1 lemma exi: "\<exists>(pi :: perm). P pi \<Longrightarrow> (\<And>(p :: perm). P p \<Longrightarrow> Q (pi \<bullet> p)) \<Longrightarrow> \<exists>pi. Q pi"

     2 apply (erule exE)

     3 apply (rule_tac x="pi \<bullet> pia" in exI)

     4 by auto

     5 

     6 ML {*

     7 fun alpha_eqvt_tac induct simps ctxt =

     8   rtac induct THEN_ALL_NEW

     9   simp_tac (HOL_basic_ss addsimps simps) THEN_ALL_NEW split_conj_tac THEN_ALL_NEW

    10   REPEAT o etac @{thm exi[of _ _ "p"]} THEN' split_conj_tac THEN_ALL_NEW

    11   asm_full_simp_tac (HOL_ss addsimps (all_eqvts ctxt @ simps)) THEN_ALL_NEW

    12   asm_full_simp_tac (HOL_ss addsimps

    13     @{thms supp_eqvt[symmetric] inter_eqvt[symmetric] empty_eqvt alphas prod_rel.simps prod_fv.simps}) THEN_ALL_NEW

    14   (split_conj_tac THEN_ALL_NEW TRY o resolve_tac

    15     @{thms fresh_star_permute_iff[of "- p", THEN iffD1] permute_eq_iff[of "- p", THEN iffD1]})

    16   THEN_ALL_NEW

    17   asm_full_simp_tac (HOL_ss addsimps (@{thms split_conv permute_minus_cancel permute_plus permute_eqvt[symmetric]} @ all_eqvts ctxt @ simps))

    18 *}

    19 

    20 ML {*

    21 fun build_alpha_eqvt alpha names =

    22 let

    23   val pi = Free ("p", @{typ perm});

    24   val (tys, _) = strip_type (fastype_of alpha)

    25   val indnames = Name.variant_list names (Datatype_Prop.make_tnames (map body_type tys));

    26   val args = map Free (indnames ~~ tys);

    27   val perm_args = map (fn x => mk_perm pi x) args

    28 in

    29   (HOLogic.mk_imp (list_comb (alpha, args), list_comb (alpha, perm_args)), indnames @ names)

    30 end

    31 *}

    32 

    33 ML {*

    34 fun build_alpha_eqvts funs tac ctxt =

    35 let

    36   val (gls, names) = fold_map build_alpha_eqvt funs ["p"]

    37   val gl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj gls)

    38   val thm = Goal.prove ctxt names [] gl tac

    39 in

    40   map (fn x => mp OF [x]) (HOLogic.conj_elims thm)

    41 end

    42 *}

    43 

    44 (* Given [fv1, fv2, fv3]

    45    produces %(x, y, z). fv1 x u fv2 y u fv3 z *)

    46 ML {*

    47 fun mk_compound_fv fvs =

    48 let

    49   val nos = (length fvs - 1) downto 0;

    50   val fvs_applied = map (fn (fv, no) => fv $ Bound no) (fvs ~~ nos);

    51   val fvs_union = mk_union fvs_applied;

    52   val (tyh :: tys) = rev (map (domain_type o fastype_of) fvs);

    53   fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))

    54 in

    55   fold fold_fun tys (Abs ("", tyh, fvs_union))

    56 end;

    57 *}

    58 

    59 (* Given [R1, R2, R3]

    60    produces %(x,x'). %(y,y'). %(z,z'). R x x' \<and> R y y' \<and> R z z' *)

    61 ML {*

    62 fun mk_compound_alpha Rs =

    63 let

    64   val nos = (length Rs - 1) downto 0;

    65   val nos2 = (2 * length Rs - 1) downto length Rs;

    66   val Rs_applied = map (fn (R, (no2, no)) => R $ Bound no2 $ Bound no)

    67     (Rs ~~ (nos2 ~~ nos));

    68   val Rs_conj = foldr1 HOLogic.mk_conj Rs_applied;

    69   val (tyh :: tys) = rev (map (domain_type o fastype_of) Rs);

    70   fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))

    71   val abs_rhs = fold fold_fun tys (Abs ("", tyh, Rs_conj))

    72 in

    73   fold fold_fun tys (Abs ("", tyh, abs_rhs))

    74 end;

    75 *}