--- a/src/CTT/Arith.thy Fri Jun 02 16:06:19 2006 +0200
+++ b/src/CTT/Arith.thy Fri Jun 02 18:15:38 2006 +0200
@@ -4,16 +4,13 @@
Copyright 1991 University of Cambridge
*)
-header {* Arithmetic operators and their definitions *}
+header {* Elementary arithmetic *}
theory Arith
imports Bool
begin
-text {*
- Proves about elementary arithmetic: addition, multiplication, etc.
- Tests definitions and simplifier.
-*}
+subsection {* Arithmetic operators and their definitions *}
consts
"#+" :: "[i,i]=>i" (infixr 65)
@@ -37,6 +34,426 @@
mod_def: "a mod b == rec(a, 0, %u v. rec(succ(v) |-| b, 0, %x y. succ(v)))"
div_def: "a div b == rec(a, 0, %u v. rec(succ(u) mod b, succ(v), %x y. v))"
-ML {* use_legacy_bindings (the_context ()) *}
+lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def
+
+
+subsection {* Proofs about elementary arithmetic: addition, multiplication, etc. *}
+
+(** Addition *)
+
+(*typing of add: short and long versions*)
+
+lemma add_typing: "[| a:N; b:N |] ==> a #+ b : N"
+apply (unfold arith_defs)
+apply (tactic "typechk_tac []")
+done
+
+lemma add_typingL: "[| a=c:N; b=d:N |] ==> a #+ b = c #+ d : N"
+apply (unfold arith_defs)
+apply (tactic "equal_tac []")
+done
+
+
+(*computation for add: 0 and successor cases*)
+
+lemma addC0: "b:N ==> 0 #+ b = b : N"
+apply (unfold arith_defs)
+apply (tactic "rew_tac []")
+done
+
+lemma addC_succ: "[| a:N; b:N |] ==> succ(a) #+ b = succ(a #+ b) : N"
+apply (unfold arith_defs)
+apply (tactic "rew_tac []")
+done
+
+
+(** Multiplication *)
+
+(*typing of mult: short and long versions*)
+
+lemma mult_typing: "[| a:N; b:N |] ==> a #* b : N"
+apply (unfold arith_defs)
+apply (tactic {* typechk_tac [thm "add_typing"] *})
+done
+
+lemma mult_typingL: "[| a=c:N; b=d:N |] ==> a #* b = c #* d : N"
+apply (unfold arith_defs)
+apply (tactic {* equal_tac [thm "add_typingL"] *})
+done
+
+(*computation for mult: 0 and successor cases*)
+
+lemma multC0: "b:N ==> 0 #* b = 0 : N"
+apply (unfold arith_defs)
+apply (tactic "rew_tac []")
+done
+
+lemma multC_succ: "[| a:N; b:N |] ==> succ(a) #* b = b #+ (a #* b) : N"
+apply (unfold arith_defs)
+apply (tactic "rew_tac []")
+done
+
+
+(** Difference *)
+
+(*typing of difference*)
+
+lemma diff_typing: "[| a:N; b:N |] ==> a - b : N"
+apply (unfold arith_defs)
+apply (tactic "typechk_tac []")
+done
+
+lemma diff_typingL: "[| a=c:N; b=d:N |] ==> a - b = c - d : N"
+apply (unfold arith_defs)
+apply (tactic "equal_tac []")
+done
+
+
+(*computation for difference: 0 and successor cases*)
+
+lemma diffC0: "a:N ==> a - 0 = a : N"
+apply (unfold arith_defs)
+apply (tactic "rew_tac []")
+done
+
+(*Note: rec(a, 0, %z w.z) is pred(a). *)
+
+lemma diff_0_eq_0: "b:N ==> 0 - b = 0 : N"
+apply (unfold arith_defs)
+apply (tactic {* NE_tac "b" 1 *})
+apply (tactic "hyp_rew_tac []")
+done
+
+
+(*Essential to simplify FIRST!! (Else we get a critical pair)
+ succ(a) - succ(b) rewrites to pred(succ(a) - b) *)
+lemma diff_succ_succ: "[| a:N; b:N |] ==> succ(a) - succ(b) = a - b : N"
+apply (unfold arith_defs)
+apply (tactic "hyp_rew_tac []")
+apply (tactic {* NE_tac "b" 1 *})
+apply (tactic "hyp_rew_tac []")
+done
+
+
+subsection {* Simplification *}
+
+lemmas arith_typing_rls = add_typing mult_typing diff_typing
+ and arith_congr_rls = add_typingL mult_typingL diff_typingL
+lemmas congr_rls = arith_congr_rls intrL2_rls elimL_rls
+
+lemmas arithC_rls =
+ addC0 addC_succ
+ multC0 multC_succ
+ diffC0 diff_0_eq_0 diff_succ_succ
+
+ML {*
+
+structure Arith_simp_data: TSIMP_DATA =
+ struct
+ val refl = thm "refl_elem"
+ val sym = thm "sym_elem"
+ val trans = thm "trans_elem"
+ val refl_red = thm "refl_red"
