(* Title: Pure/Isar/proof.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Proof states and methods.
TODO:
- assume: improve schematic Vars handling (!?);
- warn_vars;
- bind: check rhs (extra (T)Vars etc.) (How to handle these anyway?);
- prep_result: avoid duplicate asms;
- prep_result error: use error channel (!);
- check_finished: trace results (!?);
- next_block: fetch_facts (!?);
*)
signature PROOF =
sig
type context
type state
exception STATE of string * state
val context_of: state -> context
val theory_of: state -> theory
val sign_of: state -> Sign.sg
val the_facts: state -> thm list
val goal_facts: (state -> thm list) -> state -> state
val use_facts: state -> state
val reset_facts: state -> state
val assert_backward: state -> state
val enter_forward: state -> state
val print_state: state -> unit
type method
val method: (thm list -> thm ->
(thm * (indexname * term) list * (string * thm list) list) Seq.seq) -> method
val refine: (context -> method) -> state -> state Seq.seq
val bind: (indexname * string) list -> state -> state
val bind_i: (indexname * term) list -> state -> state
val match_bind: (string list * string) list -> state -> state
val match_bind_i: (term list * term) list -> state -> state
val have_thmss: string -> context attribute list ->
(thm list * context attribute list) list -> state -> state
val assume: string -> context attribute list -> (string * string list) list -> state -> state
val assume_i: string -> context attribute list -> (term * term list) list -> state -> state
val fix: (string * string option) list -> state -> state
val fix_i: (string * typ) list -> state -> state
val theorem: bstring -> theory attribute list -> string * string list -> theory -> state
val theorem_i: bstring -> theory attribute list -> term * term list -> theory -> state
val lemma: bstring -> theory attribute list -> string * string list -> theory -> state
val lemma_i: bstring -> theory attribute list -> term * term list -> theory -> state
val chain: state -> state
val from_facts: thm list -> state -> state
val show: string -> context attribute list -> string * string list -> state -> state
val show_i: string -> context attribute list -> term * term list -> state -> state
val have: string -> context attribute list -> string * string list -> state -> state
val have_i: string -> context attribute list -> term * term list -> state -> state
val begin_block: state -> state
val next_block: state -> state
val end_block: state -> state
val at_bottom: state -> bool
val local_qed: (state -> state Seq.seq) -> state -> state Seq.seq
val global_qed: (state -> state Seq.seq) -> bstring option
-> theory attribute list option -> state -> theory * (string * string * thm)
end;
signature PROOF_PRIVATE =
sig
include PROOF
val put_data: Object.kind -> ('a -> Object.T) -> 'a -> state -> state
end;
structure Proof: PROOF_PRIVATE =
struct
(** proof state **)
type context = ProofContext.context;
(* type goal *)
datatype kind =
Theorem of theory attribute list | (*top-level theorem*)
Lemma of theory attribute list | (*top-level lemma*)
Goal of context attribute list | (*intermediate result, solving subgoal*)
Aux of context attribute list ; (*intermediate result*)
val kind_name =
fn Theorem _ => "theorem" | Lemma _ => "lemma" | Goal _ => "show" | Aux _ => "have";
type goal =
(kind * (*result kind*)
string * (*result name*)
cterm list * (*result assumptions*)
term) * (*result statement*)
(thm list * (*use facts*)
thm); (*goal: subgoals ==> statement*)
(* type mode *)
datatype mode = Forward | ForwardChain | Backward;
val mode_name =
fn Forward => "state" | ForwardChain => "chain" | Backward => "prove";
(* type node *)
type node =
{context: context,
facts: thm list option,
mode: mode,
goal: goal option};
fun make_node (context, facts, mode, goal) =
{context = context, facts = facts, mode = mode, goal = goal}: node;
(* datatype state *)
datatype state =
State of
node * (*current*)
node list; (*parents wrt. block structure*)
exception STATE of string * state;
fun err_malformed name state =
raise STATE (name ^ ": internal error -- malformed proof state", state);
fun map_current f (State ({context, facts, mode, goal}, nodes)) =
State (make_node (f (context, facts, mode, goal)), nodes);
