# Theory ArithSimp

```(*  Title:      ZF/ArithSimp.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹Arithmetic with simplification›

theory ArithSimp
imports Arith
begin

subsection ‹Arithmetic simplification›

ML_file ‹~~/src/Provers/Arith/cancel_numerals.ML›
ML_file ‹~~/src/Provers/Arith/combine_numerals.ML›
ML_file ‹arith_data.ML›

simproc_setup nateq_cancel_numerals
("l #+ m = n" | "l = m #+ n" | "l #* m = n" | "l = m #* n" | "succ(m) = n" | "m = succ(n)") =
‹K ArithData.nateq_cancel_numerals_proc›

simproc_setup natless_cancel_numerals
("l #+ m < n" | "l < m #+ n" | "l #* m < n" | "l < m #* n" | "succ(m) < n" | "m < succ(n)") =
‹K ArithData.natless_cancel_numerals_proc›

simproc_setup natdiff_cancel_numerals
("(l #+ m) #- n" | "l #- (m #+ n)" | "(l #* m) #- n" | "l #- (m #* n)" |
"succ(m) #- n" | "m #- succ(n)") =
‹K ArithData.natdiff_cancel_numerals_proc›

subsubsection ‹Examples›

lemma "x #+ y = x #+ z" apply simp oops
lemma "y #+ x = x #+ z" apply simp oops
lemma "x #+ y #+ z = x #+ z" apply simp oops
lemma "y #+ (z #+ x) = z #+ x" apply simp oops
lemma "x #+ y #+ z = (z #+ y) #+ (x #+ w)" apply simp oops
lemma "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)" apply simp oops

lemma "x #+ succ(y) = x #+ z" apply simp oops
lemma "x #+ succ(y) = succ(z #+ x)" apply simp oops
lemma "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)" apply simp oops

lemma "(x #+ y) #- (x #+ z) = w" apply simp oops
lemma "(y #+ x) #- (x #+ z) = dd" apply simp oops
lemma "(x #+ y #+ z) #- (x #+ z) = dd" apply simp oops
lemma "(y #+ (z #+ x)) #- (z #+ x) = dd" apply simp oops
lemma "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd" apply simp oops
lemma "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd" apply simp oops

lemma "(x #+ succ(y)) #- (x #+ z) = dd" apply simp oops

lemma "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2" apply simp oops

lemma "(x #+ succ(y)) #- (succ(z #+ x)) = dd" apply simp oops
lemma "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd" apply simp oops

(*use of typing information*)
lemma "x : nat ==> x #+ y = x" apply simp oops
lemma "x : nat --> x #+ y = x" apply simp oops
lemma "x : nat ==> x #+ y < x" apply simp oops
lemma "x : nat ==> x < y#+x" apply simp oops
lemma "x : nat ==> x ≤ succ(x)" apply simp oops

(*fails: no typing information isn't visible*)
lemma "x #+ y = x" apply simp? oops

lemma "x #+ y < x #+ z" apply simp oops
lemma "y #+ x < x #+ z" apply simp oops
lemma "x #+ y #+ z < x #+ z" apply simp oops
lemma "y #+ z #+ x < x #+ z" apply simp oops
lemma "y #+ (z #+ x) < z #+ x" apply simp oops
lemma "x #+ y #+ z < (z #+ y) #+ (x #+ w)" apply simp oops
lemma "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)" apply simp oops

lemma "x #+ succ(y) < x #+ z" apply simp oops
lemma "x #+ succ(y) < succ(z #+ x)" apply simp oops
lemma "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)" apply simp oops

lemma "x #+ succ(y) ≤ succ(z #+ x)" apply simp oops

subsection‹Difference›

lemma diff_self_eq_0 [simp]: "m #- m = 0"
apply (subgoal_tac "natify (m) #- natify (m) = 0")
apply (rule_tac [2] natify_in_nat [THEN nat_induct], auto)
done

(**Addition is the inverse of subtraction**)

(*We need m:nat even if we replace the RHS by natify(m), for consider e.g.
