suc i) \<longrightarrow> \<not> r j)" by auto
obtain g where g: "\<And> i. g i \<equiv> f (r_index i)" ..
{
fix i
let ?ri = "r_index i"
let ?rsi = "r_index (Suc i)"
from r_index have isi: "?ri < ?rsi" by auto
obtain ri rsi where ri: "ri = ?ri" and rsi: "rsi = ?rsi" by auto
with r_index[of i] steps have inter: "\<And> j. ri < j \<and> j < rsi \<Longrightarrow> (f j, f (Suc j)) \<in> S" unfolding r by auto
from ri isi rsi have risi: "ri < rsi" by simp
{
fix n
assume "Suc n \<le> rsi - ri"
hence "(f (Suc ri), f (Suc (n + ri))) \<in> S^*"
proof (induct n, simp)
case (Suc n) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_to_SN_rel_alt | null |
SN_rel_alt_to_SN_rel: "SN_rel_alt R S \<Longrightarrow> SN_rel R S"
proof (unfold SN_rel_on_def)
assume SN: "SN_rel_alt R S"
show "SN (relto R S)"
proof
fix f
assume "chain (relto R S) f"
hence steps: "\<And>i. (f i, f (Suc i)) \<in> S^* O R O S^*" by auto
let ?prop = "\<lambda> i ai bi. (f i, bi) \<in> S^* \<and> (bi, ai) \<in> R \<and> (ai, f (Suc (i))) \<in> S^*"
{
fix i
from steps obtain bi ai where "?prop i ai bi" by blast
hence "\<exists> ai bi. ?prop i ai bi" by blast
}
hence "\<forall> i. \<exists> bi ai. ?prop i ai bi" by blast
from choice[OF this] obtain b where "\<forall> i. \<exists> ai. ?prop i ai (b i)" by blast
from choice[OF this] obtain a where steps: "\<And> i. ?prop i (a i) (b i)" by blast
let ?prop = "\<lambda> i li. (b i, a i) \<in> R \<and> (\<forall> j < length li. ((a i # li) ! j, (a i # li) ! Suc j) \<in> S) \<and> last (a i # li) = b (Suc i)"
{
fix i
from steps[of i] steps[of "Suc i"] have "(a i, f (Suc i)) \<in> S^*" and "(f (Suc i), b (Suc i)) \<in> S^*" by auto
from rtrancl_trans[OF this] steps[of i] have R: "(b i, a i) \<in> R" and S: "(a i, b (Suc i)) \<in> S^*" by blast+
from S[unfolded rtrancl_list_conv] obtain li where "last (a i # li) = b (Suc i) \<and> (\<forall> j < length li. ((a i # li) ! j, (a i # li) ! Suc j) \<in> S)" ..
with R have "?prop i li" by blast
hence "\<exists> li. ?prop i li" ..
}
hence "\<forall> i. \<exists> li. ?prop i li" ..
from choice[OF this] obtain l where steps: "\<And> i. ?prop i (l i)" by auto
let ?p = "\<lambda> i. ?prop i (l i)"
from steps have steps: "\<And> i. ?p i" by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_alt_to_SN_rel | null |
SN_rel_alt_r_empty: "SN_rel_alt {} S"
unfolding SN_rel_defs by auto | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_alt_r_empty | null |
SN_rel_alt_s_empty: "SN_rel_alt R {} = SN R"
unfolding SN_rel_defs SN_defs by auto | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_alt_s_empty | null |
SN_rel_mono':
"R \<subseteq> R' \<Longrightarrow> S \<subseteq> R' \<union> S' \<Longrightarrow> SN_rel R' S' \<Longrightarrow> SN_rel R S"
unfolding SN_rel_on_conv SN_rel_defs INFM_nat_le
by (metis contra_subsetD sup.left_idem sup.mono) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_mono' | null |
SN_rel_mono:
assumes R: "R \<subseteq> R'" and S: "S \<subseteq> S'" and SN: "SN_rel R' S'"
shows "SN_rel R S"
using SN unfolding SN_rel_defs using SN_subset[OF _ relto_mono[OF R S]] by blast
lemmas SN_rel_alt_mono = SN_rel_mono[unfolded SN_rel_on_conv] | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_mono | null |
SN_rel_imp_SN: assumes "SN_rel R S" shows "SN R"
proof
fix f
