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Partial_Fun.thy
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Partial_Fun.thy
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(******************************************************************************)
(* Project: Isabelle/UTP Toolkit *)
(* File: Partial_Fun.thy *)
(* Authors: Simon Foster and Frank Zeyda *)
(* Emails: [email protected] and [email protected] *)
(******************************************************************************)
section \<open> Partial Functions \<close>
theory Partial_Fun
imports "Optics.Optics" Map_Extra "HOL-Library.Mapping"
begin
no_notation "Stream.stream.SCons" (infixr \<open>##\<close> 65)
text \<open> I'm not completely satisfied with partial functions as provided by Map.thy, since they don't
have a unique type and so we can't instantiate classes, make use of adhoc-overloading
etc. Consequently I've created a new type and derived the laws. \<close>
subsection \<open> Partial function type and operations \<close>
typedef ('a, 'b) pfun = "UNIV :: ('a \<rightharpoonup> 'b) set"
morphisms pfun_lookup pfun_of_map ..
type_notation pfun (infixr "\<Zpfun>" 1)
setup_lifting type_definition_pfun
lemma pfun_lookup_map [simp]: "pfun_lookup (pfun_of_map f) = f"
by (simp add: pfun_of_map_inverse)
lift_bnf ('k, pran: 'v) pfun [wits: "Map.empty :: 'k \<Rightarrow> 'v option"] for map: map_pfun rel: relt_pfun
by auto
declare pfun.map_transfer [transfer_rule]
instantiation pfun :: (type, type) equal
begin
definition "HOL.equal m1 m2 \<longleftrightarrow> (\<forall>k. pfun_lookup m1 k = pfun_lookup m2 k)"
instance
by (intro_classes, auto simp add: equal_pfun_def, transfer, auto)
end
lift_definition pfun_app :: "('a, 'b) pfun \<Rightarrow> 'a \<Rightarrow> 'b" ("_'(_')\<^sub>p" [999,0] 999) is
"\<lambda> f x. if (x \<in> dom f) then the (f x) else undefined" .
lift_definition pfun_upd :: "('a, 'b) pfun \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) pfun"
is "\<lambda> f k v. f(k := Some v)" .
lift_definition pdom :: "('a, 'b) pfun \<Rightarrow> 'a set" is dom .
lemma pran_rep_eq [transfer_rule]: "pran f = ran (pfun_lookup f)"
by (transfer, auto simp add: ran_def)
lift_definition pfun_comp :: "('b, 'c) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'c) pfun" (infixl "\<circ>\<^sub>p" 55) is
"\<lambda> f g. f \<circ>\<^sub>m g" .
lift_definition map_pfun' :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('c, 'd) pfun"
is "\<lambda>f g m. (map_option g \<circ> m \<circ> f)" parametric map_parametric .
functor map_pfun'
by (transfer, auto simp add: fun_eq_iff option.map_comp option.map_id)+
lift_definition pfun_member :: "'a \<times> 'b \<Rightarrow> ('a, 'b) pfun \<Rightarrow> bool" (infix "\<in>\<^sub>p" 50) is "(\<in>\<^sub>m)" .
lift_definition pfun_inj :: "('a, 'b) pfun \<Rightarrow> bool" is "\<lambda> f. inj_on f (dom f)" .
lift_definition pfun_inv :: "('a, 'b) pfun \<Rightarrow> ('b, 'a) pfun" is map_inv .
lift_definition pId_on :: "'a set \<Rightarrow> ('a, 'a) pfun" is "\<lambda> A x. if (x \<in> A) then Some x else None" .
abbreviation pId :: "('a, 'a) pfun" where
"pId \<equiv> pId_on UNIV"
lift_definition pdom_res :: "'a set \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun" (infixr "\<lhd>\<^sub>p" 85)
is "\<lambda> A f. restrict_map f A" .
abbreviation pdom_nres (infixr "-\<lhd>\<^sub>p" 85) where "pdom_nres A P \<equiv> (- A) \<lhd>\<^sub>p P"
lift_definition pran_res :: "('a, 'b) pfun \<Rightarrow> 'b set \<Rightarrow> ('a, 'b) pfun" (infixl "\<rhd>\<^sub>p" 86)
is ran_restrict_map .
abbreviation pran_nres (infixr "\<rhd>\<^sub>p-" 66) where "pran_nres P A \<equiv> P \<rhd>\<^sub>p (- A)"
definition pfun_image :: "'a \<Zpfun> 'b \<Rightarrow> 'a set \<Rightarrow> 'b set" where
[simp]: "pfun_image f A = pran (A \<lhd>\<^sub>p f)"
lift_definition pfun_graph :: "('a, 'b) pfun \<Rightarrow> ('a \<times> 'b) set" is map_graph .
lift_definition graph_pfun :: "('a \<times> 'b) set \<Rightarrow> ('a, 'b) pfun" is "graph_map \<circ> mk_functional" .
definition pfun_pfun :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Zpfun> 'b) set" where
"pfun_pfun A B = {f :: 'a \<Zpfun> 'b. pdom(f) \<subseteq> A \<and> pran(f) \<subseteq> B}"
definition pfun_tfun :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Zpfun> 'b) set" where
"pfun_tfun A B = {f \<in> pfun_pfun A B. pdom(f) = UNIV}"
definition pfun_ffun :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Zpfun> 'b) set" where
"pfun_ffun A B = {f \<in> pfun_pfun A B. finite(pdom(f))}"
definition pfun_pinj :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Zpfun> 'b) set" where
"pfun_pinj A B = {f \<in> pfun_pfun A B. pfun_inj f}"
definition pfun_psurj :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Zpfun> 'b) set" where
"pfun_psurj A B = {f \<in> pfun_pfun A B. pran(f) = UNIV}"
definition "pfun_finj A B = pfun_ffun A B \<inter> pfun_pinj A B"
definition "pfun_tinj A B = pfun_tfun A B \<inter> pfun_pinj A B"
definition "pfun_tsurj A B = pfun_tfun A B \<inter> pfun_psurj A B"
definition "pfun_bij A B = pfun_tfun A B \<inter> pfun_pinj A B \<inter> pfun_psurj A B"
lift_definition pfun_entries :: "'k set \<Rightarrow> ('k \<Rightarrow> 'v) \<Rightarrow> ('k, 'v) pfun" is
"\<lambda> d f x. if x \<in> d then Some (f x) else None" .
definition pfuse :: "('a \<Zpfun> 'b) \<Rightarrow> ('a \<Zpfun> 'c) \<Rightarrow> ('a \<Zpfun> 'b \<times> 'c)"
where "pfuse f g = pfun_entries (pdom(f) \<inter> pdom(g)) (\<lambda> x. (pfun_app f x, pfun_app g x))"
lift_definition ptabulate :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) pfun"
is "\<lambda>ks f. (map_of (List.map (\<lambda>k. (k, f k)) ks))" .
lift_definition pcombine ::
"('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun"
is "\<lambda>f m1 m2 x. combine_options f (m1 x) (m2 x)" .
abbreviation "fun_pfun \<equiv> pfun_entries UNIV"
definition pfun_disjoint :: "'a \<Zpfun> 'b set \<Rightarrow> bool" where
"pfun_disjoint S = (\<forall> i \<in> pdom S. \<forall> j \<in> pdom S. i \<noteq> j \<longrightarrow> pfun_app S i \<inter> pfun_app S j = {})"
definition pfun_partitions :: "'a \<Zpfun> 'b set \<Rightarrow> 'b set \<Rightarrow> bool" where
"pfun_partitions S T = (pfun_disjoint S \<and> \<Union> (pran S) = T)"
no_notation disj (infixr "|" 30)
definition pabs :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Zpfun> 'b" where
"pabs A P f = (A \<inter> Collect P) \<lhd>\<^sub>p fun_pfun f"
definition pcard :: "('a, 'b) pfun \<Rightarrow> nat"
where "pcard f = card (pdom f)"
unbundle lattice_syntax
instantiation pfun :: (type, type) bot
begin
lift_definition bot_pfun :: "('a, 'b) pfun" is "Map.empty" .
