# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1498840919 -3600
# Node ID 6670f2cb5741b4e73268fbec1417313b6cbb1ec5
# Parent  9deaff82e0c57b87370fd073fb0ceb68b2410a52
added

diff -r 9deaff82e0c5 -r 6670f2cb5741 thys/Fun.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/Fun.thy	Fri Jun 30 17:41:59 2017 +0100
@@ -0,0 +1,213 @@
+theory Fun
+  imports Lexer "~~/src/HOL/Library/Infinite_Set"
+begin
+
+section {* some fun tests *}
+
+fun
+ zeroable :: "rexp ⇒ bool"
+where
+  "zeroable (ZERO) = True"
+| "zeroable (ONE) = False"
+| "zeroable (CHAR c) = False"
+| "zeroable (ALT r1 r2) = (zeroable r1 ∧ zeroable r2)"
+| "zeroable (SEQ r1 r2) = (zeroable r1 ∨ zeroable r2)"
+| "zeroable (STAR r) = False"
+
+lemma zeroable_correctness:
+  shows "zeroable r ⟷ L r = {}"
+apply(induct r rule: zeroable.induct)
+apply(auto simp add: Sequ_def)
+done
+
+fun
+ atmostempty :: "rexp ⇒ bool"
+where
+  "atmostempty (ZERO) = True"
+| "atmostempty (ONE) = True"
+| "atmostempty (CHAR c) = False"
+| "atmostempty (ALT r1 r2) = (atmostempty r1 ∧ atmostempty r2)"
+| "atmostempty (SEQ r1 r2) = ((zeroable r1) ∨ (zeroable r2) ∨ (atmostempty r1 ∧ atmostempty r2))"
+| "atmostempty (STAR r) = atmostempty r"
+
+fun
+ somechars :: "rexp ⇒ bool"
+where
+  "somechars (ZERO) = False"
+| "somechars (ONE) = False"
+| "somechars (CHAR c) = True"
+| "somechars (ALT r1 r2) = (somechars r1 ∨ somechars r2)"
+| "somechars (SEQ r1 r2) = ((¬zeroable r1 ∧ somechars r2) ∨ (¬zeroable r2 ∧ somechars r1) ∨ 
+                           (somechars r1 ∧ nullable r2) ∨ (somechars r2 ∧ nullable r1))"
+| "somechars (STAR r) = somechars r"
+
+lemma somechars_correctness:
+  shows "somechars r ⟷ (∃s. s ≠ [] ∧ s ∈ L r)"
+apply(induct r rule: somechars.induct)
+apply(simp)
+apply(simp)
+apply(simp)
+apply(auto)[1]
+prefer 2
+apply(simp)
+apply(rule iffI)
+apply(auto)[1]
+apply (metis Star_decomp neq_Nil_conv)
+apply(rule iffI)
+apply(simp add: Sequ_def zeroable_correctness nullable_correctness)
+apply(auto)[1]
+apply(simp add: Sequ_def zeroable_correctness nullable_correctness)
+apply(auto)[1]
+done
+
+lemma atmostempty_correctness_aux:
+  shows "atmostempty r ⟷ ¬somechars r"
+apply(induct r)
+apply(simp_all)
+apply(auto simp add: zeroable_correctness nullable_correctness somechars_correctness)
+done
+
+lemma atmostempty_correctness:
+  shows "atmostempty r ⟷ L r ⊆ {[]}"
+by(auto simp add: atmostempty_correctness_aux somechars_correctness)
+
+fun
+ infinitestrings :: "rexp ⇒ bool"
+where
+  "infinitestrings (ZERO) = False"
+| "infinitestrings (ONE) = False"
+| "infinitestrings (CHAR c) = False"
+| "infinitestrings (ALT r1 r2) = (infinitestrings r1 ∨ infinitestrings r2)"
+| "infinitestrings (SEQ r1 r2) = ((¬zeroable r1 ∧ infinitestrings r2) ∨ (¬zeroable r2 ∧ infinitestrings r1))"
+| "infinitestrings (STAR r) = (¬atmostempty r)"
+
+lemma Star_atmostempty:
+  assumes "A ⊆ {[]}"
+  shows "A⋆ ⊆ {[]}"
+using assms
+using Star_string concat_eq_Nil_conv empty_iff insert_iff subsetI subset_singletonD by fastforce
+
+lemma Star_empty_string_finite:
+  shows "finite ({[]}⋆)"
+using Star_atmostempty infinite_super by auto
+
+lemma Star_empty_finite:
