(* Title: HOL/SPARK/Examples/Liseq/Longest_Increasing_Subsequence.thy
Author: Stefan Berghofer
Copyright: secunet Security Networks AG
*)
theory Longest_Increasing_Subsequence
imports SPARK
begin
text {*
Set of all increasing subsequences in a prefix of an array
*}
definition iseq :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat set set" where
"iseq xs l = {is. (\<forall>i\<in>is. i < l) \<and>
(\<forall>i\<in>is. \<forall>j\<in>is. i \<le> j \<longrightarrow> xs i \<le> xs j)}"
text {*
Length of longest increasing subsequence in a prefix of an array
*}
definition liseq :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat" where
"liseq xs i = Max (card ` iseq xs i)"
text {*
Length of longest increasing subsequence ending at a particular position
*}
definition liseq' :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat" where
"liseq' xs i = Max (card ` (iseq xs (Suc i) \<inter> {is. Max is = i}))"
lemma iseq_finite: "finite (iseq xs i)"
apply (simp add: iseq_def)
apply (rule finite_subset [OF _
finite_Collect_subsets [of "{j. j < i}"]])
apply auto
done
lemma iseq_finite': "is \<in> iseq xs i \<Longrightarrow> finite is"
by (auto simp add: iseq_def bounded_nat_set_is_finite)
lemma iseq_singleton: "i < l \<Longrightarrow> {i} \<in> iseq xs l"
by (simp add: iseq_def)
lemma iseq_trivial: "{} \<in> iseq xs i"
by (simp add: iseq_def)
lemma iseq_nonempty: "iseq xs i \<noteq> {}"
by (auto intro: iseq_trivial)
lemma liseq'_ge1: "1 \<le> liseq' xs x"
apply (simp add: liseq'_def)
apply (subgoal_tac "iseq xs (Suc x) \<inter> {is. Max is = x} \<noteq> {}")
apply (simp add: Max_ge_iff iseq_finite)
apply (rule_tac x="{x}" in bexI)
apply (auto intro: iseq_singleton)
done
lemma liseq_expand:
assumes R: "\<And>is. liseq xs i = card is \<Longrightarrow> is \<in> iseq xs i \<Longrightarrow>
(\<And>js. js \<in> iseq xs i \<Longrightarrow> card js \<le> card is) \<Longrightarrow> P"
shows "P"
proof -
have "Max (card ` iseq xs i) \<in> card ` iseq xs i"
by (rule Max_in) (simp_all add: iseq_finite iseq_nonempty)
then obtain js where js: "liseq xs i = card js" and "js \<in> iseq xs i"
by (rule imageE) (simp add: liseq_def)
moreover {
fix js'
assume "js' \<in> iseq xs i"
then have "card js' \<le> card js"
by (simp add: js [symmetric] liseq_def iseq_finite iseq_trivial)
}
ultimately show ?thesis by (rule R)
qed
lemma liseq'_expand:
assumes R: "\<And>is. liseq' xs i = card is \<Longrightarrow> is \<in> iseq xs (Suc i) \<Longrightarrow>
finite is \<Longrightarrow> Max is = i \<Longrightarrow>
(\<And>js. js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow> card js \<le> card is) \<Longrightarrow>
is \<noteq> {} \<Longrightarrow> P"
shows "P"
proof -
have "Max (card ` (iseq xs (Suc i) \<inter> {is. Max is = i})) \<in>
card ` (iseq xs (Suc i) \<inter> {is. Max is = i})"
by (auto simp add: iseq_finite intro!: iseq_singleton Max_in)
then obtain js where js: "liseq' xs i = card js" and "js \<in> iseq xs (Suc i)"
and "finite js" and "Max js = i"
by (auto simp add: liseq'_def intro: iseq_finite')
moreover {
fix js'
assume "js' \<in> iseq xs (Suc i)" "Max js' = i"
then have "card js' \<le> card js"
by (auto simp add: js [symmetric] liseq'_def iseq_finite intro!: iseq_singleton)
}
note max = this
moreover have "card {i} \<le> card js"
by (rule max) (simp_all add: iseq_singleton)
then have "js \<noteq> {}" by auto
ultimately show ?thesis by (rule R)
qed
lemma liseq'_ge:
