src/ZF/OrderArith.thy

     Copyright   1994  University of Cambridge
 
 Towards ordinal arithmetic.  Also useful with wfrec.
 *)
 
-OrderArith = Order + Sum + Ordinal +
-consts
-  radd, rmult      :: [i,i,i,i]=>i
-  rvimage          :: [i,i,i]=>i
-
-defs
+theory OrderArith = Order + Sum + Ordinal:
+constdefs
+
   (*disjoint sum of two relations; underlies ordinal addition*)
-  radd_def "radd(A,r,B,s) == 
+  radd    :: "[i,i,i,i]=>i"
+    "radd(A,r,B,s) == 
                 {z: (A+B) * (A+B).  
                     (EX x y. z = <Inl(x), Inr(y)>)   |   
                     (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |      
                     (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
 
   (*lexicographic product of two relations; underlies ordinal multiplication*)
-  rmult_def "rmult(A,r,B,s) == 
+  rmult   :: "[i,i,i,i]=>i"
+    "rmult(A,r,B,s) == 
                 {z: (A*B) * (A*B).  
                     EX x' y' x y. z = <<x',y'>, <x,y>> &         
                        (<x',x>: r | (x'=x & <y',y>: s))}"
 
   (*inverse image of a relation*)
-  rvimage_def "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
-
-constdefs
-   measure :: "[i, i\\<Rightarrow>i] \\<Rightarrow> i"
-   "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
-
+  rvimage :: "[i,i,i]=>i"
+    "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
+
+  measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i"
+    "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
+
+
+(**** Addition of relations -- disjoint sum ****)
+
+(** Rewrite rules.  Can be used to obtain introduction rules **)
+
+lemma radd_Inl_Inr_iff [iff]: 
+    "<Inl(a), Inr(b)> : radd(A,r,B,s)  <->  a:A & b:B"
+apply (unfold radd_def)
+apply blast
+done
+
+lemma radd_Inl_iff [iff]: 
+    "<Inl(a'), Inl(a)> : radd(A,r,B,s)  <->  a':A & a:A & <a',a>:r"
+apply (unfold radd_def)
+apply blast
+done
+
+lemma radd_Inr_iff [iff]: 
+    "<Inr(b'), Inr(b)> : radd(A,r,B,s) <->  b':B & b:B & <b',b>:s"
+apply (unfold radd_def)
+apply blast
+done
+
+lemma radd_Inr_Inl_iff [iff]: 
+    "<Inr(b), Inl(a)> : radd(A,r,B,s) <->  False"
+apply (unfold radd_def)
+apply blast
+done
+
+(** Elimination Rule **)
+
+lemma raddE:
+    "[| <p',p> : radd(A,r,B,s);                  
+        !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q;        
+        !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q;  
+        !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q   
+     |] ==> Q"
+apply (unfold radd_def)
+apply (blast intro: elim:); 
+done
+
+(** Type checking **)
+
+lemma radd_type: "radd(A,r,B,s) <= (A+B) * (A+B)"
+apply (unfold radd_def)
+apply (rule Collect_subset)
+done
+
+lemmas field_radd = radd_type [THEN field_rel_subset]
+
+(** Linearity **)
+
+lemma linear_radd: 
+    "[| linear(A,r);  linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))"
+apply (unfold linear_def)
+apply (blast intro: elim:); 
+done
+
+
+(** Well-foundedness **)
+
+lemma wf_on_radd: "[| wf[A](r);  wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))"
+apply (rule wf_onI2)
+apply (subgoal_tac "ALL x:A. Inl (x) : Ba")
+(*Proving the lemma, which is needed twice!*)
+ prefer 2
+ apply (erule_tac V = "y : A + B" in thin_rl)
+ apply (rule_tac ballI)
+ apply (erule_tac r = "r" and a = "x" in wf_on_induct, assumption)
+ apply (blast intro: elim:); 
+(*Returning to main part of proof*)
+apply safe
+apply blast
+apply (erule_tac r = "s" and a = "ya" in wf_on_induct , assumption)
+apply (blast intro: elim:); 
+done
+
+lemma wf_radd: "[| wf(r);  wf(s) |] ==> wf(radd(field(r),r,field(s),s))"
+apply (simp add: wf_iff_wf_on_field)
+apply (rule wf_on_subset_A [OF _ field_radd])
+apply (blast intro: wf_on_radd) 
+done
+
+lemma well_ord_radd:
+     "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))"
+apply (rule well_ordI)
+apply (simp add: well_ord_def wf_on_radd)
+apply (simp add: well_ord_def tot_ord_def linear_radd)
+done
+
+(** An ord_iso congruence law **)
+
+lemma sum_bij:
+     "[| f: bij(A,C);  g: bij(B,D) |]
+      ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)"