assume "\<forall> i. (f i, f (Suc i)) \<in> R"
hence "\<And> i. (f i, f (Suc i)) \<in> relto R S" by blast
thus False using assms unfolding SN_rel_defs SN_defs by fast
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_imp_SN | null |
relto_trancl_conv: "(relto R S)^+ = ((R \<union> S))^* O R O ((R \<union> S))^*" by regexp | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_trancl_conv | null |
SN_rel_Id:
shows "SN_rel R (S \<union> Id) = SN_rel R S"
unfolding SN_rel_defs by (simp only: relto_Id) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_Id | null |
relto_rtrancl: "relto R (S^*) = relto R S" by regexp | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_rtrancl | null |
SN_rel_empty[simp]: "SN_rel R {} = SN R"
unfolding SN_rel_defs by auto | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_empty | null |
SN_rel_ideriv: "SN_rel R S = (\<not> (\<exists> as. ideriv R S as))" (is "?L = ?R")
proof
assume ?L
show ?R
proof
assume "\<exists> as. ideriv R S as"
then obtain as where id: "ideriv R S as" by auto
note id = id[unfolded ideriv_def]
from \<open>?L\<close>[unfolded SN_rel_on_conv SN_rel_defs, THEN spec[of _ as]]
id obtain i where i: "\<And> j. j \<ge> i \<Longrightarrow> (as j, as (Suc j)) \<notin> R" by auto
with id[unfolded INFM_nat, THEN conjunct2, THEN spec[of _ "Suc i"]] show False by auto
qed
next
assume ?R
show ?L
unfolding SN_rel_on_conv SN_rel_defs
proof (intro allI impI)
fix as
presume "chain (R \<union> S) as"
with \<open>?R\<close>[unfolded ideriv_def] have "\<not> (INFM i. (as i, as (Suc i)) \<in> R)" by auto
from this[unfolded INFM_nat] obtain i where i: "\<And> j. i < j \<Longrightarrow> (as j, as (Suc j)) \<notin> R" by auto
show "\<not> (INFM j. (as j, as (Suc j)) \<in> R)" unfolding INFM_nat using i by blast
qed simp
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ideriv | null |
SN_rel_map:
fixes R Rw R' Rw' :: "'a rel"
defines A: "A \<equiv> R' \<union> Rw'"
assumes SN: "SN_rel R' Rw'"
and R: "\<And>s t. (s,t) \<in> R \<Longrightarrow> (f s, f t) \<in> A^* O R' O A^*"
and Rw: "\<And>s t. (s,t) \<in> Rw \<Longrightarrow> (f s, f t) \<in> A^*"
shows "SN_rel R Rw"
unfolding SN_rel_defs
proof
fix g
assume steps: "chain (relto R Rw) g"
let ?f = "\<lambda>i. (f (g i))"
obtain h where h: "h = ?f" by auto
{
fix i
let ?m = "\<lambda> (x,y). (f x, f y)"
{
fix s t
assume "(s,t) \<in> Rw^*"
hence "?m (s,t) \<in> A^*"
proof (induct)
case base show ?case by simp
next
case (step t u)
from Rw[OF step(2)] step(3)
show ?case by auto
qed
} note Rw = this
from steps have "(g i, g (Suc i)) \<in> relto R Rw" ..
from this | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_map | null |
SN_rel_ext_type= top_s | top_ns | normal_s | normal_ns | datatype | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext_type | null |
SN_rel_ext_step:: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel \<Rightarrow> SN_rel_ext_type \<Rightarrow> 'a rel" where
"SN_rel_ext_step P Pw R Rw top_s = P"
| "SN_rel_ext_step P Pw R Rw top_ns = Pw"
| "SN_rel_ext_step P Pw R Rw normal_s = R"
| "SN_rel_ext_step P Pw R Rw normal_ns = Rw" | fun | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext_step | null |
SN_rel_ext:: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
"SN_rel_ext P Pw R Rw M \<equiv> (\<not> (\<exists>f t.
