(* Title: Pure/Pure.thy
Author: Makarius
Final stage of bootstrapping Pure, based on implicit background theory.
*)
theory Pure
keywords
"!!" "!" "%" "(" ")" "+" "," "--" ":" "::" ";" "<" "<=" "=" "=="
"=>" "?" "[" "\<equiv>" "\<leftharpoondown>" "\<rightharpoonup>"
"\<rightleftharpoons>" "\<subseteq>" "]" "and" "assumes"
"attach" "begin" "binder" "constrains" "defines" "fixes" "for"
"identifier" "if" "imports" "in" "includes" "infix" "infixl"
"infixr" "is" "keywords" "notes" "obtains" "open" "output"
"overloaded" "pervasive" "shows" "structure" "unchecked" "where" "|"
and "theory" :: thy_begin % "theory"
and "ML_file" :: thy_load % "ML"
and "header" :: diag
and "chapter" :: thy_heading1
and "section" :: thy_heading2
and "subsection" :: thy_heading3
and "subsubsection" :: thy_heading4
and "text" "text_raw" :: thy_decl
and "sect" :: prf_heading2 % "proof"
and "subsect" :: prf_heading3 % "proof"
and "subsubsect" :: prf_heading4 % "proof"
and "txt" "txt_raw" :: prf_decl % "proof"
and "default_sort" "typedecl" "type_synonym" "nonterminal" "judgment"
"consts" "syntax" "no_syntax" "translations" "no_translations" "defs"
"definition" "abbreviation" "type_notation" "no_type_notation" "notation"
"no_notation" "axiomatization" "theorems" "lemmas" "declare"
"hide_class" "hide_type" "hide_const" "hide_fact" :: thy_decl
and "ML" :: thy_decl % "ML"
and "ML_prf" :: prf_decl % "proof" (* FIXME % "ML" ?? *)
and "ML_val" "ML_command" :: diag % "ML"
and "setup" "local_setup" "attribute_setup" "method_setup"
"declaration" "syntax_declaration" "simproc_setup"
"parse_ast_translation" "parse_translation" "print_translation"
"typed_print_translation" "print_ast_translation" "oracle" :: thy_decl % "ML"
and "bundle" :: thy_decl
and "include" "including" :: prf_decl
and "print_bundles" :: diag
and "context" "locale" :: thy_decl
and "sublocale" "interpretation" :: thy_goal
and "interpret" :: prf_goal % "proof"
and "class" :: thy_decl
and "subclass" :: thy_goal
and "instantiation" :: thy_decl
and "instance" :: thy_goal
and "overloading" :: thy_decl
and "code_datatype" :: thy_decl
and "theorem" "lemma" "corollary" :: thy_goal
and "schematic_theorem" "schematic_lemma" "schematic_corollary" :: thy_goal
and "notepad" :: thy_decl
and "have" :: prf_goal % "proof"
and "hence" :: prf_goal % "proof" == "then have"
and "show" :: prf_asm_goal % "proof"
and "thus" :: prf_asm_goal % "proof" == "then show"
and "then" "from" "with" :: prf_chain % "proof"
and "note" "using" "unfolding" :: prf_decl % "proof"
and "fix" "assume" "presume" "def" :: prf_asm % "proof"
and "obtain" :: prf_asm_goal % "proof"
and "guess" :: prf_asm_goal_script % "proof"
and "let" "write" :: prf_decl % "proof"
and "case" :: prf_asm % "proof"
and "{" :: prf_open % "proof"
and "}" :: prf_close % "proof"
and "next" :: prf_block % "proof"
and "qed" :: qed_block % "proof"
and "by" ".." "." "sorry" :: "qed" % "proof"
and "done" :: "qed_script" % "proof"
and "oops" :: qed_global % "proof"
and "defer" "prefer" "apply" :: prf_script % "proof"
and "apply_end" :: prf_script % "proof" == ""
and "proof" :: prf_block % "proof"
and "also" "moreover" :: prf_decl % "proof"
and "finally" "ultimately" :: prf_chain % "proof"
and "back" :: prf_script % "proof"
and "Isabelle.command" :: control
and "help" "print_commands" "print_options"
"print_context" "print_theory" "print_syntax" "print_abbrevs" "print_defn_rules"
