src/HOL/HOL.thy
author ballarin
Tue, 02 Aug 2005 16:50:55 +0200
changeset 16999 307b2ec590ff
parent 16775 c1b87ef4a1c3
child 17197 917c6e7ca28d
permissions -rw-r--r--
Turned simp_implies into binary operator.
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     1
(*  Title:      HOL/HOL.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     2
    ID:         $Id$
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
     3
    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
     4
*)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     5
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
     6
header {* The basis of Higher-Order Logic *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     7
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15103
diff changeset
     8
theory HOL
15140
322485b816ac import -> imports
nipkow
parents: 15131
diff changeset
     9
imports CPure
16417
9bc16273c2d4 migrated theory headers to new format
haftmann
parents: 16121
diff changeset
    10
uses ("cladata.ML") ("blastdata.ML") ("simpdata.ML") ("eqrule_HOL_data.ML")
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
    11
      ("~~/src/Provers/eqsubst.ML")
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16633
diff changeset
    12
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15103
diff changeset
    13
begin
2260
b59781f2b809 added symbols syntax;
wenzelm
parents: 1674
diff changeset
    14
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
    15
subsection {* Primitive logic *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
    16
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
    17
subsubsection {* Core syntax *}
2260
b59781f2b809 added symbols syntax;
wenzelm
parents: 1674
diff changeset
    18
14854
61bdf2ae4dc5 removed obsolete sort 'logic';
wenzelm
parents: 14749
diff changeset
    19
classes type
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
    20
defaultsort type
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
    21
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
    22
global
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    23
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    24
typedecl bool
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    25
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    26
arities
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
    27
  bool :: type
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
    28
  fun :: (type, type) type
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    29
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
    30
judgment
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
    31
  Trueprop      :: "bool => prop"                   ("(_)" 5)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    32
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
    33
consts
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    34
  Not           :: "bool => bool"                   ("~ _" [40] 40)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    35
  True          :: bool
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    36
  False         :: bool
3947
eb707467f8c5 adapted to qualified names;
wenzelm
parents: 3842
diff changeset
    37
  arbitrary     :: 'a
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    38
11432
8a203ae6efe3 added "The" (definite description operator) (by Larry);
wenzelm
parents: 10489
diff changeset
    39
  The           :: "('a => bool) => 'a"
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    40
  All           :: "('a => bool) => bool"           (binder "ALL " 10)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    41
  Ex            :: "('a => bool) => bool"           (binder "EX " 10)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    42
  Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    43
  Let           :: "['a, 'a => 'b] => 'b"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    44
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    45
  "="           :: "['a, 'a] => bool"               (infixl 50)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    46
  &             :: "[bool, bool] => bool"           (infixr 35)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    47
  "|"           :: "[bool, bool] => bool"           (infixr 30)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    48
  -->           :: "[bool, bool] => bool"           (infixr 25)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    49
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
    50
local
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
    51
16587
b34c8aa657a5 Constant "If" is now local
paulson
parents: 16417
diff changeset
    52
consts
b34c8aa657a5 Constant "If" is now local
paulson
parents: 16417
diff changeset
    53
  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
2260
b59781f2b809 added symbols syntax;
wenzelm
parents: 1674
diff changeset
    54
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
    55
subsubsection {* Additional concrete syntax *}
2260
b59781f2b809 added symbols syntax;
wenzelm
parents: 1674
diff changeset
    56
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
    57
nonterminals
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    58
  letbinds  letbind
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    59
  case_syn  cases_syn
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    60
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    61
syntax
12650
fbc17f1e746b "_not_equal" dummy constant;
wenzelm
parents: 12633
diff changeset
    62
  "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)
11432
8a203ae6efe3 added "The" (definite description operator) (by Larry);
wenzelm
parents: 10489
diff changeset
    63
  "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    64
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    65
  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    66
  ""            :: "letbind => letbinds"                 ("_")
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    67
  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    68
  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    69
9060
b0dd884b1848 rename @case to _case_syntax (improves on low-level errors);
wenzelm
parents: 8959
diff changeset
    70
  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
b0dd884b1848 rename @case to _case_syntax (improves on low-level errors);
wenzelm
parents: 8959
diff changeset
    71
  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
    72
  ""            :: "case_syn => cases_syn"               ("_")
9060
b0dd884b1848 rename @case to _case_syntax (improves on low-level errors);
wenzelm
parents: 8959
diff changeset
    73
  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    74
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    75
translations
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 7220
diff changeset
    76
  "x ~= y"                == "~ (x = y)"
13764
3e180bf68496 removed some problems with print translations
nipkow
parents: 13763
diff changeset
    77
  "THE x. P"              == "The (%x. P)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    78
  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
1114
c8dfb56a7e95 Prod is now a parent of Lfp.
nipkow
parents: 1068
diff changeset
    79
  "let x = a in e"        == "Let a (%x. e)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
    80
13763
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13723
diff changeset
    81
print_translation {*
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13723
diff changeset
    82
(* To avoid eta-contraction of body: *)
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13723
diff changeset
    83
[("The", fn [Abs abs] =>
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13723
diff changeset
    84
     let val (x,t) = atomic_abs_tr' abs
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13723
diff changeset
    85
     in Syntax.const "_The" $ x $ t end)]
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13723
diff changeset
    86
*}
f94b569cd610 added print translations tha avoid eta contraction for important binders.
