-
Notifications
You must be signed in to change notification settings - Fork 12
/
index.html
1561 lines (1550 loc) · 58.7 KB
/
index.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!-- @format -->
<!doctype html>
<html lang="en">
<head>
<title>Lurch Plus!</title>
<meta charset="utf-8" />
<meta
name="description"
content="This is the home page for lurch.plus, the home of the Lurch proof assistant, plus additional content." />
<meta
name="keywords"
content="Lurch, Lurch plus, proof assistant, math academy, summer courses, math proof camp, math proofs, mathematical proof, proof class, proof course, summer math camp, summer math program, summer math class, math camp, math program, math class, MATHCOUNTS, AMC, AIME, USAMO, IMO, ARML, math contest training, math competition training, math olympiad, AP olympiad, Intel Science, ISEF, ISTS, Regeneron competition, Siemens competition, math gifted, summer school program, camps for kids, academic summer camp, summer academy, summer schools, summer school usa, summer academic programs, summer classes, academy of mathematics, university of scranton, Lehigh Valley ARML" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<link rel="shortcut icon" href="./assets/media/favicon-L+.svg" />
<link
href="//fonts.googleapis.com/css?family=Open+Sans:400,600,400italic,700,300"
rel="stylesheet"
type="text/css" />
<link
href="https://fonts.googleapis.com/css2?family=Merriweather:wght@300;400;700&display=swap"
rel="stylesheet" />
<link rel="stylesheet" href="./assets/css/monks.css" />
<link rel="stylesheet" href="./assets/css/lurchplus.css" />
<!-- <link rel='stylesheet' href='assets/css/bootstrap.min.css'> -->
</head>
<body>
<div id="leftnav">
<img
src="./assets/media/lurchlogo.png"
style="margin: 0px; border-radius: 5px; display: block"
width="190px" />
<ul>
<li><a href="#">Home</a></li>
<li><a href="#documentation">Documentation</a></li>
<li><a href="./100" target="_self">100 Theorem Challenge</a></li>
<li><a href="#libraries">Rule Libraries</a></li>
<li><a href="#contexts">Other Contexts</a></li>
<li><a href="#assignments">Example Assignments</a></li>
<li><a href="#lurchforstudents">Lurch for Students</a></li>
<li><a href="#lurchforinstructors">Lurch for Instructors</a></li>
<li><a href="#talkslides">Talk Slides</a></li>
<li><a href="#getlurch">Get Lurch!</a></li>
<li><a href="./about" target="_self">About Lurch</a></li>
</ul>
<div id="contact">
<div>
<a href="mailto:[email protected]">Ken Monks</a>
</div>
<div>
<a href="https://monks.scranton.edu" target="_blank">
monks.scranton.edu
</a>
</div>
<div>
<a href="https://proveitmath.org" target="_blank">proveitmath.org</a>
</div>
</div>
</div>
<!-- flex column for content-block and footer-block -->
<div id="wrap">
<div id="content-block">
<div class="title-box">
<h1>Proof Verification with Lurch</h1>
<h2>Building the bridge to higher mathematics</h2>
</div>
<div class="haiku">
<p>
Is my proof correct?
<br />
Lurch says it is not convinced.
<br />
I need to say more.
</p>
<p class="attribution">Anonymous, 2024</p>
</div>
<p>
Lurch is an open source math editor that can check your proofs. It was
used successfully in my four credit undergraduate bridge course for
math and math education majors. Instructors who would like to create
their own Lurch course materials using the rule libraries and content
from my course can find updated and improved supporting materials
below and on this site.
</p>
<!-- ----------------------------------------------------- -->
<div id="video-container">
<video autoplay muted loop>
<source src="./assets/media/splash.mp4" type="video/mp4" />
Your browser does not support the video tag.
</video>
</div>
<!-- ----------------------------------------------------- -->
<h2 id="documentation">Documentation and Examples</h2>
<ul class="disc strong-tags dash-after">
<li>
<strong>Introductory Worksheets</strong>
<ul>
<li>
<a href="./student.html?load=help/quick-start-guide.lurch">
1. Quick Start Guide
</a>
how to use the student version of Lurch
</li>
<li>
<a href="./student.html?load=help/proofs-worksheet.lurch">
2. Doing Proofs in Lurch
</a>
introduction to proofs in Lurch (
<a href="./student.html?load=help/proofs-worksheet-solns.lurch">
solutions
</a>
)
</li>
<li>
<a
href="./instructor.html?load=help/instructors-worksheet.lurch">
3. Lurch for instructors
</a>
additional features of Lurch for instructor
</li>
</ul>
</li>
<li>
<strong>Examples</strong>
– a few proofs done in Lurch using the topics defined below.
