(* Title: Pure/Isar/proof.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Proof states and methods.
*)
signature PROOF =
sig
type context
type state
exception STATE of string * state
val check_result: string -> state -> 'a Seq.seq -> 'a Seq.seq
val init_state: theory -> state
val context_of: state -> context
val theory_of: state -> theory
val sign_of: state -> Sign.sg
val reset_thms: string -> state -> state
val the_facts: state -> thm list
val get_goal: state -> term * (thm list * thm)
val goal_facts: (state -> thm list) -> state -> state
val use_facts: state -> state
val reset_facts: state -> state
val assert_forward: state -> state
val assert_backward: state -> state
val enter_forward: state -> state
val show_hyps: bool ref
val pretty_thm: thm -> Pretty.T
val pretty_thms: thm list -> Pretty.T
val verbose: bool ref
val print_state: int -> state -> unit
val level: state -> int
type method
val method: (thm list -> tactic) -> method
val refine: (context -> method) -> state -> state Seq.seq
val find_free: term -> string -> term option
val export_thm: context -> thm -> thm
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: thm list -> string -> context attribute list ->
(thm list * context attribute list) list -> state -> state
val simple_have_thms: string -> thm list -> state -> state
val fix: (string list * string option) list -> state -> state
val fix_i: (string list * typ option) list -> state -> state
val assm: (int -> tactic) * (int -> tactic)
-> (string * context attribute list * (string * (string list * string list)) list) list
-> state -> state
val assm_i: (int -> tactic) * (int -> tactic)
-> (string * context attribute list * (term * (term list * term list)) list) list
-> state -> state
val assume: (string * context attribute list * (string * (string list * string list)) list) list
-> state -> state
val assume_i: (string * context attribute list * (term * (term list * term list)) list) list
-> state -> state
val presume: (string * context attribute list * (string * (string list * string list)) list) list
-> state -> state
val presume_i: (string * context attribute list * (term * (term list * term list)) list) list
-> state -> state
val theorem: bstring -> theory attribute list -> string * (string list * string list)
-> theory -> state
val theorem_i: bstring -> theory attribute list -> term * (term list * term list)
-> theory -> state
val lemma: bstring -> theory attribute list -> string * (string list * string list)
-> theory -> state
val lemma_i: bstring -> theory attribute list -> term * (term list * term list)
-> theory -> state
val chain: state -> state
val export_chain: state -> state Seq.seq
val from_facts: thm list -> state -> state
val show: (state -> state Seq.seq) -> string -> context attribute list
-> string * (string list * string list) -> state -> state
val show_i: (state -> state Seq.seq) -> string -> context attribute list
-> term * (term list * term list) -> state -> state
val have: (state -> state Seq.seq) -> string -> context attribute list
-> string * (string list * string list) -> state -> state
val have_i: (state -> state Seq.seq) -> string -> context attribute list
-> term * (term list * term list) -> state -> state
val at_bottom: state -> bool
val local_qed: (state -> state Seq.seq)
-> ({kind: string, name: string, thm: thm} -> unit) * (thm -> unit) -> state -> state Seq.seq
val global_qed: (state -> state Seq.seq) -> state
-> (theory * {kind: string, name: string, thm: thm}) Seq.seq
val begin_block: state -> state
val end_block: state -> state Seq.seq
val next_block: state -> state
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*)
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");
(* datatype state *)
datatype node =
Node of
{context: context,
facts: thm list option,
mode: mode,
goal: (goal * (state -> state Seq.seq)) option}
and state =
State of
node * (*current*)
node list; (*parents wrt. block structure*)
fun make_node (context, facts, mode, goal) =
Node {context = context, facts = facts, mode = mode, goal = goal};