nipkow
parents: 13723
diff changeset
    87
12633
ad9277743664 tuned ``syntax (output)'';
wenzelm
parents: 12436
diff changeset
    88
syntax (output)
11687
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
    89
  "="           :: "['a, 'a] => bool"                    (infix 50)
12650
fbc17f1e746b "_not_equal" dummy constant;
wenzelm
parents: 12633
diff changeset
    90
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
2260
b59781f2b809 added symbols syntax;
wenzelm
parents: 1674
diff changeset
    91
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
    92
syntax (xsymbols)
11687
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
    93
  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
    94
  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
    95
  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
    96
  "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
12650
fbc17f1e746b "_not_equal" dummy constant;
wenzelm
parents: 12633
diff changeset
    97
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
11687
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
    98
  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
    99
  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
   100
  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
   101
  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
14361
ad2f5da643b4 * Support for raw latex output in control symbols: \<^raw...>
schirmer
parents: 14357
diff changeset
   102
(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
2372
a2999e19703b fixed alternative quantifier symbol syntax;
wenzelm
parents: 2368
diff changeset
   103
12114
a8e860c86252 eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents: 12023
diff changeset
   104
syntax (xsymbols output)
12650
fbc17f1e746b "_not_equal" dummy constant;
wenzelm
parents: 12633
diff changeset
   105
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
3820
46b255e140dc fixed infix syntax;
wenzelm
parents: 3370
diff changeset
   106
6340
7d5cbd5819a0 HTML output;
wenzelm
parents: 6289
diff changeset
   107
syntax (HTML output)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
   108
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
11687
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
   109
  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
14565
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
   110
  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
   111
  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
   112
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
   113
  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
   114
  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
c6dc17aab88a use more symbols in HTML output
kleing
parents: 14444
diff changeset
   115
  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
6340
7d5cbd5819a0 HTML output;
wenzelm
parents: 6289
diff changeset
   116
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 7220
diff changeset
   117
syntax (HOL)
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   118
  "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   119
  "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   120
  "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
7238
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 7220
diff changeset
   121
36e58620ffc8 replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents: 7220
diff changeset
   122
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   123
subsubsection {* Axioms and basic definitions *}
2260
b59781f2b809 added symbols syntax;
wenzelm
parents: 1674
diff changeset
   124
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   125
axioms
15380
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   126
  eq_reflection:  "(x=y) ==> (x==y)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   127
15380
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   128
  refl:           "t = (t::'a)"
6289
062aa156a300 added a commment on the "ext" rule
paulson
parents: 6027
diff changeset
   129
15380
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   130
  ext:            "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   131
    -- {*Extensionality is built into the meta-logic, and this rule expresses
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   132
         a related property.  It is an eta-expanded version of the traditional
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   133
         rule, and similar to the ABS rule of HOL*}
6289
062aa156a300 added a commment on the "ext" rule
paulson
parents: 6027
diff changeset
   134
11432
8a203ae6efe3 added "The" (definite description operator) (by Larry);
wenzelm
parents: 10489
diff changeset
   135
  the_eq_trivial: "(THE x. x = a) = (a::'a)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   136
15380
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   137
  impI:           "(P ==> Q) ==> P-->Q"
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   138
  mp:             "[| P-->Q;  P |] ==> Q"
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   139
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   140
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   141
text{*Thanks to Stephan Merz*}
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   142
theorem subst:
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   143
  assumes eq: "s = t" and p: "P(s)"
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   144
  shows "P(t::'a)"
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   145
proof -
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   146
  from eq have meta: "s \<equiv> t"
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   147
    by (rule eq_reflection)
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   148
  from p show ?thesis
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   149
    by (unfold meta)
455cfa766dad proof of subst by S Merz
paulson
parents: 15363
diff changeset
   150
qed
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   151
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   152
defs
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   153
  True_def:     "True      == ((%x::bool. x) = (%x. x))"
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   154
  All_def:      "All(P)    == (P = (%x. True))"
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents: 11438
diff changeset
   155
  Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   156
  False_def:    "False     == (!P. P)"
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   157
  not_def:      "~ P       == P-->False"
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   158
  and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   159
  or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   160
  Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   161
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   162
axioms
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   163
  iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   164
  True_or_False:  "(P=True) | (P=False)"
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   165
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   166
defs
7357
d0e16da40ea2 proper bootstrap of HOL theory and packages;
wenzelm
parents: 7238
diff changeset
   167
  Let_def:      "Let s f == f(s)"
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents: 11438
diff changeset
   168
  if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4868
diff changeset
   169
14223
0ee05eef881b Added support for making constants final, that is, ensuring that no
skalberg
parents: 14208
diff changeset
   170
finalconsts
0ee05eef881b Added support for making constants final, that is, ensuring that no
skalberg
parents: 14208
diff changeset
   171
  "op ="
0ee05eef881b Added support for making constants final, that is, ensuring that no
skalberg
parents: 14208
diff changeset
   172
  "op -->"
0ee05eef881b Added support for making constants final, that is, ensuring that no
skalberg
parents: 14208
diff changeset
   173
  The
0ee05eef881b Added support for making constants final, that is, ensuring that no
skalberg
parents: 14208
diff changeset
   174
  arbitrary
3320
3a5e4930fb77 Added `arbitrary'
nipkow
parents: 3248
diff changeset
   175
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   176
subsubsection {* Generic algebraic operations *}
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
   177
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
   178
axclass zero < type
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
   179
axclass one < type
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
   180
axclass plus < type
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
   181
axclass minus < type
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
   182
axclass times < type
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 12281
diff changeset
   183
axclass inverse < type
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   184
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   185
global
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   186
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   187
consts
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   188
  "0"           :: "'a::zero"                       ("0")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   189
  "1"           :: "'a::one"                        ("1")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   190
  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   191
  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   192
  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   193
  *             :: "['a::times, 'a] => 'a"          (infixl 70)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   194
13456
42601eb7553f special syntax for index "1" (plain numeral hidden by "1" symbol in HOL);
wenzelm
parents: 13438
diff changeset
   195
syntax
42601eb7553f special syntax for index "1" (plain numeral hidden by "1" symbol in HOL);
wenzelm
parents: 13438
diff changeset
   196
  "_index1"  :: index    ("\<^sub>1")
42601eb7553f special syntax for index "1" (plain numeral hidden by "1" symbol in HOL);
wenzelm
parents: 13438
diff changeset
   197
translations
14690
f61ea8bfa81e _index1: accomodate improved indexed syntax;
wenzelm
parents: 14590
diff changeset
   198
  (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
13456
42601eb7553f special syntax for index "1" (plain numeral hidden by "1" symbol in HOL);
wenzelm
parents: 13438
diff changeset
   199
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   200
local
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   201
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   202
typed_print_translation {*
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   203
  let
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   204
    fun tr' c = (c, fn show_sorts => fn T => fn ts =>
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   205
      if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   206
      else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   207
  in [tr' "0", tr' "1"] end;
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   208
*} -- {* show types that are presumably too general *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   209
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   210
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   211
consts
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   212
  abs           :: "'a::minus => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   213
  inverse       :: "'a::inverse => 'a"
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   214
  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   215
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   216
syntax (xsymbols)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   217
  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   218
syntax (HTML output)
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   219
  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   220
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   221
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   222
subsection {*Equality*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   223
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   224
lemma sym: "s=t ==> t=s"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   225
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   226
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   227
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   228
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   229
(*calling "standard" reduces maxidx to 0*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   230
lemmas ssubst = sym [THEN subst, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   231
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   232
lemma trans: "[| r=s; s=t |] ==> r=t"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   233
apply (erule subst , assumption)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   234
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   235
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   236
lemma def_imp_eq:  assumes meq: "A == B" shows "A = B"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   237
apply (unfold meq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   238
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   239
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   240
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   241
(*Useful with eresolve_tac for proving equalties from known equalities.
