author | wenzelm |
Thu, 14 Oct 1999 16:02:39 +0200 | |
changeset 7869 | c007f801cd59 |
parent 7860 | 7819547df4d8 |
child 7874 | 180364256231 |
permissions | -rw-r--r-- |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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(* Title: HOL/Isar_examples/Peirce.thy |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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ID: $Id$ |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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Author: Markus Wenzel, TU Muenchen |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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*) |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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header {* Peirce's Law *}; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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theory Peirce = Main:; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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text {* |
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We consider Peirce's Law: $((A \impl B) \impl A) \impl A$. This is |
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an inherently non-intuitionistic statement, so its proof will |
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certainly involve some form of classical contradiction. |
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The first proof is again a well-balanced combination of plain |
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backward and forward reasoning. The actual classical reasoning step |
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is where the negated goal is introduced as additional assumption. |
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This eventually leads to a contradiction.\footnote{The rule involved |
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here is negation elimination; it holds in intuitionistic logic as |
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well.} |
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*}; |
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theorem "((A --> B) --> A) --> A"; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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proof; |
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assume aba: "(A --> B) --> A"; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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show A; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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proof (rule classical); |
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assume "~ A"; |
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have "A --> B"; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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proof; |
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assume A; |
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thus B; by contradiction; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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qed; |
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with aba; show A; ..; |
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qed; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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qed; |
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text {* |
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The subsequent version rearranges the reasoning by means of ``weak |
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assumptions'' (as introduced by \isacommand{presume}). Before |
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assuming the negated goal $\neg A$, its intended consequence $A \impl |
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B$ is put into place in order to solve the main problem. |
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Nevertheless, we do not get anything for free, but have to establish |
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$A \impl B$ later on. The overall effect is that of a \emph{cut}. |
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Technically speaking, whenever some goal is solved by |
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\isacommand{show} in the context of weak assumptions then the latter |
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give rise to new subgoals, which may be established separately. In |
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contrast, strong assumptions (as introduced by \isacommand{assume}) |
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are solved immediately. |
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*}; |
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theorem "((A --> B) --> A) --> A"; |
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proof; |
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assume aba: "(A --> B) --> A"; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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show A; |
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proof (rule classical); |
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presume "A --> B"; |
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with aba; show A; ..; |
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next; |
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assume not_a: "~ A"; |
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show "A --> B"; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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proof; |
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assume A; |
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with not_a; show B; ..; |
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Miscellaneous Isabelle/Isar examples for Higher-Order Logic.
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qed; |
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qed; |
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qed; |
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text {* |
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Note that the goals stemming from weak assumptions may be even left |
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until qed time, where they get eventually solved ``by assumption'' as |
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well. In that case there is really no big difference between the two |
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kinds of assumptions, apart from the order of reducing the individual |
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parts of the proof configuration. |
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Nevertheless, the ``strong'' mode of plain assumptions is quite |
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important in practice to achieve robustness of proof document |
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interpretation. By forcing both the conclusion \emph{and} the |
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assumptions to unify with the pending goal to be solved, goal |
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selection becomes quite deterministic. For example, decomposition |
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with ``case-analysis'' type rules usually give rise to several goals |
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that only differ in there local contexts. With strong assumptions |
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these may be still solved in any order in a predictable way, while |
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weak ones would quickly lead to great confusion, eventually demanding |
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even some backtracking. |
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*}; |
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end; |