+ val trans_red = thm "trans_red"
+ val red_if_equal = thm "red_if_equal"
+ val default_rls = thms "arithC_rls" @ thms "comp_rls"
+ val routine_tac = routine_tac (thms "arith_typing_rls" @ thms "routine_rls")
+ end
+
+structure Arith_simp = TSimpFun (Arith_simp_data)
+
+local val congr_rls = thms "congr_rls" in
+
+fun arith_rew_tac prems = make_rew_tac
+ (Arith_simp.norm_tac(congr_rls, prems))
+
+fun hyp_arith_rew_tac prems = make_rew_tac
+ (Arith_simp.cond_norm_tac(prove_cond_tac, congr_rls, prems))
end
+*}
+
+
+subsection {* Addition *}
+
+(*Associative law for addition*)
+lemma add_assoc: "[| a:N; b:N; c:N |] ==> (a #+ b) #+ c = a #+ (b #+ c) : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic "hyp_arith_rew_tac []")
+done
+
+
+(*Commutative law for addition. Can be proved using three inductions.
+ Must simplify after first induction! Orientation of rewrites is delicate*)
+lemma add_commute: "[| a:N; b:N |] ==> a #+ b = b #+ a : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic "hyp_arith_rew_tac []")
+apply (tactic {* NE_tac "b" 2 *})
+apply (rule sym_elem)
+apply (tactic {* NE_tac "b" 1 *})
+apply (tactic "hyp_arith_rew_tac []")
+done
+
+
+subsection {* Multiplication *}
+
+(*right annihilation in product*)
+lemma mult_0_right: "a:N ==> a #* 0 = 0 : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic "hyp_arith_rew_tac []")
+done
+
+(*right successor law for multiplication*)
+lemma mult_succ_right: "[| a:N; b:N |] ==> a #* succ(b) = a #+ (a #* b) : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic {* hyp_arith_rew_tac [thm "add_assoc" RS thm "sym_elem"] *})
+apply (assumption | rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+
+done
+
+(*Commutative law for multiplication*)
+lemma mult_commute: "[| a:N; b:N |] ==> a #* b = b #* a : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic {* hyp_arith_rew_tac [thm "mult_0_right", thm "mult_succ_right"] *})
+done
+
+(*addition distributes over multiplication*)
+lemma add_mult_distrib: "[| a:N; b:N; c:N |] ==> (a #+ b) #* c = (a #* c) #+ (b #* c) : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic {* hyp_arith_rew_tac [thm "add_assoc" RS thm "sym_elem"] *})
+done
+
+(*Associative law for multiplication*)
+lemma mult_assoc: "[| a:N; b:N; c:N |] ==> (a #* b) #* c = a #* (b #* c) : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic {* hyp_arith_rew_tac [thm "add_mult_distrib"] *})
+done
+
+
+subsection {* Difference *}
+
+text {*
+Difference on natural numbers, without negative numbers
+ a - b = 0 iff a<=b a - b = succ(c) iff a>b *}
+
+lemma diff_self_eq_0: "a:N ==> a - a = 0 : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic "hyp_arith_rew_tac []")
+done
+
+
+lemma add_0_right: "[| c : N; 0 : N; c : N |] ==> c #+ 0 = c : N"
+ by (rule addC0 [THEN [3] add_commute [THEN trans_elem]])
+
+(*Addition is the inverse of subtraction: if b<=x then b#+(x-b) = x.
+ An example of induction over a quantified formula (a product).
+ Uses rewriting with a quantified, implicative inductive hypothesis.*)
+lemma add_diff_inverse_lemma: "b:N ==> ?a : PROD x:N. Eq(N, b-x, 0) --> Eq(N, b #+ (x-b), x)"
+apply (tactic {* NE_tac "b" 1 *})
+(*strip one "universal quantifier" but not the "implication"*)
+apply (rule_tac [3] intr_rls)
+(*case analysis on x in
+ (succ(u) <= x) --> (succ(u)#+(x-succ(u)) = x) *)
+apply (tactic {* NE_tac "x" 4 *}, tactic "assume_tac 4")
+(*Prepare for simplification of types -- the antecedent succ(u)<=x *)
+apply (rule_tac [5] replace_type)
+apply (rule_tac [4] replace_type)
+apply (tactic "arith_rew_tac []")
+(*Solves first 0 goal, simplifies others. Two sugbgoals remain.