instance ..
end
abbreviation pempty :: "('a, 'b) pfun" ("{}\<^sub>p")
where "pempty \<equiv> bot"
instantiation pfun :: (type, type) oplus
begin
lift_definition oplus_pfun :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun" is "(++)" .
instance ..
end
instantiation pfun :: (type, type) minus
begin
lift_definition minus_pfun :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun" is "(--)" .
instance ..
end
instantiation pfun :: (type, type) inf
begin
lift_definition inf_pfun :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun" is
"\<lambda> f g x. if (x \<in> dom(f) \<inter> dom(g) \<and> f(x) = g(x)) then f(x) else None" .
instance ..
end
abbreviation pfun_inter :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun" (infixl "\<inter>\<^sub>p" 80)
where "pfun_inter \<equiv> inf"
instantiation pfun :: (type, type) order
begin
lift_definition less_eq_pfun :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> bool" is
"\<lambda> f g. f \<subseteq>\<^sub>m g" .
lift_definition less_pfun :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> bool" is
"\<lambda> f g. f \<subseteq>\<^sub>m g \<and> f \<noteq> g" .
instance
by (intro_classes, (transfer, auto intro: map_le_trans simp add: map_le_antisym)+)
end
abbreviation pfun_subset :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> bool" (infix "\<subset>\<^sub>p" 50)
where "pfun_subset \<equiv> less"
abbreviation pfun_subset_eq :: "('a, 'b) pfun \<Rightarrow> ('a, 'b) pfun \<Rightarrow> bool" (infix "\<subseteq>\<^sub>p" 50)
where "pfun_subset_eq \<equiv> less_eq"
instance pfun :: (type, type) semilattice_inf
by (intro_classes, (transfer, auto simp add: map_le_def dom_def)+)
lemma pfun_subset_eq_least [simp]:
"{}\<^sub>p \<subseteq>\<^sub>p f"
by (transfer, auto)
syntax
"_PfunUpd" :: "[('a, 'b) pfun, maplets] => ('a, 'b) pfun" ("_'(_')\<^sub>p" [900,0]900)
"_Pfun" :: "maplets => ('a, 'b) pfun" ("(1{_}\<^sub>p)")
"_pabs" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("\<lambda> _ \<in> _ | _ \<bullet> _" [0, 0, 0, 10] 10)
"_pabs_mem" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("\<lambda> _ \<in> _ \<bullet> _" [0, 0, 10] 10)
"_pabs_pred" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic \<Rightarrow> logic" ("\<lambda> _ | _ \<bullet> _" [0, 0, 10] 10)
"_pabs_tot" :: "pttrn \<Rightarrow> logic \<Rightarrow> logic" ("\<lambda> _ \<bullet> _" [0, 10] 10)
translations
"_PfunUpd m (_Maplets xy ms)" == "_PfunUpd (_PfunUpd m xy) ms"
"_PfunUpd m (_maplet x y)" == "CONST pfun_upd m x y"
"_Pfun ms" => "_PfunUpd (CONST pempty) ms"
"_Pfun (_Maplets ms1 ms2)" <= "_PfunUpd (_Pfun ms1) ms2"
"_Pfun ms" <= "_PfunUpd (CONST pempty) ms"
"_pabs x A P f" => "CONST pabs A (\<lambda> x. P) (\<lambda> x. f)"
"_pabs x A P f" <= "CONST pabs A (\<lambda> y. P) (\<lambda> x. f)"
"_pabs x A P (f x)" <= "CONST pabs A (\<lambda> x. P) f"
"_pabs_mem x A f" == "_pabs x A (CONST True) f"
"_pabs_pred x P f" == "_pabs x (CONST UNIV) P f"
"_pabs_tot x f" == "_pabs_pred x (CONST True) f"
"_pabs_tot x f" <= "_pabs_mem x (CONST UNIV) f"
subsection \<open> Algebraic laws \<close>
lemma pfun_comp_assoc: "f \<circ>\<^sub>p (g \<circ>\<^sub>p h) = (f \<circ>\<^sub>p g) \<circ>\<^sub>p h"
by (transfer, simp add: map_comp_assoc)
lemma pfun_comp_left_id [simp]: "pId \<circ>\<^sub>p f = f"
by (transfer, auto)
lemma pfun_comp_right_id [simp]: "f \<circ>\<^sub>p pId = f"
by (transfer, auto)
lemma pfun_comp_left_zero [simp]: "{}\<^sub>p \<circ>\<^sub>p f = {}\<^sub>p"
by (transfer, auto)
lemma pfun_comp_right_zero [simp]: "f \<circ>\<^sub>p {}\<^sub>p = {}\<^sub>p"
by (transfer, auto)
lemma pfun_override_dist_comp:
"(f \<oplus> g) \<circ>\<^sub>p h = (f \<circ>\<^sub>p h) \<oplus> (g \<circ>\<^sub>p h)"
apply (transfer)
apply (rule ext)
apply (auto simp add: map_add_def)
apply (rename_tac f g h x)
apply (case_tac "h x")
apply (auto)
apply (rename_tac f g h x y)
apply (case_tac "g y")
apply (auto)
done
lemma pfun_minus_unit [simp]:
fixes f :: "('a, 'b) pfun"
shows "f - \<bottom> = f"
by (transfer, simp add: map_minus_def)
lemma pfun_minus_zero [simp]:
fixes f :: "('a, 'b) pfun"
shows "\<bottom> - f = \<bottom>"
by (transfer, simp add: map_minus_def)
lemma pfun_minus_self [simp]:
fixes f :: "('a, 'b) pfun"
shows "f - f = \<bottom>"
by (transfer, simp add: map_minus_def)
instantiation pfun :: (type, type) override
begin
definition compatible_pfun :: "'a \<Zpfun> 'b \<Rightarrow> 'a \<Zpfun> 'b \<Rightarrow> bool" where
"compatible_pfun R S = ((pdom R) \<lhd>\<^sub>p S = (pdom S) \<lhd>\<^sub>p R)"
lemma pfun_compat_add: "(P :: 'a \<Zpfun> 'b) ## Q \<Longrightarrow> P \<oplus> Q ## R \<Longrightarrow> P ## R"
apply (simp add: compatible_pfun_def oplus_pfun_def)
apply (transfer)
using map_compat_add apply auto
done
lemma pfun_compat_addI: "\<lbrakk> (P :: 'a \<Zpfun> 'b) ## Q; P ## R; Q ## R \<rbrakk> \<Longrightarrow> P \<oplus> Q ## R"
apply (simp add: compatible_pfun_def oplus_pfun_def)
apply (transfer)
apply (auto simp add: restrict_map_def fun_eq_iff dom_def map_add_def option.case_eq_if)
apply (metis option.inject)+
done
instance proof
fix P Q R :: "'a \<Zpfun> 'b"
show "P ## Q \<Longrightarrow> P \<oplus> Q ## R \<Longrightarrow> P ## R"
by (simp add: pfun_compat_add)
