author | wenzelm |
Fri, 06 May 2011 17:52:08 +0200 | |
changeset 42711 | 159c4d1d4c42 |
parent 41777 | 1f7cbe39d425 |
child 46820 | c656222c4dc1 |
permissions | -rw-r--r-- |
41777 | 1 |
(* Title: ZF/ArithSimp.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 2000 University of Cambridge |
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*) |
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header{*Arithmetic with simplification*} |
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theory ArithSimp |
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imports Arith |
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uses "~~/src/Provers/Arith/cancel_numerals.ML" |
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"~~/src/Provers/Arith/combine_numerals.ML" |
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"arith_data.ML" |
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begin |
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subsection{*Difference*} |
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lemma diff_self_eq_0 [simp]: "m #- m = 0" |
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apply (subgoal_tac "natify (m) #- natify (m) = 0") |
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apply (rule_tac [2] natify_in_nat [THEN nat_induct], auto) |
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done |
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(**Addition is the inverse of subtraction**) |
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(*We need m:nat even if we replace the RHS by natify(m), for consider e.g. |
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n=2, m=omega; then n + (m-n) = 2 + (0-2) = 2 ~= 0 = natify(m).*) |
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lemma add_diff_inverse: "[| n le m; m:nat |] ==> n #+ (m#-n) = m" |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (erule rev_mp) |
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apply (rule_tac m = m and n = n in diff_induct, auto) |
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done |
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lemma add_diff_inverse2: "[| n le m; m:nat |] ==> (m#-n) #+ n = m" |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (simp (no_asm_simp) add: add_commute add_diff_inverse) |
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done |
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(*Proof is IDENTICAL to that of add_diff_inverse*) |
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lemma diff_succ: "[| n le m; m:nat |] ==> succ(m) #- n = succ(m#-n)" |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (erule rev_mp) |
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apply (rule_tac m = m and n = n in diff_induct) |
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apply (simp_all (no_asm_simp)) |
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done |
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lemma zero_less_diff [simp]: |
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"[| m: nat; n: nat |] ==> 0 < (n #- m) <-> m<n" |
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apply (rule_tac m = m and n = n in diff_induct) |
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apply (simp_all (no_asm_simp)) |
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done |
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(** Difference distributes over multiplication **) |
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lemma diff_mult_distrib: "(m #- n) #* k = (m #* k) #- (n #* k)" |
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apply (subgoal_tac " (natify (m) #- natify (n)) #* natify (k) = (natify (m) #* natify (k)) #- (natify (n) #* natify (k))") |
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apply (rule_tac [2] m = "natify (m) " and n = "natify (n) " in diff_induct) |
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apply (simp_all add: diff_cancel) |
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done |
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lemma diff_mult_distrib2: "k #* (m #- n) = (k #* m) #- (k #* n)" |
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apply (simp (no_asm) add: mult_commute [of k] diff_mult_distrib) |
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done |
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subsection{*Remainder*} |
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(*We need m:nat even with natify*) |
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lemma div_termination: "[| 0<n; n le m; m:nat |] ==> m #- n < m" |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (erule rev_mp) |
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apply (erule rev_mp) |
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apply (rule_tac m = m and n = n in diff_induct) |
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apply (simp_all (no_asm_simp) add: diff_le_self) |
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done |
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(*for mod and div*) |
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lemmas div_rls = |
