{- |

Module : $Header$

Description : Computation of the constraints needed for free definition links.

Copyright : (c) Adrian Riesco, Facultad de Informatica UCM 2009

License : similar to LGPL, see HetCATS/LICENSE.txt or LIZENZ.txt

Maintainer : ariesco@fdi.ucm.es

Stability : experimental

Portability : portable

Computation of the constraints needed for free definition links.

-}

module CASL.Freeness where

import CASL.Sign

import CASL.Morphism

import CASL.StaticAna

import CASL.AS_Basic_CASL

import Logic.Prover ()

import Common.Id

import Common.Result

import Common.AS_Annotation

import qualified Common.Lib.Rel as Rel

import Data.Char

import Data.Maybe

import Data.List (findIndex)

import qualified Data.Map as Map

import qualified Data.Set as Set

-- | main function, in charge of computing the quotient term algebra

quotientTermAlgebra :: CASLMor -- sigma : Sigma -> SigmaM

-> [Named CASLFORMULA] -- Th(M)

-> Result (CASLSign, -- SigmaK

CASLMor, -- iota : SigmaM' -> SigmaK

[Named CASLFORMULA] -- Ax(K)

)

quotientTermAlgebra sigma sens = case horn_clauses sens of

False -> mkError "Not Horn clauses" sens

True -> let

sigma_0 = msource sigma

ss_0 = sortSet sigma_0

sigma_m = mtarget sigma

iota_mor = create_iota_mor ss_0 sigma_m

sigma_k = mtarget iota_mor

axs = create_axs ss_0 sigma_m sigma_k sens

in return (sigma_k, iota_mor, axs)

horn_clauses :: [Named CASLFORMULA] -> Bool

horn_clauses _ = True

-- -- | checks if the formulas are horn clauses

-- horn_clauses :: [Named CASLFORMULA] -> Bool

-- horn_clauses = foldr ((&&) . horn_clause) True

-- -- | check if the formula is a horn clause

-- horn_clause :: Named CASLFORMULA -> Bool

-- horn_clause sen = sl' == Atomic || sl' == Horn

-- where sl = minSublogic $ sentence sen :: CASL_SL ()

-- sl' = which_logic sl

-- | generates the axioms of the module K

create_axs :: Set.Set SORT -> CASLSign -> CASLSign -> [Named CASLFORMULA]

-> [Named CASLFORMULA]

create_axs sg_ss sg_m sg_k sens = forms

where ss_m = sortSet sg_m

ss = Set.map mkFreeName $ Set.difference ss_m sg_ss

sr = sortRel sg_k

comps = ops2comp $ opMap sg_k

ctor_sen = [freeCons (ss, sr, comps)]

make_axs = make_forms sg_ss

h_axs_ops = homomorphism_axs_ops $ opMap sg_m

h_axs_preds = homomorphism_axs_preds $ predMap sg_m

h_axs_surj = hom_surjectivity $ sortSet sg_m

q_axs = quotient_axs sens

symm_ax = symmetry_axs ss_m

tran_ax = transitivity_axs ss_m

cong_ax = congruence_axs (opMap sg_m)

satThM = sat_thm_ax sens

prems = mk_conj [symm_ax, tran_ax, cong_ax, satThM]

ltkh = larger_then_ker_h ss_m (predMap sg_m)

krnl_axs = [mkKernelAx ss_m (predMap sg_m) prems ltkh]

forms = concat [ctor_sen, make_axs, h_axs_ops, h_axs_preds,

h_axs_surj, q_axs, krnl_axs]

-- | generates the formulas relating the make functions with the homomorphism

make_forms :: Set.Set SORT -> [Named CASLFORMULA]

make_forms = Set.fold ((:) . make_form) []

-- | generates the formulas relating the make function for this sort

-- with the homomorphism

make_form :: SORT -> Named CASLFORMULA

make_form s = makeNamed "" q_eq

where free_s = mkFreeName s

v = newVar s

ot_mk = Op_type Total [s] free_s nullRange

os_mk = Qual_op_name makeId ot_mk nullRange

term_mk = Application os_mk [v] nullRange

ot_hom = Op_type Partial [free_s] s nullRange

os_hom = Qual_op_name homId ot_hom nullRange

term_hom = Application os_hom [term_mk] nullRange

eq = Strong_equation term_hom v nullRange

q_eq = quantifyUniversally eq

-- | computes the last part of the axioms to assert the kernel of h

larger_then_ker_h :: Set.Set SORT -> Map.Map Id (Set.Set PredType) -> CASLFORMULA

larger_then_ker_h ss mis = conj

where ltkhs = ltkh_sorts ss

ltkhp = ltkh_preds mis

conj = mk_conj (ltkhs ++ ltkhp)

-- | computes the second part of the conjunction of the formula "largerThanKerH"

-- from the kernel of H

ltkh_preds :: Map.Map Id (Set.Set PredType) -> [CASLFORMULA]

ltkh_preds = Map.foldWithKey ltkh_preds_set []

-- | computes the second part of the conjunction of the formula "largerThanKerH"