show "P ## Q \<Longrightarrow> P ## R \<Longrightarrow> Q ## R \<Longrightarrow> P \<oplus> Q ## R"
by (simp add: pfun_compat_addI)
qed (simp_all add: compatible_pfun_def oplus_pfun_def,
(transfer, auto simp add: map_add_subsumed2 map_add_comm_weak')+)
end
lemma pfun_indep_compat: "pdom(f) \<inter> pdom(g) = {} \<Longrightarrow> f ## g"
unfolding compatible_pfun_def
by (transfer, auto simp add: restrict_map_def fun_eq_iff)
lemma pfun_override_commute:
"pdom(f) \<inter> pdom(g) = {} \<Longrightarrow> f \<oplus> g = g \<oplus> f"
by (transfer, metis map_add_comm)
lemma pfun_override_commute_weak:
"(\<forall> k \<in> pdom(f) \<inter> pdom(g). f(k)\<^sub>p = g(k)\<^sub>p) \<Longrightarrow> f \<oplus> g = g \<oplus> f"
by (transfer, simp, metis IntD1 IntD2 domD map_add_comm_weak option.sel)
lemma pfun_override_fully: "pdom f \<subseteq> pdom g \<Longrightarrow> f \<oplus> g = g"
by (transfer, auto simp add: map_add_def option.case_eq_if fun_eq_iff)
lemma pfun_override_res: "pdom g -\<lhd>\<^sub>p f \<oplus> g = f \<oplus> g"
by (transfer, auto simp add: map_add_restrict[THEN sym])
lemma pfun_minus_override_commute:
"pdom(g) \<inter> pdom(h) = {} \<Longrightarrow> (f - g) \<oplus> h = (f \<oplus> h) - g"
by (transfer, simp add: map_minus_plus_commute)
lemma pfun_override_minus:
"f \<subseteq>\<^sub>p g \<Longrightarrow> (g - f) \<oplus> f = g"
by (transfer, rule ext, auto simp add: map_le_def map_minus_def map_add_def option.case_eq_if)
lemma pfun_minus_common_subset:
"\<lbrakk> h \<subseteq>\<^sub>p f; h \<subseteq>\<^sub>p g \<rbrakk> \<Longrightarrow> (f - h = g - h) = (f = g)"
by (transfer, simp add: map_minus_common_subset)
lemma pfun_minus_override:
"pdom(f) \<inter> pdom(g) = {} \<Longrightarrow> (f \<oplus> g) - g = f"
by (transfer, simp add: map_add_def map_minus_def option.case_eq_if, rule ext, auto)
(metis Int_commute domIff insert_disjoint(1) insert_dom)
lemma pfun_override_pos: "x \<oplus> y = {}\<^sub>p \<Longrightarrow> x = {}\<^sub>p"
by (transfer, simp)
lemma pfun_le_override: "pdom x \<inter> pdom y = {} \<Longrightarrow> x \<le> x \<oplus> y"
by (transfer, auto simp add: map_le_iff_add)
subsection \<open> Membership, application, and update \<close>
lemma pfun_ext: "\<lbrakk> \<And> x y. (x, y) \<in>\<^sub>p f \<longleftrightarrow> (x, y) \<in>\<^sub>p g \<rbrakk> \<Longrightarrow> f = g"
by (transfer, simp add: map_ext)
lemma pfun_member_alt_def:
"(x, y) \<in>\<^sub>p f \<longleftrightarrow> (x \<in> pdom f \<and> f(x)\<^sub>p = y)"
by (transfer, auto simp add: map_member_alt_def map_apply_def)
lemma pfun_member_override:
"(x, y) \<in>\<^sub>p f \<oplus> g \<longleftrightarrow> ((x \<notin> pdom(g) \<and> (x, y) \<in>\<^sub>p f) \<or> (x, y) \<in>\<^sub>p g)"
by (transfer, simp add: map_member_plus)
lemma pfun_member_minus:
"(x, y) \<in>\<^sub>p f - g \<longleftrightarrow> (x, y) \<in>\<^sub>p f \<and> (\<not> (x, y) \<in>\<^sub>p g)"
by (transfer, simp add: map_member_minus)
lemma pfun_app_in_ran [simp]: "x \<in> pdom f \<Longrightarrow> f(x)\<^sub>p \<in> pran f"
by (transfer, auto)
lemma pfun_app_map [simp]: "(pfun_of_map f)(x)\<^sub>p = (if (x \<in> dom(f)) then the (f x) else undefined)"
by (transfer, simp)
lemma pfun_app_upd_1: "x = y \<Longrightarrow> (f(x \<mapsto> v)\<^sub>p)(y)\<^sub>p = v"
by (transfer, simp)
lemma pfun_app_upd_2: "x \<noteq> y \<Longrightarrow> (f(x \<mapsto> v)\<^sub>p)(y)\<^sub>p = f(y)\<^sub>p"
by (transfer, simp)
lemma pfun_app_upd [simp]: "(f(x \<mapsto> e)\<^sub>p)(y)\<^sub>p = (if (x = y) then e else f(y)\<^sub>p)"
by (metis pfun_app_upd_1 pfun_app_upd_2)
lemma pfun_graph_apply [simp]: "rel_apply (pfun_graph f) x = f(x)\<^sub>p"
by (transfer, auto simp add: rel_apply_def map_graph_def)
lemma pfun_upd_ext [simp]: "x \<in> pdom(f) \<Longrightarrow> f(x \<mapsto> f(x)\<^sub>p)\<^sub>p = f"
by (transfer, simp add: domIff)
lemma pfun_app_add [simp]: "x \<in> pdom(g) \<Longrightarrow> (f \<oplus> g)(x)\<^sub>p = g(x)\<^sub>p"
by (transfer, auto)
lemma pfun_upd_add [simp]: "f \<oplus> g(x \<mapsto> v)\<^sub>p = (f \<oplus> g)(x \<mapsto> v)\<^sub>p"
by (transfer, simp)
lemma pfun_upd_add_left [simp]: "x \<notin> pdom(g) \<Longrightarrow> f(x \<mapsto> v)\<^sub>p \<oplus> g = (f \<oplus> g)(x \<mapsto> v)\<^sub>p"
by (transfer, auto, metis domD map_add_upd_left)
lemma pfun_app_add' [simp]: "e \<notin> pdom g \<Longrightarrow> (f \<oplus> g)(e)\<^sub>p = f(e)\<^sub>p"
by (transfer, auto)
lemma pfun_upd_twice [simp]: "f(x \<mapsto> u, x \<mapsto> v)\<^sub>p = f(x \<mapsto> v)\<^sub>p"
by (transfer, simp)
lemma pfun_upd_comm:
assumes "x \<noteq> y"
shows "f(y \<mapsto> u, x \<mapsto> v)\<^sub>p = f(x \<mapsto> v, y \<mapsto> u)\<^sub>p"
using assms by (transfer, auto)
lemma pfun_upd_comm_linorder [simp]:
fixes x y :: "'a :: linorder"
assumes "x < y"
shows "f(y \<mapsto> u, x \<mapsto> v)\<^sub>p = f(x \<mapsto> v, y \<mapsto> u)\<^sub>p"
using assms by (transfer, auto)
lemma pfun_upd_as_ovrd: "f(k \<mapsto> v)\<^sub>p = f \<oplus> {k \<mapsto> v}\<^sub>p"
by (transfer, simp)
lemma pfun_ovrd_single_upd: "x \<in> pdom(g) \<Longrightarrow> f \<oplus> ({x} \<lhd>\<^sub>p g) = f(x \<mapsto> g(x)\<^sub>p)\<^sub>p"
by (transfer, auto simp add: map_add_def restrict_map_def fun_eq_iff)
