Add support for cross-referencing PDFs.
parent
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commit
cadb07018a
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@ -6,8 +6,9 @@
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\graphicspath{{./images/}}
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\newcommand{\larea}[2]{\lean{../..}{Common/Real/Geometry/Area}{#1}{#2}}
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\newcommand{\lrect}[2]{\lean{../..}{Common/Real/Geometry/Rectangle}{#1}{#2}}
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\externaldocument[A:]
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{../../Common/Real/Geometry/Area}
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[../../Common/Real/Geometry/Area.pdf]
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\begin{document}
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@ -29,10 +30,9 @@ A set consisting of a single point.
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\begin{proof}
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Let $S$ be a set consisting of a single point.
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By definition of a \lrect{Real.Point}{Point}, $S$ is a rectangle in which all
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vertices coincide.
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By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
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width times its height.
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By definition of a Point, $S$ is a rectangle in which all vertices coincide.
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By \nameref{A:sec:choice-scale} $S$ is measurable with area its width times
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its height.
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The width and height of $S$ is trivially zero.
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Therefore $a(S) = (0)(0) = 0$.
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@ -65,7 +65,7 @@ A set consisting of a finite number of points in a plane.
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By construction, $S_{k+1} = S_k \cup T$.
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By the induction hypothesis, $S_k$ is measurable with area $0$.
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By \nameref{sub:exercise-1a}, $T$ is measurable with area $0$.
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By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
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By the \nameref{A:sec:additive-property}, $S_k \cup T$ is
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measurable, $S_k \cap T$ is measurable, and
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\begin{align}
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a(S_{k+1})
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@ -111,10 +111,10 @@ The union of a finite collection of line segments in a plane.
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\paragraph{Base Case}%
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Consider a set $S$ consisting of a single line segment in a plane.
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By definition of a \lrect{Real.LineSemgnet}{Line Segment}, $S$ is a
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rectangle in which one side has dimension $0$.
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By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
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width $w$ times its height $h$.
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By definition of a Line Segment, $S$ is a rectangle in which one side has
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dimension $0$.
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By \nameref{A:sec:choice-scale}, $S$ is measurable with area its width $w$
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times its height $h$.
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Therefore $a(S) = wh = 0$.
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Thus $P(1)$ holds.
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@ -128,8 +128,8 @@ The union of a finite collection of line segments in a plane.
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By construction, $S_{k+1} = S_k \cup T$.
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By the induction hypothesis, $S_k$ is measurable with area $0$.
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By the base case, $T$ is measurable with area $0$.
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By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
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measurable, $S_k \cap T$ is measurable, and
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By the \nameref{A:sec:additive-property}, $S_k \cup T$ is measurable,
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$S_k \cap T$ is measurable, and
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\begin{align}
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a(S_{k+1})
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& = a(S_k \cup T) \nonumber \\
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@ -190,10 +190,10 @@ Prove that every triangular region is measurable and that its area is one half
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\centering
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\end{figure}
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By \larea{Choice-of-Scale}{Choice of Scale}, both $R$ and $S$ are measurable.
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By \nameref{A:sec:choice-scale}, both $R$ and $S$ are measurable.
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By this same axiom, $a(R) = ab$ and $a(S) = ca\sin{\theta}$.
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By the \larea{Additive-Property}{Additive Property}, $R \cup S$ and $R \cap S$
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are both measurable.
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By the \nameref{A:sec:additive-property}, $R \cup S$ and $R \cap S$ are both
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measurable.
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$a(R \cap S) = a(T)$ and $a(R \cup S)$ can be determined by noting that
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$R$'s construction implies identity $a(R) = 2a(T)$.
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Therefore
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@ -206,8 +206,8 @@ Prove that every triangular region is measurable and that its area is one half
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& = ab + ca\sin{\theta} - ca\sin{\theta} - a(T).
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\end{align*}
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Solving for $a(T)$ gives the desired identity: $$a(T) = \frac{1}{2}ab.$$
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By \larea{Invariance-Under-Congruence}{Invariance Under Congruence},
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$a(T') = a(T)$, concluding our proof.
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By \nameref{A:sec:invariance-under-congruence}, $a(T') = a(T)$, concluding our
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proof.
