author | hoelzl |
Thu, 17 Jan 2013 11:59:12 +0100 | |
changeset 50936 | b28f258ebc1a |
parent 50422 | ee729dbd1b7f |
child 51299 | 30b014246e21 |
permissions | -rw-r--r-- |
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(* Authors: Jeremy Avigad and Amine Chaieb *) |
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header {* Generic transfer machinery; specific transfer from nats to ints and back. *} |
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theory Nat_Transfer |
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imports Int |
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begin |
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||
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subsection {* Generic transfer machinery *} |
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definition transfer_morphism:: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool" |
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where "transfer_morphism f A \<longleftrightarrow> True" |
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lemma transfer_morphismI[intro]: "transfer_morphism f A" |
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by (simp add: transfer_morphism_def) |
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ML_file "Tools/legacy_transfer.ML" |
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setup Legacy_Transfer.setup |
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|
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|
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subsection {* Set up transfer from nat to int *} |
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text {* set up transfer direction *} |
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|
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lemma transfer_morphism_nat_int: "transfer_morphism nat (op <= (0::int))" .. |
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|
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declare transfer_morphism_nat_int [transfer add |
29 |
mode: manual |
|
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return: nat_0_le |
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labels: nat_int |
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] |
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text {* basic functions and relations *} |
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|
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lemma transfer_nat_int_numerals [transfer key: transfer_morphism_nat_int]: |
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"(0::nat) = nat 0" |
38 |
"(1::nat) = nat 1" |
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39 |
"(2::nat) = nat 2" |
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"(3::nat) = nat 3" |
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by auto |
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||
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definition |
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tsub :: "int \<Rightarrow> int \<Rightarrow> int" |
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where |
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"tsub x y = (if x >= y then x - y else 0)" |
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lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y" |
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by (simp add: tsub_def) |
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||
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lemma transfer_nat_int_functions [transfer key: transfer_morphism_nat_int]: |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)" |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)" |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)" |
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"(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)" |
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by (auto simp add: eq_nat_nat_iff nat_mult_distrib |
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nat_power_eq tsub_def) |
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lemma transfer_nat_int_function_closures [transfer key: transfer_morphism_nat_int]: |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0" |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0" |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0" |
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"(x::int) >= 0 \<Longrightarrow> x^n >= 0" |
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"(0::int) >= 0" |
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"(1::int) >= 0" |
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"(2::int) >= 0" |
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"(3::int) >= 0" |
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"int z >= 0" |
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by (auto simp add: zero_le_mult_iff tsub_def) |
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|
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lemma transfer_nat_int_relations [transfer key: transfer_morphism_nat_int]: |
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"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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(nat (x::int) = nat y) = (x = y)" |
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"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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(nat (x::int) < nat y) = (x < y)" |
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"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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(nat (x::int) <= nat y) = (x <= y)" |
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"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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(nat (x::int) dvd nat y) = (x dvd y)" |
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by (auto simp add: zdvd_int) |
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text {* first-order quantifiers *} |
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|
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lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))" |
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by (simp split add: split_nat) |
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|
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lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))" |
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proof |
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assume "\<exists>x. P x" |
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then obtain x where "P x" .. |
