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(* Title: HOL/Library/NatTransfer.thy
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Authors: Jeremy Avigad and Amine Chaieb
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Sets up transfer from nats to ints and
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back.
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*)
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header {* NatTransfer *}
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theory NatTransfer
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imports Main Parity
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uses ("Tools/transfer_data.ML")
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begin
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subsection {* A transfer Method between isomorphic domains*}
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definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool"
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where "TransferMorphism a B = True"
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use "Tools/transfer_data.ML"
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setup TransferData.setup
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subsection {* Set up transfer from nat to int *}
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(* set up transfer direction *)
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lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))"
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by (simp add: TransferMorphism_def)
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declare TransferMorphism_nat_int[transfer
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add mode: manual
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return: nat_0_le
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labels: natint
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]
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(* basic functions and relations *)
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lemma transfer_nat_int_numerals:
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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:
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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 nat_div_distrib nat_mod_distrib tsub_def)
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lemma transfer_nat_int_function_closures:
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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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"(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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"(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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apply (auto simp add: zero_le_mult_iff tsub_def)
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apply (case_tac "y = 0")
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apply auto
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apply (subst pos_imp_zdiv_nonneg_iff, auto)
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apply (case_tac "y = 0")
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apply force
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apply (rule pos_mod_sign)
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apply arith
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done
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lemma transfer_nat_int_relations:
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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 even_nat_def)
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declare TransferMorphism_nat_int[transfer add return:
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transfer_nat_int_numerals
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transfer_nat_int_functions
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transfer_nat_int_function_closures
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transfer_nat_int_relations
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]
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(* first-order quantifiers *)
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lemma transfer_nat_int_quantifiers:
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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 TransferMorphism_nat_int[transfer add
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return: transfer_nat_int_quantifiers
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cong: all_cong ex_cong]
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(* if *)
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lemma nat_if_cong: "(if P then (nat x) else (nat y)) =
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nat (if P then x else y)"
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by auto
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declare TransferMorphism_nat_int [transfer add return: nat_if_cong]
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(* operations with sets *)
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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:
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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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"{..n} = nat ` {0..int n}"
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"{m..n} = nat ` {int m..int n}" (* need all variants of these! *)
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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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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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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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"x >= 0 \<Longrightarrow> nat_set {x..y}"
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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 expand_set_eq 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: "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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by auto
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declare TransferMorphism_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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(* 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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apply auto
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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
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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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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> setsum f A >= 0"
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"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0"
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unfolding nat_set_def simp_implies_def
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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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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 setsum_cong and setprod_cong enough,
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with the previously mentioned rule turned on? *)
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lemma transfer_nat_int_sum_prod_cong:
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"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow>
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setsum f A = setsum 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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setprod f A = setprod g B"
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unfolding nat_set_def
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apply (subst setsum_cong, assumption)
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apply auto [2]
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apply (subst setprod_cong, assumption, auto)
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done
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declare TransferMorphism_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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(* lists *)
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definition
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embed_list :: "nat list \<Rightarrow> int list"
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where
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"embed_list l = map int l";
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definition
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nat_list :: "int list \<Rightarrow> bool"
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where
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"nat_list l = nat_set (set l)";
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definition
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return_list :: "int list \<Rightarrow> nat list"
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where
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"return_list l = map nat l";
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thm nat_0_le;
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lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow>
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embed_list (return_list l) = l";
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unfolding embed_list_def return_list_def nat_list_def nat_set_def
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apply (induct l);
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apply auto;
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done;
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lemma transfer_nat_int_list_functions:
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"l @ m = return_list (embed_list l @ embed_list m)"
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"[] = return_list []";
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unfolding return_list_def embed_list_def;
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apply auto;
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apply (induct l, auto);
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apply (induct m, auto);
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done;
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(*
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lemma transfer_nat_int_fold1: "fold f l x =