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   242
        a = b
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   243
        |   |
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   244
        c = d   *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   245
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   246
apply (rule trans)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   247
apply (rule trans)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   248
apply (rule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   249
apply assumption+
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   250
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   251
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   252
text {* For calculational reasoning: *}
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   253
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   254
lemma forw_subst: "a = b ==> P b ==> P a"
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   255
  by (rule ssubst)
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   256
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   257
lemma back_subst: "P a ==> a = b ==> P b"
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   258
  by (rule subst)
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents: 15481
diff changeset
   259
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   260
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   261
subsection {*Congruence rules for application*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   262
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   263
(*similar to AP_THM in Gordon's HOL*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   264
lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   265
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   266
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   267
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   268
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   269
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   270
lemma arg_cong: "x=y ==> f(x)=f(y)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   271
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   272
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   273
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   274
15655
157f3988f775 arg_cong2 by Norbert Voelker
paulson
parents: 15570
diff changeset
   275
lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
157f3988f775 arg_cong2 by Norbert Voelker
paulson
parents: 15570
diff changeset
   276
apply (erule ssubst)+
157f3988f775 arg_cong2 by Norbert Voelker
paulson
parents: 15570
diff changeset
   277
apply (rule refl)
157f3988f775 arg_cong2 by Norbert Voelker
paulson
parents: 15570
diff changeset
   278
done
157f3988f775 arg_cong2 by Norbert Voelker
paulson
parents: 15570
diff changeset
   279
157f3988f775 arg_cong2 by Norbert Voelker
paulson
parents: 15570
diff changeset
   280
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   281
lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   282
apply (erule subst)+
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   283
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   284
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   285
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   286
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   287
subsection {*Equality of booleans -- iff*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   288
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   289
lemma iffI: assumes prems: "P ==> Q" "Q ==> P" shows "P=Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   290
apply (rules intro: iff [THEN mp, THEN mp] impI prems)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   291
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   292
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   293
lemma iffD2: "[| P=Q; Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   294
apply (erule ssubst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   295
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   296
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   297
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   298
lemma rev_iffD2: "[| Q; P=Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   299
apply (erule iffD2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   300
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   301
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   302
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   303
lemmas iffD1 = sym [THEN iffD2, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   304
lemmas rev_iffD1 = sym [THEN [2] rev_iffD2, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   305
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   306
lemma iffE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   307
  assumes major: "P=Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   308
      and minor: "[| P --> Q; Q --> P |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   309
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   310
by (rules intro: minor impI major [THEN iffD2] major [THEN iffD1])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   311
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   312
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   313
subsection {*True*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   314
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   315
lemma TrueI: "True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   316
apply (unfold True_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   317
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   318
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   319
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   320
lemma eqTrueI: "P ==> P=True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   321
by (rules intro: iffI TrueI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   322
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   323
lemma eqTrueE: "P=True ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   324
apply (erule iffD2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   325
apply (rule TrueI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   326
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   327
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   328
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   329
subsection {*Universal quantifier*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   330
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   331
lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   332
apply (unfold All_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   333
apply (rules intro: ext eqTrueI p)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   334
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   335
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   336
lemma spec: "ALL x::'a. P(x) ==> P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   337
apply (unfold All_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   338
apply (rule eqTrueE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   339
apply (erule fun_cong)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   340
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   341
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   342
lemma allE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   343
  assumes major: "ALL x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   344
      and minor: "P(x) ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   345
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   346
by (rules intro: minor major [THEN spec])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   347
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   348
lemma all_dupE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   349
  assumes major: "ALL x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   350
      and minor: "[| P(x); ALL x. P(x) |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   351
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   352
by (rules intro: minor major major [THEN spec])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   353
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   354
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   355
subsection {*False*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   356
(*Depends upon spec; it is impossible to do propositional logic before quantifiers!*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   357
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   358
lemma FalseE: "False ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   359
apply (unfold False_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   360
apply (erule spec)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   361
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   362
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   363
lemma False_neq_True: "False=True ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   364
by (erule eqTrueE [THEN FalseE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   365
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   366
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   367
subsection {*Negation*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   368
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   369
lemma notI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   370
  assumes p: "P ==> False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   371
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   372
apply (unfold not_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   373
apply (rules intro: impI p)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   374
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   375
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   376
lemma False_not_True: "False ~= True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   377
apply (rule notI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   378
apply (erule False_neq_True)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   379
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   380
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   381
lemma True_not_False: "True ~= False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   382
apply (rule notI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   383
apply (drule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   384
apply (erule False_neq_True)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   385
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   386
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   387
lemma notE: "[| ~P;  P |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   388
apply (unfold not_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   389
apply (erule mp [THEN FalseE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   390
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   391
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   392
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   393
(* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   394
lemmas notI2 = notE [THEN notI, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   395
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   396
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   397
subsection {*Implication*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   398
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   399
lemma impE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   400
  assumes "P-->Q" "P" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   401
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   402
by (rules intro: prems mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   403
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   404
(* Reduces Q to P-->Q, allowing substitution in P. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   405
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   406
by (rules intro: mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   407
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   408
lemma contrapos_nn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   409
  assumes major: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   410
      and minor: "P==>Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   411
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   412
by (rules intro: notI minor major [THEN notE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   413
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   414
(*not used at all, but we already have the other 3 combinations *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   415
lemma contrapos_pn:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   416
  assumes major: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   417
      and minor: "P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   418
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   419
by (rules intro: notI minor major notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   420
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   421
lemma not_sym: "t ~= s ==> s ~= t"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   422
apply (erule contrapos_nn)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   423
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   424
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   425
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   426