<ul class="circle">
<li>
<a href="./student.html?load=help/example-proofs.lurch">
Example Proofs
</a>
a sample of some proofs in Lurch using a variety of topics and
styles.
</li>
<li>
<a
href="./student.html?load=math/examples/Mar%2026%20-%20in%20class.lurch">
Double induction
</a>
a proof of a binomial coefficient identity done in class using
double induction (and the recursive definition of binomial
coefficients given in the course). Colored fonts are used to
help keep track of which induction is which.
</li>
<li>
<a
href="./student.html?load=math/examples/Topo-logical%20-%20soln.lurch">
Topo-logical
</a>
a proof that the composition of continuous functions is
continuous in Topology.
<br />
(spoiler alert: this is the solution to part D of the Final Exam
below)
</li>
<li>
<a href="./100">100 Famous Theorems Challenge!</a>
Lurch's status in verifying the proofs of theorems on the 100
famous theorems challenge list. Game on!
</li>
</ul>
</li>
<li>
<strong>Final Exam!</strong>
– try your hand at the Math 299 Final Exam from Spring 2024 (four
parts)
<ul>
<li class="nodash">
<a
href="./student.html?load=math/examples/A.%20The%20Logic%20of%20Love.lurch">
A. The Logic of Love
</a>
</li>
<li class="nodash">
<a
href="./student.html?load=math/examples/B.%20Nostalgia%20for%20Calculus.lurch">
B. Nostalgia for Calculus
</a>
</li>
<li class="nodash">
<a
href="./student.html?load=math/examples/C.%20Putnam%20Practice.lurch">
C. Putnam Practice
</a>
</li>
<li class="nodash">
<a
href="./student.html?load=math/examples/D.%20Topo-logical.lurch">
D. Topo-logical
</a>
</li>
</ul>
</li>
<li>
<strong>Lurch Deductive Engine</strong>
– the nitty gritty details under the hood of the validation
algorithm used by the Lurch Deductive Engine for developers (not up
to date, but mostly correct)
<ul class="circle">
<li>
<a href="./instructor.html?load=help/theory.lurch">
Validation in Lurch
</a>
introduction to the Global $n$-compact validation algorithm used
in this version of Lurch.
</li>
<li>
<a href="./lde/src/experimental/docs">
Validation Algorithm API docs
</a>
the source code documentation for the Global $n$-compact
algorithm and supporting infrastructure
</li>
<li>
<a href="lde/docs">Core API docs</a>
the LC data structure, putdown notation, Matching package, and
other core utilties
</li>
<li>
<a href="https://github.com/kenmonks/lurch">
Source Code on GitHub
</a>
the source code and content of this repository on GitHub
</li>
</ul>
</li>
<li>
<a href="./lde/src/experimental/parsers/lurch-parser-docs.html">
Lurch Syntax
</a>
a quick reference showing how to type various math expressions in
Lurch
</li>
<li>
<a href="https://monks.scranton.edu/math299">Math 299 Home Page</a>
for my Introduction to Mathematical Proof course at the University
of Scranton, Spring 2024.
</li>
</ul>
<div class="">
<!-- ----------------------------------------------------- -->
<h2 id="libraries">Math 299 Lurch Rule Libraries</h2>
<div class="blue-box">
<p>
<strong class="bigfont">Cumulative Topics</strong>
– each link one opens a blank Lurch document whose context
consists of the rules for that topic, and those from the topics
above it. The Theorem numbers refer to exercises in the draft of
the
<a
href="https://monks.scranton.edu/files/courses/Math299/math-299-lecture.pdf">
course lecture notes
</a>
.
</p>
<dl>
<dt>
<a href="./student.html?load=math/Prop.lurch">
Propositional Logic
</a>
</dt>
<dd>
defines
<code>and</code>
,
<code>or</code>
,
<code>not</code>
,
<code>implies</code>
,
<code>iff</code>
, and
<code>contradiction</code>
(
<a href="./instructor.html?load=math/Prop-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Pred.lurch">
Predicate Logic with Equality
</a>
</dt>
<dd>
defines
<code>forall</code>
$\left(\forall\right)$,
<code>exists</code>
$\left(\exists\right)$,
<code>equality</code>
$\left(=\right)$,
<code>unique existence</code>
$\left(\exists!\right)$ (
<a href="./instructor.html?load=math/Pred-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Logic-theorems.lurch">
Logic Theorems
</a>
</dt>
<dd>
provides some common theorems from Logic (
<a
href="./instructor.html?load=math/Logic-theorems-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Peano.lurch">
Natural Numbers
</a>
</dt>
<dd>
the Peano Axioms for the Natural Numbers (
<a href="./instructor.html?load=math/Peano-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Number-theory.lurch">
Number Theory Definitions
</a>
</dt>
<dd>
defines,
<code>less than</code>
$\left(\lt\right)$,
<code>divides</code>
$\left(\mid\right)$,
<code>prime</code>
,
<code>even</code>
,
<code>odd</code>
(
<a href="./instructor.html?load=math/Number-theory-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Equations.lurch">Equations</a>
</dt>
<dd>
defines a single ‘rule’ that enables Lurch to validate
transitive chains of equalities without needing substitution,
reflexivity, symmetry, or transitivity (
<a href="./instructor.html?load=math/Equations-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Number-theory-theorems.lurch">
Number Theory Theorems
</a>
</dt>
<dd>
provides some useful theorems from Number Theory that are
provable from the Peano Axioms (
<a
href="./instructor.html?load=math/Number-theory-theorems-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Sequences.lurch">
Sequences and Recursion
</a>
</dt>
<dd>
defines
<code>summation</code>
$\left(\sum\right)$,
<code>Fibonnaci numbers</code>
$\left(F_n\right)$,
<code>factorial</code>
$\left(!\right)$,
<code>multinomial coefficients</code>
$\binom{n}{k}$,
<code>binomial coefficients</code>
,
<code>exponentiation</code>
$\left(z^n\right)$ (
<a href="./instructor.html?load=math/Sequences-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Reals.lurch">Real Numbers</a>
</dt>
<dd>
defines the field axioms for the Real Numbers (
<a href="./instructor.html?load=math/Reals-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Sets.lurch">Set Theory</a>
</dt>
<dd>
defines
<code>in</code>
$\left(\in\right)$,
<code>subset</code>
$\left(\subseteq\right)$,
<code>intersection</code>
$\left(\cap\right)$,
<code>union</code>
$\left(\cup\right)$,
<code>complement</code>
$\left(‘\right)$,
<code>set difference</code>
$\left(\setminus\right)$,
<code>powerset</code>
$\left(\mathscr{P}\right)$,
<code>Cartesian product</code>
$\left(\times\right)$,
<code>finite set</code>
$\left({ \ldots }\right)$,
<code>tuple</code>
$\langle\ldots\rangle$,
<code>Indexed Union</code>
$\left(\bigcup\right)$,
<code>Indexed Intersection</code>
$\left(\bigcap\right)$,
<code>Set Builder notation</code>
$\left({ z :\ldots}\right)$ (
<a href="./instructor.html?load=math/Sets-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Functions.lurch">Functions</a>
</dt>
<dd>
defines
<code>maps</code>
$\left(f\colon A \to B\right)$,
<code>function application</code>
$\left(f(x)\right)$,
<code>maps to</code>
$\left(\mapsto\right)$,
<code>image</code>
$\left(f(U)\right)$,
<code>identity map</code>
$\left(\text{id}_A\right)$
<code>inverse image</code>
$\left(f^\text{inv}(U)\right)$,
<code>composition</code>
$\left(\circ\right)$,
<code>inverse function</code>
$\left(f^{-1}\right)$,
<code>injective</code>
,
<code>surjective</code>
,
<code>bijective</code>
(
<a href="./instructor.html?load=math/Functions-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Relations.lurch">Relations</a>
</dt>
<dd>
defines
<code>reflexive</code>
,
<code>symmetric</code>
,
<code>transitive</code>
,
<code>irreflexive</code>
,
<code>antisymmetric</code>
,
<code>total</code>
,
<code>partial order</code>
,
<code>strict partial order</code>
,
<code>total order</code>
,
<code>equivalence relation</code>
,
<code>partition</code>
,
<code>equivalence class</code>
$[a]$ (
<a href="./instructor.html?load=math/Relations-Rules.lurch">
view rules
</a>
)
</dd>
</dl>
</div>
<!-- blue-box -->
<!-- ----------------------------------------------------- -->
<h2 id="contexts">Other Useful Contexts</h2>
<div class="blue-box">
<p>
<strong>Additional Rules.</strong>
The following rules can be optionally added by the instructor to
any context above in a particular document. Currently, only one of
the three Arithmetic rules can be used in a single document.
</p>
<dl>
<dt>
<a href="./student.html?load=math/Arithmetic-naturals.lurch">
Arithmetic in the Natural Numbers
</a>
</dt>
<dd>
Allows the user to say
<code>'by</code>
<code>arithmetic'</code>
after an expression to try to justify it using a CAS. The
expression can only contain natural number constants (e.g. $0$,
$1$, $2$, etc) the operators $+$, $\cdot$, and $\text{^}$, and
the relations $=$, $\leq$, and $\lt$. (
<a
href="./instructor.html?load=math/Arithmetic-naturals-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Arithmetic-integers.lurch">
Arithmetic in the Integers
</a>
</dt>
<dd>
Allows the user to say
<code>'by</code>
<code>arithmetic'</code>
after an expression to try to justify it using a CAS. The
expression can only contain natural number constants (e.g. $0$,
$1$, $2$, etc) the operators $+$, $\cdot$, $\text{^}$, and $-$
and the relations $=$, $\leq$, and $\lt$. Exponents cannot be
negative. (
<a
href="./instructor.html?load=math/Arithmetic-integers-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Arithmetic-rationals.lurch">
Arithmetic in the Rationals
</a>
</dt>
<dd>
Allows the user to say
<code>'by</code>
<code>arithmetic'</code>
after an expression to try to justify it using a CAS. The
expression can only contain natural number constants (e.g. $0$,
$1$, $2$, etc) the operators $+$, $\cdot$, $\text{^}$, $-$, and
$/$, and the relations $=$, $\leq$, and $\lt$. Exponents must be
integers and you cannot divide by zero. (
<a
href="./instructor.html?load=math/Arithmetic-rationals-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Algebra.lurch">Algebra</a>
</dt>
<dd>
Allows the user to say
<code>'by</code>
<code>algebra'</code>
after an expression to try to justify it using a CAS. There are
no restrictions on what the expression can contain, so use at
your own discretion. The identity is evaluated using
<a href="http://algebrite.org">Algebrite</a>
. (
<a href="./instructor.html?load=math/Algebra-Rules.lurch">
view rules
</a>
)
</dd>
</dl>
<p>
<strong>Useful Cumulative Contexts.</strong>
These contexts combine some proper subsets of the cumulative Math
299 contexts listed above.
</p>
<dl>
<dt>
<a href="./student.html?load=math/Equations-Logic.lurch">
Equations and Logic Only
</a>
</dt>
<dd>
Equations, Logic Theorems, Predicate Logic, Propositional Logic.
(
<a
href="./instructor.html?load=math/Equations-Logic-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Sets-Equations-Logic.lurch">
Sets, Equations, and Logic
</a>
</dt>
<dd>
Set Theory, Equations, Logic Theorems, Predicate Logic,
Propositional Logic. This does not include Set Theory Theorems.
(
<a
href="./instructor.html?load=math/Sets-Equations-Logic-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a
href="./student.html?load=math/Functions-Sets-Equations-Logic.lurch">
Functions, Sets, Equations, and Logic
</a>
</dt>
<dd>
Functions, Set Theory Theorems, Set Theory, Equations, Logic
Theorems, Predicate Logic, Propositional Logic. (
<a
href="./instructor.html?load=math/Functions-Sets-Equations-Logic-Rules.lurch">
view rules
</a>
)
</dd>
<dt>
<a href="./student.html?load=math/Topology.lurch">Topology</a>
</dt>
<dd>
Defines
<code>topological space</code>
,
<code>open</code>
,
<code>closed</code>
, and
<code>continuous</code>
(in a toplogical space). Includes the Functions library above. (
<a href="./instructor.html?load=math/Topology-Rules.lurch">
view rules
</a>
)
</dd>
</dl>
</div>
<!-- blue-box -->
<!-- ----------------------------------------------------- -->
<h2 id="assignments">Example Assignments</h2>
<p class="smallskip">
<strong class="bigfont">Math 299 Spring 2024 Assignments</strong>
– each link one opens a Lurch document containing a homework
assignment from the course. Note that Lurch was under development
during this semester so that some assignments are not good examples
of what can be done today if these were rewritten to take advantage
of some of the new features. We are in the process of updating them.