+  shows "finite ({}⋆)"
+using Star_atmostempty infinite_super by auto
+
+lemma Star_concat_replicate:
+  assumes "s ∈ A"
+  shows "concat (replicate n s) ∈ A⋆"
+using assms
+by (induct n) (auto)
+
+
+lemma concat_replicate_inj:
+  assumes "concat (replicate n s) = concat (replicate m s)" "s ≠ []"
+  shows "n = m"
+using assms
+apply(induct n arbitrary: m)
+apply(auto)[1]
+apply(auto)
+apply(case_tac m)
+apply(clarify)
+apply(simp only: replicate.simps concat.simps)
+apply blast
+by simp
+
+lemma A0:
+  assumes "finite (A ;; B)" "B ≠ {}"
+  shows "finite A"
+apply(subgoal_tac "∃s. s ∈ B")
+apply(erule exE)
+apply(subgoal_tac "finite {s1 @ s |s1. s1 ∈ A}")
+apply(rule_tac f="λs1. s1 @ s" in finite_imageD)
+apply(simp add: image_def)
+apply(smt Collect_cong)
+apply(simp add: inj_on_def)
+apply(rule_tac B="A ;; B" in finite_subset)
+apply(auto simp add: Sequ_def)[1]
+apply(rule assms(1))
+using assms(2) by auto
+
+lemma A1:
+  assumes "finite (A ;; B)" "A ≠ {}"
+  shows "finite B"
+apply(subgoal_tac "∃s. s ∈ A")
+apply(erule exE)
+apply(subgoal_tac "finite {s @ s1 |s1. s1 ∈ B}")
+apply(rule_tac f="λs1. s @ s1" in finite_imageD)
+apply(simp add: image_def)
+apply(smt Collect_cong)
+apply(simp add: inj_on_def)
+apply(rule_tac B="A ;; B" in finite_subset)
+apply(auto simp add: Sequ_def)[1]
+apply(rule assms(1))
+using assms(2) by auto
+
+lemma Sequ_Prod_finite:
+  assumes "A ≠ {}" "B ≠ {}"
+  shows "finite (A ;; B) ⟷ (finite (A × B))"
+apply(rule iffI)
+apply(rule finite_cartesian_product)
+apply(erule A0)
+apply(rule assms(2))
+apply(erule A1)
+apply(rule assms(1))
+apply(simp add: Sequ_def)
+apply(rule finite_image_set2)
+apply(drule finite_cartesian_productD1)
+apply(rule assms(2))
+apply(simp)
+apply(drule finite_cartesian_productD2)
+apply(rule assms(1))
+apply(simp)
+done
+
+
+lemma Star_non_empty_string_infinite:
+  assumes "s ∈ A" " s ≠ []"
+  shows "infinite (A⋆)"
+proof -
+  have "inj (λn. concat (replicate n s))" 
+  using assms(2) concat_replicate_inj
+    by(auto simp add: inj_on_def)
+  moreover
+  have "infinite (UNIV::nat set)" by simp
+  ultimately
+  have "infinite ((λn. concat (replicate n s)) ` UNIV)"
+   by (simp add: range_inj_infinite)
+  moreover
+  have "((λn. concat (replicate n s)) ` UNIV) ⊆ (A⋆)"
+    using Star_concat_replicate assms(1) by auto
+  ultimately show "infinite (A⋆)" 
+  using infinite_super by auto
+qed
+
+lemma infinitestrings_correctness:
+  shows "infinitestrings r ⟷ infinite (L r)"
+apply(induct r)
+apply(simp_all)
+apply(simp add: zeroable_correctness)
+apply(rule iffI)
+apply(erule disjE)
+apply(subst Sequ_Prod_finite)
+apply(auto)[2]
+using finite_cartesian_productD2 apply blast
+apply(subst Sequ_Prod_finite)
+apply(auto)[2]
+using finite_cartesian_productD1 apply blast
+apply(subgoal_tac "L r1 ≠ {} ∧ L r2 ≠ {}")
+prefer 2
+apply(auto simp add: Sequ_def)[1]
+apply(subst (asm) Sequ_Prod_finite)
+apply(auto)[2]
+apply(auto)[1]
+apply(simp add: atmostempty_correctness)
+apply(rule iffI)
+apply (metis Star_empty_finite Star_empty_string_finite subset_singletonD)
+using Star_non_empty_string_infinite apply blast
+done
+
+
+end
\ No newline at end of file