--- a/src/HOL/Record.thy Tue Nov 10 16:11:39 2009 +0100
+++ b/src/HOL/Record.thy Tue Nov 10 16:11:43 2009 +0100
@@ -3,15 +3,386 @@
Author: Markus Wenzel, TU Muenchen
Author: Norbert Schirmer, TU Muenchen
Author: Thomas Sewell, NICTA
+ Author: Florian Haftmann, TU Muenchen
*)
header {* Extensible records with structural subtyping *}
theory Record
-imports Product_Type
+imports Datatype
uses ("Tools/record.ML")
begin
+subsection {* Introduction *}
+
+text {*
+ Records are isomorphic to compound tuple types. To implement
+ efficient records, we make this isomorphism explicit. Consider the
+ record access/update simplification @{text "alpha (beta_update f
+ rec) = alpha rec"} for distinct fields alpha and beta of some record
+ rec with n fields. There are @{text "n ^ 2"} such theorems, which
+ prohibits storage of all of them for large n. The rules can be
+ proved on the fly by case decomposition and simplification in O(n)
+ time. By creating O(n) isomorphic-tuple types while defining the
+ record, however, we can prove the access/update simplification in
+ @{text "O(log(n)^2)"} time.
+
+ The O(n) cost of case decomposition is not because O(n) steps are
+ taken, but rather because the resulting rule must contain O(n) new
+ variables and an O(n) size concrete record construction. To sidestep
+ this cost, we would like to avoid case decomposition in proving
+ access/update theorems.
+
+ Record types are defined as isomorphic to tuple types. For instance,
+ a record type with fields @{text "'a"}, @{text "'b"}, @{text "'c"}
+ and @{text "'d"} might be introduced as isomorphic to @{text "'a \<times>
+ ('b \<times> ('c \<times> 'd))"}. If we balance the tuple tree to @{text "('a \<times>
+ 'b) \<times> ('c \<times> 'd)"} then accessors can be defined by converting to the
+ underlying type then using O(log(n)) fst or snd operations.
+ Updators can be defined similarly, if we introduce a @{text
+ "fst_update"} and @{text "snd_update"} function. Furthermore, we can
+ prove the access/update theorem in O(log(n)) steps by using simple
+ rewrites on fst, snd, @{text "fst_update"} and @{text "snd_update"}.
+
+ The catch is that, although O(log(n)) steps were taken, the
+ underlying type we converted to is a tuple tree of size
+ O(n). Processing this term type wastes performance. We avoid this
+ for large n by taking each subtree of size K and defining a new type
+ isomorphic to that tuple subtree. A record can now be defined as
+ isomorphic to a tuple tree of these O(n/K) new types, or, if @{text
+ "n > K*K"}, we can repeat the process, until the record can be
+ defined in terms of a tuple tree of complexity less than the
+ constant K.
+
+ If we prove the access/update theorem on this type with the
+ analagous steps to the tuple tree, we consume @{text "O(log(n)^2)"}
+ time as the intermediate terms are @{text "O(log(n))"} in size and
+ the types needed have size bounded by K. To enable this analagous
+ traversal, we define the functions seen below: @{text
+ "istuple_fst"}, @{text "istuple_snd"}, @{text "istuple_fst_update"}
+ and @{text "istuple_snd_update"}. These functions generalise tuple
+ operations by taking a parameter that encapsulates a tuple
+ isomorphism. The rewrites needed on these functions now need an
+ additional assumption which is that the isomorphism works.
+
+ These rewrites are typically used in a structured way. They are here
+ presented as the introduction rule @{text "isomorphic_tuple.intros"}
+ rather than as a rewrite rule set. The introduction form is an
+ optimisation, as net matching can be performed at one term location
+ for each step rather than the simplifier searching the term for
+ possible pattern matches. The rule set is used as it is viewed
+ outside the locale, with the locale assumption (that the isomorphism
+ is valid) left as a rule assumption. All rules are structured to aid
+ net matching, using either a point-free form or an encapsulating
+ predicate.