n=2, m=omega; then n + (m-n) = 2 + (0-2) = 2 ≠ 0 = natify(m).*)
lemma add_diff_inverse: "⟦n ≤ m;  m:nat⟧ ⟹ n #+ (m#-n) = m"
apply (frule lt_nat_in_nat, erule nat_succI)
apply (erule rev_mp)
apply (rule_tac m = m and n = n in diff_induct, auto)
done

lemma add_diff_inverse2: "⟦n ≤ m;  m:nat⟧ ⟹ (m#-n) #+ n = m"
apply (frule lt_nat_in_nat, erule nat_succI)
done

(*Proof is IDENTICAL to that of add_diff_inverse*)
lemma diff_succ: "⟦n ≤ m;  m:nat⟧ ⟹ succ(m) #- n = succ(m#-n)"
apply (frule lt_nat_in_nat, erule nat_succI)
apply (erule rev_mp)
apply (rule_tac m = m and n = n in diff_induct)
apply (simp_all (no_asm_simp))
done

lemma zero_less_diff [simp]:
"⟦m: nat; n: nat⟧ ⟹ 0 < (n #- m)   ⟷   m<n"
apply (rule_tac m = m and n = n in diff_induct)
apply (simp_all (no_asm_simp))
done

(** Difference distributes over multiplication **)

lemma diff_mult_distrib: "(m #- n) #* k = (m #* k) #- (n #* k)"
apply (subgoal_tac " (natify (m) #- natify (n)) #* natify (k) = (natify (m) #* natify (k)) #- (natify (n) #* natify (k))")
apply (rule_tac [2] m = "natify (m) " and n = "natify (n) " in diff_induct)
done

lemma diff_mult_distrib2: "k #* (m #- n) = (k #* m) #- (k #* n)"
apply (simp (no_asm) add: mult_commute [of k] diff_mult_distrib)
done

subsection‹Remainder›

(*We need m:nat even with natify*)
lemma div_termination: "⟦0<n;  n ≤ m;  m:nat⟧ ⟹ m #- n < m"
apply (frule lt_nat_in_nat, erule nat_succI)
apply (erule rev_mp)
apply (erule rev_mp)
apply (rule_tac m = m and n = n in diff_induct)
done

(*for mod and div*)
lemmas div_rls =
nat_typechecks Ord_transrec_type apply_funtype
div_termination [THEN ltD]
nat_into_Ord not_lt_iff_le [THEN iffD1]

lemma raw_mod_type: "⟦m:nat;  n:nat⟧ ⟹ raw_mod (m, n) ∈ nat"
unfolding raw_mod_def
apply (rule Ord_transrec_type)
apply (auto simp add: nat_into_Ord [THEN Ord_0_lt_iff])
apply (blast intro: div_rls)
done

lemma mod_type [TC,iff]: "m mod n ∈ nat"
unfolding mod_def
apply (simp (no_asm) add: mod_def raw_mod_type)
done

(** Aribtrary definitions for division by zero.  Useful to simplify
certain equations **)

lemma DIVISION_BY_ZERO_DIV: "a div 0 = 0"
unfolding div_def
apply (rule raw_div_def [THEN def_transrec, THEN trans])
apply (simp (no_asm_simp))
done  (*NOT for adding to default simpset*)

lemma DIVISION_BY_ZERO_MOD: "a mod 0 = natify(a)"
unfolding mod_def
apply (rule raw_mod_def [THEN def_transrec, THEN trans])
apply (simp (no_asm_simp))
done  (*NOT for adding to default simpset*)

lemma raw_mod_less: "m<n ⟹ raw_mod (m,n) = m"
apply (rule raw_mod_def [THEN def_transrec, THEN trans])
apply (simp (no_asm_simp) add: div_termination [THEN ltD])
done

lemma mod_less [simp]: "⟦m<n; n ∈ nat⟧ ⟹ m mod n = m"
apply (frule lt_nat_in_nat, assumption)
apply (simp (no_asm_simp) add: mod_def raw_mod_less)
done

lemma raw_mod_geq:
"⟦0<n; n ≤ m;  m:nat⟧ ⟹ raw_mod (m, n) = raw_mod (m#-n, n)"
apply (frule lt_nat_in_nat, erule nat_succI)
apply (rule raw_mod_def [THEN def_transrec, THEN trans])
apply (simp (no_asm_simp) add: div_termination [THEN ltD] not_lt_iff_le [THEN iffD2], blast)
