doc-src/Logics/CTT-rules.txt
author paulson
Fri Feb 16 18:00:47 1996 +0100 (1996-02-16)
changeset 1512 ce37c64244c0
parent 104 d8205bb279a7
permissions -rw-r--r--
Elimination of fully-functorial style.
Type tactic changed to a type abbrevation (from a datatype).
Constructor tactic and function apply deleted.
     1 \idx{refl_type}         A type ==> A = A), 
     2 \idx{refl_elem}         a : A ==> a = a : A
     3 
     4 \idx{sym_type}          A = B ==> B = A
     5 \idx{sym_elem}          a = b : A ==> b = a : A
     6 
     7 \idx{trans_type}        [| A = B;  B = C |] ==> A = C
     8 \idx{trans_elem}        [| a = b : A;  b = c : A |] ==> a = c : A
     9 
    10 \idx{equal_types}       [| a : A;  A = B |] ==> a : B
    11 \idx{equal_typesL}      [| a = b : A;  A = B |] ==> a = b : B
    12 
    13 \idx{subst_type}        [| a : A;  !!z. z:A ==> B(z) type |] ==> B(a) type
    14 
    15 \idx{subst_typeL}       [| a = c : A;  !!z. z:A ==> B(z) = D(z) 
    16                   |] ==> B(a) = D(c)
    17 \idx{subst_elem}        [| a : A;  !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)
    18 
    19 \idx{subst_elemL}       [| a = c : A;  !!z. z:A ==> b(z) = d(z) : B(z) 
    20                   |] ==> b(a) = d(c) : B(a)
    21 
    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
    25 
    26 
    27 \idx{NF}        N type
    28 
    29 \idx{NI0}       0 : N
    30 \idx{NI_succ}   a : N ==> succ(a) : N
    31 \idx{NI_succL}  a = b : N ==> succ(a) = succ(b) : N
    32 
    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)
    36 
    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)
    40 
    41 \idx{NC0}       [| a: C(0);  
    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)
    44 
    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))
    49 
    50 \idx{zero_ne_succ}      [| a: N;  0 = succ(a) : N |] ==> 0: F
    51 
    52 
    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)
    56 
    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)
    61 
    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)
    64 
    65 \idx{ProdC}     [| a : A;  !!x. x:A ==> b(x) : B(x)
    66           |] ==> (lam x.b(x)) ` a = b(a) : B(a)
    67 
    68 \idx{ProdC2}    p : PROD x:A.B(x) ==> (lam x. p`x) = p : PROD x:A.B(x)
    69 
    70 
    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)
    74 
    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)
    77 
    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)
    81 
    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)
    85 
    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>)
    89 
    90 \idx{fst_def}   fst(a) == split(a, %x y.x)
    91 \idx{snd_def}   snd(a) == split(a, %x y.y)
    92 
    93 
    94 \idx{PlusF}     [| A type;  B type |] ==> A+B type
    95 \idx{PlusFL}    [| A = C;  B = D |] ==> A+B = C+D
    96 
    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
    99 
   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
   102 
   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)
   107 
   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)
   112 
   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))
   117 
   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))
   122 
   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)
   128 
   129 \idx{FF}        F type
   130 \idx{FE}        [| p: F;  C type |] ==> contr(p) : C
   131 \idx{FEL}       [| p = q : F;  C type |] ==> contr(p) = contr(q) : C
   132 
   133 \idx{TF}        T type
   134 \idx{TI}        tt : T
   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)
   138 
   139 
   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)
   142 
   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)
   145 
   146 \idx{SumIL2}    [| c=a : A;  d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)
   147 
   148 \idx{SumE_fst}  p : Sum(A,B) ==> fst(p) : A
   149 
   150 \idx{SumE_snd}  [| p: Sum(A,B);  A type;  !!x. x:A ==> B(x) type
   151           |] ==> snd(p) : B(fst(p))
   152 
   153 
   154 
   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)  
   159 
   160 \idx{mod_def}	a//b == rec(a, 0, %u v.   
   161   			rec(succ(v) |-| b, 0, %x y.succ(v)))  
   162 
   163 \idx{quo_def}	a/b == rec(a, 0, %u v.   
   164   			rec(succ(u) // b, succ(v), %x y.v))
   165 
   166 
   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
   179 
   180 \idx{diff_succ_succ}    [| a:N;  b:N |] ==> succ(a) - succ(b) = a - b : N
   181 
   182 \idx{add_assoc} [| a:N;  b:N;  c:N |] ==> (a #+ b) #+ c = a #+ (b #+ c) : N
   183 
   184 \idx{add_commute}       [| a:N;  b:N |] ==> a #+ b = b #+ a : N
   185 
   186 \idx{mult_right0}       a:N ==> a #* 0 = 0 : N
   187 
   188 \idx{mult_right_succ}   [| a:N;  b:N |] ==> a #* succ(b) = a #+ (a #* b) : N
   189 
   190 \idx{mult_commute}      [| a:N;  b:N |] ==> a #* b = b #* a : N
   191 
   192 \idx{add_mult_dist}     [| a:N;  b:N;  c:N |] ==> (a #+ b) #* c = (a #* c) #+ (b #* c) : N
   193 
   194 \idx{mult_assoc}        [| a:N;  b:N;  c:N |] ==> (a #* b) #* c = a #* (b #* c) : N
   195 
   196 \idx{diff_self_eq_0}    a:N ==> a - a = 0 : N
   197 
   198 \idx{add_inverse_diff_lemma}    
   199     b:N ==> ?a : PROD x:N. Eq(N, b-x, 0) --> Eq(N, b #+ (x-b), x)
   200 
   201 \idx{add_inverse_diff}  [| a:N;  b:N;  b-a = 0 : N |] ==> b #+ (a-b) = a : N
   202 
   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
   207 
   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
   210 
   211 \idx{add_eq0_lemma}     [| a:N;  b:N |] ==> ?c : PROD u: Eq(N,a#+b,0) .  Eq(N,a,0)
   212 
   213 \idx{add_eq0}   [| a:N;  b:N;  a #+ b = 0 : N |] ==> a = 0 : N
   214 
   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)
   218 
   219 \idx{absdiff_eq0}       [| a |-| b = 0 : N;  a:N;  b:N |] ==> a = b : N
   220 
   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
   224 \idx{modC_succ} 
   225 [| a:N; b:N |] ==> succ(a)//b = rec(succ(a//b) |-| b, 0, %x y.succ(a//b)) : N
   226 
   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
   234 
   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)))
   238 
   239 \idx{mod_quo_equality}  [| a:N;  b:N |] ==> a//b  #+  (a/b) #* b = a : N
   240 
   241