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nat_typechecks Ord_transrec_type apply_funtype |
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div_termination [THEN ltD] |
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nat_into_Ord not_lt_iff_le [THEN iffD1] |
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lemma raw_mod_type: "[| m:nat; n:nat |] ==> raw_mod (m, n) : nat" |
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apply (unfold raw_mod_def) |
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apply (rule Ord_transrec_type) |
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apply (auto simp add: nat_into_Ord [THEN Ord_0_lt_iff]) |
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apply (blast intro: div_rls) |
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done |
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lemma mod_type [TC,iff]: "m mod n : nat" |
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apply (unfold mod_def) |
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apply (simp (no_asm) add: mod_def raw_mod_type) |
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done |
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(** Aribtrary definitions for division by zero. Useful to simplify |
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certain equations **) |
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lemma DIVISION_BY_ZERO_DIV: "a div 0 = 0" |
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apply (unfold div_def) |
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apply (rule raw_div_def [THEN def_transrec, THEN trans]) |
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apply (simp (no_asm_simp)) |
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done (*NOT for adding to default simpset*) |
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lemma DIVISION_BY_ZERO_MOD: "a mod 0 = natify(a)" |
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apply (unfold mod_def) |
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apply (rule raw_mod_def [THEN def_transrec, THEN trans]) |
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apply (simp (no_asm_simp)) |
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done (*NOT for adding to default simpset*) |
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lemma raw_mod_less: "m<n ==> raw_mod (m,n) = m" |
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apply (rule raw_mod_def [THEN def_transrec, THEN trans]) |
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apply (simp (no_asm_simp) add: div_termination [THEN ltD]) |
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done |
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lemma mod_less [simp]: "[| m<n; n : nat |] ==> m mod n = m" |
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apply (frule lt_nat_in_nat, assumption) |
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apply (simp (no_asm_simp) add: mod_def raw_mod_less) |
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done |
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lemma raw_mod_geq: |
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"[| 0<n; n le m; m:nat |] ==> raw_mod (m, n) = raw_mod (m#-n, n)" |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (rule raw_mod_def [THEN def_transrec, THEN trans]) |
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apply (simp (no_asm_simp) add: div_termination [THEN ltD] not_lt_iff_le [THEN iffD2], blast) |
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done |
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lemma mod_geq: "[| n le m; m:nat |] ==> m mod n = (m#-n) mod n" |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (case_tac "n=0") |
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apply (simp add: DIVISION_BY_ZERO_MOD) |
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apply (simp add: mod_def raw_mod_geq nat_into_Ord [THEN Ord_0_lt_iff]) |
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done |
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subsection{*Division*} |
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lemma raw_div_type: "[| m:nat; n:nat |] ==> raw_div (m, n) : nat" |
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apply (unfold raw_div_def) |
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apply (rule Ord_transrec_type) |
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apply (auto simp add: nat_into_Ord [THEN Ord_0_lt_iff]) |
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apply (blast intro: div_rls) |
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done |
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lemma div_type [TC,iff]: "m div n : nat" |
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apply (unfold div_def) |
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apply (simp (no_asm) add: div_def raw_div_type) |
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done |
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lemma raw_div_less: "m<n ==> raw_div (m,n) = 0" |
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apply (rule raw_div_def [THEN def_transrec, THEN trans]) |
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apply (simp (no_asm_simp) add: div_termination [THEN ltD]) |