"j = card js \<Longrightarrow> js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow>
js \<noteq> {} \<Longrightarrow> j \<le> liseq' xs i"
by (simp add: liseq'_def iseq_finite)
lemma liseq'_eq:
"j = card js \<Longrightarrow> js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow>
js \<noteq> {} \<Longrightarrow> (\<And>js'. js' \<in> iseq xs (Suc i) \<Longrightarrow> Max js' = i \<Longrightarrow> finite js' \<Longrightarrow>
js' \<noteq> {} \<Longrightarrow> card js' \<le> card js) \<Longrightarrow>
j = liseq' xs i"
by (fastsimp simp add: liseq'_def iseq_finite
intro: Max_eqI [symmetric])
lemma liseq_ge:
"j = card js \<Longrightarrow> js \<in> iseq xs i \<Longrightarrow> j \<le> liseq xs i"
by (auto simp add: liseq_def iseq_finite)
lemma liseq_eq:
"j = card js \<Longrightarrow> js \<in> iseq xs i \<Longrightarrow>
(\<And>js'. js' \<in> iseq xs i \<Longrightarrow> finite js' \<Longrightarrow>
js' \<noteq> {} \<Longrightarrow> card js' \<le> card js) \<Longrightarrow>
j = liseq xs i"
by (fastsimp simp add: liseq_def iseq_finite
intro: Max_eqI [symmetric])
lemma max_notin: "finite xs \<Longrightarrow> Max xs < x \<Longrightarrow> x \<notin> xs"
by (cases "xs = {}") auto
lemma iseq_insert:
"xs (Max is) \<le> xs i \<Longrightarrow> is \<in> iseq xs i \<Longrightarrow>
is \<union> {i} \<in> iseq xs (Suc i)"
apply (frule iseq_finite')
apply (cases "is = {}")
apply (auto simp add: iseq_def)
apply (rule order_trans [of _ "xs (Max is)"])
apply auto
apply (thin_tac "\<forall>a\<in>is. a < i")
apply (drule_tac x=ia in bspec)
apply assumption
apply (drule_tac x="Max is" in bspec)
apply (auto intro: Max_in)
done
lemma iseq_diff: "is \<in> iseq xs (Suc (Max is)) \<Longrightarrow>
is - {Max is} \<in> iseq xs (Suc (Max (is - {Max is})))"
apply (frule iseq_finite')
apply (simp add: iseq_def less_Suc_eq_le)
done
lemma iseq_butlast:
assumes "js \<in> iseq xs (Suc i)" and "js \<noteq> {}"
and "Max js \<noteq> i"
shows "js \<in> iseq xs i"
proof -
from assms have fin: "finite js"
by (simp add: iseq_finite')
with assms have "Max js \<in> js"
by auto
with assms have "Max js < i"
by (auto simp add: iseq_def)
with fin assms have "\<forall>j\<in>js. j < i"
by simp
with assms show ?thesis
by (simp add: iseq_def)
qed
lemma iseq_mono: "is \<in> iseq xs i \<Longrightarrow> i \<le> j \<Longrightarrow> is \<in> iseq xs j"
by (auto simp add: iseq_def)
lemma diff_nonempty:
assumes "1 < card is"
shows "is - {i} \<noteq> {}"
proof -
from assms have fin: "finite is" by (auto intro: card_ge_0_finite)
with assms fin have "card is - 1 \<le> card (is - {i})"
by (simp add: card_Diff_singleton_if)
with assms have "0 < card (is - {i})" by simp
then show ?thesis by (simp add: card_gt_0_iff)
qed
lemma Max_diff:
assumes "1 < card is"
shows "Max (is - {Max is}) < Max is"
proof -
from assms have "finite is" by (auto intro: card_ge_0_finite)
moreover from assms have "is - {Max is} \<noteq> {}"
by (rule diff_nonempty)
ultimately show ?thesis using assms
apply (auto simp add: not_less)
apply (subgoal_tac "a \<le> Max is")
apply auto
done
qed
lemma iseq_nth: "js \<in> iseq xs l \<Longrightarrow> 1 < card js \<Longrightarrow>
xs (Max (js - {Max js})) \<le> xs (Max js)"
apply (auto simp add: iseq_def)
apply (subgoal_tac "Max (js - {Max js}) \<in> js")
apply (thin_tac "\<forall>i\<in>js. i < l")
apply (drule_tac x="Max (js - {Max js})" in bspec)
apply assumption
apply (drule_tac x="Max js" in bspec)
using card_gt_0_iff [of js]
apply simp
using Max_diff [of js]
apply simp
