(* Title: Pure/Isar/element.ML
Author: Makarius
Explicit data structures for some Isar language elements, with derived
logical operations.
*)
signature ELEMENT =
sig
type ('typ, 'term) obtain = binding * ((binding * 'typ option * mixfix) list * 'term list)
type obtains = (string, string) obtain list
type obtains_i = (typ, term) obtain list
datatype ('typ, 'term) stmt =
Shows of (Attrib.binding * ('term * 'term list) list) list |
Obtains of ('typ, 'term) obtain 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 * Token.src list) list) list
type context = (string, string, Facts.ref) ctxt
type context_i = (typ, term, thm list) ctxt
val map_ctxt: {binding: binding -> binding, typ: 'typ -> 'a, term: 'term -> 'b,
pattern: 'term -> 'b, fact: 'fact -> 'c, attrib: Token.src -> Token.src} ->
('typ, 'term, 'fact) ctxt -> ('a, 'b, 'c) ctxt
val map_ctxt_attrib: (Token.src -> Token.src) ->
('typ, 'term, 'fact) ctxt -> ('typ, 'term, 'fact) ctxt
val transform_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_ctxt_no_attribs: 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 transform_witness: morphism -> witness -> witness
val conclude_witness: Proof.context -> witness -> thm
val pretty_witness: Proof.context -> witness -> Pretty.T
val instT_morphism: theory -> typ Symtab.table -> morphism
val inst_morphism: theory -> typ Symtab.table * term Symtab.table -> morphism
val satisfy_morphism: witness list -> morphism
val eq_morphism: theory -> thm list -> morphism option
val init: context_i -> Context.generic -> 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 *)
type ('typ, 'term) obtain = binding * ((binding * 'typ option * mixfix) list * 'term list);
type obtains = (string, string) obtain list;
type obtains_i = (typ, term) obtain list;
datatype ('typ, 'term) stmt =
Shows of (Attrib.binding * ('term * 'term list) list) list |
Obtains of ('typ, 'term) obtain 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 * Token.src list) list) list;
type context = (string, string, Facts.ref) ctxt;
type context_i = (typ, term, thm list) ctxt;
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 transform_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 = Token.transform_src phi};
(** pretty printing **)
fun pretty_items _ _ [] = []
| pretty_items keyword sep (x :: ys) =
Pretty.block [Pretty.keyword2 keyword, Pretty.brk 1, x] ::
map (fn y => Pretty.block [Pretty.str " ", Pretty.keyword2 sep, Pretty.brk 1, y]) ys;
(* 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.keyword2 "and") o map prt_term;
val prt_binding = Attrib.pretty_binding ctxt;
fun prt_show (a, ts) =
Pretty.block (Pretty.breaks (prt_binding 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.keyword2 "and") o map prt_var;
fun prt_obtain (_, ([], props)) = Pretty.block (Pretty.breaks (prt_terms props))
| prt_obtain (_, (vars, props)) = Pretty.block (Pretty.breaks
(prt_vars vars @ [Pretty.keyword2 "where"] @ prt_terms props));
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 gen_pretty_ctxt show_attribs 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;
fun prt_binding (b, atts) =
Attrib.pretty_binding ctxt (b, if show_attribs then atts else []);
fun prt_fact (ths, atts) =
if not show_attribs orelse null atts then map prt_thm ths
else
Pretty.enclose "(" ")" (Pretty.breaks (map prt_thm ths)) ::
Attrib.pretty_attribs ctxt atts;
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_binding a ":" @ map (prt_term o fst) ts));
fun prt_def (a, (t, _)) =
Pretty.block (Pretty.breaks (prt_binding a ":" @ [prt_term t]));
fun prt_note (a, ths) =
Pretty.block (Pretty.breaks (flat (prt_binding 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;
val pretty_ctxt = gen_pretty_ctxt true;
val pretty_ctxt_no_attribs = gen_pretty_ctxt false;
(* pretty_statement *)
local
fun standard_elim ctxt th =
(case Object_Logic.elim_concl ctxt th of
SOME C =>
let
val thesis = Var ((Auto_Bind.thesisN, Thm.maxidx_of th + 1), fastype_of C);
val th' = Thm.instantiate ([], [apply2 (Thm.cterm_of ctxt) (C, 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.keyword1 kind,
Pretty.brk 1, Pretty.str (Long_Name.base_name (Thm.get_name_hint th) ^ ":")]
else Pretty.keyword1 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, NoSyn);
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 (th, is_elim) = standard_elim ctxt (Raw_Simplifier.norm_hhf ctxt 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 ctxt concl;
val fixes = fold_aterms (fn v as Free (x, T) =>
