1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/doc-src/Logics/CTT-rules.txt Wed Nov 10 05:00:57 1993 +0100
1.3 @@ -0,0 +1,241 @@
1.4 +\idx{refl_type} A type ==> A = A),
1.5 +\idx{refl_elem} a : A ==> a = a : A
1.6 +
1.7 +\idx{sym_type} A = B ==> B = A
1.8 +\idx{sym_elem} a = b : A ==> b = a : A
1.9 +
1.10 +\idx{trans_type} [| A = B; B = C |] ==> A = C
1.11 +\idx{trans_elem} [| a = b : A; b = c : A |] ==> a = c : A
1.12 +
1.13 +\idx{equal_types} [| a : A; A = B |] ==> a : B
1.14 +\idx{equal_typesL} [| a = b : A; A = B |] ==> a = b : B
1.15 +
1.16 +\idx{subst_type} [| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type
1.17 +
1.18 +\idx{subst_typeL} [| a = c : A; !!z. z:A ==> B(z) = D(z)
1.19 + |] ==> B(a) = D(c)
1.20 +\idx{subst_elem} [| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)
1.21 +
1.22 +\idx{subst_elemL} [| a = c : A; !!z. z:A ==> b(z) = d(z) : B(z)
1.23 + |] ==> b(a) = d(c) : B(a)
1.24 +
1.25 +\idx{refl_red} Reduce(a,a)
1.26 +\idx{red_if_equal} a = b : A ==> Reduce(a,b)
1.27 +\idx{trans_red} [| a = b : A; Reduce(b,c) |] ==> a = c : A
1.28 +
1.29 +
1.30 +\idx{NF} N type
1.31 +
1.32 +\idx{NI0} 0 : N
1.33 +\idx{NI_succ} a : N ==> succ(a) : N
1.34 +\idx{NI_succL} a = b : N ==> succ(a) = succ(b) : N
1.35 +
1.36 +\idx{NE} [| p: N; a: C(0);
1.37 + !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u))
1.38 + |] ==> rec(p, a, %u v.b(u,v)) : C(p)
1.39 +
1.40 +\idx{NEL} [| p = q : N; a = c : C(0);
1.41 + !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u))
1.42 + |] ==> rec(p, a, %u v.b(u,v)) = rec(q,c,d) : C(p)
1.43 +
1.44 +\idx{NC0} [| a: C(0);
1.45 + !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u))
1.46 + |] ==> rec(0, a, %u v.b(u,v)) = a : C(0)
1.47 +
1.48 +\idx{NC_succ} [| p: N; a: C(0);
1.49 + !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u))
1.50 + |] ==> rec(succ(p), a, %u v.b(u,v)) =
1.51 + b(p, rec(p, a, %u v.b(u,v))) : C(succ(p))
1.52 +
1.53 +\idx{zero_ne_succ} [| a: N; 0 = succ(a) : N |] ==> 0: F
1.54 +
1.55 +
1.56 +\idx{ProdF} [| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A.B(x) type
1.57 +\idx{ProdFL} [| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
1.58 + PROD x:A.B(x) = PROD x:C.D(x)
1.59 +
1.60 +\idx{ProdI} [| A type; !!x. x:A ==> b(x):B(x)
1.61 + |] ==> lam x.b(x) : PROD x:A.B(x)
1.62 +\idx{ProdIL} [| A type; !!x. x:A ==> b(x) = c(x) : B(x)
1.63 + |] ==> lam x.b(x) = lam x.c(x) : PROD x:A.B(x)
1.64 +