+*}
+
+subsection {* Operators and lemmas for types isomorphic to tuples *}
+
+datatype ('a, 'b, 'c) tuple_isomorphism = TupleIsomorphism "'a \<Rightarrow> 'b \<times> 'c" "'b \<times> 'c \<Rightarrow> 'a"
+
+primrec repr :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'c" where
+ "repr (TupleIsomorphism r a) = r"
+
+primrec abst :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a" where
+ "abst (TupleIsomorphism r a) = a"
+
+definition istuple_fst :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'a \<Rightarrow> 'b" where
+ "istuple_fst isom = fst \<circ> repr isom"
+
+definition istuple_snd :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'a \<Rightarrow> 'c" where
+ "istuple_snd isom = snd \<circ> repr isom"
+
+definition istuple_fst_update :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'a)" where
+ "istuple_fst_update isom f = abst isom \<circ> apfst f \<circ> repr isom"
+
+definition istuple_snd_update :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> ('c \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'a)" where
+ "istuple_snd_update isom f = abst isom \<circ> apsnd f \<circ> repr isom"
+
+definition istuple_cons :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a" where
+ "istuple_cons isom = curry (abst isom)"
+
+
+subsection {* Logical infrastructure for records *}
+
+definition istuple_surjective_proof_assist :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
+ "istuple_surjective_proof_assist x y f \<longleftrightarrow> f x = y"
+
+definition istuple_update_accessor_cong_assist :: "(('b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'a)) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
+ "istuple_update_accessor_cong_assist upd acc \<longleftrightarrow>
+ (\<forall>f v. upd (\<lambda>x. f (acc v)) v = upd f v) \<and> (\<forall>v. upd id v = v)"
+
+definition istuple_update_accessor_eq_assist :: "(('b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'a)) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ "istuple_update_accessor_eq_assist upd acc v f v' x \<longleftrightarrow>
+ upd f v = v' \<and> acc v = x \<and> istuple_update_accessor_cong_assist upd acc"
+
+lemma update_accessor_congruence_foldE:
+ assumes uac: "istuple_update_accessor_cong_assist upd acc"
+ and r: "r = r'" and v: "acc r' = v'"
+ and f: "\<And>v. v' = v \<Longrightarrow> f v = f' v"
+ shows "upd f r = upd f' r'"
+ using uac r v [symmetric]
+ apply (subgoal_tac "upd (\<lambda>x. f (acc r')) r' = upd (\<lambda>x. f' (acc r')) r'")
+ apply (simp add: istuple_update_accessor_cong_assist_def)
+ apply (simp add: f)
+ done
+
+lemma update_accessor_congruence_unfoldE:
+ "istuple_update_accessor_cong_assist upd acc \<Longrightarrow> r = r' \<Longrightarrow> acc r' = v' \<Longrightarrow> (\<And>v. v = v' \<Longrightarrow> f v = f' v)
+ \<Longrightarrow> upd f r = upd f' r'"
+ apply (erule(2) update_accessor_congruence_foldE)
+ apply simp
+ done
+
+lemma istuple_update_accessor_cong_assist_id:
+ "istuple_update_accessor_cong_assist upd acc \<Longrightarrow> upd id = id"
+ by rule (simp add: istuple_update_accessor_cong_assist_def)
+
+lemma update_accessor_noopE:
+ assumes uac: "istuple_update_accessor_cong_assist upd acc"
+ and acc: "f (acc x) = acc x"
+ shows "upd f x = x"
+using uac by (simp add: acc istuple_update_accessor_cong_assist_id [OF uac, unfolded id_def]
+ cong: update_accessor_congruence_unfoldE [OF uac])
+
+lemma update_accessor_noop_compE:
+ assumes uac: "istuple_update_accessor_cong_assist upd acc"
+ assumes acc: "f (acc x) = acc x"
+ shows "upd (g \<circ> f) x = upd g x"
+ by (simp add: acc cong: update_accessor_congruence_unfoldE[OF uac])
+
+lemma update_accessor_cong_assist_idI:
+ "istuple_update_accessor_cong_assist id id"
+ by (simp add: istuple_update_accessor_cong_assist_def)
+
+lemma update_accessor_cong_assist_triv:
+ "istuple_update_accessor_cong_assist upd acc \<Longrightarrow> istuple_update_accessor_cong_assist upd acc"
+ by assumption
+
+lemma update_accessor_accessor_eqE:
+ "istuple_update_accessor_eq_assist upd acc v f v' x \<Longrightarrow> acc v = x"
+ by (simp add: istuple_update_accessor_eq_assist_def)
+
+lemma update_accessor_updator_eqE:
+ "istuple_update_accessor_eq_assist upd acc v f v' x \<Longrightarrow> upd f v = v'"
+ by (simp add: istuple_update_accessor_eq_assist_def)
+
+lemma istuple_update_accessor_eq_assist_idI:
+ "v' = f v \<Longrightarrow> istuple_update_accessor_eq_assist id id v f v' v"
+ by (simp add: istuple_update_accessor_eq_assist_def update_accessor_cong_assist_idI)
+
+lemma istuple_update_accessor_eq_assist_triv:
+ "istuple_update_accessor_eq_assist upd acc v f v' x \<Longrightarrow> istuple_update_accessor_eq_assist upd acc v f v' x"
+ by assumption
+
+lemma istuple_update_accessor_cong_from_eq:
+ "istuple_update_accessor_eq_assist upd acc v f v' x \<Longrightarrow> istuple_update_accessor_cong_assist upd acc"
+ by (simp add: istuple_update_accessor_eq_assist_def)
+
+lemma o_eq_dest:
+ "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
+ apply (clarsimp simp: o_def)
+ apply (erule fun_cong)
+ done
+
+lemma o_eq_elim:
+ "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
+ apply (erule meta_mp)
+ apply (erule o_eq_dest)
+ done
+
+lemma istuple_surjective_proof_assistI:
+ "f x = y \<Longrightarrow> istuple_surjective_proof_assist x y f"
+ by (simp add: istuple_surjective_proof_assist_def)
+
+lemma istuple_surjective_proof_assist_idE:
+ "istuple_surjective_proof_assist x y id \<Longrightarrow> x = y"
+ by (simp add: istuple_surjective_proof_assist_def)
+
+locale isomorphic_tuple =
+ fixes isom :: "('a, 'b, 'c) tuple_isomorphism"
+ and repr and abst
+ defines "repr \<equiv> Record.repr isom"
+ defines "abst \<equiv> Record.abst isom"
+ assumes repr_inv: "\<And>x. abst (repr x) = x"
+ assumes abst_inv: "\<And>y. repr (abst y) = y"
+begin
+
+lemma repr_inj:
+ "repr x = repr y \<longleftrightarrow> x = y"
+ apply (rule iffI, simp_all)
+ apply (drule_tac f=abst in arg_cong, simp add: repr_inv)
+ done
+
+lemma abst_inj:
+ "abst x = abst y \<longleftrightarrow> x = y"
+ apply (rule iffI, simp_all)
+ apply (drule_tac f=repr in arg_cong, simp add: abst_inv)
+ done
+
+lemmas simps = Let_def repr_inv abst_inv repr_inj abst_inj repr_def [symmetric] abst_def [symmetric]
+
+lemma istuple_access_update_fst_fst:
+ "f o h g = j o f \<Longrightarrow>
+ (f o istuple_fst isom) o (istuple_fst_update isom o h) g
+ = j o (f o istuple_fst isom)"
+ by (clarsimp simp: istuple_fst_update_def istuple_fst_def simps
+ intro!: ext elim!: o_eq_elim)
+
+lemma istuple_access_update_snd_snd:
+ "f o h g = j o f \<Longrightarrow>
+ (f o istuple_snd isom) o (istuple_snd_update isom o h) g
+ = j o (f o istuple_snd isom)"
+ by (clarsimp simp: istuple_snd_update_def istuple_snd_def simps
+ intro!: ext elim!: o_eq_elim)
+
+lemma istuple_access_update_fst_snd:
+ "(f o istuple_fst isom) o (istuple_snd_update isom o h) g
+ = id o (f o istuple_fst isom)"
+ by (clarsimp simp: istuple_snd_update_def istuple_fst_def simps
+ intro!: ext elim!: o_eq_elim)
+
+lemma istuple_access_update_snd_fst:
+ "(f o istuple_snd isom) o (istuple_fst_update isom o h) g
+ = id o (f o istuple_snd isom)"
+ by (clarsimp simp: istuple_fst_update_def istuple_snd_def simps
+ intro!: ext elim!: o_eq_elim)
+
+lemma istuple_update_swap_fst_fst:
+ "h f o j g = j g o h f \<Longrightarrow>
+ (istuple_fst_update isom o h) f o (istuple_fst_update isom o j) g
+ = (istuple_fst_update isom o j) g o (istuple_fst_update isom o h) f"
+ by (clarsimp simp: istuple_fst_update_def simps apfst_compose intro!: ext)
+
+lemma istuple_update_swap_snd_snd:
+ "h f o j g = j g o h f \<Longrightarrow>
+ (istuple_snd_update isom o h) f o (istuple_snd_update isom o j) g
+ = (istuple_snd_update isom o j) g o (istuple_snd_update isom o h) f"
+ by (clarsimp simp: istuple_snd_update_def simps apsnd_compose intro!: ext)
+
+lemma istuple_update_swap_fst_snd:
+ "(istuple_snd_update isom o h) f o (istuple_fst_update isom o j) g
+ = (istuple_fst_update isom o j) g o (istuple_snd_update isom o h) f"
+ by (clarsimp simp: istuple_fst_update_def istuple_snd_update_def simps intro!: ext)
+
+lemma istuple_update_swap_snd_fst:
+ "(istuple_fst_update isom o h) f o (istuple_snd_update isom o j) g
+ = (istuple_snd_update isom o j) g o (istuple_fst_update isom o h) f"
+ by (clarsimp simp: istuple_fst_update_def istuple_snd_update_def simps intro!: ext)
+
+lemma istuple_update_compose_fst_fst:
+ "h f o j g = k (f o g) \<Longrightarrow>
+ (istuple_fst_update isom o h) f o (istuple_fst_update isom o j) g
+ = (istuple_fst_update isom o k) (f o g)"
+ by (clarsimp simp: istuple_fst_update_def simps apfst_compose intro!: ext)
+
+lemma istuple_update_compose_snd_snd:
+ "h f o j g = k (f o g) \<Longrightarrow>
+ (istuple_snd_update isom o h) f o (istuple_snd_update isom o j) g
+ = (istuple_snd_update isom o k) (f o g)"
+ by (clarsimp simp: istuple_snd_update_def simps apsnd_compose intro!: ext)
+
+lemma istuple_surjective_proof_assist_step:
+ "istuple_surjective_proof_assist v a (istuple_fst isom o f) \<Longrightarrow>
+ istuple_surjective_proof_assist v b (istuple_snd isom o f)
+ \<Longrightarrow> istuple_surjective_proof_assist v (istuple_cons isom a b) f"
+ by (clarsimp simp: istuple_surjective_proof_assist_def simps
+ istuple_fst_def istuple_snd_def istuple_cons_def)
+
+lemma istuple_fst_update_accessor_cong_assist:
+ assumes "istuple_update_accessor_cong_assist f g"
+ shows "istuple_update_accessor_cong_assist (istuple_fst_update isom o f) (g o istuple_fst isom)"
+proof -
