1.1 --- a/src/CTT/CTT.thy Thu Feb 28 13:54:45 2013 +0100
1.2 +++ b/src/CTT/CTT.thy Thu Feb 28 14:10:54 2013 +0100
1.3 @@ -93,175 +93,174 @@
1.4 "_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10)
1.5 "_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10)
1.6
1.7 -axioms
1.8 -
1.9 (*Reduction: a weaker notion than equality; a hack for simplification.
1.10 Reduce[a,b] means either that a=b:A for some A or else that "a" and "b"
1.11 are textually identical.*)
1.12
1.13 (*does not verify a:A! Sound because only trans_red uses a Reduce premise
1.14 No new theorems can be proved about the standard judgements.*)
1.15 - refl_red: "Reduce[a,a]"
1.16 - red_if_equal: "a = b : A ==> Reduce[a,b]"
1.17 - trans_red: "[| a = b : A; Reduce[b,c] |] ==> a = c : A"
1.18 +axiomatization where
1.19 + refl_red: "\<And>a. Reduce[a,a]" and
1.20 + red_if_equal: "\<And>a b A. a = b : A ==> Reduce[a,b]" and
1.21 + trans_red: "\<And>a b c A. [| a = b : A; Reduce[b,c] |] ==> a = c : A" and
1.22
1.23 (*Reflexivity*)
1.24
1.25 - refl_type: "A type ==> A = A"
1.26 - refl_elem: "a : A ==> a = a : A"
1.27 + refl_type: "\<And>A. A type ==> A = A" and
1.28 + refl_elem: "\<And>a A. a : A ==> a = a : A" and
1.29
1.30 (*Symmetry*)
1.31
1.32 - sym_type: "A = B ==> B = A"
1.33 - sym_elem: "a = b : A ==> b = a : A"
1.34 + sym_type: "\<And>A B. A = B ==> B = A" and
1.35 + sym_elem: "\<And>a b A. a = b : A ==> b = a : A" and
1.36
1.37 (*Transitivity*)
1.38
1.39 - trans_type: "[| A = B; B = C |] ==> A = C"
1.40 - trans_elem: "[| a = b : A; b = c : A |] ==> a = c : A"
1.41 + trans_type: "\<And>A B C. [| A = B; B = C |] ==> A = C" and
1.42 + trans_elem: "\<And>a b c A. [| a = b : A; b = c : A |] ==> a = c : A" and
1.43
1.44 - equal_types: "[| a : A; A = B |] ==> a : B"
1.45 - equal_typesL: "[| a = b : A; A = B |] ==> a = b : B"
1.46 + equal_types: "\<And>a A B. [| a : A; A = B |] ==> a : B" and
1.47 + equal_typesL: "\<And>a b A B. [| a = b : A; A = B |] ==> a = b : B" and
1.48
1.49 (*Substitution*)
1.50
1.51 - subst_type: "[| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type"
1.52 - subst_typeL: "[| a = c : A; !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"
1.53 + subst_type: "\<And>a A B. [| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type" and
1.54 + subst_typeL: "\<And>a c A B D. [| a = c : A; !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)" and
1.55
1.56 - subst_elem: "[| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"
1.57 + subst_elem: "\<And>a b A B. [| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)" and
1.58 subst_elemL:
1.59 - "[| a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"
1.60 + "\<And>a b c d A B. [| a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)" and
1.61
1.62
1.63 (*The type N -- natural numbers*)
1.64
1.65 - NF: "N type"
1.66 - NI0: "0 : N"
1.67 - NI_succ: "a : N ==> succ(a) : N"
1.68 - NI_succL: "a = b : N ==> succ(a) = succ(b) : N"
1.69 + NF: "N type" and
1.70 + NI0: "0 : N" and
1.71 + NI_succ: "\<And>a. a : N ==> succ(a) : N" and
1.72 + NI_succL: "\<And>a b. a = b : N ==> succ(a) = succ(b) : N" and
1.73
1.74 NE:
1.75 - "[| p: N; a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
1.76 - ==> rec(p, a, %u v. b(u,v)) : C(p)"
1.77 + "\<And>p a b C. [| p: N; a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
1.78 + ==> rec(p, a, %u v. b(u,v)) : C(p)" and
1.79
1.80 NEL:
1.81 - "[| p = q : N; a = c : C(0);
1.82 + "\<And>p q a b c d C. [| p = q : N; a = c : C(0);
1.83 !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |]
1.84 - ==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)"
1.85 + ==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)" and
1.86
1.87 NC0:
1.88 - "[| a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
1.89 - ==> rec(0, a, %u v. b(u,v)) = a : C(0)"
1.90 + "\<And>a b C. [| a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
1.91 + ==> rec(0, a, %u v. b(u,v)) = a : C(0)" and
1.92
1.93 NC_succ:
1.94 - "[| p: N; a: C(0);