+apply (rule_tac d = "case (%x. Inl (converse (f) `x) , %y. Inr (converse (g) `y))" in lam_bijective)
+apply (typecheck add: bij_is_inj inj_is_fun) 
+apply (auto simp add: left_inverse_bij right_inverse_bij) 
+done
+
+lemma sum_ord_iso_cong: 
+    "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |] ==>      
+            (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))             
+            : ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
+apply (unfold ord_iso_def)
+apply (safe intro!: sum_bij)
+(*Do the beta-reductions now*)
+apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type])
+done
+
+(*Could we prove an ord_iso result?  Perhaps 
+     ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *)
+lemma sum_disjoint_bij: "A Int B = 0 ==>      
+            (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)"
+apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective)
+apply auto
+done
+
+(** Associativity **)
+
+lemma sum_assoc_bij:
+     "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
+      : bij((A+B)+C, A+(B+C))"
+apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))" 
+       in lam_bijective)
+apply auto
+done
+
+lemma sum_assoc_ord_iso:
+     "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
+      : ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),     
+                A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
+apply (rule sum_assoc_bij [THEN ord_isoI])
+apply auto
+done
+
+
+(**** Multiplication of relations -- lexicographic product ****)
+
+(** Rewrite rule.  Can be used to obtain introduction rules **)
+
+lemma  rmult_iff [iff]: 
+    "<<a',b'>, <a,b>> : rmult(A,r,B,s) <->        
+            (<a',a>: r  & a':A & a:A & b': B & b: B) |   
+            (<b',b>: s  & a'=a & a:A & b': B & b: B)"
+
+apply (unfold rmult_def)
+apply blast
+done
+
+lemma rmultE: 
+    "[| <<a',b'>, <a,b>> : rmult(A,r,B,s);               
+        [| <a',a>: r;  a':A;  a:A;  b':B;  b:B |] ==> Q;         
+        [| <b',b>: s;  a:A;  a'=a;  b':B;  b:B |] ==> Q  
+     |] ==> Q"
+apply (blast intro: elim:); 
+done
+
+(** Type checking **)
+
+lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)"
+apply (unfold rmult_def)
+apply (rule Collect_subset)
+done
+
+lemmas field_rmult = rmult_type [THEN field_rel_subset]
+
+(** Linearity **)
+
+lemma linear_rmult:
+    "[| linear(A,r);  linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))"
+apply (simp add: linear_def); 
+apply (blast intro: elim:); 
+done
+
+(** Well-foundedness **)
+
+lemma wf_on_rmult: "[| wf[A](r);  wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))"
+apply (rule wf_onI2)
+apply (erule SigmaE)
+apply (erule ssubst)
+apply (subgoal_tac "ALL b:B. <x,b>: Ba")
+apply blast
+apply (erule_tac a = "x" in wf_on_induct , assumption)
+apply (rule ballI)
+apply (erule_tac a = "b" in wf_on_induct , assumption)
+apply (best elim!: rmultE bspec [THEN mp])
+done
+
+
+lemma wf_rmult: "[| wf(r);  wf(s) |] ==> wf(rmult(field(r),r,field(s),s))"
+apply (simp add: wf_iff_wf_on_field)
+apply (rule wf_on_subset_A [OF _ field_rmult])
+apply (blast intro: wf_on_rmult) 
+done
+
+lemma well_ord_rmult:
+     "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))"
+apply (rule well_ordI)
+apply (simp add: well_ord_def wf_on_rmult)
+apply (simp add: well_ord_def tot_ord_def linear_rmult)
+done
+
+
+(** An ord_iso congruence law **)
+
+lemma prod_bij:
+     "[| f: bij(A,C);  g: bij(B,D) |] 
+      ==> (lam <x,y>:A*B. <f`x, g`y>) : bij(A*B, C*D)"
+apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>" 
+       in lam_bijective)
+apply (typecheck add: bij_is_inj inj_is_fun) 
+apply (auto simp add: left_inverse_bij right_inverse_bij) 
+done
+
+lemma prod_ord_iso_cong: 
+    "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |]      
+     ==> (lam <x,y>:A*B. <f`x, g`y>)                                  
+         : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
+apply (unfold ord_iso_def)
+apply (safe intro!: prod_bij)
+apply (simp_all add: bij_is_fun [THEN apply_type])
+apply (blast intro: bij_is_inj [THEN inj_apply_equality])
+done
+
+lemma singleton_prod_bij: "(lam z:A. <x,z>) : bij(A, {x}*A)"