-- from the kernel of H for a concrete predicate identifier

ltkh_preds_set :: Id -> Set.Set PredType -> [CASLFORMULA] -> [CASLFORMULA]

ltkh_preds_set name sp acc = forms

where forms = Set.fold ((:) . (ltkh_preds_aux name)) acc sp

-- | computes the second part of the conjunction of the formula "largerThanKerH"

-- from the kernel of H for a concrete predicate profile

ltkh_preds_aux :: Id -> PredType -> CASLFORMULA

ltkh_preds_aux name (PredType args) = imp'

where free_name = mkFreeName name

free_args = map mkFreeName args

psi = psiName name

pt = Pred_type free_args nullRange

ps_name = Qual_pred_name free_name pt nullRange

ps_psi = Qual_pred_name psi pt nullRange

vars = createVars 1 free_args

prem = Predication ps_name vars nullRange

concl = Predication ps_psi vars nullRange

imp = Implication prem concl True nullRange

imp' = quantifyUniversally imp

-- | computes the first part of the conjunction of the formula "largerThanKerH"

-- from the kernel of H

ltkh_sorts :: Set.Set SORT -> [CASLFORMULA]

ltkh_sorts = Set.fold ((:) . ltkh_sort) []

-- | computes the first part of the conjunction of the formula "largerThanKerH"

-- from the kernel of H for a concrete sort

ltkh_sort :: SORT -> CASLFORMULA

ltkh_sort s = imp'

where free_s = mkFreeName s

v1 = newVarIndex 1 free_s

v2 = newVarIndex 2 free_s

phi = phiName s

pt = Pred_type [free_s, free_s] nullRange

ps = Qual_pred_name phi pt nullRange

ot_hom = Op_type Partial [free_s] s nullRange

name_hom = Qual_op_name homId ot_hom nullRange

t1 = Application name_hom [v1] nullRange

t2 = Application name_hom [v2] nullRange

prem = Existl_equation t1 t2 nullRange

concl = Predication ps [v1, v2] nullRange

imp = Implication prem concl True nullRange

imp' = quantifyUniversally imp

sat_thm_ax :: [Named CASLFORMULA] -> CASLFORMULA

sat_thm_ax forms = final_form

where forms' = map (free_formula . sentence) forms

final_form = mk_conj forms'

-- | computes the axiom for the congruence of the kernel of h

congruence_axs :: OpMap -> CASLFORMULA

congruence_axs om = conj

where axs = Map.foldWithKey congruence_ax [] om

conj = mk_conj axs

-- | computes the axiom for the congruence of the kernel of h

-- for a single operator id

congruence_ax :: Id -> Set.Set OpType -> [CASLFORMULA] -> [CASLFORMULA]

congruence_ax name sot acc = set_forms

where set_forms = Set.fold ((:) . (congruence_ax_aux name)) acc sot

-- | computes the axiom for the congruence of the kernel of h

-- for a single type of an operator id

congruence_ax_aux :: Id -> OpType -> CASLFORMULA

congruence_ax_aux name ot = cong_form'

where OpType _ args res = ot

free_name = mkFreeName name

free_args = map mkFreeName args

free_res = mkFreeName res

free_ot = Op_type Total free_args free_res nullRange

free_os = Qual_op_name free_name free_ot nullRange

lgth = length free_args

xs = createVars 1 free_args

ys = createVars (1 + lgth) free_args

fst_term = Application free_os xs nullRange

snd_term = Application free_os ys nullRange

phi = phiName res

pt = Pred_type [free_res, free_res] nullRange

ps = Qual_pred_name phi pt nullRange

fst_form = Predication ps [fst_term, fst_term] nullRange

snd_form = Predication ps [snd_term, snd_term] nullRange

vars_forms = congruence_ax_vars args xs ys

conj = mk_conj $ fst_form : snd_form : vars_forms

concl = Predication ps [fst_term, snd_term] nullRange

cong_form = Implication conj concl True nullRange

cong_form' = quantifyUniversally cong_form

-- | computes the formulas for the relations between variables

congruence_ax_vars :: [SORT] -> [CASLTERM] -> [CASLTERM] -> [CASLFORMULA]

congruence_ax_vars (s : ss) (x : xs) (y : ys) = form : forms

where phi = phiName s

free_s = mkFreeName s

pt = Pred_type [free_s, free_s] nullRange

ps = Qual_pred_name phi pt nullRange

form = Predication ps [x, y] nullRange

forms = congruence_ax_vars ss xs ys

congruence_ax_vars _ _ _ = []

