# Theory ComputeHOL

theory ComputeHOL
imports Complex_Main Compute_Oracle
```theory ComputeHOL
imports Complex_Main "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 = (λ x y. x)" by ((rule ext)+,auto)
lemma If_False: "If False = (λ 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 case_prod_simp: "case_prod f (x,y) = f x y" by simp

lemmas compute_pair = compute_fst compute_snd compute_pair_eq case_prod_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 case_option_compute :: "'b option ⇒ 'a ⇒ ('b ⇒ 'a) ⇒ 'a"
where "case_option_compute opt a f = case_option a f opt"

lemma case_option_compute: "case_option = (λ a f opt. case_option_compute opt a f)"

lemma case_option_compute_None: "case_option_compute None = (λ a f. a)"
apply (rule ext)+
done

lemma case_option_compute_Some: "case_option_compute (Some x) = (λ a f. f x)"
apply (rule ext)+
done

lemmas compute_case_option = case_option_compute case_option_compute_None case_option_compute_Some

lemmas compute_option = compute_the compute_None_Some_eq compute_Some_None_eq compute_None_None_eq compute_Some_Some_eq compute_case_option

(**** 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_case_list ***)

definition case_list_compute :: "'b list ⇒ 'a ⇒ ('b ⇒ 'b list ⇒ 'a) ⇒ 'a"
where "case_list_compute l a f = case_list a f l"

lemma case_list_compute: "case_list = (λ (a::'a) f (l::'b list). case_list_compute l a f)"
apply (rule ext)+
done

lemma case_list_compute_empty: "case_list_compute ([]::'b list) = (λ (a::'a) f. a)"
apply (rule ext)+
done

lemma case_list_compute_cons: "case_list_compute (u#v) = (λ (a::'a) f. (f (u::'b) v))"
apply (rule ext)+
done

lemmas compute_case_list = case_list_compute case_list_compute_empty case_list_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_case_list 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

ML ‹
signature ComputeHOL =
sig
val prep_thms : thm list -> thm list
val to_meta_eq : thm -> thm
val to_hol_eq : thm -> thm
val symmetric : thm -> thm
val trans : thm -> thm -> thm
end

structure ComputeHOL : ComputeHOL =
struct

local
fun lhs_of eq = fst (Thm.dest_equals (Thm.cprop_of eq));
in
fun rewrite_conv [] ct = raise CTERM ("rewrite_conv", [ct])
| rewrite_conv (eq :: eqs) ct =
Thm.instantiate (Thm.match (lhs_of eq, ct)) eq
handle Pattern.MATCH => rewrite_conv eqs ct;
end

val convert_conditions = Conv.fconv_rule (Conv.prems_conv ~1 (Conv.try_conv (rewrite_conv [@{thm "Trueprop_eq_eq"}])))

val eq_th = @{thm "HOL.eq_reflection"}
val meta_eq_trivial = @{thm "ComputeHOL.meta_eq_trivial"}
val bool_to_true = @{thm "ComputeHOL.bool_to_true"}

fun to_meta_eq th = eq_th OF [th] handle THM _ => meta_eq_trivial OF [th] handle THM _ => bool_to_true OF [th]

fun to_hol_eq th = @{thm "meta_eq_imp_eq"} OF [th] handle THM _ => @{thm "eq_trivial"} OF [th]

fun prep_thms ths = map (convert_conditions o to_meta_eq) ths

fun symmetric th = @{thm "HOL.sym"} OF [th] handle THM _ => @{thm "Pure.symmetric"} OF [th]

local
val trans_HOL = @{thm "HOL.trans"}
val trans_HOL_1 = @{thm "ComputeHOL.transmeta_1"}
val trans_HOL_2 = @{thm "ComputeHOL.transmeta_2"}
val trans_HOL_3 = @{thm "ComputeHOL.transmeta_3"}
fun tr [] th1 th2 = trans_HOL OF [th1, th2]
| tr (t::ts) th1 th2 = (t OF [th1, th2] handle THM _ => tr ts th1 th2)
in
fun trans th1 th2 = tr [trans_HOL, trans_HOL_1, trans_HOL_2, trans_HOL_3] th1 th2
end

end
›

end
```