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\end{proof}
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@ -233,10 +233,10 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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Suppose $S$ is a right trapezoid.
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Then $S$ is the union of non-overlapping rectangle $R$ of width $b_1$ and
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height $h$ with right triangle $T$ of base $b_2 - b_1$ and height $h$.
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By \larea{Choice-of-Scale}{Choice of Scale}, $R$ is measurable.
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By \nameref{A:sec:choice-scale}, $R$ is measurable.
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By \nameref{sec:exercise-2}, $T$ is measurable.
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By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
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are both measurable and
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By the \nameref{A:sec:additive-property}, $R \cup T$ and $R \cap T$ are both
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measurable and
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\begin{align*}
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a(S)
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& = a(R \cup T) \\
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@ -256,8 +256,8 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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Then $R$ has longer base edge of length $b_2 - c$.
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By \nameref{sec:exercise-2}, $T$ is measurable.
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By Case 1, $R$ is measurable.
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By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
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are both measurable and
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By the \nameref{A:sec:additive-property}, $R \cup T$ and $R \cap T$ are both
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measurable and
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\begin{align*}
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a(S)
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& = a(T) + a(R) - a(R \cap T) \\
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@ -274,7 +274,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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Let $c$ denote the length of base $T$.
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Reflect $T$ vertically to form another right triangle, say $T'$.
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Then $T' \cup R$ is an acute trapezoid.
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By \larea{Invariance-Under-Congruence}{Invariance Under Congruence},
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By \nameref{A:sec:invariance-under-congruence},
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\begin{equation}
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\label{par:exercise-3-case-3-eq1}
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\tag{3.1}
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@ -301,7 +301,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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Let $c$ denote the length of base $T$.
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Reflect $T$ vertically to form another right triangle, say $T'$.
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Then $T' \cup R$ is an acute trapezoid.
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By \larea{Invariance-Under-Congruence}{Invariance Under Congruence},
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By \nameref{A:sec:invariance-under-congruence},
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\begin{equation}
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\label{par:exercise-3-eq2}
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\tag{3.2}
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@ -338,8 +338,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
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We assume $P$ has three non-collinear points, ruling out any instances of
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points or line segments.
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By \larea{Choice-of-Scale}{Choice of Scale}, $P$ is measurable with area
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$a(P) = wh$.
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By \nameref{A:sec:choice-scale}, $P$ is measurable with area $a(P) = wh$.
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By construction, $P$ has $I = (w - 1)(h - 1)$ interior lattice points and
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$B = 2(w + h)$ lattice points on its boundary.
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The following shows the lattice point area formula is in agreement with
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@ -456,8 +455,8 @@ Use induction on the number of edges to construct a proof for general polygons.
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& = a(S) + (I_T + \frac{1}{2}B_T - 1) & \text{induction hypothesis} \\
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& = a(S) + a(T). & \text{base case}
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\end{align*}
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By the \larea{Additive-Property}{Additive Property}, $S \cup T$ is
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measurable, $S \cap T$ is measurable, and
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By the \nameref{A:sec:additive-property}, $S \cup T$ is measurable,
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$S \cap T$ is measurable, and
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\begin{align*}
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a(P)
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& = a(S \cup T) \\
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@ -2,11 +2,9 @@
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\input{../../preamble}
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\newcommand{\link}[1]{\lean{../..}
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{Bookshelf/Apostol/Chapter\_1\_11} % Location
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{Apostol.Chapter\_1\_11.#1} % Fragment
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{Chapter\_1\_11.#1} % Presentation
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}
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\newcommand{\lean}[1]{\href
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{./Chapter\_1\_11.html\#Apostol.Chapter\_1\_11.#1}
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{Apostol.Chapter\_1\_11.#1}}
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\begin{document}
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@ -24,7 +22,7 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
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\begin{proof}
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\link{exercise\_4a}
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\lean{exercise\_4a}
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\end{proof}
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@ -42,8 +40,8 @@ $\floor{-x} =
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\ % Force space prior to *Proof.*
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\begin{enumerate}[(a)]
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\item \link{exercise\_4b\_1}
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\item \link{exercise\_4b\_2}
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\item \lean{exercise\_4b\_1}
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\item \lean{exercise\_4b\_2}
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\end{enumerate}
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\end{proof}
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@ -55,7 +53,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
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\begin{proof}
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\link{exercise\_4c}
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\lean{exercise\_4c}
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\end{proof}
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@ -66,7 +64,7 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
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\begin{proof}
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\link{exercise\_4d}
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\lean{exercise\_4d}
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\end{proof}
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@ -77,7 +75,7 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
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\begin{proof}
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\link{exercise\_4e}
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\lean{exercise\_4e}
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\end{proof}
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@ -92,7 +90,7 @@ State and prove such a generalization.
