theory ComputeHOL imports Main "~~/src/Tools/Compute_Oracle/Compute_Oracle" begin lemma Trueprop_eq_eq: "Trueprop X == (X == True)" by (simp add: atomize_eq) lemma meta_eq_trivial: "x == y \ x == y" by simp lemma meta_eq_imp_eq: "x == y \ x = y" by auto lemma eq_trivial: "x = y \ x = y" by auto lemma bool_to_true: "x :: bool \ x == True" by simp lemma transmeta_1: "x = y \ y == z \ x = z" by simp lemma transmeta_2: "x == y \ y = z \ x = z" by simp lemma transmeta_3: "x == y \ y == z \ x = z" by simp (**** compute_if ****) lemma If_True: "If True = (\ x y. x)" by ((rule ext)+,auto) lemma If_False: "If False = (\ x y. y)" by ((rule ext)+, auto) lemmas compute_if = If_True If_False (**** compute_bool ****) lemma bool1: "(\ True) = False" by blast lemma bool2: "(\ False) = True" by blast lemma bool3: "(P \ True) = P" by blast lemma bool4: "(True \ P) = P" by blast lemma bool5: "(P \ False) = False" by blast lemma bool6: "(False \ P) = False" by blast lemma bool7: "(P \ True) = True" by blast lemma bool8: "(True \ P) = True" by blast lemma bool9: "(P \ False) = P" by blast lemma bool10: "(False \ P) = P" by blast lemma bool11: "(True \ P) = P" by blast lemma bool12: "(P \ True) = True" by blast lemma bool13: "(True \ P) = P" by blast lemma bool14: "(P \ False) = (\ P)" by blast lemma bool15: "(False \ P) = True" by blast lemma bool16: "(False = False) = True" by blast lemma bool17: "(True = True) = True" by blast lemma bool18: "(False = True) = False" by blast lemma bool19: "(True = False) = False" by blast lemmas compute_bool = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10 bool11 bool12 bool13 bool14 bool15 bool16 bool17 bool18 bool19 (*** compute_pair ***) lemma compute_fst: "fst (x,y) = x" by simp lemma compute_snd: "snd (x,y) = y" by simp lemma compute_pair_eq: "((a, b) = (c, d)) = (a = c \ b = d)" by auto lemma prod_case_simp: "prod_case f (x,y) = f x y" by simp lemmas compute_pair = compute_fst compute_snd compute_pair_eq prod_case_simp (*** compute_option ***) lemma compute_the: "the (Some x) = x" by simp lemma compute_None_Some_eq: "(None = Some x) = False" by auto lemma compute_Some_None_eq: "(Some x = None) = False" by auto lemma compute_None_None_eq: "(None = None) = True" by auto lemma compute_Some_Some_eq: "(Some x = Some y) = (x = y)" by auto definition option_case_compute :: "'b option \ 'a \ ('b \ 'a) \ 'a" where "option_case_compute opt a f = option_case a f opt" lemma option_case_compute: "option_case = (\ a f opt. option_case_compute opt a f)" by (simp add: option_case_compute_def) lemma option_case_compute_None: "option_case_compute None = (\ a f. a)" apply (rule ext)+ apply (simp add: option_case_compute_def) done lemma option_case_compute_Some: "option_case_compute (Some x) = (\ a f. f x)" apply (rule ext)+ apply (simp add: option_case_compute_def) done lemmas compute_option_case = option_case_compute option_case_compute_None option_case_compute_Some lemmas compute_option = compute_the compute_None_Some_eq compute_Some_None_eq compute_None_None_eq compute_Some_Some_eq compute_option_case (**** compute_list_length ****) lemma length_cons:"length (x#xs) = 1 + (length xs)" by simp lemma length_nil: "length [] = 0" by simp lemmas compute_list_length = length_nil length_cons (*** compute_list_case ***) definition list_case_compute :: "'b list \ 'a \ ('b \ 'b list \ 'a) \ 'a" where "list_case_compute l a f = list_case a f l" lemma list_case_compute: "list_case = (\ (a::'a) f (l::'b list). list_case_compute l a f)" apply (rule ext)+ apply (simp add: list_case_compute_def) done lemma list_case_compute_empty: "list_case_compute ([]::'b list) = (\ (a::'a) f. a)" apply (rule ext)+ apply (simp add: list_case_compute_def) done lemma list_case_compute_cons: "list_case_compute (u#v) = (\ (a::'a) f. (f (u::'b) v))" apply (rule ext)+ apply (simp add: list_case_compute_def) done lemmas compute_list_case = list_case_compute list_case_compute_empty list_case_compute_cons (*** compute_list_nth ***) (* Of course, you will need computation with nats for this to work \ *) lemma compute_list_nth: "((x#xs) ! n) = (if n = 0 then x else (xs ! (n - 1)))" by (cases n, auto) (*** compute_list ***) lemmas compute_list = compute_list_case compute_list_length compute_list_nth (*** compute_let ***) lemmas compute_let = Let_def (***********************) (* Everything together *) (***********************) lemmas compute_hol = compute_if compute_bool compute_pair compute_option compute_list compute_let end