--- a/src/HOL/NatTransfer.thy Thu Sep 10 15:57:55 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,537 +0,0 @@
-(* Title: HOL/Library/NatTransfer.thy
- Authors: Jeremy Avigad and Amine Chaieb
-
- Sets up transfer from nats to ints and
- back.
-*)
-
-
-header {* NatTransfer *}
-
-theory NatTransfer
-imports Main Parity
-uses ("Tools/transfer_data.ML")
-begin
-
-subsection {* A transfer Method between isomorphic domains*}
-
-definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool"
- where "TransferMorphism a B = True"
-
-use "Tools/transfer_data.ML"
-
-setup TransferData.setup
-
-
-subsection {* Set up transfer from nat to int *}
-
-(* set up transfer direction *)
-
-lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))"
- by (simp add: TransferMorphism_def)
-
-declare TransferMorphism_nat_int[transfer
- add mode: manual
- return: nat_0_le
- labels: natint
-]
-
-(* basic functions and relations *)
-
-lemma transfer_nat_int_numerals:
- "(0::nat) = nat 0"
- "(1::nat) = nat 1"
- "(2::nat) = nat 2"
- "(3::nat) = nat 3"
- by auto
-
-definition
- tsub :: "int \<Rightarrow> int \<Rightarrow> int"
-where
- "tsub x y = (if x >= y then x - y else 0)"
-
-lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y"
- by (simp add: tsub_def)
-
-
-lemma transfer_nat_int_functions:
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)"
- "(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
- by (auto simp add: eq_nat_nat_iff nat_mult_distrib
- nat_power_eq nat_div_distrib nat_mod_distrib tsub_def)
-
-lemma transfer_nat_int_function_closures:
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0"
- "(x::int) >= 0 \<Longrightarrow> x^n >= 0"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
- "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
- "(0::int) >= 0"
- "(1::int) >= 0"
- "(2::int) >= 0"
- "(3::int) >= 0"
- "int z >= 0"
- apply (auto simp add: zero_le_mult_iff tsub_def)
- apply (case_tac "y = 0")
- apply auto
- apply (subst pos_imp_zdiv_nonneg_iff, auto)
- apply (case_tac "y = 0")
- apply force
- apply (rule pos_mod_sign)
- apply arith
-done
-
-lemma transfer_nat_int_relations:
- "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
- (nat (x::int) = nat y) = (x = y)"
- "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
- (nat (x::int) < nat y) = (x < y)"
- "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
- (nat (x::int) <= nat y) = (x <= y)"
- "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow>
- (nat (x::int) dvd nat y) = (x dvd y)"
- by (auto simp add: zdvd_int even_nat_def)
-
-declare TransferMorphism_nat_int[transfer add return:
- transfer_nat_int_numerals
- transfer_nat_int_functions
- transfer_nat_int_function_closures
- transfer_nat_int_relations
-]
-
-
-(* first-order quantifiers *)
-
-lemma transfer_nat_int_quantifiers:
- "(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))"
- "(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))"
- by (rule all_nat, rule ex_nat)
-
-(* should we restrict these? *)
-lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>
- (ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)"
- by auto
-
-lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow>
- (EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)"
- by auto
-
-declare TransferMorphism_nat_int[transfer add
- return: transfer_nat_int_quantifiers
- cong: all_cong ex_cong]
-
-
-(* if *)
-
-lemma nat_if_cong: "(if P then (nat x) else (nat y)) =
- nat (if P then x else y)"
- by auto
-
-declare TransferMorphism_nat_int [transfer add return: nat_if_cong]
-
-
-(* operations with sets *)
-