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\begin{proof}
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\link{exercise\_5}
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\lean{exercise\_5}
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\divider
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@ -223,7 +221,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
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\begin{proof}
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\link{exercise\_7b}
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\lean{exercise\_7b}
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\end{proof}
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@ -2,11 +2,9 @@
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\input{../../preamble}
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\newcommand{\link}[1]{\lean{../..}
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{Bookshelf/Apostol/Chapter\_I\_03} % Location
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{Apostol.Chapter\_I\_03.#1} % Fragment
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{Chapter\_I\_03.#1} % Presentation
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}
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\newcommand{\lean}[1]{\href
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{./Chapter\_I\_03.html\#Apostol.Chapter\_I\_03.#1}
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{Apostol.Chapter\_I\_03.#1}}
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\begin{document}
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@ -19,7 +17,7 @@ Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
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\begin{proof}
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\link{is\_lub\_neg\_set\_iff\_is\_glb\_set\_neg}
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\lean{is\_lub\_neg\_set\_iff\_is\_glb\_set\_neg}
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\divider
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@ -40,7 +38,7 @@ Every nonempty set $S$ that is bounded below has a greatest lower bound; that
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\begin{proof}
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\link{exists\_isGLB}
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\lean{exists\_isGLB}
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\divider
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@ -59,7 +57,7 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$.
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\begin{proof}
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\link{exists\_pnat\_geq\_self}
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\lean{exists\_pnat\_geq\_self}
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\divider
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@ -82,7 +80,7 @@ If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
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\begin{proof}
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\link{exists\_pnat\_mul\_self\_geq\_of\_pos}
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\lean{exists\_pnat\_mul\_self\_geq\_of\_pos}
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\divider
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@ -101,7 +99,7 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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\begin{proof}
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\link{forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
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\lean{forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
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\divider
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@ -144,7 +142,7 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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\begin{proof}
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\link{forall\_pnat\_frac\_leq\_self\_leq\_imp\_eq}
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\lean{forall\_pnat\_frac\_leq\_self\_leq\_imp\_eq}
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\divider
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@ -191,7 +189,7 @@ If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
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\begin{proof}
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\link{sup\_imp\_exists\_gt\_sup\_sub\_delta}
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\lean{sup\_imp\_exists\_gt\_sup\_sub\_delta}
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\divider
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@ -213,7 +211,7 @@ If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
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\begin{proof}
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\link{inf\_imp\_exists\_lt\_inf\_add\_delta}
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\lean{inf\_imp\_exists\_lt\_inf\_add\_delta}
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\divider
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@ -244,7 +242,7 @@ If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
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\begin{proof}
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\link{sup\_minkowski\_sum\_eq\_sup\_add\_sup}
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\lean{sup\_minkowski\_sum\_eq\_sup\_add\_sup}
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\divider
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@ -312,7 +310,7 @@ If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
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\begin{proof}
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\link{inf\_minkowski\_sum\_eq\_inf\_add\_inf}
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\lean{inf\_minkowski\_sum\_eq\_inf\_add\_inf}
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\divider
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@ -381,7 +379,7 @@ Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that $$s \leq t$$
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\begin{proof}
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\link{forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
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\lean{forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
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\divider
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@ -2,11 +2,9 @@
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\input{../../preamble}
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\newcommand{\link}[1]{\lean{../..}
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{Bookshelf/Enderton/Chapter\_0} % Location
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{Enderton.Chapter\_0.#1} % Fragment
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{Chapter\_0.#1} % Presentation
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}
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\newcommand{\lean}[1]{\href
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{./Chapter\_0.html\#Enderton.Chapter\_0.#1}
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{Enderton.Chapter\_0.#1}}
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\begin{document}
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@ -21,7 +19,7 @@ Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$.
