Add icon to distinguish Lean definitions from custom ones.
Update to pending any proofs that were using already defined Lean proofs.finite-set-exercises
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@ -32,7 +32,7 @@ The \textbf{characteristic function} of $S$ is the function $\mathcal{X}_S$ such
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\begin{definition}
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\lean*{Common/Set/Basic}{Set.characteristic}
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\code*{Common/Set/Basic}{Set.characteristic}
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\end{definition}
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@ -160,7 +160,7 @@ A collection of points satisfying \eqref{sec:partition-eq1} is called a
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\begin{definition}
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\lean*{Common/Set/Partition}{Set.Partition}
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\code*{Common/Set/Partition}{Set.Partition}
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\end{definition}
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@ -192,7 +192,7 @@ That is to say, for each $k = 1, 2, \ldots, n$, there is a real number $s_k$
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\begin{definition}
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\lean*{Common/Geometry/StepFunction}{Geometry.StepFunction}
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\code*{Common/Geometry/StepFunction}{Geometry.StepFunction}
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\end{definition}
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@ -239,7 +239,7 @@ It is denoted as $\bar{I}(f)$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.is\_lub\_neg\_set\_iff\_is\_glb\_set\_neg}
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Suppose $L = \sup{S}$ and fix $x \in S$.
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@ -264,7 +264,7 @@ It is denoted as $\bar{I}(f)$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.exists\_isGLB}
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Let $S$ be a nonempty set bounded below by $x$.
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@ -288,7 +288,7 @@ It is denoted as $\bar{I}(f)$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.exists\_pnat\_geq\_self}
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Let $n = \abs{\ceil{x}} + 1$.
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@ -313,7 +313,7 @@ It is denoted as $\bar{I}(f)$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.exists\_pnat\_mul\_self\_geq\_of\_pos}
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Let $x > 0$ and $y$ be an arbitrary real number.
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@ -336,7 +336,7 @@ It is denoted as $\bar{I}(f)$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
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By the trichotomy of the reals, there are three cases to consider:
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@ -382,7 +382,7 @@ It is denoted as $\bar{I}(f)$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.forall\_pnat\_frac\_leq\_self\_leq\_imp\_eq}
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By the trichotomy of the reals, there are three cases to consider:
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@ -432,7 +432,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.sup\_imp\_exists\_gt\_sup\_sub\_delta}
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By definition of a \nameref{ref:supremum}, $\sup{S}$ is the least upper
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@ -458,7 +458,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.inf\_imp\_exists\_lt\_inf\_add\_delta}
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By definition of an \nameref{ref:infimum}, $\inf{S}$ is the greatest lower
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@ -496,7 +496,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.sup\_minkowski\_sum\_eq\_sup\_add\_sup}
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We prove (i) $\sup{A} + \sup{B}$ is an upper bound of $C$ and (ii)
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@ -567,7 +567,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.inf\_minkowski\_sum\_eq\_inf\_add\_inf}
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We prove (i) $\inf{A} + \inf{B}$ is a lower bound of $C$ and (ii)
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@ -639,7 +639,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\code{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
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By hypothesis, $S$ and $T$ are nonempty sets.
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@ -678,7 +678,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
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\begin{axiom}
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\leanp*{Common/Geometry/Area}{Nonnegative-Property}
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\codep*{Common/Geometry/Area}{Nonnegative-Property}
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{Nonnegative Property}
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\end{axiom}
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@ -691,7 +691,7 @@ 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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\leanp*{Common/Geometry/Area}{Additive-Property}
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\codep*{Common/Geometry/Area}{Additive-Property}
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{Additive Property}
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\end{axiom}
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@ -704,7 +704,7 @@ 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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\leanp*{Common/Geometry/Area}{Difference-Property}
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\codep*{Common/Geometry/Area}{Difference-Property}
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{Difference Property}
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\end{axiom}
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@ -717,7 +717,7 @@ 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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\leanp*{Common/Geometry/Area}{Invariance-Under-Congruence}
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\codep*{Common/Geometry/Area}{Invariance-Under-Congruence}
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{Invariance Under Congruence}
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\end{axiom}
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@ -730,7 +730,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
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\begin{axiom}
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\leanp*{Common/Geometry/Area}{Choice-of-Scale}
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\codep*{Common/Geometry/Area}{Choice-of-Scale}
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{Choice of Scale}
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\end{axiom}
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@ -750,7 +750,7 @@ If there is one and only one number $c$ which satisfies the inequalities
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\begin{axiom}
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\leanp*{Common/Geometry/Area}{Exhaustion-Property}
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\codep*{Common/Geometry/Area}{Exhaustion-Property}
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{Exhaustion Property}
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\end{axiom}
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@ -1340,7 +1340,7 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_1\_11}
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\code{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_4a}
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Let $x$ be a real number and $n$ an integer.