+ Both follow by rewriting, (2) using quantified induction hyp*)
+apply (tactic "intr_tac []") (*strips remaining PRODs*)
+apply (tactic {* hyp_arith_rew_tac [thm "add_0_right"] *})
+apply assumption
+done
+
+
+(*Version of above with premise b-a=0 i.e. a >= b.
+ Using ProdE does not work -- for ?B(?a) is ambiguous.
+ Instead, add_diff_inverse_lemma states the desired induction scheme
+ the use of RS below instantiates Vars in ProdE automatically. *)
+lemma add_diff_inverse: "[| a:N; b:N; b-a = 0 : N |] ==> b #+ (a-b) = a : N"
+apply (rule EqE)
+apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE])
+apply (assumption | rule EqI)+
+done
+
+
+subsection {* Absolute difference *}
+
+(*typing of absolute difference: short and long versions*)
+
+lemma absdiff_typing: "[| a:N; b:N |] ==> a |-| b : N"
+apply (unfold arith_defs)
+apply (tactic "typechk_tac []")
+done
+
+lemma absdiff_typingL: "[| a=c:N; b=d:N |] ==> a |-| b = c |-| d : N"
+apply (unfold arith_defs)
+apply (tactic "equal_tac []")
+done
+
+lemma absdiff_self_eq_0: "a:N ==> a |-| a = 0 : N"
+apply (unfold absdiff_def)
+apply (tactic {* arith_rew_tac [thm "diff_self_eq_0"] *})
+done
+
+lemma absdiffC0: "a:N ==> 0 |-| a = a : N"
+apply (unfold absdiff_def)
+apply (tactic "hyp_arith_rew_tac []")
+done
+
+
+lemma absdiff_succ_succ: "[| a:N; b:N |] ==> succ(a) |-| succ(b) = a |-| b : N"
+apply (unfold absdiff_def)
+apply (tactic "hyp_arith_rew_tac []")
+done
+
+(*Note how easy using commutative laws can be? ...not always... *)
+lemma absdiff_commute: "[| a:N; b:N |] ==> a |-| b = b |-| a : N"
+apply (unfold absdiff_def)
+apply (rule add_commute)
+apply (tactic {* typechk_tac [thm "diff_typing"] *})
+done
+
+(*If a+b=0 then a=0. Surprisingly tedious*)
+lemma add_eq0_lemma: "[| a:N; b:N |] ==> ?c : PROD u: Eq(N,a#+b,0) . Eq(N,a,0)"
+apply (tactic {* NE_tac "a" 1 *})
+apply (rule_tac [3] replace_type)
+apply (tactic "arith_rew_tac []")
+apply (tactic "intr_tac []") (*strips remaining PRODs*)
+apply (rule_tac [2] zero_ne_succ [THEN FE])
+apply (erule_tac [3] EqE [THEN sym_elem])
+apply (tactic {* typechk_tac [thm "add_typing"] *})
+done
+
+(*Version of above with the premise a+b=0.
+ Again, resolution instantiates variables in ProdE *)
+lemma add_eq0: "[| a:N; b:N; a #+ b = 0 : N |] ==> a = 0 : N"
+apply (rule EqE)
+apply (rule add_eq0_lemma [THEN ProdE])
+apply (rule_tac [3] EqI)
+apply (tactic "typechk_tac []")
+done
+
+(*Here is a lemma to infer a-b=0 and b-a=0 from a|-|b=0, below. *)
+lemma absdiff_eq0_lem:
+ "[| a:N; b:N; a |-| b = 0 : N |] ==>
+ ?a : SUM v: Eq(N, a-b, 0) . Eq(N, b-a, 0)"
+apply (unfold absdiff_def)
+apply (tactic "intr_tac []")
+apply (tactic eqintr_tac)
+apply (rule_tac [2] add_eq0)
+apply (rule add_eq0)
+apply (rule_tac [6] add_commute [THEN trans_elem])
+apply (tactic {* typechk_tac [thm "diff_typing"] *})
+done
+
+(*if a |-| b = 0 then a = b
+ proof: a-b=0 and b-a=0, so b = a+(b-a) = a+0 = a*)
+lemma absdiff_eq0: "[| a |-| b = 0 : N; a:N; b:N |] ==> a = b : N"
+apply (rule EqE)
+apply (rule absdiff_eq0_lem [THEN SumE])
+apply (tactic "TRYALL assume_tac")
+apply (tactic eqintr_tac)
+apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem])
+apply (rule_tac [3] EqE, tactic "assume_tac 3")
+apply (tactic {* hyp_arith_rew_tac [thm "add_0_right"] *})
