(* Title: Pure/Isar/element.ML
ID: $Id$
Author: Makarius
Explicit data structures for some Isar language elements, with derived
logical operations.
*)
signature ELEMENT =
sig
datatype ('typ, 'term) stmt =
Shows of ((string * Attrib.src list) * ('term * 'term list) list) list |
Obtains of (string * ((string * 'typ option) list * 'term list)) list
type statement (*= (string, string) stmt*)
type statement_i (*= (typ, term) stmt*)
datatype ('typ, 'term, 'fact) ctxt =
Fixes of (string * 'typ option * mixfix) list |
Constrains of (string * 'typ) list |
Assumes of ((string * Attrib.src list) * ('term * 'term list) list) list |
Defines of ((string * Attrib.src list) * ('term * 'term list)) list |
Notes of string * ((string * Attrib.src list) * ('fact * Attrib.src list) list) list
type context (*= (string, string, thmref) ctxt*)
type context_i (*= (typ, term, thm list) ctxt*)
val facts_map: (('typ, 'term, 'fact) ctxt -> ('a, 'b, 'c) ctxt) ->
((string * Attrib.src list) * ('fact * Attrib.src list) list) list ->
((string * Attrib.src list) * ('c * Attrib.src list) list) list
val map_ctxt: {name: string -> string,
var: string * mixfix -> string * mixfix,
typ: 'typ -> 'a, term: 'term -> 'b, fact: 'fact -> 'c,
attrib: Attrib.src -> Attrib.src} -> ('typ, 'term, 'fact) ctxt -> ('a, 'b, 'c) ctxt
val map_ctxt_attrib: (Attrib.src -> Attrib.src) ->
('typ, 'term, 'fact) ctxt -> ('typ, 'term, 'fact) ctxt
val morph_ctxt: morphism -> context_i -> context_i
val params_of: context_i -> (string * typ) list
val prems_of: context_i -> term list
val facts_of: theory -> context_i ->
((string * Attrib.src list) * (thm list * Attrib.src list) list) list
val pretty_stmt: Proof.context -> statement_i -> Pretty.T list
val pretty_ctxt: Proof.context -> context_i -> Pretty.T list
val pretty_statement: Proof.context -> string -> thm -> Pretty.T
type witness
val map_witness: (term * thm -> term * thm) -> witness -> witness
val morph_witness: morphism -> witness -> witness
val witness_prop: witness -> term
val witness_hyps: witness -> term list
val assume_witness: theory -> term -> witness
val prove_witness: Proof.context -> term -> tactic -> witness
val conclude_witness: witness -> thm
val mark_witness: term -> term
val make_witness: term -> thm -> witness
val dest_witness: witness -> term * thm
val transfer_witness: theory -> witness -> witness
val refine_witness: Proof.state -> Proof.state Seq.seq
val pretty_witness: Proof.context -> witness -> Pretty.T
val rename: (string * (string * mixfix option)) list -> string -> string
val rename_var: (string * (string * mixfix option)) list -> string * mixfix -> string * mixfix
val rename_term: (string * (string * mixfix option)) list -> term -> term
val rename_thm: (string * (string * mixfix option)) list -> thm -> thm
val rename_morphism: (string * (string * mixfix option)) list -> morphism
val instT_type: typ Symtab.table -> typ -> typ
val instT_term: typ Symtab.table -> term -> term
val instT_thm: theory -> typ Symtab.table -> thm -> thm
val instT_morphism: theory -> typ Symtab.table -> morphism
val inst_term: typ Symtab.table * term Symtab.table -> term -> term
val inst_thm: theory -> typ Symtab.table * term Symtab.table -> thm -> thm
val inst_morphism: theory -> typ Symtab.table * term Symtab.table -> morphism
val satisfy_thm: witness list -> thm -> thm
val satisfy_morphism: witness list -> morphism
val satisfy_facts: witness list ->
((string * Attrib.src list) * (thm list * Attrib.src list) list) list ->
((string * Attrib.src list) * (thm list * Attrib.src list) list) list
val generalize_facts: Proof.context -> Proof.context ->
((string * Attrib.src list) * (thm list * Attrib.src list) list) list ->
((string * Attrib.src list) * (thm list * Attrib.src list) list) list
end;
structure Element: ELEMENT =
struct
(** language elements **)
(* statement *)
datatype ('typ, 'term) stmt =
Shows of ((string * Attrib.src list) * ('term * 'term list) list) list |
Obtains of (string * ((string * 'typ option) list * 'term list)) list;
type statement = (string, string) stmt;
type statement_i = (typ, term) stmt;
(* context *)
datatype ('typ, 'term, 'fact) ctxt =
Fixes of (string * 'typ option * mixfix) list |
Constrains of (string * 'typ) list |
Assumes of ((string * Attrib.src list) * ('term * 'term list) list) list |
Defines of ((string * Attrib.src list) * ('term * 'term list)) list |
Notes of string * ((string * Attrib.src list) * ('fact * Attrib.src list) list) list;
type context = (string, string, thmref) ctxt;
type context_i = (typ, term, thm list) ctxt;
fun facts_map f facts = Notes ("", facts) |> f |> (fn Notes (_, facts') => facts');
fun map_ctxt {name, var, typ, term, fact, attrib} =
fn Fixes fixes => Fixes (fixes |> map (fn (x, T, mx) =>
let val (x', mx') = var (x, mx) in (x', Option.map typ T, mx') end))
| Constrains xs => Constrains (xs |> map (fn (x, T) => (#1 (var (x, NoSyn)), typ T)))
| Assumes asms => Assumes (asms |> map (fn ((a, atts), propps) =>
((name a, map attrib atts), propps |> map (fn (t, ps) => (term t, map term ps)))))
| Defines defs => Defines (defs |> map (fn ((a, atts), (t, ps)) =>
((name a, map attrib atts), (term t, map term ps))))
| Notes (kind, facts) => Notes (kind, facts |> map (fn ((a, atts), bs) =>
((name a, map attrib atts), bs |> map (fn (ths, btts) => (fact ths, map attrib btts)))));
fun map_ctxt_attrib attrib =
map_ctxt {name = I, var = I, typ = I, term = I, fact = I, attrib = attrib};
fun morph_ctxt phi = map_ctxt
{name = Morphism.name phi,
var = Morphism.var phi,
typ = Morphism.typ phi,
term = Morphism.term phi,
fact = Morphism.fact phi,
attrib = Args.morph_values phi};
(* logical content *)
fun params_of (Fixes fixes) = fixes |> map
(fn (x, SOME T, _) => (x, T)
| (x, _, _) => raise TERM ("Untyped context element parameter " ^ quote x, []))
| params_of _ = [];
fun prems_of (Assumes asms) = maps (map fst o snd) asms
| prems_of (Defines defs) = map (fst o snd) defs
| prems_of _ = [];
fun assume thy t = Assumption.assume (Thm.cterm_of thy t);
fun facts_of thy (Assumes asms) = map (apsnd (map (fn (t, _) => ([assume thy t], [])))) asms
| facts_of thy (Defines defs) = map (apsnd (fn (t, _) => [([assume thy t], [])])) defs
| facts_of _ (Notes (_, facts)) = facts
| facts_of _ _ = [];
(** pretty printing **)
fun pretty_items _ _ [] = []
| pretty_items keyword sep (x :: ys) =
Pretty.block [Pretty.keyword keyword, Pretty.brk 1, x] ::
map (fn y => Pretty.block [Pretty.str " ", Pretty.keyword sep, Pretty.brk 1, y]) ys;
fun pretty_name_atts ctxt (name, atts) sep =
if name = "" andalso null atts then []
else [Pretty.block
(Pretty.breaks (Pretty.str name :: Attrib.pretty_attribs ctxt atts @ [Pretty.str sep]))];
(* pretty_stmt *)
fun pretty_stmt ctxt =
let
val prt_typ = Pretty.quote o ProofContext.pretty_typ ctxt;
val prt_term = Pretty.quote o ProofContext.pretty_term ctxt;
val prt_terms = separate (Pretty.keyword "and") o map prt_term;
