Newer version.
theory Moment
imports Main
begin
fun firstn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
where
"firstn 0 s = []" |
"firstn (Suc n) [] = []" |
"firstn (Suc n) (e#s) = e#(firstn n s)"
fun restn :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "restn n s = rev (firstn (length s - n) (rev s))"
definition moment :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "moment n s = rev (firstn n (rev s))"
definition restm :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "restm n s = rev (restn n (rev s))"
definition from_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "from_to i j s = firstn (j - i) (restn i s)"
definition down_to :: "nat \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "down_to j i s = rev (from_to i j (rev s))"
(*
value "down_to 6 2 [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]"
value "from_to 2 6 [0, 1, 2, 3, 4, 5, 6, 7]"
*)
lemma length_eq_elim_l: "\<lbrakk>length xs = length ys; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
by auto
lemma length_eq_elim_r: "\<lbrakk>length us = length vs; xs@us = ys@vs\<rbrakk> \<Longrightarrow> xs = ys \<and> us = vs"
by simp
lemma firstn_nil [simp]: "firstn n [] = []"
by (cases n, simp+)
(*
value "from_to 0 2 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] @
from_to 2 5 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
*)
lemma firstn_le: "\<And> n s'. n \<le> length s \<Longrightarrow> firstn n (s@s') = firstn n s"
proof (induct s, simp)
fix a s n s'
assume ih: "\<And>n s'. n \<le> length s \<Longrightarrow> firstn n (s @ s') = firstn n s"
and le_n: " n \<le> length (a # s)"
show "firstn n ((a # s) @ s') = firstn n (a # s)"
proof(cases n, simp)
fix k
assume eq_n: "n = Suc k"
with le_n have "k \<le> length s" by auto
from ih [OF this] and eq_n
show "firstn n ((a # s) @ s') = firstn n (a # s)" by auto
qed
qed
lemma firstn_ge [simp]: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
proof(induct s, simp)
fix a s n
assume ih: "\<And>n. length s \<le> n \<Longrightarrow> firstn n s = s"
and le: "length (a # s) \<le> n"
show "firstn n (a # s) = a # s"
proof(cases n)
assume eq_n: "n = 0" with le show ?thesis by simp
next
fix k
assume eq_n: "n = Suc k"
with le have le_k: "length s \<le> k" by simp
from ih [OF this] have "firstn k s = s" .
from eq_n and this
show ?thesis by simp
qed
qed
lemma firstn_eq [simp]: "firstn (length s) s = s"
by simp
lemma firstn_restn_s: "(firstn n (s::'a list)) @ (restn n s) = s"
proof(induct n arbitrary:s, simp)
fix n s
assume ih: "\<And>t. firstn n (t::'a list) @ restn n t = t"
show "firstn (Suc n) (s::'a list) @ restn (Suc n) s = s"
proof(cases s, simp)
fix x xs
assume eq_s: "s = x#xs"
show "firstn (Suc n) s @ restn (Suc n) s = s"
proof -
have "firstn (Suc n) s @ restn (Suc n) s = x # (firstn n xs @ restn n xs)"
proof -
from eq_s have "firstn (Suc n) s = x # firstn n xs" by simp
moreover have "restn (Suc n) s = restn n xs"
proof -
from eq_s have "restn (Suc n) s = rev (firstn (length xs - n) (rev xs @ [x]))" by simp
also have "\<dots> = restn n xs"
proof -
have "(firstn (length xs - n) (rev xs @ [x])) = (firstn (length xs - n) (rev xs))"
by(rule firstn_le, simp)
hence "rev (firstn (length xs - n) (rev xs @ [x])) =
rev (firstn (length xs - n) (rev xs))" by simp
also have "\<dots> = rev (firstn (length (rev xs) - n) (rev xs))" by simp
finally show ?thesis by simp
qed
finally show ?thesis by simp
qed
ultimately show ?thesis by simp
qed with ih eq_s show ?thesis by simp
qed
qed
qed
lemma moment_restm_s: "(restm n s)@(moment n s) = s"
proof -
have " rev ((firstn n (rev s)) @ (restn n (rev s))) = s" (is "rev ?x = s")
proof -
have "?x = rev s" by (simp only:firstn_restn_s)
thus ?thesis by auto
qed
thus ?thesis
by (auto simp:restm_def moment_def)
qed
declare restn.simps [simp del] firstn.simps[simp del]
lemma length_firstn_ge: "length s \<le> n \<Longrightarrow> length (firstn n s) = length s"
proof(induct n arbitrary:s, simp add:firstn.simps)
case (Suc k)
assume ih: "\<And> s. length (s::'a list) \<le> k \<Longrightarrow> length (firstn k s) = length s"
and le: "length s \<le> Suc k"
show ?case
proof(cases s)
case Nil
from Nil show ?thesis by simp
next
case (Cons x xs)
from le and Cons have "length xs \<le> k" by simp
from ih [OF this] have "length (firstn k xs) = length xs" .
moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))"
by (simp add:firstn.simps)
moreover note Cons
ultimately show ?thesis by simp
qed
qed
lemma length_firstn_le: "n \<le> length s \<Longrightarrow> length (firstn n s) = n"
proof(induct n arbitrary:s, simp add:firstn.simps)
case (Suc k)
assume ih: "\<And>s. k \<le> length (s::'a list) \<Longrightarrow> length (firstn k s) = k"
and le: "Suc k \<le> length s"
show ?case
proof(cases s)
case Nil
from Nil and le show ?thesis by auto
next
case (Cons x xs)
from le and Cons have "k \<le> length xs" by simp
from ih [OF this] have "length (firstn k xs) = k" .
moreover from Cons have "length (firstn (Suc k) s) = Suc (length (firstn k xs))"
by (simp add:firstn.simps)
ultimately show ?thesis by simp
qed
qed
lemma app_firstn_restn:
fixes s1 s2
shows "s1 = firstn (length s1) (s1 @ s2) \<and> s2 = restn (length s1) (s1 @ s2)"
proof(rule length_eq_elim_l)
have "length s1 \<le> length (s1 @ s2)" by simp
from length_firstn_le [OF this]
show "length s1 = length (firstn (length s1) (s1 @ s2))" by simp
next
from firstn_restn_s
show "s1 @ s2 = firstn (length s1) (s1 @ s2) @ restn (length s1) (s1 @ s2)"
by metis
qed
lemma length_moment_le:
fixes k s
assumes le_k: "k \<le> length s"
shows "length (moment k s) = k"
proof -
have "length (rev (firstn k (rev s))) = k"
proof -
have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
also have "\<dots> = k"
proof(rule length_firstn_le)
from le_k show "k \<le> length (rev s)" by simp
qed
finally show ?thesis .
qed
thus ?thesis by (simp add:moment_def)
qed
lemma app_moment_restm:
fixes s1 s2
shows "s1 = restm (length s2) (s1 @ s2) \<and> s2 = moment (length s2) (s1 @ s2)"
proof(rule length_eq_elim_r)
have "length s2 \<le> length (s1 @ s2)" by simp
from length_moment_le [OF this]
show "length s2 = length (moment (length s2) (s1 @ s2))" by simp
next
from moment_restm_s
show "s1 @ s2 = restm (length s2) (s1 @ s2) @ moment (length s2) (s1 @ s2)"
by metis
qed
lemma length_moment_ge:
fixes k s
assumes le_k: "length s \<le> k"
shows "length (moment k s) = (length s)"
proof -
have "length (rev (firstn k (rev s))) = length s"
proof -
have "length (rev (firstn k (rev s))) = length (firstn k (rev s))" by simp
also have "\<dots> = length s"
proof -
have "\<dots> = length (rev s)"
proof(rule length_firstn_ge)
from le_k show "length (rev s) \<le> k" by simp
qed
also have "\<dots> = length s" by simp
finally show ?thesis .
qed
finally show ?thesis .