I also intend to revise the entire syllabus in like of having Lurch
available to spend less time on formal logic and more time on the
later topics in the course (and perhaps add a few new ones).
</p>
<p class="medskip">
The theorem numbers for the assigned problems below refer to the
following course lecture notes. These are not fully updated to be
consistent with the libraries used in the assignments below and may
be revised frequently.
</p>
<p>
<a
href="https://monks.scranton.edu/files/courses/Math299/math-299-lecture.pdf"
class="button medskip"
rel="noopener noreferrer">
Lecture Notes
</a>
</p>
<div class="blue-box thmlist">
<dl>
<dt>
<a href="./student.html?load=assignments/assignment-06.lurch">
Assignment #06
</a>
Propositional logic
</dt>
<dd>
<ol>
<li>
<strong>Theorem (Iff is stronger):</strong>
$(P\Leftrightarrow Q)\Rightarrow(Q\Rightarrow P)$
</li>
<li>
<strong>Theorem (the other way around):</strong>
$\neg(\neg A)\Rightarrow A$
</li>
<li>
<strong>Theorem (Modus Tollens):</strong>
$(S \Rightarrow T)\text{ and } \neg T \Rightarrow \neg S$
</li>
<li>
<strong>
Theorem (atomic statements don't have to be variables):
</strong>
$0 \lt x\text{ or }x\lt 10 \Rightarrow x\lt 10\text{ or }
0\lt x$
</li>
<li>
<strong>Theorem (excluded middle):</strong>
$K\text{ or } \neg K$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-07.lurch">
Assignment #07
</a>
Propositional logic
</dt>
<dd>
<ol>
<li>
<strong>Theorem (implies is transitive):</strong>
$((P\Rightarrow Q)\text{ and }(Q \Rightarrow R)) \Rightarrow
(P \Rightarrow R)$
</li>
<li>
<strong>Theorem (alternate or-):</strong>
$(S\text{ or } T)\text{ and } \neg S \Rightarrow T$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-08.lurch">
Assignment #08
</a>
Predicate logic with Equality
</dt>
<dd>
<ol>
<li>
<strong>Theorem (alpha equivalence warm up):</strong>
$(\forall x.P(x)) \Rightarrow (\forall y.P(y))$
</li>
<li>
<strong>Theorem (same thing but for exists):</strong>
$(\exists x.P(x)) \Rightarrow (\exists y.P(y))$
</li>
<li>
<strong>Theorem (commuting quantifiers):</strong>
$(\forall x.\forall y.x\text{ loves }y) \Rightarrow (\forall
y.\forall x.x\text{ loves }y)$
</li>
<li>
<strong>Theorem (same thing for exists):</strong>
$(\exists x.\exists y.x\text{ loves }y) \Rightarrow (\exists
y.\exists x.x\text{ loves }y)$
</li>
<li>
<strong>Theorem (DeMorgan):</strong>
$\neg (S\text{ and }T) \Rightarrow \neg S\text{ or } \neg T$
</li>
<li>
<strong>Theorem (DeMorgan):</strong>
$\neg (\forall x.R(x)) \Rightarrow (\exists x.\neg R(x))$
</li>
<li>
<strong>Theorem ($=$ is transitive):</strong>
$x=y\text{ and }y=z \Rightarrow x=z$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-09.lurch">
Assignment #09
</a>
Logic theorems
</dt>
<dd>
<ol>
<li>
<strong>Theorem 5.10 (some are the same):</strong>
$(\forall x.\forall y.P(x,y)) \Rightarrow (\forall
z.P(z,z))$
</li>
<li>
<strong>Theorem 5.16 (excluded middle):</strong>
$(\forall x.Q(x))\text{ or } (\exists x.\neg Q(x))$
</li>
<li>
<strong>Theorem 4.22 (alternate $\Rightarrow$):</strong>
$(R \Rightarrow S) \Rightarrow \neg R\text{ or }S$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-10.lurch">
Assignment #10
</a>
Logic review
</dt>
<dd>
<ol>
<li>
<strong>Theorem (ex falso quodlibet):</strong>
If Dracula fears Alice and Dracula does not fear Alice then
Bob loves Alice.