using Max_in [of "js - {Max js}", OF _ diff_nonempty] card_gt_0_iff [of js]
apply auto
done
lemma card_leq1_singleton:
assumes "finite xs" "xs \<noteq> {}" "card xs \<le> 1"
obtains x where "xs = {x}"
using assms
by induct simp_all
lemma longest_iseq1:
"liseq' xs i =
Max ({0} \<union> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}) + 1"
proof -
have "Max ({0} \<union> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}) = liseq' xs i - 1"
proof (rule Max_eqI)
fix y
assume "y \<in> {0} \<union> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}"
then show "y \<le> liseq' xs i - 1"
proof
assume "y \<in> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}"
then obtain j where j: "j < i" "xs j \<le> xs i" "y = liseq' xs j"
by auto
have "liseq' xs j + 1 \<le> liseq' xs i"
proof (rule liseq'_expand)
fix "is"
assume H: "liseq' xs j = card is" "is \<in> iseq xs (Suc j)"
"finite is" "Max is = j" "is \<noteq> {}"
from H j have "card is + 1 = card (is \<union> {i})"
by (simp add: card_insert max_notin)
moreover {
from H j have "xs (Max is) \<le> xs i" by simp
moreover from `j < i` have "Suc j \<le> i" by simp
with `is \<in> iseq xs (Suc j)` have "is \<in> iseq xs i"
by (rule iseq_mono)
ultimately have "is \<union> {i} \<in> iseq xs (Suc i)"
by (rule iseq_insert)
} moreover from H j have "Max (is \<union> {i}) = i" by simp
moreover have "is \<union> {i} \<noteq> {}" by simp
ultimately have "card is + 1 \<le> liseq' xs i"
by (rule liseq'_ge)
with H show ?thesis by simp
qed
with j show "y \<le> liseq' xs i - 1"
by simp
qed simp
next
have "liseq' xs i \<le> 1 \<or>
(\<exists>j. liseq' xs i - 1 = liseq' xs j \<and> j < i \<and> xs j \<le> xs i)"
proof (rule liseq'_expand)
fix "is"
assume H: "liseq' xs i = card is" "is \<in> iseq xs (Suc i)"
"finite is" "Max is = i" "is \<noteq> {}"
assume R: "\<And>js. js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow>
card js \<le> card is"
show ?thesis
proof (cases "card is \<le> 1")
case True with H show ?thesis by simp
next
case False
then have "1 < card is" by simp
then have "Max (is - {Max is}) < Max is"
by (rule Max_diff)
from `is \<in> iseq xs (Suc i)` `1 < card is`
have "xs (Max (is - {Max is})) \<le> xs (Max is)"
by (rule iseq_nth)
have "card is - 1 = liseq' xs (Max (is - {i}))"
proof (rule liseq'_eq)
from `Max is = i` [symmetric] `finite is` `is \<noteq> {}`
show "card is - 1 = card (is - {i})" by simp
next
from `is \<in> iseq xs (Suc i)` `Max is = i` [symmetric]
show "is - {i} \<in> iseq xs (Suc (Max (is - {i})))"
by simp (rule iseq_diff)
next
from `1 < card is`
show "is - {i} \<noteq> {}" by (rule diff_nonempty)
next
fix js
assume "js \<in> iseq xs (Suc (Max (is - {i})))"
"Max js = Max (is - {i})" "finite js" "js \<noteq> {}"
from `xs (Max (is - {Max is})) \<le> xs (Max is)`
`Max js = Max (is - {i})` `Max is = i`
have "xs (Max js) \<le> xs i" by simp
moreover from `Max is = i` `Max (is - {Max is}) < Max is`
have "Suc (Max (is - {i})) \<le> i"
by simp
with `js \<in> iseq xs (Suc (Max (is - {i})))`
have "js \<in> iseq xs i"
by (rule iseq_mono)
ultimately have "js \<union> {i} \<in> iseq xs (Suc i)"
by (rule iseq_insert)
moreover from `js \<noteq> {}` `finite js` `Max js = Max (is - {i})`
`Max is = i` [symmetric] `Max (is - {Max is}) < Max is`
have "Max (js \<union> {i}) = i"
by simp
ultimately have "card (js \<union> {i}) \<le> card is" by (rule R)
moreover from `Max is = i` [symmetric] `finite js`
`Max (is - {Max is}) < Max is` `Max js = Max (is - {i})`
