author lcp Mon, 06 Dec 1993 10:57:22 +0100 changeset 187 8729bfdcb638 parent 186 320f6bdb593a child 188 6be0856cdf49
ZF/univ/in_Vfrom_limit: new ZF/univ/sum_in_Vfrom, etc: streamlined proofs using in_Vfrom_limit
 src/ZF/Univ.ML file | annotate | diff | comparison | revisions src/ZF/univ.ML file | annotate | diff | comparison | revisions
```--- a/src/ZF/Univ.ML	Mon Dec 06 10:55:48 1993 +0100
+++ b/src/ZF/Univ.ML	Mon Dec 06 10:57:22 1993 +0100
@@ -303,14 +303,28 @@
and i is a limit ordinal
***)

-(*There are three nearly identical proofs below -- needs a general theorem
-  for proving  ...a...b : Vfrom(A,i) where i is a limit ordinal*)
+(*General theorem for membership in Vfrom(A,i) when i is a limit ordinal*)
+val [aprem,bprem,limiti,step] = goal Univ.thy
+  "[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);			\
+\     !!x y j. [| j<i; 1:j; x: Vfrom(A,j); y: Vfrom(A,j) \
+\              |] ==> EX k. h(x,y): Vfrom(A,k) & k<i |] ==> 	\
+\  h(a,b) : Vfrom(A,i)";
+(*Infer that a, b occur at ordinals x,xa < i.*)
+by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
+by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
+by (res_inst_tac [("j1", "x Un xa Un succ(1)")] (step RS exE) 1);
+by (DO_GOAL [etac conjE, etac Limit_VfromI, rtac limiti, atac] 5);
+by (etac (Vfrom_UnI2 RS Vfrom_UnI1) 4);
+by (etac (Vfrom_UnI1 RS Vfrom_UnI1) 3);
+by (rtac (succI1 RS UnI2) 2);
+by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt] 1));
+val in_Vfrom_limit = result();

(** products **)

goal Univ.thy
-    "!!A. [| a: Vfrom(A,i);  b: Vfrom(A,i);  Transset(A) |] ==> \
-\         a*b : Vfrom(A, succ(succ(succ(i))))";
+    "!!A. [| a: Vfrom(A,j);  b: Vfrom(A,j);  Transset(A) |] ==> \
+\         a*b : Vfrom(A, succ(succ(succ(j))))";
by (dtac Transset_Vfrom 1);
by (rtac subset_mem_Vfrom 1);
by (rewtac Transset_def);
@@ -320,21 +334,16 @@
val [aprem,bprem,limiti,transset] = goal Univ.thy
"[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);  Transset(A) |] ==> \
\  a*b : Vfrom(A,i)";
-(*Infer that a, b occur at ordinals x,xa < i.*)
-by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([prod_in_Vfrom, limiti] MRS Limit_VfromI) 1);
-by (etac (Vfrom_UnI1 RS Vfrom_UnI2) 1);
-by (etac (Vfrom_UnI2 RS Vfrom_UnI2) 1);
-(*Infer that succ(succ(succ(x Un xa))) < i *)
-by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt, transset] 1));
+by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1);
+by (REPEAT (ares_tac [exI, conjI, prod_in_Vfrom, transset,
+		      limiti RS Limit_has_succ] 1));
val prod_in_Vfrom_limit = result();

(** Disjoint sums, aka Quine ordered pairs **)

goalw Univ.thy [sum_def]
-    "!!A. [| a: Vfrom(A,i);  b: Vfrom(A,i);  Transset(A);  1:i |] ==> \
-\         a+b : Vfrom(A, succ(succ(succ(i))))";
+    "!!A. [| a: Vfrom(A,j);  b: Vfrom(A,j);  Transset(A);  1:j |] ==> \
+\         a+b : Vfrom(A, succ(succ(succ(j))))";
by (dtac Transset_Vfrom 1);
by (rtac subset_mem_Vfrom 1);
by (rewtac Transset_def);
@@ -345,23 +354,16 @@
val [aprem,bprem,limiti,transset] = goal Univ.thy
"[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);  Transset(A) |] ==> \
\  a+b : Vfrom(A,i)";