(*still used in HOLCF*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   427
lemma rev_contrapos:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   428
  assumes pq: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   429
      and nq: "~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   430
  shows "~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   431
apply (rule nq [THEN contrapos_nn])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   432
apply (erule pq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   433
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   434
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   435
subsection {*Existential quantifier*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   436
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   437
lemma exI: "P x ==> EX x::'a. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   438
apply (unfold Ex_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   439
apply (rules intro: allI allE impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   440
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   441
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   442
lemma exE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   443
  assumes major: "EX x::'a. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   444
      and minor: "!!x. P(x) ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   445
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   446
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   447
apply (rules intro: impI [THEN allI] minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   448
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   449
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   450
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   451
subsection {*Conjunction*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   452
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   453
lemma conjI: "[| P; Q |] ==> P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   454
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   455
apply (rules intro: impI [THEN allI] mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   456
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   457
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   458
lemma conjunct1: "[| P & Q |] ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   459
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   460
apply (rules intro: impI dest: spec mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   461
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   462
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   463
lemma conjunct2: "[| P & Q |] ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   464
apply (unfold and_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   465
apply (rules intro: impI dest: spec mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   466
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   467
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   468
lemma conjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   469
  assumes major: "P&Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   470
      and minor: "[| P; Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   471
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   472
apply (rule minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   473
apply (rule major [THEN conjunct1])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   474
apply (rule major [THEN conjunct2])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   475
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   476
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   477
lemma context_conjI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   478
  assumes prems: "P" "P ==> Q" shows "P & Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   479
by (rules intro: conjI prems)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   480
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   481
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   482
subsection {*Disjunction*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   483
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   484
lemma disjI1: "P ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   485
apply (unfold or_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   486
apply (rules intro: allI impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   487
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   488
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   489
lemma disjI2: "Q ==> P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   490
apply (unfold or_def)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   491
apply (rules intro: allI impI mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   492
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   493
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   494
lemma disjE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   495
  assumes major: "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   496
      and minorP: "P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   497
      and minorQ: "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   498
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   499
by (rules intro: minorP minorQ impI
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   500
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   501
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   502
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   503
subsection {*Classical logic*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   504
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   505
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   506
lemma classical:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   507
  assumes prem: "~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   508
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   509
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   510
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   511
apply (rule notI [THEN prem, THEN eqTrueI])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   512
apply (erule subst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   513
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   514
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   515
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   516
lemmas ccontr = FalseE [THEN classical, standard]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   517
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   518
(*notE with premises exchanged; it discharges ~R so that it can be used to
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   519
  make elimination rules*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   520
lemma rev_notE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   521
  assumes premp: "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   522
      and premnot: "~R ==> ~P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   523
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   524
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   525
apply (erule notE [OF premnot premp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   526
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   527
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   528
(*Double negation law*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   529
lemma notnotD: "~~P ==> P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   530
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   531
apply (erule notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   532
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   533
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   534
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   535
lemma contrapos_pp:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   536
  assumes p1: "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   537
      and p2: "~P ==> ~Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   538
  shows "P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   539
by (rules intro: classical p1 p2 notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   540
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   541
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   542
subsection {*Unique existence*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   543
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   544
lemma ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   545
  assumes prems: "P a" "!!x. P(x) ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   546
  shows "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   547
by (unfold Ex1_def, rules intro: prems exI conjI allI impI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   548
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   549
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   550
lemma ex_ex1I:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   551
  assumes ex_prem: "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   552
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   553
  shows "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   554
by (rules intro: ex_prem [THEN exE] ex1I eq)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   555
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   556
lemma ex1E:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   557
  assumes major: "EX! x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   558
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   559
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   560
apply (rule major [unfolded Ex1_def, THEN exE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   561
apply (erule conjE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   562
apply (rules intro: minor)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   563
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   564
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   565
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   566
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   567
apply (rule exI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   568
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   569
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   570
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   571
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   572
subsection {*THE: definite description operator*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   573
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   574
lemma the_equality:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   575
  assumes prema: "P a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   576
      and premx: "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   577
  shows "(THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   578
apply (rule trans [OF _ the_eq_trivial])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   579
apply (rule_tac f = "The" in arg_cong)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   580
apply (rule ext)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   581
apply (rule iffI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   582
 apply (erule premx)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   583
apply (erule ssubst, rule prema)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   584
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   585
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   586
lemma theI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   587
  assumes "P a" and "!!x. P x ==> x=a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   588
  shows "P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   589
by (rules intro: prems the_equality [THEN ssubst])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   590
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   591
lemma theI': "EX! x. P x ==> P (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   592
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   593
apply (erule theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   594
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   595
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   596
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   597
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   598
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   599
(*Easier to apply than theI: only one occurrence of P*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   600
lemma theI2:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   601
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   602
  shows "Q (THE x. P x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   603
by (rules intro: prems theI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   604
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   605
lemma the1_equality: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   606
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   607
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   608
apply (erule ex1E)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   609
apply (erule all_dupE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   610
apply (drule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   611
apply  assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   612
apply (erule ssubst)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   613
apply (erule allE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   614
