Theory Forward

theory Forward imports TPrimes begin

text‹\noindent
Forward proof material: of, OF, THEN, simplify, rule_format.
›

text‹\noindent
SKIP most developments...
›

(** Commutativity **)

lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
apply (auto simp add: is_gcd_def)
done

lemma gcd_commute: "gcd m n = gcd n m"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (subst is_gcd_commute)
apply (simp add: is_gcd)
done

lemma gcd_1 [simp]: "gcd m (Suc 0) = Suc 0"
apply simp
done

lemma gcd_1_left [simp]: "gcd (Suc 0) m = Suc 0"
apply (simp add: gcd_commute [of "Suc 0"])
done

text‹\noindent
as far as HERE.
›

text‹\noindent
SKIP THIS PROOF
›

lemma gcd_mult_distrib2: "k * gcd m n = gcd (k*m) (k*n)"
apply (induct_tac m n rule: gcd.induct)
apply (case_tac "n=0")
apply simp
apply (case_tac "k=0")
apply simp_all
done

text ‹
@{thm[display] gcd_mult_distrib2}
\rulename{gcd_mult_distrib2}
›

text‹\noindent
of, simplified
›


lemmas gcd_mult_0 = gcd_mult_distrib2 [of k 1] for k
lemmas gcd_mult_1 = gcd_mult_0 [simplified]

lemmas where1 = gcd_mult_distrib2 [where m=1]

lemmas where2 = gcd_mult_distrib2 [where m=1 and k=1]

lemmas where3 = gcd_mult_distrib2 [where m=1 and k="j+k"] for j k

text ‹
example using ``of'':
@{thm[display] gcd_mult_distrib2 [of _ 1]}

example using ``where'':
@{thm[display] gcd_mult_distrib2 [where m=1]}

example using ``where'', ``and'':
@{thm[display] gcd_mult_distrib2 [where m=1 and k="j+k"]}

@{thm[display] gcd_mult_0}
\rulename{gcd_mult_0}

@{thm[display] gcd_mult_1}
\rulename{gcd_mult_1}

@{thm[display] sym}
\rulename{sym}
›

lemmas gcd_mult0 = gcd_mult_1 [THEN sym]
      (*not quite right: we need ?k but this gives k*)

lemmas gcd_mult0' = gcd_mult_distrib2 [of k 1, simplified, THEN sym] for k
      (*better in one step!*)

text ‹
more legible, and variables properly generalized
›

lemma gcd_mult [simp]: "gcd k (k*n) = k"
by (rule gcd_mult_distrib2 [of k 1, simplified, THEN sym])


lemmas gcd_self0 = gcd_mult [of k 1, simplified] for k


text ‹
@{thm[display] gcd_mult}
\rulename{gcd_mult}

@{thm[display] gcd_self0}
\rulename{gcd_self0}
›

text ‹
Rules handy with THEN

@{thm[display] iffD1}
\rulename{iffD1}

@{thm[display] iffD2}
\rulename{iffD2}
›


text ‹
again: more legible, and variables properly generalized
›

lemma gcd_self [simp]: "gcd k k = k"
by (rule gcd_mult [of k 1, simplified])


text‹
NEXT SECTION: Methods for Forward Proof

NEW

theorem arg_cong, useful in forward steps
@{thm[display] arg_cong[no_vars]}
\rulename{arg_cong}
›

lemma "2 ≤ u ⟹ u*m ≠ Suc(u*n)"
apply (intro notI)
txt‹
before using arg_cong
@{subgoals[display,indent=0,margin=65]}
›
apply (drule_tac f="λx. x mod u" in arg_cong)
txt‹
after using arg_cong
@{subgoals[display,indent=0,margin=65]}
›
apply (simp add: mod_Suc)
done

text‹
have just used this rule:
@{thm[display] mod_Suc[no_vars]}
\rulename{mod_Suc}

@{thm[display] mult_le_mono1[no_vars]}
\rulename{mult_le_mono1}
›


text‹
example of "insert"
›

lemma relprime_dvd_mult:
      "⟦ gcd k n = 1; k dvd m*n ⟧ ⟹ k dvd m"
apply (insert gcd_mult_distrib2 [of m k n])
txt‹@{subgoals[display,indent=0,margin=65]}›
apply simp
txt‹@{subgoals[display,indent=0,margin=65]}›
apply (erule_tac t="m" in ssubst)
apply simp
done


text ‹
@{thm[display] relprime_dvd_mult}
\rulename{relprime_dvd_mult}

Another example of "insert"

@{thm[display] div_mult_mod_eq}
\rulename{div_mult_mod_eq}
›

(*MOVED to Force.thy, which now depends only on Divides.thy
lemma div_mult_self_is_m: "0<n ⟹ (m*n) div n = (m::nat)"
*)

lemma relprime_dvd_mult_iff: "gcd k n = 1 ⟹ (k dvd m*n) = (k dvd m)"
by (auto intro: relprime_dvd_mult elim: dvdE)

lemma relprime_20_81: "gcd 20 81 = 1"
by (simp add: gcd.simps)

text ‹
Examples of 'OF'

@{thm[display] relprime_dvd_mult}
\rulename{relprime_dvd_mult}

@{thm[display] relprime_dvd_mult [OF relprime_20_81]}

@{thm[display] dvd_refl}
\rulename{dvd_refl}

@{thm[display] dvd_add}
\rulename{dvd_add}

@{thm[display] dvd_add [OF dvd_refl dvd_refl]}

@{thm[display] dvd_add [OF _ dvd_refl]}
›

lemma "⟦(z::int) < 37; 66 < 2*z; z*z ≠ 1225; Q(34); Q(36)⟧ ⟹ Q(z)"
apply (subgoal_tac "z = 34 ∨ z = 36")
txt‹
the tactic leaves two subgoals:
@{subgoals[display,indent=0,margin=65]}
›
apply blast
apply (subgoal_tac "z ≠ 35")
txt‹
the tactic leaves two subgoals:
@{subgoals[display,indent=0,margin=65]}
›
apply arith
apply force
done


end