src/CCL/ex/List.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/CCL/ex/List.ML	Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,108 @@
+(*  Title: 	CCL/ex/list
+    ID:         $Id$
+    Author: 	Martin Coen, Cambridge University Computer Laboratory
+    Copyright   1993  University of Cambridge
+
+For list.thy.
+*)
+
+open List;
+
+val list_defs = [map_def,comp_def,append_def,filter_def,flat_def,
+                 insert_def,isort_def,partition_def,qsort_def];
+
+(****)
+
+val listBs = map (fn s=>prove_goalw List.thy list_defs s (fn _ => [SIMP_TAC term_ss 1]))
+     ["(f o g) = (%a.f(g(a)))",
+      "(f o g)(a) = f(g(a))",
+      "map(f,[]) = []",
+      "map(f,x.xs) = f(x).map(f,xs)",
+      "[] @ m = m",
+      "x.xs @ m = x.(xs @ m)",
+      "filter(f,[]) = []",
+      "filter(f,x.xs) = if f`x then x.filter(f,xs) else filter(f,xs)",
+      "flat([]) = []",
+      "flat(x.xs) = x @ flat(xs)",
+      "insert(f,a,[]) = a.[]",
+      "insert(f,a,x.xs) = if f`a`x then a.x.xs else x.insert(f,a,xs)"];
+
+val list_congs = ccl_mk_congs List.thy ["map","op @","filter","flat","insert","napply"];
+
+val list_ss = nat_ss addrews listBs addcongs list_congs;
+
+(****)
+
+val [prem] = goal List.thy "n:Nat ==> map(f) ^ n ` [] = []";
+br (prem RS Nat_ind) 1;
+by (ALLGOALS (ASM_SIMP_TAC list_ss));
+val nmapBnil = result();
+
+val [prem] = goal List.thy "n:Nat ==> map(f)^n`(x.xs) = f^n`x.map(f)^n`xs";
+br (prem RS Nat_ind) 1;
+by (ALLGOALS (ASM_SIMP_TAC list_ss));
+val nmapBcons = result();
+
+(***)
+
+val prems = goalw List.thy [map_def]
+  "[| !!x.x:A==>f(x):B;  l : List(A) |] ==> map(f,l) : List(B)";
+by (typechk_tac prems 1);
+val mapT = result();
+
+val prems = goalw List.thy [append_def]
+  "[| l : List(A);  m : List(A) |] ==> l @ m : List(A)";
+by (typechk_tac prems 1);
+val appendT = result();
+
+val prems = goal List.thy
+  "[| l : {l:List(A). m : {m:List(A).P(l @ m)}} |] ==> l @ m : {x:List(A). P(x)}";
+by (cut_facts_tac prems 1);
+by (fast_tac (set_cs addSIs [SubtypeI,appendT] addSEs [SubtypeE]) 1);
+val appendTS = result();
+
+val prems = goalw List.thy [filter_def]
+  "[| f:A->Bool;   l : List(A) |] ==> filter(f,l) : List(A)";
+by (typechk_tac prems 1);
+val filterT = result();
+
+val prems = goalw List.thy [flat_def]
+  "l : List(List(A)) ==> flat(l) : List(A)";
+by (typechk_tac (appendT::prems) 1);
+val flatT = result();
+
+val prems = goalw List.thy [insert_def]
+  "[|  f : A->A->Bool; a:A; l : List(A) |] ==> insert(f,a,l) : List(A)";
+by (typechk_tac prems 1);
+val insertT = result();
+
+val prems = goal List.thy
+  "[| f : {f:A->A->Bool. a : {a:A. l : {l:List(A).P(insert(f,a,l))}}} |] ==> \
+\  insert(f,a,l)  : {x:List(A). P(x)}";
+by (cut_facts_tac prems 1);
+by (fast_tac (set_cs addSIs [SubtypeI,insertT] addSEs [SubtypeE]) 1);
+val insertTS = result();
+
+val prems = goalw List.thy [partition_def]
+  "[| f:A->Bool;  l : List(A) |] ==> partition(f,l) : List(A)*List(A)";
+by (typechk_tac prems 1);
+by clean_ccs_tac;
+br (ListPRI RS wfstI RS (ListPR_wf RS wmap_wf RS wfI)) 2;
+br (ListPRI RS wfstI RS (ListPR_wf RS wmap_wf RS wfI)) 1;
+by (REPEAT (atac 1));
+val partitionT = result();
+
+(*** Correctness Conditions for Insertion Sort ***)
+
+
+val prems = goalw List.thy [isort_def] 
+    "f:A->A->Bool ==> isort(f) : PROD l:List(A).{x: List(A). Ord(f,x) & Perm(x,l)}";
+by (gen_ccs_tac  ([insertTS,insertT]@prems) 1);
+
+
+(*** Correctness Conditions for Quick Sort ***)
+
+val prems = goalw List.thy [qsort_def] 
+    "f:A->A->Bool ==> qsort(f) : PROD l:List(A).{x: List(A). Ord(f,x) & Perm(x,l)}";
+by (gen_ccs_tac  ([partitionT,appendTS,appendT]@prems) 1);
+