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| -rw-r--r-- | doc/faq/FAQ.tex | 29 | ||||
| -rw-r--r-- | doc/faq/axioms.eps | 413 | ||||
| -rw-r--r-- | doc/faq/axioms.eps_t | 43 | ||||
| -rw-r--r-- | doc/faq/axioms.fig | 100 | ||||
| -rw-r--r-- | doc/faq/axioms.pdf | bin | 0 -> 4974 bytes | |||
| -rw-r--r-- | doc/faq/axioms.pdf_t | 43 | ||||
| -rw-r--r-- | doc/faq/axioms.png | bin | 10075 -> 32222 bytes |
7 files changed, 381 insertions, 247 deletions
diff --git a/doc/faq/FAQ.tex b/doc/faq/FAQ.tex index d264ac62a..8495156ca 100644 --- a/doc/faq/FAQ.tex +++ b/doc/faq/FAQ.tex @@ -503,19 +503,34 @@ $\forall A \forall x y:A \forall p_1 p_2:x=y, p_1=p_2$ \item Extensionality of functions: $\forall f g:A\rightarrow B, (\forall x, f(x)=g(x)) \rightarrow f=g$ \end{itemize} -Here is a summary of the relative strength of these axioms, most -proofs can be found in directory {\tt Logic} of the standard library. -The justification of their validity relies on the interpretability in -set theory. - +Figure~\ref{fig:axioms} is a summary of the relative strength of these +axioms, most proofs can be found in directory {\tt Logic} of the standard +library. (Statements in boldface are the most ``interesting'' ones for +Coq.) The justification of their validity relies on the interpretability +in set theory. + +% fig2dev -m 2 -L png axioms.fig axioms.png +% fig2dev -L pdftex axioms.fig axioms.pdf +% fig2dev -L pdftex_t -p axioms.pdf axioms.fig axioms.pdf_t +% fig2dev -L pstex axioms.fig axioms.eps +% fig2dev -L pstex_t -p axioms.eps axioms.fig axioms.eps_t + +\begin{figure}[htbp] %HEVEA\imgsrc{axioms.png} %BEGIN LATEX +\begin{center} \ifpdf % si on est en pdflatex -\includegraphics[width=1.0\textwidth]{axioms.png} +\scalebox{0.65}{\input{axioms.pdf_t}} +%\includegraphics[width=1.0\textwidth]{axioms.png} \else -\includegraphics[width=1.0\textwidth]{axioms.eps} +\scalebox{0.65}{\input{axioms.eps_t}} +%\includegraphics[width=1.0\textwidth]{axioms.eps} \fi +\end{center} %END LATEX +\caption{The dependency graph of axioms in the Calculus of Inductive Constructions} +\label{fig:axioms} +\end{figure} \Question{What standard axioms are inconsistent with {\Coq}?} diff --git a/doc/faq/axioms.eps b/doc/faq/axioms.eps index 3f3c01c43..836afc347 100644 --- a/doc/faq/axioms.eps +++ b/doc/faq/axioms.eps @@ -1,11 +1,11 @@ -%!PS-Adobe-2.0 EPSF-2.0 +%!PS-Adobe-3.0 EPSF-3.0 %%Title: axioms.fig -%%Creator: fig2dev Version 3.2 Patchlevel 4 -%%CreationDate: Wed May 5 18:30:03 2004 -%%For: herbelin@limoux.polytechnique.fr (Hugo Herbelin) -%%BoundingBox: 0 0 437 372 -%%Magnification: 1.0000 +%%Creator: fig2dev Version 3.2 Patchlevel 5e +%%CreationDate: Sun Jul 26 08:23:54 2015 +%%BoundingBox: 0 0 647 514 +%Magnification: 1.0000 %%EndComments +%%BeginProlog /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put @@ -44,10 +44,6 @@ $F2psDict /mtrx matrix put /col31 {1.000 0.840 0.000 srgb} bind def end -save -newpath 0 372 moveto 0 0 lineto 437 0 lineto 437 372 lineto closepath clip newpath --90.0 435.2 translate -1 -1 scale /cp {closepath} bind def /ef {eofill} bind def @@ -80,51 +76,25 @@ newpath 0 372 moveto 0 0 lineto 437 0 lineto 437 372 lineto closepath clip newpa bind def /shd {dup dup 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+\end{picture}% diff --git a/doc/faq/axioms.fig b/doc/faq/axioms.fig index 78e448865..963178503 100644 --- a/doc/faq/axioms.fig +++ b/doc/faq/axioms.fig @@ -1,4 +1,4 @@ -#FIG 3.2 +#FIG 3.2 Produced by xfig version 3.2.5c Landscape Center Inches @@ -27,14 +27,6 @@ Single 1 1 1.00 60.00 120.00 5 1 0 1 0 7 50 -1 -1 0.000 0 1 1 0 6309.515 5767.724 4200 3825 3450 5550 3825 7200 1 1 1.00 60.00 120.00 -6 2025 300 7725 525 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 - 7725 525 2025 525 -4 0 0 50 -1 0 12 0.0000 4 180 5700 2025 450 The dependency graph of axioms in the Calculus of Inductive Constructions\001 --6 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 4200 6225 4200 7200 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 1 1 1.00 60.00 120.00 7725 3900 7200 6000 @@ -45,8 +37,6 @@ Single 1 1 1.00 60.00 120.00 1 1 1.00 60.00 120.00 5550 5625 5550 6000 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 - 4575 6000 6450 6000 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 1 2 1 1 1.00 60.00 120.00 1 1 1.00 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8175 Axiom K on A\001 +4 0 -1 50 -1 0 12 0.0000 2 180 4035 3600 8475 Uniqueness of reflexivity proofs for equality on A\001 +4 0 -1 50 -1 0 12 0.0000 2 180 2865 3600 8775 Uniqueness of equality proofs on A\001 +4 0 -1 50 -1 0 12 0.0000 2 180 5220 3600 9375 Invariance by substitution of reflexivity proofs for equality on A\001 +4 0 -1 50 -1 2 12 0.0000 2 180 2145 9000 5175 Functional extensionality\001 +4 0 -1 50 -1 2 12 0.0000 2 180 3585 3600 9075 Injectivity of equality on Sigma-types on A\001 +4 0 -1 50 -1 2 12 0.0000 2 135 1515 6450 7275 Proof-irrelevance\001 +4 0 -1 50 -1 2 12 0.0000 2 180 1440 3150 1050 Operator epsilon\001 +4 0 -1 50 -1 2 12 0.0000 2 135 1080 3150 1650 Constructive\001 +4 0 -1 50 -1 2 12 0.0000 2 180 1785 3150 1875 indefinite description\001 +4 0 -1 50 -1 2 12 0.0000 2 135 2085 3150 3150 Functional choice axiom\001 +4 0 -1 50 -1 2 12 0.0000 2 135 1080 450 1650 Constructive\001 +4 0 -1 50 -1 2 12 0.0000 2 180 1620 450 1875 definite description\001 +4 0 -1 50 -1 2 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