lemma pfun_app_minus [simp]: "x \<notin> pdom g \<Longrightarrow> (f - g)(x)\<^sub>p = f(x)\<^sub>p"
by (transfer, auto simp add: map_minus_def)
lemma pfun_app_empty [simp]: "{}\<^sub>p(x)\<^sub>p = undefined"
by (transfer, simp)
lemma pfun_app_not_in_dom:
"x \<notin> pdom(f) \<Longrightarrow> f(x)\<^sub>p = undefined"
by (transfer, simp)
lemma pfun_upd_minus [simp]:
"x \<notin> pdom g \<Longrightarrow> (f - g)(x \<mapsto> v)\<^sub>p = (f(x \<mapsto> v)\<^sub>p - g)"
by (transfer, auto simp add: map_minus_def)
lemma pdom_member_minus_iff [simp]:
"x \<notin> pdom g \<Longrightarrow> x \<in> pdom(f - g) \<longleftrightarrow> x \<in> pdom(f)"
by (transfer, simp add: domIff map_minus_def)
lemma psubseteq_pfun_upd1 [intro]:
"\<lbrakk> f \<subseteq>\<^sub>p g; x \<notin> pdom(g) \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>p g(x \<mapsto> v)\<^sub>p"
by (transfer, auto simp add: map_le_def dom_def)
lemma psubseteq_pfun_upd2 [intro]:
"\<lbrakk> f \<subseteq>\<^sub>p g; x \<notin> pdom(f) \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>p g(x \<mapsto> v)\<^sub>p"
by (transfer, auto simp add: map_le_def dom_def)
lemma psubseteq_pfun_upd3 [intro]:
"\<lbrakk> f \<subseteq>\<^sub>p g; g(x)\<^sub>p = v \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>p g(x \<mapsto> v)\<^sub>p"
by (transfer, auto simp add: map_le_def dom_def)
lemma psubseteq_dom_subset:
"f \<subseteq>\<^sub>p g \<Longrightarrow> pdom(f) \<subseteq> pdom(g)"
by (transfer, auto simp add: map_le_def dom_def)
lemma psubseteq_ran_subset:
"f \<subseteq>\<^sub>p g \<Longrightarrow> pran(f) \<subseteq> pran(g)"
by (transfer, auto simp add: map_le_def dom_def ran_def)
lemma pfun_eq_iff: "f = g \<longleftrightarrow> (pdom(f) = pdom(g) \<and> (\<forall> x \<in> pdom(f). f(x)\<^sub>p = g(x)\<^sub>p))"
by (auto, transfer, simp add: map_eq_iff, metis domD option.sel)
lemma pfun_leI: "\<lbrakk> pdom f \<subseteq> pdom g; \<forall>x\<in>pdom f. f(x)\<^sub>p = g(x)\<^sub>p \<rbrakk> \<Longrightarrow> f \<subseteq>\<^sub>p g"
by (transfer, auto simp add: map_le_def)
(metis domD domI option.sel subsetD)
lemma pfun_le_iff: "(f \<subseteq>\<^sub>p g) = (pdom f \<subseteq> pdom g \<and> (\<forall>x\<in>pdom f. f(x)\<^sub>p = g(x)\<^sub>p))"
by (metis pfun_app_add pfun_leI pfun_override_minus psubseteq_dom_subset)
subsection \<open> Map laws \<close>
lemma map_pfun_empty [simp]: "map_pfun f {}\<^sub>p = {}\<^sub>p"
by (transfer, simp)
lemma map_pfun'_empty [simp]: "map_pfun' f g {}\<^sub>p = {}\<^sub>p"
unfolding map_pfun'_def by (transfer, simp add: comp_def)
lemma map_pfun_upd [simp]: "map_pfun f (g(x \<mapsto> v)\<^sub>p) = (map_pfun f g)(x \<mapsto> f v)\<^sub>p"
by (simp add: map_pfun_def pfun_upd.rep_eq pfun_upd.abs_eq)
lemma map_pfun_apply [simp]: "x \<in> pdom G \<Longrightarrow> (map_pfun F G)(x)\<^sub>p = F(G(x)\<^sub>p)"
unfolding map_pfun_def by (auto simp add: pfun_app.rep_eq domD pdom.rep_eq)
lemma map_pfun_as_pabs: "map_pfun f g = (\<lambda> x \<in> pdom(g) \<bullet> f(g(x)\<^sub>p))"
by (simp add: pabs_def, transfer, auto simp add: fun_eq_iff restrict_map_def)
lemma map_pfun_ovrd [simp]: "map_pfun f (g \<oplus> h) = (map_pfun f g) \<oplus> (map_pfun f h)"
by (simp add: map_pfun_def, transfer, auto simp add: map_add_def fun_eq_iff)
(metis bind.bind_lunit comp_apply map_conv_bind_option option.case_eq_if)
lemma map_pfun_dres [simp]: "map_pfun f (A \<lhd>\<^sub>p g) = A \<lhd>\<^sub>p map_pfun f g"
by (simp add: map_pfun_def, transfer, auto simp add: restrict_map_def)
subsection \<open> Domain laws \<close>
lemma pdom_zero [simp]: "pdom \<bottom> = {}"
by (transfer, simp)
lemma pdom_pId_on [simp]: "pdom (pId_on A) = A"
by (transfer, auto)
lemma pdom_plus [simp]: "pdom (f \<oplus> g) = pdom f \<union> pdom g"
by (transfer, auto)
lemma pdom_minus [simp]: "g \<le> f \<Longrightarrow> pdom (f - g) = pdom f - pdom g"
apply (transfer, auto simp add: map_minus_def)
apply (meson option.distinct(1))
apply (metis domIff map_le_def option.simps(3))
done
lemma pdom_inter: "pdom (f \<inter>\<^sub>p g) \<subseteq> pdom f \<inter> pdom g"
by (transfer, auto simp add: dom_def)
lemma pdom_comp [simp]: "pdom (g \<circ>\<^sub>p f) = pdom (f \<rhd>\<^sub>p pdom g)"
by (transfer, auto simp add: ran_restrict_map_def)
lemma pdom_upd [simp]: "pdom (f(k \<mapsto> v)\<^sub>p) = insert k (pdom f)"
by (transfer, simp)
lemma pdom_pdom_res [simp]: "pdom (A \<lhd>\<^sub>p f) = A \<inter> pdom(f)"
by (transfer, auto)
lemma pdom_graph_pfun: "pdom (graph_pfun R) \<subseteq> Domain R"
by (transfer, simp add: graph_map_dom fst_eq_Domain Domain_mk_functional)
lemma pdom_functional_graph_pfun [simp]:
"functional R \<Longrightarrow> pdom (graph_pfun R) = Domain R"
by (transfer, simp add: dom_map_graph mk_functional_fp)
lemma pdom_pran_res_finite [simp]:
"finite (pdom f) \<Longrightarrow> finite (pdom (f \<rhd>\<^sub>p A))"
by (transfer, auto)
lemma pdom_pfun_graph_finite [simp]:
"finite (pdom f) \<Longrightarrow> finite (pfun_graph f)"
by (transfer, simp add: finite_dom_graph)
lemma pdom_map_pfun [simp]: "pdom (map_pfun F G) = pdom G"
unfolding map_pfun_def by (auto; metis dom_map_option_comp pdom.abs_eq pdom.rep_eq)
lemma rel_comp_pfun: "R O pfun_graph f = (\<lambda> p. (fst p, pfun_app f (snd p))) ` (R \<rhd>\<^sub>r pdom(f))"