+ from assms have "f id = id" by (rule istuple_update_accessor_cong_assist_id)
+ with assms show ?thesis by (clarsimp simp: istuple_update_accessor_cong_assist_def simps
+ istuple_fst_update_def istuple_fst_def)
+qed
+
+lemma istuple_snd_update_accessor_cong_assist:
+ assumes "istuple_update_accessor_cong_assist f g"
+ shows "istuple_update_accessor_cong_assist (istuple_snd_update isom o f) (g o istuple_snd isom)"
+proof -
+ from assms have "f id = id" by (rule istuple_update_accessor_cong_assist_id)
+ with assms show ?thesis by (clarsimp simp: istuple_update_accessor_cong_assist_def simps
+ istuple_snd_update_def istuple_snd_def)
+qed
+
+lemma istuple_fst_update_accessor_eq_assist:
+ assumes "istuple_update_accessor_eq_assist f g a u a' v"
+ shows "istuple_update_accessor_eq_assist (istuple_fst_update isom o f) (g o istuple_fst isom)
+ (istuple_cons isom a b) u (istuple_cons isom a' b) v"
+proof -
+ from assms have "f id = id"
+ by (auto simp add: istuple_update_accessor_eq_assist_def intro: istuple_update_accessor_cong_assist_id)
+ with assms show ?thesis by (clarsimp simp: istuple_update_accessor_eq_assist_def
+ istuple_fst_update_def istuple_fst_def istuple_update_accessor_cong_assist_def istuple_cons_def simps)
+qed
+
+lemma istuple_snd_update_accessor_eq_assist:
+ assumes "istuple_update_accessor_eq_assist f g b u b' v"
+ shows "istuple_update_accessor_eq_assist (istuple_snd_update isom o f) (g o istuple_snd isom)
+ (istuple_cons isom a b) u (istuple_cons isom a b') v"
+proof -
+ from assms have "f id = id"
+ by (auto simp add: istuple_update_accessor_eq_assist_def intro: istuple_update_accessor_cong_assist_id)
+ with assms show ?thesis by (clarsimp simp: istuple_update_accessor_eq_assist_def
+ istuple_snd_update_def istuple_snd_def istuple_update_accessor_cong_assist_def istuple_cons_def simps)
+qed
+
+lemma istuple_cons_conj_eqI:
+ "a = c \<and> b = d \<and> P \<longleftrightarrow> Q \<Longrightarrow>
+ istuple_cons isom a b = istuple_cons isom c d \<and> P \<longleftrightarrow> Q"
+ by (clarsimp simp: istuple_cons_def simps)
+
+lemmas intros =
+ istuple_access_update_fst_fst
+ istuple_access_update_snd_snd
+ istuple_access_update_fst_snd
+ istuple_access_update_snd_fst
+ istuple_update_swap_fst_fst
+ istuple_update_swap_snd_snd
+ istuple_update_swap_fst_snd
+ istuple_update_swap_snd_fst
+ istuple_update_compose_fst_fst
+ istuple_update_compose_snd_snd
+ istuple_surjective_proof_assist_step
+ istuple_fst_update_accessor_eq_assist
+ istuple_snd_update_accessor_eq_assist
+ istuple_fst_update_accessor_cong_assist
+ istuple_snd_update_accessor_cong_assist
+ istuple_cons_conj_eqI
+
+end
+
+lemma isomorphic_tuple_intro:
+ fixes repr abst
+ assumes repr_inj: "\<And>x y. repr x = repr y \<longleftrightarrow> x = y"
+ and abst_inv: "\<And>z. repr (abst z) = z"
+ assumes v: "v \<equiv> TupleIsomorphism repr abst"
+ shows "isomorphic_tuple v"
+ apply (rule isomorphic_tuple.intro)
+ apply (simp_all add: abst_inv v)
+ apply (cut_tac x="abst (repr x)" and y="x" in repr_inj)
+ apply (simp add: abst_inv)
+ done
+
+definition
+ "tuple_istuple \<equiv> TupleIsomorphism id id"
+
+lemma tuple_istuple:
+ "isomorphic_tuple tuple_istuple"
+ by (simp add: isomorphic_tuple_intro [OF _ _ reflexive] tuple_istuple_def)
+
+lemma refl_conj_eq:
+ "Q = R \<Longrightarrow> P \<and> Q \<longleftrightarrow> P \<and> R"
+ by simp
+
+lemma istuple_UNIV_I: "x \<in> UNIV \<equiv> True"
+ by simp
+
+lemma istuple_True_simp: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
+ by simp
+
lemma prop_subst: "s = t \<Longrightarrow> PROP P t \<Longrightarrow> PROP P s"
by simp
@@ -22,14 +393,11 @@
"a o b = c \<Longrightarrow> a (b v) = c v"
by clarsimp
-lemma id_o_refl:
- "id o f = f o id"
- by simp
-
lemma o_eq_id_dest:
"a o b = id o c \<Longrightarrow> a (b v) = c v"
by clarsimp
+
subsection {* Concrete record syntax *}
nonterminals
@@ -63,400 +431,15 @@
"_record_scheme" :: "[fields, 'a] => 'a" ("(3\<lparr>_,/ (2\<dots> =/ _)\<rparr>)")
"_record_update" :: "['a, updates] => 'b" ("_/(3\<lparr>_\<rparr>)" [900,0] 900)
-subsection {* Operators and lemmas for types isomorphic to tuples *}
-text {*
-Records are isomorphic to compound tuple types. To implement efficient
-records, we make this isomorphism explicit. Consider the record
-access/update simplification @{text "alpha (beta_update f rec) = alpha rec"} for
-distinct fields alpha and beta of some record rec with n fields. There
-are @{text "n ^ 2"} such theorems, which prohibits storage of all of them for
-large n. The rules can be proved on the fly by case decomposition and
-simplification in O(n) time. By creating O(n) isomorphic-tuple types
-while defining the record, however, we can prove the access/update
-simplification in @{text "O(log(n)^2)"} time.