(\<forall> i. (f i, f (Suc i)) \<in> SN_rel_ext_step P Pw R Rw (t i))
\<and> (\<forall> i. M (f i))
\<and> (INFM i. t i \<in> {top_s,top_ns})
\<and> (INFM i. t i \<in> {top_s,normal_s})))" | definition | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext | relative termination with four relations as required in DP-framework |
SN_rel_ext_step_mono: assumes "P \<subseteq> P'" "Pw \<subseteq> Pw'" "R \<subseteq> R'" "Rw \<subseteq> Rw'"
shows "SN_rel_ext_step P Pw R Rw t \<subseteq> SN_rel_ext_step P' Pw' R' Rw' t"
using assms
by (cases t, auto) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext_step_mono | null |
SN_rel_ext_mono: assumes subset: "P \<subseteq> P'" "Pw \<subseteq> Pw'" "R \<subseteq> R'" "Rw \<subseteq> Rw'" and
SN: "SN_rel_ext P' Pw' R' Rw' M" shows "SN_rel_ext P Pw R Rw M"
using SN_rel_ext_step_mono[OF subset] SN unfolding SN_rel_ext_def by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext_mono | null |
SN_rel_ext_trans:
fixes P Pw R Rw :: "'a rel" and M :: "'a \<Rightarrow> bool"
defines M': "M' \<equiv> {(s,t). M t}"
defines A: "A \<equiv> (P \<union> Pw \<union> R \<union> Rw) \<inter> M'"
assumes "SN_rel_ext P Pw R Rw M"
shows "SN_rel_ext (A^* O (P \<inter> M') O A^*) (A^* O ((P \<union> Pw) \<inter> M') O A^*) (A^* O ((P \<union> R) \<inter> M') O A^*) (A^*) M" (is "SN_rel_ext ?P ?Pw ?R ?Rw M")
proof (rule ccontr)
let ?relt = "SN_rel_ext_step ?P ?Pw ?R ?Rw"
let ?rel = "SN_rel_ext_step P Pw R Rw"
assume "\<not> ?thesis"
from this[unfolded SN_rel_ext_def]
obtain f ty
where steps: "\<And> i. (f i, f (Suc i)) \<in> ?relt (ty i)"
and min: "\<And> i. M (f i)"
and inf1: "INFM i. ty i \<in> {top_s, top_ns}"
and inf2: "INFM i. ty i \<in> {top_s, normal_s}"
by auto
let ?Un = "\<lambda> tt. \<Union> (?rel ` tt)"
let ?UnM = "\<lambda> tt. (\<Union> (?rel ` tt)) \<inter> M'"
let ?A = "?UnM {top_s,top_ns,normal_s,normal_ns}"
let ?P' = "?UnM {top_s}"
let ?Pw' = "?UnM {top_s,top_ns}"
let ?R' = "?UnM {top_s,normal_s}"
let ?Rw' = "?UnM {top_s,top_ns,normal_s,normal_ns}"
have A: "A = ?A" unfolding A by auto
have P: "(P \<inter> M') = ?P'" by auto
have Pw: "(P \<union> Pw) \<inter> M' = ?Pw'" by auto
have R: "(P \<union> R) \<inter> M' = ?R'" by auto
have Rw: "A = ?Rw'" unfolding A ..
{ | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext_trans | null |
SN_rel_ext_map: fixes P Pw R Rw P' Pw' R' Rw' :: "'a rel" and M M' :: "'a \<Rightarrow> bool"
defines Ms: "Ms \<equiv> {(s,t). M' t}"
defines A: "A \<equiv> (P' \<union> Pw' \<union> R' \<union> Rw') \<inter> Ms"
assumes SN: "SN_rel_ext P' Pw' R' Rw' M'"
and P: "\<And> s t. M s \<Longrightarrow> M t \<Longrightarrow> (s,t) \<in> P \<Longrightarrow> (f s, f t) \<in> (A^* O (P' \<inter> Ms) O A^*) \<and> I t"
and Pw: "\<And> s t. M s \<Longrightarrow> M t \<Longrightarrow> (s,t) \<in> Pw \<Longrightarrow> (f s, f t) \<in> (A^* O ((P' \<union> Pw') \<inter> Ms) O A^*) \<and> I t"
and R: "\<And> s t. I s \<Longrightarrow> M s \<Longrightarrow> M t \<Longrightarrow> (s,t) \<in> R \<Longrightarrow> (f s, f t) \<in> (A^* O ((P' \<union> R') \<inter> Ms) O A^*) \<and> I t"
and Rw: "\<And> s t. I s \<Longrightarrow> M s \<Longrightarrow> M t \<Longrightarrow> (s,t) \<in> Rw \<Longrightarrow> (f s, f t) \<in> A^* \<and> I t"
shows "SN_rel_ext P Pw R Rw M"
proof -
note SN = SN_rel_ext_trans[OF SN]
let ?P = "(A^* O (P' \<inter> Ms) O A^*)"
let ?Pw = "(A^* O ((P' \<union> Pw') \<inter> Ms) O A^*)"
let ?R = "(A^* O ((P' \<union> R') \<inter> Ms) O A^*)"
let ?Rw = "A^*"