by (transfer, auto simp add: rel_comp_map rel_ranres_def)
lemma pdom_empty_iff_dom_empty: "f = {}\<^sub>p \<longleftrightarrow> pdom f = {}"
by (transfer, simp)
lemma empty_map_pfunD [dest!]: "{}\<^sub>p = map_pfun f F \<Longrightarrow> F = {}\<^sub>p"
by (metis pdom_empty_iff_dom_empty pdom_map_pfun)
subsection \<open> Range laws \<close>
lemma pran_zero [simp]: "pran \<bottom> = {}"
by (transfer, simp)
lemma pran_pId_on [simp]: "pran (pId_on A) = A"
by (transfer, auto simp add: ran_def)
lemma pran_upd [simp]: "pran (f(k \<mapsto> v)\<^sub>p) = insert v (pran ((- {k}) \<lhd>\<^sub>p f))"
by (transfer, auto simp add: ran_def restrict_map_def)
lemma pran_pran_res [simp]: "pran (f \<rhd>\<^sub>p A) = pran(f) \<inter> A"
by (transfer, auto simp add: ran_restrict_map_def)
lemma pran_comp [simp]: "pran (g \<circ>\<^sub>p f) = pran (pran f \<lhd>\<^sub>p g)"
by (transfer, auto simp add: ran_def restrict_map_def)
lemma pran_finite [simp]: "finite (pdom f) \<Longrightarrow> finite (pran f)"
by (simp add: pdom.rep_eq pran_rep_eq)
lemma pran_pdom: "pran F = pfun_app F ` pdom F"
by (transfer, force simp add: dom_def)
lemma pran_override [simp]: "pran (f \<oplus> g) = pran(g) \<union> pran(pdom(g) -\<lhd>\<^sub>p f)"
by (transfer, auto simp add: restrict_map_def dom_def map_add_def option.case_eq_if)
subsection \<open> Graph laws \<close>
lemma pfun_graph_inv [code_unfold]: "graph_pfun (pfun_graph f) = f"
by (transfer, simp add: mk_functional_fp)
lemma pfun_eq_graph: "f = g \<longleftrightarrow> pfun_graph f = pfun_graph g"
by (metis pfun_graph_inv)
lemma Dom_pfun_graph: "Domain (pfun_graph f) = pdom f"
by (transfer, simp add: dom_map_graph)
lemma Range_pfun_graph: "Range (pfun_graph f) = pran f"
by (transfer, auto simp add: ran_map_graph[THEN sym] ran_def)
lemma card_pfun_graph: "finite (pdom f) \<Longrightarrow> card (pfun_graph f) = card (pdom f)"
by (transfer, simp add: card_map_graph dom_map_graph finite_dom_graph)
lemma functional_pfun_graph [simp]: "functional (pfun_graph f)"
by (transfer, simp)
lemma pfun_graph_zero: "pfun_graph \<bottom> = {}"
by (transfer, simp add: map_graph_def)
lemma pfun_graph_pId_on: "pfun_graph (pId_on A) = Id_on A"
by (transfer, auto simp add: map_graph_def)
lemma pfun_graph_minus: "pfun_graph (f - g) = pfun_graph f - pfun_graph g"
by (transfer, simp add: map_graph_minus)
lemma pfun_graph_inter: "pfun_graph (f \<inter>\<^sub>p g) = pfun_graph f \<inter> pfun_graph g"
apply (transfer, auto simp add: map_graph_def)
apply (metis option.discI)+
done
lemma pfun_graph_domres: "pfun_graph (A \<lhd>\<^sub>p f) = (A \<lhd>\<^sub>r pfun_graph f)"
by (transfer, simp add: rel_domres_math_def map_graph_def restrict_map_def, metis option.simps(3))
lemma pfun_graph_override: "pfun_graph (f \<oplus> g) = pfun_graph f \<oplus> pfun_graph g"
by (transfer, auto simp add: map_add_def oplus_set_def rel_domres_def map_graph_def option.case_eq_if)
(metis option.collapse)+
lemma pfun_graph_update: "pfun_graph (f(k \<mapsto> v)\<^sub>p) = insert (k, v) ((- {k}) \<lhd>\<^sub>r pfun_graph f)"
by (transfer, simp add: map_graph_update)
lemma pfun_graph_comp: "pfun_graph (f \<circ>\<^sub>p g) = pfun_graph g O pfun_graph f"
by (transfer, simp add: map_graph_comp)
lemma comp_pfun_graph: "pfun_graph f O pfun_graph g = pfun_graph (g \<circ>\<^sub>p f)"
by (simp add: pfun_graph_comp)
lemma pfun_graph_pfun_inv: "pfun_inj f \<Longrightarrow> pfun_graph (pfun_inv f) = (pfun_graph f)\<inverse>"
by (transfer, simp add: map_graph_map_inv)
lemma pfun_graph_pabs: "pfun_graph (\<lambda> x \<in> A | P x \<bullet> f x) = {(k, v). k \<in> A \<and> P k \<and> v = f k}"
unfolding pabs_def by (transfer, auto simp add: map_graph_def restrict_map_def)
lemma pfun_graph_le_iff:
"pfun_graph f \<subseteq> pfun_graph g \<longleftrightarrow> f \<subseteq>\<^sub>p g"
by (simp add: inf.order_iff pfun_eq_graph pfun_graph_inter)
lemma pfun_member_iff [simp]: "(k, v) \<in> pfun_graph f \<longleftrightarrow> (k \<in> pdom(f) \<and> pfun_app f k = v)"
by (transfer, auto simp add: map_graph_def)
lemma pfun_graph_rres: "pfun_graph (f \<rhd>\<^sub>p A) = pfun_graph f \<rhd>\<^sub>r A"
by (transfer, auto simp add: map_graph_def rel_ranres_def ran_restrict_map_def)
subsection \<open> Graph Transfer Setup \<close>
definition cr_pfung :: "('a \<leftrightarrow> 'b) \<Rightarrow> 'a \<Zpfun> 'b \<Rightarrow> bool" where
"cr_pfung f g = (f = pfun_graph g)"
lemma Domainp_cr_pfung [transfer_domain_rule]: "Domainp cr_pfung = functional"
unfolding cr_pfung_def Domainp_iff[abs_def]
by (auto simp add: fun_eq_iff, metis graph_map_inv pfun_graph.abs_eq)
bundle pfun_graph_lifting
begin
unbundle lifting_syntax
lemma bi_unique_cr_pfung [transfer_rule]: "bi_unique cr_pfung"
unfolding cr_pfung_def bi_unique_def by (auto simp add: pfun_eq_graph)
lemma right_total_cr_pfung [transfer_rule]: "right_total cr_pfung"
unfolding cr_pfung_def right_total_def by simp
lemma cr_pfung_empty [transfer_rule]: "cr_pfung {} {}\<^sub>p"
unfolding cr_pfung_def by (simp add: pfun_graph_zero)
lemma cr_pfung_dom [transfer_rule]: "(cr_pfung ===> (=)) Domain pdom"
unfolding rel_fun_def cr_pfung_def by (simp add: Dom_pfun_graph)
lemma cr_pfung_ran [transfer_rule]: "(cr_pfung ===> (=)) Range pran"
