(* Title: Pure/Isar/element.ML
Author: Makarius
Explicit data structures for some Isar language elements, with derived
logical operations.
*)
signature ELEMENT =
sig
datatype ('typ, 'term) stmt =
Shows of (Attrib.binding * ('term * 'term list) list) list |
Obtains of (binding * ((binding * 'typ option) list * 'term list)) list
type statement = (string, string) stmt
type statement_i = (typ, term) stmt
datatype ('typ, 'term, 'fact) ctxt =
Fixes of (binding * 'typ option * mixfix) list |
Constrains of (string * 'typ) list |
Assumes of (Attrib.binding * ('term * 'term list) list) list |
Defines of (Attrib.binding * ('term * 'term list)) list |
Notes of string * (Attrib.binding * ('fact * Attrib.src list) list) list
type context = (string, string, Facts.ref) ctxt
type context_i = (typ, term, thm list) ctxt
val facts_map: (('typ, 'term, 'fact) ctxt -> ('a, 'b, 'c) ctxt) ->
(Attrib.binding * ('fact * Attrib.src list) list) list ->
(Attrib.binding * ('c * Attrib.src list) list) list
val map_ctxt: {binding: binding -> binding, typ: 'typ -> 'a, term: 'term -> 'b,
pattern: '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 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 prove_witness: Proof.context -> term -> tactic -> witness
val witness_proof: (witness list list -> Proof.context -> Proof.context) ->
term list list -> Proof.context -> Proof.state
val witness_proof_eqs: (witness list list -> thm list -> Proof.context -> Proof.context) ->
term list list -> term list -> Proof.context -> Proof.state
val witness_local_proof: (witness list list -> Proof.state -> Proof.state) ->
string -> term list list -> Proof.context -> bool -> Proof.state -> Proof.state
val witness_local_proof_eqs: (witness list list -> thm list -> Proof.state -> Proof.state) ->
string -> term list list -> term list -> Proof.context -> bool -> Proof.state ->
Proof.state
val morph_witness: morphism -> witness -> witness
val conclude_witness: witness -> thm
val pretty_witness: Proof.context -> witness -> Pretty.T
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 ->
(Attrib.binding * (thm list * Attrib.src list) list) list ->
(Attrib.binding * (thm list * Attrib.src list) list) list
val eq_morphism: theory -> thm list -> morphism option
val transfer_morphism: theory -> morphism
val generic_note_thmss: string -> (Attrib.binding * (thm list * Attrib.src list) list) list ->
Context.generic -> (string * thm list) list * Context.generic
val init: context_i -> Context.generic -> Context.generic
val activate_i: context_i -> Proof.context -> context_i * Proof.context
val activate: (typ, term, Facts.ref) ctxt -> Proof.context -> context_i * Proof.context
end;
structure Element: ELEMENT =
struct
(** language elements **)
(* statement *)
datatype ('typ, 'term) stmt =
Shows of (Attrib.binding * ('term * 'term list) list) list |
Obtains of (binding * ((binding * '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 (binding * 'typ option * mixfix) list |
Constrains of (string * 'typ) list |
Assumes of (Attrib.binding * ('term * 'term list) list) list |
Defines of (Attrib.binding * ('term * 'term list)) list |
Notes of string * (Attrib.binding * ('fact * Attrib.src list) list) list;
type context = (string, string, Facts.ref) ctxt;
type context_i = (typ, term, thm list) ctxt;
fun facts_map f facts = Notes ("", facts) |> f |> (fn Notes (_, facts') => facts');
fun map_ctxt {binding, typ, term, pattern, fact, attrib} =
fn Fixes fixes => Fixes (fixes |> map (fn (x, T, mx) => (binding x, Option.map typ T, mx)))
| Constrains xs => Constrains (xs |> map (fn (x, T) =>
(Variable.check_name (binding (Binding.name x)), typ T)))
| Assumes asms => Assumes (asms |> map (fn ((a, atts), propps) =>
((binding a, map attrib atts), propps |> map (fn (t, ps) => (term t, map pattern ps)))))
| Defines defs => Defines (defs |> map (fn ((a, atts), (t, ps)) =>
((binding a, map attrib atts), (term t, map pattern ps))))
| Notes (kind, facts) => Notes (kind, facts |> map (fn ((a, atts), bs) =>
((binding a, map attrib atts), bs |> map (fn (ths, btts) => (fact ths, map attrib btts)))));
fun map_ctxt_attrib attrib =
