# Theory Fixrec

theory Fixrec
imports Cprod Sprod Ssum One Fix
```(*  Title:      HOL/HOLCF/Fixrec.thy
Author:     Amber Telfer and Brian Huffman
*)

section "Package for defining recursive functions in HOLCF"

theory Fixrec
imports Cprod Sprod Ssum Up One Tr Fix
keywords "fixrec" :: thy_decl
begin

default_sort cpo

pcpodef 'a match = "UNIV::(one ++ 'a u) set"
by simp_all

definition
fail :: "'a match" where
"fail = Abs_match (sinl⋅ONE)"

definition
succeed :: "'a → 'a match" where
"succeed = (Λ x. Abs_match (sinr⋅(up⋅x)))"

lemma matchE [case_names bottom fail succeed, cases type: match]:
"⟦p = ⊥ ⟹ Q; p = fail ⟹ Q; ⋀x. p = succeed⋅x ⟹ Q⟧ ⟹ Q"
unfolding fail_def succeed_def
apply (cases p, rename_tac r)
apply (rule_tac p=r in ssumE, simp add: Abs_match_strict)
apply (rule_tac p=x in oneE, simp, simp)
apply (rule_tac p=y in upE, simp, simp add: cont_Abs_match)
done

lemma succeed_defined [simp]: "succeed⋅x ≠ ⊥"
by (simp add: succeed_def cont_Abs_match Abs_match_bottom_iff)

lemma fail_defined [simp]: "fail ≠ ⊥"

lemma succeed_eq [simp]: "(succeed⋅x = succeed⋅y) = (x = y)"
by (simp add: succeed_def cont_Abs_match Abs_match_inject)

lemma succeed_neq_fail [simp]:
"succeed⋅x ≠ fail" "fail ≠ succeed⋅x"
by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject)

subsubsection ‹Run operator›

definition
run :: "'a match → 'a::pcpo" where
"run = (Λ m. sscase⋅⊥⋅(fup⋅ID)⋅(Rep_match m))"

text ‹rewrite rules for run›

lemma run_strict [simp]: "run⋅⊥ = ⊥"
unfolding run_def

lemma run_fail [simp]: "run⋅fail = ⊥"
unfolding run_def fail_def

lemma run_succeed [simp]: "run⋅(succeed⋅x) = x"
unfolding run_def succeed_def
by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)

definition
mplus :: "'a match → 'a match → 'a match" where
"mplus = (Λ m1 m2. sscase⋅(Λ _. m2)⋅(Λ _. m1)⋅(Rep_match m1))"

abbreviation
mplus_syn :: "['a match, 'a match] ⇒ 'a match"  (infixr "+++" 65)  where
"m1 +++ m2 == mplus⋅m1⋅m2"

text ‹rewrite rules for mplus›

lemma mplus_strict [simp]: "⊥ +++ m = ⊥"
unfolding mplus_def

lemma mplus_fail [simp]: "fail +++ m = m"
unfolding mplus_def fail_def

lemma mplus_succeed [simp]: "succeed⋅x +++ m = succeed⋅x"
unfolding mplus_def succeed_def
by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)

lemma mplus_fail2 [simp]: "m +++ fail = m"
by (cases m, simp_all)

lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
by (cases x, simp_all)

subsection ‹Match functions for built-in types›

default_sort pcpo

definition
match_bottom :: "'a → 'c match → 'c match"
where
"match_bottom = (Λ x k. seq⋅x⋅fail)"

definition
match_Pair :: "'a::cpo × 'b::cpo → ('a → 'b → 'c match) → 'c match"
where
"match_Pair = (Λ x k. csplit⋅k⋅x)"

definition
match_spair :: "'a ⊗ 'b → ('a → 'b → 'c match) → 'c match"
where
"match_spair = (Λ x k. ssplit⋅k⋅x)"

definition