let ?relt = "SN_rel_ext_step ?P ?Pw ?R ?Rw"
let ?rel = "SN_rel_ext_step P Pw R Rw"
show ?thesis
proof (rule ccontr)
assume "\<not> ?thesis"
from this[unfolded SN_rel_ext_def]
obtain g ty
where steps: "\<And> i. (g i, g (Suc i)) \<in> ?rel (ty i)"
and min: "\<And> i. M (g i)"
and inf1: "INFM i. ty i \<in> {top_s, top_ns}"
and inf2: "INFM i. ty i \<in> {top_s, normal_s}"
by auto
from inf1[unfolded INFM_nat] obtain k where k: "ty k \<in> {top_s, top_ns}" by auto
let ?k = "Suc k"
let ?i = "shift id ?k" | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext_map | null |
SN_rel_ext_map_min: fixes P Pw R Rw P' Pw' R' Rw' :: "'a rel" and M M' :: "'a \<Rightarrow> bool"
defines Ms: "Ms \<equiv> {(s,t). M' t}"
defines A: "A \<equiv> P' \<inter> Ms \<union> Pw' \<inter> Ms \<union> R' \<union> Rw'"
assumes SN: "SN_rel_ext P' Pw' R' Rw' M'"
and M: "\<And> t. M t \<Longrightarrow> M' (f t)"
and M': "\<And> s t. M' s \<Longrightarrow> (s,t) \<in> R' \<union> Rw' \<Longrightarrow> M' t"
and P: "\<And> s t. M s \<Longrightarrow> M t \<Longrightarrow> M' (f s) \<Longrightarrow> M' (f t) \<Longrightarrow> (s,t) \<in> P \<Longrightarrow> (f s, f t) \<in> (A^* O (P' \<inter> Ms) O A^*) \<and> I t"
and Pw: "\<And> s t. M s \<Longrightarrow> M t \<Longrightarrow> M' (f s) \<Longrightarrow> M' (f t) \<Longrightarrow> (s,t) \<in> Pw \<Longrightarrow> (f s, f t) \<in> (A^* O (P' \<inter> Ms \<union> Pw' \<inter> Ms) O A^*) \<and> I t"
and R: "\<And> s t. I s \<Longrightarrow> M s \<Longrightarrow> M t \<Longrightarrow> M' (f s) \<Longrightarrow> M' (f t) \<Longrightarrow> (s,t) \<in> R \<Longrightarrow> (f s, f t) \<in> (A^* O (P' \<inter> Ms \<union> R') O A^*) \<and> I t"
and Rw: "\<And> s t. I s \<Longrightarrow> M s \<Longrightarrow> M t \<Longrightarrow> M' (f s) \<Longrightarrow> M' (f t) \<Longrightarrow> (s,t) \<in> Rw \<Longrightarrow> (f s, f t) \<in> A^* \<and> I t"
shows "SN_rel_ext P Pw R Rw M"
proof -
let ?Ms = "{(s,t). M' t}"
let ?A = "(P' \<union> Pw' \<union> R' \<union> Rw') \<inter> ?Ms"
{
fix s t
assume s: "M' s" and "(s,t) \<in> A"
with M'[OF s, of t] have "(s,t) \<in> ?A \<and> M' t" unfolding Ms A by auto
} note Aone = this
{
fix s t
assume s: "M' s" and steps: "(s,t) \<in> A^*"
from steps have "(s,t) \<in> ?A^* \<and> M' t"
proof (induct)
case base from s show ?case by simp
next
case (step t u)
note one = Aone[OF step(3)[THEN conjunct2] step(2)]
from step(3) one
have steps: "(s,u) \<in> ?A^* O ?A" by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_ext_map_min | and a version where it is assumed that f always preserves M and that R' and Rw' preserve M' |
SN_relto_imp_SN_rel: "SN (relto R S) \<Longrightarrow> SN_rel R S"
proof -
assume SN: "SN (relto R S)"
show ?thesis
proof (simp only: SN_rel_on_conv SN_rel_defs, intro allI impI)
fix f
presume steps: "chain (R \<union> S) f"
obtain r where r: "\<And> j. r j \<equiv> (f j, f (Suc j)) \<in> R" by auto
show "\<not> (INFM j. (f j, f (Suc j)) \<in> R)"
proof (rule ccontr)
assume "\<not> ?thesis"
hence ih: "infinitely_many r" unfolding infinitely_many_def r INFM_nat_le by blast
obtain r_index where "r_index = infinitely_many.index r" by simp
with infinitely_many.index_p[OF ih] infinitely_many.index_ordered[OF ih] infinitely_many.index_not_p_between[OF ih]
have r_index: "\<And> i. r (r_index i) \<and> r_index i < r_index (Suc i) \<and> (\<forall> j. r_index i < j \<and> j < r_index (Suc i) \<longrightarrow> \<not> r j)" by auto
obtain g where g: "\<And> i. g i \<equiv> f (r_index i)" ..