val prt_name_atts = pretty_name_atts ctxt;
fun prt_show (a, ts) =
Pretty.block (Pretty.breaks (prt_name_atts a ":" @ prt_terms (map fst ts)));
fun prt_var (x, SOME T) = Pretty.block [Pretty.str (x ^ " ::"), Pretty.brk 1, prt_typ T]
| prt_var (x, NONE) = Pretty.str x;
val prt_vars = separate (Pretty.keyword "and") o map prt_var;
fun prt_obtain (_, ([], ts)) = Pretty.block (Pretty.breaks (prt_terms ts))
| prt_obtain (_, (xs, ts)) = Pretty.block (Pretty.breaks
(prt_vars xs @ [Pretty.keyword "where"] @ prt_terms ts));
in
fn Shows shows => pretty_items "shows" "and" (map prt_show shows)
| Obtains obtains => pretty_items "obtains" "|" (map prt_obtain obtains)
end;
(* pretty_ctxt *)
fun pretty_ctxt ctxt =
let
val prt_typ = Pretty.quote o ProofContext.pretty_typ ctxt;
val prt_term = Pretty.quote o ProofContext.pretty_term ctxt;
val prt_thm = Pretty.backquote o ProofContext.pretty_thm ctxt;
val prt_name_atts = pretty_name_atts ctxt;
fun prt_mixfix NoSyn = []
| prt_mixfix mx = [Pretty.brk 2, Syntax.pretty_mixfix mx];
fun prt_fix (x, SOME T, mx) = Pretty.block (Pretty.str (x ^ " ::") :: Pretty.brk 1 ::
prt_typ T :: Pretty.brk 1 :: prt_mixfix mx)
| prt_fix (x, NONE, mx) = Pretty.block (Pretty.str x :: Pretty.brk 1 :: prt_mixfix mx);
fun prt_constrain (x, T) = prt_fix (x, SOME T, NoSyn);
fun prt_asm (a, ts) =
Pretty.block (Pretty.breaks (prt_name_atts a ":" @ map (prt_term o fst) ts));
fun prt_def (a, (t, _)) =
Pretty.block (Pretty.breaks (prt_name_atts a ":" @ [prt_term t]));
fun prt_fact (ths, []) = map prt_thm ths
| prt_fact (ths, atts) = Pretty.enclose "(" ")"
(Pretty.breaks (map prt_thm ths)) :: Attrib.pretty_attribs ctxt atts;
fun prt_note (a, ths) =
Pretty.block (Pretty.breaks (flat (prt_name_atts a "=" :: map prt_fact ths)));
in
fn Fixes fixes => pretty_items "fixes" "and" (map prt_fix fixes)
| Constrains xs => pretty_items "constrains" "and" (map prt_constrain xs)
| Assumes asms => pretty_items "assumes" "and" (map prt_asm asms)
| Defines defs => pretty_items "defines" "and" (map prt_def defs)
| Notes ("", facts) => pretty_items "notes" "and" (map prt_note facts)
| Notes (kind, facts) => pretty_items ("notes " ^ kind) "and" (map prt_note facts)
end;
(* pretty_statement *)
local
fun thm_name kind th prts =
let val head =
if PureThy.has_name_hint th then
Pretty.block [Pretty.command kind,
Pretty.brk 1, Pretty.str (Sign.base_name (PureThy.get_name_hint th) ^ ":")]
else Pretty.command kind
in Pretty.block (Pretty.fbreaks (head :: prts)) end;
fun obtain prop ctxt =
let
val ((xs, prop'), ctxt') = Variable.focus prop ctxt;
val As = Logic.strip_imp_prems (Thm.term_of prop');
fun var (x, T) = (ProofContext.revert_skolem ctxt' x, SOME T);
in (("", (map (var o Term.dest_Free o Thm.term_of) xs, As)), ctxt') end;
in
fun pretty_statement ctxt kind raw_th =
let
val thy = ProofContext.theory_of ctxt;
val cert = Thm.cterm_of thy;
val th = MetaSimplifier.norm_hhf raw_th;
val is_elim = ObjectLogic.is_elim th;
val ((_, [th']), ctxt') = Variable.import_thms true [th] ctxt;
val prop = Thm.prop_of th';
val (prems, concl) = Logic.strip_horn prop;
val concl_term = ObjectLogic.drop_judgment thy concl;
val fixes = fold_aterms (fn v as Free (x, T) =>
if Variable.newly_fixed ctxt' ctxt x andalso not (v aconv concl_term)
then insert (op =) (x, T) else I | _ => I) prop []
|> rev |> map (apfst (ProofContext.revert_skolem ctxt'));
val (assumes, cases) = take_suffix (fn prem =>