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@ -1366,10 +1366,10 @@ $\floor{-x} =
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\statementpadding
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\lean*{Bookshelf/Apostol/Chapter\_1\_11}
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\code*{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_4b\_1}
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\lean{Bookshelf/Apostol/Chapter\_1\_11}
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\code{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_4b\_2}
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There are two cases to consider:
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@ -1409,7 +1409,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_1\_11}
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\code{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_4c}
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Rewrite $x$ and $y$ as the sum of their floor and fractional components:
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@ -1453,7 +1453,7 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_1\_11}
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\code{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_4d}
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This is immediately proven by applying \nameref{sub:hermites-identity}.
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@ -1467,7 +1467,7 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_1\_11}
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\code{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_4e}
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This is immediately proven by applying \nameref{sub:hermites-identity}.
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@ -1484,7 +1484,7 @@ State and prove such a generalization.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_1\_11}
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\code{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_5}
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We prove that for all natural numbers $n$ and real numbers $x$, the following
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@ -1710,7 +1710,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
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\begin{proof}
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\lean{Bookshelf/Apostol/Chapter\_1\_11}
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\code{Bookshelf/Apostol/Chapter\_1\_11}
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{Apostol.Chapter\_1\_11.exercise\_7b}
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Let $n = 1, \ldots, b - 1$.
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File diff suppressed because it is too large
Load Diff
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@ -9,6 +9,34 @@ Axioms and Operations
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namespace Enderton.Set.Chapter_2
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/-! #### Commutative Laws
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For any sets `A` and `B`,
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```
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A ∪ B = B ∪ A
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A ∩ B = B ∩ A
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```
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-/
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#check Set.union_comm
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theorem commutative_law_i (A B : Set α)
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: A ∪ B = B ∪ A := calc A ∪ B
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_ = { x | x ∈ A ∨ x ∈ B } := rfl
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_ = { x | x ∈ B ∨ x ∈ A } := by
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ext
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exact or_comm
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_ = B ∪ A := rfl
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#check Set.inter_comm
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theorem commutative_law_ii (A B : Set α)
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: A ∩ B = B ∩ A := calc A ∩ B
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_ = { x | x ∈ A ∧ x ∈ B } := rfl
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_ = { x | x ∈ B ∧ x ∈ A } := by
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ext
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exact and_comm
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_ = B ∩ A := rfl
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/-- #### Exercise 2.1
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@ -24,7 +24,7 @@ Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
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\begin{proof}
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\lean{Common/Real/Sequence/Arithmetic}
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\code{Common/Real/Sequence/Arithmetic}
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{Real.Arithmetic.sum\_recursive\_closed}
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Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
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@ -84,7 +84,7 @@ Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
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\begin{proof}
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\lean{Common/Real/Sequence/Geometric}
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\code{Common/Real/Sequence/Geometric}
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{Real.Geometric.sum\_recursive\_closed}
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Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
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@ -40,28 +40,26 @@ def index : BaseHtmlM Html := do templateExtends (baseHtml "Index") <|
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<li>
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<span style="color:darkgray">Dark gray statements </span> are
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reserved for definitions and axioms that have been encoded in LaTeX.
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A corresponding reference to a definition/axiom in Lean may also be
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provided.
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A reference to a definition in Lean may also be provided.
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</li>
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<li>
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<span style="color:teal">Teal statements </span> are reserved for
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statements, theorems, lemmas, etc. that have been proven in both
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LaTeX and Lean.
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statements, theorems, lemmas, etc. that have been proven in LaTeX
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and have a corresponding proof in Lean.
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</li>
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<li>
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<span style="color:olive">Olive statements </span> are reserved for
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statements, theorems, lemmas, etc. that have been proven in LaTeX
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and will <i>not </i> be proven in Lean.