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done |
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lemma div_less [simp]: "[| m<n; n : nat |] ==> m div n = 0" |
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apply (frule lt_nat_in_nat, assumption) |
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apply (simp (no_asm_simp) add: div_def raw_div_less) |
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done |
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lemma raw_div_geq: "[| 0<n; n le m; m:nat |] ==> raw_div(m,n) = succ(raw_div(m#-n, n))" |
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apply (subgoal_tac "n ~= 0") |
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prefer 2 apply blast |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (rule raw_div_def [THEN def_transrec, THEN trans]) |
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apply (simp (no_asm_simp) add: div_termination [THEN ltD] not_lt_iff_le [THEN iffD2] ) |
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done |
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lemma div_geq [simp]: |
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"[| 0<n; n le m; m:nat |] ==> m div n = succ ((m#-n) div n)" |
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apply (frule lt_nat_in_nat, erule nat_succI) |
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apply (simp (no_asm_simp) add: div_def raw_div_geq) |
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done |
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declare div_less [simp] div_geq [simp] |
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(*A key result*) |
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lemma mod_div_lemma: "[| m: nat; n: nat |] ==> (m div n)#*n #+ m mod n = m" |
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apply (case_tac "n=0") |
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apply (simp add: DIVISION_BY_ZERO_MOD) |
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apply (simp add: nat_into_Ord [THEN Ord_0_lt_iff]) |
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apply (erule complete_induct) |
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apply (case_tac "x<n") |
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txt{*case x<n*} |
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apply (simp (no_asm_simp)) |
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txt{*case n le x*} |
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apply (simp add: not_lt_iff_le add_assoc mod_geq div_termination [THEN ltD] add_diff_inverse) |
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done |
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lemma mod_div_equality_natify: "(m div n)#*n #+ m mod n = natify(m)" |
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apply (subgoal_tac " (natify (m) div natify (n))#*natify (n) #+ natify (m) mod natify (n) = natify (m) ") |
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apply force |
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apply (subst mod_div_lemma, auto) |
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done |
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lemma mod_div_equality: "m: nat ==> (m div n)#*n #+ m mod n = m" |
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apply (simp (no_asm_simp) add: mod_div_equality_natify) |
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done |
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subsection{*Further Facts about Remainder*} |
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text{*(mainly for mutilated chess board)*} |
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lemma mod_succ_lemma: |
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"[| 0<n; m:nat; n:nat |] |
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==> succ(m) mod n = (if succ(m mod n) = n then 0 else succ(m mod n))" |
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apply (erule complete_induct) |
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apply (case_tac "succ (x) <n") |
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txt{* case succ(x) < n *} |
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apply (simp (no_asm_simp) add: nat_le_refl [THEN lt_trans] succ_neq_self) |
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apply (simp add: ltD [THEN mem_imp_not_eq]) |
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txt{* case n le succ(x) *} |
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apply (simp add: mod_geq not_lt_iff_le) |
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apply (erule leE) |
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apply (simp (no_asm_simp) add: mod_geq div_termination [THEN ltD] diff_succ) |
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txt{*equality case*} |
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apply (simp add: diff_self_eq_0) |
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done |
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lemma mod_succ: |
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"n:nat ==> succ(m) mod n = (if succ(m mod n) = n then 0 else succ(m mod n))" |