apply (erule mp)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   615
apply assumption
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   616
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   617
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   618
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   619
apply (rule the_equality)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   620
apply (rule refl)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   621
apply (erule sym)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   622
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   623
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   624
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   625
subsection {*Classical intro rules for disjunction and existential quantifiers*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   626
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   627
lemma disjCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   628
  assumes "~Q ==> P" shows "P|Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   629
apply (rule classical)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   630
apply (rules intro: prems disjI1 disjI2 notI elim: notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   631
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   632
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   633
lemma excluded_middle: "~P | P"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   634
by (rules intro: disjCI)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   635
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   636
text{*case distinction as a natural deduction rule. Note that @{term "~P"}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   637
   is the second case, not the first.*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   638
lemma case_split_thm:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   639
  assumes prem1: "P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   640
      and prem2: "~P ==> Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   641
  shows "Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   642
apply (rule excluded_middle [THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   643
apply (erule prem2)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   644
apply (erule prem1)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   645
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   646
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   647
(*Classical implies (-->) elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   648
lemma impCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   649
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   650
      and minor: "~P ==> R" "Q ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   651
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   652
apply (rule excluded_middle [of P, THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   653
apply (rules intro: minor major [THEN mp])+
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   654
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   655
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   656
(*This version of --> elimination works on Q before P.  It works best for
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   657
  those cases in which P holds "almost everywhere".  Can't install as
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   658
  default: would break old proofs.*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   659
lemma impCE':
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   660
  assumes major: "P-->Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   661
      and minor: "Q ==> R" "~P ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   662
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   663
apply (rule excluded_middle [of P, THEN disjE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   664
apply (rules intro: minor major [THEN mp])+
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   665
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   666
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   667
(*Classical <-> elimination. *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   668
lemma iffCE:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   669
  assumes major: "P=Q"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   670
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   671
  shows "R"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   672
apply (rule major [THEN iffE])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   673
apply (rules intro: minor elim: impCE notE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   674
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   675
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   676
lemma exCI:
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   677
  assumes "ALL x. ~P(x) ==> P(a)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   678
  shows "EX x. P(x)"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   679
apply (rule ccontr)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   680
apply (rules intro: prems exI allI notI notE [of "\<exists>x. P x"])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   681
done
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   682
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   683
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   684
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   685
subsection {* Theory and package setup *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   686
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   687
ML
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   688
{*
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   689
val plusI = thm "plusI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   690
val minusI = thm "minusI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   691
val timesI = thm "timesI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   692
val eq_reflection = thm "eq_reflection"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   693
val refl = thm "refl"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   694
val subst = thm "subst"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   695
val ext = thm "ext"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   696
val impI = thm "impI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   697
val mp = thm "mp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   698
val True_def = thm "True_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   699
val All_def = thm "All_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   700
val Ex_def = thm "Ex_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   701
val False_def = thm "False_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   702
val not_def = thm "not_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   703
val and_def = thm "and_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   704
val or_def = thm "or_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   705
val Ex1_def = thm "Ex1_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   706
val iff = thm "iff"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   707
val True_or_False = thm "True_or_False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   708
val Let_def = thm "Let_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   709
val if_def = thm "if_def"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   710
val sym = thm "sym"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   711
val ssubst = thm "ssubst"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   712
val trans = thm "trans"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   713
val def_imp_eq = thm "def_imp_eq"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   714
val box_equals = thm "box_equals"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   715
val fun_cong = thm "fun_cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   716
val arg_cong = thm "arg_cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   717
val cong = thm "cong"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   718
val iffI = thm "iffI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   719
val iffD2 = thm "iffD2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   720
val rev_iffD2 = thm "rev_iffD2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   721
val iffD1 = thm "iffD1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   722
val rev_iffD1 = thm "rev_iffD1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   723
val iffE = thm "iffE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   724
val TrueI = thm "TrueI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   725
val eqTrueI = thm "eqTrueI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   726
val eqTrueE = thm "eqTrueE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   727
val allI = thm "allI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   728
val spec = thm "spec"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   729
val allE = thm "allE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   730
val all_dupE = thm "all_dupE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   731
val FalseE = thm "FalseE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   732
val False_neq_True = thm "False_neq_True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   733
val notI = thm "notI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   734
val False_not_True = thm "False_not_True"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   735
val True_not_False = thm "True_not_False"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   736
val notE = thm "notE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   737
val notI2 = thm "notI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   738
val impE = thm "impE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   739
val rev_mp = thm "rev_mp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   740
val contrapos_nn = thm "contrapos_nn"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   741
val contrapos_pn = thm "contrapos_pn"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   742
val not_sym = thm "not_sym"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   743
val rev_contrapos = thm "rev_contrapos"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   744
val exI = thm "exI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   745
val exE = thm "exE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   746
val conjI = thm "conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   747
val conjunct1 = thm "conjunct1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   748
val conjunct2 = thm "conjunct2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   749
val conjE = thm "conjE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   750
val context_conjI = thm "context_conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   751
val disjI1 = thm "disjI1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   752
val disjI2 = thm "disjI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   753
val disjE = thm "disjE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   754
val classical = thm "classical"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   755
val ccontr = thm "ccontr"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   756
val rev_notE = thm "rev_notE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   757
val notnotD = thm "notnotD"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   758
val contrapos_pp = thm "contrapos_pp"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   759
val ex1I = thm "ex1I"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   760
val ex_ex1I = thm "ex_ex1I"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   761
val ex1E = thm "ex1E"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   762
val ex1_implies_ex = thm "ex1_implies_ex"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   763
val the_equality = thm "the_equality"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   764
val theI = thm "theI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   765
val theI' = thm "theI'"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   766
val theI2 = thm "theI2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   767
val the1_equality = thm "the1_equality"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   768
val the_sym_eq_trivial = thm "the_sym_eq_trivial"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   769
val disjCI = thm "disjCI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   770
val excluded_middle = thm "excluded_middle"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   771
val case_split_thm = thm "case_split_thm"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   772
val impCE = thm "impCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   773
val impCE = thm "impCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   774
val iffCE = thm "iffCE"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   775
val exCI = thm "exCI"
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
   776
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   777
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   778
local
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   779
  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   780
  |   wrong_prem (Bound _) = true
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   781
  |   wrong_prem _ = false
15570
8d8c70b41bab Move towards standard functions.
