author | wenzelm |
Mon, 03 Dec 2001 21:01:11 +0100 | |
changeset 12345 | 5d1436d1c268 |
parent 12114 | a8e860c86252 |
child 12349 | 94e812f9683e |
permissions | -rw-r--r-- |
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(* Title: FOL/IFOL.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson and Markus Wenzel |
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*) |
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header {* Intuitionistic first-order logic *} |
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theory IFOL = Pure |
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files ("IFOL_lemmas.ML") ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML"): |
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subsection {* Syntax and axiomatic basis *} |
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global |
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classes "term" < logic |
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defaultsort "term" |
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typedecl o |
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judgment |
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Trueprop :: "o => prop" ("(_)" 5) |
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consts |
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True :: o |
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False :: o |
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(* Connectives *) |
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"=" :: "['a, 'a] => o" (infixl 50) |
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Not :: "o => o" ("~ _" [40] 40) |
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& :: "[o, o] => o" (infixr 35) |
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"|" :: "[o, o] => o" (infixr 30) |
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--> :: "[o, o] => o" (infixr 25) |
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<-> :: "[o, o] => o" (infixr 25) |
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(* Quantifiers *) |
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All :: "('a => o) => o" (binder "ALL " 10) |
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Ex :: "('a => o) => o" (binder "EX " 10) |
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Ex1 :: "('a => o) => o" (binder "EX! " 10) |
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syntax |
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"~=" :: "['a, 'a] => o" (infixl 50) |
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translations |
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"x ~= y" == "~ (x = y)" |
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syntax (xsymbols) |
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Not :: "o => o" ("\<not> _" [40] 40) |
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"op &" :: "[o, o] => o" (infixr "\<and>" 35) |
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"op |" :: "[o, o] => o" (infixr "\<or>" 30) |
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"ALL " :: "[idts, o] => o" ("(3\<forall>_./ _)" [0, 10] 10) |
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"EX " :: "[idts, o] => o" ("(3\<exists>_./ _)" [0, 10] 10) |
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"EX! " :: "[idts, o] => o" ("(3\<exists>!_./ _)" [0, 10] 10) |
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"op ~=" :: "['a, 'a] => o" (infixl "\<noteq>" 50) |
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"op -->" :: "[o, o] => o" (infixr "\<longrightarrow>" 25) |
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"op <->" :: "[o, o] => o" (infixr "\<longleftrightarrow>" 25) |
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syntax (HTML output) |
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Not :: "o => o" ("\<not> _" [40] 40) |
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local |
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axioms |
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(* Equality *) |
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refl: "a=a" |
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subst: "[| a=b; P(a) |] ==> P(b)" |
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(* Propositional logic *) |
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conjI: "[| P; Q |] ==> P&Q" |
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conjunct1: "P&Q ==> P" |
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conjunct2: "P&Q ==> Q" |
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disjI1: "P ==> P|Q" |
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disjI2: "Q ==> P|Q" |
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disjE: "[| P|Q; P ==> R; Q ==> R |] ==> R" |
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impI: "(P ==> Q) ==> P-->Q" |
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mp: "[| P-->Q; P |] ==> Q" |
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FalseE: "False ==> P" |
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(* Definitions *) |
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True_def: "True == False-->False" |
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not_def: "~P == P-->False" |
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iff_def: "P<->Q == (P-->Q) & (Q-->P)" |
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(* Unique existence *) |
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ex1_def: "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)" |
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(* Quantifiers *) |
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allI: "(!!x. P(x)) ==> (ALL x. P(x))" |
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spec: "(ALL x. P(x)) ==> P(x)" |
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exI: "P(x) ==> (EX x. P(x))" |
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exE: "[| EX x. P(x); !!x. P(x) ==> R |] ==> R" |
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(* Reflection *) |
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eq_reflection: "(x=y) ==> (x==y)" |
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iff_reflection: "(P<->Q) ==> (P==Q)" |
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subsection {* Lemmas and proof tools *} |
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setup Simplifier.setup |
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use "IFOL_lemmas.ML" |
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declare impE [Pure.elim] iffD1 [Pure.elim] iffD2 [Pure.elim] |
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use "fologic.ML" |
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use "hypsubstdata.ML" |
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setup hypsubst_setup |
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use "intprover.ML" |
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subsection {* Atomizing meta-level rules *} |
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lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))" |
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proof |
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assume "!!x. P(x)" |
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show "ALL x. P(x)" by (rule allI) |
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next |
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assume "ALL x. P(x)" |
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thus "!!x. P(x)" by (rule allE) |
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qed |
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lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)" |
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proof |
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assume r: "A ==> B" |
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show "A --> B" by (rule impI) (rule r) |
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next |
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assume "A --> B" and A |
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thus B by (rule mp) |
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qed |
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lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)" |
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proof |
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assume "x == y" |
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show "x = y" by (unfold prems) (rule refl) |
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next |
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assume "x = y" |
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thus "x == y" by (rule eq_reflection) |
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qed |
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lemma atomize_conj [atomize]: "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)" |
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proof |
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assume "!!C. (A ==> B ==> PROP C) ==> PROP C" |
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show "A & B" by (rule conjI) |
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next |
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fix C |
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assume "A & B" |
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assume "A ==> B ==> PROP C" |
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thus "PROP C" |
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proof this |
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show A by (rule conjunct1) |
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show B by (rule conjunct2) |
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qed |
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qed |
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declare atomize_all [symmetric, rulify] atomize_imp [symmetric, rulify] |
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subsection {* Calculational rules *} |
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lemma forw_subst: "a = b ==> P(b) ==> P(a)" |
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by (rule ssubst) |
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lemma back_subst: "P(a) ==> a = b ==> P(b)" |
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by (rule subst) |
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text {* |
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Note that this list of rules is in reverse order of priorities. |
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*} |
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lemmas basic_trans_rules [trans] = |
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forw_subst |
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back_subst |
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rev_mp |
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mp |
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trans |
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lemmas [Pure.elim] = sym |
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end |