</li>
<li>
<strong>Theorem 4.29 (the most beautiful?):</strong>
$S \Rightarrow (S \Leftrightarrow S\text{ or }(\neg S \text{
and } S))$
</li>
<li>
<strong>Theorem 4.25 (shunting):</strong>
$((P \text{ and } Q) \Rightarrow R) \Leftrightarrow (P
\Rightarrow (Q \Rightarrow R))$
</li>
<li>
<strong>Theorem 5.21 (avoiding vacuous domains):</strong>
$(\exists x. x = x)\text{ and }(\forall y. T(y)) \Rightarrow
(\exists z. T(z))$
</li>
<li>
<strong>Theorem 5.5 (distributivity):</strong>
$(\exists x. A(x) \Rightarrow B(x)) \Leftrightarrow (\forall
y. A(y)) \Rightarrow (\exists z. B(z))$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-11.lurch">
Assignment #11
</a>
Logic review
</dt>
<dd>
<ol>
<li>
<strong>Theorem (Pierce's Law):</strong>
$((S \Rightarrow T) \Rightarrow S) \Rightarrow S$
</li>
<li>
<strong>Theorem (quantifier fun):</strong>
$(\forall x.\exists y.\forall z.A(x,y,z)) \Rightarrow
(\forall z.\forall x.\exists y.A(x,y,z))$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-12.lurch">
Assignment #12
</a>
Peano arithmetic and Induction
</dt>
<dd>
<ol>
<li>
<strong>
Theorem 7.3 (no number is it's own successor):
</strong>
$n\neq\sigma(n)$
</li>
<li>
<strong>
Theorem 7.4 (alternate definition of $\sigma$):
</strong>
$\sigma(n)=n+1$
</li>
<li>
<strong>Theorem 7.5 (associativity of addition):</strong>
$(m+n)+p=m+(n+p)$
</li>
<li>
<strong>Theorem 7.6 (additive identity, part 2):</strong>
$0+n=n$
</li>
<li>
<strong>Theorem 7.7 (commutativity of adding $1$):</strong>
$1+n=n+1$
</li>
<li>
<strong>Theorem 7.8 (commutativity of addition):</strong>
$m+n=n+m$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-13.lurch">
Assignment #13
</a>
Peano arithmetic and Induction
</dt>
<dd>
<ol>
<li>
<strong>Theorem 7.11 (left multiplication by zero):</strong>
$0\cdot m=0$
</li>
<li>
<strong>
Theorem 7.19 (nonzero naturals are positive):
</strong>
<ul>
<li>(a) $0\leq n$</li>
<li>(b) $0\lt n \Leftrightarrow n\neq 0$</li>
</ul>
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-14.lurch">
Assignment #14
</a>
Sequences, Recursion, and Induction
</dt>
<dd>
<ol>
<li>
<strong>Theorem 8.3(a) (a useful identity):</strong>
$n^2=n\cdot n$
</li>
<li>
<strong>Theorem 8.3(b) (useful identity):</strong>
$2\cdot n=n+n$
</li>
<li>
<strong>Theorem 8.2(b) (power law):</strong>
$z^m\cdot z^n=z^{m+n}$
</li>
<li>
<strong>Theorem 8.2(c) (power law):</strong>
$(z^m)^n=z^{m\cdot n}$
</li>
<li>
<strong>Theorem 8.4 (quadratic beats linear):</strong>
If $a+b\leq n$ then $a\cdot n+b\leq n^2$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-15.lurch">
Assignment #15
</a>
Sequences, Recursion, and Induction
</dt>
<dd>
<ol>
<li>
<strong>Theorem 8.7 (Fib fun):</strong>
$F_{n+3}+F_n=2\cdot F_{n+2}$
</li>
<li>
<strong>Theorem 8.5(b) (basic sum property):</strong>
$\displaystyle \sum_{i=0}^n s\cdot a_i = s\cdot \sum_{i=0}^n
a_i$
</li>
<li>
<strong>
Theorem 8.11 (closed formula for binomial coefficients):
</strong>
$\displaystyle(m!\cdot n!)\cdot \binom{n+m}{m}=(n+m)!$
</li>
</ol>
</dd>
<dt>
<a href="./student.html?load=assignments/assignment-16.lurch">
Assignment #16
</a>
Real numbers and Field axioms
</dt>
<dd>
<ol>
<li>
<strong>Theorem 9.3 (cancellation for addition):</strong>