{
fix i
let ?ri = "r_index i"
let ?rsi = "r_index (Suc i)"
from r_index have isi: "?ri < ?rsi" by auto
obtain ri rsi where ri: "ri = ?ri" and rsi: "rsi = ?rsi" by auto
with r_index[of i] steps have inter: "\<And> j. ri < j \<and> j < rsi \<Longrightarrow> (f j, f (Suc j)) \<in> S" unfolding r by auto
from ri isi rsi have risi: "ri < rsi" by simp
{
fix n
assume "Suc n \<le> rsi - ri"
hence "(f (Suc ri), f (Suc (n + ri))) \<in> S^*"
proof (induct n, simp)
case (Suc n) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_relto_imp_SN_rel | OLD PART |
rtrancl_list_conv:
"((s,t) \<in> R^*) =
(\<exists>list. last (s # list) = t \<and> (\<forall>i. i < length list \<longrightarrow> ((s # list) ! i, (s # list) ! Suc i) \<in> R))" (is "?l = ?r")
proof
assume ?r
then obtain list where "last (s # list) = t \<and> (\<forall> i. i < length list \<longrightarrow> ((s # list) ! i, (s # list) ! Suc i) \<in> R)" ..
thus ?l
proof (induct list arbitrary: s, simp)
case (Cons u ll)
hence "last (u # ll) = t \<and> (\<forall> i. i < length ll \<longrightarrow> ((u # ll) ! i, (u # ll) ! Suc i) \<in> R)" by auto
from Cons(1)[OF this] have rec: "(u,t) \<in> R^*" .
from Cons have "(s, u) \<in> R" by auto
with rec show ?case by auto
qed
next
assume ?l
from rtrancl_imp_seq[OF this]
obtain S n where s: "S 0 = s" and t: "S n = t" and steps: "\<forall> i<n. (S i, S (Suc i)) \<in> R" by auto
let ?list = "map (\<lambda> i. S (Suc i)) [0 ..< n]"
show ?r
proof (rule exI[of _ ?list], intro conjI,
cases n, simp add: s[symmetric] t[symmetric], simp add: t[symmetric])
show "\<forall> i < length ?list. ((s # ?list) ! i, (s # ?list) ! Suc i) \<in> R"
proof (intro allI impI)
fix i
assume i: "i < length ?list"
thus "((s # ?list) ! i, (s # ?list) ! Suc i) \<in> R"
proof (cases i, simp add: s[symmetric] steps)
case (Suc j)
with i steps show ?thesis by simp | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | rtrancl_list_conv | FIXME: move |
choice:: "(nat \<Rightarrow> 'a list) \<Rightarrow> nat \<Rightarrow> (nat \<times> nat)" where
"choice f 0 = (0,0)"
| "choice f (Suc n) = (let (i, j) = choice f n in
if Suc j < length (f i)
then (i, Suc j)
else (Suc i, 0))" | fun | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | choice | null |
SN_rel_imp_SN_relto: "SN_rel R S \<Longrightarrow> SN (relto R S)"
proof -
assume SN: "SN_rel R S"
show "SN (relto R S)"
proof
fix f
assume "\<forall> i. (f i, f (Suc i)) \<in> relto R S"
hence steps: "\<And> i. (f i, f (Suc i)) \<in> S^* O R O S^*" by auto
let ?prop = "\<lambda> i ai bi. (f i, bi) \<in> S^* \<and> (bi, ai) \<in> R \<and> (ai, f (Suc (i))) \<in> S^*"
{
fix i
from steps obtain bi ai where "?prop i ai bi" by blast
hence "\<exists> ai bi. ?prop i ai bi" by blast
}
hence "\<forall> i. \<exists> bi ai. ?prop i ai bi" by blast
from choice[OF this] obtain b where "\<forall> i. \<exists> ai. ?prop i ai (b i)" by blast
from choice[OF this] obtain a where steps: "\<And> i. ?prop i (a i) (b i)" by blast
let ?prop = "\<lambda> i li. (b i, a i) \<in> R \<and> (\<forall> j < length li. ((a i # li) ! j, (a i # li) ! Suc j) \<in> S) \<and> last (a i # li) = b (Suc i)"
{
fix i
from steps[of i] steps[of "Suc i"] have "(a i, f (Suc i)) \<in> S^*" and "(f (Suc i), b (Suc i)) \<in> S^*" by auto
from rtrancl_trans[OF this] steps[of i] have R: "(b i, a i) \<in> R" and S: "(a i, b (Suc i)) \<in> S^*" by blast+
from S[unfolded rtrancl_list_conv] obtain li where "last (a i # li) = b (Suc i) \<and> (\<forall> j < length li. ((a i # li) ! j, (a i # li) ! Suc j) \<in> S)" ..
with R have "?prop i li" by blast
hence "\<exists> li. ?prop i li" ..