have "i \<notin> js" by (simp add: max_notin)
with `finite js`
have "card (js \<union> {i}) = card ((js \<union> {i}) - {i}) + 1"
by simp
ultimately show "card js \<le> card (is - {i})"
using `i \<notin> js` `Max is = i` [symmetric] `is \<noteq> {}` `finite is`
by simp
qed simp
with H `Max (is - {Max is}) < Max is`
`xs (Max (is - {Max is})) \<le> xs (Max is)`
show ?thesis by auto
qed
qed
then show "liseq' xs i - 1 \<in> {0} \<union>
{liseq' xs j |j. j < i \<and> xs j \<le> xs i}" by simp
qed simp
moreover have "1 \<le> liseq' xs i" by (rule liseq'_ge1)
ultimately show ?thesis by simp
qed
lemma longest_iseq2': "liseq xs i < liseq' xs i \<Longrightarrow>
liseq xs (Suc i) = liseq' xs i"
apply (rule_tac xs=xs and i=i in liseq'_expand)
apply simp
apply (rule liseq_eq [symmetric])
apply (rule refl)
apply assumption
apply (case_tac "Max js' = i")
apply simp
apply (drule_tac js=js' in iseq_butlast)
apply assumption+
apply (drule_tac js=js' in liseq_ge [OF refl])
apply simp
done
lemma longest_iseq2: "liseq xs i < liseq' xs i \<Longrightarrow>
liseq xs i + 1 = liseq' xs i"
apply (rule_tac xs=xs and i=i in liseq'_expand)
apply simp
apply (rule_tac xs=xs and i=i in liseq_expand)
apply (drule_tac s="Max is" in sym)
apply simp
apply (case_tac "card is \<le> 1")
apply simp
apply (drule iseq_diff)
apply (drule_tac i="Suc (Max (is - {Max is}))" and j="Max is" in iseq_mono)
apply (simp add: less_eq_Suc_le [symmetric])
apply (rule Max_diff)
apply simp
apply (drule_tac x="is - {Max is}" in meta_spec,
drule meta_mp, assumption)
apply simp
done
lemma longest_iseq3:
"liseq xs j = liseq' xs i \<Longrightarrow> xs i \<le> xs j \<Longrightarrow> i < j \<Longrightarrow>
liseq xs (Suc j) = liseq xs j + 1"
apply (rule_tac xs=xs and i=j in liseq_expand)
apply simp
apply (rule_tac xs=xs and i=i in liseq'_expand)
apply simp
apply (rule_tac js="isa \<union> {j}" in liseq_eq [symmetric])
apply (simp add: card_insert card_Diff_singleton_if max_notin)
apply (rule iseq_insert)
apply simp
apply (erule iseq_mono)
apply simp
apply (case_tac "j = Max js'")
apply simp
apply (drule iseq_diff)
apply (drule_tac x="js' - {j}" in meta_spec)
apply (drule meta_mp)
apply simp
apply (case_tac "card js' \<le> 1")
apply (erule_tac xs=js' in card_leq1_singleton)
apply assumption+
apply (simp add: iseq_trivial)
apply (erule iseq_mono)
apply (simp add: less_eq_Suc_le [symmetric])
apply (rule Max_diff)
apply simp
apply (rule le_diff_iff [THEN iffD1, of 1])
apply (simp add: card_0_eq [symmetric] del: card_0_eq)
apply (simp add: card_insert)
apply (subgoal_tac "card (js' - {j}) = card js' - 1")
apply (simp add: card_insert card_Diff_singleton_if max_notin)
apply (frule_tac A=js' in Max_in)
apply assumption
apply (simp add: card_Diff_singleton_if)
apply (drule_tac js=js' in iseq_butlast)
apply assumption
apply (erule not_sym)
apply (drule_tac x=js' in meta_spec)
apply (drule meta_mp)
apply assumption
apply (simp add: card_insert_disjoint max_notin)
done
lemma longest_iseq4:
"liseq xs j = liseq' xs i \<Longrightarrow> xs i \<le> xs j \<Longrightarrow> i < j \<Longrightarrow>
liseq' xs j = liseq' xs i + 1"
apply (rule_tac xs=xs and i=j in liseq_expand)
apply simp
apply (rule_tac xs=xs and i=i in liseq'_expand)
apply simp
apply (rule_tac js="isa \<union> {j}" in liseq'_eq [symmetric])
apply (simp add: card_insert card_Diff_singleton_if max_notin)
apply (rule iseq_insert)
apply simp
apply (erule iseq_mono)
apply simp
apply simp
apply simp
apply (drule_tac s="Max js'" in sym)
