author | haftmann |
Sun, 08 Oct 2017 22:28:22 +0200 | |
changeset 66817 | 0b12755ccbb2 |
parent 66795 | 420d0080545f |
child 66836 | 4eb431c3f974 |
permissions | -rw-r--r-- |
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(* Authors: Jeremy Avigad and Amine Chaieb *) |
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section \<open>Generic transfer machinery; specific transfer from nats to ints and back.\<close> |
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theory Nat_Transfer |
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imports Int Divides |
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begin |
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subsection \<open>Generic transfer machinery\<close> |
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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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subsection \<open>Set up transfer from nat to int\<close> |
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text \<open>set up transfer direction\<close> |
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lemma transfer_morphism_nat_int [no_atp]: |
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"transfer_morphism nat (op <= (0::int))" .. |
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declare transfer_morphism_nat_int [transfer add |
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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 \<open>basic functions and relations\<close> |
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lemma transfer_nat_int_numerals [no_atp, transfer key: transfer_morphism_nat_int]: |
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"(0::nat) = nat 0" |
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"(1::nat) = nat 1" |
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"(2::nat) = nat 2" |
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"(3::nat) = nat 3" |
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by auto |
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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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lemma transfer_nat_int_functions [no_atp, 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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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)" |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)" |
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by (auto simp add: eq_nat_nat_iff nat_mult_distrib |
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nat_power_eq tsub_def nat_div_distrib nat_mod_distrib) |
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|
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lemma transfer_nat_int_function_closures [no_atp, 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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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0" |
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"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0" |
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apply (auto simp add: zero_le_mult_iff tsub_def pos_imp_zdiv_nonneg_iff) |
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apply (cases "y = 0") |
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apply (auto simp add: pos_imp_zdiv_nonneg_iff) |
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apply (cases "y = 0") |
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apply auto |
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done |
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lemma transfer_nat_int_relations [no_atp, 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 \<open>first-order quantifiers\<close> |
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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: split_nat) |
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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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lemma transfer_nat_int_quantifiers [no_atp, 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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declare transfer_morphism_nat_int [transfer add |
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cong: all_cong ex_cong] |
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text \<open>if\<close> |
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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 \<open>operations with sets\<close> |
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definition |
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nat_set :: "int set \<Rightarrow> bool" |
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where |
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"nat_set S = (ALL x:S. x >= 0)" |
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lemma transfer_nat_int_set_functions [no_atp]: |
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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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apply auto |
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apply (rule_tac x = "int x" in bexI) |
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apply auto |
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apply (rule_tac x = "int x" in bexI) |
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apply auto |
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apply (rule_tac x = "int x" in exI) |
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apply auto |
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done |
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lemma transfer_nat_int_set_function_closures [no_atp]: |
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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 [no_atp]: |
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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) |
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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 [no_atp]: "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 [no_atp]: "(!!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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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 \<open>sum and prod\<close> |
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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 [no_atp]: |
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"sum f A = sum (%x. f (nat x)) (int ` A)" |
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"prod f A = prod (%x. f (nat x)) (int ` A)" |
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apply (subst sum.reindex) |