}
hence "\<forall> i. \<exists> li. ?prop i li" ..
from choice[OF this] obtain l where steps: "\<And> i. ?prop i (l i)" by auto
let ?p = "\<lambda> i. ?prop i (l i)"
from steps have steps: "\<And> i. ?p i" by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_imp_SN_relto | null |
SN_relto_SN_rel_conv: "SN (relto R S) = SN_rel R S"
by (blast intro: SN_relto_imp_SN_rel SN_rel_imp_SN_relto) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_relto_SN_rel_conv | null |
SN_rel_empty1: "SN_rel {} S"
unfolding SN_rel_defs by auto | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_empty1 | null |
SN_rel_empty2: "SN_rel R {} = SN R"
unfolding SN_rel_defs SN_defs by auto | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_rel_empty2 | null |
SN_relto_mono:
assumes R: "R \<subseteq> R'" and S: "S \<subseteq> S'"
and SN: "SN (relto R' S')"
shows "SN (relto R S)"
using SN SN_subset[OF _ relto_mono[OF R S]] by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_relto_mono | null |
SN_relto_imp_SN:
assumes "SN (relto R S)" shows "SN R"
proof
fix f
assume "\<forall>i. (f i, f (Suc i)) \<in> R"
hence "\<And>i. (f i, f (Suc i)) \<in> relto R S" by blast
thus False using assms unfolding SN_defs by blast
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_relto_imp_SN | null |
SN_relto_Id:
"SN (relto R (S \<union> Id)) = SN (relto R S)"
by (simp only: relto_Id) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_relto_Id | null |
trans_subset_SN:
assumes "trans R" and "R \<subseteq> (r \<union> s)" and "SN r" and "SN s"
shows "SN R"
proof
fix f :: "nat \<Rightarrow> 'a"
assume "f 0 \<in> UNIV"
and chain: "chain R f"
have *: "\<And>i j. i < j \<Longrightarrow> (f i, f j) \<in> r \<union> s"
using assms and chain_imp_trancl [OF chain] by auto
let ?M = "{i. \<forall>j>i. (f i, f j) \<notin> r}"
show False
proof (cases "finite ?M")
let ?n = "Max ?M"
assume "finite ?M"
with Max_ge have "\<forall>i\<in>?M. i \<le> ?n" by simp
then have "\<forall>k\<ge>Suc ?n. \<exists>k'>k. (f k, f k') \<in> r" by auto
with steps_imp_chainp [of "Suc ?n" "\<lambda>x y. (x, y) \<in> r"] and assms
show False by auto
next
assume "infinite ?M"
then have "INFM j. j \<in> ?M" by (simp add: Inf_many_def)
then interpret infinitely_many "\<lambda>i. i \<in> ?M" by (unfold_locales) assumption
define g where [simp]: "g = index"
have "\<forall>i. (f (g i), f (g (Suc i))) \<in> s"
proof
fix i
have less: "g i < g (Suc i)" using index_ordered_less [of i "Suc i"] by simp
have "g i \<in> ?M" using index_p by simp
then have "(f (g i), f (g (Suc i))) \<notin> r" using less by simp
moreover have "(f (g i), f (g (Suc i))) \<in> r \<union> s" using * [OF less] by simp | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | trans_subset_SN | Termination inheritance by transitivity (see, e.g., Geser's thesis). |
SN_Un_conv:
assumes "trans (r \<union> s)"
shows "SN (r \<union> s) \<longleftrightarrow> SN r \<and> SN s"
(is "SN ?r \<longleftrightarrow> ?rhs")
proof
assume "SN (r \<union> s)" thus "SN r \<and> SN s"
using SN_subset[of ?r] by blast
next
assume "SN r \<and> SN s"
with trans_subset_SN[OF assms subset_refl] show "SN ?r" by simp
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_Un_conv | SN s |
SN_relto_Un:
"SN (relto (R \<union> S) Q) \<longleftrightarrow> SN (relto R (S \<union> Q)) \<and> SN (relto S Q)"
(is "SN ?a \<longleftrightarrow> SN ?b \<and> SN ?c")
proof -
have eq: "?a^+ = ?b^+ \<union> ?c^+" by regexp
from SN_Un_conv[of "?b^+" "?c^+", unfolded eq[symmetric]]
show ?thesis unfolding SN_trancl_SN_conv by simp
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_relto_Un | null |
SN_relto_split:
assumes "SN (relto r (s \<union> q2) \<union> relto q1 (s \<union> q2))" (is "SN ?a")
and "SN (relto s q2)" (is "SN ?b")
shows "SN (relto r (q1 \<union> q2) \<union> relto s (q1 \<union> q2))" (is "SN ?c")
proof -
have "?c^+ \<subseteq> ?a^+ \<union> ?b^+" by regexp
from trans_subset_SN[OF _ this, unfolded SN_trancl_SN_conv, OF _ assms]
show ?thesis by simp
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | SN_relto_split | null |
relto_trancl_subset: assumes "a \<subseteq> c" and "b \<subseteq> c" shows "relto a b \<subseteq> c^+"
proof -
have "relto a b \<subseteq> (a \<union> b)^+" by regexp
also have "\<dots> \<subseteq> c^+"
by (rule trancl_mono_set, insert assms, auto)
finally show ?thesis .