if Variable.is_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 transform_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 _ => resolve_tac ctxt [Drule.protectI] 1 THEN tac)));
local
val refine_witness =
Proof.refine (Method.Basic (fn ctxt => EMPTY_CASES o
K (ALLGOALS (CONJUNCTS (ALLGOALS (CONJUNCTS (TRYALL (resolve_tac ctxt [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 propp =
let
fun after_qed' (result_ctxt, results) state' =
after_qed (burrow (Proof_Context.export result_ctxt (Proof.context_of state')) results) state';
in
Proof.map_context (K goal_ctxt) #>
Proof.internal_goal (K (K ())) (Proof_Context.get_mode goal_ctxt) cmd
NONE after_qed' [] [] (map (pair Thm.empty_binding) propp) #> snd
end;
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 ctxt (Witness (_, th)) =
Thm.close_derivation (Raw_Simplifier.norm_hhf_protect ctxt (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 idx = Thm.maxidx_of th + 1;
fun cert_inst (a, (S, T)) = apply2 (Thm.global_ctyp_of thy) (TVar ((a, idx), S), 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 o 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 =
Drule.forall_intr_list (map (Thm.global_cterm_of thy o Free o fst) subst) #>
Drule.forall_elim_list (map (Thm.global_cterm_of thy o snd) subst);
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_same env =
if Symtab.is_empty env then Same.same
else
Term_Subst.map_atypsT_same
(fn TFree (a, _) => (case Symtab.lookup env a of SOME T => T | NONE => raise Same.SAME)
| _ => raise Same.SAME);
fun instT_term_same env =
if Symtab.is_empty env then Same.same
else Term_Subst.map_types_same (instT_type_same env);
val instT_type = Same.commit o instT_type_same;
val instT_term = Same.commit o instT_term_same;
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 (eq_fst (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 =
Morphism.morphism "Element.instT"
{binding = [],
typ = [instT_type env],
term = [instT_term env],
fact = [map (instT_thm thy env)]};
(* instantiate types and terms *)
fun inst_term (envT, env) =
if Symtab.is_empty env then instT_term envT
else
instT_term envT #>
Same.commit (Term_Subst.map_aterms_same
(fn Free (x, _) => (case Symtab.lookup env x of SOME t => t | NONE => raise Same.SAME)
| _ => raise Same.SAME)) #>
Envir.beta_norm;
fun inst_subst (envT, env) th =
(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 [];
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 = inst_subst (envT, env) 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 (envT, env) =
Morphism.morphism "Element.inst"
{binding = [],
typ = [instT_type envT],
term = [inst_term (envT, env)],
fact = [map (inst_thm thy (envT, env))]};
(* 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 "Element.satisfy" o satisfy_thm;
(* rewriting with equalities *)
fun eq_morphism _ [] = NONE
| eq_morphism thy thms =
let
(* FIXME proper context!? *)
fun rewrite th = rewrite_rule (Proof_Context.init_global (Thm.theory_of_thm th)) thms th;
val phi =
Morphism.morphism "Element.eq_morphism"
{binding = [],
typ = [],
term = [Raw_Simplifier.rewrite_term thy thms []],
fact = [map rewrite]};
in SOME phi end;
(** activate in context **)
(* init *)
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 (map (Attrib.attribute ctxt)) asms;
val (_, ctxt') = ctxt
|> fold Variable.auto_fixes (maps (map #1 o #2) asms')
|> Proof_Context.add_assms Assumption.assume_export asms';
in ctxt' end)
| init (Defines defs) = Context.map_proof (fn ctxt =>
let
val defs' = Attrib.map_specs (map (Attrib.attribute ctxt)) defs;
val asms = defs' |> map (fn (b, (t, ps)) =>
let val (_, t') = Local_Defs.cert_def ctxt t (* FIXME adapt ps? *)
in (t', (b, [(t', ps)])) end);
val (_, ctxt') = ctxt
|> fold Variable.auto_fixes (map #1 asms)
|> Proof_Context.add_assms Local_Defs.def_export (map #2 asms);
in ctxt' end)
| init (Notes (kind, facts)) = Attrib.generic_notes kind facts #> #2;
fun init' elem context =
context
|> Context.mapping I (Thm.unchecked_hyps #> Context_Position.not_really)
|> init elem
|> Context.mapping I (fn ctxt =>
let val ctxt0 = Context.proof_of context
in ctxt |> Context_Position.restore_visible ctxt0 |> Thm.restore_hyps ctxt0 end);
(* activate *)
fun activate_i elem ctxt =
let
val elem' =
(case map_ctxt_attrib Token.init_assignable_src elem of
Defines defs =>
Defines (defs |> map (fn ((a, atts), (t, ps)) =>
((Thm.def_binding_optional (Binding.name (#1 (#1 (Local_Defs.cert_def ctxt t)))) a, atts),
(t, ps))))
| e => e);
val ctxt' = Context.proof_map (init elem') ctxt;
in (map_ctxt_attrib Token.closure_src 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.check_src ctxt}
in activate_i elem ctxt end;
end;