1 \idx{refl_type} A type ==> A = A),
2 \idx{refl_elem} a : A ==> a = a : A
4 \idx{sym_type} A = B ==> B = A
5 \idx{sym_elem} a = b : A ==> b = a : A
7 \idx{trans_type} [| A = B; B = C |] ==> A = C
8 \idx{trans_elem} [| a = b : A; b = c : A |] ==> a = c : A
10 \idx{equal_types} [| a : A; A = B |] ==> a : B
11 \idx{equal_typesL} [| a = b : A; A = B |] ==> a = b : B
13 \idx{subst_type} [| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type
15 \idx{subst_typeL} [| a = c : A; !!z. z:A ==> B(z) = D(z)
17 \idx{subst_elem} [| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)
19 \idx{subst_elemL} [| a = c : A; !!z. z:A ==> b(z) = d(z) : B(z)
20 |] ==> b(a) = d(c) : B(a)
22 \idx{refl_red} Reduce(a,a)
23 \idx{red_if_equal} a = b : A ==> Reduce(a,b)
24 \idx{trans_red} [| a = b : A; Reduce(b,c) |] ==> a = c : A
30 \idx{NI_succ} a : N ==> succ(a) : N
31 \idx{NI_succL} a = b : N ==> succ(a) = succ(b) : N
33 \idx{NE} [| p: N; a: C(0);
34 !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u))
35 |] ==> rec(p, a, %u v.b(u,v)) : C(p)
37 \idx{NEL} [| p = q : N; a = c : C(0);
38 !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u))
39 |] ==> rec(p, a, %u v.b(u,v)) = rec(q,c,d) : C(p)
42 !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u))
43 |] ==> rec(0, a, %u v.b(u,v)) = a : C(0)
45 \idx{NC_succ} [| p: N; a: C(0);
46 !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u))
47 |] ==> rec(succ(p), a, %u v.b(u,v)) =
48 b(p, rec(p, a, %u v.b(u,v))) : C(succ(p))
50 \idx{zero_ne_succ} [| a: N; 0 = succ(a) : N |] ==> 0: F
53 \idx{ProdF} [| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A.B(x) type
54 \idx{ProdFL} [| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
55 PROD x:A.B(x) = PROD x:C.D(x)
57 \idx{ProdI} [| A type; !!x. x:A ==> b(x):B(x)
58 |] ==> lam x.b(x) : PROD x:A.B(x)
59 \idx{ProdIL} [| A type; !!x. x:A ==> b(x) = c(x) : B(x)
60 |] ==> lam x.b(x) = lam x.c(x) : PROD x:A.B(x)
62 \idx{ProdE} [| p : PROD x:A.B(x); a : A |] ==> p`a : B(a)
63 \idx{ProdEL} [| p=q: PROD x:A.B(x); a=b : A |] ==> p`a = q`b : B(a)
65 \idx{ProdC} [| a : A; !!x. x:A ==> b(x) : B(x)
66 |] ==> (lam x.b(x)) ` a = b(a) : B(a)
68 \idx{ProdC2} p : PROD x:A.B(x) ==> (lam x. p`x) = p : PROD x:A.B(x)
71 \idx{SumF} [| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A.B(x) type
72 \idx{SumFL} [| A = C; !!x. x:A ==> B(x) = D(x)
73 |] ==> SUM x:A.B(x) = SUM x:C.D(x)
75 \idx{SumI} [| a : A; b : B(a) |] ==> <a,b> : SUM x:A.B(x)
76 \idx{SumIL} [| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A.B(x)
78 \idx{SumE} [| p: SUM x:A.B(x);
79 !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>)
80 |] ==> split(p, %x y.c(x,y)) : C(p)
82 \idx{SumEL} [| p=q : SUM x:A.B(x);
