--- a/src/HOL/Nat_Transfer.thy Thu Oct 29 08:14:39 2009 +0100
+++ b/src/HOL/Nat_Transfer.thy Thu Oct 29 11:41:36 2009 +0100
@@ -1,15 +1,26 @@
(* Authors: Jeremy Avigad and Amine Chaieb *)
-header {* Sets up transfer from nats to ints and back. *}
+header {* Generic transfer machinery; specific transfer from nats to ints and back. *}
theory Nat_Transfer
-imports Main Parity
+imports Nat_Numeral
+uses ("Tools/transfer.ML")
begin
+subsection {* Generic transfer machinery *}
+
+definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool"
+ where "TransferMorphism a B \<longleftrightarrow> True"
+
+use "Tools/transfer.ML"
+
+setup Transfer.setup
+
+
subsection {* Set up transfer from nat to int *}
-(* set up transfer direction *)
+text {* set up transfer direction *}
lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))"
by (simp add: TransferMorphism_def)
@@ -20,7 +31,7 @@
labels: natint
]
-(* basic functions and relations *)
+text {* basic functions and relations *}
lemma transfer_nat_int_numerals:
"(0::nat) = nat 0"
@@ -43,31 +54,20 @@
"(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)
+ nat_power_eq 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:
@@ -89,7 +89,21 @@
]
-(* first-order quantifiers *)
+text {* first-order quantifiers *}
+
+lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"
+ by (simp split add: split_nat)
+
+lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"
+proof
+ assume "\<exists>x. P x"
+ then obtain x where "P x" ..
+ then have "int x \<ge> 0 \<and> P (nat (int x))" by simp
+ then show "\<exists>x\<ge>0. P (nat x)" ..
+next
+ assume "\<exists>x\<ge>0. P (nat x)"
+ then show "\<exists>x. P x" by auto
+qed
lemma transfer_nat_int_quantifiers:
"(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))"
@@ -110,7 +124,7 @@
cong: all_cong ex_cong]
-(* if *)
+text {* if *}
lemma nat_if_cong: "(if P then (nat x) else (nat y)) =
nat (if P then x else y)"
@@ -119,7 +133,7 @@
declare TransferMorphism_nat_int [transfer add return: nat_if_cong]
-(* operations with sets *)
+text {* operations with sets *}
definition
nat_set :: "int set \<Rightarrow> bool"
@@ -132,8 +146,6 @@
"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)
@@ -144,17 +156,12 @@
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? *)
@@ -167,7 +174,6 @@
"(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)
@@ -194,7 +200,7 @@
]
-(* setsum and setprod *)
+text {* setsum and setprod *}
(* this handles the case where the *domain* of f is nat *)
lemma transfer_nat_int_sum_prod:
@@ -262,52 +268,10 @@
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 *)
+text {* set up transfer direction *}
lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)"
by (simp add: TransferMorphism_def)
@@ -319,7 +283,11 @@
]
-(* basic functions and relations *)
+text {* basic functions and relations *}
+
+lemma UNIV_apply:
+ "UNIV x = True"
+ by (simp add: top_fun_eq top_bool_eq)
definition
is_nat :: "int \<Rightarrow> bool"
@@ -338,17 +306,13 @@
"(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)
+ by (auto simp add: int_mult tsub_def int_power)
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"
@@ -361,12 +325,7 @@
"(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)
+ by (auto simp add: zdvd_int)
declare TransferMorphism_int_nat[transfer add return:
transfer_int_nat_numerals
@@ -377,7 +336,7 @@
]
-(* first-order quantifiers *)
+text {* first-order quantifiers *}
lemma transfer_int_nat_quantifiers:
"(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))"
@@ -392,7 +351,7 @@
return: transfer_int_nat_quantifiers]
-(* if *)
+text {* if *}
lemma int_if_cong: "(if P then (int x) else (int y)) =
int (if P then x else y)"
@@ -402,7 +361,7 @@
-(* operations with sets *)
+text {* operations with sets *}
lemma transfer_int_nat_set_functions:
"nat_set A \<Longrightarrow> card A = card (nat ` A)"
@@ -410,7 +369,6 @@
"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
@@ -421,7 +379,6 @@
"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"
@@ -454,7 +411,7 @@
]
-(* setsum and setprod *)
+text {* setsum and setprod *}
(* this handles the case where the *domain* of f is int *)
lemma transfer_int_nat_sum_prod: