theory Tacs

imports Main

begin

(* General not-nominal/quotient functionality useful for proving *)

(* A version of case_rule_tac that takes more exhaust rules *)

ML {*

fun case_rules_tac ctxt0 s rules i st =

let

  val (_, ctxt) = Variable.focus_subgoal i st ctxt0;

  val ty = fastype_of (ProofContext.read_term_schematic ctxt s)

  fun exhaust_ty thm = fastype_of (hd (Induct.vars_of (Thm.term_of (Thm.cprem_of thm 1))));

  val ty_rules = filter (fn x => exhaust_ty x = ty) rules;

in

  InductTacs.case_rule_tac ctxt0 s (hd ty_rules) i st

end

*}

ML {*

fun mk_conjl props =

  fold (fn a => fn b =>

    if a = @{term True} then b else

    if b = @{term True} then a else

    HOLogic.mk_conj (a, b)) (rev props) @{term True};

*}

ML {*

val split_conj_tac = REPEAT o etac conjE THEN' TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI)

*}

(* Given function for buildng a goal for an input, prepares a

   one common goals for all the inputs and proves it by induction

   together *)

ML {*

fun prove_by_induct tys build_goal ind utac inputs ctxt =

let

  val names = Datatype_Prop.make_tnames tys;

  val (names', ctxt') = Variable.variant_fixes names ctxt;

  val frees = map Free (names' ~~ tys);

  val (gls_lists, ctxt'') = fold_map (build_goal (tys ~~ frees)) inputs ctxt';

  val gls = flat gls_lists;

  fun trm_gls_map t = filter (exists_subterm (fn s => s = t)) gls;

  val trm_gl_lists = map trm_gls_map frees;

  val trm_gl_insts = map2 (fn n => fn l => [NONE, if l = [] then NONE else SOME n]) names' trm_gl_lists

  val trm_gls = map mk_conjl trm_gl_lists;

  val gl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj trm_gls);

  fun tac {context,...} = (

    InductTacs.induct_rules_tac context [(flat trm_gl_insts)] [ind]

    THEN_ALL_NEW split_conj_tac THEN_ALL_NEW utac) 1

  val th_loc = Goal.prove ctxt'' [] [] gl tac

  val ths_loc = HOLogic.conj_elims th_loc

  val ths = Variable.export ctxt'' ctxt ths_loc

in

  filter (fn x => not (prop_of x = prop_of @{thm TrueI})) ths

end

*}

(* An induction for a single relation is "R x y \<Longrightarrow> P x y"

   but for multiple relations is "(R1 x y \<longrightarrow> P x y) \<and> (R2 a b \<longrightarrow> P2 a b)" *)

ML {*

fun rel_indtac induct = (rtac impI THEN' etac induct) ORELSE' rtac induct

*}

ML {*

fun prove_by_rel_induct alphas build_goal ind utac inputs ctxt =

let

  val tys = map (domain_type o fastype_of) alphas;

  val names = Datatype_Prop.make_tnames tys;

  val (namesl, ctxt') = Variable.variant_fixes names ctxt;

  val (namesr, ctxt'') = Variable.variant_fixes names ctxt';

  val freesl = map Free (namesl ~~ tys);

  val freesr = map Free (namesr ~~ tys);

  val (gls_lists, ctxt'') = fold_map (build_goal (tys ~~ (freesl ~~ freesr))) inputs ctxt'';

  val gls = flat gls_lists;

  fun trm_gls_map t = filter (exists_subterm (fn s => s = t)) gls;

  val trm_gl_lists = map trm_gls_map freesl;

  val trm_gls = map mk_conjl trm_gl_lists;

  val pgls = map

    (fn ((alpha, gl), (l, r)) => HOLogic.mk_imp (alpha $ l $ r, gl)) 

    ((alphas ~~ trm_gls) ~~ (freesl ~~ freesr))

  val gl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj pgls);

  fun tac {context,...} = (rel_indtac ind THEN_ALL_NEW split_conj_tac THEN_ALL_NEW

    TRY o rtac @{thm TrueI} THEN_ALL_NEW utac context) 1

  val th_loc = Goal.prove ctxt'' [] [] gl tac

  val ths_loc = HOLogic.conj_elims th_loc

  val ths = Variable.export ctxt'' ctxt ths_loc

in

  filter (fn x => not (prop_of x = prop_of @{thm TrueI})) ths

end

*}

(* Code for transforming an inductive relation to a function *)

ML {*

fun rel_inj_tac dist_inj intrs elims =

  SOLVED' (asm_full_simp_tac (HOL_ss addsimps intrs)) ORELSE'

  (rtac @{thm iffI} THEN' RANGE [

     (eresolve_tac elims THEN_ALL_NEW

       asm_full_simp_tac (HOL_ss addsimps dist_inj)

     ),

     asm_full_simp_tac (HOL_ss addsimps intrs)])

*}

ML {*

fun build_rel_inj_gl thm =

  let

    val prop = prop_of thm;

    val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop);

    val hyps = map HOLogic.dest_Trueprop (Logic.strip_imp_prems prop);

    fun list_conj l = foldr1 HOLogic.mk_conj l;

  in

    if hyps = [] then concl

    else HOLogic.mk_eq (concl, list_conj hyps)

  end;

*}

ML {*

fun build_rel_inj intrs dist_inj elims ctxt =

let

  val ((_, thms_imp), ctxt') = Variable.import false intrs ctxt;

  val gls = map (HOLogic.mk_Trueprop o build_rel_inj_gl) thms_imp;

  fun tac _ = rel_inj_tac dist_inj intrs elims 1;

  val thms = map (fn gl => Goal.prove ctxt' [] [] gl tac) gls;

in

  Variable.export ctxt' ctxt thms

end

*}

end