-
-The O(n) cost of case decomposition is not because O(n) steps are taken,
-but rather because the resulting rule must contain O(n) new variables and
-an O(n) size concrete record construction. To sidestep this cost, we would
-like to avoid case decomposition in proving access/update theorems.
-
-Record types are defined as isomorphic to tuple types. For instance, a
-record type with fields 'a, 'b, 'c and 'd might be introduced as
-isomorphic to @{text "'a \<times> ('b \<times> ('c \<times> 'd))"}. If we balance the tuple tree to
-@{text "('a \<times> 'b) \<times> ('c \<times> 'd)"} then accessors can be defined by converting to
-the underlying type then using O(log(n)) fst or snd operations.
-Updators can be defined similarly, if we introduce a @{text "fst_update"} and
-@{text "snd_update"} function. Furthermore, we can prove the access/update
-theorem in O(log(n)) steps by using simple rewrites on fst, snd,
-@{text "fst_update"} and @{text "snd_update"}.
-
-The catch is that, although O(log(n)) steps were taken, the underlying
-type we converted to is a tuple tree of size O(n). Processing this term
-type wastes performance. We avoid this for large n by taking each
-subtree of size K and defining a new type isomorphic to that tuple
-subtree. A record can now be defined as isomorphic to a tuple tree
-of these O(n/K) new types, or, if @{text "n > K*K"}, we can repeat the process,
-until the record can be defined in terms of a tuple tree of complexity
-less than the constant K.
-
-If we prove the access/update theorem on this type with the analagous
-steps to the tuple tree, we consume @{text "O(log(n)^2)"} time as the intermediate
-terms are O(log(n)) in size and the types needed have size bounded by K.
-To enable this analagous traversal, we define the functions seen below:
-@{text "istuple_fst"}, @{text "istuple_snd"}, @{text "istuple_fst_update"}
-and @{text "istuple_snd_update"}. These functions generalise tuple
-operations by taking a parameter that encapsulates a tuple isomorphism.
-The rewrites needed on these functions now need an additional assumption
-which is that the isomorphism works.
-
-These rewrites are typically used in a structured way. They are here
-presented as the introduction rule @{text "isomorphic_tuple.intros"} rather than
-as a rewrite rule set. The introduction form is an optimisation, as net
-matching can be performed at one term location for each step rather than
-the simplifier searching the term for possible pattern matches. The rule
-set is used as it is viewed outside the locale, with the locale assumption
-(that the isomorphism is valid) left as a rule assumption. All rules are
-structured to aid net matching, using either a point-free form or an
-encapsulating predicate.
-*}
-
-typedef ('a, 'b, 'c) tuple_isomorphism
- = "UNIV :: (('a \<Rightarrow> ('b \<times> 'c)) \<times> (('b \<times> 'c) \<Rightarrow> 'a)) set"
- by simp
-
-definition
- "TupleIsomorphism repr abst = Abs_tuple_isomorphism (repr, abst)"
-
-definition
- istuple_fst :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'a \<Rightarrow> 'b"
-where
- "istuple_fst isom \<equiv> let (repr, abst) = Rep_tuple_isomorphism isom in fst \<circ> repr"
-
-definition
- istuple_snd :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'a \<Rightarrow> 'c"
-where
- "istuple_snd isom \<equiv> let (repr, abst) = Rep_tuple_isomorphism isom in snd \<circ> repr"
-
-definition
- istuple_fst_update :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'a)"
-where
- "istuple_fst_update isom \<equiv>
- let (repr, abst) = Rep_tuple_isomorphism isom in
- (\<lambda>f v. abst (f (fst (repr v)), snd (repr v)))"
-
-definition
- istuple_snd_update :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> ('c \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'a)"
-where
- "istuple_snd_update isom \<equiv>
- let (repr, abst) = Rep_tuple_isomorphism isom in
- (\<lambda>f v. abst (fst (repr v), f (snd (repr v))))"
-
-definition
- istuple_cons :: "('a, 'b, 'c) tuple_isomorphism \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"
-where
- "istuple_cons isom \<equiv> let (repr, abst) = Rep_tuple_isomorphism isom in curry abst"
-
-text {*
-These predicates are used in the introduction rule set to constrain
-matching appropriately. The elimination rules for them produce the
-desired theorems once they are proven. The final introduction rules are
-used when no further rules from the introduction rule set can apply.