+apply (rule_tac d = "snd" in lam_bijective)
+apply auto
+done
+
+(*Used??*)
+lemma singleton_prod_ord_iso:
+     "well_ord({x},xr) ==>   
+          (lam z:A. <x,z>) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
+apply (rule singleton_prod_bij [THEN ord_isoI])
+apply (simp (no_asm_simp))
+apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])
+done
+
+(*Here we build a complicated function term, then simplify it using
+  case_cong, id_conv, comp_lam, case_case.*)
+lemma prod_sum_singleton_bij:
+     "a~:C ==>  
+       (lam x:C*B + D. case(%x. x, %y.<a,y>, x))  
+       : bij(C*B + D, C*B Un {a}*D)"
+apply (rule subst_elem)
+apply (rule id_bij [THEN sum_bij, THEN comp_bij])
+apply (rule singleton_prod_bij)
+apply (rule sum_disjoint_bij)
+apply blast
+apply (simp (no_asm_simp) cong add: case_cong)
+apply (rule comp_lam [THEN trans, symmetric])
+apply (fast elim!: case_type)
+apply (simp (no_asm_simp) add: case_case)
+done
+
+lemma prod_sum_singleton_ord_iso:
+ "[| a:A;  well_ord(A,r) |] ==>  
+    (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))  
+    : ord_iso(pred(A,a,r)*B + pred(B,b,s),               
+                  radd(A*B, rmult(A,r,B,s), B, s),       
+              pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))"
+apply (rule prod_sum_singleton_bij [THEN ord_isoI])
+apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
+apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)
+done
+
+(** Distributive law **)
+
+lemma sum_prod_distrib_bij:
+     "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))  
+      : bij((A+B)*C, (A*C)+(B*C))"
+apply (rule_tac d = "case (%<x,y>.<Inl (x) ,y>, %<x,y>.<Inr (x) ,y>) " 
+       in lam_bijective)
+apply auto
+done
+
+lemma sum_prod_distrib_ord_iso:
+ "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))  
+  : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),  
+            (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
+apply (rule sum_prod_distrib_bij [THEN ord_isoI])
+apply auto
+done
+
+(** Associativity **)
+
+lemma prod_assoc_bij:
+     "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : bij((A*B)*C, A*(B*C))"
+apply (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective)
+apply auto
+done
+
+lemma prod_assoc_ord_iso:
+ "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)                    
+  : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),   
+            A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
+apply (rule prod_assoc_bij [THEN ord_isoI])
+apply auto
+done
+
+(**** Inverse image of a relation ****)
+
+(** Rewrite rule **)
+
+lemma rvimage_iff: "<a,b> : rvimage(A,f,r)  <->  <f`a,f`b>: r & a:A & b:A"
+apply (unfold rvimage_def)
+apply blast
+done
+
+(** Type checking **)
+
+lemma rvimage_type: "rvimage(A,f,r) <= A*A"
+apply (unfold rvimage_def)
+apply (rule Collect_subset)
+done
+
+lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
+
+lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
+apply (unfold rvimage_def)
+apply blast
+done
+
+
+(** Partial Ordering Properties **)
+
+lemma irrefl_rvimage: 
+    "[| f: inj(A,B);  irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
+apply (unfold irrefl_def rvimage_def)
+apply (blast intro: inj_is_fun [THEN apply_type])
+done
+
+lemma trans_on_rvimage: 
+    "[| f: inj(A,B);  trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
+apply (unfold trans_on_def rvimage_def)
+apply (blast intro: inj_is_fun [THEN apply_type])
+done
+
+lemma part_ord_rvimage: 
+    "[| f: inj(A,B);  part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
+apply (unfold part_ord_def)
+apply (blast intro!: irrefl_rvimage trans_on_rvimage)
+done
+
+(** Linearity **)
+
+lemma linear_rvimage:
+    "[| f: inj(A,B);  linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
+apply (simp add: inj_def linear_def rvimage_iff) 
+apply (blast intro: apply_funtype); 
+done
+
+lemma tot_ord_rvimage: 
+    "[| f: inj(A,B);  tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
+apply (unfold tot_ord_def)
+apply (blast intro!: part_ord_rvimage linear_rvimage)
+done
+
+
+(** Well-foundedness **)
+