qed
thus ?thesis by (simp add:moment_def)
qed
lemma length_firstn: "(length (firstn n s) = length s) \<or> (length (firstn n s) = n)"
proof(cases "n \<le> length s")
case True
from length_firstn_le [OF True] show ?thesis by auto
next
case False
from False have "length s \<le> n" by simp
from firstn_ge [OF this] show ?thesis by auto
qed
lemma firstn_conc:
fixes m n
assumes le_mn: "m \<le> n"
shows "firstn m s = firstn m (firstn n s)"
proof(cases "m \<le> length s")
case True
have "s = (firstn n s) @ (restn n s)" by (simp add:firstn_restn_s)
hence "firstn m s = firstn m \<dots>" by simp
also have "\<dots> = firstn m (firstn n s)"
proof -
from length_firstn [of n s]
have "m \<le> length (firstn n s)"
proof
assume "length (firstn n s) = length s" with True show ?thesis by simp
next
assume "length (firstn n s) = n " with le_mn show ?thesis by simp
qed
from firstn_le [OF this, of "restn n s"]
show ?thesis .
qed
finally show ?thesis by simp
next
case False
from False and le_mn have "length s \<le> n" by simp
from firstn_ge [OF this] show ?thesis by simp
qed
lemma restn_conc:
fixes i j k s
assumes eq_k: "j + i = k"
shows "restn k s = restn j (restn i s)"
proof -
have "(firstn (length s - k) (rev s)) =
(firstn (length (rev (firstn (length s - i) (rev s))) - j)
(rev (rev (firstn (length s - i) (rev s)))))"
proof -
have "(firstn (length s - k) (rev s)) =
(firstn (length (rev (firstn (length s - i) (rev s))) - j)
(firstn (length s - i) (rev s)))"
proof -
have " (length (rev (firstn (length s - i) (rev s))) - j) = length s - k"
proof -
have "(length (rev (firstn (length s - i) (rev s))) - j) = (length s - i) - j"
proof -
have "(length (rev (firstn (length s - i) (rev s))) - j) =
length ((firstn (length s - i) (rev s))) - j"
by simp
also have "\<dots> = length ((firstn (length (rev s) - i) (rev s))) - j" by simp
also have "\<dots> = (length (rev s) - i) - j"
proof -
have "length ((firstn (length (rev s) - i) (rev s))) = (length (rev s) - i)"
by (rule length_firstn_le, simp)
thus ?thesis by simp
qed
also have "\<dots> = (length s - i) - j" by simp
finally show ?thesis .
qed
with eq_k show ?thesis by auto
qed
moreover have "(firstn (length s - k) (rev s)) =
(firstn (length s - k) (firstn (length s - i) (rev s)))"
proof(rule firstn_conc)
from eq_k show "length s - k \<le> length s - i" by simp
qed
ultimately show ?thesis by simp
qed
thus ?thesis by simp
qed
thus ?thesis by (simp only:restn.simps)
qed
(*
value "down_to 2 0 [5, 4, 3, 2, 1, 0]"
value "moment 2 [5, 4, 3, 2, 1, 0]"
*)
lemma from_to_firstn: "from_to 0 k s = firstn k s"
by (simp add:from_to_def restn.simps)
lemma moment_app [simp]:
assumes
ile: "i \<le> length s"
shows "moment i (s'@s) = moment i s"
proof -
have "moment i (s'@s) = rev (firstn i (rev (s'@s)))" by (simp add:moment_def)
moreover have "firstn i (rev (s'@s)) = firstn i (rev s @ rev s')" by simp
moreover have "\<dots> = firstn i (rev s)"
proof(rule firstn_le)
have "length (rev s) = length s" by simp
with ile show "i \<le> length (rev s)" by simp
qed
ultimately show ?thesis by (simp add:moment_def)
qed
lemma moment_eq [simp]: "moment (length s) (s'@s) = s"
proof -
have "length s \<le> length s" by simp
from moment_app [OF this, of s']
have " moment (length s) (s' @ s) = moment (length s) s" .