-*}
-
-definition
- istuple_surjective_proof_assist :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
-where
- "istuple_surjective_proof_assist x y f \<equiv> (f x = y)"
-
-definition
- istuple_update_accessor_cong_assist :: "(('b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'a))
- \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
-where
- "istuple_update_accessor_cong_assist upd acc
- \<equiv> (\<forall>f v. upd (\<lambda>x. f (acc v)) v = upd f v)
- \<and> (\<forall>v. upd id v = v)"
-
-definition
- istuple_update_accessor_eq_assist :: "(('b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'a)) \<Rightarrow> ('a \<Rightarrow> 'b)
- \<Rightarrow> 'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
-where
- "istuple_update_accessor_eq_assist upd acc v f v' x
- \<equiv> upd f v = v' \<and> acc v = x
- \<and> istuple_update_accessor_cong_assist upd acc"
-
-lemma update_accessor_congruence_foldE:
- assumes uac: "istuple_update_accessor_cong_assist upd acc"
- and r: "r = r'" and v: "acc r' = v'"
- and f: "\<And>v. v' = v \<Longrightarrow> f v = f' v"
- shows "upd f r = upd f' r'"
- using uac r v[symmetric]
- apply (subgoal_tac "upd (\<lambda>x. f (acc r')) r' = upd (\<lambda>x. f' (acc r')) r'")
- apply (simp add: istuple_update_accessor_cong_assist_def)
- apply (simp add: f)
- done
-
-lemma update_accessor_congruence_unfoldE:
- "\<lbrakk> istuple_update_accessor_cong_assist upd acc;
- r = r'; acc r' = v'; \<And>v. v = v' \<Longrightarrow> f v = f' v \<rbrakk>
- \<Longrightarrow> upd f r = upd f' r'"
- apply (erule(2) update_accessor_congruence_foldE)
- apply simp
- done
-
-lemma istuple_update_accessor_cong_assist_id:
- "istuple_update_accessor_cong_assist upd acc \<Longrightarrow> upd id = id"
- by (rule ext, simp add: istuple_update_accessor_cong_assist_def)
-
-lemma update_accessor_noopE:
- assumes uac: "istuple_update_accessor_cong_assist upd acc"
- and acc: "f (acc x) = acc x"
- shows "upd f x = x"
- using uac
- by (simp add: acc istuple_update_accessor_cong_assist_id[OF uac, unfolded id_def]
- cong: update_accessor_congruence_unfoldE[OF uac])
-
-lemma update_accessor_noop_compE:
- assumes uac: "istuple_update_accessor_cong_assist upd acc"
- assumes acc: "f (acc x) = acc x"
- shows "upd (g \<circ> f) x = upd g x"
- by (simp add: acc cong: update_accessor_congruence_unfoldE[OF uac])
-
-lemma update_accessor_cong_assist_idI:
- "istuple_update_accessor_cong_assist id id"
- by (simp add: istuple_update_accessor_cong_assist_def)
-
-lemma update_accessor_cong_assist_triv:
- "istuple_update_accessor_cong_assist upd acc
- \<Longrightarrow> istuple_update_accessor_cong_assist upd acc"
- by assumption
-
-lemma update_accessor_accessor_eqE:
- "\<lbrakk> istuple_update_accessor_eq_assist upd acc v f v' x \<rbrakk> \<Longrightarrow> acc v = x"
- by (simp add: istuple_update_accessor_eq_assist_def)
-
-lemma update_accessor_updator_eqE:
- "\<lbrakk> istuple_update_accessor_eq_assist upd acc v f v' x \<rbrakk> \<Longrightarrow> upd f v = v'"
- by (simp add: istuple_update_accessor_eq_assist_def)
-
-lemma istuple_update_accessor_eq_assist_idI:
- "v' = f v \<Longrightarrow> istuple_update_accessor_eq_assist id id v f v' v"
- by (simp add: istuple_update_accessor_eq_assist_def
- update_accessor_cong_assist_idI)
-
-lemma istuple_update_accessor_eq_assist_triv:
- "istuple_update_accessor_eq_assist upd acc v f v' x
- \<Longrightarrow> istuple_update_accessor_eq_assist upd acc v f v' x"
- by assumption
-
-lemma istuple_update_accessor_cong_from_eq:
- "istuple_update_accessor_eq_assist upd acc v f v' x
- \<Longrightarrow> istuple_update_accessor_cong_assist upd acc"
- by (simp add: istuple_update_accessor_eq_assist_def)
-
-lemma o_eq_dest:
- "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
- apply (clarsimp simp: o_def)
- apply (erule fun_cong)
- done
-
-lemma o_eq_elim:
- "\<lbrakk> a o b = c o d; \<lbrakk> \<And>v. a (b v) = c (d v) \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
