src/HOL/Hoare/hoare_tac.ML
 author haftmann Wed May 26 16:28:55 2010 +0200 (2010-05-26) changeset 37138 ee23611b6bf2 parent 37135 636e6d8645d6 child 37391 476270a6c2dc permissions -rw-r--r--
dropped legacy theorem bindings
```     1 (*  Title:      HOL/Hoare/hoare_tac.ML
```
```     2     Author:     Leonor Prensa Nieto & Tobias Nipkow
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```     3
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```     4 Derivation of the proof rules and, most importantly, the VCG tactic.
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```     5 *)
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```     6
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```     7 (*** The tactics ***)
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```     8
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```     9 (*****************************************************************************)
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```    10 (** The function Mset makes the theorem                                     **)
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```    11 (** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
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```    12 (** where (x1,...,xn) are the variables of the particular program we are    **)
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```    13 (** working on at the moment of the call                                    **)
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```    14 (*****************************************************************************)
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```    15
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```    16 local open HOLogic in
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```    17
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```    18 (** maps (%x1 ... xn. t) to [x1,...,xn] **)
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```    19 fun abs2list (Const (@{const_name split}, _) \$ Abs (x, T, t)) = Free (x, T) :: abs2list t
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```    20   | abs2list (Abs (x, T, t)) = [Free (x, T)]
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```    21   | abs2list _ = [];
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```    22
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```    23 (** maps {(x1,...,xn). t} to [x1,...,xn] **)
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```    24 fun mk_vars (Const ("Collect",_) \$ T) = abs2list T
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```    25   | mk_vars _ = [];
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```    26
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```    27 (** abstraction of body over a tuple formed from a list of free variables.
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```    28 Types are also built **)
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```    29 fun mk_abstupleC []     body = absfree ("x", unitT, body)
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```    30   | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
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```    31                                in if w=[] then absfree (n, T, body)
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```    32         else let val z  = mk_abstupleC w body;
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```    33                  val T2 = case z of Abs(_,T,_) => T
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```    34                         | Const (_, Type (_,[_, Type (_,[T,_])])) \$ _ => T;
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```    35        in Const (@{const_name split}, (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT)
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```    36           \$ absfree (n, T, z) end end;
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```    37
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```    38 (** maps [x1,...,xn] to (x1,...,xn) and types**)
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```    39 fun mk_bodyC []      = HOLogic.unit
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```    40   | mk_bodyC (x::xs) = if xs=[] then x
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```    41                else let val (n, T) = dest_Free x ;
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```    42                         val z = mk_bodyC xs;
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```    43                         val T2 = case z of Free(_, T) => T
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```    44                                          | Const ("Pair", Type ("fun", [_, Type
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```    45                                             ("fun", [_, T])])) \$ _ \$ _ => T;
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```    46                  in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) \$ x \$ z end;
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```    47
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```    48 (** maps a subgoal of the form:
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```    49         VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**)
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```    50 fun get_vars c =
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```    51   let
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```    52     val d = Logic.strip_assums_concl c;
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```    53     val Const _ \$ pre \$ _ \$ _ = dest_Trueprop d;
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```    54   in mk_vars pre end;
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```    55
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```    56 fun mk_CollectC trm =
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```    57   let val T as Type ("fun",[t,_]) = fastype_of trm
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```    58   in Collect_const t \$ trm end;
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```    59
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```    60 fun inclt ty = Const (@{const_name Orderings.less_eq}, [ty,ty] ---> boolT);
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```    61
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```    62
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```    63 fun Mset ctxt prop =
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```    64   let
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```    65     val [(Mset, _), (P, _)] = Variable.variant_frees ctxt [] [("Mset", ()), ("P", ())];
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```    66
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```    67     val vars = get_vars prop;
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```    68     val varsT = fastype_of (mk_bodyC vars);
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```    69     val big_Collect = mk_CollectC (mk_abstupleC vars (Free (P, varsT --> boolT) \$ mk_bodyC vars));
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```    70     val small_Collect = mk_CollectC (Abs ("x", varsT, Free (P, varsT --> boolT) \$ Bound 0));
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```    71
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```    72     val MsetT = fastype_of big_Collect;
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```    73     fun Mset_incl t = mk_Trueprop (inclt MsetT \$ Free (Mset, MsetT) \$ t);
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```    74     val impl = Logic.mk_implies (Mset_incl big_Collect, Mset_incl small_Collect);
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```    75     val th = Goal.prove ctxt [Mset, P] [] impl (fn _ => blast_tac (claset_of ctxt) 1);
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```    76  in (vars, th) end;
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```    77
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```    78 end;
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```    79
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```    80
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```    81 (*****************************************************************************)
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```    82 (** Simplifying:                                                            **)
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```    83 (** Some useful lemmata, lists and simplification tactics to control which  **)