done

lemma mod_geq: "⟦n ≤ m;  m:nat⟧ ⟹ m mod n = (m#-n) mod n"
apply (frule lt_nat_in_nat, erule nat_succI)
apply (case_tac "n=0")
apply (simp add: mod_def raw_mod_geq nat_into_Ord [THEN Ord_0_lt_iff])
done

subsection‹Division›

lemma raw_div_type: "⟦m:nat;  n:nat⟧ ⟹ raw_div (m, n) ∈ nat"
unfolding raw_div_def
apply (rule Ord_transrec_type)
apply (auto simp add: nat_into_Ord [THEN Ord_0_lt_iff])
apply (blast intro: div_rls)
done

lemma div_type [TC,iff]: "m div n ∈ nat"
unfolding div_def
apply (simp (no_asm) add: div_def raw_div_type)
done

lemma raw_div_less: "m<n ⟹ raw_div (m,n) = 0"
apply (rule raw_div_def [THEN def_transrec, THEN trans])
apply (simp (no_asm_simp) add: div_termination [THEN ltD])
done

lemma div_less [simp]: "⟦m<n; n ∈ nat⟧ ⟹ m div n = 0"
apply (frule lt_nat_in_nat, assumption)
apply (simp (no_asm_simp) add: div_def raw_div_less)
done

lemma raw_div_geq: "⟦0<n;  n ≤ m;  m:nat⟧ ⟹ raw_div(m,n) = succ(raw_div(m#-n, n))"
apply (subgoal_tac "n ≠ 0")
prefer 2 apply blast
apply (frule lt_nat_in_nat, erule nat_succI)
apply (rule raw_div_def [THEN def_transrec, THEN trans])
apply (simp (no_asm_simp) add: div_termination [THEN ltD] not_lt_iff_le [THEN iffD2] )
done

lemma div_geq [simp]:
"⟦0<n;  n ≤ m;  m:nat⟧ ⟹ m div n = succ ((m#-n) div n)"
apply (frule lt_nat_in_nat, erule nat_succI)
apply (simp (no_asm_simp) add: div_def raw_div_geq)
done

declare div_less [simp] div_geq [simp]

(*A key result*)
lemma mod_div_lemma: "⟦m: nat;  n: nat⟧ ⟹ (m div n)#*n #+ m mod n = m"
apply (case_tac "n=0")
apply (simp add: nat_into_Ord [THEN Ord_0_lt_iff])
apply (erule complete_induct)
apply (case_tac "x<n")
txt‹case x<n›
apply (simp (no_asm_simp))
txt‹case \<^term>‹n ≤ x››
done

lemma mod_div_equality_natify: "(m div n)#*n #+ m mod n = natify(m)"
apply (subgoal_tac " (natify (m) div natify (n))#*natify (n) #+ natify (m) mod natify (n) = natify (m) ")
apply force
apply (subst mod_div_lemma, auto)
done

lemma mod_div_equality: "m: nat ⟹ (m div n)#*n #+ m mod n = m"
done

text‹(mainly for mutilated chess board)›

lemma mod_succ_lemma:
"⟦0<n;  m:nat;  n:nat⟧
⟹ succ(m) mod n = (if succ(m mod n) = n then 0 else succ(m mod n))"
apply (erule complete_induct)
apply (case_tac "succ (x) <n")
txt‹case succ(x) < n›
apply (simp (no_asm_simp) add: nat_le_refl [THEN lt_trans] succ_neq_self)
apply (simp add: ltD [THEN mem_imp_not_eq])
txt‹case \<^term>‹n ≤ succ(x)››
apply (erule leE)
apply (simp (no_asm_simp) add: mod_geq div_termination [THEN ltD] diff_succ)
txt‹equality case›
done

lemma mod_succ:
"n:nat ⟹ succ(m) mod n = (if succ(m mod n) = n then 0 else succ(m mod n))"
apply (case_tac "n=0")
apply (simp (no_asm_simp) add: natify_succ DIVISION_BY_ZERO_MOD)
apply (subgoal_tac "natify (succ (m)) mod n = (if succ (natify (m) mod n) = n then 0 else succ (natify (m) mod n))")
prefer 2
apply (subst natify_succ)
apply (rule mod_succ_lemma)
apply (auto simp del: natify_succ simp add: nat_into_Ord [THEN Ord_0_lt_iff])
done

lemma mod_less_divisor: "⟦0<n;  n:nat⟧ ⟹ m mod n < n"
