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
|
; $Id: align.scm 2156 2008-01-25 13:25:12Z schimans $
; (load "~/minlog/init.scm")
(set! COMMENT-FLAG #f)
(libload "nat.scm")
(libload "listrev.scm")
(set! COMMENT-FLAG #t)
(add-alg "char" '("A" "char") '("B" "char") '("C" "char"))
; We want to use (unit yplus char) as the type characters including space.
(add-param-alg "yplus" 'sum-typeop
'("Inleft" "alpha1=>yplus")
'("Inright" "alpha2=>yplus"))
(type-to-string
(const-to-type (car (type-info-to-rec-consts (py "unit yplus char=>alpha")))))
; "(unit=>alpha)=>(char=>alpha)=>unit yplus char=>alpha"
; (pp (pt "(Inleft unit char)Dummy"))
; (pp (pt "(Inright char unit) A"))
; (pp (pt "(Inright char unit) B"))
(add-token "Spc" 'const (pt "(Inleft unit char)Dummy"))
(add-token
"I" 'prefix-op
(lambda (x) (mk-term-in-app-form (pt "(Inright char unit)") x)))
(add-display
(py "unit yplus char")
(lambda (x)
(let ((op (term-in-app-form-to-final-op x))
(args (term-in-app-form-to-args x)))
(cond ((and (term-in-const-form? op)
(string=? "Inleft"
(const-to-name (term-in-const-form-to-const op)))
(= 1 (length args))
(let* ((arg (car args))
(argop (term-in-app-form-to-final-op arg))
(argargs (term-in-app-form-to-args arg)))
(and (term-in-const-form? argop)
(string=? "Dummy"
(const-to-name
(term-in-const-form-to-const argop)))
(null? argargs))))
(list 'const "Spc"))
((and (term-in-const-form? op)
(string=? "Inright"
(const-to-name (term-in-const-form-to-const op)))
(= 1 (length args)))
(list 'prefix-op "I" (term-to-token-tree (car args))))
(else #f)))))
; (pp (pt "I A"))
; (pp (pt "I B"))
; (pp (pt "Spc"))
; We use (tensor) pairs of characters, to work with alignments.
(add-param-alg "ytensor" 'tensor-typeop
'("TensorPair" "alpha1=>alpha2=>ytensor"))
(type-to-string
(const-to-type
(car (type-info-to-rec-consts (py "alpha1 ytensor alpha2=>alpha")))))
; "(alpha1=>alpha2=>alpha)=>alpha1 ytensor alpha2=>alpha"
; We want the display r#s for terms, and (Lft r) and (Rht r) for the
; components.
(add-token
"#" 'pair-op ;hence right associative
(lambda (x y)
(mk-term-in-app-form
(make-term-in-const-form
(let* ((const (constr-name-to-constr "TensorPair"))
(tvars (const-to-tvars const))
(type1 (term-to-type x))
(type2 (term-to-type y))
(subst (make-substitution tvars (list type1 type2))))
(const-substitute const subst #f)))
x y)))
(add-display
(py "alpha1 ytensor alpha2")
(lambda (x)
(if (term-in-app-form? x)
(let ((op (term-in-app-form-to-final-op x))
(args (term-in-app-form-to-args x)))
(if (and (term-in-const-form? op)
(string=? "TensorPair"
(const-to-name (term-in-const-form-to-const op)))
(= 2 (length args)))
(list 'pair-op "#"
(term-to-token-tree (car args))
(term-to-token-tree (cadr args)))
#f))
#f)))
; (pp (pt "alpha1#boole"))
(add-program-constant
"TensorLft" (py "alpha1 ytensor alpha2=>alpha1") t-deg-one)
(add-program-constant
"TensorRht" (py "alpha1 ytensor alpha2=>alpha2") t-deg-one)
(add-token
"Lft" 'prefix-op
(lambda (x)
(mk-term-in-app-form
(make-term-in-const-form
(let* ((const (pconst-name-to-pconst "TensorLft"))
(tvars (const-to-tvars const))
(tensortype (term-to-type x))
(types (alg-form-to-types tensortype))
(subst (make-substitution tvars types)))
(const-substitute const subst #f)))
x)))
(add-display
(py "alpha")
(lambda (x)
(if (term-in-app-form? x)
(let ((op (term-in-app-form-to-final-op x))
(args (term-in-app-form-to-args x)))
(if (and (term-in-const-form? op)
(string=? "TensorLft"
(const-to-name (term-in-const-form-to-const op)))
(= 1 (length args)))
(list 'prefix-op "Lft"
(term-to-token-tree (car args)))
#f))
#f)))
(add-token
"Rht" 'prefix-op
(lambda (x)
(mk-term-in-app-form
(make-term-in-const-form
(let* ((const (pconst-name-to-pconst "TensorRht"))
(tvars (const-to-tvars const))
(tensortype (term-to-type x))
(types (alg-form-to-types tensortype))
(subst (make-substitution tvars types)))
(const-substitute const subst #f)))
x)))
(add-display
(py "alpha")
(lambda (x)
(if (term-in-app-form? x)
(let ((op (term-in-app-form-to-final-op x))
(args (term-in-app-form-to-args x)))
(if (and (term-in-const-form? op)
(string=? "TensorRht"
(const-to-name (term-in-const-form-to-const op)))
(= 1 (length args)))
(list 'prefix-op "Rht"
(term-to-token-tree (car args)))
#f))
#f)))
(add-computation-rule (pt "Lft(alpha1#alpha2)") (pt "alpha1"))
(add-computation-rule (pt "Rht(alpha1#alpha2)") (pt "alpha2"))
(pp (nt (pt "Rht(nat#alpha)")))
(add-var-name "c" (py "char")) ;character
(add-var-name "d" (py "unit yplus char")) ;extended character
(add-var-name "cc" (py "char ytensor char")) ;pair of characters
(add-var-name "dd" (py "(unit yplus char) ytensor (unit yplus char)"))
; pair of extended characters
(add-var-name "s" (py "list char")) ;list of characters, string
(add-var-name "t" (py "list(unit yplus char)"))
; list of extended characters, extended string
(add-var-name "u" (py "list(char ytensor char)"))
; list of pairs of characters
(add-var-name "v" (py "list((unit yplus char) ytensor (unit yplus char))"))
; list of pairs of extended characters
; "NoDSpc" expresses that in v there are no double spaces.