-- | computes the transitivity axioms for the kernel of h

transitivity_axs :: Set.Set SORT -> CASLFORMULA

transitivity_axs ss = conj

where axs = Set.fold ((:) . transitivity_ax) [] ss

conj = mk_conj axs

-- | computes the transitivity axiom of a concrete sort for

-- the kernel of h

transitivity_ax :: SORT -> CASLFORMULA

transitivity_ax s = quant

where free_sort = mkFreeName s

v1@(Qual_var n1 _ _) = newVarIndex 1 free_sort

v2@(Qual_var n2 _ _) = newVarIndex 2 free_sort

v3@(Qual_var n3 _ _) = newVarIndex 3 free_sort

pt = Pred_type [free_sort, free_sort] nullRange

name = phiName s

ps = Qual_pred_name name pt nullRange

fst_form = Predication ps [v1, v2] nullRange

snd_form = Predication ps [v2, v3] nullRange

thr_form = Predication ps [v1, v3] nullRange

conj = mk_conj [fst_form, snd_form]

imp = Implication conj thr_form True nullRange

vd = [Var_decl [n1, n2, n3] free_sort nullRange]

quant = Quantification Universal vd imp nullRange

-- | computes the symmetry axioms for the kernel of h

symmetry_axs :: Set.Set SORT -> CASLFORMULA

symmetry_axs ss = conj

where axs = Set.fold ((:) . symmetry_ax) [] ss

conj = mk_conj axs

-- | computes the symmetry axiom of a concrete sort for the kernel of h

symmetry_ax :: SORT -> CASLFORMULA

symmetry_ax s = quant

where free_sort = mkFreeName s

v1@(Qual_var n1 _ _) = newVarIndex 1 free_sort

v2@(Qual_var n2 _ _) = newVarIndex 2 free_sort

pt = Pred_type [free_sort, free_sort] nullRange

name = phiName s

ps = Qual_pred_name name pt nullRange

lhs = Predication ps [v1, v2] nullRange

rhs = Predication ps [v2, v1] nullRange

inner_form = Implication lhs rhs True nullRange

vd = [Var_decl [n1, n2] free_sort nullRange]

quant = Quantification Universal vd inner_form nullRange

-- | generates the name of the phi variable of a concrete sort

phiName :: SORT -> Id

phiName s = mkId [mkSimpleId $ "Phi_" ++ show s]

-- | generates the name of the phi variable of a concrete predicate

psiName :: Id -> Id

psiName s = mkId [mkSimpleId $ "Psi_" ++ show s]

-- | creates the axiom for the kernel of h given the sorts and the predicates

-- in M, the premises and the conclusion

mkKernelAx :: Set.Set SORT -> Map.Map Id (Set.Set PredType) -> CASLFORMULA

-> CASLFORMULA -> Named CASLFORMULA

mkKernelAx ss preds prem conc = makeNamed "freeness_kernel" q2

where imp = Implication prem conc True nullRange

q1 = quantifyPredsSorts ss imp

q2 = quantifyPredsPreds preds q1

-- | applies the second order quantification to the formula for the given

-- set of sorts

quantifyPredsSorts :: Set.Set SORT -> CASLFORMULA -> CASLFORMULA

quantifyPredsSorts ss f = Set.fold quantifyPredsSort f ss

-- | applies the second order quantification to the formula for the given

-- sort

quantifyPredsSort :: SORT -> CASLFORMULA -> CASLFORMULA

quantifyPredsSort s f = q_form

where phi = phiName s

free_s = mkFreeName s

pt = Pred_type [free_s, free_s] nullRange

q_form = QuantPred phi pt f

-- | applies the second order quantification to the formula for the given

-- predicates

quantifyPredsPreds :: Map.Map Id (Set.Set PredType) -> CASLFORMULA -> CASLFORMULA

quantifyPredsPreds preds f = Map.foldWithKey quantifyPredsPredTypes f preds

-- | applies the second order quantification to the formula for the given

-- profiles

quantifyPredsPredTypes :: Id -> Set.Set PredType -> CASLFORMULA -> CASLFORMULA

quantifyPredsPredTypes name spt f = Set.fold (quantifyPredsPred name) f spt

-- | applies the second order quantification to the formula for the given

-- predicate

quantifyPredsPred :: Id -> PredType -> CASLFORMULA -> CASLFORMULA

quantifyPredsPred name (PredType args) f = q_form

where psi = psiName name

free_args = map mkFreeName args

pt = Pred_type free_args nullRange

q_form = QuantPred psi pt f

-- | creates a conjunction distinguishing cases on the size of the list

mk_conj :: [CASLFORMULA] -> CASLFORMULA

mk_conj [] = True_atom nullRange

mk_conj [f] = f

mk_conj fs = Conjunction fs nullRange

-- | given the axioms in the module M, the function computes the

-- axioms obtained for the homomorphisms

quotient_axs :: [Named CASLFORMULA] -> [Named CASLFORMULA]

quotient_axs = map quotient_ax

-- | given an axiom in the module M, the function computes the

-- axioms obtained for the homomorphisms

quotient_ax :: Named CASLFORMULA -> Named CASLFORMULA

quotient_ax nsen = nsen'