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apply (case_tac "n=0") |
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apply (simp (no_asm_simp) add: natify_succ DIVISION_BY_ZERO_MOD) |
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apply (subgoal_tac "natify (succ (m)) mod n = (if succ (natify (m) mod n) = n then 0 else succ (natify (m) mod n))") |
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prefer 2 |
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apply (subst natify_succ) |
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apply (rule mod_succ_lemma) |
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apply (auto simp del: natify_succ simp add: nat_into_Ord [THEN Ord_0_lt_iff]) |
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done |
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lemma mod_less_divisor: "[| 0<n; n:nat |] ==> m mod n < n" |
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apply (subgoal_tac "natify (m) mod n < n") |
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apply (rule_tac [2] i = "natify (m) " in complete_induct) |
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apply (case_tac [3] "x<n", auto) |
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txt{* case n le x*} |
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apply (simp add: mod_geq not_lt_iff_le div_termination [THEN ltD]) |
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done |
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lemma mod_1_eq [simp]: "m mod 1 = 0" |
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by (cut_tac n = 1 in mod_less_divisor, auto) |
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lemma mod2_cases: "b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)" |
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apply (subgoal_tac "k mod 2: 2") |
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prefer 2 apply (simp add: mod_less_divisor [THEN ltD]) |
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apply (drule ltD, auto) |
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done |
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lemma mod2_succ_succ [simp]: "succ(succ(m)) mod 2 = m mod 2" |
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apply (subgoal_tac "m mod 2: 2") |
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prefer 2 apply (simp add: mod_less_divisor [THEN ltD]) |
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apply (auto simp add: mod_succ) |
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done |
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lemma mod2_add_more [simp]: "(m#+m#+n) mod 2 = n mod 2" |
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apply (subgoal_tac " (natify (m) #+natify (m) #+n) mod 2 = n mod 2") |
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apply (rule_tac [2] n = "natify (m) " in nat_induct) |
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apply auto |
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done |
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lemma mod2_add_self [simp]: "(m#+m) mod 2 = 0" |
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by (cut_tac n = 0 in mod2_add_more, auto) |
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subsection{*Additional theorems about @{text "\<le>"}*} |
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lemma add_le_self: "m:nat ==> m le (m #+ n)" |
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apply (simp (no_asm_simp)) |
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done |
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lemma add_le_self2: "m:nat ==> m le (n #+ m)" |
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apply (simp (no_asm_simp)) |
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done |
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(*** Monotonicity of Multiplication ***) |
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lemma mult_le_mono1: "[| i le j; j:nat |] ==> (i#*k) le (j#*k)" |
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apply (subgoal_tac "natify (i) #*natify (k) le j#*natify (k) ") |
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apply (frule_tac [2] lt_nat_in_nat) |
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apply (rule_tac [3] n = "natify (k) " in nat_induct) |
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apply (simp_all add: add_le_mono) |
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done |
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(* le monotonicity, BOTH arguments*) |
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lemma mult_le_mono: "[| i le j; k le l; j:nat; l:nat |] ==> i#*k le j#*l" |
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apply (rule mult_le_mono1 [THEN le_trans], assumption+) |
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apply (subst mult_commute, subst mult_commute, rule mult_le_mono1, assumption+) |
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done |
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(*strict, in 1st argument; proof is by induction on k>0. |
