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factorial.thy
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factorial.thy
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(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
*
* SPDX-License-Identifier: BSD-2-Clause
*)
theory factorial
imports "CParser.CTranslation" "ARM/imports/MachineWords"
begin
declare hrs_simps [simp add]
declare sep_conj_ac [simp add]
consts free_pool :: "nat \<Rightarrow> heap_assert"
primrec
fac :: "nat \<Rightarrow> machine_word"
where
"fac 0 = 1"
| "fac (Suc n) = of_nat (Suc n) * fac n"
lemma fac_unfold:
"unat n \<noteq> 0 \<Longrightarrow> fac (unat n) = n * fac (unat (n - 1))"
apply(case_tac "unat n")
apply simp
apply(subst unat_minus_one)
apply(simp only: unat_simps)
apply(clarify)
apply simp
apply clarsimp
apply(simp only: unat_simps)
apply(subst mod_less)
apply (fold len_of_addr_card)
apply unat_arith
apply (clarsimp simp: distrib_right split: unat_splits)
done
primrec
fac_list :: "nat \<Rightarrow> machine_word list"
where
"fac_list 0 = [1]"
| "fac_list (Suc n) = fac (Suc n) # fac_list n"
lemma fac_list_length [simp]:
"length (fac_list n) = n + 1"
by (induct n) auto
lemma fac_list_unfold:
"unat n \<noteq> 0 \<Longrightarrow> fac_list (unat n) = fac (unat n) # fac_list (unat (n - 1))"
by (metis Suc_unat_minus_one fac_list.simps(2) unat_eq_zero)
primrec
sep_list :: "machine_word list \<Rightarrow> machine_word ptr \<Rightarrow> heap_assert"
where
"sep_list [] p = (\<lambda>s. p=NULL \<and> \<box> s)"
| "sep_list (x#xs) p = (\<lambda>s. \<exists>j. ((p \<mapsto> x) \<and>\<^sup>* (p +\<^sub>p 1) \<mapsto> j \<and>\<^sup>*
sep_list xs (Ptr j)) s)"
lemma sep_list_NULL [simp]:
"sep_list xs NULL = (\<lambda>s. xs = [] \<and> \<box> s)"
by (case_tac xs) auto
definition
sep_fac_list :: "machine_word \<Rightarrow> machine_word ptr \<Rightarrow> heap_assert"
where
"sep_fac_list n p \<equiv> sep_list (fac_list (unat n)) p"
lemma sep_fac_list_unfold:
"((\<lambda>s. unat n \<noteq> 0 \<and> (\<exists>q. (p \<mapsto> fac (unat n) \<and>\<^sup>* (p +\<^sub>p 1) \<mapsto> q \<and>\<^sup>*
sep_fac_list (n - 1) (Ptr q)) s)) \<and>\<^sup>* R) s \<Longrightarrow>
(sep_fac_list n p \<and>\<^sup>* R) s"
apply (erule sep_globalise)
apply (simp add: sep_fac_list_def fac_list_unfold)
done
lemma sep_fac_list_points:
"sep_points (sep_fac_list n p) s \<Longrightarrow> (p \<hookrightarrow> fac (unat n)) s"
apply(unfold sep_points_def)
apply(subst sep_map'_unfold)
apply(erule sep_conj_impl)
apply(unfold sep_fac_list_def)
apply(case_tac "unat n")
apply simp
apply(unfold sep_map'_old)
apply(erule (1) sep_conj_impl)
apply simp
apply clarsimp
apply(subst (asm) sep_conj_assoc [symmetric])+
apply(erule sep_conj_impl)
apply simp+
done
external_file "factorial.c"
install_C_file memsafe "factorial.c"
thm factorial_global_addresses.factorial_body_def
lemma (in factorial_global_addresses) mem_safe_alloc:
"mem_safe (\<acute>ret__ptr_to_unsigned_long :== PROC alloc()) \<Gamma>"
apply(insert alloc_impl)
apply(unfold alloc_body_def)
apply(subst mem_safe_restrict)
apply(rule intra_mem_safe)
apply(simp_all add: restrict_map_def split: if_split_asm)
