theory QuotTest
imports QuotMain
begin


section {* various tests for quotient types*}
datatype trm =
  var  "nat"
| app  "trm" "trm"
| lam  "nat" "trm"

axiomatization
  RR :: "trm \<Rightarrow> trm \<Rightarrow> bool"
where
  r_eq: "EQUIV RR"

print_quotients

quotient qtrm = trm / "RR"
  apply(rule r_eq)
  done

print_quotients

typ qtrm
term Rep_qtrm
term REP_qtrm
term Abs_qtrm
term ABS_qtrm
thm QUOT_TYPE_qtrm
thm QUOTIENT_qtrm
thm REP_qtrm_def

(* Test interpretation *)
thm QUOT_TYPE_I_qtrm.thm11
thm QUOT_TYPE.thm11

print_theorems

thm Rep_qtrm

text {* another test *}
datatype 'a trm' =
  var'  "'a"
| app'  "'a trm'" "'a trm'"
| lam'  "'a" "'a trm'"

consts R' :: "'a trm' \<Rightarrow> 'a trm' \<Rightarrow> bool"
axioms r_eq': "EQUIV R'"

quotient qtrm' = "'a trm'" / "R'"
  apply(rule r_eq')
  done

print_quotients
print_theorems

term ABS_qtrm'
term REP_qtrm'
thm QUOT_TYPE_qtrm'
thm QUOTIENT_qtrm'
thm Rep_qtrm'


text {* a test with lists of terms *}
datatype t =
  vr "string"
| ap "t list"
| lm "string" "t"

consts Rt :: "t \<Rightarrow> t \<Rightarrow> bool"
axioms t_eq: "EQUIV Rt"

quotient qt = "t" / "Rt"
  by (rule t_eq)

print_quotients

ML {*
  get_fun repF [(@{typ qt}, @{typ qt})] @{context} @{typ "((((qt \<Rightarrow> qt) \<Rightarrow> qt) \<Rightarrow> qt) list) * nat"}
  |> Syntax.string_of_term @{context}
  |> writeln
*}

ML {*
  get_fun absF [(@{typ qt}, @{typ t})] @{context} @{typ "qt * nat"}
  |> Syntax.string_of_term @{context}
  |> writeln
*}

ML {*
  get_fun absF [(@{typ qt}, @{typ t})] @{context} @{typ "(qt \<Rightarrow> qt) \<Rightarrow> qt"}
  |> Syntax.pretty_term @{context}
  |> Pretty.string_of
  |> writeln
*}

(* A test whether get_fun works properly
consts bla :: "(t \<Rightarrow> t) \<Rightarrow> t"
local_setup {*
    make_const_def @{binding bla'} @{term "bla"} NoSyn @{typ "t"} @{typ "qt"} lthy |> snd
*}
*)

consts Rl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
axioms Rl_eq: "EQUIV Rl"

quotient ql = "'a list" / "Rl"
  by (rule Rl_eq)

ML {* val al = snd (dest_Free (term_of @{cpat "f :: ?'a list"})) *}
ML {* val aq = snd (dest_Free (term_of @{cpat "f :: ?'a ql"})) *}
ML {* val ttt = snd (dest_Free (term_of @{cpat "f :: ?'a ql \<Rightarrow> ?'a ql"})) *}

ML {*
  fst (get_fun absF [(aq, al)] @{context} ttt)
  |> cterm_of @{theory}
*}

ML {* val ttt2 = snd (dest_Free (term_of @{cpat "f :: nat ql \<Rightarrow> bool ql \<Rightarrow> 'a ql"})) *}

ML {*
  fst (get_fun absF [(aq, al), (@{typ "nat ql"}, @{typ "nat list"}), (@{typ "bool ql"}, @{typ "bool list"})] @{context} ttt2)
  |> cterm_of @{theory}
*}

quotient_def (for qt)
  VR ::"string \<Rightarrow> qt"
where
  "VR \<equiv> vr"

quotient_def (for qt)
  AP ::"qt list \<Rightarrow> qt"
where
  "AP \<equiv> ap"

quotient_def (for qt)
  LM ::"string \<Rightarrow> qt \<Rightarrow> qt"
where
  "LM \<equiv> lm"

term vr
term ap
term lm
thm VR_def AP_def LM_def
term LM
term VR
term AP

text {* a test with functions *}
datatype 'a t' =
  vr' "string"
| ap' "('a t') * ('a t')"
| lm' "'a" "string \<Rightarrow> ('a t')"

consts Rt' :: "('a t') \<Rightarrow> ('a t') \<Rightarrow> bool"
axioms t_eq': "EQUIV Rt'"

quotient qt' = "'a t'" / "Rt'"
  apply(rule t_eq')
  done

print_quotients
print_theorems


quotient_def (for "'a qt'")
  VR' ::"string \<Rightarrow> 'a qt"
where
  "VR' \<equiv> vr'"

quotient_def (for "'a qt'")
  AP' ::"('a qt') * ('a qt') \<Rightarrow> 'a qt"
where
  "AP' \<equiv> ap'"

quotient_def (for "'a qt'")
  LM' ::"'a \<Rightarrow> string \<Rightarrow> 'a qt'"
where
  "LM' \<equiv> lm'"

term vr'
term ap'
term ap'
thm VR'_def AP'_def LM'_def
term LM'
term VR'
term AP'

text {* Tests of regularise *}
ML {*
  cterm_of @{theory} (regularise @{term "\<lambda>x :: int. x"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (regularise @{term "\<lambda>x :: trm. x"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (regularise @{term "\<forall>x :: trm. P x"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (regularise @{term "\<exists>x :: trm. P x"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (regularise @{term "All (P :: trm \<Rightarrow> bool)"} @{typ "trm"} @{term "RR"} @{context});
*}

ML {*
  cterm_of @{theory} (regularise @{term "\<exists>(y::trm). P (\<lambda>(x::trm). y)"} @{typ "trm"}
     @{term "RR"} @{context});
  cterm_of @{theory} (my_reg @{term "RR"} @{term "\<exists>(y::trm). P (\<lambda>(x::trm). y)"})
*}

ML {*
  cterm_of @{theory} (regularise @{term "\<lambda>x::trm. x"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (my_reg @{term "RR"} @{term "\<lambda>x::trm. x"})
*}

ML {*
  cterm_of @{theory} (regularise @{term "\<forall>(x::trm) (y::trm). P x y"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (my_reg @{term "RR"} @{term "\<forall>(x::trm) (y::trm). P x y"})
*}

ML {*
  cterm_of @{theory} (regularise @{term "\<forall>x::trm. P x"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (my_reg @{term "RR"} @{term "\<forall>x::trm. P x"})
*}

ML {*
  cterm_of @{theory} (regularise @{term "\<exists>x::trm. P x"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (my_reg @{term "RR"} @{term "\<exists>x::trm. P x"})
*}

(* my version is not eta-expanded, but that should be OK *)
ML {*
  cterm_of @{theory} (regularise @{term "All (P::trm \<Rightarrow> bool)"} @{typ "trm"} @{term "RR"} @{context});
  cterm_of @{theory} (my_reg @{term "RR"} @{term "All (P::trm \<Rightarrow> bool)"})
*}