author | haftmann |
Tue, 19 Sep 2006 15:22:03 +0200 | |
changeset 20596 | 3950e65f48f8 |
parent 19800 | 5f764272183e |
child 20627 | 30da2841553e |
permissions | -rw-r--r-- |
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(* Title: Pure/Pure.thy |
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ID: $Id$ |
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*) |
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header {* The Pure theory *} |
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theory Pure |
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imports ProtoPure |
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begin |
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setup -- {* Common setup of internal components *} |
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subsection {* Meta-level connectives in assumptions *} |
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lemma meta_mp: |
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assumes "PROP P ==> PROP Q" and "PROP P" |
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shows "PROP Q" |
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by (rule `PROP P ==> PROP Q` [OF `PROP P`]) |
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lemma meta_spec: |
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assumes "!!x. PROP P(x)" |
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shows "PROP P(x)" |
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by (rule `!!x. PROP P(x)`) |
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lemmas meta_allE = meta_spec [elim_format] |
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||
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subsection {* Meta-level conjunction *} |
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locale (open) meta_conjunction_syntax = |
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fixes meta_conjunction :: "prop => prop => prop" (infixr "&&" 2) |
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parse_translation {* |
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[("\<^fixed>meta_conjunction", fn [t, u] => Logic.mk_conjunction (t, u))] |
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*} |
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lemma all_conjunction: |
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includes meta_conjunction_syntax |
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shows "(!!x. PROP A(x) && PROP B(x)) == ((!!x. PROP A(x)) && (!!x. PROP B(x)))" |
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proof |
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assume conj: "!!x. PROP A(x) && PROP B(x)" |
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show "(\<And>x. PROP A(x)) && (\<And>x. PROP B(x))" |
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proof - |
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fix x |
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from conj show "PROP A(x)" by (rule conjunctionD1) |
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from conj show "PROP B(x)" by (rule conjunctionD2) |
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qed |
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next |
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assume conj: "(!!x. PROP A(x)) && (!!x. PROP B(x))" |
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fix x |
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show "PROP A(x) && PROP B(x)" |
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proof - |
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show "PROP A(x)" by (rule conj [THEN conjunctionD1, rule_format]) |
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show "PROP B(x)" by (rule conj [THEN conjunctionD2, rule_format]) |
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qed |
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qed |
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lemma imp_conjunction: |
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includes meta_conjunction_syntax |
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shows "(PROP A ==> PROP B && PROP C) == (PROP A ==> PROP B) && (PROP A ==> PROP C)" |
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proof |
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assume conj: "PROP A ==> PROP B && PROP C" |
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show "(PROP A ==> PROP B) && (PROP A ==> PROP C)" |
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proof - |
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assume "PROP A" |
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from conj [OF `PROP A`] show "PROP B" by (rule conjunctionD1) |
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from conj [OF `PROP A`] show "PROP C" by (rule conjunctionD2) |
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qed |
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next |
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assume conj: "(PROP A ==> PROP B) && (PROP A ==> PROP C)" |
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assume "PROP A" |
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show "PROP B && PROP C" |
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proof - |
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from `PROP A` show "PROP B" by (rule conj [THEN conjunctionD1]) |
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from `PROP A` show "PROP C" by (rule conj [THEN conjunctionD2]) |
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qed |
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qed |
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lemma conjunction_imp: |
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includes meta_conjunction_syntax |
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shows "(PROP A && PROP B ==> PROP C) == (PROP A ==> PROP B ==> PROP C)" |
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proof |
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assume r: "PROP A && PROP B ==> PROP C" |
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assume "PROP A" and "PROP B" |
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show "PROP C" by (rule r) - |
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next |
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assume r: "PROP A ==> PROP B ==> PROP C" |
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assume conj: "PROP A && PROP B" |
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show "PROP C" |
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proof (rule r) |
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from conj show "PROP A" by (rule conjunctionD1) |
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from conj show "PROP B" by (rule conjunctionD2) |
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qed |
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qed |
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end |