is_elim andalso concl aconv Logic.strip_assums_concl prem) prems;
in
pretty_ctxt ctxt' (Fixes (map (fn (x, T) => (x, SOME T, NoSyn)) fixes)) @
pretty_ctxt ctxt' (Assumes (map (fn t => (("", []), [(t, [])])) assumes)) @
pretty_stmt ctxt'
(if null cases then Shows [(("", []), [(concl, [])])]
else Obtains (#1 (fold_map (obtain o cert) cases ctxt')))
end |> thm_name kind raw_th;
end;
(** logical operations **)
(* witnesses -- hypotheses as protected facts *)
datatype witness = Witness of term * thm;
fun map_witness f (Witness witn) = Witness (f witn);
fun morph_witness phi = map_witness (fn (t, th) => (Morphism.term phi t, Morphism.thm phi th));
fun witness_prop (Witness (t, _)) = t;
fun witness_hyps (Witness (_, th)) = #hyps (Thm.rep_thm th);
fun assume_witness thy t =
Witness (t, Goal.protect (Thm.assume (Thm.cterm_of thy t)));
fun prove_witness ctxt t tac =
Witness (t, Goal.prove ctxt [] [] (Logic.protect t) (fn _ =>
Tactic.rtac Drule.protectI 1 THEN tac));
fun conclude_witness (Witness (_, th)) = MetaSimplifier.norm_hhf_protect (Goal.conclude th);
val mark_witness = Logic.protect;
fun make_witness t th = Witness (t, th);
fun dest_witness (Witness w) = w;
fun transfer_witness thy (Witness (t, th)) = Witness (t, Thm.transfer thy th);
val refine_witness =
Proof.refine (Method.Basic (K (Method.RAW_METHOD
(K (ALLGOALS
(PRECISE_CONJUNCTS ~1 (ALLGOALS
(PRECISE_CONJUNCTS ~1 (TRYALL (Tactic.rtac Drule.protectI))))))))));
fun pretty_witness ctxt witn =
let val prt_term = Pretty.quote o ProofContext.pretty_term ctxt in
Pretty.block (prt_term (witness_prop witn) ::
(if ! show_hyps then [Pretty.brk 2, Pretty.list "[" "]"
(map prt_term (witness_hyps witn))] else []))
end;
(* derived rules *)
fun instantiate_tfrees thy subst th =
let
val certT = Thm.ctyp_of thy;
val idx = Thm.maxidx_of th + 1;
fun cert_inst (a, (S, T)) = (certT (TVar ((a, idx), S)), certT T);
fun add_inst (a, S) insts =
if AList.defined (op =) insts a then insts
else (case AList.lookup (op =) subst a of NONE => insts | SOME T => (a, (S, T)) :: insts);
val insts =
Term.fold_types (Term.fold_atyps (fn TFree v => add_inst v | _ => I))
(Thm.full_prop_of th) [];
in
th
|> Thm.generalize (map fst insts, []) idx
|> Thm.instantiate (map cert_inst insts, [])
end;
fun instantiate_frees thy subst =
let val cert = Thm.cterm_of thy in
Drule.forall_intr_list (map (cert o Free o fst) subst) #>
Drule.forall_elim_list (map (cert o snd) subst)
end;
fun hyps_rule rule th =
let val {hyps, ...} = Thm.crep_thm th in
Drule.implies_elim_list
(rule (Drule.implies_intr_list hyps th))
(map (Thm.assume o Drule.cterm_rule rule) hyps)
end;
(* rename *)
fun rename ren x =
(case AList.lookup (op =) ren (x: string) of
NONE => x
| SOME (x', _) => x');
fun rename_var ren (x, mx) =
(case (AList.lookup (op =) ren (x: string), mx) of
(NONE, _) => (x, mx)
| (SOME (x', NONE), Structure) => (x', mx)
| (SOME (x', SOME _), Structure) =>
error ("Attempt to change syntax of structure parameter " ^ quote x)
| (SOME (x', NONE), _) => (x', NoSyn)
| (SOME (x', SOME mx'), _) => (x', mx'));
fun rename_term ren (Free (x, T)) = Free (rename ren x, T)
| rename_term ren (t $ u) = rename_term ren t $ rename_term ren u
| rename_term ren (Abs (x, T, t)) = Abs (x, T, rename_term ren t)
| rename_term _ a = a;
fun rename_thm ren th =
let
val thy = Thm.theory_of_thm th;
val subst = (Thm.fold_terms o Term.fold_aterms)
(fn Free (x, T) =>
let val x' = rename ren x