fun init_state thy =
State (make_node (ProofContext.init thy, None, Forward, None), []);
(** basic proof state operations **)
(* context *)
fun context_of (State ({context, ...}, _)) = context;
val theory_of = ProofContext.theory_of o context_of;
val sign_of = ProofContext.sign_of o context_of;
fun map_context f = map_current (fn (ctxt, facts, mode, goal) => (f ctxt, facts, mode, goal));
fun map_context_result f (state as State ({context, facts, mode, goal}, nodes)) =
let val (context', result) = f context
in (State (make_node (context', facts, mode, goal), nodes), result) end;
fun put_data kind f = map_context o ProofContext.put_data kind f;
val declare_term = map_context o ProofContext.declare_term;
val add_binds = map_context o ProofContext.add_binds_i;
val put_thms = map_context o ProofContext.put_thms;
val put_thmss = map_context o ProofContext.put_thmss;
(* bind statements *)
fun bind_props bs state =
let val mk_bind = map (fn (x, t) => ((Syntax.binding x, 0), t)) o ObjectLogic.dest_statement
in state |> add_binds (flat (map mk_bind bs)) end;
fun bind_thms (name, thms) state =
let
val props = map (#prop o Thm.rep_thm) thms;
val named_props =
(case props of
[prop] => [(name, prop)]
| props => map2 (fn (i, t) => (name ^ string_of_int i, t)) (1 upto length props, props));
in state |> bind_props named_props end;
fun let_thms name_thms state =
state
|> put_thms name_thms
|> bind_thms name_thms;
(* facts *)
fun the_facts (State ({facts = Some facts, ...}, _)) = facts
| the_facts state = raise STATE ("No current facts available", state);
fun put_facts facts state =
state
|> map_current (fn (ctxt, _, mode, goal) => (ctxt, facts, mode, goal))
|> let_thms ("facts", if_none facts []);
val reset_facts = put_facts None;
fun have_facts (name, facts) state =
state
|> put_facts (Some facts)
|> let_thms (name, facts);
fun these_facts (state, ths) = have_facts ths state;
fun fetch_facts (State ({facts, ...}, _)) = put_facts facts;
(* goal *)
fun find_goal i (State ({goal = Some goal, ...}, _)) = (i, goal)
| find_goal i (State ({goal = None, ...}, node :: nodes)) =
find_goal (i + 1) (State (node, nodes))
| find_goal _ (state as State (_, [])) = err_malformed "find_goal" state;
fun put_goal goal = map_current (fn (ctxt, facts, mode, _) => (ctxt, facts, mode, goal));
fun map_goal f (State ({context, facts, mode, goal = Some goal}, nodes)) =
State (make_node (context, facts, mode, Some (f goal)), nodes)
| map_goal f (State (nd, node :: nodes)) =
let val State (node', nodes') = map_goal f (State (node, nodes))
in State (nd, node' :: nodes') end
| map_goal _ state = state;
fun goal_facts get state =
state
|> map_goal (fn (result, (_, thm)) => (result, (get state, thm)));
fun use_facts state =
state
|> goal_facts the_facts
|> reset_facts;
(* mode *)
fun get_mode (State ({mode, ...}, _)) = mode;
fun put_mode mode = map_current (fn (ctxt, facts, _, goal) => (ctxt, facts, mode, goal));
val enter_forward = put_mode Forward;
val enter_forward_chain = put_mode ForwardChain;
val enter_backward = put_mode Backward;
fun assert_mode pred state =
let val mode = get_mode state in
if pred mode then state
else raise STATE ("Illegal application of command in " ^ mode_name mode ^ " mode", state)
end;
fun is_chain state = get_mode state = ForwardChain;
val assert_forward = assert_mode (equal Forward);
val assert_forward_or_chain = assert_mode (equal Forward orf equal ForwardChain);
val assert_backward = assert_mode (equal Backward);
(* blocks *)
fun open_block (State (node, nodes)) = State (node, node :: nodes);
fun new_block state =
state
|> open_block
|> put_goal None;
fun close_block (State (_, node :: nodes)) = State (node, nodes)
| close_block state = raise STATE ("Unbalanced block parentheses", state);
(** print_state **)
fun print_state (state as State ({context, facts, mode, goal = _}, nodes)) =
let
val ref (_, _, begin_goal) = Goals.current_goals_markers;
fun print_facts None = ()
| print_facts (Some ths) =
Pretty.writeln (Pretty.big_list "Current facts:" (map Display.pretty_thm ths));
fun levels_up 0 = ""
| levels_up i = " (" ^ string_of_int i ^ " levels up)";
fun print_goal (i, ((kind, name, _, _), (_, thm))) =