-definition
- nat_set :: "int set \<Rightarrow> bool"
-where
- "nat_set S = (ALL x:S. x >= 0)"
-
-lemma transfer_nat_int_set_functions:
- "card A = card (int ` A)"
- "{} = nat ` ({}::int set)"
- "A Un B = nat ` (int ` A Un int ` B)"
- "A Int B = nat ` (int ` A Int int ` B)"
- "{x. P x} = nat ` {x. x >= 0 & P(nat x)}"
- "{..n} = nat ` {0..int n}"
- "{m..n} = nat ` {int m..int n}" (* need all variants of these! *)
- apply (rule card_image [symmetric])
- apply (auto simp add: inj_on_def image_def)
- apply (rule_tac x = "int x" in bexI)
- apply auto
- apply (rule_tac x = "int x" in bexI)
- apply auto
- apply (rule_tac x = "int x" in bexI)
- apply auto
- apply (rule_tac x = "int x" in exI)
- apply auto
- apply (rule_tac x = "int x" in bexI)
- apply auto
- apply (rule_tac x = "int x" in bexI)
- apply auto
-done
-
-lemma transfer_nat_int_set_function_closures:
- "nat_set {}"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"
- "x >= 0 \<Longrightarrow> nat_set {x..y}"
- "nat_set {x. x >= 0 & P x}"
- "nat_set (int ` C)"
- "nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *)
- unfolding nat_set_def apply auto
-done
-
-lemma transfer_nat_int_set_relations:
- "(finite A) = (finite (int ` A))"
- "(x : A) = (int x : int ` A)"
- "(A = B) = (int ` A = int ` B)"
- "(A < B) = (int ` A < int ` B)"
- "(A <= B) = (int ` A <= int ` B)"
-
- apply (rule iffI)
- apply (erule finite_imageI)
- apply (erule finite_imageD)
- apply (auto simp add: image_def expand_set_eq inj_on_def)
- apply (drule_tac x = "int x" in spec, auto)
- apply (drule_tac x = "int x" in spec, auto)
- apply (drule_tac x = "int x" in spec, auto)
-done
-
-lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow>
- (int ` nat ` A = A)"
- by (auto simp add: nat_set_def image_def)
-
-lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow>
- {(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}"
- by auto
-
-declare TransferMorphism_nat_int[transfer add
- return: transfer_nat_int_set_functions
- transfer_nat_int_set_function_closures
- transfer_nat_int_set_relations
- transfer_nat_int_set_return_embed
- cong: transfer_nat_int_set_cong
-]
-
-
-(* setsum and setprod *)
-
-(* this handles the case where the *domain* of f is nat *)
-lemma transfer_nat_int_sum_prod:
- "setsum f A = setsum (%x. f (nat x)) (int ` A)"
- "setprod f A = setprod (%x. f (nat x)) (int ` A)"
- apply (subst setsum_reindex)
- apply (unfold inj_on_def, auto)
- apply (subst setprod_reindex)
- apply (unfold inj_on_def o_def, auto)
-done
-
-(* this handles the case where the *range* of f is nat *)
-lemma transfer_nat_int_sum_prod2:
- "setsum f A = nat(setsum (%x. int (f x)) A)"
- "setprod f A = nat(setprod (%x. int (f x)) A)"
- apply (subst int_setsum [symmetric])
- apply auto
- apply (subst int_setprod [symmetric])
- apply auto
-done
-
-lemma transfer_nat_int_sum_prod_closure:
- "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"
- "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"
- unfolding nat_set_def
- apply (rule setsum_nonneg)
- apply auto
- apply (rule setprod_nonneg)
- apply auto
-done
-
-(* this version doesn't work, even with nat_set A \<Longrightarrow>
- x : A \<Longrightarrow> x >= 0 turned on. Why not?
-
- also: what does =simp=> do?
-
-lemma transfer_nat_int_sum_prod_closure:
- "(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0"
- "(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"
- unfolding nat_set_def simp_implies_def
- apply (rule setsum_nonneg)
- apply auto
- apply (rule setprod_nonneg)
- apply auto
-done
-*)
-
-(* Making A = B in this lemma doesn't work. Why not?