where sen = sentence nsen

sen' = homomorphy_form sen

nsen' = nsen { sentence = sen' }

-- | applies the homomorphism operator to the terms of the given formula

homomorphy_form :: CASLFORMULA -> CASLFORMULA

homomorphy_form (Quantification q vd f r) = Quantification q vd f' r

where f' = homomorphy_form f

homomorphy_form (Conjunction fs r) = Conjunction fs' r

where fs' = map homomorphy_form fs

homomorphy_form (Disjunction fs r) = Disjunction fs' r

where fs' = map homomorphy_form fs

homomorphy_form (Implication f1 f2 b r) = Implication f1' f2' b r

where f1' = homomorphy_form f1

f2' = homomorphy_form f2

homomorphy_form (Equivalence f1 f2 r) = Equivalence f1' f2' r

where f1' = homomorphy_form f1

f2' = homomorphy_form f2

homomorphy_form (Negation f r) = Negation f' r

where f' = homomorphy_form f

homomorphy_form (Predication ps ts r) = Predication ps ts' r

where ts' = map homomorphy_term ts

homomorphy_form (Definedness t r) = Definedness t' r

where t' = homomorphy_term t

homomorphy_form (Existl_equation t1 t2 r) = Existl_equation t1' t2' r

where t1' = homomorphy_term t1

t2' = homomorphy_term t2

homomorphy_form (Strong_equation t1 t2 r) = Strong_equation t1' t2' r

where t1' = homomorphy_term t1

t2' = homomorphy_term t2

homomorphy_form (Membership t s r) = Membership t' s r

where t' = homomorphy_term t

homomorphy_form (Mixfix_formula t) = Mixfix_formula t'

where t' = homomorphy_term t

homomorphy_form f = f

-- | applies the homomorphism operator to the term when possible

homomorphy_term :: CASLTERM -> CASLTERM

homomorphy_term t@(Qual_var _ s _) = t'

where ot_hom = Op_type Partial [mkFreeName s] s nullRange

name_hom = Qual_op_name homId ot_hom nullRange

t' = Application name_hom [t] nullRange

homomorphy_term t@(Application os _ _) = t'

where Qual_op_name _ ot _ = os

Op_type _ _ s _ = ot

ot_hom = Op_type Partial [mkFreeName s] s nullRange

name_hom = Qual_op_name homId ot_hom nullRange

t' = Application name_hom [t] nullRange

homomorphy_term t = t

hom_surjectivity :: Set.Set SORT -> [Named CASLFORMULA]

hom_surjectivity = Set.fold f []

where f = \ x y -> (sort_surj x) : y

-- | generates the formula to state the homomorphism is surjective

sort_surj :: SORT -> Named CASLFORMULA

sort_surj s = form'

where v1 = newVarIndex 0 $ mkFreeName s

id_v1 = mkSimpleId "V0"

vd1 = Var_decl [id_v1] (mkFreeName s) nullRange

v2 = newVarIndex 1 s

id_v2 = mkSimpleId "V1"

vd2 = Var_decl [id_v2] s nullRange

ot_hom = Op_type Partial [mkFreeName s] s nullRange

name_hom = Qual_op_name homId ot_hom nullRange

lhs = Application name_hom [v1] nullRange

inner_form = Existl_equation lhs v2 nullRange

inner_form' = Quantification Existential [vd1] inner_form nullRange

form = Quantification Universal [vd2] inner_form' nullRange

form' = makeNamed "" form

-- | generates the axioms for the homomorphisms applied to the predicates

homomorphism_axs_preds :: Map.Map Id (Set.Set PredType) -> [Named CASLFORMULA]

homomorphism_axs_preds = Map.foldWithKey g []

where f = \ p_name pt sens -> (homomorphism_form_pred p_name pt) : sens

g = \ p_name set_pt sens -> Set.fold (f p_name) sens set_pt

-- | generates the axioms for the homomorphisms applied to a predicate

homomorphism_form_pred :: Id -> PredType -> Named CASLFORMULA

homomorphism_form_pred name (PredType args) = named_form

where free_args = map mkFreeName args

vars_lhs = createVars 0 free_args

lhs_pt = Pred_type free_args nullRange

lhs_pred_name = Qual_pred_name (mkFreeName name) lhs_pt nullRange

lhs = Predication lhs_pred_name vars_lhs nullRange

inner_rhs = apply_hom_vars args vars_lhs

pt_rhs = Pred_type args nullRange

name_rhs = Qual_pred_name name pt_rhs nullRange

rhs = Predication name_rhs inner_rhs nullRange

form = Equivalence lhs rhs nullRange

form' = quantifyUniversally form

named_form = makeNamed "" form'

-- | generates the axioms for the homomorphisms applied to the operators

homomorphism_axs_ops :: OpMap -> [Named CASLFORMULA]

homomorphism_axs_ops = Map.foldWithKey g []

where f = \ op_name ot sens -> (homomorphism_form_op op_name ot) : sens

g = \ op_name set_ot sens -> Set.fold (f op_name) sens set_ot

-- | generates the axiom for the homomorphism applied to a concrete op

homomorphism_form_op :: Id -> OpType -> Named CASLFORMULA

homomorphism_form_op name (OpType _ args res) = named_form

where free_args = map mkFreeName args

vars_lhs = createVars 0 free_args

ot_lhs = Op_type Total free_args (mkFreeName res) nullRange

ot_hom = Op_type Partial [mkFreeName res] res nullRange

name_hom = Qual_op_name homId ot_hom nullRange

name_lhs = Qual_op_name (mkFreeName name) ot_lhs nullRange

inner_lhs = Application name_lhs vars_lhs nullRange

lhs = Application name_hom [inner_lhs] nullRange

ot_rhs = Op_type Total args res nullRange

name_rhs = Qual_op_name name ot_rhs nullRange

inner_rhs = apply_hom_vars args vars_lhs

rhs = Application name_rhs inner_rhs nullRange

form = Strong_equation lhs rhs nullRange

form' = quantifyUniversally form

named_form = makeNamed "" form'