exception STATE of string * state;
fun err_malformed name state =
raise STATE (name ^ ": internal error -- malformed proof state", state);
fun check_result msg state sq =
(case Seq.pull sq of
None => raise STATE (msg, state)
| Some s_sq => Seq.cons s_sq);
fun map_current f (State (Node {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 (Node {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 (Node {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 auto_bind_goal = map_context o ProofContext.auto_bind_goal;
val auto_bind_facts = map_context oo ProofContext.auto_bind_facts;
val put_thms = map_context o ProofContext.put_thms;
val put_thmss = map_context o ProofContext.put_thmss;
val reset_thms = map_context o ProofContext.reset_thms;
val assumptions = ProofContext.assumptions o context_of;
(* facts *)
fun the_facts (State (Node {facts = Some facts, ...}, _)) = facts
| the_facts state = raise STATE ("No current facts available", state);
fun assert_facts state = (the_facts state; state);
fun get_facts (State (Node {facts, ...}, _)) = facts;
val thisN = "this";
fun put_facts facts state =
state
|> map_current (fn (ctxt, _, mode, goal) => (ctxt, facts, mode, goal))
|> (case facts of None => reset_thms thisN | Some ths => put_thms (thisN, ths));
val reset_facts = put_facts None;
fun have_facts (name, facts) state =
state
|> put_facts (Some facts)
|> put_thms (name, facts);
fun these_facts (state, ths) = have_facts ths state;
(* goal *)
fun find_goal i (state as State (Node {goal = Some goal, ...}, _)) = (context_of state, (i, goal))
| find_goal i (State (Node {goal = None, ...}, node :: nodes)) =
find_goal (i + 1) (State (node, nodes))
| find_goal _ (state as State (_, [])) = err_malformed "find_goal" state;
fun get_goal state =
let val (_, (_, (((_, _, t), goal), _))) = find_goal 0 state
in (t, goal) end;
fun put_goal goal = map_current (fn (ctxt, facts, mode, _) => (ctxt, facts, mode, goal));
fun map_goal f (State (Node {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)), f) => ((result, (get state, thm)), f));
fun use_facts state =
state
|> goal_facts the_facts
|> reset_facts;
(* mode *)
fun get_mode (State (Node {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 proof 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 level (State (_, nodes)) = length nodes;
fun open_block (State (node, nodes)) = State (node, node :: nodes);
fun new_block state =
state
|> open_block
|> put_goal None;
fun close_block (state as State (_, node :: nodes)) = State (node, nodes)
| close_block state = raise STATE ("Unbalanced block parentheses", state);
(** print_state **)
val show_hyps = ProofContext.show_hyps;
val pretty_thm = ProofContext.pretty_thm;
fun pretty_thms [th] = pretty_thm th
| pretty_thms ths = Pretty.blk (0, Pretty.fbreaks (map pretty_thm ths));
val verbose = ProofContext.verbose;
fun print_facts _ None = ()
| print_facts s (Some ths) =
(Pretty.writeln (Pretty.big_list (s ^ "this:") (map pretty_thm ths)); writeln "");
fun print_state nr (state as State (Node {context, facts, mode, goal = _}, nodes)) =
let
val ref (_, _, begin_goal) = Goals.current_goals_markers;
fun levels_up 0 = ""
| levels_up 1 = ", 1 level up"
| levels_up i = ", " ^ string_of_int i ^ " levels up";
fun print_goal (_, (i, (((kind, name, _), (goal_facts, thm)), _))) =
(print_facts "using "
(if mode <> Backward orelse null goal_facts then None else Some goal_facts);
writeln ("goal (" ^ kind_name kind ^ (if name = "" then "" else " " ^ name) ^
levels_up (i div 2) ^ "):");
Locale.print_goals_marker begin_goal (! goals_limit) thm);
val ctxt_strings =
if ! verbose orelse mode = Forward then ProofContext.strings_of_context context
else if mode = Backward then ProofContext.strings_of_prems context
else [];
in
writeln ("Proof(" ^ mode_name mode ^ "): step " ^ string_of_int nr ^
", depth " ^ string_of_int (length nodes div 2));
writeln "";
if null ctxt_strings then () else (seq writeln ctxt_strings; writeln "");
if ! verbose orelse mode = Forward then