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_trancl_subset | null |
relto_fun:: "'a rel \<Rightarrow> 'a rel \<Rightarrow> nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a \<times> 'a \<Rightarrow> bool" where
relto_fun: "as 0 = a \<Longrightarrow> as m = b \<Longrightarrow>
(\<And> i. i < m \<Longrightarrow>
(sel i \<longrightarrow> (as i, as (Suc i)) \<in> A) \<and> (\<not> sel i \<longrightarrow> (as i, as (Suc i)) \<in> B))
\<Longrightarrow> n = card { i . i < m \<and> sel i}
\<Longrightarrow> (n = 0 \<longleftrightarrow> m = 0) \<Longrightarrow> relto_fun A B n as sel m (a,b)" | inductive | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_fun | An explicit version of @{const relto} which mentions all intermediate terms |
relto_funD: assumes "relto_fun A B n as sel m (a,b)"
shows "as 0 = a" "as m = b"
"\<And> i. i < m \<Longrightarrow> sel i \<Longrightarrow> (as i, as (Suc i)) \<in> A"
"\<And> i. i < m \<Longrightarrow> \<not> sel i \<Longrightarrow> (as i, as (Suc i)) \<in> B"
"n = card { i . i < m \<and> sel i}"
"n = 0 \<longleftrightarrow> m = 0"
using assms[unfolded relto_fun.simps] by blast+ | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_funD | null |
relto_fun_refl: "\<exists> as sel. relto_fun A B 0 as sel 0 (a,a)"
by (rule exI[of _ "\<lambda> _. a"], rule exI, rule relto_fun, auto) | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_fun_refl | null |
relto_into_relto_fun: assumes "(a,b) \<in> relto A B"
shows "\<exists> as sel m. relto_fun A B (Suc 0) as sel m (a,b)"
proof -
from assms obtain a' b' where aa: "(a,a') \<in> B^*" and ab: "(a',b') \<in> A"
and bb: "(b',b) \<in> B^*" by auto
from aa[unfolded rtrancl_fun_conv] obtain f1 n1 where
f1: "f1 0 = a" "f1 n1 = a'" "\<And> i. i<n1 \<Longrightarrow> (f1 i, f1 (Suc i)) \<in> B" by auto
from bb[unfolded rtrancl_fun_conv] obtain f2 n2 where
f2: "f2 0 = b'" "f2 n2 = b" "\<And> i. i<n2 \<Longrightarrow> (f2 i, f2 (Suc i)) \<in> B" by auto
let ?gen = "\<lambda> aa ab bb i. if i < n1 then aa i else if i = n1 then ab else bb (i - Suc n1)"
let ?f = "?gen f1 a' f2"
let ?sel = "?gen (\<lambda> _. False) True (\<lambda> _. False)"
let ?m = "Suc (n1 + n2)"
show ?thesis
proof (rule exI[of _ ?f], rule exI[of _ ?sel], rule exI[of _ ?m], rule relto_fun)
fix i
assume i: "i < ?m"
show "(?sel i \<longrightarrow> (?f i, ?f (Suc i)) \<in> A) \<and> (\<not> ?sel i \<longrightarrow> (?f i, ?f (Suc i)) \<in> B)"
proof (cases "i < n1")
case True
with f1(3)[OF this] f1(2) show ?thesis by (cases "Suc i = n1", auto)
next
case False note nle = this
show ?thesis
proof (cases "i > n1")
case False
with nle have "i = n1" by auto
thus ?thesis using f1 f2 ab by auto
next
case True | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_into_relto_fun | null |
relto_fun_trans: assumes ab: "relto_fun A B n1 as1 sel1 m1 (a,b)"
and bc: "relto_fun A B n2 as2 sel2 m2 (b,c)"
shows "\<exists> as sel. relto_fun A B (n1 + n2) as sel (m1 + m2) (a,c)"
proof -
from relto_funD[OF ab]
have 1: "as1 0 = a" "as1 m1 = b"
"\<And> i. i < m1 \<Longrightarrow> (sel1 i \<longrightarrow> (as1 i, as1 (Suc i)) \<in> A) \<and> (\<not> sel1 i \<longrightarrow> (as1 i, as1 (Suc i)) \<in> B)"
"n1 = 0 \<longleftrightarrow> m1 = 0" and card1: "n1 = card {i. i < m1 \<and> sel1 i}" by blast+
from relto_funD[OF bc]
have 2: "as2 0 = b" "as2 m2 = c"
"\<And> i. i < m2 \<Longrightarrow> (sel2 i \<longrightarrow> (as2 i, as2 (Suc i)) \<in> A) \<and> (\<not> sel2 i \<longrightarrow> (as2 i, as2 (Suc i)) \<in> B)"