-- | generates the variables for the homomorphisms

apply_hom_vars :: [SORT] -> [CASLTERM] -> [CASLTERM]

apply_hom_vars (s : ss) (t : ts) = t' : ts'

where ot_hom = Op_type Partial [mkFreeName s] s nullRange

name_hom = Qual_op_name homId ot_hom nullRange

t' = Application name_hom [t] nullRange

ts' = apply_hom_vars ss ts

apply_hom_vars _ _ = []

-- | generates a list of differents variables of the given sorts

createVars :: Int -> [SORT] -> [CASLTERM]

createVars _ [] = []

createVars i (s : ss) = var : ts

where var = newVarIndex i s

ts = createVars (i + 1) ss

-- | computes the set of components from the map of operators

ops2comp :: OpMap -> Set.Set Component

ops2comp = Map.foldWithKey g Set.empty

where f = \ n ot s -> Set.insert (Component n ot) s

g = \ name sot s -> Set.fold (f name) s sot

-- | computes the sentence for the constructors

freeCons :: GenAx -> Named CASLFORMULA

freeCons (sorts, rel, ops) = do

let sortList = Set.toList sorts

opSyms = map ( \ c -> let iden = compId c in Qual_op_name iden

(toOP_TYPE $ compType c) $ posOfId iden) $ Set.toList ops

injSyms = map ( \ (s, t) -> let p = posOfId s in

Qual_op_name (mkUniqueInjName s t)

(Op_type Total [s] t p) p) $ Rel.toList

$ Rel.irreflex rel

allSyms = opSyms ++ injSyms

resType _ (Op_name _) = False

resType s (Qual_op_name _ t _) = res_OP_TYPE t == s

getIndex s = maybe (-1) id $ findIndex (== s) sortList

addIndices (Op_name _) =

error "CASL/StaticAna: Internal error in function addIndices"

addIndices os@(Qual_op_name _ t _) =

(os,map getIndex $ args_OP_TYPE t)

collectOps s =

Constraint s (map addIndices $ filter (resType s) allSyms) s

constrs = map collectOps sortList

f = Sort_gen_ax constrs True

makeNamed ("ga_generated_" ++

showSepList (showString "_") showId sortList "") f

-- | given the signature in M the function computes the morphism from M' to K

create_iota_mor :: Set.Set SORT -> CASLSign -> CASLMor

create_iota_mor ss sg_m = Morphism sg_m usg' sm om pm ()

where iota_sg = totalSignCopy sg_m

usg = addSig const sg_m iota_sg

sm = iota_sort_map_mor $ sortSet sg_m

om = iota_op_map_mor $ opMap sg_m

pm = iota_pred_map_mor $ predMap sg_m

om' = homomorphism_ops (sortSet sg_m) (opMap usg)

om'' = make_ops ss om'

usg' = usg { opMap = om'' }

-- | adds the make functions for the sorts in the initial module to the

-- operator map

make_ops :: Set.Set SORT -> OpMap -> OpMap

make_ops ss om = Set.fold make_op om ss

-- | adds the make functions for the sort to the operator map

make_op :: SORT -> OpMap -> OpMap

make_op s om = Map.insertWith (Set.union) makeId (Set.singleton $ ot) om

where ot = OpType Total [s] $ mkFreeName s

-- | identifier of the make function

makeId :: Id

makeId = mkId [mkSimpleId "make"]

-- | identifier of the homomorphism function

homId :: Id

homId = mkId [mkSimpleId "hom"]