unfolding rel_fun_def cr_pfung_def by (simp add: Range_pfun_graph)
lemma cr_pfung_id [transfer_rule]: "((=) ===> cr_pfung) Id_on pId_on"
unfolding rel_fun_def cr_pfung_def by (simp add: pfun_graph_pId_on)
lemma cr_pfung_ovrd [transfer_rule]: "(cr_pfung ===> cr_pfung ===> cr_pfung) (\<oplus>) (\<oplus>)"
unfolding rel_fun_def cr_pfung_def by (simp add: pfun_graph_override)
lemma cr_pfung_ovrd [transfer_rule]: "(cr_pfung ===> cr_pfung ===> cr_pfung) (O) (\<lambda> x y. y \<circ>\<^sub>p x)"
unfolding rel_fun_def cr_pfung_def by (simp add: pfun_graph_comp)
lemma cr_pfung_dres [transfer_rule]: "((=) ===> cr_pfung ===> cr_pfung) (\<lhd>\<^sub>r) (\<lhd>\<^sub>p)"
unfolding rel_fun_def cr_pfung_def by (simp add: pfun_graph_domres)
lemma cr_pfung_rres [transfer_rule]: "(cr_pfung ===> (=) ===> cr_pfung) (\<rhd>\<^sub>r) (\<rhd>\<^sub>p)"
unfolding rel_fun_def cr_pfung_def by (simp add: pfun_graph_rres)
lemma cr_pfung_le [transfer_rule]: "(cr_pfung ===> cr_pfung ===> (=)) (\<le>) (\<le>)"
unfolding rel_fun_def cr_pfung_def by (simp add: pfun_graph_le_iff)
lemma cr_pfung_update [transfer_rule]: "(cr_pfung ===> (=) ===> (=) ===> cr_pfung) (\<lambda> f k v. insert (k, v) ((- {k}) \<lhd>\<^sub>r f)) pfun_upd"
unfolding rel_fun_def cr_pfung_def by (simp add: pfun_graph_update)
end
subsection \<open> Partial Injections \<close>
lemma pfun_inj_empty [simp]: "pfun_inj {}\<^sub>p"
by (transfer, simp)
lemma pinj_pId_on [simp]: "pfun_inj (pId_on A)"
by (transfer, auto simp add: inj_on_def)
lemma pfun_inj_inv_inv: "pfun_inj f \<Longrightarrow> pfun_inv (pfun_inv f) = f"
by (transfer, simp)
lemma pfun_inj_inv: "pfun_inj f \<Longrightarrow> pfun_inj (pfun_inv f)"
by (transfer, simp add: inj_map_inv)
lemma f_pfun_inv_f_apply: "\<lbrakk> pfun_inj f; x \<in> pran f \<rbrakk> \<Longrightarrow> f(pfun_inv f(x)\<^sub>p)\<^sub>p = x"
by (transfer, auto simp add: ranI)
lemma pfun_inv_f_f_apply: "\<lbrakk> pfun_inj f; x \<in> pdom f \<rbrakk> \<Longrightarrow> pfun_inv f(f(x)\<^sub>p)\<^sub>p = x"
by (transfer, auto simp add: ranI)
lemma pfun_inj_upd: "\<lbrakk> pfun_inj f; v \<notin> pran f \<rbrakk> \<Longrightarrow> pfun_inj (f(k \<mapsto> v)\<^sub>p)"
by (transfer, auto, meson f_the_inv_into_f inj_on_fun_updI)
lemma pfun_inj_dres: "pfun_inj f \<Longrightarrow> pfun_inj (A \<lhd>\<^sub>p f)"
by (transfer, auto simp add: inj_on_def)
lemma pfun_inj_rres: "pfun_inj f \<Longrightarrow> pfun_inj (f \<rhd>\<^sub>p A)"
by (transfer, auto simp add: inj_on_def ran_restrict_map_def, metis domI option.simps(3))
lemma pfun_inj_comp: "\<lbrakk> pfun_inj f; pfun_inj g \<rbrakk> \<Longrightarrow> pfun_inj (f \<circ>\<^sub>p g)"
by (transfer, auto simp add: inj_on_def map_comp_def option.case_eq_if dom_def)
lemma pfun_inj_ovrd: "\<lbrakk> pfun_inj f; pfun_inj g; pran f \<inter> pran g = {} \<rbrakk> \<Longrightarrow> pfun_inj (f \<oplus> g)"
by (transfer, force simp add: inj_on_def map_add_def option.case_eq_if dom_def)
lemma pfun_inv_dres: "pfun_inj f \<Longrightarrow> pfun_inv (A \<lhd>\<^sub>p f) = (pfun_inv f) \<rhd>\<^sub>p A"
by (transfer, simp add: map_inv_dom_res)
lemma pfun_inv_rres: "pfun_inj f \<Longrightarrow> pfun_inv (f \<rhd>\<^sub>p A) = A \<lhd>\<^sub>p (pfun_inv f)"
by (transfer, simp add: map_inv_ran_res)
lemma pfun_inv_empty [simp]: "pfun_inv {}\<^sub>p = {}\<^sub>p"
by (transfer, simp)
lemma pdom_pfun_inv [simp]: "pdom (pfun_inv f) = pran f"
by (simp add: pran_rep_eq, transfer, simp)
lemma pfun_inv_add:
assumes "pfun_inj f" "pfun_inj g" "pran f \<inter> pran g = {}"
shows "pfun_inv (f \<oplus> g) = (pfun_inv f \<rhd>\<^sub>p (- pdom g)) \<oplus> pfun_inv g"
using assms by (simp add: pran_rep_eq, transfer, auto, meson map_inv_add)
lemma pfun_inv_upd:
assumes "pfun_inj f" "v \<notin> pran f"
shows "pfun_inv (f(k \<mapsto> v)\<^sub>p) = (pfun_inv ((- {k}) \<lhd>\<^sub>p f))(v \<mapsto> k)\<^sub>p"
using assms by (simp add: pran_rep_eq, transfer, meson map_inv_upd)
subsection \<open> Domain restriction laws \<close>
lemma pdom_res_zero [simp]: "A \<lhd>\<^sub>p {}\<^sub>p = {}\<^sub>p"
by (transfer, auto)
lemma pdom_res_empty [simp]:
"({} \<lhd>\<^sub>p f) = {}\<^sub>p"
by (transfer, auto)
lemma pdom_res_pdom [simp]:
"pdom(f) \<lhd>\<^sub>p f = f"
by (transfer, auto)
lemma pdom_res_UNIV [simp]: "UNIV \<lhd>\<^sub>p f = f"
by (transfer, auto)
lemma pdom_res_alt_def: "A \<lhd>\<^sub>p f = f \<circ>\<^sub>p pId_on A"
by (transfer, rule ext, auto simp add: restrict_map_def)
lemma pdom_res_upd_in [simp]:
"k \<in> A \<Longrightarrow> A \<lhd>\<^sub>p f(k \<mapsto> v)\<^sub>p = (A \<lhd>\<^sub>p f)(k \<mapsto> v)\<^sub>p"
by (transfer, auto)
lemma pdom_res_upd_out [simp]:
"k \<notin> A \<Longrightarrow> A \<lhd>\<^sub>p f(k \<mapsto> v)\<^sub>p = A \<lhd>\<^sub>p f"
by (transfer, auto)
lemma pfun_pdom_antires_upd [simp]:
"k \<in> A \<Longrightarrow> ((- A) \<lhd>\<^sub>p m)(k \<mapsto> v)\<^sub>p = ((- (A - {k})) \<lhd>\<^sub>p m)(k \<mapsto> v)\<^sub>p"
by (transfer, simp)
lemma pdom_antires_insert_notin [simp]:
"k \<notin> pdom(f) \<Longrightarrow> (- insert k A) \<lhd>\<^sub>p f = (- A) \<lhd>\<^sub>p f"
by (transfer, auto simp add: restrict_map_def)
lemma pdom_res_override [simp]: "A \<lhd>\<^sub>p (f \<oplus> g) = (A \<lhd>\<^sub>p f) \<oplus> (A \<lhd>\<^sub>p g)"