- apply (erule meta_mp)
- apply (erule o_eq_dest)
- done
-
-lemma istuple_surjective_proof_assistI:
- "f x = y \<Longrightarrow>
- istuple_surjective_proof_assist x y f"
- by (simp add: istuple_surjective_proof_assist_def)
-
-lemma istuple_surjective_proof_assist_idE:
- "istuple_surjective_proof_assist x y id \<Longrightarrow> x = y"
- by (simp add: istuple_surjective_proof_assist_def)
-
-locale isomorphic_tuple =
- fixes isom :: "('a, 'b, 'c) tuple_isomorphism"
- and repr and abst
- defines "repr \<equiv> fst (Rep_tuple_isomorphism isom)"
- defines "abst \<equiv> snd (Rep_tuple_isomorphism isom)"
- assumes repr_inv: "\<And>x. abst (repr x) = x"
- assumes abst_inv: "\<And>y. repr (abst y) = y"
-
-begin
-
-lemma repr_inj:
- "(repr x = repr y) = (x = y)"
- apply (rule iffI, simp_all)
- apply (drule_tac f=abst in arg_cong, simp add: repr_inv)
- done
-
-lemma abst_inj:
- "(abst x = abst y) = (x = y)"
- apply (rule iffI, simp_all)
- apply (drule_tac f=repr in arg_cong, simp add: abst_inv)
- done
-
-lemma split_Rep:
- "split f (Rep_tuple_isomorphism isom)
- = f repr abst"
- by (simp add: split_def repr_def abst_def)
-
-lemmas simps = Let_def split_Rep repr_inv abst_inv repr_inj abst_inj
-
-lemma istuple_access_update_fst_fst:
- "\<lbrakk> f o h g = j o f \<rbrakk> \<Longrightarrow>
- (f o istuple_fst isom) o (istuple_fst_update isom o h) g
- = j o (f o istuple_fst isom)"
- by (clarsimp simp: istuple_fst_update_def istuple_fst_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_access_update_snd_snd:
- "\<lbrakk> f o h g = j o f \<rbrakk> \<Longrightarrow>
- (f o istuple_snd isom) o (istuple_snd_update isom o h) g
- = j o (f o istuple_snd isom)"
- by (clarsimp simp: istuple_snd_update_def istuple_snd_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_access_update_fst_snd:
- "(f o istuple_fst isom) o (istuple_snd_update isom o h) g
- = id o (f o istuple_fst isom)"
- by (clarsimp simp: istuple_snd_update_def istuple_fst_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_access_update_snd_fst:
- "(f o istuple_snd isom) o (istuple_fst_update isom o h) g
- = id o (f o istuple_snd isom)"
- by (clarsimp simp: istuple_fst_update_def istuple_snd_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_update_swap_fst_fst:
- "\<lbrakk> h f o j g = j g o h f \<rbrakk> \<Longrightarrow>
- (istuple_fst_update isom o h) f o (istuple_fst_update isom o j) g
- = (istuple_fst_update isom o j) g o (istuple_fst_update isom o h) f"
- by (clarsimp simp: istuple_fst_update_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_update_swap_snd_snd:
- "\<lbrakk> h f o j g = j g o h f \<rbrakk> \<Longrightarrow>
- (istuple_snd_update isom o h) f o (istuple_snd_update isom o j) g
- = (istuple_snd_update isom o j) g o (istuple_snd_update isom o h) f"
- by (clarsimp simp: istuple_snd_update_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_update_swap_fst_snd:
- "(istuple_snd_update isom o h) f o (istuple_fst_update isom o j) g
- = (istuple_fst_update isom o j) g o (istuple_snd_update isom o h) f"
- by (clarsimp simp: istuple_fst_update_def istuple_snd_update_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_update_swap_snd_fst:
- "(istuple_fst_update isom o h) f o (istuple_snd_update isom o j) g
- = (istuple_snd_update isom o j) g o (istuple_fst_update isom o h) f"
- by (clarsimp simp: istuple_fst_update_def istuple_snd_update_def simps
- intro!: ext elim!: o_eq_elim)
-
-lemma istuple_update_compose_fst_fst:
- "\<lbrakk> h f o j g = k (f o g) \<rbrakk> \<Longrightarrow>
- (istuple_fst_update isom o h) f o (istuple_fst_update isom o j) g
- = (istuple_fst_update isom o k) (f o g)"
- by (fastsimp simp: istuple_fst_update_def simps
- intro!: ext elim!: o_eq_elim dest: fun_cong)
-
-lemma istuple_update_compose_snd_snd:
- "\<lbrakk> h f o j g = k (f o g) \<rbrakk> \<Longrightarrow>
- (istuple_snd_update isom o h) f o (istuple_snd_update isom o j) g
- = (istuple_snd_update isom o k) (f o g)"