apply simp
apply (drule iseq_diff)
apply (drule_tac x="js' - {j}" in meta_spec)
apply (drule meta_mp)
apply simp
apply (case_tac "card js' \<le> 1")
apply (erule_tac xs=js' in card_leq1_singleton)
apply assumption+
apply (simp add: iseq_trivial)
apply (erule iseq_mono)
apply (simp add: less_eq_Suc_le [symmetric])
apply (rule Max_diff)
apply simp
apply (rule le_diff_iff [THEN iffD1, of 1])
apply (simp add: card_0_eq [symmetric] del: card_0_eq)
apply (simp add: card_insert)
apply (subgoal_tac "card (js' - {j}) = card js' - 1")
apply (simp add: card_insert card_Diff_singleton_if max_notin)
apply (frule_tac A=js' in Max_in)
apply assumption
apply (simp add: card_Diff_singleton_if)
done
lemma longest_iseq5: "liseq' xs i \<le> liseq xs i \<Longrightarrow>
liseq xs (Suc i) = liseq xs i"
apply (rule_tac i=i and xs=xs in liseq'_expand)
apply simp
apply (rule_tac xs=xs and i=i in liseq_expand)
apply simp
apply (rule liseq_eq [symmetric])
apply (rule refl)
apply (erule iseq_mono)
apply simp
apply (case_tac "Max js' = i")
apply (drule_tac x=js' in meta_spec)
apply simp
apply (drule iseq_butlast, assumption, assumption)
apply simp
done
lemma liseq_empty: "liseq xs 0 = 0"
apply (rule_tac js="{}" in liseq_eq [symmetric])
apply simp
apply (rule iseq_trivial)
apply (simp add: iseq_def)
done
lemma liseq'_singleton: "liseq' xs 0 = 1"
by (simp add: longest_iseq1 [of _ 0])
lemma liseq_singleton: "liseq xs (Suc 0) = Suc 0"
by (simp add: longest_iseq2' liseq_empty liseq'_singleton)
lemma liseq'_Suc_unfold:
"A j \<le> x \<Longrightarrow>
(insert 0 {liseq' A j' |j'. j' < Suc j \<and> A j' \<le> x}) =
(insert 0 {liseq' A j' |j'. j' < j \<and> A j' \<le> x}) \<union>
{liseq' A j}"
by (auto simp add: less_Suc_eq)
lemma liseq'_Suc_unfold':
"\<not> (A j \<le> x) \<Longrightarrow>
{liseq' A j' |j'. j' < Suc j \<and> A j' \<le> x} =
{liseq' A j' |j'. j' < j \<and> A j' \<le> x}"
by (auto simp add: less_Suc_eq)
lemma iseq_card_limit:
assumes "is \<in> iseq A i"
shows "card is \<le> i"
proof -
from assms have "is \<subseteq> {0..<i}"
by (auto simp add: iseq_def)
with finite_atLeastLessThan have "card is \<le> card {0..<i}"
by (rule card_mono)
with card_atLeastLessThan show ?thesis by simp
qed
lemma liseq_limit: "liseq A i \<le> i"
by (rule_tac xs=A and i=i in liseq_expand)
(simp add: iseq_card_limit)
lemma liseq'_limit: "liseq' A i \<le> i + 1"
by (rule_tac xs=A and i=i in liseq'_expand)
(simp add: iseq_card_limit)
definition max_ext :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" where
"max_ext A i j = Max ({0} \<union> {liseq' A j' |j'. j' < j \<and> A j' \<le> A i})"
lemma max_ext_limit: "max_ext A i j \<le> j"
apply (auto simp add: max_ext_def)
apply (drule Suc_leI)
apply (cut_tac i=j' and A=A in liseq'_limit)
apply simp
done
text {* Proof functions *}
abbreviation (input)
"arr_conv a \<equiv> (\<lambda>n. a (int n))"
lemma idx_conv_suc:
"0 \<le> i \<Longrightarrow> nat (i + 1) = nat i + 1"
by simp
abbreviation liseq_ends_at' :: "(int \<Rightarrow> 'a::linorder) \<Rightarrow> int \<Rightarrow> int" where
"liseq_ends_at' A i \<equiv> int (liseq' (\<lambda>l. A (int l)) (nat i))"
abbreviation liseq_prfx' :: "(int \<Rightarrow> 'a::linorder) \<Rightarrow> int \<Rightarrow> int" where
"liseq_prfx' A i \<equiv> int (liseq (\<lambda>l. A (int l)) (nat i))"
abbreviation max_ext' :: "(int \<Rightarrow> 'a::linorder) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int" where