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\begin{proof}
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\link{lemma\_0a}
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\lean{lemma\_0a}
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\end{proof}
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@ -1,12 +1,8 @@
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\documentclass{article}
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\input{../../preamble}
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\input{../../../preamble}
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\newcommand{\link}[2]{\lean{../..}
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{Common/Real/Geometry/Area} % Location
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{#1} % Fragment
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{#2} % Presentation
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}
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\newcommand{\lean}[2]{\href{./Area.html\##1}{#2}}
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\begin{document}
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@ -23,7 +19,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
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\begin{axiom}
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\link{Nonnegative-Property}{Nonnegative Property}
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\lean{Nonnegative-Property}{Nonnegative Property}
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\end{axiom}
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@ -35,7 +31,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
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\begin{axiom}
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\link{Additive-Property}{Additive Property}
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\lean{Additive-Property}{Additive Property}
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\end{axiom}
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@ -47,7 +43,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
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\begin{axiom}
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\link{Difference-Property}{Difference Property}
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\lean{Difference-Property}{Difference Property}
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\end{axiom}
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@ -59,7 +55,7 @@ If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
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\begin{axiom}
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\link{Invariant-Under-Congruence}{Invariance Under Congruence}
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\lean{Invariant-Under-Congruence}{Invariance Under Congruence}
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\end{axiom}
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@ -71,7 +67,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
|
|||
|
||||
\begin{axiom}
|
||||
|
||||
\link{Choice-of-Scale}{Choice of Scale}
|
||||
\lean{Choice-of-Scale}{Choice of Scale}
|
||||
|
||||
\end{axiom}
|
||||
|
||||
|
|
@ -90,7 +86,7 @@ If there is one and only one number $c$ which satisfies the inequalities
|
|||
|
||||
\begin{axiom}
|
||||
|
||||
\link{Exhaustion-Property}{Exhaustion Property}
|
||||
\lean{Exhaustion-Property}{Exhaustion Property}
|
||||
|
||||
\end{axiom}
|
||||
|
||||
|
|
|
|||
|
|
@ -2,11 +2,9 @@
|
|||
|
||||
\input{../../../preamble}
|
||||
|
||||
\newcommand{\link}[1]{\lean{../../..}
|
||||
{Common/Real/Sequence/Arithmetic} % Location
|
||||
{Real.Arithmetic.#1} % Fragment
|
||||
{Real.Arithmetic.#1} % Presentation
|
||||
}
|
||||
\newcommand{\lean}[1]{\href
|
||||
{./Arithmetic.html\#Real.Arithmetic.#1}
|
||||
{Real.Arithmetic.#1}}
|
||||
|
||||
\begin{document}
|
||||
|
||||
|
|
@ -19,7 +17,7 @@ $$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
|
|||
|
||||
\begin{proof}
|
||||
|
||||
\link{sum\_recursive\_closed}
|
||||
\lean{sum\_recursive\_closed}
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
|
|
|||
|
|
@ -2,11 +2,9 @@
|
|||
|
||||
\input{../../../preamble}
|
||||
|
||||
\newcommand{\link}[1]{\lean{../../..}
|
||||
{Common/Real/Sequence/Geometric} % Location
|
||||
{Real.Geometric.#1} % Fragment
|
||||
{Real.Geometric.#1} % Presentation
|
||||
}
|
||||
\newcommand{\lean}[1]{\href
|
||||
{./Geometric.html\#Real.Geometric.#1}
|
||||
{Real.Geometric.#1}}
|
||||
|
||||
\begin{document}
|
||||
|
||||
|
|
@ -19,7 +17,7 @@ $$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
|
|||
|
||||
\begin{proof}
|
||||
|
||||
\link{sum\_recursive\_closed}
|
||||
\lean{sum\_recursive\_closed}
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
|
|
|||
15
README.md
15
README.md
|
|
@ -30,11 +30,10 @@ project.
|
|||
|
||||
A color/symbol code is used on generated PDF headers to indicate their status:
|
||||
|
||||
* Teal coloring (with a checkmark) indicates the corresponding proof is
|
||||
complete. That is, the proof has been written in TeX and also formally
|
||||
verified in Lean.