-(*Infer that a, b occur at ordinals x,xa < i.*)
-by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([sum_in_Vfrom, limiti] MRS Limit_VfromI) 1);
-by (rtac (succI1 RS UnI1) 4);
-(*Infer that  succ(succ(succ(succ(1) Un (x Un xa)))) < i *)
-by (etac (Vfrom_UnI1 RS Vfrom_UnI2) 1);
-by (etac (Vfrom_UnI2 RS Vfrom_UnI2) 1);
-by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt,
-		      transset] 1));
+by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1);
+by (REPEAT (ares_tac [exI, conjI, sum_in_Vfrom, transset,
+		      limiti RS Limit_has_succ] 1));
val sum_in_Vfrom_limit = result();

(** function space! **)

goalw Univ.thy [Pi_def]
-    "!!A. [| a: Vfrom(A,i);  b: Vfrom(A,i);  Transset(A) |] ==> \
-\         a->b : Vfrom(A, succ(succ(succ(succ(i)))))";
+    "!!A. [| a: Vfrom(A,j);  b: Vfrom(A,j);  Transset(A) |] ==> \
+\         a->b : Vfrom(A, succ(succ(succ(succ(j)))))";
by (dtac Transset_Vfrom 1);
by (rtac subset_mem_Vfrom 1);
by (rtac (Collect_subset RS subset_trans) 1);
@@ -377,14 +379,9 @@
val [aprem,bprem,limiti,transset] = goal Univ.thy
"[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);  Transset(A) |] ==> \
\  a->b : Vfrom(A,i)";
-(*Infer that a, b occur at ordinals x,xa < i.*)
-by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([fun_in_Vfrom, limiti] MRS Limit_VfromI) 1);
-(*Infer that succ(succ(succ(x Un xa))) < i *)
-by (etac (Vfrom_UnI1 RS Vfrom_UnI2) 1);
-by (etac (Vfrom_UnI2 RS Vfrom_UnI2) 1);
-by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt, transset] 1));
+by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1);
+by (REPEAT (ares_tac [exI, conjI, fun_in_Vfrom, transset,
+		      limiti RS Limit_has_succ] 1));
val fun_in_Vfrom_limit = result();

```
```--- a/src/ZF/univ.ML	Mon Dec 06 10:55:48 1993 +0100
+++ b/src/ZF/univ.ML	Mon Dec 06 10:57:22 1993 +0100
@@ -303,14 +303,28 @@
and i is a limit ordinal
***)

-(*There are three nearly identical proofs below -- needs a general theorem
-  for proving  ...a...b : Vfrom(A,i) where i is a limit ordinal*)
+(*General theorem for membership in Vfrom(A,i) when i is a limit ordinal*)
+val [aprem,bprem,limiti,step] = goal Univ.thy
+  "[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);			\
+\     !!x y j. [| j<i; 1:j; x: Vfrom(A,j); y: Vfrom(A,j) \
+\              |] ==> EX k. h(x,y): Vfrom(A,k) & k<i |] ==> 	\
+\  h(a,b) : Vfrom(A,i)";
+(*Infer that a, b occur at ordinals x,xa < i.*)
+by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
+by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
+by (res_inst_tac [("j1", "x Un xa Un succ(1)")] (step RS exE) 1);
+by (DO_GOAL [etac conjE, etac Limit_VfromI, rtac limiti, atac] 5);
+by (etac (Vfrom_UnI2 RS Vfrom_UnI1) 4);
+by (etac (Vfrom_UnI1 RS Vfrom_UnI1) 3);
+by (rtac (succI1 RS UnI2) 2);
+by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt] 1));
+val in_Vfrom_limit = result();

(** products **)

goal Univ.thy
-    "!!A. [| a: Vfrom(A,i);  b: Vfrom(A,i);  Transset(A) |] ==> \
-\         a*b : Vfrom(A, succ(succ(succ(i))))";
+    "!!A. [| a: Vfrom(A,j);  b: Vfrom(A,j);  Transset(A) |] ==> \
+\         a*b : Vfrom(A, succ(succ(succ(j))))";
by (dtac Transset_Vfrom 1);