match_sinl :: "'a ⊕ 'b → ('a → 'c match) → 'c match"
where
"match_sinl = (Λ x k. sscase⋅k⋅(Λ b. fail)⋅x)"

definition
match_sinr :: "'a ⊕ 'b → ('b → 'c match) → 'c match"
where
"match_sinr = (Λ x k. sscase⋅(Λ a. fail)⋅k⋅x)"

definition
match_up :: "'a::cpo u → ('a → 'c match) → 'c match"
where
"match_up = (Λ x k. fup⋅k⋅x)"

definition
match_ONE :: "one → 'c match → 'c match"
where
"match_ONE = (Λ ONE k. k)"

definition
match_TT :: "tr → 'c match → 'c match"
where
"match_TT = (Λ x k. If x then k else fail)"

definition
match_FF :: "tr → 'c match → 'c match"
where
"match_FF = (Λ x k. If x then fail else k)"

lemma match_bottom_simps [simp]:
"match_bottom⋅x⋅k = (if x = ⊥ then ⊥ else fail)"

lemma match_Pair_simps [simp]:
"match_Pair⋅(x, y)⋅k = k⋅x⋅y"

lemma match_spair_simps [simp]:
"⟦x ≠ ⊥; y ≠ ⊥⟧ ⟹ match_spair⋅(:x, y:)⋅k = k⋅x⋅y"
"match_spair⋅⊥⋅k = ⊥"

lemma match_sinl_simps [simp]:
"x ≠ ⊥ ⟹ match_sinl⋅(sinl⋅x)⋅k = k⋅x"
"y ≠ ⊥ ⟹ match_sinl⋅(sinr⋅y)⋅k = fail"
"match_sinl⋅⊥⋅k = ⊥"

lemma match_sinr_simps [simp]:
"x ≠ ⊥ ⟹ match_sinr⋅(sinl⋅x)⋅k = fail"
"y ≠ ⊥ ⟹ match_sinr⋅(sinr⋅y)⋅k = k⋅y"
"match_sinr⋅⊥⋅k = ⊥"

lemma match_up_simps [simp]:
"match_up⋅(up⋅x)⋅k = k⋅x"
"match_up⋅⊥⋅k = ⊥"

lemma match_ONE_simps [simp]:
"match_ONE⋅ONE⋅k = k"
"match_ONE⋅⊥⋅k = ⊥"

lemma match_TT_simps [simp]:
"match_TT⋅TT⋅k = k"
"match_TT⋅FF⋅k = fail"
"match_TT⋅⊥⋅k = ⊥"

lemma match_FF_simps [simp]:
"match_FF⋅FF⋅k = k"
"match_FF⋅TT⋅k = fail"
"match_FF⋅⊥⋅k = ⊥"

subsection ‹Mutual recursion›

text ‹
The following rules are used to prove unfolding theorems from
fixed-point definitions of mutually recursive functions.
›

lemma Pair_equalI: "⟦x ≡ fst p; y ≡ snd p⟧ ⟹ (x, y) ≡ p"
by simp

lemma Pair_eqD1: "(x, y) = (x', y') ⟹ x = x'"
by simp

lemma Pair_eqD2: "(x, y) = (x', y') ⟹ y = y'"
by simp

lemma def_cont_fix_eq:
"⟦f ≡ fix⋅(Abs_cfun F); cont F⟧ ⟹ f = F f"
by (simp, subst fix_eq, simp)

lemma def_cont_fix_ind:
"⟦f ≡ fix⋅(Abs_cfun F); cont F; adm P; P ⊥; ⋀x. P x ⟹ P (F x)⟧ ⟹ P f"

text ‹lemma for proving rewrite rules›

lemma ssubst_lhs: "⟦t = s; P s = Q⟧ ⟹ P t = Q"
by simp

subsection ‹Initializing the fixrec package›

ML_file "Tools/holcf_library.ML"
ML_file "Tools/fixrec.ML"

method_setup fixrec_simp = ‹
Scan.succeed (SIMPLE_METHOD' o Fixrec.fixrec_simp_tac)
› "pattern prover for fixrec constants"

setup ‹
[ (@{const_name up}, @{const_name match_up}),
(@{const_name sinl}, @{const_name match_sinl}),
(@{const_name sinr}, @{const_name match_sinr}),
(@{const_name spair}, @{const_name match_spair}),
(@{const_name Pair}, @{const_name match_Pair}),
(@{const_name ONE}, @{const_name match_ONE}),
(@{const_name TT}, @{const_name match_TT}),
(@{const_name FF}, @{const_name match_FF}),
(@{const_name bottom}, @{const_name match_bottom}) ]
›

hide_const (open) succeed fail run

end
```