# HG changeset patch
# User wenzelm
# Date 1491760612 -7200
# Node ID fae6051ec19242de361a7625fc9fb815c2d26a30
# Parent  ed18feb34c071f0626a106473925b94f697b62c6
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;

diff -r ed18feb34c07 -r fae6051ec192 src/CTT/Arith.thy
--- a/src/CTT/Arith.thy	Sun Apr 09 19:03:55 2017 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,415 +0,0 @@
-(*  Title:      CTT/Arith.thy
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1991  University of Cambridge
-*)
-
-section \<open>Elementary arithmetic\<close>
-
-theory Arith
-  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 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 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 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>
-
-subsubsection \<open>Addition\<close>
-
-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"
-  unfolding arith_defs by typechk
-
-lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N"
-  unfolding arith_defs by equal
-
-
-text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close>
-
-lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N"
-  unfolding arith_defs by rew
-
-lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N"
-  unfolding arith_defs by rew
-
-
-subsubsection \<open>Multiplication\<close>
-
-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"
-  unfolding arith_defs by (typechk add_typing)
-
-lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N"
-  unfolding arith_defs by (equal add_typingL)
-
-
-text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close>
-
-lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N"
-  unfolding arith_defs by rew
-
-lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N"
-  unfolding arith_defs by rew
-
-
-subsubsection \<open>Difference\<close>
-
-text \<open>Typing of difference.\<close>
-
-lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a - b : N"
-  unfolding arith_defs by typechk
-
-lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a - b = c - d : N"
-  unfolding arith_defs by equal
-
-
-text \<open>Computation for difference: 0 and successor cases.\<close>
-
-lemma diffC0: "a:N \<Longrightarrow> a - 0 = a : N"
-  unfolding arith_defs by rew
-
-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"
-  unfolding arith_defs
-  apply (NE b)
-    apply hyp_rew
-  done
-
-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"
-  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 =
-  addC0 addC_succ
-  multC0 multC_succ
-  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}
-  )
-
-  fun arith_rew_tac ctxt prems =
-    make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems))
-
-  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>
-  Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths))
-\<close>
-
-method_setup hyp_arith_rew = \<open>
-  Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths))
-\<close>
-
-
-subsection \<open>Addition\<close>
-
-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
-
-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
-
-
-subsection \<open>Multiplication\<close>
-
-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
-
-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
-
-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
-
-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
-
-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
-
-
-subsection \<open>Difference\<close>
-
-text \<open>
-  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
-
-
-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]])
-
-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.
-\<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)
-    \<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
-
-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
-
-
-subsection \<open>Absolute difference\<close>
-
-text \<open>Typing of absolute difference: short and long versions.\<close>
-
-lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b : N"
-  unfolding arith_defs by typechk
-
-lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a |-| b = c |-| d : N"
-  unfolding arith_defs by equal
-
-lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a |-| a = 0 : N"
-  unfolding absdiff_def by (arith_rew diff_self_eq_0)
-
-lemma absdiffC0: "a:N \<Longrightarrow> 0 |-| a = a : N"
-  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"
-  unfolding absdiff_def by hyp_arith_rew
-
-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"
-  unfolding absdiff_def
-  apply (rule add_commute)
-   apply (typechk diff_typing)
-  done
-
-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 : Eq(N,a#+b,0) \<longrightarrow> Eq(N,a,0)"
-  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
-
-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
-
-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 : Eq(N, a-b, 0) \<times> 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
-
-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
-
-
-subsection \<open>Remainder and Quotient\<close>
-
-text \<open>Typing of remainder: short and long versions.\<close>
-
-lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N"
-  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"
-  unfolding mod_def by (equal absdiff_typingL)
-
-
-text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close>
-
-lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N"
-  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"
-  unfolding mod_def by (rew absdiff_typing)
-
-
-text \<open>Typing of quotient: short and long versions.\<close>
-
-lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N"
-  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"
-  unfolding div_def by (equal absdiff_typingL mod_typingL)
-
-lemmas div_typing_rls = mod_typing div_typing absdiff_typing
-
-
-text \<open>Computation for quotient: 0 and successor cases.\<close>
-
-lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N"
-  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"
-  unfolding div_def by (rew mod_typing)
-
-
-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
-
-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
-
-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)
-    \<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
diff -r ed18feb34c07 -r fae6051ec192 src/CTT/Bool.thy
--- a/src/CTT/Bool.thy	Sun Apr 09 19:03:55 2017 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,73 +0,0 @@
-(*  Title:      CTT/Bool.thy
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1991  University of Cambridge
-*)
-
-section \<open>The two-element type (booleans and conditionals)\<close>
-
-theory Bool
-  imports CTT
-begin
-
-definition Bool :: "t"
-  where "Bool \<equiv> T+T"
-
-definition true :: "i"
-  where "true \<equiv> inl(tt)"
-
-definition false :: "i"
-  where "false \<equiv> inr(tt)"
-
-definition cond :: "[i,i,i]\<Rightarrow>i"
-  where "cond(a,b,c) \<equiv> when(a, \<lambda>_. b, \<lambda>_. c)"
-
-lemmas bool_defs = Bool_def true_def false_def cond_def
-
-
-subsection \<open>Derivation of rules for the type \<open>Bool\<close>\<close>
-
-text \<open>Formation rule.\<close>
-lemma boolF: "Bool type"
-  unfolding bool_defs by typechk
-
-text \<open>Introduction rules for \<open>true\<close>, \<open>false\<close>.\<close>
-
-lemma boolI_true: "true : Bool"
-  unfolding bool_defs by typechk
-
-lemma boolI_false: "false : Bool"
-  unfolding bool_defs by typechk
-
-text \<open>Elimination rule: typing of \<open>cond\<close>.\<close>
-lemma boolE: "\<lbrakk>p:Bool; a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(p,a,b) : C(p)"
-  unfolding bool_defs
-  apply (typechk; erule TE)
-   apply typechk
-  done
-
-lemma boolEL: "\<lbrakk>p = q : Bool; a = c : C(true); b = d : C(false)\<rbrakk>
-  \<Longrightarrow> cond(p,a,b) = cond(q,c,d) : C(p)"
-  unfolding bool_defs
-  apply (rule PlusEL)
-    apply (erule asm_rl refl_elem [THEN TEL])+
-  done
-
-text \<open>Computation rules for \<open>true\<close>, \<open>false\<close>.\<close>
-
-lemma boolC_true: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(true,a,b) = a : C(true)"
-  unfolding bool_defs
-  apply (rule comp_rls)
-    apply typechk
-   apply (erule_tac [!] TE)
-   apply typechk
-  done
-
-lemma boolC_false: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(false,a,b) = b : C(false)"
-  unfolding bool_defs
-  apply (rule comp_rls)
-    apply typechk
-   apply (erule_tac [!] TE)
-   apply typechk
-  done
-
-end
diff -r ed18feb34c07 -r fae6051ec192 src/CTT/CTT.thy
--- a/src/CTT/CTT.thy	Sun Apr 09 19:03:55 2017 +0200
+++ b/src/CTT/CTT.thy	Sun Apr 09 19:56:52 2017 +0200
@@ -3,12 +3,12 @@
     Copyright   1993  University of Cambridge
 *)
 