-- | creates the map between sorts in the morphism iota

iota_sort_map_mor :: Set.Set SORT -> Sort_map

iota_sort_map_mor = Set.fold f Map.empty

where f = \ x y -> Map.insert x (mkFreeName x) y

-- | creates the map between operators in the morphism iota

iota_op_map_mor :: OpMap -> Op_map

iota_op_map_mor = Map.foldWithKey g Map.empty

where f = \ name ot om -> Map.insert (name, ot) (mkFreeName name, Total) om

g = \ k sot m -> Set.fold (f k) m sot

-- | creates the map between predicates in the morphism iota

iota_pred_map_mor :: Map.Map Id (Set.Set PredType) -> Pred_map

iota_pred_map_mor = Map.foldWithKey g Map.empty

where f = \ name pt pm -> Map.insert (name, pt) (mkFreeName name) pm

g = \ k spt m -> Set.fold (f k) m spt

-- | creates the homomorphism operators and adds it to the given operator map

homomorphism_ops :: Set.Set SORT -> OpMap -> OpMap

homomorphism_ops ss om = Set.fold f om ss

where ot = \ sort -> OpType Partial [mkFreeName sort] sort

f = \ x y -> Map.insertWith (Set.union) homId (Set.singleton $ ot x) y

-- | applies the iota renaming to a signature

totalSignCopy :: CASLSign -> CASLSign

totalSignCopy sg = sg {

sortSet = ss,

emptySortSet = ess,

sortRel = sr,

opMap = om,

assocOps = aom,

predMap = pm,

varMap = vm,

sentences = [],

declaredSymbols = sms,

annoMap = am

}

where ss = iota_sort_set $ sortSet sg

ess = iota_sort_set $ emptySortSet sg

sr = iota_sort_rel $ sortRel sg

om = iota_op_map $ opMap sg

aom = iota_op_map $ assocOps sg

pm = iota_pred_map $ predMap sg

vm = iota_var_map $ varMap sg

sms = iota_syms $ declaredSymbols sg

am = iota_anno_map $ annoMap sg

-- | applies the iota renaming to a set of sorts

iota_sort_set :: Set.Set SORT -> Set.Set SORT

iota_sort_set = Set.map mkFreeName

-- | applies the iota renaming to a sort relation

iota_sort_rel :: Rel.Rel SORT -> Rel.Rel SORT

iota_sort_rel = Rel.fromList . map f . Rel.toList

where f = \ (x, y) -> (mkFreeName x, mkFreeName y)

-- | applies the iota renaming to an operator map

iota_op_map :: OpMap -> OpMap

iota_op_map = Map.fromList . map g . Map.toList

where f = \ (OpType _ args res) -> OpType Total (map mkFreeName args) (mkFreeName res)

g = \ (op, ot) -> (mkFreeName op, Set.map f ot)

-- | applies the iota renaming to a predicate map

iota_pred_map :: Map.Map Id (Set.Set PredType) -> Map.Map Id (Set.Set PredType)

iota_pred_map = Map.fromList . map g . Map.toList

where f = \ (PredType args) -> PredType $ map mkFreeName args

g = \ (op, pt) -> (mkFreeName op, Set.map f pt)

-- | applies the iota renaming to a variable map

iota_var_map :: Map.Map SIMPLE_ID SORT -> Map.Map SIMPLE_ID SORT

iota_var_map = Map.map mkFreeName

-- | applies the iota renaming to symbols

iota_syms :: Set.Set Symbol -> Set.Set Symbol

iota_syms = Set.map iota_symbol

-- | applies the iota renaming to a symbol

iota_symbol :: Symbol -> Symbol

iota_symbol (Symbol name SortAsItemType) = Symbol (mkFreeName name) SortAsItemType

iota_symbol (Symbol name (OpAsItemType ot)) = Symbol (mkFreeName name) oait

where f = \ (OpType _ args res) -> OpType Total (map mkFreeName args) (mkFreeName res)

oait = OpAsItemType $ f ot

iota_symbol (Symbol name (PredAsItemType pt)) = Symbol (mkFreeName name) pait

where f = \ (PredType args) -> PredType $ map mkFreeName args

pait = PredAsItemType $ f pt

iota_symbol sym = sym

-- | applies the iota renaming to the annotations

iota_anno_map :: Map.Map Symbol (Set.Set Annotation) -> Map.Map Symbol (Set.Set Annotation)

iota_anno_map = Map.mapKeys iota_symbol

-- Some auxiliary functions

-- | create a new name for the iota morphism

mkFreeName :: Id -> Id

mkFreeName i@(Id ts cs r) = case ts of

t : s -> let st = tokStr t in case st of

c : _ | isAlpha c || isDigit c -> Id (freeToken st : s) cs r

| isPlace t -> Id (mkSimpleId "free" : ts) cs r

| c == '\'' -> i

_ -> Id (mkSimpleId "free_f" : ts) cs r

_ -> i

-- | a prefix for free names

freeNamePrefix :: String

freeNamePrefix = "free_"

-- | create a generated simple identifier

freeToken :: String -> Token

freeToken str = mkSimpleId $ freeNamePrefix ++ str

-- | obtains the sorts of the given list of term

getSorts :: [CASLTERM] -> [SORT]

getSorts = mapMaybe getSort

-- | compute the sort of the term, if possible

getSort :: CASLTERM -> Maybe SORT

getSort (Qual_var _ kind _) = Just kind

getSort (Application op _ _) = case op of

Qual_op_name _ (Op_type _ _ kind _) _ -> Just kind

_ -> Nothing

getSort _ = Nothing

-- | extracts the predicate name from the predicate symbol

pred_symb_name :: PRED_SYMB -> PRED_NAME

pred_symb_name (Pred_name pn) = pn

pred_symb_name (Qual_pred_name pn _ _) = pn

-- | extract the variables from a CASL formula and put them in a map

-- with keys the sort of the variables and value the set of variables

-- in this sort

getVars :: CASLFORMULA -> Map.Map Id (Set.Set Token)