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apply (unfold inj_on_def, auto) |
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apply (subst prod.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 [no_atp]: |
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"sum f A = nat(sum (%x. int (f x)) A)" |
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"prod f A = nat(prod (%x. int (f x)) A)" |
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apply (simp only: int_sum [symmetric] nat_int) |
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apply (simp only: int_prod [symmetric] nat_int) |
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done |
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lemma transfer_nat_int_sum_prod_closure [no_atp]: |
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"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> sum f A >= 0" |
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"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> prod f A >= 0" |
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unfolding nat_set_def |
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apply (rule sum_nonneg) |
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apply auto |
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apply (rule prod_nonneg) |
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apply auto |
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done |
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(* this version doesn't work, even with nat_set A \<Longrightarrow> |
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x : A \<Longrightarrow> x >= 0 turned on. Why not? |
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also: what does =simp=> do? |
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lemma transfer_nat_int_sum_prod_closure: |
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"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> sum f A >= 0" |
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"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> prod f A >= 0" |
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unfolding nat_set_def simp_implies_def |
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apply (rule sum_nonneg) |
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apply auto |
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apply (rule prod_nonneg) |
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apply auto |
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done |
|
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*) |
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(* Making A = B in this lemma doesn't work. Why not? |
|
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Also, why aren't sum.cong and prod.cong enough, |
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with the previously mentioned rule turned on? *) |
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lemma transfer_nat_int_sum_prod_cong [no_atp]: |
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"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
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sum f A = sum g B" |
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"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
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prod f A = prod g B" |
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unfolding nat_set_def |
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apply (subst sum.cong, assumption) |
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apply auto [2] |
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apply (subst prod.cong, assumption, auto) |
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done |
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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 |
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transfer_nat_int_sum_prod_closure |
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cong: transfer_nat_int_sum_prod_cong] |
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subsection \<open>Set up transfer from int to nat\<close> |
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text \<open>set up transfer direction\<close> |
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lemma transfer_morphism_int_nat [no_atp]: "transfer_morphism int (\<lambda>n. True)" .. |
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declare transfer_morphism_int_nat [transfer add |
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mode: manual |
274 |
return: nat_int |
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labels: int_nat |
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] |
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text \<open>basic functions and relations\<close> |
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|
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definition |
282 |
is_nat :: "int \<Rightarrow> bool" |
|
283 |
where |
|
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"is_nat x = (x >= 0)" |
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||
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lemma transfer_int_nat_numerals [no_atp]: |
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"0 = int 0" |
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"1 = int 1" |
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"2 = int 2" |
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"3 = int 3" |
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by auto |
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||
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lemma transfer_int_nat_functions [no_atp]: |
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"(int x) + (int y) = int (x + y)" |
295 |
"(int x) * (int y) = int (x * y)" |
|
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"tsub (int x) (int y) = int (x - y)" |
|
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"(int x)^n = int (x^n)" |
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"(int x) div (int y) = int (x div y)" |
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"(int x) mod (int y) = int (x mod y)" |
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by (auto simp add: zdiv_int zmod_int tsub_def) |
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|
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lemma transfer_int_nat_function_closures [no_atp]: |
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"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)" |
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"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)" |