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I can't see how to relax the typing conditions.*) |
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lemma mult_lt_mono2: "[| i<j; 0<k; j:nat; k:nat |] ==> k#*i < k#*j" |
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apply (erule zero_lt_natE) |
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apply (frule_tac [2] lt_nat_in_nat) |
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apply (simp_all (no_asm_simp)) |
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apply (induct_tac "x") |
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apply (simp_all (no_asm_simp) add: add_lt_mono) |
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done |
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lemma mult_lt_mono1: "[| i<j; 0<k; j:nat; k:nat |] ==> i#*k < j#*k" |
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apply (simp (no_asm_simp) add: mult_lt_mono2 mult_commute [of _ k]) |
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done |
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lemma add_eq_0_iff [iff]: "m#+n = 0 <-> natify(m)=0 & natify(n)=0" |
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apply (subgoal_tac "natify (m) #+ natify (n) = 0 <-> natify (m) =0 & natify (n) =0") |
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apply (rule_tac [2] n = "natify (m) " in natE) |
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apply (rule_tac [4] n = "natify (n) " in natE) |
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apply auto |
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done |
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lemma zero_lt_mult_iff [iff]: "0 < m#*n <-> 0 < natify(m) & 0 < natify(n)" |
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apply (subgoal_tac "0 < natify (m) #*natify (n) <-> 0 < natify (m) & 0 < natify (n) ") |
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apply (rule_tac [2] n = "natify (m) " in natE) |
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apply (rule_tac [4] n = "natify (n) " in natE) |
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apply (rule_tac [3] n = "natify (n) " in natE) |
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apply auto |
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done |
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lemma mult_eq_1_iff [iff]: "m#*n = 1 <-> natify(m)=1 & natify(n)=1" |
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apply (subgoal_tac "natify (m) #* natify (n) = 1 <-> natify (m) =1 & natify (n) =1") |
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apply (rule_tac [2] n = "natify (m) " in natE) |
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apply (rule_tac [4] n = "natify (n) " in natE) |
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apply auto |
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done |
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lemma mult_is_zero: "[|m: nat; n: nat|] ==> (m #* n = 0) <-> (m = 0 | n = 0)" |
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apply auto |
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apply (erule natE) |
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apply (erule_tac [2] natE, auto) |
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done |
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lemma mult_is_zero_natify [iff]: |
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"(m #* n = 0) <-> (natify(m) = 0 | natify(n) = 0)" |
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apply (cut_tac m = "natify (m) " and n = "natify (n) " in mult_is_zero) |
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apply auto |
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done |
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subsection{*Cancellation Laws for Common Factors in Comparisons*} |
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lemma mult_less_cancel_lemma: |
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"[| k: nat; m: nat; n: nat |] ==> (m#*k < n#*k) <-> (0<k & m<n)" |
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apply (safe intro!: mult_lt_mono1) |
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apply (erule natE, auto) |
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apply (rule not_le_iff_lt [THEN iffD1]) |
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apply (drule_tac [3] not_le_iff_lt [THEN [2] rev_iffD2]) |
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prefer 5 apply (blast intro: mult_le_mono1, auto) |
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done |
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lemma mult_less_cancel2 [simp]: |
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"(m#*k < n#*k) <-> (0 < natify(k) & natify(m) < natify(n))" |
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apply (rule iff_trans) |
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apply (rule_tac [2] mult_less_cancel_lemma, auto) |
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done |
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lemma mult_less_cancel1 [simp]: |
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"(k#*m < k#*n) <-> (0 < natify(k) & natify(m) < natify(n))" |