1.95 + "\<And>p a b C. [| p: N; a: C(0);
1.96 !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==>
1.97 - rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"
1.98 + rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))" and
1.99
1.100 (*The fourth Peano axiom. See page 91 of Martin-Lof's book*)
1.101 zero_ne_succ:
1.102 - "[| a: N; 0 = succ(a) : N |] ==> 0: F"
1.103 + "\<And>a. [| a: N; 0 = succ(a) : N |] ==> 0: F" and
1.104
1.105
1.106 (*The Product of a family of types*)
1.107
1.108 - ProdF: "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type"
1.109 + ProdF: "\<And>A B. [| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type" and
1.110
1.111 ProdFL:
1.112 - "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
1.113 - PROD x:A. B(x) = PROD x:C. D(x)"
1.114 + "\<And>A B C D. [| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
1.115 + PROD x:A. B(x) = PROD x:C. D(x)" and
1.116
1.117 ProdI:
1.118 - "[| A type; !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)"
1.119 + "\<And>b A B. [| A type; !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)" and
1.120
1.121 ProdIL:
1.122 - "[| A type; !!x. x:A ==> b(x) = c(x) : B(x)|] ==>
1.123 - lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"
1.124 + "\<And>b c A B. [| A type; !!x. x:A ==> b(x) = c(x) : B(x)|] ==>
1.125 + lam x. b(x) = lam x. c(x) : PROD x:A. B(x)" and
1.126
1.127 - ProdE: "[| p : PROD x:A. B(x); a : A |] ==> p`a : B(a)"
1.128 - ProdEL: "[| p=q: PROD x:A. B(x); a=b : A |] ==> p`a = q`b : B(a)"
1.129 + ProdE: "\<And>p a A B. [| p : PROD x:A. B(x); a : A |] ==> p`a : B(a)" and
1.130 + ProdEL: "\<And>p q a b A B. [| p=q: PROD x:A. B(x); a=b : A |] ==> p`a = q`b : B(a)" and
1.131
1.132 ProdC:
1.133 - "[| a : A; !!x. x:A ==> b(x) : B(x)|] ==>
1.134 - (lam x. b(x)) ` a = b(a) : B(a)"
1.135 + "\<And>a b A B. [| a : A; !!x. x:A ==> b(x) : B(x)|] ==>
1.136 + (lam x. b(x)) ` a = b(a) : B(a)" and
1.137
1.138 ProdC2:
1.139 - "p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"
1.140 + "\<And>p A B. p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)" and
1.141
1.142
1.143 (*The Sum of a family of types*)
1.144
1.145 - SumF: "[| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type"
1.146 + SumF: "\<And>A B. [| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type" and
1.147 SumFL:
1.148 - "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)"
1.149 + "\<And>A B C D. [| A = C; !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)" and
1.150
1.151 - SumI: "[| a : A; b : B(a) |] ==> <a,b> : SUM x:A. B(x)"
1.152 - SumIL: "[| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)"
1.153 + SumI: "\<And>a b A B. [| a : A; b : B(a) |] ==> <a,b> : SUM x:A. B(x)" and
1.154 + SumIL: "\<And>a b c d A B. [| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)" and
1.155
1.156 SumE:
1.157 - "[| p: SUM x:A. B(x); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
1.158 - ==> split(p, %x y. c(x,y)) : C(p)"
1.159 + "\<And>p c A B C. [| p: SUM x:A. B(x); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
1.160 + ==> split(p, %x y. c(x,y)) : C(p)" and
1.161
1.162 SumEL:
1.163 - "[| p=q : SUM x:A. B(x);
1.164 + "\<And>p q c d A B C. [| p=q : SUM x:A. B(x);
1.165 !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|]
1.166 - ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"
1.167 + ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)" and
1.168
1.169 SumC:
1.170 - "[| a: A; b: B(a); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
1.171 - ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"
1.172 + "\<And>a b c A B C. [| a: A; b: B(a); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
1.173 + ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)" and
1.174
1.175 - fst_def: "fst(a) == split(a, %x y. x)"
1.176 - snd_def: "snd(a) == split(a, %x y. y)"
1.177 + fst_def: "\<And>a. fst(a) == split(a, %x y. x)" and
1.178 + snd_def: "\<And>a. snd(a) == split(a, %x y. y)" and
1.179
1.180
1.181 (*The sum of two types*)
1.182