moreover have "\<dots> = s" by (simp add:moment_def)
ultimately show ?thesis by simp
qed
lemma moment_ge [simp]: "length s \<le> n \<Longrightarrow> moment n s = s"
by (unfold moment_def, simp)
lemma moment_zero [simp]: "moment 0 s = []"
by (simp add:moment_def firstn.simps)
lemma p_split_gen:
"\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk> \<Longrightarrow>
(\<exists> i. i < length s \<and> k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
proof (induct s, simp)
fix a s
assume ih: "\<lbrakk>Q s; \<not> Q (moment k s)\<rbrakk>
\<Longrightarrow> \<exists>i<length s. k \<le> i \<and> \<not> Q (moment i s) \<and> (\<forall>i'>i. Q (moment i' s))"
and nq: "\<not> Q (moment k (a # s))" and qa: "Q (a # s)"
have le_k: "k \<le> length s"
proof -
{ assume "length s < k"
hence "length (a#s) \<le> k" by simp
from moment_ge [OF this] and nq and qa
have "False" by auto
} thus ?thesis by arith
qed
have nq_k: "\<not> Q (moment k s)"
proof -
have "moment k (a#s) = moment k s"
proof -
from moment_app [OF le_k, of "[a]"] show ?thesis by simp
qed
with nq show ?thesis by simp
qed
show "\<exists>i<length (a # s). k \<le> i \<and> \<not> Q (moment i (a # s)) \<and> (\<forall>i'>i. Q (moment i' (a # s)))"
proof -
{ assume "Q s"
from ih [OF this nq_k]
obtain i where lti: "i < length s"
and nq: "\<not> Q (moment i s)"
and rst: "\<forall>i'>i. Q (moment i' s)"
and lki: "k \<le> i" by auto
have ?thesis
proof -
from lti have "i < length (a # s)" by auto
moreover have " \<not> Q (moment i (a # s))"
proof -
from lti have "i \<le> (length s)" by simp
from moment_app [OF this, of "[a]"]
have "moment i (a # s) = moment i s" by simp
with nq show ?thesis by auto
qed
moreover have " (\<forall>i'>i. Q (moment i' (a # s)))"
proof -
{
fix i'
assume lti': "i < i'"
have "Q (moment i' (a # s))"
proof(cases "length (a#s) \<le> i'")
case True
from True have "moment i' (a#s) = a#s" by simp
with qa show ?thesis by simp
next
case False
from False have "i' \<le> length s" by simp
from moment_app [OF this, of "[a]"]
have "moment i' (a#s) = moment i' s" by simp
with rst lti' show ?thesis by auto
qed
} thus ?thesis by auto
qed
moreover note lki
ultimately show ?thesis by auto
qed
} moreover {
assume ns: "\<not> Q s"
have ?thesis
proof -
let ?i = "length s"
have "\<not> Q (moment ?i (a#s))"
proof -
have "?i \<le> length s" by simp
from moment_app [OF this, of "[a]"]
have "moment ?i (a#s) = moment ?i s" by simp
moreover have "\<dots> = s" by simp
ultimately show ?thesis using ns by auto
qed
moreover have "\<forall> i' > ?i. Q (moment i' (a#s))"
proof -
{ fix i'
assume "i' > ?i"
hence "length (a#s) \<le> i'" by simp
from moment_ge [OF this]
have " moment i' (a # s) = a # s" .