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apply (simp (no_asm) add: mult_less_cancel2 mult_commute [of k]) |
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done |
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||
362 |
lemma mult_le_cancel2 [simp]: "(m#*k le n#*k) <-> (0 < natify(k) --> natify(m) le natify(n))" |
|
363 |
apply (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym]) |
|
364 |
apply auto |
|
365 |
done |
|
366 |
||
367 |
lemma mult_le_cancel1 [simp]: "(k#*m le k#*n) <-> (0 < natify(k) --> natify(m) le natify(n))" |
|
368 |
apply (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym]) |
|
369 |
apply auto |
|
370 |
done |
|
371 |
||
372 |
lemma mult_le_cancel_le1: "k : nat ==> k #* m le k \<longleftrightarrow> (0 < k \<longrightarrow> natify(m) le 1)" |
|
13784 | 373 |
by (cut_tac k = k and m = m and n = 1 in mult_le_cancel1, auto) |
13259 | 374 |
|
375 |
lemma Ord_eq_iff_le: "[| Ord(m); Ord(n) |] ==> m=n <-> (m le n & n le m)" |
|
376 |
by (blast intro: le_anti_sym) |
|
377 |
||
378 |
lemma mult_cancel2_lemma: |
|
379 |
"[| k: nat; m: nat; n: nat |] ==> (m#*k = n#*k) <-> (m=n | k=0)" |
|
380 |
apply (simp (no_asm_simp) add: Ord_eq_iff_le [of "m#*k"] Ord_eq_iff_le [of m]) |
|
381 |
apply (auto simp add: Ord_0_lt_iff) |
|
382 |
done |
|
383 |
||
384 |
lemma mult_cancel2 [simp]: |
|
385 |
"(m#*k = n#*k) <-> (natify(m) = natify(n) | natify(k) = 0)" |
|
386 |
apply (rule iff_trans) |
|
387 |
apply (rule_tac [2] mult_cancel2_lemma, auto) |
|
388 |
done |
|
389 |
||
390 |
lemma mult_cancel1 [simp]: |
|
391 |
"(k#*m = k#*n) <-> (natify(m) = natify(n) | natify(k) = 0)" |
|
392 |
apply (simp (no_asm) add: mult_cancel2 mult_commute [of k]) |
|
393 |
done |
|
394 |
||
395 |
||
396 |
(** Cancellation law for division **) |
|
397 |
||
398 |
lemma div_cancel_raw: |
|
399 |
"[| 0<n; 0<k; k:nat; m:nat; n:nat |] ==> (k#*m) div (k#*n) = m div n" |
|
13784 | 400 |
apply (erule_tac i = m in complete_induct) |
13259 | 401 |
apply (case_tac "x<n") |
402 |
apply (simp add: div_less zero_lt_mult_iff mult_lt_mono2) |
|
403 |
apply (simp add: not_lt_iff_le zero_lt_mult_iff le_refl [THEN mult_le_mono] |
|
404 |
div_geq diff_mult_distrib2 [symmetric] div_termination [THEN ltD]) |
|
405 |
done |
|
406 |
||
407 |
lemma div_cancel: |
|
408 |
"[| 0 < natify(n); 0 < natify(k) |] ==> (k#*m) div (k#*n) = m div n" |
|
409 |
apply (cut_tac k = "natify (k) " and m = "natify (m)" and n = "natify (n)" |
|
410 |
in div_cancel_raw) |
|
411 |
apply auto |
|
412 |
done |
|
413 |
||
414 |
||
13356 | 415 |
subsection{*More Lemmas about Remainder*} |
13259 | 416 |
|
417 |
lemma mult_mod_distrib_raw: |
|
418 |
"[| k:nat; m:nat; n:nat |] ==> (k#*m) mod (k#*n) = k #* (m mod n)" |
|
419 |
apply (case_tac "k=0") |
|
420 |
apply (simp add: DIVISION_BY_ZERO_MOD) |
|
421 |
apply (case_tac "n=0") |
|
422 |
apply (simp add: DIVISION_BY_ZERO_MOD) |
|
423 |
apply (simp add: nat_into_Ord [THEN Ord_0_lt_iff]) |
|
13784 | 424 |
apply (erule_tac i = m in complete_induct) |
13259 | 425 |
apply (case_tac "x<n") |
426 |
apply (simp (no_asm_simp) add: mod_less zero_lt_mult_iff mult_lt_mono2) |
|
427 |
apply (simp add: not_lt_iff_le zero_lt_mult_iff le_refl [THEN mult_le_mono] |
|
428 |
mod_geq diff_mult_distrib2 [symmetric] div_termination [THEN ltD]) |
|
429 |
done |
|
430 |
||
431 |
lemma mod_mult_distrib2: "k #* (m mod n) = (k#*m) mod (k#*n)" |
|
432 |
apply (cut_tac k = "natify (k) " and m = "natify (m)" and n = "natify (n)" |
|
433 |
in mult_mod_distrib_raw) |
|
434 |
apply auto |
|
435 |
done |
|
436 |
||
437 |
lemma mult_mod_distrib: "(m mod n) #* k = (m#*k) mod (n#*k)" |
|
438 |
apply (simp (no_asm) add: mult_commute mod_mult_distrib2) |
|
439 |
done |
|
440 |
||
441 |
lemma mod_add_self2_raw: "n \<in> nat ==> (m #+ n) mod n = m mod n" |
|
442 |
apply (subgoal_tac " (n #+ m) mod n = (n #+ m #- n) mod n") |
|
443 |
apply (simp add: add_commute) |
|
444 |
apply (subst mod_geq [symmetric], auto) |
|
445 |
done |
|
446 |
||
447 |
lemma mod_add_self2 [simp]: "(m #+ n) mod n = m mod n" |
|
448 |
apply (cut_tac n = "natify (n) " in mod_add_self2_raw) |
|
449 |
apply auto |
|
450 |
done |
|
451 |
||
452 |
lemma mod_add_self1 [simp]: "(n#+m) mod n = m mod n" |
|
453 |
apply (simp (no_asm_simp) add: add_commute mod_add_self2) |
|
454 |
done |
|
455 |
||
456 |
lemma mod_mult_self1_raw: "k \<in> nat ==> (m #+ k#*n) mod n = m mod n" |
|
457 |
apply (erule nat_induct) |
|
458 |
apply (simp_all (no_asm_simp) add: add_left_commute [of _ n]) |
|
459 |
done |
|
460 |
||
461 |
lemma mod_mult_self1 [simp]: "(m #+ k#*n) mod n = m mod n" |
|
462 |
apply (cut_tac k = "natify (k) " in mod_mult_self1_raw) |
|
463 |
apply auto |
|
464 |
done |
|
465 |
||
466 |
lemma mod_mult_self2 [simp]: "(m #+ n#*k) mod n = m mod n" |
|
467 |
apply (simp (no_asm) add: mult_commute mod_mult_self1) |
|
468 |
done |
|
469 |
||
470 |
(*Lemma for gcd*) |
|
471 |
lemma mult_eq_self_implies_10: "m = m#*n ==> natify(n)=1 | m=0" |
|
472 |
apply (subgoal_tac "m: nat") |
|
473 |
prefer 2 |
|
474 |
apply (erule ssubst) |
|
475 |
apply simp |
|
476 |
apply (rule disjCI) |
|
477 |
apply (drule sym) |
|
478 |
apply (rule Ord_linear_lt [of "natify(n)" 1]) |
|