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then have "int x \<ge> 0 \<and> P (nat (int x))" by simp |
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then show "\<exists>x\<ge>0. P (nat x)" .. |
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next |
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assume "\<exists>x\<ge>0. P (nat x)" |
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then show "\<exists>x. P x" by auto |
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qed |
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|
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lemma transfer_nat_int_quantifiers [transfer key: transfer_morphism_nat_int]: |
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"(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))" |
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"(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))" |
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by (rule all_nat, rule ex_nat) |
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(* should we restrict these? *) |
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lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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(ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)" |
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by auto |
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lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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(EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)" |
|
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by auto |
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||
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declare transfer_morphism_nat_int [transfer add |
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cong: all_cong ex_cong] |
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||
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text {* if *} |
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lemma nat_if_cong [transfer key: transfer_morphism_nat_int]: |
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"(if P then (nat x) else (nat y)) = nat (if P then x else y)" |
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by auto |
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text {* operations with sets *} |
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|
126 |
definition |
|
127 |
nat_set :: "int set \<Rightarrow> bool" |
|
128 |
where |
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129 |
"nat_set S = (ALL x:S. x >= 0)" |
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lemma transfer_nat_int_set_functions: |
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"card A = card (int ` A)" |
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"{} = nat ` ({}::int set)" |
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"A Un B = nat ` (int ` A Un int ` B)" |
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"A Int B = nat ` (int ` A Int int ` B)" |
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"{x. P x} = nat ` {x. x >= 0 & P(nat x)}" |
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apply (rule card_image [symmetric]) |
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apply (auto simp add: inj_on_def image_def) |
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apply (rule_tac x = "int x" in bexI) |
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140 |
apply auto |
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apply (rule_tac x = "int x" in bexI) |
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142 |
apply auto |
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143 |
apply (rule_tac x = "int x" in bexI) |
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144 |
apply auto |
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apply (rule_tac x = "int x" in exI) |
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apply auto |
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147 |
done |
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lemma transfer_nat_int_set_function_closures: |
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"nat_set {}" |
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"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
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"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
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"nat_set {x. x >= 0 & P x}" |
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"nat_set (int ` C)" |
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"nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *) |
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unfolding nat_set_def apply auto |
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done |
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lemma transfer_nat_int_set_relations: |
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"(finite A) = (finite (int ` A))" |
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"(x : A) = (int x : int ` A)" |
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"(A = B) = (int ` A = int ` B)" |
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"(A < B) = (int ` A < int ` B)" |
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"(A <= B) = (int ` A <= int ` B)" |
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apply (rule iffI) |
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apply (erule finite_imageI) |
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apply (erule finite_imageD) |
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apply (auto simp add: image_def set_eq_iff inj_on_def) |
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apply (drule_tac x = "int x" in spec, auto) |
170 |
apply (drule_tac x = "int x" in spec, auto) |
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apply (drule_tac x = "int x" in spec, auto) |
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done |
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lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow> |
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(int ` nat ` A = A)" |
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by (auto simp add: nat_set_def image_def) |
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lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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{(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}" |
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180 |
by auto |
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declare transfer_morphism_nat_int [transfer add |
