--- a/src/CTT/Arith.thy Fri Jul 15 11:26:40 2016 +0200
+++ b/src/CTT/Arith.thy Fri Jul 15 15:19:04 2016 +0200
@@ -6,141 +6,115 @@
section \<open>Elementary arithmetic\<close>
theory Arith
-imports Bool
+ imports Bool
begin
subsection \<open>Arithmetic operators and their definitions\<close>
-definition
- add :: "[i,i]\<Rightarrow>i" (infixr "#+" 65) where
- "a#+b \<equiv> rec(a, b, \<lambda>u v. succ(v))"
+definition add :: "[i,i]\<Rightarrow>i" (infixr "#+" 65)
+ where "a#+b \<equiv> rec(a, b, \<lambda>u v. succ(v))"
-definition
- diff :: "[i,i]\<Rightarrow>i" (infixr "-" 65) where
- "a-b \<equiv> rec(b, a, \<lambda>u v. rec(v, 0, \<lambda>x y. x))"
+definition diff :: "[i,i]\<Rightarrow>i" (infixr "-" 65)
+ where "a-b \<equiv> rec(b, a, \<lambda>u v. rec(v, 0, \<lambda>x y. x))"
-definition
- absdiff :: "[i,i]\<Rightarrow>i" (infixr "|-|" 65) where
- "a|-|b \<equiv> (a-b) #+ (b-a)"
+definition absdiff :: "[i,i]\<Rightarrow>i" (infixr "|-|" 65)
+ where "a|-|b \<equiv> (a-b) #+ (b-a)"
-definition
- mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 70) where
- "a#*b \<equiv> rec(a, 0, \<lambda>u v. b #+ v)"
+definition mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 70)
+ where "a#*b \<equiv> rec(a, 0, \<lambda>u v. b #+ v)"
-definition
- mod :: "[i,i]\<Rightarrow>i" (infixr "mod" 70) where
- "a mod b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(v) |-| b, 0, \<lambda>x y. succ(v)))"
+definition mod :: "[i,i]\<Rightarrow>i" (infixr "mod" 70)
+ where "a mod b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(v) |-| b, 0, \<lambda>x y. succ(v)))"
-definition
- div :: "[i,i]\<Rightarrow>i" (infixr "div" 70) where
- "a div b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(u) mod b, succ(v), \<lambda>x y. v))"
-
+definition div :: "[i,i]\<Rightarrow>i" (infixr "div" 70)
+ where "a div b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(u) mod b, succ(v), \<lambda>x y. v))"
lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def
subsection \<open>Proofs about elementary arithmetic: addition, multiplication, etc.\<close>
-(** Addition *)
+subsubsection \<open>Addition\<close>
-(*typing of add: short and long versions*)
+text \<open>Typing of \<open>add\<close>: short and long versions.\<close>
lemma add_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b : N"
-apply (unfold arith_defs)
-apply typechk
-done
+ unfolding arith_defs by typechk
lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N"
-apply (unfold arith_defs)
-apply equal
-done
+ unfolding arith_defs by equal
-(*computation for add: 0 and successor cases*)
+text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close>
lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N"
-apply (unfold arith_defs)
-apply rew
-done
+ unfolding arith_defs by rew
lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N"
-apply (unfold arith_defs)
-apply rew
-done
+ unfolding arith_defs by rew
-(** Multiplication *)
+subsubsection \<open>Multiplication\<close>
-(*typing of mult: short and long versions*)
+text \<open>Typing of \<open>mult\<close>: short and long versions.\<close>
lemma mult_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b : N"
-apply (unfold arith_defs)
-apply (typechk add_typing)
-done
+ unfolding arith_defs by (typechk add_typing)
lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N"
-apply (unfold arith_defs)
-apply (equal add_typingL)
-done
+ unfolding arith_defs by (equal add_typingL)
-(*computation for mult: 0 and successor cases*)
+
+text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close>
lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N"
-apply (unfold arith_defs)
-apply rew
-done
+ unfolding arith_defs by rew
lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N"
-apply (unfold arith_defs)
-apply rew
-done
+ unfolding arith_defs by rew
-(** Difference *)
+subsubsection \<open>Difference\<close>
-(*typing of difference*)
+text \<open>Typing of difference.\<close>
lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a - b : N"
-apply (unfold arith_defs)
-apply typechk
-done
+ unfolding arith_defs by typechk
lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a - b = c - d : N"
-apply (unfold arith_defs)
-apply equal
-done
+ unfolding arith_defs by equal
-(*computation for difference: 0 and successor cases*)
+text \<open>Computation for difference: 0 and successor cases.\<close>
lemma diffC0: "a:N \<Longrightarrow> a - 0 = a : N"
-apply (unfold arith_defs)
-apply rew
-done
+ unfolding arith_defs by rew
-(*Note: rec(a, 0, \<lambda>z w.z) is pred(a). *)
+text \<open>Note: \<open>rec(a, 0, \<lambda>z w.z)\<close> is \<open>pred(a).\<close>\<close>
lemma diff_0_eq_0: "b:N \<Longrightarrow> 0 - b = 0 : N"
-apply (unfold arith_defs)
-apply (NE b)
-apply hyp_rew
-done
-
+ unfolding arith_defs
+ apply (NE b)
+ apply hyp_rew
+ done
-(*Essential to simplify FIRST!! (Else we get a critical pair)
- succ(a) - succ(b) rewrites to pred(succ(a) - b) *)
+text \<open>
+ Essential to simplify FIRST!! (Else we get a critical pair)
+ \<open>succ(a) - succ(b)\<close> rewrites to \<open>pred(succ(a) - b)\<close>.
+\<close>
lemma diff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) - succ(b) = a - b : N"
-apply (unfold arith_defs)
-apply hyp_rew
-apply (NE b)
-apply hyp_rew
-done
+ unfolding arith_defs
+ apply hyp_rew
+ apply (NE b)
+ apply hyp_rew
+ done
subsection \<open>Simplification\<close>
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 =
@@ -149,30 +123,23 @@
diffC0 diff_0_eq_0 diff_succ_succ
ML \<open>
+ structure Arith_simp = TSimpFun(
+ 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 comp_rls}
+ val routine_tac = routine_tac @{thms arith_typing_rls routine_rls}
+ )
-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
+ fun arith_rew_tac ctxt prems =
+ make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems))
-structure Arith_simp = TSimpFun (Arith_simp_data)
-
-local val congr_rls = @{thms congr_rls} in
-
-fun arith_rew_tac ctxt prems = make_rew_tac ctxt
- (Arith_simp.norm_tac ctxt (congr_rls, prems))
-
-fun hyp_arith_rew_tac ctxt prems = make_rew_tac ctxt
- (Arith_simp.cond_norm_tac ctxt (prove_cond_tac ctxt, congr_rls, prems))
-
-end
+ fun hyp_arith_rew_tac ctxt prems =
+ make_rew_tac ctxt
+ (Arith_simp.cond_norm_tac ctxt (prove_cond_tac ctxt, @{thms congr_rls}, prems))
\<close>
method_setup arith_rew = \<open>
@@ -186,284 +153,263 @@
subsection \<open>Addition\<close>
-(*Associative law for addition*)
+text \<open>Associative law for addition.\<close>
lemma add_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #+ c = a #+ (b #+ c) : N"
-apply (NE a)
-apply hyp_arith_rew
-done
+ apply (NE a)
+ apply hyp_arith_rew
+ done
-
-(*Commutative law for addition. Can be proved using three inductions.
- Must simplify after first induction! Orientation of rewrites is delicate*)
+text \<open>Commutative law for addition. Can be proved using three inductions.
+ Must simplify after first induction! Orientation of rewrites is delicate.\<close>
lemma add_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b = b #+ a : N"
-apply (NE a)
-apply hyp_arith_rew
-apply (rule sym_elem)
-prefer 2
-apply (NE b)
-prefer 4
-apply (NE b)
-apply hyp_arith_rew
-done
+ apply (NE a)
+ apply hyp_arith_rew
+ apply (rule sym_elem)
+ prefer 2
+ apply (NE b)
+ prefer 4
+ apply (NE b)
+ apply hyp_arith_rew
+ done
subsection \<open>Multiplication\<close>
-(*right annihilation in product*)
+text \<open>Right annihilation in product.\<close>
lemma mult_0_right: "a:N \<Longrightarrow> a #* 0 = 0 : N"
-apply (NE a)
-apply hyp_arith_rew
-done
+ apply (NE a)
+ apply hyp_arith_rew
+ done
-(*right successor law for multiplication*)
+text \<open>Right successor law for multiplication.\<close>
lemma mult_succ_right: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* succ(b) = a #+ (a #* b) : N"
-apply (NE a)
-apply (hyp_arith_rew add_assoc [THEN sym_elem])
-apply (assumption | rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+
-done
+ apply (NE a)
+ apply (hyp_arith_rew add_assoc [THEN sym_elem])
+ apply (assumption | rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+
+ done
-(*Commutative law for multiplication*)
+text \<open>Commutative law for multiplication.\<close>
lemma mult_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b = b #* a : N"
-apply (NE a)
-apply (hyp_arith_rew mult_0_right mult_succ_right)
-done
+ apply (NE a)
+ apply (hyp_arith_rew mult_0_right mult_succ_right)
+ done
-(*addition distributes over multiplication*)
+text \<open>Addition distributes over multiplication.\<close>
lemma add_mult_distrib: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #* c = (a #* c) #+ (b #* c) : N"
-apply (NE a)
-apply (hyp_arith_rew add_assoc [THEN sym_elem])
-done
+ apply (NE a)
+ apply (hyp_arith_rew add_assoc [THEN sym_elem])
+ done
-(*Associative law for multiplication*)
+text \<open>Associative law for multiplication.\<close>
lemma mult_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #* b) #* c = a #* (b #* c) : N"
-apply (NE a)
-apply (hyp_arith_rew add_mult_distrib)
-done
+ apply (NE a)
+ apply (hyp_arith_rew add_mult_distrib)
+ done
subsection \<open>Difference\<close>
text \<open>
-Difference on natural numbers, without negative numbers
- a - b = 0 iff a<=b a - b = succ(c) iff a>b\<close>
+ Difference on natural numbers, without negative numbers
+ \<^item> \<open>a - b = 0\<close> iff \<open>a \<le> b\<close>
+ \<^item> \<open>a - b = succ(c)\<close> iff \<open>a > b\<close>
+\<close>
lemma diff_self_eq_0: "a:N \<Longrightarrow> a - a = 0 : N"
-apply (NE a)
-apply hyp_arith_rew
-done
+ apply (NE a)
+ apply hyp_arith_rew
+ done
lemma add_0_right: "\<lbrakk>c : N; 0 : N; c : N\<rbrakk> \<Longrightarrow> 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.
+text \<open>
+ Addition is the inverse of subtraction: if \<open>b \<le> x\<close> then \<open>b #+ (x - b) = x\<close>.
An example of induction over a quantified formula (a product).
- Uses rewriting with a quantified, implicative inductive hypothesis.*)
+ Uses rewriting with a quantified, implicative inductive hypothesis.
+\<close>
schematic_goal add_diff_inverse_lemma:
"b:N \<Longrightarrow> ?a : \<Prod>x:N. Eq(N, b-x, 0) \<longrightarrow> Eq(N, b #+ (x-b), x)"
-apply (NE b)
-(*strip one "universal quantifier" but not the "implication"*)
-apply (rule_tac [3] intr_rls)
-(*case analysis on x in
- (succ(u) <= x) \<longrightarrow> (succ(u)#+(x-succ(u)) = x) *)
-prefer 4
-apply (NE x)
-apply assumption
-(*Prepare for simplification of types -- the antecedent succ(u)<=x *)
-apply (rule_tac [2] replace_type)
-apply (rule_tac [1] replace_type)
-apply arith_rew
-(*Solves first 0 goal, simplifies others. Two sugbgoals remain.
- Both follow by rewriting, (2) using quantified induction hyp*)
-apply intr (*strips remaining PRODs*)
-apply (hyp_arith_rew add_0_right)
-apply assumption
-done
+ apply (NE b)
+ \<comment> \<open>strip one "universal quantifier" but not the "implication"\<close>
+ apply (rule_tac [3] intr_rls)
+ \<comment> \<open>case analysis on \<open>x\<close> in \<open>succ(u) \<le> x \<longrightarrow> succ(u) #+ (x - succ(u)) = x\<close>\<close>
+ prefer 4
+ apply (NE x)
+ apply assumption
+ \<comment> \<open>Prepare for simplification of types -- the antecedent \<open>succ(u) \<le> x\<close>\<close>
+ apply (rule_tac [2] replace_type)
+ apply (rule_tac [1] replace_type)
+ apply arith_rew
+ \<comment> \<open>Solves first 0 goal, simplifies others. Two sugbgoals remain.
+ Both follow by rewriting, (2) using quantified induction hyp.\<close>
+ apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close>
+ apply (hyp_arith_rew 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. *)
+text \<open>
+ Version of above with premise \<open>b - a = 0\<close> i.e. \<open>a \<ge> b\<close>.
+ Using @{thm ProdE} does not work -- for \<open>?B(?a)\<close> is ambiguous.
+ Instead, @{thm add_diff_inverse_lemma} states the desired induction scheme;
+ the use of \<open>THEN\<close> below instantiates Vars in @{thm ProdE} automatically.
+\<close>
lemma add_diff_inverse: "\<lbrakk>a:N; b:N; b - a = 0 : N\<rbrakk> \<Longrightarrow> b #+ (a-b) = a : N"
-apply (rule EqE)
-apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE])
-apply (assumption | rule EqI)+
-done
+ apply (rule EqE)
+ apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE])
+ apply (assumption | rule EqI)+
+ done
subsection \<open>Absolute difference\<close>
-(*typing of absolute difference: short and long versions*)
+text \<open>Typing of absolute difference: short and long versions.\<close>
lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b : N"
-apply (unfold arith_defs)
-apply typechk
-done
+ unfolding arith_defs by typechk
lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a |-| b = c |-| d : N"
-apply (unfold arith_defs)
-apply equal
-done
+ unfolding arith_defs by equal
lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a |-| a = 0 : N"
-apply (unfold absdiff_def)
-apply (arith_rew diff_self_eq_0)
-done
+ unfolding absdiff_def by (arith_rew diff_self_eq_0)
lemma absdiffC0: "a:N \<Longrightarrow> 0 |-| a = a : N"
-apply (unfold absdiff_def)
-apply hyp_arith_rew
-done
-
+ unfolding absdiff_def by hyp_arith_rew
lemma absdiff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) |-| succ(b) = a |-| b : N"
-apply (unfold absdiff_def)
-apply hyp_arith_rew
-done
+ unfolding absdiff_def by hyp_arith_rew
-(*Note how easy using commutative laws can be? ...not always... *)
+text \<open>Note how easy using commutative laws can be? ...not always...\<close>
lemma absdiff_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b = b |-| a : N"
-apply (unfold absdiff_def)
-apply (rule add_commute)
-apply (typechk diff_typing)
-done
+ unfolding absdiff_def
+ apply (rule add_commute)
+ apply (typechk diff_typing)
+ done
-(*If a+b=0 then a=0. Surprisingly tedious*)
+text \<open>If \<open>a + b = 0\<close> then \<open>a = 0\<close>. Surprisingly tedious.\<close>
schematic_goal add_eq0_lemma: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> ?c : \<Prod>u: Eq(N,a#+b,0) . Eq(N,a,0)"
-apply (NE a)
-apply (rule_tac [3] replace_type)
-apply arith_rew
-apply intr (*strips remaining PRODs*)
-apply (rule_tac [2] zero_ne_succ [THEN FE])
-apply (erule_tac [3] EqE [THEN sym_elem])
-apply (typechk add_typing)
-done
+ apply (NE a)
+ apply (rule_tac [3] replace_type)
+ apply arith_rew
+ apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close>
+ apply (rule_tac [2] zero_ne_succ [THEN FE])
+ apply (erule_tac [3] EqE [THEN sym_elem])
+ apply (typechk add_typing)
+ done
-(*Version of above with the premise a+b=0.
- Again, resolution instantiates variables in ProdE *)
+text \<open>
+ Version of above with the premise \<open>a + b = 0\<close>.
+ Again, resolution instantiates variables in @{thm ProdE}.
+\<close>
lemma add_eq0: "\<lbrakk>a:N; b:N; a #+ b = 0 : N\<rbrakk> \<Longrightarrow> a = 0 : N"
-apply (rule EqE)
-apply (rule add_eq0_lemma [THEN ProdE])
-apply (rule_tac [3] EqI)
-apply typechk
-done
+ apply (rule EqE)
+ apply (rule add_eq0_lemma [THEN ProdE])
+ apply (rule_tac [3] EqI)
+ apply typechk
+ done
-(*Here is a lemma to infer a-b=0 and b-a=0 from a|-|b=0, below. *)
+text \<open>Here is a lemma to infer \<open>a - b = 0\<close> and \<open>b - a = 0\<close> from \<open>a |-| b = 0\<close>, below.\<close>
schematic_goal absdiff_eq0_lem:
"\<lbrakk>a:N; b:N; a |-| b = 0 : N\<rbrakk> \<Longrightarrow> ?a : \<Sum>v: Eq(N, a-b, 0) . Eq(N, b-a, 0)"
-apply (unfold absdiff_def)
-apply intr
-apply eqintr
-apply (rule_tac [2] add_eq0)
-apply (rule add_eq0)
-apply (rule_tac [6] add_commute [THEN trans_elem])
-apply (typechk diff_typing)
-done
+ apply (unfold absdiff_def)
+ apply intr
+ apply eqintr
+ apply (rule_tac [2] add_eq0)
+ apply (rule add_eq0)
+ apply (rule_tac [6] add_commute [THEN trans_elem])
+ apply (typechk 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*)
+text \<open>If \<open>a |-| b = 0\<close> then \<open>a = b\<close>
+ proof: \<open>a - b = 0\<close> and \<open>b - a = 0\<close>, so \<open>b = a + (b - a) = a + 0 = a\<close>.
+\<close>
lemma absdiff_eq0: "\<lbrakk>a |-| b = 0 : N; a:N; b:N\<rbrakk> \<Longrightarrow> a = b : N"
-apply (rule EqE)
-apply (rule absdiff_eq0_lem [THEN SumE])
-apply eqintr
-apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem])
-apply (erule_tac [3] EqE)
-apply (hyp_arith_rew add_0_right)
-done
+ apply (rule EqE)
+ apply (rule absdiff_eq0_lem [THEN SumE])
+ apply eqintr
+ apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem])
+ apply (erule_tac [3] EqE)
+ apply (hyp_arith_rew add_0_right)
+ done
subsection \<open>Remainder and Quotient\<close>
-(*typing of remainder: short and long versions*)
+text \<open>Typing of remainder: short and long versions.\<close>
lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N"
-apply (unfold mod_def)
-apply (typechk absdiff_typing)
-done
+ unfolding mod_def by (typechk absdiff_typing)
lemma mod_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a mod b = c mod d : N"
-apply (unfold mod_def)
-apply (equal absdiff_typingL)
-done
+ unfolding mod_def by (equal absdiff_typingL)
-(*computation for mod : 0 and successor cases*)
+text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close>
lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N"
-apply (unfold mod_def)
-apply (rew absdiff_typing)
-done
+ unfolding mod_def by (rew absdiff_typing)
lemma modC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
succ(a) mod b = rec(succ(a mod b) |-| b, 0, \<lambda>x y. succ(a mod b)) : N"
-apply (unfold mod_def)
-apply (rew absdiff_typing)
-done
+ unfolding mod_def by (rew absdiff_typing)
-(*typing of quotient: short and long versions*)
+text \<open>Typing of quotient: short and long versions.\<close>
lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N"
-apply (unfold div_def)
-apply (typechk absdiff_typing mod_typing)
-done
+ unfolding div_def by (typechk absdiff_typing mod_typing)