by (rtac subset_mem_Vfrom 1);
by (rewtac Transset_def);
@@ -320,21 +334,16 @@
val [aprem,bprem,limiti,transset] = goal Univ.thy
"[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);  Transset(A) |] ==> \
\  a*b : Vfrom(A,i)";
-(*Infer that a, b occur at ordinals x,xa < i.*)
-by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([prod_in_Vfrom, limiti] MRS Limit_VfromI) 1);
-by (etac (Vfrom_UnI1 RS Vfrom_UnI2) 1);
-by (etac (Vfrom_UnI2 RS Vfrom_UnI2) 1);
-(*Infer that succ(succ(succ(x Un xa))) < i *)
-by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt, transset] 1));
+by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1);
+by (REPEAT (ares_tac [exI, conjI, prod_in_Vfrom, transset,
+		      limiti RS Limit_has_succ] 1));
val prod_in_Vfrom_limit = result();

(** Disjoint sums, aka Quine ordered pairs **)

goalw Univ.thy [sum_def]
-    "!!A. [| a: Vfrom(A,i);  b: Vfrom(A,i);  Transset(A);  1:i |] ==> \
-\         a+b : Vfrom(A, succ(succ(succ(i))))";
+    "!!A. [| a: Vfrom(A,j);  b: Vfrom(A,j);  Transset(A);  1:j |] ==> \
+\         a+b : Vfrom(A, succ(succ(succ(j))))";
by (dtac Transset_Vfrom 1);
by (rtac subset_mem_Vfrom 1);
by (rewtac Transset_def);
@@ -345,23 +354,16 @@
val [aprem,bprem,limiti,transset] = goal Univ.thy
"[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);  Transset(A) |] ==> \
\  a+b : Vfrom(A,i)";
-(*Infer that a, b occur at ordinals x,xa < i.*)
-by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([sum_in_Vfrom, limiti] MRS Limit_VfromI) 1);
-by (rtac (succI1 RS UnI1) 4);
-(*Infer that  succ(succ(succ(succ(1) Un (x Un xa)))) < i *)
-by (etac (Vfrom_UnI1 RS Vfrom_UnI2) 1);
-by (etac (Vfrom_UnI2 RS Vfrom_UnI2) 1);
-by (REPEAT (ares_tac [limiti, Limit_has_0, Limit_has_succ, Un_least_lt,
-		      transset] 1));
+by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1);
+by (REPEAT (ares_tac [exI, conjI, sum_in_Vfrom, transset,
+		      limiti RS Limit_has_succ] 1));
val sum_in_Vfrom_limit = result();

(** function space! **)

goalw Univ.thy [Pi_def]
-    "!!A. [| a: Vfrom(A,i);  b: Vfrom(A,i);  Transset(A) |] ==> \
-\         a->b : Vfrom(A, succ(succ(succ(succ(i)))))";
+    "!!A. [| a: Vfrom(A,j);  b: Vfrom(A,j);  Transset(A) |] ==> \
+\         a->b : Vfrom(A, succ(succ(succ(succ(j)))))";
by (dtac Transset_Vfrom 1);
by (rtac subset_mem_Vfrom 1);
by (rtac (Collect_subset RS subset_trans) 1);
@@ -377,14 +379,9 @@
val [aprem,bprem,limiti,transset] = goal Univ.thy
"[| a: Vfrom(A,i);  b: Vfrom(A,i);  Limit(i);  Transset(A) |] ==> \
\  a->b : Vfrom(A,i)";
-(*Infer that a, b occur at ordinals x,xa < i.*)
-by (rtac ([aprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([bprem,limiti] MRS Limit_VfromE) 1);
-by (rtac ([fun_in_Vfrom, limiti] MRS Limit_VfromI) 1);
-(*Infer that succ(succ(succ(x Un xa))) < i *)
-by (etac (Vfrom_UnI1 RS Vfrom_UnI2) 1);
-by (etac (Vfrom_UnI2 RS Vfrom_UnI2) 1);
-by (REPEAT (ares_tac [limiti, Limit_has_succ, Un_least_lt, transset] 1));
+by (rtac ([aprem,bprem,limiti] MRS in_Vfrom_limit) 1);
+by (REPEAT (ares_tac [exI, conjI, fun_in_Vfrom, transset,
+		      limiti RS Limit_has_succ] 1));
val fun_in_Vfrom_limit = result();

```