by (simp add: pdom_res_alt_def pfun_override_dist_comp)
lemma pdom_res_minus [simp]: "A \<lhd>\<^sub>p (f - g) = (A \<lhd>\<^sub>p f) - g"
by (transfer, auto simp add: map_minus_def restrict_map_def)
lemma pdom_res_swap: "(A \<lhd>\<^sub>p f) \<rhd>\<^sub>p B = A \<lhd>\<^sub>p (f \<rhd>\<^sub>p B)"
by (transfer, auto simp add: restrict_map_def ran_restrict_map_def)
lemma pdom_res_twice [simp]: "A \<lhd>\<^sub>p (B \<lhd>\<^sub>p f) = (A \<inter> B) \<lhd>\<^sub>p f"
by (transfer, auto simp add: Int_commute)
lemma pdom_res_comp [simp]: "A \<lhd>\<^sub>p (g \<circ>\<^sub>p f) = g \<circ>\<^sub>p (A \<lhd>\<^sub>p f)"
by (simp add: pdom_res_alt_def pfun_comp_assoc)
lemma pdom_res_apply [simp]:
"x \<in> A \<Longrightarrow> (A \<lhd>\<^sub>p f)(x)\<^sub>p = f(x)\<^sub>p"
by (transfer, auto)
lemma pdom_res_frame_update [simp]:
"\<lbrakk> x \<in> pdom(f); (-{x}) \<lhd>\<^sub>p f = (-{x}) \<lhd>\<^sub>p g \<rbrakk> \<Longrightarrow> g(x \<mapsto> f(x)\<^sub>p)\<^sub>p = f"
by (transfer, auto, metis fun_upd_triv fun_upd_upd restrict_complement_singleton_eq)
lemma pdres_rres_commute: "A \<lhd>\<^sub>p (P \<rhd>\<^sub>p B) = (A \<lhd>\<^sub>p P) \<rhd>\<^sub>p B"
by (transfer, simp add: map_dres_rres_commute)
lemma pdom_nres_disjoint: "pdom(f) \<inter> A = {} \<Longrightarrow> (- A) \<lhd>\<^sub>p f = f"
by (metis disjoint_eq_subset_Compl inf.absorb2 pdom_res_pdom pdom_res_twice)
lemma pranres_pdom [simp]: "pdom (f \<rhd>\<^sub>p A) \<lhd>\<^sub>p f = f \<rhd>\<^sub>p A"
by (transfer, auto simp add: restrict_map_def fun_eq_iff ran_restrict_map_def option.case_eq_if)
(metis not_None_eq)
lemma pdom_pranres [simp]: "pdom (f \<rhd>\<^sub>p A) \<subseteq> pdom f"
by (metis inf.absorb_iff1 inf.commute pdom_pdom_res pdom_res_pdom pdom_res_swap)
lemma pfun_split_domain: "A \<lhd>\<^sub>p f \<oplus> (- A) \<lhd>\<^sub>p f = f"
by (transfer, auto simp add: restrict_map_def map_add_def fun_eq_iff option.case_eq_if)
subsection \<open> Range restriction laws \<close>
lemma pran_res_UNIV [simp]: "f \<rhd>\<^sub>p UNIV = f"
by (transfer, simp add: ran_restrict_map_def)
lemma pran_res_empty [simp]: "f \<rhd>\<^sub>p {} = {}\<^sub>p"
by (transfer, auto simp add: ran_restrict_map_def)
lemma pran_res_zero [simp]: "{}\<^sub>p \<rhd>\<^sub>p A = {}\<^sub>p"
by (transfer, auto simp add: ran_restrict_map_def)
lemma pran_res_upd_1 [simp]: "v \<in> A \<Longrightarrow> f(x \<mapsto> v)\<^sub>p \<rhd>\<^sub>p A = (f \<rhd>\<^sub>p A)(x \<mapsto> v)\<^sub>p"
by (transfer, auto simp add: ran_restrict_map_def)
lemma pran_res_upd_2 [simp]: "v \<notin> A \<Longrightarrow> f(x \<mapsto> v)\<^sub>p \<rhd>\<^sub>p A = ((- {x}) \<lhd>\<^sub>p f) \<rhd>\<^sub>p A"
by (transfer, auto simp add: ran_restrict_map_def)
lemma pran_res_twice [simp]: "f \<rhd>\<^sub>p A \<rhd>\<^sub>p B = f \<rhd>\<^sub>p (A \<inter> B)"
by (transfer, simp)
lemma pran_res_alt_def: "f \<rhd>\<^sub>p A = pId_on A \<circ>\<^sub>p f"
by (transfer, rule ext, auto simp add: ran_restrict_map_def)
lemma pran_res_override: "(f \<oplus> g) \<rhd>\<^sub>p A \<subseteq>\<^sub>p (f \<rhd>\<^sub>p A) \<oplus> (g \<rhd>\<^sub>p A)"
apply (transfer, auto simp add: map_add_def ran_restrict_map_def map_le_def)
apply (rename_tac f g A a y x)
apply (case_tac "g a")
apply (auto)
done
lemma pcomp_ranres [simp]: "(f \<circ>\<^sub>p g) \<rhd>\<^sub>p A = (f \<rhd>\<^sub>p A) \<circ>\<^sub>p g"
by (simp add: pfun_comp_assoc pran_res_alt_def)
lemma pranres_le: "A \<subseteq> B \<Longrightarrow> f \<rhd>\<^sub>p A \<le> f \<rhd>\<^sub>p B"
by (simp add: pfun_graph_le_iff[THEN sym] pfun_graph_comp pfun_graph_rres relcomp_mono rel_ranres_le)
lemma pranres_neg_ran [simp]: "P \<rhd>\<^sub>p- pran P = {}\<^sub>p"
by (transfer, auto simp add: ran_restrict_map_def fun_eq_iff option.case_eq_if bind_eq_None_conv, meson option.exhaust_sel)
subsection \<open> Preimage Laws \<close>
lemma ppreimageI [intro!]: "\<lbrakk> x \<in> pdom(f); f(x)\<^sub>p \<in> A \<rbrakk> \<Longrightarrow> x \<in> pdom (f \<rhd>\<^sub>p A)"
by (metis (full_types) insertI1 pdom_upd pfun_upd_ext pran_res_upd_1)
lemma ppreimageD: "x \<in> pdom (f \<rhd>\<^sub>p A) \<Longrightarrow> \<exists> y \<in> A. f(x)\<^sub>p = y"
by (transfer, auto simp add: ran_restrict_map_def)
lemma ppreimageE [elim!]: "\<lbrakk> x \<in> pdom (f \<rhd>\<^sub>p A); \<And> y. \<lbrakk> x \<in> pdom(f); y \<in> A; f(x)\<^sub>p = y \<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
by (metis (no_types) pdom_pranres ppreimageD subsetD)
lemma mem_pimage_iff: "x \<in> pran (A \<lhd>\<^sub>p f) \<longleftrightarrow> (\<exists> y \<in> A \<inter> pdom(f). f(y)\<^sub>p = x)"
by (auto simp add: pran_pdom)
lemma ppreimage_inter [simp]: "pdom (f \<rhd>\<^sub>p (A \<inter> B)) = pdom (f \<rhd>\<^sub>p A) \<inter> pdom (f \<rhd>\<^sub>p B)"
by fastforce
subsection \<open> Composition \<close>
lemma pcomp_apply [simp]: "\<lbrakk> x \<in> pdom(g) \<rbrakk> \<Longrightarrow> (f \<circ>\<^sub>p g)(x)\<^sub>p = f(g(x)\<^sub>p)\<^sub>p"
by (transfer, auto)
lemma pcomp_mono: "\<lbrakk> f \<le> f'; g \<le> g' \<rbrakk> \<Longrightarrow> f \<circ>\<^sub>p g \<le> f' \<circ>\<^sub>p g'"
by (simp add: pfun_graph_le_iff[THEN sym] pfun_graph_comp relcomp_mono)
lemma pdom_UNIV_comp: "pdom f = UNIV \<Longrightarrow> pdom (f \<circ>\<^sub>p g) = pdom g"