|
||||
* Magenta coloring (with a spinner) indicates the corresponding proof is in
|
||||
progress. That is, a proof in both TeX and Lean have not yet been finished,
|
||||
but is actively being worked on.
|
||||
* Red coloring (with a warning) indicates the formal Lean proof has not yet been
|
||||
started. It may or may not also indicate the TeX proof has been written.
|
||||
* Teal coloring indicates the corresponding proof is complete. That is, the
|
||||
proof has been written in TeX and also formally verified in Lean.
|
||||
* Magenta coloring indicates the corresponding proof is in progress. That is, a
|
||||
proof in both TeX and Lean have not yet been finished, but is actively being
|
||||
worked on.
|
||||
* Red coloring indicates the formal Lean proof has not yet been started. It may
|
||||
or may not also indicate the TeX proof has been written.
|
||||
|
|
|
|||
|
|
@ -16,9 +16,9 @@
|
|||
{"git":
|
||||
{"url": "https://github.com/jrpotter/bookshelf-docgen.git",
|
||||
"subDir?": null,
|
||||
"rev": "80a4dc8d508161c859dbbb455f3855e051a28890",
|
||||
"rev": "3f941dc8a814321498082da49f4a8430bbfbbb6c",
|
||||
"name": "doc-gen4",
|
||||
"inputRev?": "80a4dc8d508161c859dbbb455f3855e051a28890"}},
|
||||
"inputRev?": "3f941dc8a814321498082da49f4a8430bbfbbb6c"}},
|
||||
{"git":
|
||||
{"url": "https://github.com/mhuisi/lean4-cli",
|
||||
"subDir?": null,
|
||||
|
|
|
|||
|
|
@ -12,7 +12,7 @@ require std4 from git
|
|||
"6006307d2ceb8743fea7e00ba0036af8654d0347"
|
||||
require «doc-gen4» from git
|
||||
"https://github.com/jrpotter/bookshelf-docgen.git" @
|
||||
"80a4dc8d508161c859dbbb455f3855e051a28890"
|
||||
"3f941dc8a814321498082da49f4a8430bbfbbb6c"
|
||||
|
||||
@[default_target]
|
||||
lean_lib «Bookshelf» {
|
||||
|
|
|
|||
17
preamble.tex
17
preamble.tex
|
|
@ -3,23 +3,18 @@
|
|||
\usepackage{environ}
|
||||
\usepackage{fancybox}
|
||||
\usepackage{fontawesome5}
|
||||
\usepackage{hyperref}
|
||||
\usepackage{mathrsfs}
|
||||
\usepackage{soul, xcolor}
|
||||
\usepackage{soul}
|
||||
\usepackage{xcolor}
|
||||
% `hyperref` comes after `xr-hyper`.
|
||||
\usepackage{xr-hyper}
|
||||
\usepackage{hyperref}
|
||||
|
||||
% ========================================
|
||||
% Linking
|
||||
% ========================================
|
||||
|
||||
\hypersetup{colorlinks=true, urlcolor=blue}
|
||||
|
||||
% The first argument refers to a relative path upward from a current file to
|
||||
% the root of the workspace (i.e. where this `preamble.tex` file is located).
|
||||
% #1 - Path to root
|
||||
% #2 - Location
|
||||
% #3 - Fragment
|
||||
% #4 - Presentation
|
||||
\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}
|
||||
\newcommand{\hyperlabel}[1]{%
|
||||
\label{#1}%
|
||||
\hypertarget{#1}{}}
|
||||
|
|
@ -56,7 +51,7 @@
|
|||
\DeclareRobustCommand{\verified}[1]{%
|
||||
\texorpdfstring{\color{teal}#1\ \faCheckCircle}{#1}}
|
||||
\DeclareRobustCommand{\proceeding}[1]{%
|
||||
\texorpdfstring{\color{magenta}#1\ \faSpinner}{#1}}
|
||||
\texorpdfstring{\color{magenta}#1\ \faDotCircle[regular]}{#1}}
|
||||
\DeclareRobustCommand{\unverified}[1]{%
|
||||
\texorpdfstring{\color{red}#1\ \faExclamationCircle}{#1}}
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue