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373 The induction hypothesis contains an application of \isa{lists}. Using a |
373 The induction hypothesis contains an application of \isa{lists}. Using a |
374 monotone function in the inductive definition always has this effect. The |
374 monotone function in the inductive definition always has this effect. The |
375 subgoal may look uninviting, but fortunately |
375 subgoal may look uninviting, but fortunately |
376 \isa{lists} distributes over intersection: |
376 \isa{lists} distributes over intersection: |
377 \begin{isabelle}% |
377 \begin{isabelle}% |
378 listsp\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ B{\isaliteral{29}{\isacharparenright}}\rulename{lists{\isaliteral{5F}{\isacharunderscore}}Int{\isaliteral{5F}{\isacharunderscore}}eq}% |
378 lists\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ lists\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ lists\ B\rulename{lists{\isaliteral{5F}{\isacharunderscore}}Int{\isaliteral{5F}{\isacharunderscore}}eq}% |
379 \end{isabelle} |
379 \end{isabelle} |
380 Thanks to this default simplification rule, the induction hypothesis |
380 Thanks to this default simplification rule, the induction hypothesis |
381 is quickly replaced by its two parts: |
381 is quickly replaced by its two parts: |
382 \begin{trivlist} |
382 \begin{trivlist} |
383 \item \isa{args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ {\isaliteral{28}{\isacharparenleft}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{29}{\isacharparenright}}} |
383 \item \isa{args\ {\isaliteral{5C3C696E3E}{\isasymin}}\ lists\ {\isaliteral{28}{\isacharparenleft}}well{\isaliteral{5F}{\isacharunderscore}}formed{\isaliteral{5F}{\isacharunderscore}}gterm{\isaliteral{27}{\isacharprime}}\ arity{\isaliteral{29}{\isacharparenright}}} |