1.65 +\idx{ProdE} [| p : PROD x:A.B(x); a : A |] ==> p`a : B(a)
1.66 +\idx{ProdEL} [| p=q: PROD x:A.B(x); a=b : A |] ==> p`a = q`b : B(a)
1.67 +
1.68 +\idx{ProdC} [| a : A; !!x. x:A ==> b(x) : B(x)
1.69 + |] ==> (lam x.b(x)) ` a = b(a) : B(a)
1.70 +
1.71 +\idx{ProdC2} p : PROD x:A.B(x) ==> (lam x. p`x) = p : PROD x:A.B(x)
1.72 +
1.73 +
1.74 +\idx{SumF} [| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A.B(x) type
1.75 +\idx{SumFL} [| A = C; !!x. x:A ==> B(x) = D(x)
1.76 + |] ==> SUM x:A.B(x) = SUM x:C.D(x)
1.77 +
1.78 +\idx{SumI} [| a : A; b : B(a) |] ==> <a,b> : SUM x:A.B(x)
1.79 +\idx{SumIL} [| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A.B(x)
1.80 +
1.81 +\idx{SumE} [| p: SUM x:A.B(x);
1.82 + !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>)
1.83 + |] ==> split(p, %x y.c(x,y)) : C(p)
1.84 +
1.85 +\idx{SumEL} [| p=q : SUM x:A.B(x);
1.86 + !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)
1.87 + |] ==> split(p, %x y.c(x,y)) = split(q, % x y.d(x,y)) : C(p)
1.88 +
1.89 +\idx{SumC} [| a: A; b: B(a);
1.90 + !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>)
1.91 + |] ==> split(<a,b>, %x y.c(x,y)) = c(a,b) : C(<a,b>)
1.92 +
1.93 +\idx{fst_def} fst(a) == split(a, %x y.x)
1.94 +\idx{snd_def} snd(a) == split(a, %x y.y)
1.95 +
1.96 +
1.97 +\idx{PlusF} [| A type; B type |] ==> A+B type
1.98 +\idx{PlusFL} [| A = C; B = D |] ==> A+B = C+D
1.99 +
1.100 +\idx{PlusI_inl} [| a : A; B type |] ==> inl(a) : A+B
1.101 +\idx{PlusI_inlL} [| a = c : A; B type |] ==> inl(a) = inl(c) : A+B
1.102 +
1.103 +\idx{PlusI_inr} [| A type; b : B |] ==> inr(b) : A+B
1.104 +\idx{PlusI_inrL} [| A type; b = d : B |] ==> inr(b) = inr(d) : A+B
1.105 +
1.106 +\idx{PlusE} [| p: A+B;
1.107 + !!x. x:A ==> c(x): C(inl(x));
1.108 + !!y. y:B ==> d(y): C(inr(y))
1.109 + |] ==> when(p, %x.c(x), %y.d(y)) : C(p)
1.110 +
1.111 +\idx{PlusEL} [| p = q : A+B;
1.112 + !!x. x: A ==> c(x) = e(x) : C(inl(x));
1.113 + !!y. y: B ==> d(y) = f(y) : C(inr(y))
1.114 + |] ==> when(p, %x.c(x), %y.d(y)) = when(q, %x.e(x), %y.f(y)) : C(p)
1.115 +
1.116 +\idx{PlusC_inl} [| a: A;
1.117 + !!x. x:A ==> c(x): C(inl(x));
1.118 + !!y. y:B ==> d(y): C(inr(y))
1.119 + |] ==> when(inl(a), %x.c(x), %y.d(y)) = c(a) : C(inl(a))
1.120 +
1.121 +\idx{PlusC_inr} [| b: B;
1.122 + !!x. x:A ==> c(x): C(inl(x));
1.123 + !!y. y:B ==> d(y): C(inr(y))
1.124 + |] ==> when(inr(b), %x.c(x), %y.d(y)) = d(b) : C(inr(b))
1.125 +
1.126 +\idx{EqF} [| A type; a : A; b : A |] ==> Eq(A,a,b) type