by simp
subsection \<open> Entries \<close>
lemma pfun_entries_empty [simp]: "pfun_entries {} f = {}\<^sub>p"
by (transfer, simp)
lemma pdom_pfun_entries [simp]: "pdom (pfun_entries A f) = A"
by (transfer, auto)
lemma pran_pfun_entries [simp]: "pran (pfun_entries A f) = f ` A"
by (transfer, simp add: ran_def, auto)
lemma pfun_entries_apply_1 [simp]:
"x \<in> d \<Longrightarrow> (pfun_entries d f)(x)\<^sub>p = f x"
by (transfer, auto)
lemma pfun_entries_apply_2 [simp]:
"x \<notin> d \<Longrightarrow> (pfun_entries d f)(x)\<^sub>p = undefined"
by (transfer, auto)
lemma pdom_res_entries: "A \<lhd>\<^sub>p pfun_entries B f = pfun_entries (A \<inter> B) f"
by (transfer, auto simp add: fun_eq_iff restrict_map_def)
lemma pfuse_empty [simp]: "pfuse {}\<^sub>p g = {}\<^sub>p"
by (simp add: pfuse_def)
lemma pfuse_app [simp]:
"\<lbrakk> e \<in> pdom F; e \<in> pdom G \<rbrakk> \<Longrightarrow> (pfuse F G)(e)\<^sub>p = (F(e)\<^sub>p, G(e)\<^sub>p)"
by (metis (no_types, lifting) IntI pfun_entries_apply_1 pfuse_def)
lemma pdom_pfuse [simp]: "pdom (pfuse f g) = pdom(f) \<inter> pdom(g)"
by (auto simp add: pfuse_def)
lemma pfuse_upd:
"pfuse (f(k \<mapsto> v)\<^sub>p) g =
(if k \<in> pdom g then (pfuse ((-{k}) \<lhd>\<^sub>p f) g)(k \<mapsto> (v, pfun_app g k))\<^sub>p else pfuse f g)"
by (simp add: pfuse_def, transfer, auto simp add: fun_eq_iff)
subsection \<open> Lambda abstraction \<close>
lemma pabs_cong:
assumes "A = B" "\<And> x. x \<in> A \<Longrightarrow> P(x) = Q(x)" "\<And> x. \<lbrakk> x \<in> A; P x \<rbrakk> \<Longrightarrow> F(x) = G(x)"
shows "(\<lambda> x \<in> A | P x \<bullet> F(x)) = (\<lambda> x \<in> B | Q x \<bullet> G(x))"
using assms unfolding pabs_def
by (transfer, auto simp add: restrict_map_def fun_eq_iff)
lemma pabs_apply [simp]: "\<lbrakk> y \<in> A; P y \<rbrakk> \<Longrightarrow> (\<lambda> x \<in> A | P x \<bullet> f x) (y)\<^sub>p = f y"
by (simp add: pabs_def)
lemma pdom_pabs [simp]: "pdom (\<lambda> x \<in> A | P x \<bullet> f x) = A \<inter> Collect P"
by (simp add: pabs_def)
lemma pran_pabs [simp]: "pran (\<lambda> x \<in> A | P x \<bullet> f x) = {f x | x. x \<in> A \<and> P x}"
unfolding pabs_def
by (transfer, auto simp add: ran_def restrict_map_def)
lemma pabs_eta [simp]: "(\<lambda> x \<in> pdom(f) \<bullet> f(x)\<^sub>p) = f"
by (simp add: pabs_def, transfer, auto simp add: fun_eq_iff domIff restrict_map_def)
lemma pabs_id [simp]: "(\<lambda> x \<in> A | P x \<bullet> x) = pId_on {x\<in>A. P x}"
unfolding pabs_def by (transfer, simp add: restrict_map_def)
lemma pfun_entries_pabs: "pfun_entries A f = (\<lambda> x \<in> A \<bullet> f x)"
by (simp add: pabs_def, transfer, auto)
lemma pabs_empty [simp]: "(\<lambda> x\<in>{} \<bullet> f(x)) = {}\<^sub>p"
by (simp add: pabs_def)
lemma pabs_insert_maplet: "(\<lambda> x\<in>insert y A \<bullet> f(x)) = (\<lambda> x\<in>A \<bullet> f(x)) \<oplus> {y \<mapsto> f(y)}\<^sub>p"
by (simp add: pabs_def, transfer, auto simp add: restrict_map_insert)
text \<open> This rule can perhaps be simplified \<close>
lemma pcomp_pabs:
"(\<lambda> x \<in> A | P x \<bullet> f x) \<circ>\<^sub>p (\<lambda> x \<in> B | Q x \<bullet> g x)
= (\<lambda> x \<in> pdom (pabs B Q g \<rhd>\<^sub>p (A \<inter> Collect P)) \<bullet> (f (g x)))"
apply (subst pabs_eta[THEN sym, of "(\<lambda> x \<in> A | P x \<bullet> f x) \<circ>\<^sub>p (\<lambda> x \<in> B | Q x \<bullet> g x)"])
apply (simp)
apply (simp add: pabs_def)
apply (transfer, auto simp add: restrict_map_def map_comp_def ran_restrict_map_def fun_eq_iff)
done
lemma pabs_rres [simp]: "pabs A P f \<rhd>\<^sub>p B = pabs A (\<lambda> x. P x \<and> f x \<in> B) f"
by (simp add: pabs_def, transfer, auto simp add: ran_restrict_map_def restrict_map_def)
(* This law should be generalised *)
lemma pabs_simple_comp [simp]: "(\<lambda> x \<bullet> f x) \<circ>\<^sub>p g(k \<mapsto> v)\<^sub>p = ((\<lambda> x \<bullet> f x) \<circ>\<^sub>p g)(k \<mapsto> f v)\<^sub>p"
by (simp add: pabs_def, transfer, auto)
lemma pabs_comp: "(\<lambda> x \<in> A \<bullet> f x) \<circ>\<^sub>p g = (\<lambda> x \<in> pdom (g \<rhd>\<^sub>p A) \<bullet> f (pfun_app g x))"
by (metis pabs_eta pcomp_pabs pdom_pId_on pdom_pabs)
lemma map_pfun_pabs [simp]: "map_pfun f (\<lambda> x\<in>A | B(x) \<bullet> g(x)) = (\<lambda> x\<in>A | B(x) \<bullet> f(g(x)))"
by (simp add: pfun_eq_iff)
subsection \<open> Summation \<close>
definition pfun_sum :: "('k, 'v::comm_monoid_add) pfun \<Rightarrow> 'v" where
"pfun_sum f = sum (pfun_app f) (pdom f)"
lemma pfun_sum_empty [simp]: "pfun_sum {}\<^sub>p = 0"
by (simp add: pfun_sum_def)
lemma pfun_sum_upd_1:
assumes "finite(pdom(m))" "k \<notin> pdom(m)"
shows "pfun_sum (m(k \<mapsto> v)\<^sub>p) = pfun_sum m + v"
proof -
from assms(2) have "(\<Sum>x\<in>pdom m. if k = x then v else m(x)\<^sub>p) = sum (pfun_app m) (pdom m)"
by (auto intro!: sum.cong)
thus ?thesis
by (simp_all add: pfun_sum_def assms add.commute cong: sum.cong)
qed
lemma pfun_sums_upd_2:
assumes "finite(pdom(m))"
shows "pfun_sum (m(k \<mapsto> v)\<^sub>p) = pfun_sum ((- {k}) \<lhd>\<^sub>p m) + v"
proof (cases "k \<notin> pdom(m)")
case True
then show ?thesis
by (simp add: pfun_sum_upd_1 assms)
next
case False
then show ?thesis