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21 | 21 | <pre class="alectryon-io"><!-- Generator: Alectryon --><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk0" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk0"><span class="highlight"><span class="kn">Lemma</span> <span class="nf">Gauss</span>: <span class="kr">∀</span> <span class="nv">n</span>, |
22 | 22 | <span class="mi">2</span> * (nsum n (<span class="kr">fun</span> <span class="nv">x</span> => x)) = |
23 | 23 | n * (n + <span class="mi">1</span>).</span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="kr">∀</span> <span class="nv">n</span> : nat, <span class="mi">2</span> * nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x) = n * (n + <span class="mi">1</span>)</span></div></blockquote></div></div></small><span class="alectryon-wsp"> |
24 | | -</span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk1" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk1"><span class="highlight"><span class="nb">induction</span> n; <span class="nb">cbn</span> [nsum].</span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * <span class="mi">0</span> = <span class="mi">0</span> * (<span class="mi">0</span> + <span class="mi">1</span>)</span></div></blockquote><div class="alectryon-extra-goals"><input class="alectryon-extra-goal-toggle" id="hyps-rst-chk2" style="display: none" type="checkbox"><blockquote class="alectryon-goal"><div class="goal-hyps"><div><var>n</var><span><span class="hyp-type"><b>:</b><span class="highlight">nat</span></span></span></div><div><var>IHn</var><span><span class="hyp-type"><b>:</b><span class="highlight"><span class="mi">2</span> * nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x) = n * (n + <span class="mi">1</span>)</span></span></span></div></div><label class="goal-separator" for="hyps-rst-chk2"><hr></label><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * (S n + nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x)) = S n * (S n + <span class="mi">1</span>)</span></div></blockquote></div></div></div></small><span class="alectryon-wsp"> |
| 24 | +</span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk1" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk1"><span class="highlight"><span class="nb">induction</span> n; <span class="nb">cbn</span> [nsum].</span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * <span class="mi">0</span> = <span class="mi">0</span> * (<span class="mi">0</span> + <span class="mi">1</span>)</span></div></blockquote><div class="alectryon-extra-goals"><input class="alectryon-extra-goal-toggle" id="hyps-rst-chk2" style="display: none" type="checkbox"><blockquote class="alectryon-goal"><div class="goal-hyps"><div><var>n</var><span class="hyp-type"><b>:</b><span class="highlight">nat</span></span></div><div><var>IHn</var><span class="hyp-type"><b>:</b><span class="highlight"><span class="mi">2</span> * nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x) = n * (n + <span class="mi">1</span>)</span></span></div></div><label class="goal-separator" for="hyps-rst-chk2"><hr></label><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * (S n + nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x)) = S n * (S n + <span class="mi">1</span>)</span></div></blockquote></div></div></div></small><span class="alectryon-wsp"> |
25 | 25 | </span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk3" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk3"><span class="highlight">- <span class="c">(* n ← 0 *)</span></span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * <span class="mi">0</span> = <span class="mi">0</span> * (<span class="mi">0</span> + <span class="mi">1</span>)</span></div></blockquote></div></div></small><span class="alectryon-wsp"> |
26 | 26 | </span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><span class="alectryon-input"><span class="highlight"><span class="bp">reflexivity</span>.</span></span><span class="alectryon-wsp"> |
27 | | -</span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk4" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk4"><span class="highlight">- <span class="c">(* n ← S _ *)</span></span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><div class="goal-hyps"><div><var>n</var><span><span class="hyp-type"><b>:</b><span class="highlight">nat</span></span></span></div><div><var>IHn</var><span><span class="hyp-type"><b>:</b><span class="highlight"><span class="mi">2</span> * nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x) = n * (n + <span class="mi">1</span>)</span></span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * (S n + nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x)) = S n * (S n + <span class="mi">1</span>)</span></div></blockquote></div></div></small><span class="alectryon-wsp"> |
| 27 | +</span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk4" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk4"><span class="highlight">- <span class="c">(* n ← S _ *)</span></span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><div class="goal-hyps"><div><var>n</var><span class="hyp-type"><b>:</b><span class="highlight">nat</span></span></div><div><var>IHn</var><span class="hyp-type"><b>:</b><span class="highlight"><span class="mi">2</span> * nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x) = n * (n + <span class="mi">1</span>)</span></span></div></div><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * (S n + nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x)) = S n * (S n + <span class="mi">1</span>)</span></div></blockquote></div></div></small><span class="alectryon-wsp"> |
28 | 28 | </span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk5" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk5"><span class="highlight"><span class="nb">rewrite</span> Mult.mult_plus_distr_l.</span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * S n + <span class="mi">2</span> * nsum n (<span class="kr">λ</span> <span class="nv">x</span> : nat, x) = S n * (S n + <span class="mi">1</span>)</span></div></blockquote></div></div></small><span class="alectryon-wsp"> |
29 | 29 | </span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><input checked="checked" class="alectryon-toggle" id="hyps-rst-chk6" style="display: none" type="checkbox"><label class="alectryon-input" for="hyps-rst-chk6"><span class="highlight"><span class="nb">rewrite</span> IHn.</span></label><small class="alectryon-output"><div><div class="alectryon-goals"><blockquote class="alectryon-goal"><span class="goal-separator"><hr></span><div class="goal-conclusion"><span class="highlight"><span class="mi">2</span> * S n + n * (n + <span class="mi">1</span>) = S n * (S n + <span class="mi">1</span>)</span></div></blockquote></div></div></small><span class="alectryon-wsp"> |
30 | 30 | </span></span><span class="alectryon-wsp"> </span><span class="alectryon-sentence"><span class="alectryon-input"><span class="highlight"><span class="bp">ring</span>.</span></span><span class="alectryon-wsp"> |
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