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return: transfer_nat_int_set_functions |
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transfer_nat_int_set_function_closures |
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transfer_nat_int_set_relations |
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transfer_nat_int_set_return_embed |
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cong: transfer_nat_int_set_cong |
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] |
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text {* setsum and setprod *} |
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(* this handles the case where the *domain* of f is nat *) |
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lemma transfer_nat_int_sum_prod: |
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"setsum f A = setsum (%x. f (nat x)) (int ` A)" |
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"setprod f A = setprod (%x. f (nat x)) (int ` A)" |
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apply (subst setsum_reindex) |
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apply (unfold inj_on_def, auto) |
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apply (subst setprod_reindex) |
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apply (unfold inj_on_def o_def, auto) |
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done |
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(* this handles the case where the *range* of f is nat *) |
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lemma transfer_nat_int_sum_prod2: |
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"setsum f A = nat(setsum (%x. int (f x)) A)" |
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"setprod f A = nat(setprod (%x. int (f x)) A)" |
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apply (subst int_setsum [symmetric]) |
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apply auto |
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apply (subst int_setprod [symmetric]) |
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210 |
apply auto |
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211 |
done |
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lemma transfer_nat_int_sum_prod_closure: |
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"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
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"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
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unfolding nat_set_def |
|
217 |
apply (rule setsum_nonneg) |
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apply auto |
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apply (rule setprod_nonneg) |
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apply auto |
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221 |
done |
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222 |
||
223 |
(* this version doesn't work, even with nat_set A \<Longrightarrow> |
|
224 |
x : A \<Longrightarrow> x >= 0 turned on. Why not? |
|
225 |
||
226 |
also: what does =simp=> do? |
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227 |
||
228 |
lemma transfer_nat_int_sum_prod_closure: |
|
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"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
|
230 |
"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
|
231 |
unfolding nat_set_def simp_implies_def |
|
232 |
apply (rule setsum_nonneg) |
|
233 |
apply auto |
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234 |
apply (rule setprod_nonneg) |
|
235 |
apply auto |
|
236 |
done |
|
237 |
*) |
|
238 |
||
239 |
(* Making A = B in this lemma doesn't work. Why not? |
|
240 |
Also, why aren't setsum_cong and setprod_cong enough, |
|
241 |
with the previously mentioned rule turned on? *) |
|
242 |
||
243 |
lemma transfer_nat_int_sum_prod_cong: |
|
244 |
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
|
245 |
setsum f A = setsum g B" |
|
246 |
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
|
247 |
setprod f A = setprod g B" |
|
248 |
unfolding nat_set_def |
|
249 |
apply (subst setsum_cong, assumption) |
|
250 |
apply auto [2] |
|
251 |
apply (subst setprod_cong, assumption, auto) |
|
252 |
done |
|
253 |
||
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declare transfer_morphism_nat_int [transfer add |
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return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2 |
256 |
transfer_nat_int_sum_prod_closure |
|
257 |
cong: transfer_nat_int_sum_prod_cong] |
|
258 |
||
259 |
||
260 |
subsection {* Set up transfer from int to nat *} |
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261 |
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text {* set up transfer direction *} |
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|
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lemma transfer_morphism_int_nat: "transfer_morphism int (\<lambda>n. True)" .. |
31708 | 265 |
|
35644 | 266 |
declare transfer_morphism_int_nat [transfer add |
31708 | 267 |
mode: manual |
268 |
return: nat_int |
|
35683 | 269 |
labels: int_nat |
31708 | 270 |
] |
271 |
||
272 |
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text {* basic functions and relations *} |
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274 |
|
31708 | 275 |
definition |
276 |
is_nat :: "int \<Rightarrow> bool" |
|
277 |
where |
|
278 |
"is_nat x = (x >= 0)" |
|
279 |
||
280 |
lemma transfer_int_nat_numerals: |
|
281 |
"0 = int 0" |
|
282 |
"1 = int 1" |
|
283 |
"2 = int 2" |
|
284 |
"3 = int 3" |
|
285 |
by auto |
|
286 |
||
287 |
lemma transfer_int_nat_functions: |
|
288 |
"(int x) + (int y) = int (x + y)" |
|
289 |
"(int x) * (int y) = int (x * y)" |
|
290 |
"tsub (int x) (int y) = int (x - y)" |
|
291 |
"(int x)^n = int (x^n)" |
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292 |
by (auto simp add: int_mult tsub_def int_power) |
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|
294 |
lemma transfer_int_nat_function_closures: |
|
295 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)" |
|
296 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)" |
|
297 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)" |
|
298 |
"is_nat x \<Longrightarrow> is_nat (x^n)" |
|
299 |
"is_nat 0" |
|
300 |
"is_nat 1" |
|
301 |
"is_nat 2" |
|
302 |
"is_nat 3" |
|
303 |
"is_nat (int z)" |
|
304 |
by (simp_all only: is_nat_def transfer_nat_int_function_closures) |
|
305 |
||
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lemma transfer_int_nat_relations: |
|
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"(int x = int y) = (x = y)" |
|
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"(int x < int y) = (x < y)" |
|
