aboutsummaryrefslogtreecommitdiffhomepage
path: root/doc/refman/RefMan-decl.tex
blob: aae10e323ceb9c4a5db4ae6f51825a8871cc0765 (plain)
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
\newcommand{\DPL}{Mathematical Proof Language}

\chapter{The \DPL\label{DPL}\index{DPL}}

\section{Introduction}

\subsection{Foreword}

In this chapter, we describe an alternative language that may be used
to do proofs using the Coq proof assistant.  The language described
here uses the same objects (proof-terms) as Coq, but it differs in the
way proofs are described. This language was created by Pierre
Corbineau at the Radboud University of Nijmegen, The Netherlands.

The intent is to provide language where proofs are less formalism-{}
and implementation-{}sensitive, and in the process to ease a bit the
learning of computer-{}aided proof verification.

\subsection{What is a declarative proof?}
In vanilla Coq, proofs are written in the imperative style: the user
issues commands that transform a so called proof state until it
reaches a state where the proof is completed. In the process, the user
mostly described the transitions of this system rather than the
intermediate states it goes through.

The purpose of a declarative proof language is to take the opposite
approach where intermediate states are always given by the user, but
the transitions of the system are automated as much as possible.

\subsection{Well-formedness and Completeness}

The \DPL{} introduces a notion of well-formed
proofs which are weaker than correct (and complete)
proofs. Well-formed proofs are actually proof script where only the
reasoning is incomplete. All the other aspects of the proof are
correct:
\begin{itemize}
\item All objects referred to exist where they are used
\item Conclusion steps actually prove something related to the
  conclusion of the theorem (the {\tt thesis}.
\item Hypothesis introduction steps are done when the goal is an
  implication with a corresponding assumption.
\item Sub-objects in the elimination steps for tuples are correct
  sub-objects of the tuple being decomposed.
\item Patterns in case analysis are type-correct, and induction is well guarded.
\end{itemize}

\subsection{Note for tactics users}

This section explain what differences the casual Coq user will
experience using the \DPL.
\begin{enumerate}
\item The focusing mechanism is constrained so that only one goal at
  a time is visible.
\item Giving a statement that Coq cannot prove does not produce an
  error, only a warning: this allows going on with the proof and fill
  the gap later.
\item Tactics can still be used for justifications and after
{\texttt{escape}}.
\end{enumerate}

\subsection{Compatibility}

The \DPL{} is available for all Coq interfaces that use
text-based interaction, including:
\begin{itemize}
\item the command-{}line toplevel {\texttt{coqtop}}
\item the native GUI {\CoqIDE}
\item the {\ProofGeneral} Emacs mode
\item Cezary Kaliszyk'{}s Web interface
\item L.E. Mamane'{}s tmEgg TeXmacs plugin
\end{itemize}

However it is not supported by structured editors such as PCoq.



\section{Syntax}

Here is a complete formal description of the syntax for \DPL{} commands.