+(*Not sure if wf_on_rvimage could be proved from this!*)
+lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
+apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
+apply clarify
+apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }")
+ apply (erule allE)
+ apply (erule impE)
+ apply assumption; 
+ apply blast
+apply (blast intro: elim:); 
+done
+
+lemma wf_on_rvimage: "[| f: A->B;  wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
+apply (rule wf_onI2)
+apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba")
+ apply blast
+apply (erule_tac a = "f`y" in wf_on_induct)
+ apply (blast intro!: apply_funtype)
+apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])
+done
+
+(*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
+lemma well_ord_rvimage:
+     "[| f: inj(A,B);  well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
+apply (rule well_ordI)
+apply (unfold well_ord_def tot_ord_def)
+apply (blast intro!: wf_on_rvimage inj_is_fun)
+apply (blast intro!: linear_rvimage)
+done
+
+lemma ord_iso_rvimage: 
+    "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)"
+apply (unfold ord_iso_def)
+apply (simp add: rvimage_iff)
+done
+
+lemma ord_iso_rvimage_eq: 
+    "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A"
+apply (unfold ord_iso_def rvimage_def)
+apply blast
+done
+
+
+(** The "measure" relation is useful with wfrec **)
+
+lemma measure_eq_rvimage_Memrel:
+     "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
+apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
+apply (rule equalityI)
+apply auto
+apply (auto intro: Ord_in_Ord simp add: lt_def)
+done
+
+lemma wf_measure [iff]: "wf(measure(A,f))"
+apply (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
+done
+
+lemma measure_iff [iff]: "<x,y> : measure(A,f) <-> x:A & y:A & f(x)<f(y)"
+apply (simp (no_asm) add: measure_def)
+done
+
+ML {*
+val measure_def = thm "measure_def";
+val radd_Inl_Inr_iff = thm "radd_Inl_Inr_iff";
+val radd_Inl_iff = thm "radd_Inl_iff";
+val radd_Inr_iff = thm "radd_Inr_iff";
+val radd_Inr_Inl_iff = thm "radd_Inr_Inl_iff";
+val raddE = thm "raddE";
+val radd_type = thm "radd_type";
+val field_radd = thm "field_radd";
+val linear_radd = thm "linear_radd";
+val wf_on_radd = thm "wf_on_radd";
+val wf_radd = thm "wf_radd";
+val well_ord_radd = thm "well_ord_radd";
+val sum_bij = thm "sum_bij";
+val sum_ord_iso_cong = thm "sum_ord_iso_cong";
+val sum_disjoint_bij = thm "sum_disjoint_bij";
+val sum_assoc_bij = thm "sum_assoc_bij";
+val sum_assoc_ord_iso = thm "sum_assoc_ord_iso";
+val rmult_iff = thm "rmult_iff";
+val rmultE = thm "rmultE";
+val rmult_type = thm "rmult_type";
+val field_rmult = thm "field_rmult";
+val linear_rmult = thm "linear_rmult";
+val wf_on_rmult = thm "wf_on_rmult";
+val wf_rmult = thm "wf_rmult";
+val well_ord_rmult = thm "well_ord_rmult";
+val prod_bij = thm "prod_bij";
+val prod_ord_iso_cong = thm "prod_ord_iso_cong";
+val singleton_prod_bij = thm "singleton_prod_bij";
+val singleton_prod_ord_iso = thm "singleton_prod_ord_iso";
+val prod_sum_singleton_bij = thm "prod_sum_singleton_bij";
+val prod_sum_singleton_ord_iso = thm "prod_sum_singleton_ord_iso";
+val sum_prod_distrib_bij = thm "sum_prod_distrib_bij";
+val sum_prod_distrib_ord_iso = thm "sum_prod_distrib_ord_iso";
+val prod_assoc_bij = thm "prod_assoc_bij";
+val prod_assoc_ord_iso = thm "prod_assoc_ord_iso";
+val rvimage_iff = thm "rvimage_iff";
+val rvimage_type = thm "rvimage_type";
+val field_rvimage = thm "field_rvimage";
+val rvimage_converse = thm "rvimage_converse";
+val irrefl_rvimage = thm "irrefl_rvimage";
+val trans_on_rvimage = thm "trans_on_rvimage";
+val part_ord_rvimage = thm "part_ord_rvimage";
+val linear_rvimage = thm "linear_rvimage";
+val tot_ord_rvimage = thm "tot_ord_rvimage";
+val wf_rvimage = thm "wf_rvimage";
+val wf_on_rvimage = thm "wf_on_rvimage";
+val well_ord_rvimage = thm "well_ord_rvimage";
+val ord_iso_rvimage = thm "ord_iso_rvimage";
+val ord_iso_rvimage_eq = thm "ord_iso_rvimage_eq";
+val measure_eq_rvimage_Memrel = thm "measure_eq_rvimage_Memrel";
+val wf_measure = thm "wf_measure";
+val measure_iff = thm "measure_iff";
+*}
 
 end