"n2 = 0 \<longleftrightarrow> m2 = 0" and card2: "n2 = card {i. i < m2 \<and> sel2 i}" by blast+
let ?as = "\<lambda> i. if i < m1 then as1 i else as2 (i - m1)"
let ?sel = "\<lambda> i. if i < m1 then sel1 i else sel2 (i - m1)"
let ?m = "m1 + m2"
let ?n = "n1 + n2"
show ?thesis
proof (rule exI[of _ ?as], rule exI[of _ ?sel], rule relto_fun)
have id: "{ i . i < ?m \<and> ?sel i} = { i . i < m1 \<and> sel1 i} \<union> ((+) m1) ` { i. i < m2 \<and> sel2 i}"
(is "_ = ?A \<union> ?f ` ?B")
by force
have "card (?A \<union> ?f ` ?B) = card ?A + card (?f ` ?B)"
by (rule card_Un_disjoint, auto)
also have "card (?f ` ?B) = card ?B"
by (rule card_image, auto simp: inj_on_def)
finally show "?n = card { i . i < ?m \<and> ?sel i}" unfolding card1 card2 id by simp
next
fix i
assume i: "i < ?m"
show "(?sel i \<longrightarrow> (?as i, ?as (Suc i)) \<in> A) \<and> (\<not> ?sel i \<longrightarrow> (?as i, ?as (Suc i)) \<in> B)" | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_fun_trans | null |
reltos_into_relto_fun: assumes "(a,b) \<in> (relto A B)^^n"
shows "\<exists> as sel m. relto_fun A B n as sel m (a,b)"
using assms
proof (induct n arbitrary: b)
case (0 b)
hence b: "b = a" by auto
show ?case unfolding b using relto_fun_refl[of A B a] by blast
next
case (Suc n c)
from relpow_Suc_E[OF Suc(2)]
obtain b where ab: "(a,b) \<in> (relto A B)^^n" and bc: "(b,c) \<in> relto A B" by auto
from Suc(1)[OF ab] obtain as sel m where
IH: "relto_fun A B n as sel m (a, b)" by auto
from relto_into_relto_fun[OF bc] obtain as sel m where "relto_fun A B (Suc 0) as sel m (b,c)" by blast
from relto_fun_trans[OF IH this] show ?case by auto
qed | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | reltos_into_relto_fun | null |
relto_fun_into_reltos: assumes "relto_fun A B n as sel m (a,b)"
shows "(a,b) \<in> (relto A B)^^n"
proof -
note * = relto_funD[OF assms]
{
fix m'
let ?c = "\<lambda> m'. card {i. i < m' \<and> sel i}"
assume "m' \<le> m"
hence "(?c m' > 0 \<longrightarrow> (as 0, as m') \<in> (relto A B)^^ ?c m') \<and> (?c m' = 0 \<longrightarrow> (as 0, as m') \<in> B^*)"
proof (induct m')
case (Suc m')
let ?x = "as 0"
let ?y = "as m'"
let ?z = "as (Suc m')"
let ?C = "?c (Suc m')"
have C: "?C = ?c m' + (if (sel m') then 1 else 0)"
proof -
have id: "{i. i < Suc m' \<and> sel i} = {i. i < m' \<and> sel i} \<union> (if sel m' then {m'} else {})"
by (cases "sel m'", auto, case_tac "x = m'", auto)
show ?thesis unfolding id by auto
qed
from Suc(2) have m': "m' \<le> m" and lt: "m' < m" by auto
from Suc(1)[OF m'] have IH: "?c m' > 0 \<Longrightarrow> (?x, ?y) \<in> (relto A B) ^^ ?c m'"
"?c m' = 0 \<Longrightarrow> (?x, ?y) \<in> B^*" by auto
from *(3-4)[OF lt] have yz: "sel m' \<Longrightarrow> (?y, ?z) \<in> A" "\<not> sel m' \<Longrightarrow> (?y, ?z) \<in> B" by auto
show ?case
proof (cases "?c m' = 0")
case True note c = this
from IH(2)[OF this] have xy: "(?x, ?y) \<in> B^*" by auto
show ?thesis | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_fun_into_reltos | null |
relto_relto_fun_conv: "((a,b) \<in> (relto A B)^^n) = (\<exists> as sel m. relto_fun A B n as sel m (a,b))"
using relto_fun_into_reltos[of A B n _ _ _ a b] reltos_into_relto_fun[of a b n B A] by blast | lemma | Abstract-Rewriting | [
"Abstract_Rewriting"
] | Abstract-Rewriting/Relative_Rewriting.thy | relto_relto_fun_conv | null |
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