\begin{figure}[htbp]
\begin{centerframe}
\begin{tabular}{lcl@{\qquad}r}
  instruction & ::= & {\tt proof} \\ 
  &  $|$ & {\tt assume } \nelist{statement}{\tt and}
  \zeroone{[{\tt and } \{{\tt we have}\}-clause]} \\
  &  $|$ & \{{\tt let},{\tt be}\}-clause \\
  &  $|$ & \{{\tt given}\}-clause \\
  &  $|$ & \{{\tt consider}\}-clause {\tt  from} term \\
  &  $|$ & ({\tt have} $|$ {\tt then} $|$ {\tt thus} $|$ {\tt hence}]) statement
  justification \\
  &  $|$ &  \zeroone{\tt thus} ($\sim${\tt =}|{\tt =}$\sim$) \zeroone{\ident{\tt :}}\term\relax justification \\  &  $|$ & {\tt suffices} (\{{\tt to have}\}-clause $|$
  \nelist{statement}{\tt and } \zeroone{{\tt and} \{{\tt to have}\}-clause})\\
  & & {\tt to show} statement justification \\ 
  &  $|$ & ({\tt claim} $|$ {\tt focus on}) statement \\
  &  $|$ & {\tt take} \term \\
  &  $|$ & {\tt define} \ident \sequence{var}{,}  {\tt  as} \term\\
  &  $|$ & {\tt reconsider} (\ident $|$ {\tt thesis}) {\tt  as} type\\
  &  $|$ & 
  {\tt per} ({\tt cases}$|${\tt induction}) {\tt on} \term \\
  &  $|$ & {\tt per cases of} type justification \\
  &  $|$ & {\tt suppose} \zeroone{\nelist{ident}{,} {\tt and}}~
  {\tt it is }pattern\\
  &          & \zeroone{{\tt such that} \nelist{statement} {\tt and} \zeroone{{\tt and} \{{\tt we have}\}-clause}} \\
  &  $|$ & {\tt end} 
  ({\tt proof} $|$ {\tt claim} $|$ {\tt focus} $|$ {\tt cases} $|$ {\tt induction}) \\
  & $|$ & {\tt escape} \\
  & $|$ & {\tt return} \medskip \\
  \{$\alpha,\beta$\}-clause & ::=& $\alpha$ \nelist{var}{,}~
  $\beta$ {\tt such that} \nelist{statement}{\tt and } \\
  & & \zeroone{{\tt and } \{$\alpha,\beta$\}-clause} \medskip\\
  statement   & ::= & \zeroone{\ident {\tt :}} type  \\
  & $|$ & {\tt thesis} \\
  & $|$ & {\tt thesis for} \ident \medskip \\
  var    & ::= & \ident \zeroone{{\tt :} type} \medskip \\
  justification & ::= & 
  \zeroone{{\tt by} ({\tt *} | \nelist{\term}{,})}
  ~\zeroone{{\tt using} tactic} \\
\end{tabular}
\end{centerframe}
\caption{Syntax of mathematical proof commands}
\end{figure}

The lexical conventions used here follows those of section \ref{lexical}.


Conventions:\begin{itemize}

 \item {\texttt{<{}tactic>{}}} stands for a Coq tactic.

 \end{itemize}

\subsection{Temporary names}

In proof commands where an optional name is asked for, omitting the
name will trigger the creation of a fresh temporary name (e.g. for a
hypothesis). Temporary names always start with an underscore `\_'
character (e.g. {\tt \_hyp0}). Temporary names have a lifespan of one
command: they get erased after the next command. They can however be safely in the step after their creation.

\section{Language description}

\subsection{Starting and Ending a mathematical proof}

The standard way to use the \DPL{} is to first state a \texttt{Lemma} /
\texttt{Theorem} / \texttt{Definition} and then use the \texttt{proof}
command to switch the current subgoal to mathematical mode.  After the
proof is completed, the \texttt{end proof} command will close the
mathematical proof. If any subgoal remains to be proved, they will be
displayed using the usual Coq display.

\begin{coq_example}
Theorem this_is_trivial: True.
proof.
  thus thesis.
end proof.
Qed.
\end{coq_example}

The {\texttt{proof}} command only applies to \emph{one subgoal}, thus
if several sub-goals are already present, the {\texttt{proof ... end
    proof}} sequence has to be used several times.

\begin{coq_example*}
Theorem T: (True /\ True) /\ True.
  split. split.
\end{coq_example*}
\begin{coq_example}
  Show.
  proof. (* first subgoal *)
    thus thesis.
  end proof.
  trivial. (* second subgoal *)
  proof. (* third subgoal *)
    thus thesis.
  end proof.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

As with all other block structures, the {\texttt{end proof}} command
assumes that your proof is complete. If not, executing it will be
equivalent to admitting that the statement is proved: A warning will
be issued and you will not be able to run the {\texttt{Qed}}
command. Instead, you can run {\texttt{Admitted}} if you wish to start
another theorem and come back
later.