apply(auto simp: whileAnno_def comp_def hrs_comm split_def mono_guard_def)
done
lemma (in factorial_global_addresses) sep_frame_alloc:
"\<lbrakk> \<forall>\<sigma>. \<Gamma> \<turnstile> \<lbrace>\<sigma>. (P (f \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace> \<acute>ret__ptr_to_unsigned_long :== PROC alloc() \<lbrace> (Q (g \<sigma> \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace>;
htd_ind f; htd_ind g; \<forall>s. htd_ind (g s) \<rbrakk> \<Longrightarrow>
\<forall>\<sigma>. \<Gamma> \<turnstile> \<lbrace>\<sigma>. (P (f \<acute>(\<lambda>x. x)) \<and>\<^sup>* R (h \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace>
\<acute>ret__ptr_to_unsigned_long :== PROC alloc()
\<lbrace> (Q (g \<sigma> \<acute>(\<lambda>x. x)) \<and>\<^sup>* R (h \<sigma>))\<^bsup>sep\<^esup> \<rbrace>"
unfolding sep_app_def
by (rule sep_frame, auto intro!: mem_safe_alloc)
lemma (in alloc_spec) alloc_spec':
shows "\<forall>\<sigma> k R f. factorial_global_addresses.\<Gamma> \<turnstile>
\<lbrace>\<sigma>. ((\<lambda>x. free_pool k) ((\<lambda>x. undefined) \<acute>(\<lambda>x. x)) \<and>\<^sup>* R (f \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace>
\<acute>ret__ptr_to_unsigned_long :== PROC alloc()
\<lbrace> ((\<lambda>p s. if k > 0 then (\<turnstile>\<^sub>s p \<and>\<^sup>* \<turnstile>\<^sub>s (p +\<^sub>p 1) \<and>\<^sup>*
free_pool (k - 1)) s else (free_pool k) s \<and> p = NULL) \<acute>ret__ptr_to_unsigned_long
\<and>\<^sup>* R (f \<sigma>))\<^bsup>sep\<^esup> \<rbrace>"
apply clarify
apply(insert alloc_spec)
apply(rule_tac x=\<sigma> in spec)
apply(rule sep_frame_alloc)
apply(clarsimp simp: sep_app_def split: if_split_asm)
apply simp+
done
lemma (in factorial_global_addresses) mem_safe_free:
"mem_safe (PROC free(\<acute>p)) \<Gamma>"
apply(insert free_impl)
apply(unfold free_body_def)
apply(subst mem_safe_restrict)
apply(auto simp: whileAnno_def)
apply(rule intra_mem_safe)
apply(auto simp: restrict_map_def split: if_split_asm)
done
lemma (in factorial_global_addresses) sep_frame_free:
"\<lbrakk> \<forall>\<sigma>. \<Gamma> \<turnstile> \<lbrace>\<sigma>. (P (f \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace> PROC free(\<acute>p) \<lbrace> (Q (g \<sigma> \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace>;
htd_ind f; htd_ind g; \<forall>s. htd_ind (g s) \<rbrakk> \<Longrightarrow>
\<forall>\<sigma>. \<Gamma> \<turnstile> \<lbrace>\<sigma>. (P (f \<acute>(\<lambda>x. x)) \<and>\<^sup>* R (h \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace>
PROC free(\<acute>p)
\<lbrace> (Q (g \<sigma> \<acute>(\<lambda>x. x)) \<and>\<^sup>* R (h \<sigma>))\<^bsup>sep\<^esup> \<rbrace>"
apply(simp only: sep_app_def)
apply(rule sep_frame, auto intro!: mem_safe_free)
done
lemma (in free_spec) free_spec':
shows "\<forall>\<sigma> k R f. factorial_global_addresses.\<Gamma> \<turnstile>
\<lbrace>\<sigma>. ((\<lambda>p. sep_cut' (ptr_val p) (2 * size_of TYPE(machine_word)) \<and>\<^sup>* free_pool k) \<acute>p \<and>\<^sup>* R (f \<acute>(\<lambda>x. x)))\<^bsup>sep\<^esup> \<rbrace>
PROC free(\<acute>p)