apply (subgoal_tac "natify (m) mod n < n")
apply (rule_tac [2] i = "natify (m) " in complete_induct)
apply (case_tac [3] "x<n", auto)
txt‹case \<^term>‹n ≤ x››
apply (simp add: mod_geq not_lt_iff_le div_termination [THEN ltD])
done

lemma mod_1_eq [simp]: "m mod 1 = 0"
by (cut_tac n = 1 in mod_less_divisor, auto)

lemma mod2_cases: "b<2 ⟹ k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)"
apply (subgoal_tac "k mod 2: 2")
prefer 2 apply (simp add: mod_less_divisor [THEN ltD])
apply (drule ltD, auto)
done

lemma mod2_succ_succ [simp]: "succ(succ(m)) mod 2 = m mod 2"
apply (subgoal_tac "m mod 2: 2")
prefer 2 apply (simp add: mod_less_divisor [THEN ltD])
done

lemma mod2_add_more [simp]: "(m#+m#+n) mod 2 = n mod 2"
apply (subgoal_tac " (natify (m) #+natify (m) #+n) mod 2 = n mod 2")
apply (rule_tac [2] n = "natify (m) " in nat_induct)
apply auto
done

lemma mod2_add_self [simp]: "(m#+m) mod 2 = 0"
by (cut_tac n = 0 in mod2_add_more, auto)

lemma add_le_self: "m:nat ⟹ m ≤ (m #+ n)"
apply (simp (no_asm_simp))
done

lemma add_le_self2: "m:nat ⟹ m ≤ (n #+ m)"
apply (simp (no_asm_simp))
done

(*** Monotonicity of Multiplication ***)

lemma mult_le_mono1: "⟦i ≤ j; j:nat⟧ ⟹ (i#*k) ≤ (j#*k)"
apply (subgoal_tac "natify (i) #*natify (k) ≤ j#*natify (k) ")
apply (frule_tac [2] lt_nat_in_nat)
apply (rule_tac [3] n = "natify (k) " in nat_induct)
done

(* @{text"≤"} monotonicity, BOTH arguments*)
lemma mult_le_mono: "⟦i ≤ j; k ≤ l; j:nat; l:nat⟧ ⟹ i#*k ≤ j#*l"
apply (rule mult_le_mono1 [THEN le_trans], assumption+)
apply (subst mult_commute, subst mult_commute, rule mult_le_mono1, assumption+)
done

(*strict, in 1st argument; proof is by induction on k>0.
I can't see how to relax the typing conditions.*)
lemma mult_lt_mono2: "⟦i<j; 0<k; j:nat; k:nat⟧ ⟹ k#*i < k#*j"
apply (erule zero_lt_natE)
apply (frule_tac [2] lt_nat_in_nat)
apply (simp_all (no_asm_simp))
apply (induct_tac "x")
done

lemma mult_lt_mono1: "⟦i<j; 0<k; j:nat; k:nat⟧ ⟹ i#*k < j#*k"
apply (simp (no_asm_simp) add: mult_lt_mono2 mult_commute [of _ k])
done

lemma add_eq_0_iff [iff]: "m#+n = 0 ⟷ natify(m)=0 ∧ natify(n)=0"
apply (subgoal_tac "natify (m) #+ natify (n) = 0 ⟷ natify (m) =0 ∧ natify (n) =0")
apply (rule_tac [2] n = "natify (m) " in natE)
apply (rule_tac [4] n = "natify (n) " in natE)
apply auto
done

lemma zero_lt_mult_iff [iff]: "0 < m#*n ⟷ 0 < natify(m) ∧ 0 < natify(n)"
apply (subgoal_tac "0 < natify (m) #*natify (n) ⟷ 0 < natify (m) ∧ 0 < natify (n) ")
apply (rule_tac [2] n = "natify (m) " in natE)
apply (rule_tac [4] n = "natify (n) " in natE)
apply (rule_tac [3] n = "natify (n) " in natE)
apply auto
done

lemma mult_eq_1_iff [iff]: "m#*n = 1 ⟷ natify(m)=1 ∧ natify(n)=1"
apply (subgoal_tac "natify (m) #* natify (n) = 1 ⟷ natify (m) =1 ∧ natify (n) =1")
apply (rule_tac [2] n = "natify (m) " in natE)
apply (rule_tac [4] n = "natify (n) " in natE)
apply auto
done

lemma mult_is_zero: "⟦m: nat; n: nat⟧ ⟹ (m #* n = 0) ⟷ (m = 0 | n = 0)"
apply auto
apply (erule natE)