(print_facts "" facts; print_goal (find_goal 0 state))
else if mode = ForwardChain then print_facts "picking " facts
else print_goal (find_goal 0 state)
end;
(** proof steps **)
(* datatype method *)
datatype method = Method of thm list -> tactic;
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 =
let
val Method meth = meth_fun (context_of state);
val (_, (_, ((result, (facts, thm)), f))) = find_goal 0 state;
fun refn thm' =
state
|> check_sign (Thm.sign_of_thm thm')
|> map_goal (K ((result, (facts, thm')), f));
in Seq.map refn (meth facts thm) end;
(* export *)
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);
local
fun varify_frees fixes thm =
let
val {sign, prop, ...} = Thm.rep_thm thm;
val frees = map (Thm.cterm_of sign) (mapfilter (find_free prop) fixes);
in
thm
|> Drule.forall_intr_list frees
|> Drule.forall_elim_vars 0
end;
fun most_general_varify_tfrees thm =
let
val {hyps, prop, ...} = Thm.rep_thm thm;
val frees = foldr Term.add_term_frees (prop :: hyps, []);
val leave_tfrees = foldr Term.add_term_tfree_names (frees, []);
in thm |> Thm.varifyT' leave_tfrees end;
fun diff_context inner None = (ProofContext.fixed_names inner, ProofContext.assumptions inner)
| diff_context inner (Some outer) =
(ProofContext.fixed_names inner \\ ProofContext.fixed_names outer,
Library.drop (length (ProofContext.assumptions outer), ProofContext.assumptions inner));
in
fun export fixes casms thm =
thm
|> Drule.implies_intr_list casms
|> varify_frees fixes
|> most_general_varify_tfrees;
fun export_wrt inner opt_outer =
let
val (fixes, asms) = diff_context inner opt_outer;
val casms = map (Drule.mk_cgoal o #1) asms;
val tacs = map #2 asms;
in (export fixes casms, tacs) end;
end;
(* export results *)
fun RANGE [] _ = all_tac
| RANGE (tac :: tacs) i = RANGE tacs (i + 1) THEN tac i;
fun export_goal print_rule raw_rule inner state =
let
val (outer, (_, ((result, (facts, thm)), f))) = find_goal 0 state;
val (exp, tacs) = export_wrt inner (Some outer);
val rule = exp raw_rule;
val _ = print_rule rule;
val thmq = FIRSTGOAL (Tactic.rtac rule THEN' RANGE (map #1 tacs)) thm;
in Seq.map (fn th => map_goal (K ((result, (facts, th)), f)) state) thmq end;
fun export_thm inner thm =
let val (exp, tacs) = export_wrt inner None in
(case Seq.chop (2, RANGE (map #2 tacs) 1 (exp thm)) of
([thm'], _) => thm'
| ([], _) => raise THM ("export: failed", 0, [thm])
| _ => raise THM ("export: more than one result", 0, [thm]))
end;
fun export_facts inner_state opt_outer_state state =
let
val thms = the_facts inner_state;
val (exp, tacs) = export_wrt (context_of inner_state) (apsome context_of opt_outer_state);
val thmqs = map (RANGE (map #2 tacs) 1 o exp) thms;
in Seq.map (fn ths => put_facts (Some ths) state) (Seq.commute thmqs) end;
fun transfer_facts inner_state state =
(case get_facts inner_state of
None => Seq.single (reset_facts state)
| Some ths => export_facts inner_state (Some state) state);
(* prepare result *)
fun prep_result state 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;
val {hyps, prop, sign, maxidx, ...} = Thm.rep_thm thm;
val tsig = Sign.tsig_of sign;
val casms = map #1 (assumptions state);
val bad_hyps = Library.gen_rems Term.aconv (hyps, map (Thm.term_of o Drule.mk_cgoal) casms);
in
if not (null bad_hyps) then
err ("Additional hypotheses:\n" ^ cat_lines (map (Sign.string_of_term sign) bad_hyps))
else if not (t aconv prop) then
err ("Proved a different theorem: " ^ Sign.string_of_term sign prop)
else Drule.forall_elim_vars (maxidx + 1) thm
end;
(* prepare final result *)
fun strip_flexflex thm =
Seq.hd (Thm.flexflex_rule thm) handle THM _ => thm;
fun final_result state thm =
thm
|> strip_flexflex
|> Drule.standard;
(*** 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 o map (apsnd Some))
val bind_i = gen_bind (ProofContext.add_binds_i o map (apsnd Some))
val match_bind = gen_bind ProofContext.match_binds;
val match_bind_i = gen_bind ProofContext.match_binds_i;
(* have_thmss *)
fun have_thmss ths name atts ths_atts state =
state
|> assert_forward
|> map_context_result (ProofContext.have_thmss ths name atts ths_atts)
