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reference:attribute_examples [2011/07/06 12:36] 131.246.161.187 [OF] |
reference:attribute_examples [2011/07/25 12:32] (current) 131.246.41.159 [rotated] |
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==== of ==== | ==== of ==== | ||
+ | Consider rule/theorem ''mp'': | ||
+ | |||
+ | $\quad [|\ ?P \longrightarrow ?Q; ?P\ |] \Longrightarrow ?Q$. | ||
+ | |||
+ | Using | ||
+ | mp [of _ "C ∨ D"] | ||
+ | yields the more specific theorem | ||
+ | |||
+ | $\quad [|\ ?P \longrightarrow C \vee D; ?P\ |] \Longrightarrow C \vee D$. | ||
+ | |||
+ | Note that $C, D$ are no longer schematic variables but specific terms. | ||
==== where ==== | ==== where ==== | ||
+ | |||
+ | Consider rule/theorem ''mp'': | ||
+ | |||
+ | $\quad [|\ ?P \longrightarrow ?Q; ?P\ |] \Longrightarrow ?Q$. | ||
+ | |||
+ | Using | ||
+ | mp [where Q="C ∨ D"] | ||
+ | yields the more specific theorem | ||
+ | |||
+ | $\quad [|\ ?P \longrightarrow C \vee D; ?P\ |] \Longrightarrow C \vee D$. | ||
+ | |||
+ | Note that $C, D$ are no longer schematic variables but specific terms. | ||
==== THEN ==== | ==== THEN ==== | ||
+ | |||
+ | Consider the "composite" theorem | ||
+ | |||
+ | $\quad t : (?A \vee \neg ?A) \wedge (\text{False} \longrightarrow ?A)$ | ||
+ | |||
+ | Then, | ||
+ | |||
+ | t [THEN conjunct1] | ||
+ | |||
+ | yields the new theorem | ||
+ | |||
+ | $\quad ?A \vee \neg ?A$ | ||
==== OF ==== | ==== OF ==== | ||
Consider rule ''subst'': | Consider rule ''subst'': | ||
Line 21: | Line 56: | ||
which allows substitutions from right to left. | which allows substitutions from right to left. | ||
==== simplified ==== | ==== simplified ==== | ||
+ | Consider theorem ''Nat.le_square'': | ||
+ | |||
+ | $\quad ?m \leq\ ?m\ \cdot\ ?m$ | ||
+ | |||
+ | Using | ||
+ | |||
+ | Nat.le_square [simplified] | ||
+ | you get | ||
+ | |||
+ | $\quad 0 <\ ?m \longrightarrow Suc\ 0 \leq\ ?m$ | ||
+ | |||
+ | Wether or not this is actually "simpler" is certainly open to discussion, especially because the original form does say something about $?m=0$ while the ''simplified'' version does not. | ||
==== rotated ==== | ==== rotated ==== | ||
+ | Consider rule ''disjE'': | ||
+ | |||
+ | $\quad [|\ ?P\;\vee\; ?Q; ?P \Longrightarrow ?R; ?Q \Longrightarrow ?R\ |] \Longrightarrow ?R$. | ||
+ | |||
+ | Then, | ||
+ | |||
+ | disjE [rotated 2] | ||
+ | yields | ||
+ | |||
+ | $\quad [|\ ?Q \Longrightarrow ?R; ?P\;\vee\; ?Q; ?P \Longrightarrow ?R\ |] \Longrightarrow ?R$. | ||
==== symmetric ===== | ==== symmetric ===== | ||
+ | Consider the definition of set membership ''Set.mem_def'', | ||
+ | |||
+ | $\quad (?x \in ?A) = ?A ?x$. | ||
+ | |||
+ | Then, | ||
+ | |||
+ | test [symmetric] | ||
+ | yields | ||
+ | |||
+ | $\quad ?X ?x = (?x \in ?X)$. |