-section \<open>Constructive Type Theory\<close>
-
 theory CTT
 imports Pure
 begin
 
+section \<open>Constructive Type Theory: axiomatic basis\<close>
+
 ML_file "~~/src/Provers/typedsimp.ML"
 setup Pure_Thy.old_appl_syntax_setup
 
@@ -482,4 +482,473 @@
       apply (typechk assms)
   done
 
+
+section \<open>The two-element type (booleans and conditionals)\<close>
+
+definition Bool :: "t"
+  where "Bool \<equiv> T+T"
+
+definition true :: "i"
+  where "true \<equiv> inl(tt)"
+
+definition false :: "i"
+  where "false \<equiv> inr(tt)"
+
+definition cond :: "[i,i,i]\<Rightarrow>i"
+  where "cond(a,b,c) \<equiv> when(a, \<lambda>_. b, \<lambda>_. c)"
+
+lemmas bool_defs = Bool_def true_def false_def cond_def
+
+
+subsection \<open>Derivation of rules for the type \<open>Bool\<close>\<close>
+
+text \<open>Formation rule.\<close>
+lemma boolF: "Bool type"
+  unfolding bool_defs by typechk
+
+text \<open>Introduction rules for \<open>true\<close>, \<open>false\<close>.\<close>
+
+lemma boolI_true: "true : Bool"
+  unfolding bool_defs by typechk
+
+lemma boolI_false: "false : Bool"
+  unfolding bool_defs by typechk
+
+text \<open>Elimination rule: typing of \<open>cond\<close>.\<close>
+lemma boolE: "\<lbrakk>p:Bool; a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(p,a,b) : C(p)"
+  unfolding bool_defs
+  apply (typechk; erule TE)
+   apply typechk
+  done
+
+lemma boolEL: "\<lbrakk>p = q : Bool; a = c : C(true); b = d : C(false)\<rbrakk>
+  \<Longrightarrow> cond(p,a,b) = cond(q,c,d) : C(p)"
+  unfolding bool_defs
+  apply (rule PlusEL)
+    apply (erule asm_rl refl_elem [THEN TEL])+
+  done
+
+text \<open>Computation rules for \<open>true\<close>, \<open>false\<close>.\<close>
+
+lemma boolC_true: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(true,a,b) = a : C(true)"
+  unfolding bool_defs
+  apply (rule comp_rls)
+    apply typechk
+   apply (erule_tac [!] TE)
+   apply typechk
+  done
+
+lemma boolC_false: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(false,a,b) = b : C(false)"
+  unfolding bool_defs
+  apply (rule comp_rls)
+    apply typechk
+   apply (erule_tac [!] TE)
+   apply typechk
+  done
+
+section \<open>Elementary arithmetic\<close>
+
+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 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 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 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>
+
+subsubsection \<open>Addition\<close>
+
+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"
+  unfolding arith_defs by typechk
+
+lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N"
+  unfolding arith_defs by equal
+
+
+text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close>
+
+lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N"
+  unfolding arith_defs by rew
+
+lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N"
+  unfolding arith_defs by rew
+
+
+subsubsection \<open>Multiplication\<close>
+
+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"
+  unfolding arith_defs by (typechk add_typing)
+
+lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N"
+  unfolding arith_defs by (equal add_typingL)
+
+
+text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close>
+
+lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N"
+  unfolding arith_defs by rew
+
+lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N"
+  unfolding arith_defs by rew
+
+
+subsubsection \<open>Difference\<close>
+
+text \<open>Typing of difference.\<close>
+
+lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a - b : N"
+  unfolding arith_defs by typechk
+
+lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a - b = c - d : N"
+  unfolding arith_defs by equal
+
+
+text \<open>Computation for difference: 0 and successor cases.\<close>
+
+lemma diffC0: "a:N \<Longrightarrow> a - 0 = a : N"
+  unfolding arith_defs by rew
+
+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"
+  unfolding arith_defs
+  apply (NE b)
+    apply hyp_rew
+  done
+
+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"
+  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 =
+  addC0 addC_succ
+  multC0 multC_succ
+  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}
+  )
+
+  fun arith_rew_tac ctxt prems =
+    make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems))
+
+  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>
+  Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths))