1.127 +\idx{EqFL} [| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)
1.128 +\idx{EqI} a = b : A ==> eq : Eq(A,a,b)
1.129 +\idx{EqE} p : Eq(A,a,b) ==> a = b : A
1.130 +\idx{EqC} p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)
1.131 +
1.132 +\idx{FF} F type
1.133 +\idx{FE} [| p: F; C type |] ==> contr(p) : C
1.134 +\idx{FEL} [| p = q : F; C type |] ==> contr(p) = contr(q) : C
1.135 +
1.136 +\idx{TF} T type
1.137 +\idx{TI} tt : T
1.138 +\idx{TE} [| p : T; c : C(tt) |] ==> c : C(p)
1.139 +\idx{TEL} [| p = q : T; c = d : C(tt) |] ==> c = d : C(p)
1.140 +\idx{TC} p : T ==> p = tt : T)
1.141 +
1.142 +
1.143 +\idx{replace_type} [| B = A; a : A |] ==> a : B
1.144 +\idx{subst_eqtyparg} [| a=c : A; !!z. z:A ==> B(z) type |] ==> B(a)=B(c)
1.145 +
1.146 +\idx{subst_prodE} [| p: Prod(A,B); a: A; !!z. z: B(a) ==> c(z): C(z)
1.147 + |] ==> c(p`a): C(p`a)
1.148 +
1.149 +\idx{SumIL2} [| c=a : A; d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)
1.150 +
1.151 +\idx{SumE_fst} p : Sum(A,B) ==> fst(p) : A
1.152 +
1.153 +\idx{SumE_snd} [| p: Sum(A,B); A type; !!x. x:A ==> B(x) type
1.154 + |] ==> snd(p) : B(fst(p))
1.155 +
1.156 +
1.157 +
1.158 +\idx{add_def} a#+b == rec(a, b, %u v.succ(v))
1.159 +\idx{diff_def} a-b == rec(b, a, %u v.rec(v, 0, %x y.x))
1.160 +\idx{absdiff_def} a|-|b == (a-b) #+ (b-a)
1.161 +\idx{mult_def} a#*b == rec(a, 0, %u v. b #+ v)
1.162 +
1.163 +\idx{mod_def} a//b == rec(a, 0, %u v.
1.164 + rec(succ(v) |-| b, 0, %x y.succ(v)))
1.165 +
1.166 +\idx{quo_def} a/b == rec(a, 0, %u v.
1.167 + rec(succ(u) // b, succ(v), %x y.v))
1.168 +
1.169 +
1.170 +\idx{add_typing} [| a:N; b:N |] ==> a #+ b : N
1.171 +\idx{add_typingL} [| a=c:N; b=d:N |] ==> a #+ b = c #+ d : N
1.172 +\idx{addC0} b:N ==> 0 #+ b = b : N
1.173 +\idx{addC_succ} [| a:N; b:N |] ==> succ(a) #+ b = succ(a #+ b) : N
1.174 +\idx{mult_typing} [| a:N; b:N |] ==> a #* b : N
1.175 +\idx{mult_typingL} [| a=c:N; b=d:N |] ==> a #* b = c #* d : N
1.176 +\idx{multC0} b:N ==> 0 #* b = 0 : N
1.177 +\idx{multC_succ} [| a:N; b:N |] ==> succ(a) #* b = b #+ (a #* b) : N
1.178 +\idx{diff_typing} [| a:N; b:N |] ==> a - b : N
1.179 +\idx{diff_typingL} [| a=c:N; b=d:N |] ==> a - b = c - d : N
1.180 +\idx{diffC0} a:N ==> a - 0 = a : N
1.181 +\idx{diff_0_eq_0} b:N ==> 0 - b = 0 : N
1.182 +
1.183 +\idx{diff_succ_succ} [| a:N; b:N |] ==> succ(a) - succ(b) = a - b : N
1.184 +
1.185 +\idx{add_assoc} [| a:N; b:N; c:N |] ==> (a #+ b) #+ c = a #+ (b #+ c) : N