\begin{coq_example}
Theorem this_is_not_so_trivial: False.
proof.
end proof. (* here a warning is issued *)
Fail Qed. (* fails: the proof in incomplete *)
Admitted. (* Oops! *)
\end{coq_example}
\begin{coq_eval}
Reset this_is_not_so_trivial.   
\end{coq_eval}

\subsection{Switching modes}

When writing a mathematical proof, you may wish to use procedural
tactics at some point. One way to do so is to write a using-{}phrase
in a deduction step (see section~\ref{justifications}). The other way
is to use an {\texttt{escape...return}} block.

\begin{coq_eval}  
Theorem T: True.
proof.
\end{coq_eval}
\begin{coq_example}
 Show.
 escape.
 auto.
 return.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

The return statement expects all subgoals to be closed, otherwise a
warning is issued and the proof cannot be saved anymore.

It is possible to use the {\texttt{proof}} command inside an
{\texttt{escape...return}} block, thus nesting a mathematical proof
inside a procedural proof inside a mathematical proof...

\subsection{Computation steps}

The {\tt reconsider ... as} command allows changing the type of a hypothesis or of {\tt thesis} to a convertible one.

\begin{coq_eval}
Theorem T: let a:=false in let b:= true in ( if a then True else False -> if b then True else False).
intros a b.
proof.
assume H:(if a then True else False).
\end{coq_eval}
\begin{coq_example}
 Show.
 reconsider H as False.
 reconsider thesis as True.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}


\subsection{Deduction steps}

The most common instruction in a mathematical proof is the deduction
step: it asserts a new statement (a formula/type of the \CIC) and tries
to prove it using a user-provided indication: the justification. The
asserted statement is then added as a hypothesis to the proof context.

\begin{coq_eval}
Theorem T: forall x, x=2 -> 2+x=4.
proof.
let x be such that H:(x=2).
\end{coq_eval} 
\begin{coq_example}
 Show.
 have H':(2+x=2+2) by H.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

It is often the case that the justifications uses the last hypothesis
introduced in the context, so the {\tt then} keyword can be used as a
shortcut, e.g. if we want to do the same as the last example:
 
\begin{coq_eval}
Theorem T: forall x, x=2 -> 2+x=4.
proof.
let x be such that H:(x=2).
\end{coq_eval} 
\begin{coq_example}
 Show.
 then (2+x=2+2).
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

In this example, you can also see the creation of a temporary name {\tt \_fact}.

\subsection{Iterated equalities}

A common proof pattern when doing a chain of deductions is to do
multiple rewriting steps over the same term, thus proving the
corresponding equalities. The iterated equalities are a syntactic
support for this kind of reasoning:

\begin{coq_eval}
Theorem T: forall x, x=2 -> x + x = x * x.
proof.
let x be such that H:(x=2).
\end{coq_eval} 
\begin{coq_example}
 Show.
 have (4 = 4).
        ~= (2 * 2).
        ~= (x * x) by H.
        =~ (2 + 2).
        =~ H':(x + x) by H.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

Notice that here we use temporary names heavily.

\subsection{Subproofs}

When an intermediate step in a proof gets too complicated or involves a
well contained set of intermediate deductions, it can be useful to insert
its proof as a subproof of the current proof. This is done by using the
{\tt claim ... end claim} pair of commands.

\begin{coq_eval}
Theorem T: forall x, x + x = x * x -> x = 0 \/ x = 2.
proof.
let x be such that H:(x + x = x * x).
\end{coq_eval} 
\begin{coq_example}
Show.
claim H':((x - 2) * x = 0).
\end{coq_example}

A few steps later...

\begin{coq_example}
thus thesis. 
end claim.
\end{coq_example}

Now the rest of the proof can happen.