- Also, why aren't setsum_cong and setprod_cong enough,
- with the previously mentioned rule turned on? *)
-
-lemma transfer_nat_int_sum_prod_cong:
- "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
- setsum f A = setsum g B"
- "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
- setprod f A = setprod g B"
- unfolding nat_set_def
- apply (subst setsum_cong, assumption)
- apply auto [2]
- apply (subst setprod_cong, assumption, auto)
-done
-
-declare TransferMorphism_nat_int[transfer add
- return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2
- transfer_nat_int_sum_prod_closure
- cong: transfer_nat_int_sum_prod_cong]
-
-(* lists *)
-
-definition
- embed_list :: "nat list \<Rightarrow> int list"
-where
- "embed_list l = map int l";
-
-definition
- nat_list :: "int list \<Rightarrow> bool"
-where
- "nat_list l = nat_set (set l)";
-
-definition
- return_list :: "int list \<Rightarrow> nat list"
-where
- "return_list l = map nat l";
-
-thm nat_0_le;
-
-lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
- embed_list (return_list l) = l";
- unfolding embed_list_def return_list_def nat_list_def nat_set_def
- apply (induct l);
- apply auto;
-done;
-
-lemma transfer_nat_int_list_functions:
- "l @ m = return_list (embed_list l @ embed_list m)"
- "[] = return_list []";
- unfolding return_list_def embed_list_def;
- apply auto;
- apply (induct l, auto);
- apply (induct m, auto);
-done;
-
-(*
-lemma transfer_nat_int_fold1: "fold f l x =
- fold (%x. f (nat x)) (embed_list l) x";
-*)
-
-
-
-
-subsection {* Set up transfer from int to nat *}
-
-(* set up transfer direction *)
-
-lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)"
- by (simp add: TransferMorphism_def)
-
-declare TransferMorphism_int_nat[transfer add
- mode: manual
-(* labels: int-nat *)
- return: nat_int
-]
-
-
-(* basic functions and relations *)
-
-definition
- is_nat :: "int \<Rightarrow> bool"
-where
- "is_nat x = (x >= 0)"
-
-lemma transfer_int_nat_numerals:
- "0 = int 0"
- "1 = int 1"
- "2 = int 2"
- "3 = int 3"
- by auto
-
-lemma transfer_int_nat_functions:
- "(int x) + (int y) = int (x + y)"
- "(int x) * (int y) = int (x * y)"
- "tsub (int x) (int y) = int (x - y)"
- "(int x)^n = int (x^n)"
- "(int x) div (int y) = int (x div y)"
- "(int x) mod (int y) = int (x mod y)"
- by (auto simp add: int_mult tsub_def int_power zdiv_int zmod_int)
-
-lemma transfer_int_nat_function_closures:
- "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)"
- "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)"
- "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)"
- "is_nat x \<Longrightarrow> is_nat (x^n)"
- "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
- "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
- "is_nat 0"
- "is_nat 1"
- "is_nat 2"
- "is_nat 3"
- "is_nat (int z)"
- by (simp_all only: is_nat_def transfer_nat_int_function_closures)
-
-lemma transfer_int_nat_relations:
- "(int x = int y) = (x = y)"
- "(int x < int y) = (x < y)"
- "(int x <= int y) = (x <= y)"
- "(int x dvd int y) = (x dvd y)"
- "(even (int x)) = (even x)"
- by (auto simp add: zdvd_int even_nat_def)
-
-lemma UNIV_apply:
- "UNIV x = True"
- by (simp add: top_fun_eq top_bool_eq)
-
-declare TransferMorphism_int_nat[transfer add return:
- transfer_int_nat_numerals
- transfer_int_nat_functions
- transfer_int_nat_function_closures
- transfer_int_nat_relations
- UNIV_apply
-]
-
-
-(* first-order quantifiers *)
-
-lemma transfer_int_nat_quantifiers:
- "(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))"
- "(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))"
- apply (subst all_nat)
- apply auto [1]
- apply (subst ex_nat)
- apply auto
-done
-
-declare TransferMorphism_int_nat[transfer add