apply (erule_tac [2] natE, auto)
done

lemma mult_is_zero_natify [iff]:
"(m #* n = 0) ⟷ (natify(m) = 0 | natify(n) = 0)"
apply (cut_tac m = "natify (m) " and n = "natify (n) " in mult_is_zero)
apply auto
done

subsection‹Cancellation Laws for Common Factors in Comparisons›

lemma mult_less_cancel_lemma:
"⟦k: nat; m: nat; n: nat⟧ ⟹ (m#*k < n#*k) ⟷ (0<k ∧ m<n)"
apply (safe intro!: mult_lt_mono1)
apply (erule natE, auto)
apply (rule not_le_iff_lt [THEN iffD1])
apply (drule_tac [3] not_le_iff_lt [THEN [2] rev_iffD2])
prefer 5 apply (blast intro: mult_le_mono1, auto)
done

lemma mult_less_cancel2 [simp]:
"(m#*k < n#*k) ⟷ (0 < natify(k) ∧ natify(m) < natify(n))"
apply (rule iff_trans)
apply (rule_tac [2] mult_less_cancel_lemma, auto)
done

lemma mult_less_cancel1 [simp]:
"(k#*m < k#*n) ⟷ (0 < natify(k) ∧ natify(m) < natify(n))"
apply (simp (no_asm) add: mult_less_cancel2 mult_commute [of k])
done

lemma mult_le_cancel2 [simp]: "(m#*k ≤ n#*k) ⟷ (0 < natify(k) ⟶ natify(m) ≤ natify(n))"
apply (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
apply auto
done

lemma mult_le_cancel1 [simp]: "(k#*m ≤ k#*n) ⟷ (0 < natify(k) ⟶ natify(m) ≤ natify(n))"
apply (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
apply auto
done

lemma mult_le_cancel_le1: "k ∈ nat ⟹ k #* m ≤ k ⟷ (0 < k ⟶ natify(m) ≤ 1)"
by (cut_tac k = k and m = m and n = 1 in mult_le_cancel1, auto)

lemma Ord_eq_iff_le: "⟦Ord(m); Ord(n)⟧ ⟹ m=n ⟷ (m ≤ n ∧ n ≤ m)"
by (blast intro: le_anti_sym)

lemma mult_cancel2_lemma:
"⟦k: nat; m: nat; n: nat⟧ ⟹ (m#*k = n#*k) ⟷ (m=n | k=0)"
apply (simp (no_asm_simp) add: Ord_eq_iff_le [of "m#*k"] Ord_eq_iff_le [of m])
done

lemma mult_cancel2 [simp]:
"(m#*k = n#*k) ⟷ (natify(m) = natify(n) | natify(k) = 0)"
apply (rule iff_trans)
apply (rule_tac [2] mult_cancel2_lemma, auto)
done

lemma mult_cancel1 [simp]:
"(k#*m = k#*n) ⟷ (natify(m) = natify(n) | natify(k) = 0)"
apply (simp (no_asm) add: mult_cancel2 mult_commute [of k])
done

(** Cancellation law for division **)

lemma div_cancel_raw:
"⟦0<n; 0<k; k:nat; m:nat; n:nat⟧ ⟹ (k#*m) div (k#*n) = m div n"
apply (erule_tac i = m in complete_induct)
apply (case_tac "x<n")
apply (simp add: div_less zero_lt_mult_iff mult_lt_mono2)
apply (simp add: not_lt_iff_le zero_lt_mult_iff le_refl [THEN mult_le_mono]
div_geq diff_mult_distrib2 [symmetric] div_termination [THEN ltD])
done

lemma div_cancel:
"⟦0 < natify(n);  0 < natify(k)⟧ ⟹ (k#*m) div (k#*n) = m div n"
apply (cut_tac k = "natify (k) " and m = "natify (m)" and n = "natify (n)"
in div_cancel_raw)
apply auto
done

lemma mult_mod_distrib_raw:
"⟦k:nat; m:nat; n:nat⟧ ⟹ (k#*m) mod (k#*n) = k #* (m mod n)"
apply (case_tac "k=0")
apply (case_tac "n=0")
apply (simp add: nat_into_Ord [THEN Ord_0_lt_iff])
apply (erule_tac i = m in complete_induct)
apply (case_tac "x<n")
apply (simp (no_asm_simp) add: mod_less zero_lt_mult_iff mult_lt_mono2)
apply (simp add: not_lt_iff_le zero_lt_mult_iff le_refl [THEN mult_le_mono]
mod_geq diff_mult_distrib2 [symmetric] div_termination [THEN ltD])
done