"print_theorems" "print_locales" "print_classes" "print_locale"
"print_interps" "print_dependencies" "print_attributes"
"print_simpset" "print_rules" "print_trans_rules" "print_methods"
"print_antiquotations" "thy_deps" "locale_deps" "class_deps" "thm_deps"
"print_binds" "print_facts" "print_cases" "print_statement" "thm"
"prf" "full_prf" "prop" "term" "typ" "print_codesetup" "unused_thms"
:: diag
and "cd" :: control
and "pwd" :: diag
and "use_thy" "remove_thy" "kill_thy" :: control
and "display_drafts" "print_state" "pr" :: diag
and "pretty_setmargin" "disable_pr" "enable_pr" "commit" "quit" "exit" :: control
and "welcome" :: diag
and "init_toplevel" "linear_undo" "undo" "undos_proof" "cannot_undo" "kill" :: control
and "end" :: thy_end % "theory"
and "realizers" "realizability" "extract_type" "extract" :: thy_decl
and "find_theorems" "find_consts" :: diag
and "ProofGeneral.process_pgip" "ProofGeneral.pr" "ProofGeneral.undo"
"ProofGeneral.restart" "ProofGeneral.kill_proof" "ProofGeneral.inform_file_processed"
"ProofGeneral.inform_file_retracted" :: control
begin
ML_file "Isar/isar_syn.ML"
ML_file "Isar/calculation.ML"
ML_file "Tools/rail.ML"
ML_file "Tools/rule_insts.ML";
ML_file "Tools/find_theorems.ML"
ML_file "Tools/find_consts.ML"
ML_file "Tools/proof_general_pure.ML"
ML_file "Tools/simplifier_trace.ML"
section {* Basic attributes *}
attribute_setup tagged =
"Scan.lift (Args.name -- Args.name) >> Thm.tag"
"tagged theorem"
attribute_setup untagged =
"Scan.lift Args.name >> Thm.untag"
"untagged theorem"
attribute_setup kind =
"Scan.lift Args.name >> Thm.kind"
"theorem kind"
attribute_setup THEN =
"Scan.lift (Scan.optional (Args.bracks Parse.nat) 1) -- Attrib.thm
>> (fn (i, B) => Thm.rule_attribute (fn _ => fn A => A RSN (i, B)))"
"resolution with rule"
attribute_setup OF =
"Attrib.thms >> (fn Bs => Thm.rule_attribute (fn _ => fn A => A OF Bs))"
"rule resolved with facts"
attribute_setup rename_abs =
"Scan.lift (Scan.repeat (Args.maybe Args.name)) >> (fn vs =>
Thm.rule_attribute (K (Drule.rename_bvars' vs)))"
"rename bound variables in abstractions"
attribute_setup unfolded =
"Attrib.thms >> (fn ths =>
Thm.rule_attribute (fn context => Local_Defs.unfold (Context.proof_of context) ths))"
"unfolded definitions"
attribute_setup folded =
"Attrib.thms >> (fn ths =>
Thm.rule_attribute (fn context => Local_Defs.fold (Context.proof_of context) ths))"
"folded definitions"
attribute_setup consumes =
"Scan.lift (Scan.optional Parse.int 1) >> Rule_Cases.consumes"
"number of consumed facts"
attribute_setup constraints =
"Scan.lift Parse.nat >> Rule_Cases.constraints"
"number of equality constraints"
attribute_setup case_names = {*
Scan.lift (Scan.repeat1 (Args.name --
Scan.optional (@{keyword "["} |-- Scan.repeat1 (Args.maybe Args.name) --| @{keyword "]"}) []))
>> (fn cs =>
Rule_Cases.cases_hyp_names
(map #1 cs)
(map (map (the_default Rule_Cases.case_hypsN) o #2) cs))
*} "named rule cases"
attribute_setup case_conclusion =
"Scan.lift (Args.name -- Scan.repeat Args.name) >> Rule_Cases.case_conclusion"
"named conclusion of rule cases"
attribute_setup params =
"Scan.lift (Parse.and_list1 (Scan.repeat Args.name)) >> Rule_Cases.params"
"named rule parameters"
attribute_setup rule_format = {*
Scan.lift (Args.mode "no_asm")
>> (fn true => Object_Logic.rule_format_no_asm | false => Object_Logic.rule_format)