\<lbrace> ((\<lambda>x. free_pool (k + 1)) ((\<lambda>x. ()) \<acute>(\<lambda>x. x)) \<and>\<^sup>* R (f \<sigma>))\<^bsup>sep\<^esup> \<rbrace>"
apply clarify
apply(insert free_spec)
apply(rule_tac x=\<sigma> in spec)
apply(rule sep_frame_free)
apply(clarsimp simp: sep_app_def split: if_split_asm)
apply simp+
done
lemma double_word_sep_cut':
"(p \<mapsto> x \<and>\<^sup>* (p +\<^sub>p 1) \<mapsto> y) s \<Longrightarrow> sep_cut' (ptr_val (p::machine_word ptr)) (2*machine_word_bytes) s"
apply(clarsimp simp: sep_conj_def sep_cut'_def dest!: sep_map_dom)
apply(subgoal_tac "{ptr_val p..+machine_word_bytes} \<subseteq> {ptr_val p..+(2*machine_word_bytes)}")
apply(subgoal_tac "{ptr_val p+(of_nat machine_word_bytes)..+machine_word_bytes} \<subseteq> {ptr_val p..+(2*machine_word_bytes)}")
apply rule
apply (fastforce simp: ptr_add_def)
apply clarsimp
apply(drule intvlD)
apply clarsimp
apply(case_tac "k < machine_word_bytes")
apply(erule intvlI)
apply(erule notE)
apply(clarsimp simp: intvl_def)
apply(rule_tac x="k - machine_word_bytes" in exI)
apply rule
apply(simp only: unat_simps)
apply(subst mod_less)
apply(simp add: addr_card)
apply (simp add: ptr_add_def addr_card)
apply simp
apply(clarsimp simp: intvl_def)
apply(rule_tac x="machine_word_bytes+k" in exI)
apply simp
apply(rule intvl_start_le)
apply simp
done
locale specs = factorial_global_addresses + alloc_spec + free_spec
lemma (in specs) factorial_spec:
shows "factorial_spec"
apply unfold_locales
apply(hoare_rule HoarePartial.ProcRec1)
apply (hoare_rule anno = "factorial_invs_body k" in HoarePartial.annotateI)
prefer 2
apply (simp add: whileAnno_def factorial_invs_body_def)
apply(subst factorial_invs_body_def)
apply(unfold sep_app_def)
apply (vcg exspec=alloc_spec' exspec=free_spec')
apply (fold lift_def)
apply(clarsimp simp: sep_app_def)
apply (rule conjI)
apply clarsimp
apply(rule_tac x=k in exI)
apply(rule_tac x="\<lambda>p. \<box>" in exI)
apply(rule_tac x="\<lambda>s. undefined" in exI)
apply clarsimp
apply (rule conjI)
apply clarsimp
apply clarsimp
apply (rename_tac a b ret__ptr_to_unsigned_long)
apply(subgoal_tac "(\<turnstile>\<^sub>s ret__ptr_to_unsigned_long \<and>\<^sup>* sep_true) (lift_state (a,b))")
prefer 2
apply(drule sep_conj_sep_true_right)
apply simp
apply(subgoal_tac "(\<turnstile>\<^sub>s (ret__ptr_to_unsigned_long +\<^sub>p 1) \<and>\<^sup>* sep_true) (lift_state (a,b))")
prefer 2
apply(drule sep_conj_sep_true_left)
apply simp
apply(subst (asm) sep_conj_assoc [symmetric])+
apply(drule_tac Q="free_pool (k - Suc 0)" in sep_conj_sep_true_right)
apply simp
apply(simp add: tagd_ptr_safe tagd_g c_guard_ptr_aligned c_guard_NULL)
apply(simp add: sep_fac_list_def)
apply(sep_select_tac "(_ +\<^sub>p _) \<mapsto> _")
apply(rule sep_heap_update_global')
apply simp
apply(rule sep_heap_update_global')
apply simp
apply clarsimp
apply(rule_tac x=k in exI)
apply clarsimp
apply(rule_tac x="k - Suc (unat (n - 1))" in exI)
apply clarsimp
apply(rule_tac x="\<lambda>(n,p). sep_fac_list (n - 1) p" in exI)
apply(rule_tac x="\<lambda>s. (n_' s,q_' s)" in exI)
apply (rule conjI, clarsimp)
apply clarsimp
apply (rule conjI)
apply clarsimp
apply(simp add: sep_fac_list_def)
apply(rule_tac x="fac_list (unat (n - 1))" in exI)