+done
+
+
+subsection {* Remainder and Quotient *}
+
+(*typing of remainder: short and long versions*)
+
+lemma mod_typing: "[| a:N; b:N |] ==> a mod b : N"
+apply (unfold mod_def)
+apply (tactic {* typechk_tac [thm "absdiff_typing"] *})
+done
+
+lemma mod_typingL: "[| a=c:N; b=d:N |] ==> a mod b = c mod d : N"
+apply (unfold mod_def)
+apply (tactic {* equal_tac [thm "absdiff_typingL"] *})
+done
+
+
+(*computation for mod : 0 and successor cases*)
+
+lemma modC0: "b:N ==> 0 mod b = 0 : N"
+apply (unfold mod_def)
+apply (tactic {* rew_tac [thm "absdiff_typing"] *})
+done
+
+lemma modC_succ:
+"[| a:N; b:N |] ==> succ(a) mod b = rec(succ(a mod b) |-| b, 0, %x y. succ(a mod b)) : N"
+apply (unfold mod_def)
+apply (tactic {* rew_tac [thm "absdiff_typing"] *})
+done
+
+
+(*typing of quotient: short and long versions*)
+
+lemma div_typing: "[| a:N; b:N |] ==> a div b : N"
+apply (unfold div_def)
+apply (tactic {* typechk_tac [thm "absdiff_typing", thm "mod_typing"] *})
+done
+
+lemma div_typingL: "[| a=c:N; b=d:N |] ==> a div b = c div d : N"
+apply (unfold div_def)
+apply (tactic {* equal_tac [thm "absdiff_typingL", thm "mod_typingL"] *})
+done
+
+lemmas div_typing_rls = mod_typing div_typing absdiff_typing
+
+
+(*computation for quotient: 0 and successor cases*)
+
+lemma divC0: "b:N ==> 0 div b = 0 : N"
+apply (unfold div_def)
+apply (tactic {* rew_tac [thm "mod_typing", thm "absdiff_typing"] *})
+done
+
+lemma divC_succ:
+ "[| a:N; b:N |] ==> succ(a) div b =
+ rec(succ(a) mod b, succ(a div b), %x y. a div b) : N"
+apply (unfold div_def)
+apply (tactic {* rew_tac [thm "mod_typing"] *})
+done
+
+
+(*Version of above with same condition as the mod one*)
+lemma divC_succ2: "[| a:N; b:N |] ==>
+ succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), %x y. a div b) : N"
+apply (rule divC_succ [THEN trans_elem])
+apply (tactic {* rew_tac (thms "div_typing_rls" @ [thm "modC_succ"]) *})
+apply (tactic {* NE_tac "succ (a mod b) |-|b" 1 *})
+apply (tactic {* rew_tac [thm "mod_typing", thm "div_typing", thm "absdiff_typing"] *})
+done
+
+(*for case analysis on whether a number is 0 or a successor*)
+lemma iszero_decidable: "a:N ==> rec(a, inl(eq), %ka kb. inr(<ka, eq>)) :
+ Eq(N,a,0) + (SUM x:N. Eq(N,a, succ(x)))"
+apply (tactic {* NE_tac "a" 1 *})
+apply (rule_tac [3] PlusI_inr)
+apply (rule_tac [2] PlusI_inl)
+apply (tactic eqintr_tac)
+apply (tactic "equal_tac []")
+done
+
+(*Main Result. Holds when b is 0 since a mod 0 = a and a div 0 = 0 *)
+lemma mod_div_equality: "[| a:N; b:N |] ==> a mod b #+ (a div b) #* b = a : N"
+apply (tactic {* NE_tac "a" 1 *})
+apply (tactic {* arith_rew_tac (thms "div_typing_rls" @
+ [thm "modC0", thm "modC_succ", thm "divC0", thm "divC_succ2"]) *})
+apply (rule EqE)
+(*case analysis on succ(u mod b)|-|b *)
+apply (rule_tac a1 = "succ (u mod b) |-| b" in iszero_decidable [THEN PlusE])
+apply (erule_tac [3] SumE)
+apply (tactic {* hyp_arith_rew_tac (thms "div_typing_rls" @
+ [thm "modC0", thm "modC_succ", thm "divC0", thm "divC_succ2"]) *})
+(*Replace one occurence of b by succ(u mod b). Clumsy!*)
+apply (rule add_typingL [THEN trans_elem])
+apply (erule EqE [THEN absdiff_eq0, THEN sym_elem])
+apply (rule_tac [3] refl_elem)
+apply (tactic {* hyp_arith_rew_tac (thms "div_typing_rls") *})
+done
+
+end