lemma div_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a div b = c div d : N"
-apply (unfold div_def)
-apply (equal absdiff_typingL mod_typingL)
-done
+ unfolding div_def by (equal absdiff_typingL mod_typingL)
lemmas div_typing_rls = mod_typing div_typing absdiff_typing
-(*computation for quotient: 0 and successor cases*)
+text \<open>Computation for quotient: 0 and successor cases.\<close>
lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N"
-apply (unfold div_def)
-apply (rew mod_typing absdiff_typing)
-done
+ unfolding div_def by (rew mod_typing absdiff_typing)
lemma divC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
succ(a) div b = rec(succ(a) mod b, succ(a div b), \<lambda>x y. a div b) : N"
-apply (unfold div_def)
-apply (rew mod_typing)
-done
+ unfolding div_def by (rew mod_typing)
-(*Version of above with same condition as the mod one*)
+text \<open>Version of above with same condition as the \<open>mod\<close> one.\<close>
lemma divC_succ2: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow>
succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), \<lambda>x y. a div b) : N"
-apply (rule divC_succ [THEN trans_elem])
-apply (rew div_typing_rls modC_succ)
-apply (NE "succ (a mod b) |-|b")
-apply (rew mod_typing div_typing absdiff_typing)
-done
+ apply (rule divC_succ [THEN trans_elem])
+ apply (rew div_typing_rls modC_succ)
+ apply (NE "succ (a mod b) |-|b")
+ apply (rew mod_typing div_typing absdiff_typing)
+ done
-(*for case analysis on whether a number is 0 or a successor*)
+text \<open>For case analysis on whether a number is 0 or a successor.\<close>
lemma iszero_decidable: "a:N \<Longrightarrow> rec(a, inl(eq), \<lambda>ka kb. inr(<ka, eq>)) :
- Eq(N,a,0) + (\<Sum>x:N. Eq(N,a, succ(x)))"
-apply (NE a)
-apply (rule_tac [3] PlusI_inr)
-apply (rule_tac [2] PlusI_inl)
-apply eqintr
-apply equal
-done
+ Eq(N,a,0) + (\<Sum>x:N. Eq(N,a, succ(x)))"
+ apply (NE a)
+ apply (rule_tac [3] PlusI_inr)
+ apply (rule_tac [2] PlusI_inl)
+ apply eqintr
+ apply equal
+ done
-(*Main Result. Holds when b is 0 since a mod 0 = a and a div 0 = 0 *)
+text \<open>Main Result. Holds when \<open>b\<close> is 0 since \<open>a mod 0 = a\<close> and \<open>a div 0 = 0\<close>.\<close>
lemma mod_div_equality: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b #+ (a div b) #* b = a : N"
-apply (NE a)
-apply (arith_rew div_typing_rls modC0 modC_succ divC0 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 (hyp_arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2)
-(*Replace one occurrence 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 (hyp_arith_rew div_typing_rls)
-done
+ apply (NE a)
+ apply (arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2)
+ apply (rule EqE)
+ \<comment> \<open>case analysis on \<open>succ(u mod b) |-| b\<close>\<close>
+ apply (rule_tac a1 = "succ (u mod b) |-| b" in iszero_decidable [THEN PlusE])
+ apply (erule_tac [3] SumE)
+ apply (hyp_arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2)
+ \<comment> \<open>Replace one occurrence of \<open>b\<close> by \<open>succ(u mod b)\<close>. Clumsy!\<close>
+ apply (rule add_typingL [THEN trans_elem])
+ apply (erule EqE [THEN absdiff_eq0, THEN sym_elem])
+ apply (rule_tac [3] refl_elem)
+ apply (hyp_arith_rew div_typing_rls)
+ done
end