getVars (Quantification _ _ f _) = getVars f

getVars (QuantOp _ _ f) = getVars f

getVars (QuantPred _ _ f) = getVars f

getVars (Conjunction fs _) = foldr (Map.unionWith (Set.union) . getVars) Map.empty fs

getVars (Disjunction fs _) = foldr (Map.unionWith (Set.union) . getVars) Map.empty fs

getVars (Implication f1 f2 _ _) = Map.unionWith (Set.union) v1 v2

where v1 = getVars f1

v2 = getVars f2

getVars (Equivalence f1 f2 _) = Map.unionWith (Set.union) v1 v2

where v1 = getVars f1

v2 = getVars f2

getVars (Negation f _) = getVars f

getVars (Predication _ ts _) = foldr (Map.unionWith (Set.union) . getVarsTerm) Map.empty ts

getVars (Definedness t _) = getVarsTerm t

getVars (Existl_equation t1 t2 _) = Map.unionWith (Set.union) v1 v2

where v1 = getVarsTerm t1

v2 = getVarsTerm t2

getVars (Strong_equation t1 t2 _) = Map.unionWith (Set.union) v1 v2

where v1 = getVarsTerm t1

v2 = getVarsTerm t2

getVars (Membership t _ _) = getVarsTerm t

getVars (Mixfix_formula t) = getVarsTerm t

getVars _ = Map.empty

-- | extract the variables of a CASL term

getVarsTerm :: CASLTERM -> Map.Map Id (Set.Set Token)

getVarsTerm (Qual_var var sort _) = Map.insert sort (Set.singleton var) Map.empty

getVarsTerm (Application _ ts _) = foldr (Map.unionWith (Set.union) . getVarsTerm) Map.empty ts

getVarsTerm (Sorted_term t _ _) = getVarsTerm t

getVarsTerm (Cast t _ _) = getVarsTerm t

getVarsTerm (Conditional t1 f t2 _) = Map.unionWith (Set.union) v3 m

where v1 = getVarsTerm t1

v2 = getVarsTerm t2

v3 = getVars f

m = Map.unionWith (Set.union) v1 v2

getVarsTerm (Mixfix_term ts) = foldr (Map.unionWith (Set.union) . getVarsTerm) Map.empty ts

getVarsTerm (Mixfix_parenthesized ts _) =

foldr (Map.unionWith (Set.union) . getVarsTerm) Map.empty ts

getVarsTerm (Mixfix_bracketed ts _) =

foldr (Map.unionWith (Set.union) . getVarsTerm) Map.empty ts

getVarsTerm (Mixfix_braced ts _) =

foldr (Map.unionWith (Set.union) . getVarsTerm) Map.empty ts

getVarsTerm _ = Map.empty

-- | add universal quantification of all variables in the formula

quantifyUniversally :: CASLFORMULA -> CASLFORMULA

quantifyUniversally form = if null var_decl

then form

else Quantification Universal var_decl form nullRange

where vars = getVars form

var_decl = listVarDecl vars

-- | traverses a map with sorts as keys and sets of variables as value and creates

-- a list of variable declarations

listVarDecl :: Map.Map Id (Set.Set Token) -> [VAR_DECL]

listVarDecl = Map.foldWithKey f []

where f = \ sort var_set acc -> Var_decl (Set.toList var_set) sort nullRange : acc

-- | removes a quantification from a formula

noQuantification :: CASLFORMULA -> (CASLFORMULA, [VAR_DECL])

noQuantification (Quantification _ vars form _) = (form, vars)

noQuantification form = (form, [])

-- | generates a new variable qualified with the given number

newVarIndex :: Int -> SORT -> CASLTERM

newVarIndex i sort = Qual_var var sort nullRange

where var = mkSimpleId $ "V" ++ show i

-- | generates a new variable

newVar :: SORT -> CASLTERM

newVar sort = Qual_var var sort nullRange

where var = mkSimpleId "V"

-- | deletes the repetitions from a list of sorts

delete_reps :: [SORT] -> [SORT]