(add-program-constant
"NoDSpc" (py "list((unit yplus char) ytensor (unit yplus char))=>boole")
t-deg-one)
(add-computation-rule
(pt "NoDSpc(Nil (unit yplus char) ytensor (unit yplus char))")
(pt "True"))
(add-computation-rule (pt "NoDSpc(v::(Spc#Spc))") (pt "False"))
(add-computation-rule (pt "NoDSpc(v::(I c#d))") (pt "True"))
(add-computation-rule (pt "NoDSpc(v::(Spc#I c))") (pt "True"))
(add-rewrite-rule (pt "NoDSpc(v::(d#I c))") (pt "True"))
; We take some canonical inhabitants.
(add-computation-rule (pt "(Inhab char)") (pt "A"))
(pp (nt (pt "(Inhab char)")))
(pp (nt (pt "(Inhab alpha)")))
(pp (nt (pt "(Inhab alpha1)")))
(add-computation-rule (pt "(Inhab (unit yplus char))") (pt "Spc"))
(add-computation-rule (pt "(Inhab char ytensor char)") (pt "A#A"))
(add-computation-rule
(pt "(Inhab (unit yplus char) ytensor (unit yplus char))")
(pt "Spc#Spc"))
; "CharEProj"
(set-goal (pf "all n,u E(u__n)"))
(assume "n")
(ind)
(use "Truth-Axiom")
(assume "u" "cc" "H1")
(ng)
(cases 'auto)
(assume "H2")
(use "H1")
(assume "H2")
(use "Truth-Axiom")
(save "CharEProj")
; "ExtCharEProj"
(set-goal (pf "all n,v E(v__n)"))
(assume "n")
(ind)
(use "Truth-Axiom")
(assume "v" "dd" "H1")
(ng)
(cases 'auto)
(assume "H2")
(use "H1")
(assume "H2")
(use "Truth-Axiom")
(save "ExtCharEProj")
(add-var-name "a" (py "list char=>list char=>
list((unit yplus char) ytensor (unit yplus char))"))
; alignment
; "Cp" compresses a list of extended characters, by removing spaces.
(add-program-constant "Cp" (py "list(unit yplus char)=>list char") t-deg-one)
(add-computation-rule (pt "Cp(Nil unit yplus char)") (pt "(Nil char)"))
(add-computation-rule (pt "Cp(t::I c)") (pt "Cp t::c"))
(add-computation-rule (pt "Cp(t::Spc)") (pt "Cp t"))
(pp (nt (pt "Cp(:I A::Spc::I A::I B::Spc)"))) ;:A::A::B
; To express that "a" is a correct alignment we use an inductive definition.
(add-ids
(list (list "Align"
(make-arity
(py "list char=>list char=>
list((unit yplus char) ytensor (unit yplus char))"))))
'("all a.(all s1,s2 Cp(([dd]Lft dd)map a s1 s2)=s1) ->
(all s1,s2 Cp(([dd]Rht dd)map a s1 s2)=s2) ->
(all s1,s2 NoDSpc(a s1 s2)) -> Align a" "DefAlign"))
; (pp "DefAlign")
; For simplicity we assume that scores have nat values.
(add-var-name
"sc" (py "(unit yplus char) ytensor (unit yplus char)=>nat")) ;score
(add-program-constant "Sum" (py "list nat=>nat") t-deg-one)
(add-computation-rule (pt "Sum(Nil nat)") (pt "Zero"))
(add-computation-rule (pt "Sum(list nat::nat)") (pt "Sum list nat+nat"))
; (pp (nt (pt "Sum(:1::2::3::4)")))
; The recursively defined V function. Computation rules to be added.
(add-program-constant
"V" (py "((unit yplus char) ytensor (unit yplus char)=>nat)=>nat=>nat=>nat")
t-deg-one)
; Theorem 1 consists of two parts:
(set-goal
(pf "all sc ex a.Align a & all s1,s2.V sc Lh s1 Lh s2=Sum(sc map a s1 s2)"))
(set-goal
(pf "all sc,a,s1,s2.Align a -> V sc Lh s1 Lh s2<=Sum(sc map a s1 s2)"))
(assume "sc" "a" "s1" "s2" "H1")
(inversion "H1")
(assume "a1")
|