skalberg
parents: 15524
diff changeset
   782
  val filter_right = List.filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t)))))
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   783
in
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   784
  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp])
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   785
  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac]
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   786
end
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   787
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   788
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   789
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   790
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   791
(*Obsolete form of disjunctive case analysis*)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   792
fun excluded_middle_tac sP =
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   793
    res_inst_tac [("Q",sP)] (excluded_middle RS disjE)
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   794
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   795
fun case_tac a = res_inst_tac [("P",a)] case_split_thm
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   796
*}
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
   797
11687
b0fe6e522559 non-oriented infix = and ~= (output only);
wenzelm
parents: 11451
diff changeset
   798
theorems case_split = case_split_thm [case_names True False]
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   799
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   800
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   801
subsubsection {* Intuitionistic Reasoning *}
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   802
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   803
lemma impE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   804
  assumes 1: "P --> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   805
    and 2: "Q ==> R"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   806
    and 3: "P --> Q ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   807
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   808
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   809
  from 3 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   810
  with 1 have Q by (rule impE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   811
  with 2 show R .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   812
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   813
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   814
lemma allE':
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   815
  assumes 1: "ALL x. P x"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   816
    and 2: "P x ==> ALL x. P x ==> Q"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   817
  shows Q
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   818
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   819
  from 1 have "P x" by (rule spec)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   820
  from this and 1 show Q by (rule 2)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   821
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   822
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   823
lemma notE':
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   824
  assumes 1: "~ P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   825
    and 2: "~ P ==> P"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   826
  shows R
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   827
proof -
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   828
  from 2 and 1 have P .
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   829
  with 1 show R by (rule notE)
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   830
qed
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   831
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   832
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   833
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   834
  and [Pure.elim 2] = allE notE' impE'
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   835
  and [Pure.intro] = exI disjI2 disjI1
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   836
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   837
lemmas [trans] = trans
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   838
  and [sym] = sym not_sym
15801
d2f5ca3c048d superceded by Pure.thy and CPure.thy;
wenzelm
parents: 15676
diff changeset
   839
  and [Pure.elim?] = iffD1 iffD2 impE
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   840
11438
3d9222b80989 declare trans [trans] (*overridden in theory Calculation*);
wenzelm
parents: 11432
diff changeset
   841
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   842
subsubsection {* Atomizing meta-level connectives *}
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   843
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   844
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   845
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   846
  assume "!!x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   847
  show "ALL x. P x" by (rule allI)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   848
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   849
  assume "ALL x. P x"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   850
  thus "!!x. P x" by (rule allE)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   851
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   852
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   853
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   854
proof
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   855
  assume r: "A ==> B"
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   856
  show "A --> B" by (rule impI) (rule r)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   857
next
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   858
  assume "A --> B" and A
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   859
  thus B by (rule mp)
9488
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   860
qed
f11bece4e2db added all_eq, imp_eq (for blast);
wenzelm
parents: 9352
diff changeset
   861
14749
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   862
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   863
proof
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   864
  assume r: "A ==> False"
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   865
  show "~A" by (rule notI) (rule r)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   866
next
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   867
  assume "~A" and A
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   868
  thus False by (rule notE)
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   869
qed
9ccfd0f59e11 new atomize theorem
paulson
parents: 14690
diff changeset
   870
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   871
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   872
proof
10432
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   873
  assume "x == y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   874
  show "x = y" by (unfold prems) (rule refl)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   875
next
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   876
  assume "x = y"
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   877
  thus "x == y" by (rule eq_reflection)
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   878
qed
3dfbc913d184 added axclass inverse and consts inverse, divide (infix "/");
wenzelm
parents: 10383
diff changeset
   879
12023
wenzelm
parents: 12003
diff changeset
   880
lemma atomize_conj [atomize]:
wenzelm
parents: 12003
diff changeset
   881
  "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
12003
c09427e5f554 removed obsolete (rule equal_intr_rule);
wenzelm
parents: 11989
diff changeset
   882
proof
11953
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   883
  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   884
  show "A & B" by (rule conjI)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   885
next
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   886
  fix C
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   887
  assume "A & B"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   888
  assume "A ==> B ==> PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   889
  thus "PROP C"
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   890
  proof this
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   891
    show A by (rule conjunct1)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   892
    show B by (rule conjunct2)
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   893
  qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   894
qed
f98623fdf6ef atomize_conj;
wenzelm
parents: 11824
diff changeset
   895
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   896
lemmas [symmetric, rulify] = atomize_all atomize_imp
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   897
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   898
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   899
subsubsection {* Classical Reasoner setup *}
9529
d9434a9277a4 lemmas atomize = all_eq imp_eq;
wenzelm
parents: 9488
diff changeset
   900
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   901
use "cladata.ML"
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   902
setup hypsubst_setup
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   903
16121
wenzelm
parents: 16019
diff changeset
   904
setup {*
wenzelm
parents: 16019
diff changeset
   905
  [ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)]
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   906
*}
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   907
10383
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   908
setup Classical.setup
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   909
setup clasetup
a092ae7bb2a6 "atomize" for classical tactics;
wenzelm
parents: 9970
diff changeset
   910
12386
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   911
lemmas [intro?] = ext
9c38ec9eca1c tuned declarations (rules, sym, etc.);
wenzelm
parents: 12354
diff changeset
   912
  and [elim?] = ex1_implies_ex
11977
2e7c54b86763 tuned declaration of rules;
wenzelm
parents: 11953
diff changeset
   913
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   914
use "blastdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
   915
setup Blast.setup
4868
843a9f5b3c3d nonterminals;
wenzelm
parents: 4793
diff changeset
   916
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   917
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
   918
subsection {* Simplifier setup *}
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
   919
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   920
lemma meta_eq_to_obj_eq: "x == y ==> x = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   921
proof -
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   922
  assume r: "x == y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   923
  show "x = y" by (unfold r) (rule refl)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   924
qed
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   925
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   926
lemma eta_contract_eq: "(%s. f s) = f" ..