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fold (%x. f (nat x)) (embed_list l) x";
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*)
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subsection {* Set up transfer from int to nat *}
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(* set up transfer direction *)
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lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)"
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by (simp add: TransferMorphism_def)
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declare TransferMorphism_int_nat[transfer add
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mode: manual
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(* labels: int-nat *)
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return: nat_int
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]
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(* basic functions and relations *)
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definition
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is_nat :: "int \<Rightarrow> bool"
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where
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"is_nat x = (x >= 0)"
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lemma transfer_int_nat_numerals:
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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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lemma transfer_int_nat_functions:
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"(int x) + (int y) = int (x + y)"
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"(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: int_mult tsub_def int_power zdiv_int zmod_int)
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lemma transfer_int_nat_function_closures:
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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 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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"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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by (simp_all only: is_nat_def transfer_nat_int_function_closures)
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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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"(even (int x)) = (even x)"
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by (auto simp add: zdvd_int even_nat_def)
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declare TransferMorphism_int_nat[transfer add return:
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transfer_int_nat_numerals
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385 |
transfer_int_nat_functions
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386 |
transfer_int_nat_function_closures
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387 |
transfer_int_nat_relations
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388 |
UNIV_code
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389 |
]
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390 |
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391 |
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392 |
(* first-order quantifiers *)
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393 |
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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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398 |
apply auto [1]
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apply (subst ex_nat)
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apply auto
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done
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402 |
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declare TransferMorphism_int_nat[transfer add
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return: transfer_int_nat_quantifiers]
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405 |
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406 |
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407 |
(* if *)
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408 |
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lemma int_if_cong: "(if P then (int x) else (int y)) =
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int (if P then x else y)"
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by auto
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412 |
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declare TransferMorphism_int_nat [transfer add return: int_if_cong]
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414 |
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415 |
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416 |
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417 |
(* operations with sets *)
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418 |
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419 |
lemma transfer_int_nat_set_functions:
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"nat_set A \<Longrightarrow> card A = card (nat ` A)"
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421 |
"{} = int ` ({}::nat set)"
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"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)"
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"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)"
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"{x. x >= 0 & P x} = int ` {x. P(int x)}"
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"is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
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(* need all variants of these! *)
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427 |
by (simp_all only: is_nat_def transfer_nat_int_set_functions
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428 |
transfer_nat_int_set_function_closures
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429 |
transfer_nat_int_set_return_embed nat_0_le
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430 |
cong: transfer_nat_int_set_cong)
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431 |
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432 |
lemma transfer_int_nat_set_function_closures:
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"nat_set {}"
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434 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)"
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435 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)"
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436 |
"is_nat x \<Longrightarrow> nat_set {x..y}"
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437 |
"nat_set {x. x >= 0 & P x}"
|
|
438 |
"nat_set (int ` C)"
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439 |
"nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x"
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|
440 |
by (simp_all only: transfer_nat_int_set_function_closures is_nat_def)
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|
441 |
|
|
442 |
lemma transfer_int_nat_set_relations:
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|
443 |
"nat_set A \<Longrightarrow> finite A = finite (nat ` A)"
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|
444 |
"is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)"
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|
445 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)"
|
|
446 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)"
|
|
447 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)"
|
|
448 |
by (simp_all only: is_nat_def transfer_nat_int_set_relations
|
|
449 |
transfer_nat_int_set_return_embed nat_0_le)
|
|
450 |
|
|
451 |
lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A"
|
|
452 |
by (simp only: transfer_nat_int_set_relations
|
|
453 |
transfer_nat_int_set_function_closures
|
|
454 |
transfer_nat_int_set_return_embed nat_0_le)
|
|
455 |
|
|
456 |
lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow>
|
|
457 |
{(x::nat). P x} = {x. P' x}"
|
|
458 |
by auto
|
|
459 |
|
|
460 |
declare TransferMorphism_int_nat[transfer add
|
|
461 |
return: transfer_int_nat_set_functions
|
|
462 |
transfer_int_nat_set_function_closures
|
|
463 |
transfer_int_nat_set_relations
|
|
464 |
transfer_int_nat_set_return_embed
|
|
465 |
cong: transfer_int_nat_set_cong
|
|
466 |
]
|
|
467 |
|
|
468 |
|
|
469 |
(* setsum and setprod *)
|
|
470 |
|
|
471 |
(* this handles the case where the *domain* of f is int *)
|
|
472 |
lemma transfer_int_nat_sum_prod:
|
|
473 |
"nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)"
|
|
474 |
"nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)"
|
|
475 |
apply (subst setsum_reindex)
|
|
476 |
apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff)
|
|
477 |
apply (subst setprod_reindex)
|
|
478 |
apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff
|
|
479 |
cong: setprod_cong)
|
|
480 |
done
|
|
481 |
|
|
482 |
(* this handles the case where the *range* of f is int *)
|
|
483 |
lemma transfer_int_nat_sum_prod2:
|
|
484 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)"
|
|
485 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow>
|
|
486 |
setprod f A = int(setprod (%x. nat (f x)) A)"
|
|
487 |
unfolding is_nat_def
|
|
488 |
apply (subst int_setsum, auto)
|
|
489 |
apply (subst int_setprod, auto simp add: cong: setprod_cong)
|
|
490 |
done
|
|
491 |
|
|
492 |
declare TransferMorphism_int_nat[transfer add
|
|
493 |
return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2
|
|
494 |
cong: setsum_cong setprod_cong]
|
|
495 |
|
|
496 |
|
|
497 |
subsection {* Test it out *}
|
|
498 |
|
|
499 |
(* nat to int *)
|
|
500 |
|
|
501 |
lemma ex1: "(x::nat) + y = y + x"
|
|
502 |
by auto
|
|
503 |
|
|
504 |
thm ex1 [transferred]
|
|
505 |
|
|
506 |
lemma ex2: "(a::nat) div b * b + a mod b = a"
|
|
507 |
by (rule mod_div_equality)
|
|
508 |
|
|
509 |
thm ex2 [transferred]
|
|
510 |
|
|
511 |
lemma ex3: "ALL (x::nat). ALL y. EX z. z >= x + y"
|
|
512 |
by auto
|
|
513 |
|
|
514 |
thm ex3 [transferred natint]
|
|
515 |
|
|
516 |
lemma ex4: "(x::nat) >= y \<Longrightarrow> (x - y) + y = x"
|
|
517 |
by auto
|
|
518 |
|
|
519 |
thm ex4 [transferred]
|
|
520 |
|
|
521 |
lemma ex5: "(2::nat) * (SUM i <= n. i) = n * (n + 1)"
|
|
522 |
by (induct n rule: nat_induct, auto)
|
|
523 |
|
|
524 |
thm ex5 [transferred]
|
|
525 |
|
|
526 |
theorem ex6: "0 <= (n::int) \<Longrightarrow> 2 * \<Sum>{0..n} = n * (n + 1)"
|
|
527 |
by (rule ex5 [transferred])
|
|
528 |
|
|
529 |
thm ex6 [transferred]
|
|
530 |
|
|
531 |
thm ex5 [transferred, transferred]
|
|
532 |
|
|
533 |
end
|