- return: transfer_int_nat_quantifiers]
-
-
-(* if *)
-
-lemma int_if_cong: "(if P then (int x) else (int y)) =
- int (if P then x else y)"
- by auto
-
-declare TransferMorphism_int_nat [transfer add return: int_if_cong]
-
-
-
-(* operations with sets *)
-
-lemma transfer_int_nat_set_functions:
- "nat_set A \<Longrightarrow> card A = card (nat ` A)"
- "{} = int ` ({}::nat set)"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)"
- "{x. x >= 0 & P x} = int ` {x. P(int x)}"
- "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
- (* need all variants of these! *)
- by (simp_all only: is_nat_def transfer_nat_int_set_functions
- transfer_nat_int_set_function_closures
- transfer_nat_int_set_return_embed nat_0_le
- cong: transfer_nat_int_set_cong)
-
-lemma transfer_int_nat_set_function_closures:
- "nat_set {}"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"
- "is_nat x \<Longrightarrow> nat_set {x..y}"
- "nat_set {x. x >= 0 & P x}"
- "nat_set (int ` C)"
- "nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x"
- by (simp_all only: transfer_nat_int_set_function_closures is_nat_def)
-
-lemma transfer_int_nat_set_relations:
- "nat_set A \<Longrightarrow> finite A = finite (nat ` A)"
- "is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)"
- "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)"
- by (simp_all only: is_nat_def transfer_nat_int_set_relations
- transfer_nat_int_set_return_embed nat_0_le)
-
-lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A"
- by (simp only: transfer_nat_int_set_relations
- transfer_nat_int_set_function_closures
- transfer_nat_int_set_return_embed nat_0_le)
-
-lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow>
- {(x::nat). P x} = {x. P' x}"
- by auto
-
-declare TransferMorphism_int_nat[transfer add
- return: transfer_int_nat_set_functions
- transfer_int_nat_set_function_closures
- transfer_int_nat_set_relations
- transfer_int_nat_set_return_embed
- cong: transfer_int_nat_set_cong
-]
-
-
-(* setsum and setprod *)
-
-(* this handles the case where the *domain* of f is int *)
-lemma transfer_int_nat_sum_prod:
- "nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)"
- "nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)"
- apply (subst setsum_reindex)
- apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff)
- apply (subst setprod_reindex)
- apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff
- cong: setprod_cong)
-done
-
-(* this handles the case where the *range* of f is int *)
-lemma transfer_int_nat_sum_prod2:
- "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)"
- "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow>
- setprod f A = int(setprod (%x. nat (f x)) A)"
- unfolding is_nat_def
- apply (subst int_setsum, auto)
- apply (subst int_setprod, auto simp add: cong: setprod_cong)
-done
-
-declare TransferMorphism_int_nat[transfer add
- return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2
- cong: setsum_cong setprod_cong]
-
-
-subsection {* Test it out *}
-
-(* nat to int *)
-
-lemma ex1: "(x::nat) + y = y + x"
- by auto
-
-thm ex1 [transferred]
-
-lemma ex2: "(a::nat) div b * b + a mod b = a"
- by (rule mod_div_equality)
-
-thm ex2 [transferred]
-
-lemma ex3: "ALL (x::nat). ALL y. EX z. z >= x + y"
- by auto
-
-thm ex3 [transferred natint]
-
-lemma ex4: "(x::nat) >= y \<Longrightarrow> (x - y) + y = x"
- by auto
-
-thm ex4 [transferred]
-
-lemma ex5: "(2::nat) * (SUM i <= n. i) = n * (n + 1)"
- by (induct n rule: nat_induct, auto)
-
-thm ex5 [transferred]
-
-theorem ex6: "0 <= (n::int) \<Longrightarrow> 2 * \<Sum>{0..n} = n * (n + 1)"
- by (rule ex5 [transferred])
-
-thm ex6 [transferred]
-
-thm ex5 [transferred, transferred]
-
-end