with qa have "Q (moment i' (a#s))" by simp
} thus ?thesis by auto
qed
moreover have "?i < length (a#s)" by simp
moreover note le_k
ultimately show ?thesis by auto
qed
} ultimately show ?thesis by auto
qed
qed
lemma p_split:
"\<And> s Q. \<lbrakk>Q s; \<not> Q []\<rbrakk> \<Longrightarrow>
(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
proof -
fix s Q
assume qs: "Q s" and nq: "\<not> Q []"
from nq have "\<not> Q (moment 0 s)" by simp
from p_split_gen [of Q s 0, OF qs this]
show "(\<exists> i. i < length s \<and> \<not> Q (moment i s) \<and> (\<forall> i' > i. Q (moment i' s)))"
by auto
qed
lemma moment_plus:
"Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = (hd (moment (Suc i) s)) # (moment i s)"
proof(induct s, simp+)
fix a s
assume ih: "Suc i \<le> length s \<Longrightarrow> moment (Suc i) s = hd (moment (Suc i) s) # moment i s"
and le_i: "i \<le> length s"
show "moment (Suc i) (a # s) = hd (moment (Suc i) (a # s)) # moment i (a # s)"
proof(cases "i= length s")
case True
hence "Suc i = length (a#s)" by simp
with moment_eq have "moment (Suc i) (a#s) = a#s" by auto
moreover have "moment i (a#s) = s"
proof -
from moment_app [OF le_i, of "[a]"]
and True show ?thesis by simp
qed
ultimately show ?thesis by auto
next
case False
from False and le_i have lti: "i < length s" by arith
hence les_i: "Suc i \<le> length s" by arith
show ?thesis
proof -
from moment_app [OF les_i, of "[a]"]
have "moment (Suc i) (a # s) = moment (Suc i) s" by simp
moreover have "moment i (a#s) = moment i s"
proof -
from lti have "i \<le> length s" by simp
from moment_app [OF this, of "[a]"] show ?thesis by simp
qed
moreover note ih [OF les_i]
ultimately show ?thesis by auto
qed
qed
qed
lemma from_to_conc:
fixes i j k s
assumes le_ij: "i \<le> j"
and le_jk: "j \<le> k"
shows "from_to i j s @ from_to j k s = from_to i k s"
proof -
let ?ris = "restn i s"
have "firstn (j - i) (restn i s) @ firstn (k - j) (restn j s) =
firstn (k - i) (restn i s)" (is "?x @ ?y = ?z")
proof -
let "firstn (k-j) ?u" = "?y"
let ?rst = " restn (k - j) (restn (j - i) ?ris)"
let ?rst' = "restn (k - i) ?ris"
have "?u = restn (j-i) ?ris"
proof(rule restn_conc)
from le_ij show "j - i + i = j" by simp
qed
hence "?x @ ?y = ?x @ firstn (k-j) (restn (j-i) ?ris)" by simp
moreover have "firstn (k - j) (restn (j - i) (restn i s)) @ ?rst =
restn (j-i) ?ris" by (simp add:firstn_restn_s)
ultimately have "?x @ ?y @ ?rst = ?x @ (restn (j-i) ?ris)" by simp
also have "\<dots> = ?ris" by (simp add:firstn_restn_s)
finally have "?x @ ?y @ ?rst = ?ris" .
moreover have "?z @ ?rst = ?ris"
proof -
have "?z @ ?rst' = ?ris" by (simp add:firstn_restn_s)
moreover have "?rst' = ?rst"
proof(rule restn_conc)
from le_ij le_jk show "k - j + (j - i) = k - i" by auto
qed
ultimately show ?thesis by simp
qed
ultimately have "?x @ ?y @ ?rst = ?z @ ?rst" by simp
thus ?thesis by auto
qed
thus ?thesis by (simp only:from_to_def)
qed
lemma down_to_conc:
fixes i j k s
assumes le_ij: "i \<le> j"
and le_jk: "j \<le> k"
shows "down_to k j s @ down_to j i s = down_to k i s"
proof -
have "rev (from_to j k (rev s)) @ rev (from_to i j (rev s)) = rev (from_to i k (rev s))"
(is "?L = ?R")
proof -
have "?L = rev (from_to i j (rev s) @ from_to j k (rev s))" by simp
also have "\<dots> = ?R" (is "rev ?x = rev ?y")
proof -
have "?x = ?y" by (rule from_to_conc[OF le_ij le_jk])
thus ?thesis by simp
qed
finally show ?thesis .
qed
thus ?thesis by (simp add:down_to_def)
qed
lemma restn_ge:
fixes s k
assumes le_k: "length s \<le> k"
shows "restn k s = []"
proof -
from firstn_restn_s [of k s, symmetric] have "s = (firstn k s) @ (restn k s)" .
hence "length s = length \<dots>" by simp
also have "\<dots> = length (firstn k s) + length (restn k s)" by simp
finally have "length s = ..." by simp
moreover from length_firstn_ge and le_k
have "length (firstn k s) = length s" by simp
ultimately have "length (restn k s) = 0" by auto
thus ?thesis by auto
qed
lemma from_to_ge: "length s \<le> k \<Longrightarrow> from_to k j s = []"
proof(simp only:from_to_def)
assume "length s \<le> k"
from restn_ge [OF this]
show "firstn (j - k) (restn k s) = []" by simp
qed
(*
value "from_to 2 5 [0, 1, 2, 3, 4]"
value "restn 2 [0, 1, 2, 3, 4]"
*)
lemma from_to_restn:
fixes k j s
assumes le_j: "length s \<le> j"
shows "from_to k j s = restn k s"
proof -
have "from_to 0 k s @ from_to k j s = from_to 0 j s"
proof(cases "k \<le> j")
case True
from from_to_conc True show ?thesis by auto
next
case False
from False le_j have lek: "length s \<le> k" by auto
from from_to_ge [OF this] have "from_to k j s = []" .
hence "from_to 0 k s @ from_to k j s = from_to 0 k s" by simp
also have "\<dots> = s"
proof -
from from_to_firstn [of k s]
have "\<dots> = firstn k s" .
also have "\<dots> = s" by (rule firstn_ge [OF lek])
finally show ?thesis .
qed
finally have "from_to 0 k s @ from_to k j s = s" .
moreover have "from_to 0 j s = s"
proof -
have "from_to 0 j s = firstn j s" by (simp add:from_to_firstn)
also have "\<dots> = s"
proof(rule firstn_ge)
from le_j show "length s \<le> j " by simp
qed
finally show ?thesis .
qed
ultimately show ?thesis by auto
qed
also have "\<dots> = s"
proof -
from from_to_firstn have "\<dots> = firstn j s" .
also have "\<dots> = s"
proof(rule firstn_ge)
from le_j show "length s \<le> j" by simp
qed
finally show ?thesis .
qed
finally have "from_to 0 k s @ from_to k j s = s" .
moreover have "from_to 0 k s @ restn k s = s"
proof -
from from_to_firstn [of k s]
have "from_to 0 k s = firstn k s" .
thus ?thesis by (simp add:firstn_restn_s)
qed
ultimately have "from_to 0 k s @ from_to k j s =
from_to 0 k s @ restn k s" by simp
thus ?thesis by auto
qed
lemma down_to_moment: "down_to k 0 s = moment k s"
proof -
have "rev (from_to 0 k (rev s)) = rev (firstn k (rev s))"
using from_to_firstn by metis
thus ?thesis by (simp add:down_to_def moment_def)
qed
lemma down_to_restm:
assumes le_s: "length s \<le> j"
shows "down_to j k s = restm k s"
proof -
have "rev (from_to k j (rev s)) = rev (restn k (rev s))" (is "?L = ?R")
proof -
from le_s have "length (rev s) \<le> j" by simp
from from_to_restn [OF this, of k] show ?thesis by simp
qed
thus ?thesis by (simp add:down_to_def restm_def)
qed
lemma moment_split: "moment (m+i) s = down_to (m+i) i s @down_to i 0 s"
proof -
have "moment (m + i) s = down_to (m+i) 0 s" using down_to_moment by metis
also have "\<dots> = (down_to (m+i) i s) @ (down_to i 0 s)"
by(rule down_to_conc[symmetric], auto)
finally show ?thesis .
qed
lemma length_restn: "length (restn i s) = length s - i"
proof(cases "i \<le> length s")
case True
from length_firstn_le [OF this] have "length (firstn i s) = i" .
moreover have "length s = length (firstn i s) + length (restn i s)"
proof -
have "s = firstn i s @ restn i s" using firstn_restn_s by metis
hence "length s = length \<dots>" by simp
thus ?thesis by simp
qed
ultimately show ?thesis by simp
next
case False
hence "length s \<le> i" by simp
from restn_ge [OF this] have "restn i s = []" .
with False show ?thesis by simp
qed
lemma length_from_to_in:
fixes i j s
assumes le_ij: "i \<le> j"
and le_j: "j \<le> length s"
shows "length (from_to i j s) = j - i"
proof -
have "from_to 0 j s = from_to 0 i s @ from_to i j s"
by (rule from_to_conc[symmetric, OF _ le_ij], simp)
moreover have "length (from_to 0 j s) = j"
proof -
have "from_to 0 j s = firstn j s" using from_to_firstn by metis
moreover have "length \<dots> = j" by (rule length_firstn_le [OF le_j])
ultimately show ?thesis by simp
qed
moreover have "length (from_to 0 i s) = i"
proof -
have "from_to 0 i s = firstn i s" using from_to_firstn by metis
moreover have "length \<dots> = i"
proof (rule length_firstn_le)
from le_ij le_j show "i \<le> length s" by simp
qed
ultimately show ?thesis by simp
qed
ultimately show ?thesis by auto
qed
lemma firstn_restn_from_to: "from_to i (m + i) s = firstn m (restn i s)"
proof(cases "m+i \<le> length s")
case True
have "restn i s = from_to i (m+i) s @ from_to (m+i) (length s) s"
proof -
have "restn i s = from_to i (length s) s"
by(rule from_to_restn[symmetric], simp)
also have "\<dots> = from_to i (m+i) s @ from_to (m+i) (length s) s"
by(rule from_to_conc[symmetric, OF _ True], simp)
finally show ?thesis .
qed
hence "firstn m (restn i s) = firstn m \<dots>" by simp
moreover have "\<dots> = firstn (length (from_to i (m+i) s))
(from_to i (m+i) s @ from_to (m+i) (length s) s)"
proof -
have "length (from_to i (m+i) s) = m"
proof -
have "length (from_to i (m+i) s) = (m+i) - i"
by(rule length_from_to_in [OF _ True], simp)
thus ?thesis by simp
qed
thus ?thesis by simp
qed
ultimately show ?thesis using app_firstn_restn by metis
next
case False
hence "length s \<le> m + i" by simp
from from_to_restn [OF this]
have "from_to i (m + i) s = restn i s" .
moreover have "firstn m (restn i s) = restn i s"
proof(rule firstn_ge)
show "length (restn i s) \<le> m"
proof -
have "length (restn i s) = length s - i" using length_restn by metis
with False show ?thesis by simp
qed
qed
ultimately show ?thesis by simp
qed
lemma down_to_moment_restm:
fixes m i s
shows "down_to (m + i) i s = moment m (restm i s)"
by (simp add:firstn_restn_from_to down_to_def moment_def restm_def)
lemma moment_plus_split:
fixes m i s
shows "moment (m + i) s = moment m (restm i s) @ moment i s"
proof -
from moment_split [of m i s]
have "moment (m + i) s = down_to (m + i) i s @ down_to i 0 s" .
also have "\<dots> = down_to (m+i) i s @ moment i s" using down_to_moment by simp
also from down_to_moment_restm have "\<dots> = moment m (restm i s) @ moment i s"
by simp
finally show ?thesis .
qed
lemma length_restm: "length (restm i s) = length s - i"
proof -
have "length (rev (restn i (rev s))) = length s - i" (is "?L = ?R")
proof -
have "?L = length (restn i (rev s))" by simp
also have "\<dots> = length (rev s) - i" using length_restn by metis
also have "\<dots> = ?R" by simp
finally show ?thesis .
qed
thus ?thesis by (simp add:restm_def)
qed
end