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src/FOL/IFOL.thy

author | paulson |

Tue, 07 Dec 2004 11:31:14 +0100 | |

changeset 15377 | 3d99eea28a9b |

parent 14854 | 61bdf2ae4dc5 |

child 15481 | fc075ae929e4 |

permissions | -rw-r--r-- |

proof of subst by S Merz

(* Title: FOL/IFOL.thy ID: $Id$ Author: Lawrence C Paulson and Markus Wenzel *) header {* Intuitionistic first-order logic *} theory IFOL = Pure files ("IFOL_lemmas.ML") ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML"): subsection {* Syntax and axiomatic basis *} global classes "term" defaultsort "term" typedecl o judgment Trueprop :: "o => prop" ("(_)" 5) consts True :: o False :: o (* Connectives *) "=" :: "['a, 'a] => o" (infixl 50) Not :: "o => o" ("~ _" [40] 40) & :: "[o, o] => o" (infixr 35) "|" :: "[o, o] => o" (infixr 30) --> :: "[o, o] => o" (infixr 25) <-> :: "[o, o] => o" (infixr 25) (* Quantifiers *) All :: "('a => o) => o" (binder "ALL " 10) Ex :: "('a => o) => o" (binder "EX " 10) Ex1 :: "('a => o) => o" (binder "EX! " 10) syntax "_not_equal" :: "['a, 'a] => o" (infixl "~=" 50) translations "x ~= y" == "~ (x = y)" syntax (xsymbols) Not :: "o => o" ("\<not> _" [40] 40) "op &" :: "[o, o] => o" (infixr "\<and>" 35) "op |" :: "[o, o] => o" (infixr "\<or>" 30) "ALL " :: "[idts, o] => o" ("(3\<forall>_./ _)" [0, 10] 10) "EX " :: "[idts, o] => o" ("(3\<exists>_./ _)" [0, 10] 10) "EX! " :: "[idts, o] => o" ("(3\<exists>!_./ _)" [0, 10] 10) "_not_equal" :: "['a, 'a] => o" (infixl "\<noteq>" 50) "op -->" :: "[o, o] => o" (infixr "\<longrightarrow>" 25) "op <->" :: "[o, o] => o" (infixr "\<longleftrightarrow>" 25) syntax (HTML output) Not :: "o => o" ("\<not> _" [40] 40) "op &" :: "[o, o] => o" (infixr "\<and>" 35) "op |" :: "[o, o] => o" (infixr "\<or>" 30) "ALL " :: "[idts, o] => o" ("(3\<forall>_./ _)" [0, 10] 10) "EX " :: "[idts, o] => o" ("(3\<exists>_./ _)" [0, 10] 10) "EX! " :: "[idts, o] => o" ("(3\<exists>!_./ _)" [0, 10] 10) "_not_equal" :: "['a, 'a] => o" (infixl "\<noteq>" 50) local finalconsts False All Ex "op =" "op &" "op |" "op -->" axioms (* Equality *) refl: "a=a" (* Propositional logic *) conjI: "[| P; Q |] ==> P&Q" conjunct1: "P&Q ==> P" conjunct2: "P&Q ==> Q" disjI1: "P ==> P|Q" disjI2: "Q ==> P|Q" disjE: "[| P|Q; P ==> R; Q ==> R |] ==> R" impI: "(P ==> Q) ==> P-->Q" mp: "[| P-->Q; P |] ==> Q" FalseE: "False ==> P" (* Quantifiers *) allI: "(!!x. P(x)) ==> (ALL x. P(x))" spec: "(ALL x. P(x)) ==> P(x)" exI: "P(x) ==> (EX x. P(x))" exE: "[| EX x. P(x); !!x. P(x) ==> R |] ==> R" (* Reflection *) eq_reflection: "(x=y) ==> (x==y)" iff_reflection: "(P<->Q) ==> (P==Q)" text{*Thanks to Stephan Merz*} theorem subst: assumes eq: "a = b" and p: "P(a)" shows "P(b)" proof - from eq have meta: "a \<equiv> b" by (rule eq_reflection) from p show ?thesis by (unfold meta) qed defs (* Definitions *) True_def: "True == False-->False" not_def: "~P == P-->False" iff_def: "P<->Q == (P-->Q) & (Q-->P)" (* Unique existence *) ex1_def: "Ex1(P) == EX x. P(x) & (ALL y. P(y) --> y=x)" subsection {* Lemmas and proof tools *} setup Simplifier.setup use "IFOL_lemmas.ML" use "fologic.ML" use "hypsubstdata.ML" setup hypsubst_setup use "intprover.ML" subsection {* Intuitionistic Reasoning *} lemma impE': assumes 1: "P --> Q" and 2: "Q ==> R" and 3: "P --> Q ==> P" shows R proof - from 3 and 1 have P . with 1 have Q by (rule impE) with 2 show R . qed lemma allE': assumes 1: "ALL x. P(x)" and 2: "P(x) ==> ALL x. P(x) ==> Q" shows Q proof - from 1 have "P(x)" by (rule spec) from this and 1 show Q by (rule 2) qed lemma notE': assumes 1: "~ P" and 2: "~ P ==> P" shows R proof - from 2 and 1 have P . with 1 show R by (rule notE) qed lemmas [Pure.elim!] = disjE iffE FalseE conjE exE and [Pure.intro!] = iffI conjI impI TrueI notI allI refl and [Pure.elim 2] = allE notE' impE' and [Pure.intro] = exI disjI2 disjI1 ML_setup {* Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac)); *} lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)" by rules lemmas [sym] = sym iff_sym not_sym iff_not_sym and [Pure.elim?] = iffD1 iffD2 impE lemma eq_commute: "a=b <-> b=a" apply (rule iffI) apply (erule sym)+ done subsection {* Atomizing meta-level rules *} lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))" proof assume "!!x. P(x)" show "ALL x. P(x)" .. next assume "ALL x. P(x)" thus "!!x. P(x)" .. qed lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)" proof assume "A ==> B" thus "A --> B" .. next assume "A --> B" and A thus B by (rule mp) qed lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)" proof assume "x == y" show "x = y" by (unfold prems) (rule refl) next assume "x = y" thus "x == y" by (rule eq_reflection) qed lemma atomize_conj [atomize]: "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)" proof assume "!!C. (A ==> B ==> PROP C) ==> PROP C" show "A & B" by (rule conjI) next fix C assume "A & B" assume "A ==> B ==> PROP C" thus "PROP C" proof this show A by (rule conjunct1) show B by (rule conjunct2) qed qed lemmas [symmetric, rulify] = atomize_all atomize_imp subsection {* Calculational rules *} lemma forw_subst: "a = b ==> P(b) ==> P(a)" by (rule ssubst) lemma back_subst: "P(a) ==> a = b ==> P(b)" by (rule subst) text {* Note that this list of rules is in reverse order of priorities. *} lemmas basic_trans_rules [trans] = forw_subst back_subst rev_mp mp trans subsection {* ``Let'' declarations *} nonterminals letbinds letbind constdefs Let :: "['a::{}, 'a => 'b] => ('b::{})" "Let(s, f) == f(s)" syntax "_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10) "" :: "letbind => letbinds" ("_") "_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _") "_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10) translations "_Let(_binds(b, bs), e)" == "_Let(b, _Let(bs, e))" "let x = a in e" == "Let(a, %x. e)" lemma LetI: assumes prem: "(!!x. x=t ==> P(u(x)))" shows "P(let x=t in u(x))" apply (unfold Let_def) apply (rule refl [THEN prem]) done ML {* val Let_def = thm "Let_def"; val LetI = thm "LetI"; *} end