- by (fastsimp simp: istuple_snd_update_def simps
- intro!: ext elim!: o_eq_elim dest: fun_cong)
-
-lemma istuple_surjective_proof_assist_step:
- "\<lbrakk> istuple_surjective_proof_assist v a (istuple_fst isom o f);
- istuple_surjective_proof_assist v b (istuple_snd isom o f) \<rbrakk>
- \<Longrightarrow> istuple_surjective_proof_assist v (istuple_cons isom a b) f"
- by (clarsimp simp: istuple_surjective_proof_assist_def simps
- istuple_fst_def istuple_snd_def istuple_cons_def)
-
-lemma istuple_fst_update_accessor_cong_assist:
- "istuple_update_accessor_cong_assist f g \<Longrightarrow>
- istuple_update_accessor_cong_assist (istuple_fst_update isom o f) (g o istuple_fst isom)"
- by (clarsimp simp: istuple_update_accessor_cong_assist_def simps
- istuple_fst_update_def istuple_fst_def)
-
-lemma istuple_snd_update_accessor_cong_assist:
- "istuple_update_accessor_cong_assist f g \<Longrightarrow>
- istuple_update_accessor_cong_assist (istuple_snd_update isom o f) (g o istuple_snd isom)"
- by (clarsimp simp: istuple_update_accessor_cong_assist_def simps
- istuple_snd_update_def istuple_snd_def)
-
-lemma istuple_fst_update_accessor_eq_assist:
- "istuple_update_accessor_eq_assist f g a u a' v \<Longrightarrow>
- istuple_update_accessor_eq_assist (istuple_fst_update isom o f) (g o istuple_fst isom)
- (istuple_cons isom a b) u (istuple_cons isom a' b) v"
- by (clarsimp simp: istuple_update_accessor_eq_assist_def istuple_fst_update_def istuple_fst_def
- istuple_update_accessor_cong_assist_def istuple_cons_def simps)
-
-lemma istuple_snd_update_accessor_eq_assist:
- "istuple_update_accessor_eq_assist f g b u b' v \<Longrightarrow>
- istuple_update_accessor_eq_assist (istuple_snd_update isom o f) (g o istuple_snd isom)
- (istuple_cons isom a b) u (istuple_cons isom a b') v"
- by (clarsimp simp: istuple_update_accessor_eq_assist_def istuple_snd_update_def istuple_snd_def
- istuple_update_accessor_cong_assist_def istuple_cons_def simps)
-
-lemma istuple_cons_conj_eqI:
- "\<lbrakk> (a = c \<and> b = d \<and> P) = Q \<rbrakk> \<Longrightarrow>
- (istuple_cons isom a b = istuple_cons isom c d \<and> P) = Q"
- by (clarsimp simp: istuple_cons_def simps)
-
-lemmas intros =
- istuple_access_update_fst_fst
- istuple_access_update_snd_snd
- istuple_access_update_fst_snd
- istuple_access_update_snd_fst
- istuple_update_swap_fst_fst
- istuple_update_swap_snd_snd
- istuple_update_swap_fst_snd
- istuple_update_swap_snd_fst
- istuple_update_compose_fst_fst
- istuple_update_compose_snd_snd
- istuple_surjective_proof_assist_step
- istuple_fst_update_accessor_eq_assist
- istuple_snd_update_accessor_eq_assist
- istuple_fst_update_accessor_cong_assist
- istuple_snd_update_accessor_cong_assist
- istuple_cons_conj_eqI
-
-end
-
-lemma isomorphic_tuple_intro:
- assumes repr_inj: "\<And>x y. (repr x = repr y) = (x = y)"
- and abst_inv: "\<And>z. repr (abst z) = z"
- shows "v \<equiv> TupleIsomorphism repr abst \<Longrightarrow> isomorphic_tuple v"
- apply (rule isomorphic_tuple.intro,
- simp_all add: TupleIsomorphism_def Abs_tuple_isomorphism_inverse
- tuple_isomorphism_def abst_inv)
- apply (cut_tac x="abst (repr x)" and y="x" in repr_inj)
- apply (simp add: abst_inv)
- done
-
-definition
- "tuple_istuple \<equiv> TupleIsomorphism id id"
-
-lemma tuple_istuple:
- "isomorphic_tuple tuple_istuple"
- by (simp add: isomorphic_tuple_intro[OF _ _ reflexive] tuple_istuple_def)
-
-lemma refl_conj_eq:
- "Q = R \<Longrightarrow> (P \<and> Q) = (P \<and> R)"
- by simp
-
-lemma istuple_UNIV_I: "\<And>x. x\<in>UNIV \<equiv> True"
- by simp
-
-lemma istuple_True_simp: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
- by simp
+subsection {* Record package *}
use "Tools/record.ML"
setup Record.setup
+hide (open) const TupleIsomorphism repr abst istuple_fst istuple_snd
+ istuple_fst_update istuple_snd_update istuple_cons
+ istuple_surjective_proof_assist istuple_update_accessor_cong_assist
+ istuple_update_accessor_eq_assist tuple_istuple
+
end