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"(int x <= int y) = (x <= y)" |
|
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"(int x dvd int y) = (x dvd y)" |
|
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311 |
by (auto simp add: zdvd_int) |
32121 | 312 |
|
35644 | 313 |
declare transfer_morphism_int_nat [transfer add return: |
31708 | 314 |
transfer_int_nat_numerals |
315 |
transfer_int_nat_functions |
|
316 |
transfer_int_nat_function_closures |
|
317 |
transfer_int_nat_relations |
|
318 |
] |
|
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||
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||
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text {* first-order quantifiers *} |
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|
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lemma transfer_int_nat_quantifiers: |
|
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"(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))" |
|
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"(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))" |
|
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apply (subst all_nat) |
|
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apply auto [1] |
|
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apply (subst ex_nat) |
|
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apply auto |
|
330 |
done |
|
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||
35644 | 332 |
declare transfer_morphism_int_nat [transfer add |
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return: transfer_int_nat_quantifiers] |
334 |
||
335 |
||
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|
336 |
text {* if *} |
31708 | 337 |
|
338 |
lemma int_if_cong: "(if P then (int x) else (int y)) = |
|
339 |
int (if P then x else y)" |
|
340 |
by auto |
|
341 |
||
35644 | 342 |
declare transfer_morphism_int_nat [transfer add return: int_if_cong] |
31708 | 343 |
|
344 |
||
345 |
||
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changeset
|
346 |
text {* operations with sets *} |
31708 | 347 |
|
348 |
lemma transfer_int_nat_set_functions: |
|
349 |
"nat_set A \<Longrightarrow> card A = card (nat ` A)" |
|
350 |
"{} = int ` ({}::nat set)" |
|
351 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)" |
|
352 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)" |
|
353 |
"{x. x >= 0 & P x} = int ` {x. P(int x)}" |
|
354 |
(* need all variants of these! *) |
|
355 |
by (simp_all only: is_nat_def transfer_nat_int_set_functions |
|
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transfer_nat_int_set_function_closures |
|
357 |
transfer_nat_int_set_return_embed nat_0_le |
|
358 |
cong: transfer_nat_int_set_cong) |
|
359 |
||
360 |
lemma transfer_int_nat_set_function_closures: |
|
361 |
"nat_set {}" |
|
362 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
|
363 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
|
364 |
"nat_set {x. x >= 0 & P x}" |
|
365 |
"nat_set (int ` C)" |
|
366 |
"nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x" |
|
367 |
by (simp_all only: transfer_nat_int_set_function_closures is_nat_def) |
|
368 |
||
369 |
lemma transfer_int_nat_set_relations: |
|
370 |
"nat_set A \<Longrightarrow> finite A = finite (nat ` A)" |
|
371 |
"is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)" |
|
372 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)" |
|
373 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)" |
|
374 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)" |
|
375 |
by (simp_all only: is_nat_def transfer_nat_int_set_relations |
|
376 |
transfer_nat_int_set_return_embed nat_0_le) |
|
377 |
||
378 |
lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A" |
|
379 |
by (simp only: transfer_nat_int_set_relations |
|
380 |
transfer_nat_int_set_function_closures |
|
381 |
transfer_nat_int_set_return_embed nat_0_le) |
|
382 |
||
383 |
lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow> |
|
384 |
{(x::nat). P x} = {x. P' x}" |
|
385 |
by auto |
|
386 |
||
35644 | 387 |
declare transfer_morphism_int_nat [transfer add |
31708 | 388 |
return: transfer_int_nat_set_functions |
389 |
transfer_int_nat_set_function_closures |
|
390 |
transfer_int_nat_set_relations |
|
391 |
transfer_int_nat_set_return_embed |
|
392 |
cong: transfer_int_nat_set_cong |
|
393 |
] |
|
394 |
||
395 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
396 |
text {* setsum and setprod *} |
31708 | 397 |
|
398 |
(* this handles the case where the *domain* of f is int *) |
|
399 |
lemma transfer_int_nat_sum_prod: |
|
400 |
"nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)" |
|
401 |
"nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)" |
|
402 |
apply (subst setsum_reindex) |
|
403 |
apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff) |
|
404 |
apply (subst setprod_reindex) |
|
405 |
apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff |
|
406 |
cong: setprod_cong) |
|
407 |
done |
|
408 |
||
409 |
(* this handles the case where the *range* of f is int *) |
|
410 |
lemma transfer_int_nat_sum_prod2: |
|
411 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)" |
|
412 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> |
|
413 |
setprod f A = int(setprod (%x. nat (f x)) A)" |
|
414 |
unfolding is_nat_def |
|
415 |
apply (subst int_setsum, auto) |
|
416 |
apply (subst int_setprod, auto simp add: cong: setprod_cong) |
|
417 |
done |
|
418 |
||
35644 | 419 |
declare transfer_morphism_int_nat [transfer add |
31708 | 420 |
return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2 |
421 |
cong: setsum_cong setprod_cong] |
|
422 |
||
50422
ee729dbd1b7f
avoid ML_file in large theory files to improve performance of dependency discovery of main HOL (approx. 1s CPU time) -- relevant for any application using it, e.g. small paper sessions;
wenzelm
parents:
48891
diff
changeset
|
423 |
|
ee729dbd1b7f
avoid ML_file in large theory files to improve performance of dependency discovery of main HOL (approx. 1s CPU time) -- relevant for any application using it, e.g. small paper sessions;
wenzelm
parents:
48891
diff
changeset
|
424 |
(*belongs to Divides.thy, but slows down dependency discovery*) |
ee729dbd1b7f
avoid ML_file in large theory files to improve performance of dependency discovery of main HOL (approx. 1s CPU time) -- relevant for any application using it, e.g. small paper sessions;
wenzelm
parents:
48891
diff
changeset
|
425 |
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML" |
ee729dbd1b7f
avoid ML_file in large theory files to improve performance of dependency discovery of main HOL (approx. 1s CPU time) -- relevant for any application using it, e.g. small paper sessions;
wenzelm
parents:
48891
diff
changeset
|
426 |
|
31708 | 427 |
end |