1.183 - PlusF: "[| A type; B type |] ==> A+B type"
1.184 - PlusFL: "[| A = C; B = D |] ==> A+B = C+D"
1.185 + PlusF: "\<And>A B. [| A type; B type |] ==> A+B type" and
1.186 + PlusFL: "\<And>A B C D. [| A = C; B = D |] ==> A+B = C+D" and
1.187
1.188 - PlusI_inl: "[| a : A; B type |] ==> inl(a) : A+B"
1.189 - PlusI_inlL: "[| a = c : A; B type |] ==> inl(a) = inl(c) : A+B"
1.190 + PlusI_inl: "\<And>a A B. [| a : A; B type |] ==> inl(a) : A+B" and
1.191 + PlusI_inlL: "\<And>a c A B. [| a = c : A; B type |] ==> inl(a) = inl(c) : A+B" and
1.192
1.193 - PlusI_inr: "[| A type; b : B |] ==> inr(b) : A+B"
1.194 - PlusI_inrL: "[| A type; b = d : B |] ==> inr(b) = inr(d) : A+B"
1.195 + PlusI_inr: "\<And>b A B. [| A type; b : B |] ==> inr(b) : A+B" and
1.196 + PlusI_inrL: "\<And>b d A B. [| A type; b = d : B |] ==> inr(b) = inr(d) : A+B" and
1.197
1.198 PlusE:
1.199 - "[| p: A+B; !!x. x:A ==> c(x): C(inl(x));
1.200 + "\<And>p c d A B C. [| p: A+B; !!x. x:A ==> c(x): C(inl(x));
1.201 !!y. y:B ==> d(y): C(inr(y)) |]
1.202 - ==> when(p, %x. c(x), %y. d(y)) : C(p)"
1.203 + ==> when(p, %x. c(x), %y. d(y)) : C(p)" and
1.204
1.205 PlusEL:
1.206 - "[| p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x));
1.207 + "\<And>p q c d e f A B C. [| p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x));
1.208 !!y. y: B ==> d(y) = f(y) : C(inr(y)) |]
1.209 - ==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"
1.210 + ==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)" and
1.211
1.212 PlusC_inl:
1.213 - "[| a: A; !!x. x:A ==> c(x): C(inl(x));
1.214 + "\<And>a c d A C. [| a: A; !!x. x:A ==> c(x): C(inl(x));
1.215 !!y. y:B ==> d(y): C(inr(y)) |]
1.216 - ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"
1.217 + ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))" and
1.218
1.219 PlusC_inr:
1.220 - "[| b: B; !!x. x:A ==> c(x): C(inl(x));
1.221 + "\<And>b c d A B C. [| b: B; !!x. x:A ==> c(x): C(inl(x));
1.222 !!y. y:B ==> d(y): C(inr(y)) |]
1.223 - ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"
1.224 + ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))" and
1.225
1.226
1.227 (*The type Eq*)
1.228
1.229 - EqF: "[| A type; a : A; b : A |] ==> Eq(A,a,b) type"
1.230 - EqFL: "[| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"
1.231 - EqI: "a = b : A ==> eq : Eq(A,a,b)"
1.232 - EqE: "p : Eq(A,a,b) ==> a = b : A"
1.233 + EqF: "\<And>a b A. [| A type; a : A; b : A |] ==> Eq(A,a,b) type" and
1.234 + EqFL: "\<And>a b c d A B. [| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)" and
1.235 + EqI: "\<And>a b A. a = b : A ==> eq : Eq(A,a,b)" and
1.236 + EqE: "\<And>p a b A. p : Eq(A,a,b) ==> a = b : A" and
1.237
1.238 (*By equality of types, can prove C(p) from C(eq), an elimination rule*)
1.239 - EqC: "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"
1.240 + EqC: "\<And>p a b A. p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)" and
1.241
1.242 (*The type F*)
1.243
1.244 - FF: "F type"
1.245 - FE: "[| p: F; C type |] ==> contr(p) : C"
1.246 - FEL: "[| p = q : F; C type |] ==> contr(p) = contr(q) : C"
1.247 + FF: "F type" and
1.248 + FE: "\<And>p C. [| p: F; C type |] ==> contr(p) : C" and
1.249 + FEL: "\<And>p q C. [| p = q : F; C type |] ==> contr(p) = contr(q) : C" and
1.250
1.251 (*The type T
1.252 Martin-Lof's book (page 68) discusses elimination and computation.
1.253 @@ -269,11 +268,11 @@
1.254 but with an extra premise C(x) type x:T.
1.255 Also computation can be derived from elimination. *)
1.256
1.257 - TF: "T type"
1.258 - TI: "tt : T"
1.259 - TE: "[| p : T; c : C(tt) |] ==> c : C(p)"
1.260 - TEL: "[| p = q : T; c = d : C(tt) |] ==> c = d : C(p)"
1.261 - TC: "p : T ==> p = tt : T"
1.262 + TF: "T type" and
1.263 + TI: "tt : T" and
1.264 + TE: "\<And>p c C. [| p : T; c : C(tt) |] ==> c : C(p)" and
1.265 + TEL: "\<And>p q c d C. [| p = q : T; c = d : C(tt) |] ==> c = d : C(p)" and
1.266 + TC: "\<And>p. p : T ==> p = tt : T"
1.267
1.268
1.269 subsection "Tactics and derived rules for Constructive Type Theory"