(writeln (kind_name kind ^ " " ^ quote name ^ levels_up (i div 2) ^ ":");
Locale.print_goals_marker begin_goal (! goals_limit) thm);
in
writeln ("Nesting level: " ^ string_of_int (length nodes div 2));
writeln "";
writeln (mode_name mode ^ " mode");
writeln "";
ProofContext.print_context context;
writeln "";
print_facts facts;
print_goal (find_goal 0 state)
end;
(** proof steps **)
(* datatype method *)
datatype method = Method of
thm list -> (*use facts*)
thm (*goal: subgoals ==> statement*)
-> (thm * (*refined goal*)
(indexname * term) list * (*new bindings*)
(string * thm list) list) (*new thms*)
Seq.seq;
val method = Method;
(* refine goal *)
fun check_sign sg state =
if Sign.subsig (sg, sign_of state) then state
else raise STATE ("Bad signature of result: " ^ Sign.str_of_sg sg, state);
fun refine meth_fun (state as State ({context, ...}, _)) =
let
val Method meth = meth_fun context;
val (_, (result, (facts, thm))) = find_goal 0 state;
fun refn (thm', new_binds, new_thms) =
state
|> check_sign (sign_of_thm thm')
|> map_goal (K (result, (facts, thm')))
|> add_binds new_binds
|> put_thmss new_thms;
in Seq.map refn (meth facts thm) end;
(* prepare result *)
fun varify_frees names thm =
let
fun get_free x (None, t as Free (y, _)) = if x = y then Some t else None
| get_free _ (opt, _) = opt;
fun find_free t x = foldl_aterms (get_free x) (None, t);
val {sign, maxidx, prop, ...} = Thm.rep_thm thm;
val frees = map (Thm.cterm_of sign) (mapfilter (find_free prop) names);
in
thm
|> Drule.forall_intr_list frees
|> Drule.forall_elim_vars (maxidx + 1)
end;
fun implies_elim_hyps thm =
foldl (uncurry Thm.implies_elim) (thm, map Thm.assume (Drule.cprems_of thm));
fun prep_result state asms t raw_thm =
let
val ctxt = context_of state;
fun err msg = raise STATE (msg, state);
val ngoals = Thm.nprems_of raw_thm;
val _ =
if ngoals = 0 then ()
else (Locale.print_goals ngoals raw_thm; err (string_of_int ngoals ^ " unsolved goal(s)!"));
val thm =
raw_thm RS Drule.rev_triv_goal
|> implies_elim_hyps
|> Drule.implies_intr_list asms
|> varify_frees (ProofContext.fixed_names ctxt);
val {hyps, prop, sign, ...} = Thm.rep_thm thm;
val tsig = Sign.tsig_of sign;
in
(* FIXME
if not (Pattern.matches tsig (t, Logic.skip_flexpairs prop)) then
warning ("Proved a different theorem: " ^ Sign.string_of_term sign prop)
else ();
*)
if not (null hyps) then
err ("Additional hypotheses:\n" ^ cat_lines (map (Sign.string_of_term sign) hyps))
(* FIXME else if not (Pattern.matches tsig (t, Logic.skip_flexpairs prop)) then
err ("Proved a different theorem: " ^ Sign.string_of_term sign prop) *)
else thm
end;
(* prepare final result *)
fun strip_flexflex thm =
Seq.hd (Thm.flexflex_rule thm) handle THM _ => thm;
fun final_result state pre_thm =
let
val thm =
pre_thm
|> strip_flexflex
|> Thm.strip_shyps
|> Drule.standard;
val str_of_sort = Sign.str_of_sort (Thm.sign_of_thm thm);
val xshyps = Thm.extra_shyps thm;
in
if not (null xshyps) then
raise STATE ("Extra sort hypotheses: " ^ commas (map str_of_sort xshyps), state)
else thm
end;
(* solve goal *)
fun solve_goal rule state =
let
val (_, (result, (facts, thm))) = find_goal 0 state;
val thms' = FIRSTGOAL (rtac rule THEN_ALL_NEW (TRY o assume_tac)) thm;
in Seq.map (fn thm' => map_goal (K (result, (facts, thm'))) state) thms' end;
(*** structured proof commands ***)
(** context **)
(* bind *)
fun gen_bind f x state =
state
|> assert_forward
|> map_context (f x)
|> reset_facts;
val bind = gen_bind ProofContext.add_binds;
val bind_i = gen_bind ProofContext.add_binds_i;
val match_bind = gen_bind ProofContext.match_binds;
val match_bind_i = gen_bind ProofContext.match_binds_i;
(* have_thmss *)
fun have_thmss name atts ths_atts state =
state
|> assert_forward
|> map_context_result (ProofContext.have_thmss (PureThy.default_name name) atts ths_atts)
|> these_facts;
(* fix *)
fun gen_fix f xs state =
state
|> assert_forward
|> map_context (f xs)
|> reset_facts;
val fix = gen_fix ProofContext.fix;
val fix_i = gen_fix ProofContext.fix_i;
(* assume *)
fun gen_assume f name atts props state =
state
|> assert_forward
|> map_context_result (f (PureThy.default_name name) atts props)
|> these_facts
|> (fn st => let_thms ("prems", ProofContext.assumptions (context_of st)) st);