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"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)" |
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"is_nat x \<Longrightarrow> is_nat (x^n)" |
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"is_nat 0" |
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"is_nat 1" |
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"is_nat 2" |
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"is_nat 3" |
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"is_nat (int z)" |
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"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)" |
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"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)" |
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by (simp_all only: is_nat_def transfer_nat_int_function_closures) |
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||
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lemma transfer_int_nat_relations [no_atp]: |
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"(int x = int y) = (x = y)" |
318 |
"(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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by (auto simp add: zdvd_int) |
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declare transfer_morphism_int_nat [transfer add return: |
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transfer_int_nat_numerals |
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transfer_int_nat_functions |
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transfer_int_nat_function_closures |
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transfer_int_nat_relations |
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] |
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||
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text \<open>first-order quantifiers\<close> |
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|
66817 | 333 |
lemma transfer_int_nat_quantifiers [no_atp]: |
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"(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))" |
335 |
"(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))" |
|
336 |
apply (subst all_nat) |
|
337 |
apply auto [1] |
|
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apply (subst ex_nat) |
|
339 |
apply auto |
|
340 |
done |
|
341 |
||
35644 | 342 |
declare transfer_morphism_int_nat [transfer add |
31708 | 343 |
return: transfer_int_nat_quantifiers] |
344 |
||
345 |
||
60758 | 346 |
text \<open>if\<close> |
31708 | 347 |
|
348 |
lemma int_if_cong: "(if P then (int x) else (int y)) = |
|
349 |
int (if P then x else y)" |
|
350 |
by auto |
|
351 |
||
35644 | 352 |
declare transfer_morphism_int_nat [transfer add return: int_if_cong] |
31708 | 353 |
|
354 |
||
355 |
||
60758 | 356 |
text \<open>operations with sets\<close> |
31708 | 357 |
|
66817 | 358 |
lemma transfer_int_nat_set_functions [no_atp]: |
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"nat_set A \<Longrightarrow> card A = card (nat ` A)" |
360 |
"{} = int ` ({}::nat set)" |
|
361 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)" |
|
362 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)" |
|
363 |
"{x. x >= 0 & P x} = int ` {x. P(int x)}" |
|
364 |
(* need all variants of these! *) |
|
365 |
by (simp_all only: is_nat_def transfer_nat_int_set_functions |
|
366 |
transfer_nat_int_set_function_closures |
|
367 |
transfer_nat_int_set_return_embed nat_0_le |
|
368 |
cong: transfer_nat_int_set_cong) |
|
369 |
||
66817 | 370 |
lemma transfer_int_nat_set_function_closures [no_atp]: |
31708 | 371 |
"nat_set {}" |
372 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
|
373 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
|
374 |
"nat_set {x. x >= 0 & P x}" |
|
375 |
"nat_set (int ` C)" |
|
376 |
"nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x" |
|
377 |
by (simp_all only: transfer_nat_int_set_function_closures is_nat_def) |
|
378 |
||
66817 | 379 |
lemma transfer_int_nat_set_relations [no_atp]: |
31708 | 380 |
"nat_set A \<Longrightarrow> finite A = finite (nat ` A)" |
381 |
"is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)" |
|
382 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)" |
|
383 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)" |
|
384 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)" |
|
385 |
by (simp_all only: is_nat_def transfer_nat_int_set_relations |
|
386 |
transfer_nat_int_set_return_embed nat_0_le) |
|
387 |
||
66817 | 388 |
lemma transfer_int_nat_set_return_embed [no_atp]: "nat ` int ` A = A" |
31708 | 389 |
by (simp only: transfer_nat_int_set_relations |
390 |
transfer_nat_int_set_function_closures |
|
391 |
transfer_nat_int_set_return_embed nat_0_le) |
|
392 |
||
66817 | 393 |
lemma transfer_int_nat_set_cong [no_atp]: "(!!x. P x = P' x) \<Longrightarrow> |
31708 | 394 |
{(x::nat). P x} = {x. P' x}" |
395 |
by auto |
|
396 |
||
35644 | 397 |
declare transfer_morphism_int_nat [transfer add |
31708 | 398 |
return: transfer_int_nat_set_functions |
399 |
transfer_int_nat_set_function_closures |
|
400 |
transfer_int_nat_set_relations |
|
401 |
transfer_int_nat_set_return_embed |
|
402 |
cong: transfer_int_nat_set_cong |
|
403 |
] |
|
404 |
||
405 |
||
64272 | 406 |
text \<open>sum and prod\<close> |
31708 | 407 |
|
408 |
(* this handles the case where the *domain* of f is int *) |
|
66817 | 409 |
lemma transfer_int_nat_sum_prod [no_atp]: |
64267 | 410 |
"nat_set A \<Longrightarrow> sum f A = sum (%x. f (int x)) (nat ` A)" |
64272 | 411 |
"nat_set A \<Longrightarrow> prod f A = prod (%x. f (int x)) (nat ` A)" |
64267 | 412 |
apply (subst sum.reindex) |
31708 | 413 |
apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff) |
64272 | 414 |
apply (subst prod.reindex) |
31708 | 415 |
apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff |
64272 | 416 |
cong: prod.cong) |
31708 | 417 |
done |
418 |
||
419 |
(* this handles the case where the *range* of f is int *) |
|
66817 | 420 |
lemma transfer_int_nat_sum_prod2 [no_atp]: |
64267 | 421 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> sum f A = int(sum (%x. nat (f x)) A)" |
31708 | 422 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> |
64272 | 423 |
prod f A = int(prod (%x. nat (f x)) A)" |
31708 | 424 |
unfolding is_nat_def |
64267 | 425 |
by (subst int_sum) auto |
31708 | 426 |
|
35644 | 427 |
declare transfer_morphism_int_nat [transfer add |
31708 | 428 |
return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2 |
64272 | 429 |
cong: sum.cong prod.cong] |
31708 | 430 |
|
66817 | 431 |
declare transfer_morphism_int_nat [transfer add return: even_int_iff] |
432 |
||
31708 | 433 |
end |