\begin{coq_eval}
Abort.
\end{coq_eval}

\subsection{Conclusion steps}

The commands described above have a conclusion counterpart, where the
new hypothesis is used to refine the conclusion.

\begin{figure}[b]
 \centering
\begin{tabular}{c|c|c|c|c|}
        X       & \,simple\, & \,with previous step\, & 
               \,opens sub-proof\, & \,iterated equality\, \\
\hline
intermediate step & {\tt have} & {\tt then} & 
               {\tt claim} & {\tt $\sim$=/=$\sim$}\\ 
conclusion step & {\tt thus} & {\tt hence} & 
                {\tt focus on} & {\tt thus $\sim$=/=$\sim$}\\ 
\hline
\end{tabular}
\caption{Correspondence between basic forward steps and conclusion steps}
\end{figure}

Let us begin with simple examples:

\begin{coq_eval}
Theorem T: forall (A B:Prop), A -> B -> A /\ B.
intros A B HA HB.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
hence B.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

In the next example, we have to use {\tt thus} because {\tt HB} is no longer
the last hypothesis.

\begin{coq_eval}
Theorem T: forall (A B C:Prop), A -> B -> C -> A /\ B /\ C.
intros A B C HA HB HC.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
thus B by HB.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

The command fails if the refinement process cannot find a place to fit
the object in a proof of the conclusion.


\begin{coq_eval}
Theorem T: forall (A B C:Prop), A -> B -> C -> A /\ B.
intros A B C HA HB HC.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
Fail hence C. (* fails *)
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

The refinement process may induce non
reversible choices, e.g. when proving a disjunction it may {\it
  choose} one side of the disjunction.

\begin{coq_eval}
Theorem T: forall (A B:Prop), B -> A \/ B.
intros A B HB.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
hence B.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

In this example you can see that the right branch was chosen since {\tt D} remains to be proved.

\begin{coq_eval}
Theorem T: forall (A B C D:Prop), C -> D -> (A /\ B) \/ (C /\ D).
intros A B C D HC HD.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
thus C by HC.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

Now for existential statements, we can use the {\tt take} command to
choose {\tt 2} as an explicit witness of existence.

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop), P 2 -> exists x,P x.
intros P HP.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
take 2.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

It is also possible to prove the existence directly.

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop), P 2 -> exists x,P x.
intros P HP.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
hence (P 2).
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

Here a more involved example where the choice of {\tt P 2} propagates
the choice of {\tt 2} to another part of the formula.

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop) (R:nat -> nat -> Prop), P 2 -> R 0 2 -> exists x, exists y, P y /\ R x y.
intros P R HP HR.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
thus (P 2) by HP.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

Now, an example with the {\tt suffices} command. {\tt suffices}
is a sort of dual for {\tt have}: it allows replacing the conclusion
(or part of it) by a sufficient condition. 

\begin{coq_eval}
Theorem T: forall (A B:Prop) (P:nat -> Prop), (forall x, P x -> B) -> A -> A /\ B.
intros A B P HP HA.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
suffices to have x such that HP':(P x) to show B by HP,HP'.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

Finally, an example where {\tt focus} is handy: local assumptions.

\begin{coq_eval}
Theorem T: forall (A:Prop) (P:nat -> Prop), P 2 -> A -> A /\ (forall x, x = 2 -> P x).
intros A P HP HA.
proof.
\end{coq_eval} 
\begin{coq_example}
Show.
focus on (forall x, x = 2 -> P x).
let x be such that (x = 2).
hence thesis by HP.
end focus.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

\subsection{Declaring an Abbreviation}

In order to shorten long expressions, it is possible to use the {\tt
  define ... as ...} command to give a name to recurring expressions.

\begin{coq_eval}
Theorem T: forall x, x = 0 -> x + x = x * x.
proof.
let x be such that H:(x = 0).
\end{coq_eval} 
\begin{coq_example}
Show.
define sqr x as (x * x).
reconsider thesis as (x + x = sqr x).
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

\subsection{Introduction steps}

When the {\tt thesis} consists of a hypothetical formula (implication
or universal quantification (e.g. \verb+A -> B+), it is possible to
assume the hypothetical part {\tt A} and then prove {\tt B}. In the
\DPL{}, this comes in two syntactic flavors that are semantically
equivalent: {\tt let} and {\tt assume}. Their syntax is designed so that
{\tt let} works better for universal quantifiers and {\tt assume} for
implications.