val assume = gen_assume ProofContext.assume;
val assume_i = gen_assume ProofContext.assume_i;
(** goals **)
(* forward chaining *)
fun chain state =
state
|> assert_forward
|> enter_forward_chain;
fun from_facts facts state =
state
|> put_facts (Some facts)
|> chain;
(* setup goals *)
fun setup_goal opt_block prepp kind name atts raw_propp state =
let
val (state', concl) =
state
|> assert_forward_or_chain
|> enter_forward
|> opt_block
|> map_context_result (fn c => prepp (c, raw_propp));
val casms = map (#prop o Thm.crep_thm) (ProofContext.assumptions (context_of state'));
val asms = map Thm.term_of casms;
val prop = Logic.list_implies (asms, concl);
val cprop = Thm.cterm_of (sign_of state') prop;
val thm = Drule.mk_triv_goal cprop;
in
state'
|> put_goal (Some ((kind atts, (PureThy.default_name name), casms, prop), ([], thm)))
|> bind_props [("thesis", prop)]
|> (if is_chain state then use_facts else reset_facts)
|> new_block
|> enter_backward
end;
(*global goals*)
fun global_goal prep kind name atts x thy =
setup_goal I prep kind name atts x (init_state thy);
val theorem = global_goal ProofContext.bind_propp Theorem;
val theorem_i = global_goal ProofContext.bind_propp_i Theorem;
val lemma = global_goal ProofContext.bind_propp Lemma;
val lemma_i = global_goal ProofContext.bind_propp_i Lemma;
(*local goals*)
fun local_goal prep kind name atts x =
setup_goal open_block prep kind name atts x;
val show = local_goal ProofContext.bind_propp Goal;
val show_i = local_goal ProofContext.bind_propp_i Goal;
val have = local_goal ProofContext.bind_propp Aux;
val have_i = local_goal ProofContext.bind_propp_i Aux;
(** blocks **)
(* begin_block *)
fun begin_block state =
state
|> assert_forward
|> new_block
|> reset_facts
|> open_block;
(* next_block *)
fun next_block state =
state
|> assert_forward
|> close_block
|> new_block;
(** conclusions **)
(* current goal *)
fun current_goal (State ({goal = Some goal, ...}, _)) = goal
| current_goal state = raise STATE ("No current goal!", state);
fun assert_current_goal true (state as State ({goal = None, ...}, _)) =
raise STATE ("No goal in this block!", state)
| assert_current_goal false (state as State ({goal = Some _, ...}, _)) =
raise STATE ("Goal present in this block!", state)
| assert_current_goal _ state = state;
fun assert_bottom true (state as State (_, _ :: _)) =
raise STATE ("Not at bottom of proof!", state)
| assert_bottom false (state as State (_, [])) =
raise STATE ("Already at bottom of proof!", state)
| assert_bottom _ state = state;
val at_bottom = can (assert_bottom true o close_block);
(* finish proof *)
fun check_finished state states =
(case Seq.pull states of
None => raise STATE ("Failed to finish proof", state)
| Some s_sq => Seq.cons s_sq);
fun finish_proof bot finalize state =
state
|> assert_forward
|> close_block
|> assert_bottom bot
|> assert_current_goal true
|> finalize
|> check_finished state;
(* end_block *)
fun end_block state =
state
|> assert_forward
|> close_block
|> assert_current_goal false
|> close_block
|> fetch_facts state;
(* local_qed *)
fun finish_local state =
let
val ((kind, name, asms, t), (_, raw_thm)) = current_goal state;
val result = prep_result state asms t raw_thm;
val (atts, opt_solve) =
(case kind of
Goal atts => (atts, solve_goal result)
| Aux atts => (atts, Seq.single)
| _ => raise STATE ("No local goal!", state));
in
state
|> close_block
|> have_thmss name atts [Thm.no_attributes [result]]
|> opt_solve
end;
fun local_qed finalize state =
state
|> finish_proof false finalize
|> (Seq.flat o Seq.map finish_local);
(* global_qed *)
fun finish_global alt_name alt_atts state =
let
val ((kind, def_name, asms, t), (_, raw_thm)) = current_goal state;
val result = final_result state (prep_result state asms t raw_thm);
val name = if_none alt_name def_name;
val atts =
(case kind of
Theorem atts => if_none alt_atts atts
| Lemma atts => (if_none alt_atts atts) @ [Drule.tag_lemma]
| _ => raise STATE ("No global goal!", state));
val (thy', result') = PureThy.store_thm ((name, result), atts) (theory_of state);
in (thy', (kind_name kind, name, result')) end;
fun global_qed finalize alt_name alt_atts state =
state
|> finish_proof true finalize
|> Seq.hd
|> finish_global alt_name alt_atts;
end;