"max_ext' A i j \<equiv> int (max_ext (\<lambda>l. A (int l)) (nat i) (nat j))"
spark_proof_functions
liseq_ends_at = "liseq_ends_at' :: (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> int"
liseq_prfx = "liseq_prfx' :: (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> int"
max_ext = "max_ext' :: (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int"
text {* The verification conditions *}
spark_open "liseq/liseq_length.siv"
spark_vc procedure_liseq_length_5
by (simp_all add: liseq_singleton liseq'_singleton)
spark_vc procedure_liseq_length_6
proof -
from H1 H2 H3 H4
have eq: "liseq (arr_conv a) (nat i) =
liseq' (arr_conv a) (nat pmax)"
by simp
from H14 H3 H4
have pmax1: "arr_conv a (nat pmax) \<le> arr_conv a (nat i)"
by simp
from H3 H4 have pmax2: "nat pmax < nat i"
by simp
{
fix i2
assume i2: "0 \<le> i2" "i2 \<le> i"
have "(l(i := l pmax + 1)) i2 =
int (liseq' (arr_conv a) (nat i2))"
proof (cases "i2 = i")
case True
from eq pmax1 pmax2 have "liseq' (arr_conv a) (nat i) =
liseq' (arr_conv a) (nat pmax) + 1"
by (rule longest_iseq4)
with True H1 H3 H4 show ?thesis
by simp
next
case False
with H1 i2 show ?thesis
by simp
qed
}
then show ?C1 by simp
from eq pmax1 pmax2
have "liseq (arr_conv a) (Suc (nat i)) =
liseq (arr_conv a) (nat i) + 1"
by (rule longest_iseq3)
with H2 H3 H4 show ?C2
by (simp add: idx_conv_suc)
qed
spark_vc procedure_liseq_length_7
proof -
from H1 show ?C1
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"])
from H6
have m: "max_ext (arr_conv a) (nat i) (nat i) + 1 =
liseq' (arr_conv a) (nat i)"
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"])
with H2 H18
have gt: "liseq (arr_conv a) (nat i) < liseq' (arr_conv a) (nat i)"
by simp
then have "liseq' (arr_conv a) (nat i) = liseq (arr_conv a) (nat i) + 1"
by (rule longest_iseq2 [symmetric])
with H2 m show ?C2 by simp
from gt have "liseq (arr_conv a) (Suc (nat i)) = liseq' (arr_conv a) (nat i)"
by (rule longest_iseq2')
with m H6 show ?C3 by (simp add: idx_conv_suc)
qed
spark_vc procedure_liseq_length_8
proof -
{
fix i2
assume i2: "0 \<le> i2" "i2 \<le> i"
have "(l(i := max_ext' a i i + 1)) i2 =
int (liseq' (arr_conv a) (nat i2))"
proof (cases "i2 = i")
case True
with H1 show ?thesis
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"])
next
case False
with H1 i2 show ?thesis by simp
qed
}
then show ?C1 by simp
from H2 H6 H18
have "liseq' (arr_conv a) (nat i) \<le> liseq (arr_conv a) (nat i)"
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"])
then have "liseq (arr_conv a) (Suc (nat i)) = liseq (arr_conv a) (nat i)"
by (rule longest_iseq5)
with H2 H6 show ?C2 by (simp add: idx_conv_suc)
qed
spark_vc procedure_liseq_length_12
by (simp add: max_ext_def)
spark_vc procedure_liseq_length_13
using H1 H6 H13 H21 H22
by (simp add: max_ext_def
idx_conv_suc liseq'_Suc_unfold max_def del: Max_less_iff)
spark_vc procedure_liseq_length_14
using H1 H6 H13 H21
by (cases "a j \<le> a i")
(simp_all add: max_ext_def
idx_conv_suc liseq'_Suc_unfold liseq'_Suc_unfold')
spark_vc procedure_liseq_length_19
using H3 H4 H5 H8 H9
apply (rule_tac y="int (nat i)" in order_trans)
apply (cut_tac A="arr_conv a" and i="nat i" and j="nat i" in max_ext_limit)
apply simp_all
done
spark_vc procedure_liseq_length_23
using H2 H3 H4 H7 H8 H11
apply (rule_tac y="int (nat i)" in order_trans)
apply (cut_tac A="arr_conv a" and i="nat i" in liseq_limit)
apply simp_all
done
spark_vc procedure_liseq_length_29
using H2 H3 H8 H13
by (simp add: add1_zle_eq [symmetric])
spark_end
end