|> these_facts;
fun simple_have_thms name thms = have_thmss [] name [] [(thms, [])];
(* 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 *)
local
fun gen_assume f tac args state =
state
|> assert_forward
|> map_context_result (f tac args)
|> (fn (st, (factss, prems)) =>
foldl these_facts (st, factss)
|> put_thms ("prems", prems)
|> put_facts (Some (flat (map #2 factss))));
val hard_asm_tac = Tactic.etac Drule.triv_goal;
val soft_asm_tac = Tactic.rtac Drule.triv_goal;
in
val assm = gen_assume ProofContext.assume;
val assm_i = gen_assume ProofContext.assume_i;
val assume = assm (hard_asm_tac, soft_asm_tac);
val assume_i = assm_i (hard_asm_tac, soft_asm_tac);
val presume = assm (soft_asm_tac, soft_asm_tac);
val presume_i = assm_i (soft_asm_tac, soft_asm_tac);
end;
(** goals **)
(* forward chaining *)
fun chain state =
state
|> assert_forward
|> assert_facts
|> enter_forward_chain;
fun export_chain state =
state
|> assert_forward
|> export_facts state None
|> Seq.map chain;
fun from_facts facts state =
state
|> put_facts (Some facts)
|> chain;
(* setup goals *)
fun setup_goal opt_block prepp kind after_qed name atts raw_propp state =
let
val (state', prop) =
state
|> assert_forward_or_chain
|> enter_forward
|> map_context_result (fn c => prepp (c, raw_propp));
val cprop = Thm.cterm_of (sign_of state') prop;
val goal = Drule.mk_triv_goal cprop;
in
state'
|> opt_block
|> put_goal (Some (((kind atts, name, prop), ([], goal)), after_qed))
|> auto_bind_goal 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 Seq.single 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 f name atts x =
setup_goal open_block prep kind f 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;
(** conclusions **)
(* current goal *)
fun current_goal (State (Node {context, goal = Some goal, ...}, _)) = (context, goal)
| current_goal state = raise STATE ("No current goal!", state);
fun assert_current_goal true (state as State (Node {goal = None, ...}, _)) =
raise STATE ("No goal in this block!", state)
| assert_current_goal false (state as State (Node {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);
fun end_proof bot state =
state
|> assert_forward
|> close_block
|> assert_bottom bot
|> assert_current_goal true
|> goal_facts (K []);
(* local_qed *)
fun finish_local (print_result, print_rule) state =
let
val (ctxt, (((kind, name, t), (_, raw_thm)), after_qed)) = current_goal state;
val result = prep_result state t raw_thm;
val (atts, opt_solve) =
(case kind of
Goal atts => (atts, export_goal print_rule result ctxt)
| Aux atts => (atts, Seq.single)
| _ => err_malformed "finish_local" state);
in
print_result {kind = kind_name kind, name = name, thm = result};
state
|> close_block
|> auto_bind_facts name [t]
|> have_thmss [] name atts [Thm.no_attributes [result]]
|> opt_solve
|> (Seq.flat o Seq.map after_qed)
end;
fun local_qed finalize print state =
state
|> end_proof false
|> finalize
|> (Seq.flat o Seq.map (finish_local print));
(* global_qed *)
fun finish_global state =
let
val (_, (((kind, name, t), (_, raw_thm)), _)) = current_goal state; (*ignores after_qed!*)
val result = final_result state (prep_result state t raw_thm);
val atts =
(case kind of
Theorem atts => atts
| Lemma atts => atts @ [Drule.tag_lemma]
| _ => err_malformed "finish_global" state);
val (thy', result') = PureThy.store_thm ((name, result), atts) (theory_of state);
in (thy', {kind = kind_name kind, name = name, thm = result'}) end;
(*Note: should inspect first result only, backtracking may destroy theory*)
fun global_qed finalize state =
state
|> end_proof true
|> finalize
|> Seq.map finish_global;
(** blocks **)
(* begin_block *)
fun begin_block state =
state
|> assert_forward
|> new_block
|> open_block;
(* end_block *)
fun end_block state =
state
|> assert_forward
|> close_block
|> assert_current_goal false
|> close_block
|> transfer_facts state;
(* next_block *)
fun next_block state =
state
|> assert_forward
|> close_block
|> assert_current_goal true
|> new_block;
end;