+\<close>
+
+method_setup hyp_arith_rew = \<open>
+  Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths))
+\<close>
+
+
+subsection \<open>Addition\<close>
+
+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
+
+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
+
+
+subsection \<open>Multiplication\<close>
+
+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
+
+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
+
+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
+
+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
+
+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
+
+
+subsection \<open>Difference\<close>
+
+text \<open>
+  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
+
+
+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]])
+
+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.
+\<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)
+    \<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
+
+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
+
+
+subsection \<open>Absolute difference\<close>
+
+text \<open>Typing of absolute difference: short and long versions.\<close>
+
+lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b : N"
+  unfolding arith_defs by typechk
+
+lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a |-| b = c |-| d : N"
+  unfolding arith_defs by equal
+
+lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a |-| a = 0 : N"
+  unfolding absdiff_def by (arith_rew diff_self_eq_0)
+
+lemma absdiffC0: "a:N \<Longrightarrow> 0 |-| a = a : N"
+  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"
+  unfolding absdiff_def by hyp_arith_rew
+
+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"
+  unfolding absdiff_def
+  apply (rule add_commute)
+   apply (typechk diff_typing)
+  done
+
+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 : Eq(N,a#+b,0) \<longrightarrow> Eq(N,a,0)"
+  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
+
+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
+
+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 : Eq(N, a-b, 0) \<times> 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
+
+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
+
+
+subsection \<open>Remainder and Quotient\<close>
+
+text \<open>Typing of remainder: short and long versions.\<close>
+
+lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N"
+  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"
+  unfolding mod_def by (equal absdiff_typingL)
+
+
+text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close>
+
+lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N"
+  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"
+  unfolding mod_def by (rew absdiff_typing)
+
+
+text \<open>Typing of quotient: short and long versions.\<close>
+
+lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N"
+  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"
+  unfolding div_def by (equal absdiff_typingL mod_typingL)
+
+lemmas div_typing_rls = mod_typing div_typing absdiff_typing
+
+
+text \<open>Computation for quotient: 0 and successor cases.\<close>
+
+lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N"
+  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"
+  unfolding div_def by (rew mod_typing)
+
+
+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
+
+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
+
+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)
+    \<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
diff -r ed18feb34c07 -r fae6051ec192 src/CTT/Main.thy
--- a/src/CTT/Main.thy	Sun Apr 09 19:03:55 2017 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,6 +0,0 @@
-section \<open>Main includes everything\<close>
-
-theory Main
-  imports CTT Bool Arith
-begin
-end
diff -r ed18feb34c07 -r fae6051ec192 src/CTT/ROOT
--- a/src/CTT/ROOT	Sun Apr 09 19:03:55 2017 +0200
+++ b/src/CTT/ROOT	Sun Apr 09 19:56:52 2017 +0200
@@ -18,7 +18,7 @@
   *}
   options [thy_output_source]
   theories
-    Main
+    CTT
     "ex/Typechecking"
     "ex/Elimination"
     "ex/Equality"
diff -r ed18feb34c07 -r fae6051ec192 src/CTT/ex/Synthesis.thy
--- a/src/CTT/ex/Synthesis.thy	Sun Apr 09 19:03:55 2017 +0200
+++ b/src/CTT/ex/Synthesis.thy	Sun Apr 09 19:56:52 2017 +0200
@@ -6,7 +6,7 @@
 section "Synthesis examples, using a crude form of narrowing"
 
 theory Synthesis
-imports "../Arith"
+imports "../CTT"
 begin
 
 text "discovery of predecessor function"