*} "result put into canonical rule format"
attribute_setup elim_format =
"Scan.succeed (Thm.rule_attribute (K Tactic.make_elim))"
"destruct rule turned into elimination rule format"
attribute_setup no_vars = {*
Scan.succeed (Thm.rule_attribute (fn context => fn th =>
let
val ctxt = Variable.set_body false (Context.proof_of context);
val ((_, [th']), _) = Variable.import true [th] ctxt;
in th' end))
*} "imported schematic variables"
attribute_setup eta_long =
"Scan.succeed (Thm.rule_attribute (fn _ => Conv.fconv_rule Drule.eta_long_conversion))"
"put theorem into eta long beta normal form"
attribute_setup atomize =
"Scan.succeed Object_Logic.declare_atomize"
"declaration of atomize rule"
attribute_setup rulify =
"Scan.succeed Object_Logic.declare_rulify"
"declaration of rulify rule"
attribute_setup rotated =
"Scan.lift (Scan.optional Parse.int 1
>> (fn n => Thm.rule_attribute (fn _ => rotate_prems n)))"
"rotated theorem premises"
attribute_setup defn =
"Attrib.add_del Local_Defs.defn_add Local_Defs.defn_del"
"declaration of definitional transformations"
attribute_setup abs_def =
"Scan.succeed (Thm.rule_attribute (fn context =>
Local_Defs.meta_rewrite_rule (Context.proof_of context) #> Drule.abs_def))"
"abstract over free variables of definitional theorem"
section {* Further content for the Pure theory *}
subsection {* Meta-level connectives in assumptions *}
lemma meta_mp:
assumes "PROP P ==> PROP Q" and "PROP P"
shows "PROP Q"
by (rule `PROP P ==> PROP Q` [OF `PROP P`])
lemmas meta_impE = meta_mp [elim_format]
lemma meta_spec:
assumes "!!x. PROP P x"
shows "PROP P x"
by (rule `!!x. PROP P x`)
lemmas meta_allE = meta_spec [elim_format]
lemma swap_params:
"(!!x y. PROP P x y) == (!!y x. PROP P x y)" ..
subsection {* Meta-level conjunction *}
lemma all_conjunction:
"(!!x. PROP A x &&& PROP B x) == ((!!x. PROP A x) &&& (!!x. PROP B x))"
proof
assume conj: "!!x. PROP A x &&& PROP B x"
show "(!!x. PROP A x) &&& (!!x. PROP B x)"
proof -
fix x
from conj show "PROP A x" by (rule conjunctionD1)
from conj show "PROP B x" by (rule conjunctionD2)
qed
next
assume conj: "(!!x. PROP A x) &&& (!!x. PROP B x)"
fix x
show "PROP A x &&& PROP B x"
proof -
show "PROP A x" by (rule conj [THEN conjunctionD1, rule_format])
show "PROP B x" by (rule conj [THEN conjunctionD2, rule_format])
qed
qed
lemma imp_conjunction:
"(PROP A ==> PROP B &&& PROP C) == ((PROP A ==> PROP B) &&& (PROP A ==> PROP C))"
proof
assume conj: "PROP A ==> PROP B &&& PROP C"
show "(PROP A ==> PROP B) &&& (PROP A ==> PROP C)"
proof -
assume "PROP A"
from conj [OF `PROP A`] show "PROP B" by (rule conjunctionD1)
from conj [OF `PROP A`] show "PROP C" by (rule conjunctionD2)
qed
next
assume conj: "(PROP A ==> PROP B) &&& (PROP A ==> PROP C)"
assume "PROP A"
show "PROP B &&& PROP C"
proof -
from `PROP A` show "PROP B" by (rule conj [THEN conjunctionD1])
from `PROP A` show "PROP C" by (rule conj [THEN conjunctionD2])
qed
qed
lemma conjunction_imp:
"(PROP A &&& PROP B ==> PROP C) == (PROP A ==> PROP B ==> PROP C)"
proof
assume r: "PROP A &&& PROP B ==> PROP C"
assume ab: "PROP A" "PROP B"
show "PROP C"
proof (rule r)
from ab show "PROP A &&& PROP B" .
qed
next
assume r: "PROP A ==> PROP B ==> PROP C"
assume conj: "PROP A &&& PROP B"
show "PROP C"
proof (rule r)
from conj show "PROP A" by (rule conjunctionD1)
from conj show "PROP B" by (rule conjunctionD2)
qed
qed
end