map_ctxt {binding = I, typ = I, term = I, pattern = I, fact = I, attrib = attrib};
fun morph_ctxt phi = map_ctxt
{binding = Morphism.binding phi,
typ = Morphism.typ phi,
term = Morphism.term phi,
pattern = Morphism.term phi,
fact = Morphism.fact phi,
attrib = Args.morph_values phi};
(** 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 (b, atts) sep =
if Binding.is_empty b andalso null atts then []
else
[Pretty.block (Pretty.breaks
(Binding.pretty b :: Attrib.pretty_attribs ctxt atts @ [Pretty.str sep]))];
(* pretty_stmt *)
fun pretty_stmt ctxt =
let
val prt_typ = Pretty.quote o Syntax.pretty_typ ctxt;
val prt_term = Pretty.quote o Syntax.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 (Binding.name_of x ^ " ::"), Pretty.brk 1, prt_typ T]
| prt_var (x, NONE) = Pretty.str (Binding.name_of 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 Syntax.pretty_typ ctxt;
val prt_term = Pretty.quote o Syntax.pretty_term ctxt;
val prt_thm = Pretty.backquote o Display.pretty_thm ctxt;
val prt_name_atts = pretty_name_atts ctxt;
fun prt_mixfix NoSyn = []
| prt_mixfix mx = [Pretty.brk 2, Mixfix.pretty_mixfix mx];
fun prt_fix (x, SOME T, mx) = Pretty.block (Pretty.str (Binding.name_of x ^ " ::") ::
Pretty.brk 1 :: prt_typ T :: Pretty.brk 1 :: prt_mixfix mx)
| prt_fix (x, NONE, mx) = Pretty.block (Pretty.str (Binding.name_of x) ::
Pretty.brk 1 :: prt_mixfix mx);
fun prt_constrain (x, T) = prt_fix (Binding.name 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 standard_elim th =
(case Object_Logic.elim_concl th of
SOME C =>
let
val cert = Thm.cterm_of (Thm.theory_of_thm th);
val thesis = Var ((Auto_Bind.thesisN, Thm.maxidx_of th + 1), fastype_of C);
val th' = Thm.instantiate ([], [(cert C, cert thesis)]) th;
in (th', true) end
| NONE => (th, false));
fun thm_name kind th prts =
let val head =
if Thm.has_name_hint th then
Pretty.block [Pretty.command kind,
Pretty.brk 1, Pretty.str (Long_Name.base_name (Thm.get_name_hint th) ^ ":")]
else Pretty.command kind
in Pretty.block (Pretty.fbreaks (head :: prts)) end;
fun obtain prop ctxt =
let
val ((ps, prop'), ctxt') = Variable.focus prop ctxt;
fun fix (x, T) = (Binding.name (Variable.revert_fixed ctxt' x), SOME T);
val xs = map (fix o #2) ps;
val As = Logic.strip_imp_prems prop';
in ((Binding.empty, (xs, As)), ctxt') end;
in
fun pretty_statement ctxt kind raw_th =
let
val thy = Proof_Context.theory_of ctxt;
val (th, is_elim) = standard_elim (Raw_Simplifier.norm_hhf raw_th);
val ((_, [th']), ctxt') = Variable.import true [th] (Variable.set_body true ctxt);
val prop = Thm.prop_of th';
val (prems, concl) = Logic.strip_horn prop;
val concl_term = Object_Logic.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 =) (Variable.revert_fixed ctxt' x, T) else I | _ => I) prop [] |> rev;
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) => (Binding.name x, SOME T, NoSyn)) fixes)) @
pretty_ctxt ctxt' (Assumes (map (fn t => (Attrib.empty_binding, [(t, [])])) assumes)) @
(if null cases then pretty_stmt ctxt' (Shows [(Attrib.empty_binding, [(concl, [])])])
else
let val (clauses, ctxt'') = fold_map obtain cases ctxt'
in pretty_stmt ctxt'' (Obtains clauses) end)
end |> thm_name kind raw_th;
end;
(** logical operations **)
(* witnesses -- hypotheses as protected facts *)
datatype witness = Witness of term * thm;
val mark_witness = Logic.protect;
fun witness_prop (Witness (t, _)) = t;
fun witness_hyps (Witness (_, th)) = Thm.hyps_of th;
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 prove_witness ctxt t tac =
Witness (t, Thm.close_derivation (Goal.prove ctxt [] [] (mark_witness t) (fn _ =>
Tactic.rtac Drule.protectI 1 THEN tac)));
local
val refine_witness =
Proof.refine (Method.Basic (K (RAW_METHOD
(K (ALLGOALS
(CONJUNCTS (ALLGOALS
(CONJUNCTS (TRYALL (Tactic.rtac Drule.protectI))))))))));
fun gen_witness_proof proof after_qed wit_propss eq_props =
let
val propss = (map o map) (fn prop => (mark_witness prop, [])) wit_propss
@ [map (rpair []) eq_props];
fun after_qed' thmss =
let val (wits, eqs) = split_last ((map o map) Thm.close_derivation thmss);
in after_qed ((map2 o map2) (curry Witness) wit_propss wits) eqs end;