in if x = x' then I else insert (eq_fst (op =)) ((x, T), Free (x', T)) end
| _ => I) th [];
in
if null subst then th
else th |> hyps_rule (instantiate_frees thy subst)
end;
fun rename_morphism ren = Morphism.morphism
{name = I, var = rename_var ren, typ = I, term = rename_term ren, fact = map (rename_thm ren)};
(* instantiate types *)
fun instT_type env =
if Symtab.is_empty env then I
else Term.map_type_tfree (fn (x, S) => the_default (TFree (x, S)) (Symtab.lookup env x));
fun instT_term env =
if Symtab.is_empty env then I
else Term.map_types (instT_type env);
fun instT_subst env th = (Thm.fold_terms o Term.fold_types o Term.fold_atyps)
(fn T as TFree (a, _) =>
let val T' = the_default T (Symtab.lookup env a)
in if T = T' then I else insert (op =) (a, T') end
| _ => I) th [];
fun instT_thm thy env th =
if Symtab.is_empty env then th
else
let val subst = instT_subst env th
in if null subst then th else th |> hyps_rule (instantiate_tfrees thy subst) end;
fun instT_morphism thy env =
let val thy_ref = Theory.self_ref thy in
Morphism.morphism
{name = I, var = I,
typ = instT_type env,
term = instT_term env,
fact = map (fn th => instT_thm (Theory.deref thy_ref) env th)}
end;
(* instantiate types and terms *)
fun inst_term (envT, env) =
if Symtab.is_empty env then instT_term envT
else
let
val instT = instT_type envT;
fun inst (Const (x, T)) = Const (x, instT T)
| inst (Free (x, T)) =
(case Symtab.lookup env x of
NONE => Free (x, instT T)
| SOME t => t)
| inst (Var (xi, T)) = Var (xi, instT T)
| inst (b as Bound _) = b
| inst (Abs (x, T, t)) = Abs (x, instT T, inst t)
| inst (t $ u) = inst t $ inst u;
in Envir.beta_norm o inst end;
fun inst_thm thy (envT, env) th =
if Symtab.is_empty env then instT_thm thy envT th
else
let
val substT = instT_subst envT th;
val subst = (Thm.fold_terms o Term.fold_aterms)
(fn Free (x, T) =>
let
val T' = instT_type envT T;
val t = Free (x, T');
val t' = the_default t (Symtab.lookup env x);
in if t aconv t' then I else insert (eq_fst (op =)) ((x, T'), t') end
| _ => I) th [];
in
if null substT andalso null subst then th
else th |> hyps_rule
(instantiate_tfrees thy substT #>
instantiate_frees thy subst #>
Drule.fconv_rule (Thm.beta_conversion true))
end;
fun inst_morphism thy envs =
let val thy_ref = Theory.self_ref thy in
Morphism.morphism
{name = I, var = I,
typ = instT_type (#1 envs),
term = inst_term envs,
fact = map (fn th => inst_thm (Theory.deref thy_ref) envs th)}
end;
(* satisfy hypotheses *)
fun satisfy_thm witns thm = thm |> fold (fn hyp =>
(case find_first (fn Witness (t, _) => Thm.term_of hyp aconv t) witns of
NONE => I
| SOME (Witness (_, th)) => Drule.implies_intr_protected [hyp] #> Goal.comp_hhf th))
(#hyps (Thm.crep_thm thm));
fun satisfy_morphism witns = Morphism.thm_morphism (satisfy_thm witns);
fun satisfy_facts witns = facts_map (morph_ctxt (satisfy_morphism witns));
(* generalize type/term parameters *)
val maxidx_atts = fold Args.maxidx_values;
fun generalize_facts inner outer facts =
let
val thy = ProofContext.theory_of inner;
val maxidx =
fold (fn ((_, atts), bs) => maxidx_atts atts #> fold (maxidx_atts o #2) bs) facts ~1;
val exp_fact = map (Thm.adjust_maxidx_thm maxidx) #> Variable.export inner outer;
val exp_term = Drule.term_rule thy (singleton exp_fact);
val exp_typ = Logic.type_map exp_term;
val morphism =
Morphism.morphism {name = I, var = I, typ = exp_typ, term = exp_term, fact = exp_fact};
in facts_map (morph_ctxt morphism) facts end;
end;