Révision 196 CSL17/completeness.tex
| completeness.tex (revision 196) | ||
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& |
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%Seq(z) \cand |
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\exists y_0 . ( A_g (\vec u , \vec x , y_0) \cand \beta(0, w , y_0) ) \cand \beta(|u|, w,y ) \\ |
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\cand & \forall k < |u| . \exists y_k , y_{k+1} . ( \beta (k, w, y_k) \cand \beta (k+1 ,w, y_{k+1}) \cand A_{h_i} (u , \vec u ; \vec x , y_k , y_{k+1}) )
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\cand & \forall k < |u| . \exists y_k , y_{k+1} . ( \beta (k, w, y_k) \cand \beta (k+1 ,w, y_{k+1}) \cand A_{h_i} (u , \vec u ; \vec x , y_k , y_{k+1}) )\\
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\cand & \forall k < |u| . \exists y_k , y_{k+1} . ( \beta (k, w, y_k) \cand \beta (k+1 ,w, y_{k+1}) \cand B (u , \vec u ; \vec x , y_k , y_{k+1}) )
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\end{array}
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\right) |
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\] |
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where |
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%where |
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% |
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%\[ |
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%B (u , \vec u ; \vec x , y_k , y_{k+1}) =
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%\begin{array}{ll}
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%& \\ |
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%\cand & |
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% |
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%\end{array}
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% |
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%\] |
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%which is $\Sigma^\safe_i$ by inspection, and indeed defines the graph of $f$. |
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\[ |
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B (u , \vec u ; \vec x , y_k , y_{k+1}) =
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\begin{array}{ll}
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& \\ |
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\cand & |
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\end{array}
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\] |
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which is $\Sigma^\safe_i$ by inspection, and indeed defines the graph of $f$. |
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To show totality, let $\vec u \in \normal, \vec x \in \safe$ and proceed by induction on $u \in \normal$. |
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The base case, when $u=0$, is immediate from the totality of $A_g$, so for the inductive case we need to show: |
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\[ |
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