83 !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)
84 |] ==> split(p, %x y.c(x,y)) = split(q, % x y.d(x,y)) : C(p)
86 \idx{SumC} [| a: A; b: B(a);
87 !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>)
88 |] ==> split(<a,b>, %x y.c(x,y)) = c(a,b) : C(<a,b>)
90 \idx{fst_def} fst(a) == split(a, %x y.x)
91 \idx{snd_def} snd(a) == split(a, %x y.y)
94 \idx{PlusF} [| A type; B type |] ==> A+B type
95 \idx{PlusFL} [| A = C; B = D |] ==> A+B = C+D
97 \idx{PlusI_inl} [| a : A; B type |] ==> inl(a) : A+B
98 \idx{PlusI_inlL} [| a = c : A; B type |] ==> inl(a) = inl(c) : A+B
100 \idx{PlusI_inr} [| A type; b : B |] ==> inr(b) : A+B
101 \idx{PlusI_inrL} [| A type; b = d : B |] ==> inr(b) = inr(d) : A+B
103 \idx{PlusE} [| p: A+B;
104 !!x. x:A ==> c(x): C(inl(x));
105 !!y. y:B ==> d(y): C(inr(y))
106 |] ==> when(p, %x.c(x), %y.d(y)) : C(p)
108 \idx{PlusEL} [| p = q : A+B;
109 !!x. x: A ==> c(x) = e(x) : C(inl(x));
110 !!y. y: B ==> d(y) = f(y) : C(inr(y))
111 |] ==> when(p, %x.c(x), %y.d(y)) = when(q, %x.e(x), %y.f(y)) : C(p)
113 \idx{PlusC_inl} [| a: A;
114 !!x. x:A ==> c(x): C(inl(x));
115 !!y. y:B ==> d(y): C(inr(y))
116 |] ==> when(inl(a), %x.c(x), %y.d(y)) = c(a) : C(inl(a))
118 \idx{PlusC_inr} [| b: B;
119 !!x. x:A ==> c(x): C(inl(x));
120 !!y. y:B ==> d(y): C(inr(y))
121 |] ==> when(inr(b), %x.c(x), %y.d(y)) = d(b) : C(inr(b))
123 \idx{EqF} [| A type; a : A; b : A |] ==> Eq(A,a,b) type
124 \idx{EqFL} [| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)
125 \idx{EqI} a = b : A ==> eq : Eq(A,a,b)
126 \idx{EqE} p : Eq(A,a,b) ==> a = b : A
127 \idx{EqC} p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)
130 \idx{FE} [| p: F; C type |] ==> contr(p) : C
131 \idx{FEL} [| p = q : F; C type |] ==> contr(p) = contr(q) : C
135 \idx{TE} [| p : T; c : C(tt) |] ==> c : C(p)
136 \idx{TEL} [| p = q : T; c = d : C(tt) |] ==> c = d : C(p)
137 \idx{TC} p : T ==> p = tt : T)
140 \idx{replace_type} [| B = A; a : A |] ==> a : B
141 \idx{subst_eqtyparg} [| a=c : A; !!z. z:A ==> B(z) type |] ==> B(a)=B(c)
143 \idx{subst_prodE} [| p: Prod(A,B); a: A; !!z. z: B(a) ==> c(z): C(z)
144 |] ==> c(p`a): C(p`a)
146 \idx{SumIL2} [| c=a : A; d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)
148 \idx{SumE_fst} p : Sum(A,B) ==> fst(p) : A
150 \idx{SumE_snd} [| p: Sum(A,B); A type; !!x. x:A ==> B(x) type
151 |] ==> snd(p) : B(fst(p))
155 \idx{add_def} a#+b == rec(a, b, %u v.succ(v))
156 \idx{diff_def} a-b == rec(b, a, %u v.rec(v, 0, %x y.x))
157 \idx{absdiff_def} a|-|b == (a-b) #+ (b-a)
158 \idx{mult_def} a#*b == rec(a, 0, %u v. b #+ v)
160 \idx{mod_def} a//b == rec(a, 0, %u v.
161 rec(succ(v) |-| b, 0, %x y.succ(v)))
163 \idx{quo_def} a/b == rec(a, 0, %u v.
164 rec(succ(u) // b, succ(v), %x y.v))
167 \idx{add_typing} [| a:N; b:N |] ==> a #+ b : N
168 \idx{add_typingL} [| a=c:N; b=d:N |] ==> a #+ b = c #+ d : N
169 \idx{addC0} b:N ==> 0 #+ b = b : N
170 \idx{addC_succ} [| a:N; b:N |] ==> succ(a) #+ b = succ(a #+ b) : N
171 \idx{mult_typing} [| a:N; b:N |] ==> a #* b : N
172 \idx{mult_typingL} [| a=c:N; b=d:N |] ==> a #* b = c #* d : N
173 \idx{multC0} b:N ==> 0 #* b = 0 : N
174 \idx{multC_succ} [| a:N; b:N |] ==> succ(a) #* b = b #+ (a #* b) : N
175 \idx{diff_typing} [| a:N; b:N |] ==> a - b : N
176 \idx{diff_typingL} [| a=c:N; b=d:N |] ==> a - b = c - d : N
177 \idx{diffC0} a:N ==> a - 0 = a : N
178 \idx{diff_0_eq_0} b:N ==> 0 - b = 0 : N
180 \idx{diff_succ_succ} [| a:N; b:N |] ==> succ(a) - succ(b) = a - b : N
182 \idx{add_assoc} [| a:N; b:N; c:N |] ==> (a #+ b) #+ c = a #+ (b #+ c) : N
184 \idx{add_commute} [| a:N; b:N |] ==> a #+ b = b #+ a : N
186 \idx{mult_right0} a:N ==> a #* 0 = 0 : N
188 \idx{mult_right_succ} [| a:N; b:N |] ==> a #* succ(b) = a #+ (a #* b) : N
190 \idx{mult_commute} [| a:N; b:N |] ==> a #* b = b #* a : N
192 \idx{add_mult_dist} [| a:N; b:N; c:N |] ==> (a #+ b) #* c = (a #* c) #+ (b #* c) : N
194 \idx{mult_assoc} [| a:N; b:N; c:N |] ==> (a #* b) #* c = a #* (b #* c) : N
196 \idx{diff_self_eq_0} a:N ==> a - a = 0 : N
198 \idx{add_inverse_diff_lemma}
199 b:N ==> ?a : PROD x:N. Eq(N, b-x, 0) --> Eq(N, b #+ (x-b), x)
201 \idx{add_inverse_diff} [| a:N; b:N; b-a = 0 : N |] ==> b #+ (a-b) = a : N
203 \idx{absdiff_typing} [| a:N; b:N |] ==> a |-| b : N
204 \idx{absdiff_typingL} [| a=c:N; b=d:N |] ==> a |-| b = c |-| d : N
205 \idx{absdiff_self_eq_0} a:N ==> a |-| a = 0 : N
206 \idx{absdiffC0} a:N ==> 0 |-| a = a : N
208 \idx{absdiff_succ_succ} [| a:N; b:N |] ==> succ(a) |-| succ(b) = a |-| b : N
209 \idx{absdiff_commute} [| a:N; b:N |] ==> a |-| b = b |-| a : N
211 \idx{add_eq0_lemma} [| a:N; b:N |] ==> ?c : PROD u: Eq(N,a#+b,0) . Eq(N,a,0)
213 \idx{add_eq0} [| a:N; b:N; a #+ b = 0 : N |] ==> a = 0 : N
215 \idx{absdiff_eq0_lem}
216 [| a:N; b:N; a |-| b = 0 : N |] ==>
217 ?a : SUM v: Eq(N, a-b, 0) . Eq(N, b-a, 0)
219 \idx{absdiff_eq0} [| a |-| b = 0 : N; a:N; b:N |] ==> a = b : N
221 \idx{mod_typing} [| a:N; b:N |] ==> a//b : N
222 \idx{mod_typingL} [| a=c:N; b=d:N |] ==> a//b = c//d : N
223 \idx{modC0} b:N ==> 0//b = 0 : N
225 [| a:N; b:N |] ==> succ(a)//b = rec(succ(a//b) |-| b, 0, %x y.succ(a//b)) : N
227 \idx{quo_typing} [| a:N; b:N |] ==> a / b : N
228 \idx{quo_typingL} [| a=c:N; b=d:N |] ==> a / b = c / d : N
229 \idx{quoC0} b:N ==> 0 / b = 0 : N
230 [| a:N; b:N |] ==> succ(a) / b =
231 rec(succ(a)//b, succ(a / b), %x y. a / b) : N
232 \idx{quoC_succ2} [| a:N; b:N |] ==>
233 succ(a) / b =rec(succ(a//b) |-| b, succ(a / b), %x y. a / b) : N
235 \idx{iszero_decidable}
236 a:N ==> rec(a, inl(eq), %ka kb.inr(<ka, eq>)) :
237 Eq(N,a,0) + (SUM x:N. Eq(N,a, succ(x)))
239 \idx{mod_quo_equality} [| a:N; b:N |] ==> a//b #+ (a/b) #* b = a : N