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop), forall x, P x -> P x.
proof.
let P:(nat -> Prop).
\end{coq_eval} 
\begin{coq_example}
Show.
let x:nat.
assume HP:(P x).
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

In the {\tt let} variant, the type of the assumed object is optional
provided it can be deduced from the command. The objects introduced by
let can be followed by assumptions using {\tt such that}.

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop), forall x, P x -> P x.
proof.
let P:(nat -> Prop).
\end{coq_eval} 
\begin{coq_example}
Show.
Fail let x. (* fails because x's type is not clear *)
let x be such that HP:(P x). (* here x's type is inferred from (P x) *)
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

In the {\tt assume } variant, the type of the assumed object is mandatory
but the name is optional:

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop), forall x, P x -> P x -> P x.
proof.
let P:(nat -> Prop).
let x:nat.
\end{coq_eval} 
\begin{coq_example}
Show.
assume (P x). (* temporary name created *)
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

After {\tt such that}, it is also the case:

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop), forall x, P x -> P x.
proof.
let P:(nat -> Prop).
\end{coq_eval} 
\begin{coq_example}
Show.
let x be such that (P x). (* temporary name created *)
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

\subsection{Tuple elimination steps}

In the \CIC, many objects dealt with in simple proofs are tuples:
pairs, records, existentially quantified formulas. These are so
common that the \DPL{} provides a mechanism to extract members of
those tuples, and also objects in tuples within tuples within
tuples...

\begin{coq_eval}
Theorem T: forall (P:nat -> Prop) (A:Prop), (exists x, (P x /\ A)) -> A.
proof.
let P:(nat -> Prop),A:Prop be such that H:(exists x, P x /\ A).
\end{coq_eval} 
\begin{coq_example}
Show.
consider x such that HP:(P x) and HA:A from H.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

Here is an example with pairs:

\begin{coq_eval}
Theorem T: forall p:(nat * nat)%type, (fst p >= snd p) \/ (fst p < snd p).
proof.
let p:(nat * nat)%type.
\end{coq_eval} 
\begin{coq_example}
Show.
consider x:nat,y:nat from p.
reconsider thesis as (x >= y \/ x < y). 
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

It is sometimes desirable to combine assumption and tuple
decomposition. This can be done using the {\tt given} command.

\begin{coq_eval}
Theorem T: forall P:(nat -> Prop), (forall n, P n -> P (n - 1)) -> 
(exists m, P m) -> P 0.
proof.
let P:(nat -> Prop) be such that HP:(forall n, P n -> P (n - 1)).
\end{coq_eval} 
\begin{coq_example}
Show.
given m such that Hm:(P m).  
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}
 
\subsection{Disjunctive reasoning}

In some proofs (most of them usually) one has to consider several
cases and prove that the {\tt thesis} holds in all the cases. This is
done by first specifying which object will be subject to case
distinction (usually a disjunction) using {\tt per cases}, and then specifying which case is being proved by using {\tt suppose}.


\begin{coq_eval}
Theorem T: forall (A B C:Prop), (A -> C) -> (B -> C) -> (A \/ B) -> C.
proof.
let A:Prop,B:Prop,C:Prop be such that HAC:(A -> C) and HBC:(B -> C).
assume HAB:(A \/ B).
\end{coq_eval} 
\begin{coq_example}
per cases on HAB.
suppose A.
  hence thesis by HAC.
suppose HB:B.
  thus thesis by HB,HBC.
end cases.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

The proof is well formed (but incomplete) even if you type {\tt end
 cases} or the next {\tt suppose} before the previous case is proved.