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   927
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   928
lemma simp_thms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   929
  shows not_not: "(~ ~ P) = P"
15354
9234f5765d9c Added > and >= sugar
nipkow
parents: 15288
diff changeset
   930
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   931
  and
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   932
    "(P ~= Q) = (P = (~Q))"
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   933
    "(P | ~P) = True"    "(~P | P) = True"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   934
    "(x = x) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   935
    "(~True) = False"  "(~False) = True"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   936
    "(~P) ~= P"  "P ~= (~P)"
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   937
    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   938
    "(True --> P) = P"  "(False --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   939
    "(P --> True) = True"  "(P --> P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   940
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   941
    "(P & True) = P"  "(True & P) = P"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   942
    "(P & False) = False"  "(False & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   943
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   944
    "(P & ~P) = False"    "(~P & P) = False"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   945
    "(P | True) = True"  "(True | P) = True"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   946
    "(P | False) = P"  "(False | P) = P"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   947
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   948
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   949
    -- {* needed for the one-point-rule quantifier simplification procs *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   950
    -- {* essential for termination!! *} and
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   951
    "!!P. (EX x. x=t & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   952
    "!!P. (EX x. t=x & P(x)) = P(t)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   953
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   954
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   955
  by (blast, blast, blast, blast, blast, rules+)
13421
8fcdf4a26468 simplified locale predicates;
wenzelm
parents: 13412
diff changeset
   956
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   957
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
   958
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   959
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   960
lemma ex_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   961
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   962
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   963
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   964
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   965
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   966
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   967
  -- {* Miniscoping: pushing in existential quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   968
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   969
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   970
lemma all_simps:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   971
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   972
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   973
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   974
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   975
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   976
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   977
  -- {* Miniscoping: pushing in universal quantifiers. *}
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   978
  by (rules | blast)+
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   979
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   980
lemma disj_absorb: "(A | A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   981
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   982
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   983
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   984
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   985
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   986
lemma conj_absorb: "(A & A) = A"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   987
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   988
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   989
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   990
  by blast
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
   991
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   992
lemma eq_ac:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   993
  shows eq_commute: "(a=b) = (b=a)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   994
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   995
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
   996
lemma neq_commute: "(a~=b) = (b~=a)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   997
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
   998
lemma conj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
   999
  shows conj_commute: "(P&Q) = (Q&P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1000
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1001
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1002
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1003
lemma disj_comms:
12937
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1004
  shows disj_commute: "(P|Q) = (Q|P)"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
wenzelm
parents: 12892
diff changeset
  1005
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1006
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1007
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1008
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1009
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1010
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1011
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1012
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1013
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1014
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1015
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1016
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1017
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1018
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1019
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1020
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1021
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1022
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1023
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1024
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1025
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1026
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1027
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1028
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1029
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1030
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1031
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1032
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1033
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1034
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1035
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1036
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1037
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1038
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1039
  -- {* cases boil down to the same thing. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1040
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1041
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1042
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1043
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1044
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1045
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1046
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1047
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1048
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1049
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1050
text {*
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1051
  \medskip The @{text "&"} congruence rule: not included by default!
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1052
  May slow rewrite proofs down by as much as 50\% *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1053
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1054
lemma conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1055
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1056
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1057
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1058
lemma rev_conj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1059
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1060
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1061
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1062
text {* The @{text "|"} congruence rule: not included by default! *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1063
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1064
lemma disj_cong:
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1065
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1066
  by blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1067
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1068
lemma eq_sym_conv: "(x = y) = (y = x)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1069
  by rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1070
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1071
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1072
text {* \medskip if-then-else rules *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1073
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1074
lemma if_True: "(if True then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1075
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1076
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1077
lemma if_False: "(if False then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1078
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1079
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1080
lemma if_P: "P ==> (if P then x else y) = x"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1081
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1082
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1083
lemma if_not_P: "~P ==> (if P then x else y) = y"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1084
  by (unfold if_def) blast
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1085
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1086
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1087
  apply (rule case_split [of Q])
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1088
   apply (simplesubst if_P)
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1089
    prefer 3 apply (simplesubst if_not_P, blast+)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1090
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1091
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1092
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1093
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1094
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1095
lemmas if_splits = split_if split_if_asm
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1096
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1097
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1098
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1099
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1100
lemma if_cancel: "(if c then x else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1101
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1102
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1103
lemma if_eq_cancel: "(if x = y then y else x) = x"
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1104
by (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1105
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1106
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1107
  -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1108
  by (rule split_if)
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1109
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1110
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1111
  -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1112
  apply (simplesubst split_if, blast)
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1113
  done
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1114
12436
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1115
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
a2df07fefed7 Replaced several occurrences of "blast" by "rules".
berghofe
parents: 12386
diff changeset
  1116
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
12281
3bd113b8f7a6 converted simp lemmas;
wenzelm
parents: 12256
diff changeset
  1117
15423
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1118
text {* \medskip let rules for simproc *}
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1119
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1120
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1121
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1122
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1123
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1124
  by (unfold Let_def)
761a4f8e6ad6 added simproc for Let
schirmer
parents: 15411
diff changeset
  1125
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1126
text {*
16999
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1127
  The following copy of the implication operator is useful for
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1128
  fine-tuning congruence rules.  It instructs the simplifier to simplify
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1129
  its premise.
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1130
*}
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1131
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1132
consts
16999
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1133
  "=simp=>" :: "[prop, prop] => prop"  (infixr 1)
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1134
(*
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1135
  "op =simp=>" :: "[prop, prop] => prop"  ("(_/ =simp=> _)" [2, 1] 1)
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1136
syntax
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1137
  "op =simp=>" :: "[prop, prop] => prop"  ("op =simp=>")
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1138
*)
307b2ec590ff Turned simp_implies into binary operator.