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```    84 (** theorems are used to simplify at each moment, so that the original      **)
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```    85 (** input does not suffer any unexpected transformation                     **)
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```    86 (*****************************************************************************)
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```    87
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```    88 (**Simp_tacs**)
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```    89
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```    90 val before_set2pred_simp_tac =
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```    91   (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym, @{thm Compl_Collect}]));
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```    92
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```    93 val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [@{thm split_conv}]));
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```    94
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```    95 (*****************************************************************************)
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```    96 (** set2pred_tac transforms sets inclusion into predicates implication,     **)
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```    97 (** maintaining the original variable names.                                **)
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```    98 (** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
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```    99 (** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
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```   100 (** are first simplified by "before_set2pred_simp_tac", that returns only   **)
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```   101 (** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
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```   102 (** transformed.                                                            **)
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```   103 (** This transformation may solve very easy subgoals due to a ligth         **)
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```   104 (** simplification done by (split_all_tac)                                  **)
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```   105 (*****************************************************************************)
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```   106
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```   107 fun set2pred_tac var_names = SUBGOAL (fn (goal, i) =>
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```   108   before_set2pred_simp_tac i THEN_MAYBE
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```   109   EVERY [
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```   110     rtac subsetI i,
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```   111     rtac CollectI i,
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```   112     dtac CollectD i,
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```   113     TRY (split_all_tac i) THEN_MAYBE
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```   114      (rename_tac var_names i THEN full_simp_tac (HOL_basic_ss addsimps [@{thm split_conv}]) i)]);
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```   115
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```   116 (*****************************************************************************)
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```   117 (** BasicSimpTac is called to simplify all verification conditions. It does **)
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```   118 (** a light simplification by applying "mem_Collect_eq", then it calls      **)
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```   119 (** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
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```   120 (** and transforms any other into predicates, applying then                 **)
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```   121 (** the tactic chosen by the user, which may solve the subgoal completely.  **)
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```   122 (*****************************************************************************)
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```   123
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```   124 fun MaxSimpTac var_names tac = FIRST'[rtac subset_refl, set2pred_tac var_names THEN_MAYBE' tac];
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```   125
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```   126 fun BasicSimpTac var_names tac =
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```   127   simp_tac
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```   128     (HOL_basic_ss addsimps [mem_Collect_eq, @{thm split_conv}] addsimprocs [record_simproc])
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```   129   THEN_MAYBE' MaxSimpTac var_names tac;
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```   130
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```   131
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```   132 (** hoare_rule_tac **)
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```   133
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```   134 fun hoare_rule_tac (vars, Mlem) tac =
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```   135   let
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```   136     val var_names = map (fst o dest_Free) vars;
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```   137     fun wlp_tac i =
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```   138       rtac @{thm SeqRule} i THEN rule_tac false (i + 1)
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```   139     and rule_tac pre_cond i st = st |> (*abstraction over st prevents looping*)
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```   140       ((wlp_tac i THEN rule_tac pre_cond i)
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```   141         ORELSE
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```   142         (FIRST [
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```   143           rtac @{thm SkipRule} i,
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```   144           rtac @{thm AbortRule} i,
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```   145           EVERY [
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```   146             rtac @{thm BasicRule} i,
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```   147             rtac Mlem i,
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```   148             split_simp_tac i],
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```   149           EVERY [
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```   150             rtac @{thm CondRule} i,
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```   151             rule_tac false (i + 2),
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```   152             rule_tac false (i + 1)],
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```   153           EVERY [
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```   154             rtac @{thm WhileRule} i,
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```   155             BasicSimpTac var_names tac (i + 2),
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```   156             rule_tac true (i + 1)]]
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```   157          THEN (if pre_cond then BasicSimpTac var_names tac i else rtac subset_refl i)));
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```   158   in rule_tac end;
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```   159
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```   160
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```   161 (** tac is the tactic the user chooses to solve or simplify **)
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```   162 (** the final verification conditions                       **)
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```   163
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```   164 fun hoare_tac ctxt (tac: int -> tactic) = SUBGOAL (fn (goal, i) =>
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```   165   SELECT_GOAL (hoare_rule_tac (Mset ctxt goal) tac true 1) i);
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```   166
```