1.186 +
1.187 +\idx{add_commute} [| a:N; b:N |] ==> a #+ b = b #+ a : N
1.188 +
1.189 +\idx{mult_right0} a:N ==> a #* 0 = 0 : N
1.190 +
1.191 +\idx{mult_right_succ} [| a:N; b:N |] ==> a #* succ(b) = a #+ (a #* b) : N
1.192 +
1.193 +\idx{mult_commute} [| a:N; b:N |] ==> a #* b = b #* a : N
1.194 +
1.195 +\idx{add_mult_dist} [| a:N; b:N; c:N |] ==> (a #+ b) #* c = (a #* c) #+ (b #* c) : N
1.196 +
1.197 +\idx{mult_assoc} [| a:N; b:N; c:N |] ==> (a #* b) #* c = a #* (b #* c) : N
1.198 +
1.199 +\idx{diff_self_eq_0} a:N ==> a - a = 0 : N
1.200 +
1.201 +\idx{add_inverse_diff_lemma}
1.202 + b:N ==> ?a : PROD x:N. Eq(N, b-x, 0) --> Eq(N, b #+ (x-b), x)
1.203 +
1.204 +\idx{add_inverse_diff} [| a:N; b:N; b-a = 0 : N |] ==> b #+ (a-b) = a : N
1.205 +
1.206 +\idx{absdiff_typing} [| a:N; b:N |] ==> a |-| b : N
1.207 +\idx{absdiff_typingL} [| a=c:N; b=d:N |] ==> a |-| b = c |-| d : N
1.208 +\idx{absdiff_self_eq_0} a:N ==> a |-| a = 0 : N
1.209 +\idx{absdiffC0} a:N ==> 0 |-| a = a : N
1.210 +
1.211 +\idx{absdiff_succ_succ} [| a:N; b:N |] ==> succ(a) |-| succ(b) = a |-| b : N
1.212 +\idx{absdiff_commute} [| a:N; b:N |] ==> a |-| b = b |-| a : N
1.213 +
1.214 +\idx{add_eq0_lemma} [| a:N; b:N |] ==> ?c : PROD u: Eq(N,a#+b,0) . Eq(N,a,0)
1.215 +
1.216 +\idx{add_eq0} [| a:N; b:N; a #+ b = 0 : N |] ==> a = 0 : N
1.217 +
1.218 +\idx{absdiff_eq0_lem}
1.219 + [| a:N; b:N; a |-| b = 0 : N |] ==>
1.220 + ?a : SUM v: Eq(N, a-b, 0) . Eq(N, b-a, 0)
1.221 +
1.222 +\idx{absdiff_eq0} [| a |-| b = 0 : N; a:N; b:N |] ==> a = b : N
1.223 +
1.224 +\idx{mod_typing} [| a:N; b:N |] ==> a//b : N
1.225 +\idx{mod_typingL} [| a=c:N; b=d:N |] ==> a//b = c//d : N
1.226 +\idx{modC0} b:N ==> 0//b = 0 : N
1.227 +\idx{modC_succ}
1.228 +[| a:N; b:N |] ==> succ(a)//b = rec(succ(a//b) |-| b, 0, %x y.succ(a//b)) : N
1.229 +
1.230 +\idx{quo_typing} [| a:N; b:N |] ==> a / b : N
1.231 +\idx{quo_typingL} [| a=c:N; b=d:N |] ==> a / b = c / d : N
1.232 +\idx{quoC0} b:N ==> 0 / b = 0 : N
1.233 +[| a:N; b:N |] ==> succ(a) / b =
1.234 + rec(succ(a)//b, succ(a / b), %x y. a / b) : N
1.235 +\idx{quoC_succ2} [| a:N; b:N |] ==>
1.236 + succ(a) / b =rec(succ(a//b) |-| b, succ(a / b), %x y. a / b) : N
1.237 +
1.238 +\idx{iszero_decidable}
1.239 + a:N ==> rec(a, inl(eq), %ka kb.inr(<ka, eq>)) :
1.240 + Eq(N,a,0) + (SUM x:N. Eq(N,a, succ(x)))
1.241 +
1.242 +\idx{mod_quo_equality} [| a:N; b:N |] ==> a//b #+ (a/b) #* b = a : N
1.243 +
1.244 +