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statements, theorems, lemmas, etc. that have been proven in LaTeX.
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A reference to a statement in Lean may also be provided.
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</li>
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<li>
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<span style="color:fuchsia">Fuchsia statements </span> are reserved
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for definitions, axioms, statements, theorems, lemmas, etc. that
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have been proven or encoded in LaTeX and <i>will </i> be encoded in
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Lean.
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for statements, theorems, lemmas, etc. that have been proven in
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LaTeX and <i>will </i> have a corresponding proof in Lean.
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</li>
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<li>
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<span style="color:maroon">Maroon </span> serves as a catch-all for
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all statements that don't fit the above categorizations. Incomplete
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statements that don't fit the above categorizations. Incomplete
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definitions, statements without proof, etc. belong here.
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</li>
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</ul>
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38
preamble.tex
38
preamble.tex
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@ -9,7 +9,9 @@
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\usepackage{mathabx, mathrsfs}
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\usepackage{soul}
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\usepackage{stmaryrd}
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\usepackage[usenames,dvipsnames]{xcolor}
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% Must load `xcolor` before `tikz`.
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\usepackage[dvipsnames]{xcolor}
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\usepackage{tikz}
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% `hyperref` comes after `xr-hyper`.
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\usepackage{xr-hyper}
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\usepackage{hyperref}
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@ -42,8 +44,17 @@
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\label{#1}%
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\hypertarget{#1}{}}
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% Denote whether we are working with a standard/Mathlib statement (lean) or a
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% custom one (code).
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\newcommand\@leanlink[4]{%
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\textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1/#2.html\##3}{#4}}
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\textcolor{blue}{\raisebox{-4.5pt}{%
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\tikz{\draw (0, 0) node[yscale=-1,xscale=1] {\faFont};}}}%
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{-\;}\href{#1/#2.html\##3}{#4}}
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\newcommand\@codelink[4]{%
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\textcolor{blue}{\raisebox{-4.5pt}{%
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\tikz{\draw (0, 0) node[] {\faCodeBranch};}}}%
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{-\;}\href{#1/#2.html\##3}{#4}}
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% Reference to an anchor of Lean documentation.
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\newcommand\leanref[3]{%
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@ -51,22 +62,43 @@
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\WithSuffix\newcommand\leanref*[3]{%
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\@leanlink{#1}{#2}{#3}{#3}}
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% Variant that allows customizing display text.
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\newcommand\coderef[3]{%
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\@codelink{#1}{#2}{#3}{#3}\vspace{10pt}}
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\WithSuffix\newcommand\coderef*[3]{%
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\@codelink{#1}{#2}{#3}{#3}}
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% Variants that allows customizing display text.
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\newcommand\leanpref[4]{%
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\@leanlink{#1}{#2}{#3}{#4}\vspace{10pt}}
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\WithSuffix\newcommand\leanpref*[4]{%
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\@leanlink{#1}{#2}{#3}{#4}}
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\newcommand\codepref[4]{%
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\@codelink{#1}{#2}{#3}{#4}\vspace{10pt}}
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\WithSuffix\newcommand\codepref*[4]{%
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\@codelink{#1}{#2}{#3}{#4}}
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% Macro to build all Lean related commands relative to a specified directory.
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\newcommand\makeleancommands[1]{%
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\newcommand\lean[2]{%
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\leanref{#1}{##1}{##2}}
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\WithSuffix\newcommand\lean*[2]{%
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\leanref*{#1}{##1}{##2}}
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\newcommand\code[2]{%
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\coderef{#1}{##1}{##2}}
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\WithSuffix\newcommand\code*[2]{%
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\coderef*{#1}{##1}{##2}}
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\newcommand\leanp[3]{%
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\leanpref{#1}{##1}{##2}{##3}}
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\WithSuffix\newcommand\leanp*[3]{%
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\leanpref*{#1}{##1}{##2}{##3}}
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\newcommand\codep[3]{%
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\codepref{#1}{##1}{##2}{##3}}
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\WithSuffix\newcommand\codep*[3]{%
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\codepref*{#1}{##1}{##2}{##3}}
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}
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||||
|
||||
% ========================================
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|
|
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Loading…
Reference in New Issue