apply clarsimp
apply clarsimp
apply(subgoal_tac "(\<turnstile>\<^sub>s ret__ptr_to_unsigned_long \<and>\<^sup>* sep_true) (lift_state (ab,bb))")
prefer 2
apply(erule (1) sep_conj_impl)
apply simp
apply(subgoal_tac "(\<turnstile>\<^sub>s (ret__ptr_to_unsigned_long +\<^sub>p 1) \<and>\<^sup>* sep_true) (lift_state (ab,bb))")
prefer 2
apply(subgoal_tac "(\<turnstile>\<^sub>s ret__ptr_to_unsigned_long \<and>\<^sup>*
\<turnstile>\<^sub>s (ret__ptr_to_unsigned_long +\<^sub>p 1) \<and>\<^sup>*
sep_fac_list (n - 1) ret__ptr_to_unsigned_longa \<and>\<^sup>*
free_pool (k - Suc (unat n))) =
(\<turnstile>\<^sub>s (ret__ptr_to_unsigned_long +\<^sub>p 1) \<and>\<^sup>* (\<turnstile>\<^sub>s ret__ptr_to_unsigned_long \<and>\<^sup>*
sep_fac_list (n - 1) ret__ptr_to_unsigned_longa \<and>\<^sup>*
free_pool (k - Suc (unat n))))")
prefer 2
apply simp
apply (simp only:)
apply(erule (1) sep_conj_impl)
apply simp
apply(sep_point_tac sep_fac_list_points)
apply(simp add: tagd_ptr_safe tagd_g sep_map'_g c_guard_ptr_aligned c_guard_NULL sep_map'_lift)
apply(rule sep_fac_list_unfold)
apply clarsimp
apply (rule conjI, unat_arith)
apply sep_exists_tac
apply(rule_tac x="ptr_val ret__ptr_to_unsigned_longa" in exI)
apply clarsimp
apply(subst fac_unfold)
apply unat_arith
apply clarsimp
apply(sep_select_tac "(_ +\<^sub>p _) \<mapsto> _")
apply(rule sep_heap_update_global')
apply(sep_select_tac "_ \<mapsto> _")
apply(rule sep_heap_update_global')
apply(erule (1) sep_conj_impl)+
apply clarsimp
apply clarsimp
apply(case_tac xs)
apply simp
apply clarsimp
apply sep_exists_tac
apply clarsimp
apply sep_point_tac
apply(simp add: sep_map'_g c_guard_ptr_aligned c_guard_NULL sep_map'_lift)
apply(rule_tac x="k - Suc (length list)" in exI)
apply(rule_tac x="\<lambda>p. sep_list list (Ptr j)" in exI)
apply(rule_tac x="\<lambda>x. ()" in exI)
apply(clarsimp simp: sep_app_def)
apply (rule conjI)
apply(subgoal_tac "(q \<mapsto> aa \<and>\<^sup>* sep_list list (Ptr j) \<and>\<^sup>*
(q +\<^sub>p 1) \<mapsto> j \<and>\<^sup>* free_pool (k - Suc (length list))) =
(sep_list list (Ptr j) \<and>\<^sup>* (q \<mapsto> aa \<and>\<^sup>*
(q +\<^sub>p 1) \<mapsto> j \<and>\<^sup>* free_pool (k - Suc (length list))))")
apply(simp only:)
apply(erule (1) sep_conj_impl)
apply simp
apply(subst sep_conj_com)
apply(subst (asm) sep_conj_assoc [symmetric])+
apply(erule sep_conj_impl)
apply simp
apply(erule double_word_sep_cut'[simplified])
apply assumption
apply simp
apply clarsimp
apply(subgoal_tac "Suc (k - Suc (length list)) = k - length list")
apply force
apply arith
apply clarsimp
done
declare hrs_simps [simp del]
lemma (in factorial_global_addresses) mem_safe_factorial:
shows "mem_safe (\<acute>ret__ptr_to_unsigned_long :== PROC factorial(\<acute>n)) \<Gamma>"
apply(subst mem_safe_restrict)
apply(rule intra_mem_safe)
apply (insert factorial_impl free_impl alloc_impl)[1]
apply(drule_tac t="Some C" in sym)
apply(simp add: restrict_map_def call_def block_def whileAnno_def block_exn_def
free_body_def alloc_body_def factorial_body_def creturn_def
split: if_split_asm option.splits)
subgoal by (force simp: intra_sc)
apply clarsimp
done
end