lemma mod_mult_distrib2: "k #* (m mod n) = (k#*m) mod (k#*n)"
apply (cut_tac k = "natify (k) " and m = "natify (m)" and n = "natify (n)"
in mult_mod_distrib_raw)
apply auto
done

lemma mult_mod_distrib: "(m mod n) #* k = (m#*k) mod (n#*k)"
apply (simp (no_asm) add: mult_commute mod_mult_distrib2)
done

lemma mod_add_self2_raw: "n ∈ nat ⟹ (m #+ n) mod n = m mod n"
apply (subgoal_tac " (n #+ m) mod n = (n #+ m #- n) mod n")
apply (subst mod_geq [symmetric], auto)
done

lemma mod_add_self2 [simp]: "(m #+ n) mod n = m mod n"
apply (cut_tac n = "natify (n) " in mod_add_self2_raw)
apply auto
done

lemma mod_add_self1 [simp]: "(n#+m) mod n = m mod n"
done

lemma mod_mult_self1_raw: "k ∈ nat ⟹ (m #+ k#*n) mod n = m mod n"
apply (erule nat_induct)
done

lemma mod_mult_self1 [simp]: "(m #+ k#*n) mod n = m mod n"
apply (cut_tac k = "natify (k) " in mod_mult_self1_raw)
apply auto
done

lemma mod_mult_self2 [simp]: "(m #+ n#*k) mod n = m mod n"
apply (simp (no_asm) add: mult_commute mod_mult_self1)
done

(*Lemma for gcd*)
lemma mult_eq_self_implies_10: "m = m#*n ⟹ natify(n)=1 | m=0"
apply (subgoal_tac "m: nat")
prefer 2
apply (erule ssubst)
apply simp
apply (rule disjCI)
apply (drule sym)
apply (rule Ord_linear_lt [of "natify(n)" 1])
apply simp_all
apply (subgoal_tac "m #* n = 0", simp)
apply (subst mult_natify2 [symmetric])
apply (simp del: mult_natify2)
apply (drule nat_into_Ord [THEN Ord_0_lt, THEN [2] mult_lt_mono2], auto)
done

"⟦m<n; n: nat⟧ ⟹ ∃k∈nat. n = succ(m#+k)"
apply (frule lt_nat_in_nat, assumption)
apply (erule rev_mp)
apply (induct_tac "n")
done

"⟦m: nat; n: nat⟧ ⟹ (m<n) ⟷ (∃k∈nat. n = succ(m#+k))"

"⟦a #+ d = b #+ c; a < b; b ∈ nat; c ∈ nat; d ∈ nat⟧ ⟹ c < d"

"⟦a #+ d = b #+ c; a ≤ b; b ∈ nat; c ∈ nat; d ∈ nat⟧ ⟹ c ≤ d"

lemma diff_is_0_lemma:
"⟦m: nat; n: nat⟧ ⟹ m #- n = 0 ⟷ m ≤ n"
apply (rule_tac m = m and n = n in diff_induct, simp_all)
done

lemma diff_is_0_iff: "m #- n = 0 ⟷ natify(m) ≤ natify(n)"

lemma nat_lt_imp_diff_eq_0:
"⟦a:nat; b:nat; a<b⟧ ⟹ a #- b = 0"

lemma raw_nat_diff_split:
"⟦a:nat; b:nat⟧ ⟹
(P(a #- b)) ⟷ ((a < b ⟶P(0)) ∧ (∀d∈nat. a = b #+ d ⟶ P(d)))"
apply (case_tac "a < b")
apply (rule iffI, force, simp)
apply (drule_tac x="a#-b" in bspec)
done

lemma nat_diff_split:
"(P(a #- b)) ⟷
(natify(a) < natify(b) ⟶P(0)) ∧ (∀d∈nat. natify(a) = b #+ d ⟶ P(d))"
apply (cut_tac P=P and a="natify(a)" and b="natify(b)" in raw_nat_diff_split)
apply simp_all
done

text‹Difference and less-than›

lemma diff_lt_imp_lt: "⟦(k#-i) < (k#-j); i∈nat; j∈nat; k∈nat⟧ ⟹ j<i"
apply (erule rev_mp)
apply (simp split: nat_diff_split, auto)
apply (rule not_le_iff_lt [THEN iffD1], auto)
apply (subgoal_tac "i #+ da < j #+ d", force)
done

lemma lt_imp_diff_lt: "⟦j<i; i≤k; k∈nat⟧ ⟹ (k#-i) < (k#-j)"
apply (frule le_in_nat, assumption)
apply (frule lt_nat_in_nat, assumption)
apply (simp split: nat_diff_split, auto)
apply (blast intro: lt_asym lt_trans2)
apply (blast intro: lt_irrefl lt_trans2)
apply (rule not_le_iff_lt [THEN iffD1], auto)
apply (subgoal_tac "j #+ d < i #+ da", force)