If the disjunction is derived from a more general principle, e.g. the
excluded middle axiom), it is desirable to just specify which instance
of it is being used:

\begin{coq_eval}
Section Coq.
\end{coq_eval}
\begin{coq_example}
Hypothesis EM : forall P:Prop, P \/ ~ P.
\end{coq_example} 
\begin{coq_eval}
Theorem T: forall (A C:Prop), (A -> C) -> (~A -> C) -> C.
proof.
let A:Prop,C:Prop be such that HAC:(A -> C) and HNAC:(~A -> C).
\end{coq_eval} 
\begin{coq_example}
per cases of (A \/ ~A) by EM.
suppose (~A).
  hence thesis by HNAC.
suppose A.
  hence thesis by HAC.
end cases.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

\subsection{Proofs per cases}

If the case analysis is to be made on a particular object, the script
is very similar: it starts with {\tt per cases on }\emph{object} instead.

\begin{coq_eval}
Theorem T: forall (A C:Prop), (A -> C) -> (~A -> C) -> C.
proof.
let A:Prop,C:Prop be such that HAC:(A -> C) and HNAC:(~A -> C).
\end{coq_eval} 
\begin{coq_example}
per cases on (EM A).
suppose (~A).
\end{coq_example}
\begin{coq_eval}
Abort.
End Coq.
\end{coq_eval}

If the object on which a case analysis occurs in the statement to be
proved, the command {\tt suppose it is }\emph{pattern} is better
suited than {\tt suppose}. \emph{pattern} may contain nested patterns
with {\tt as} clauses. A detailed description of patterns is to be
found in figure \ref{term-syntax-aux}. here is an example.

\begin{coq_eval}
Theorem T: forall (A B:Prop) (x:bool), (if x then A else B) -> A \/ B.
proof.
let A:Prop,B:Prop,x:bool.
\end{coq_eval} 
\begin{coq_example}
per cases on x.
suppose it is true.
  assume A.
  hence A.
suppose it is false.
  assume B.
  hence B.
end cases.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

\subsection{Proofs by induction} 

Proofs by induction are very similar to proofs per cases: they start
with {\tt per induction on }{\tt object} and proceed with {\tt suppose
  it is }\emph{pattern}{\tt and }\emph{induction hypothesis}. The
induction hypothesis can be given explicitly or identified by the
sub-object $m$ it refers to using {\tt thesis for }\emph{m}. 

\begin{coq_eval}
Theorem T: forall (n:nat), n + 0 = n.
proof.
let n:nat.
\end{coq_eval} 
\begin{coq_example}
per induction on n.
suppose it is 0.
  thus (0 + 0 = 0).
suppose it is (S m) and H:thesis for m.
  then (S (m + 0) = S m).
  thus =~ (S m + 0).
end induction.
\end{coq_example}
\begin{coq_eval}
Abort.
\end{coq_eval}

\subsection{Justifications}\label{justifications}


Intuitively, justifications are hints for the system to understand how
to prove the statements the user types in. In the case of this
language justifications are made of two components:

Justification objects: {\texttt{by}} followed by a comma-{}separated
list of objects that will be used by a selected tactic to prove the
statement. This defaults to the empty list (the statement should then
be tautological). The * wildcard provides the usual tactics behavior:
use all statements in local context. However, this wildcard should be
avoided since it reduces the robustness of the script.

Justification tactic: {\texttt{using}} followed by a Coq tactic that
is executed to prove the statement. The default is a solver for
(intuitionistic) first-{}order with equality.

\section{More details and Formal Semantics}

I suggest the users looking for more information have a look at the
paper \cite{corbineau08types}. They will find in that paper a formal
semantics of the proof state transition induces by mathematical
commands.