ballarin
parents: 16775
diff changeset
  1139
defs simp_implies_def: "op =simp=> \<equiv> op ==>"
16633
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1140
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1141
lemma simp_impliesI: 
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1142
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1143
  shows "PROP P =simp=> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1144
  apply (unfold simp_implies_def)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1145
  apply (rule PQ)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1146
  apply assumption
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1147
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1148
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1149
lemma simp_impliesE:
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1150
  assumes PQ:"PROP P =simp=> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1151
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1152
  and QR: "PROP Q \<Longrightarrow> PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1153
  shows "PROP R"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1154
  apply (rule QR)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1155
  apply (rule PQ [unfolded simp_implies_def])
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1156
  apply (rule P)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1157
  done
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1158
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1159
lemma simp_implies_cong:
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1160
  assumes PP' :"PROP P == PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1161
  and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1162
  shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1163
proof (unfold simp_implies_def, rule equal_intr_rule)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1164
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1165
  and P': "PROP P'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1166
  from PP' [symmetric] and P' have "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1167
    by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1168
  hence "PROP Q" by (rule PQ)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1169
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1170
next
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1171
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1172
  and P: "PROP P"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1173
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1174
  hence "PROP Q'" by (rule P'Q')
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1175
  with P'QQ' [OF P', symmetric] show "PROP Q"
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1176
    by (rule equal_elim_rule1)
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1177
qed
208ebc9311f2 Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
berghofe
parents: 16587
diff changeset
  1178
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1179
subsubsection {* Actual Installation of the Simplifier *}
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1180
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1181
use "simpdata.ML"
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1182
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1183
setup Splitter.setup setup Clasimp.setup
95dca9f991f2 improved meson setup;
wenzelm
parents: 9852
diff changeset
  1184
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1185
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1186
subsubsection {* Lucas Dixon's eqstep tactic *}
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1187
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1188
use "~~/src/Provers/eqsubst.ML";
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1189
use "eqrule_HOL_data.ML";
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1190
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1191
setup EQSubstTac.setup
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1192
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1193
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1194
subsection {* Other simple lemmas *}
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1195
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1196
declare disj_absorb [simp] conj_absorb [simp]
14201
7ad7ab89c402 some basic new lemmas
paulson
parents: 13764
diff changeset
  1197
13723
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1198
lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1199
by blast+
c7d104550205 *** empty log message ***
nipkow
parents: 13638
diff changeset
  1200
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1201
13638
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1202
theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1203
  apply (rule iffI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1204
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1205
  apply (fast dest!: theI')
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1206
  apply (fast intro: ext the1_equality [symmetric])
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1207
  apply (erule ex1E)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1208
  apply (rule allI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1209
  apply (rule ex1I)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1210
  apply (erule spec)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1211
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1212
  apply (erule impE)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1213
  apply (rule allI)
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1214
  apply (rule_tac P = "xa = x" in case_split_thm)
14208
144f45277d5a misc tidying
paulson
parents: 14201
diff changeset
  1215
  apply (drule_tac [3] x = x in fun_cong, simp_all)
13638
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1216
  done
2b234b079245 Added choice_eq.
berghofe
parents: 13598
diff changeset
  1217
13438
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1218
text{*Needs only HOL-lemmas:*}
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1219
lemma mk_left_commute:
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1220
  assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1221
          c: "\<And>x y. f x y = f y x"
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1222
  shows "f x (f y z) = f y (f x z)"
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1223
by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
527811f00c56 added mk_left_commute to HOL.thy and used it "everywhere"
nipkow
parents: 13421
diff changeset
  1224
11750
3e400964893e judgment Trueprop;
wenzelm
parents: 11724
diff changeset
  1225
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15423
diff changeset
  1226
subsection {* Generic cases and induction *}
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1227
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1228
constdefs
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1229
  induct_forall :: "('a => bool) => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1230
  "induct_forall P == \<forall>x. P x"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1231
  induct_implies :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1232
  "induct_implies A B == A --> B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1233
  induct_equal :: "'a => 'a => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1234
  "induct_equal x y == x = y"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1235
  induct_conj :: "bool => bool => bool"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1236
  "induct_conj A B == A & B"
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1237
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1238
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1239
  by (simp only: atomize_all induct_forall_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1240
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1241
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1242
  by (simp only: atomize_imp induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1243
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1244
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1245
  by (simp only: atomize_eq induct_equal_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1246
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1247
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1248
    induct_conj (induct_forall A) (induct_forall B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1249
  by (unfold induct_forall_def induct_conj_def) rules
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1250
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1251
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1252
    induct_conj (induct_implies C A) (induct_implies C B)"
12354
5f5ee25513c5 setup "rules" method;
wenzelm
parents: 12338
diff changeset
  1253
  by (unfold induct_implies_def induct_conj_def) rules
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1254
13598
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1255
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1256
proof
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1257
  assume r: "induct_conj A B ==> PROP C" and A B
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1258
  show "PROP C" by (rule r) (simp! add: induct_conj_def)
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1259
next
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1260
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1261
  show "PROP C" by (rule r) (simp! add: induct_conj_def)+
8bc77b17f59f Fixed problem with induct_conj_curry: variable C should have type prop.
berghofe
parents: 13596
diff changeset
  1262
qed
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1263
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1264
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1265
  by (simp add: induct_implies_def)
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1266
12161
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
  1267
lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
  1268
lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
ea4fbf26a945 lemmas induct_atomize = atomize_conj ...;
wenzelm
parents: 12114
diff changeset
  1269
lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1270
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1271
11989
d4bcba4e080e renamed inductive_XXX to induct_XXX;
wenzelm
parents: 11977
diff changeset
  1272
hide const induct_forall induct_implies induct_equal induct_conj
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1273
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1274
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1275
text {* Method setup. *}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1276
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1277
ML {*
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1278
  structure InductMethod = InductMethodFun
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1279
  (struct
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1280
    val dest_concls = HOLogic.dest_concls
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1281
    val cases_default = thm "case_split"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1282
    val local_impI = thm "induct_impliesI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1283
    val conjI = thm "conjI"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1284
    val atomize = thms "induct_atomize"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1285
    val rulify1 = thms "induct_rulify1"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1286
    val rulify2 = thms "induct_rulify2"
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1287
    val localize = [Thm.symmetric (thm "induct_implies_def")]
11824
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1288
  end);
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1289
*}
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1290
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1291
setup InductMethod.setup
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1292
f4c1882dde2c setup generic cases and induction (from Inductive.thy);
wenzelm
parents: 11770
diff changeset
  1293
14357
e49d5d5ae66a print translation for ALL x <= n. P x
kleing
parents: 14295
diff changeset
  1294
end
15411
1d195de59497 removal of HOL_Lemmas
paulson
parents: 15380
diff changeset
  1295