479 |
apply simp_all |
|
480 |
apply (subgoal_tac "m #* n = 0", simp) |
|
481 |
apply (subst mult_natify2 [symmetric]) |
|
482 |
apply (simp del: mult_natify2) |
|
483 |
apply (drule nat_into_Ord [THEN Ord_0_lt, THEN [2] mult_lt_mono2], auto) |
|
484 |
done |
|
485 |
||
486 |
lemma less_imp_succ_add [rule_format]: |
|
487 |
"[| m<n; n: nat |] ==> EX k: nat. n = succ(m#+k)" |
|
488 |
apply (frule lt_nat_in_nat, assumption) |
|
489 |
apply (erule rev_mp) |
|
490 |
apply (induct_tac "n") |
|
491 |
apply (simp_all (no_asm) add: le_iff) |
|
492 |
apply (blast elim!: leE intro!: add_0_right [symmetric] add_succ_right [symmetric]) |
|
493 |
done |
|
494 |
||
495 |
lemma less_iff_succ_add: |
|
496 |
"[| m: nat; n: nat |] ==> (m<n) <-> (EX k: nat. n = succ(m#+k))" |
|
497 |
by (auto intro: less_imp_succ_add) |
|
498 |
||
14055 | 499 |
lemma add_lt_elim2: |
500 |
"\<lbrakk>a #+ d = b #+ c; a < b; b \<in> nat; c \<in> nat; d \<in> nat\<rbrakk> \<Longrightarrow> c < d" |
|
501 |
by (drule less_imp_succ_add, auto) |
|
502 |
||
503 |
lemma add_le_elim2: |
|
504 |
"\<lbrakk>a #+ d = b #+ c; a le b; b \<in> nat; c \<in> nat; d \<in> nat\<rbrakk> \<Longrightarrow> c le d" |
|
505 |
by (drule less_imp_succ_add, auto) |
|
506 |
||
13356 | 507 |
|
508 |
subsubsection{*More Lemmas About Difference*} |
|
13259 | 509 |
|
510 |
lemma diff_is_0_lemma: |
|
511 |
"[| m: nat; n: nat |] ==> m #- n = 0 <-> m le n" |
|
13784 | 512 |
apply (rule_tac m = m and n = n in diff_induct, simp_all) |
13259 | 513 |
done |
514 |
||
515 |
lemma diff_is_0_iff: "m #- n = 0 <-> natify(m) le natify(n)" |
|
516 |
by (simp add: diff_is_0_lemma [symmetric]) |
|
517 |
||
518 |
lemma nat_lt_imp_diff_eq_0: |
|
519 |
"[| a:nat; b:nat; a<b |] ==> a #- b = 0" |
|
520 |
by (simp add: diff_is_0_iff le_iff) |
|
521 |
||
14055 | 522 |
lemma raw_nat_diff_split: |
13259 | 523 |
"[| a:nat; b:nat |] ==> |
524 |
(P(a #- b)) <-> ((a < b -->P(0)) & (ALL d:nat. a = b #+ d --> P(d)))" |
|
525 |
apply (case_tac "a < b") |
|
526 |
apply (force simp add: nat_lt_imp_diff_eq_0) |
|
13784 | 527 |
apply (rule iffI, force, simp) |
13259 | 528 |
apply (drule_tac x="a#-b" in bspec) |
529 |
apply (simp_all add: Ordinal.not_lt_iff_le add_diff_inverse) |
|
530 |
done |
|
531 |
||
14055 | 532 |
lemma nat_diff_split: |
533 |
"(P(a #- b)) <-> |
|
534 |
(natify(a) < natify(b) -->P(0)) & (ALL d:nat. natify(a) = b #+ d --> P(d))" |
|
535 |
apply (cut_tac P=P and a="natify(a)" and b="natify(b)" in raw_nat_diff_split) |
|
536 |
apply simp_all |
|
537 |
done |
|
538 |
||
14060
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
539 |
text{*Difference and less-than*} |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
540 |
|
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
541 |
lemma diff_lt_imp_lt: "[|(k#-i) < (k#-j); i\<in>nat; j\<in>nat; k\<in>nat|] ==> j<i" |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
542 |
apply (erule rev_mp) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
543 |
apply (simp split add: nat_diff_split, auto) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
544 |
apply (blast intro: add_le_self lt_trans1) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
545 |
apply (rule not_le_iff_lt [THEN iffD1], auto) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
546 |
apply (subgoal_tac "i #+ da < j #+ d", force) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
547 |
apply (blast intro: add_le_lt_mono) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
548 |
done |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
549 |
|
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
550 |
lemma lt_imp_diff_lt: "[|j<i; i\<le>k; k\<in>nat|] ==> (k#-i) < (k#-j)" |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
551 |
apply (frule le_in_nat, assumption) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
552 |
apply (frule lt_nat_in_nat, assumption) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
553 |
apply (simp split add: nat_diff_split, auto) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
554 |
apply (blast intro: lt_asym lt_trans2) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
555 |
apply (blast intro: lt_irrefl lt_trans2) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
556 |
apply (rule not_le_iff_lt [THEN iffD1], auto) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
557 |
apply (subgoal_tac "j #+ d < i #+ da", force) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
558 |
apply (blast intro: add_lt_le_mono) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
559 |
done |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
560 |
|
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
561 |
|
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
562 |
lemma diff_lt_iff_lt: "[|i\<le>k; j\<in>nat; k\<in>nat|] ==> (k#-i) < (k#-j) <-> j<i" |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
563 |
apply (frule le_in_nat, assumption) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
564 |
apply (blast intro: lt_imp_diff_lt diff_lt_imp_lt) |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
565 |
done |
c0c4af41fa3b
Adding the theory UNITY/AllocImpl.thy, with supporting lemmas
paulson
parents:
14055
diff
changeset
|
566 |
|
9548 | 567 |
end |