in proof after_qed' propss #> refine_witness #> Seq.hd end;
fun proof_local cmd goal_ctxt int after_qed' propss =
Proof.map_context (K goal_ctxt)
#> Proof.local_goal (Proof_Display.print_results int) (K I) Proof_Context.bind_propp_i
cmd NONE after_qed' (map (pair Thm.empty_binding) propss);
in
fun witness_proof after_qed wit_propss =
gen_witness_proof (Proof.theorem NONE) (fn wits => fn _ => after_qed wits)
wit_propss [];
val witness_proof_eqs = gen_witness_proof (Proof.theorem NONE);
fun witness_local_proof after_qed cmd wit_propss goal_ctxt int =
gen_witness_proof (proof_local cmd goal_ctxt int)
(fn wits => fn _ => after_qed wits) wit_propss [];
fun witness_local_proof_eqs after_qed cmd wit_propss eq_props goal_ctxt int =
gen_witness_proof (proof_local cmd goal_ctxt int) after_qed wit_propss eq_props;
end;
fun compose_witness (Witness (_, th)) r =
let
val th' = Goal.conclude th;
val A = Thm.cprem_of r 1;
in
Thm.implies_elim
(Conv.gconv_rule Drule.beta_eta_conversion 1 r)
(Conv.fconv_rule Drule.beta_eta_conversion
(Thm.instantiate (Thm.match (Thm.cprop_of th', A)) th'))
end;
fun conclude_witness (Witness (_, th)) =
Thm.close_derivation (Raw_Simplifier.norm_hhf_protect (Goal.conclude th));
fun pretty_witness ctxt witn =
let val prt_term = Pretty.quote o Syntax.pretty_term ctxt in
Pretty.block (prt_term (witness_prop witn) ::
(if Config.get ctxt 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;
(* 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.check_thy thy in
Morphism.morphism
{binding = [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 #>
Conv.fconv_rule (Thm.beta_conversion true))
end;
fun inst_morphism thy envs =
let val thy_ref = Theory.check_thy thy in
Morphism.morphism
{binding = [],
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 w => Thm.implies_intr hyp #> compose_witness w)) (#hyps (Thm.crep_thm thm));
val satisfy_morphism = Morphism.thm_morphism o satisfy_thm;
val satisfy_facts = facts_map o morph_ctxt o satisfy_morphism;
(* rewriting with equalities *)
fun eq_morphism thy thms = if null thms then NONE else SOME (Morphism.morphism
{binding = [],
typ = [],
term = [Raw_Simplifier.rewrite_term thy thms []],
fact = [map (Raw_Simplifier.rewrite_rule thms)]});
(* transfer to theory using closure *)
fun transfer_morphism thy =
let val thy_ref = Theory.check_thy thy
in Morphism.thm_morphism (fn th => Thm.transfer (Theory.deref thy_ref) th) end;
(** activate in context **)
(* init *)
fun generic_note_thmss kind facts context =
let
val facts' = Attrib.map_facts (Attrib.attribute_i (Context.theory_of context)) facts;
in
context |> Context.mapping_result
(Global_Theory.note_thmss kind facts')
(Proof_Context.note_thmss kind facts')
end;
fun init (Fixes fixes) = Context.map_proof (Proof_Context.add_fixes fixes #> #2)
| init (Constrains _) = I
| init (Assumes asms) = Context.map_proof (fn ctxt =>
let
val asms' = Attrib.map_specs (Attrib.attribute_i (Proof_Context.theory_of ctxt)) asms;
val (_, ctxt') = ctxt
|> fold Variable.auto_fixes (maps (map #1 o #2) asms')
|> Proof_Context.add_assms_i Assumption.assume_export asms';
in ctxt' end)
| init (Defines defs) = Context.map_proof (fn ctxt =>
let
val defs' = Attrib.map_specs (Attrib.attribute_i (Proof_Context.theory_of ctxt)) defs;
val asms = defs' |> map (fn ((name, atts), (t, ps)) =>
let val ((c, _), t') = Local_Defs.cert_def ctxt t (* FIXME adapt ps? *)
in (t', ((Thm.def_binding_optional (Binding.name c) name, atts), [(t', ps)])) end);
val (_, ctxt') = ctxt
|> fold Variable.auto_fixes (map #1 asms)
|> Proof_Context.add_assms_i Local_Defs.def_export (map #2 asms);
in ctxt' end)
| init (Notes (kind, facts)) = generic_note_thmss kind facts #> #2;
(* activate *)
fun activate_i elem ctxt =
let
val elem' = map_ctxt_attrib Args.assignable elem;
val ctxt' = Context.proof_map (init elem') ctxt;
in (map_ctxt_attrib Args.closure elem', ctxt') end;
fun activate raw_elem ctxt =
let val elem = raw_elem |> map_ctxt
{binding = I,
typ = I,
term = I,
pattern = I,
fact = Proof_Context.get_fact ctxt,
attrib = Attrib.intern_src (Proof_Context.theory_of ctxt)}
in activate_i elem ctxt end;
end;