delete_reps = Set.toList . Set.fromList

-- | generates the free representation of an OP_SYMB

free_op_sym :: OP_SYMB -> OP_SYMB

free_op_sym (Op_name on) = Op_name $ mkFreeName on

free_op_sym (Qual_op_name on ot r) = Qual_op_name on' ot' r

where on' = mkFreeName on

ot' = free_op_type ot

-- | generates the free representation of an OP_TYPE

free_op_type :: OP_TYPE -> OP_TYPE

free_op_type (Op_type _ args res r) = Op_type Total args' res' r

where args' = map mkFreeName args

res' = mkFreeName res

-- | generates the free representation of a PRED_SYMB

free_pred_sym :: PRED_SYMB -> PRED_SYMB

free_pred_sym (Pred_name pn) = Pred_name $ mkFreeName pn

free_pred_sym (Qual_pred_name pn pt r) = Qual_pred_name pn' pt' r

where pn' = mkFreeName pn

pt' = free_pred_type pt

-- | generates the free representation of a PRED_TYPE

free_pred_type :: PRED_TYPE -> PRED_TYPE

free_pred_type (Pred_type args r) = Pred_type args' r

where args' = map mkFreeName args

-- | generates the free representation of a CASLTERM

free_term :: CASLTERM -> CASLTERM

free_term (Qual_var v s r) = Qual_var v (mkFreeName s) r

free_term (Application os ts r) = Application os' ts' r

where ts' = map free_term ts

os' = free_op_sym os

free_term (Sorted_term t s r) = Sorted_term t' s' r

where t' = free_term t

s' = mkFreeName s

free_term (Cast t s r) = Cast t' s' r

where t' = free_term t

s' = mkFreeName s

free_term (Conditional t1 f t2 r) = Conditional t1' f' t2' r

where t1' = free_term t1

t2' = free_term t2

f' = free_formula f

free_term (Mixfix_qual_pred ps) = Mixfix_qual_pred ps'

where ps' = free_pred_sym ps

free_term (Mixfix_term ts) = Mixfix_term ts'

where ts' = map free_term ts

free_term (Mixfix_sorted_term s r) = Mixfix_sorted_term s' r

where s' = mkFreeName s

free_term (Mixfix_cast s r) = Mixfix_cast s' r

where s' = mkFreeName s

free_term (Mixfix_parenthesized ts r) = Mixfix_parenthesized ts' r

where ts' = map free_term ts

free_term (Mixfix_bracketed ts r) = Mixfix_bracketed ts' r

where ts' = map free_term ts

free_term (Mixfix_braced ts r) = Mixfix_braced ts' r

where ts' = map free_term ts

free_term t = t

-- | generates the free representation of a list of variable declarations

free_var_decls :: [VAR_DECL] -> [VAR_DECL]

free_var_decls = map free_var_decl

-- | generates the free representation of a variable declaration

free_var_decl :: VAR_DECL -> VAR_DECL

free_var_decl (Var_decl vs s r) = Var_decl vs s' r

where s' = mkFreeName s

-- | computes the substitution needed for the kernel of h to the

-- sentences of the theory of M

free_formula :: CASLFORMULA -> CASLFORMULA

free_formula (Quantification q vs f r) = Quantification q vs' f' r

where vs' = free_var_decls vs

f' = free_formula f

free_formula (Conjunction fs r) = Conjunction fs' r

where fs' = map free_formula fs

free_formula (Disjunction fs r) = Disjunction fs' r

where fs' = map free_formula fs

free_formula (Implication f1 f2 b r) = Implication f1' f2' b r

where f1' = free_formula f1

f2' = free_formula f2

free_formula (Equivalence f1 f2 r) = Equivalence f1' f2' r

where f1' = free_formula f1

f2' = free_formula f2

free_formula (Negation f r) = Negation f' r

where f' = free_formula f

free_formula (Predication ps ts r) = pr

where ss = getSorts ts

free_ss = map mkFreeName ss

ts' = map free_term ts

psi = psiName $ pred_symb_name ps

pt = Pred_type free_ss nullRange

ps' = Qual_pred_name psi pt nullRange

pr = Predication ps' ts' r

free_formula (Definedness t r) = case sort of

Nothing -> Definedness t' r

Just s -> let free_s = mkFreeName s

phi = phiName s

pt = Pred_type [free_s, free_s] nullRange

ps = Qual_pred_name phi pt nullRange

in Predication ps [t', t'] nullRange

where t' = free_term t

sort = getSort t

free_formula (Existl_equation t1 t2 r) = case sort of

Nothing -> Existl_equation t1' t2' r

Just s -> let free_s = mkFreeName s

phi = phiName s

pt = Pred_type [free_s, free_s] nullRange

ps = Qual_pred_name phi pt nullRange

in Predication ps [t1', t2'] nullRange

where t1' = free_term t1

t2' = free_term t2

sort = getSort t1

free_formula (Strong_equation t1 t2 r) = case sort of

Nothing -> Strong_equation t1' t2' r

Just s -> let free_s = mkFreeName s

phi = phiName s

pt = Pred_type [free_s, free_s] nullRange

ps = Qual_pred_name phi pt nullRange

pred1 = Predication ps [t1', t2'] nullRange

pred2 = Negation (Predication ps [t1', t1'] nullRange) nullRange

pred3 = Negation (Predication ps [t2', t2'] nullRange) nullRange

pred4 = Conjunction [pred2, pred3] nullRange

pred5 = Disjunction [pred1, pred4] nullRange

in pred5

where t1' = free_term t1

t2' = free_term t2

sort = getSort t1

free_formula (Membership t s r) = Membership t' s' r

where t' = free_term t

s' = mkFreeName s

free_formula (Mixfix_formula t) = Mixfix_formula t'

where t' = free_term t

free_formula (QuantOp on ot f) = QuantOp on' ot' f'

where on' = mkFreeName on

ot' = free_op_type ot

f' = free_formula f

free_formula (QuantPred pn pt f) = QuantPred pn' pt' f'

where pn' = mkFreeName pn

pt' = free_pred_type pt

f' = free_formula f

free_formula f = f