(* Title: HOL/Fun_Def.thy
Author: Alexander Krauss, TU Muenchen
*)
section ‹Function Definitions
and Termination Proofs
›
theory Fun_Def
imports Basic_BNF_LFPs Partial_Function SAT
keywords
"function" "termination" :: thy_goal_defn
and
"fun" "fun_cases" :: thy_defn
begin
subsection ‹Definitions
with default
value›
definition THE_default ::
"'a \ ('a \ bool) \ 'a"
where "THE_default d P = (if (\!x. P x) then (THE x. P x) else d)"
lemma THE_defaultI
': "\!x. P x \ P (THE_default d P)"
by (simp add: theI
' THE_default_def)
lemma THE_default1_equality:
"\!x. P x \ P a \ THE_default d P = a"
by (simp add: the1_equality THE_default_def)
lemma THE_default_none:
"\ (\!x. P x) \ THE_default d P = d"
by (simp add: THE_default_def)
lemma fundef_ex1_existence:
assumes f_def:
"f \ (\x::'a. THE_default (d x) (\y. G x y))"
assumes ex1:
"\!y. G x y"
shows "G x (f x)"
apply (simp only: f_def)
apply (rule THE_defaultI
')
apply (rule ex1)
done
lemma fundef_ex1_uniqueness:
assumes f_def:
"f \ (\x::'a. THE_default (d x) (\y. G x y))"
assumes ex1:
"\!y. G x y"
assumes elm:
"G x (h x)"
shows "h x = f x"
by (auto simp add: f_def ex1 elm THE_default1_equality[symmetric])
lemma fundef_ex1_iff:
assumes f_def:
"f \ (\x::'a. THE_default (d x) (\y. G x y))"
assumes ex1:
"\!y. G x y"
shows "(G x y) = (f x = y)"
by (auto simp add: ex1 f_def THE_default1_equality THE_defaultI
')
lemma fundef_default_value:
assumes f_def:
"f \ (\x::'a. THE_default (d x) (\y. G x y))"
assumes graph:
"\x y. G x y \ D x"
assumes "\ D x"
shows "f x = d x"
proof -
have "\(\y. G x y)"
proof
assume "\y. G x y"
then have "D x" using graph ..
with ‹¬ D x
› show False ..
qed
then have "\(\!y. G x y)" by blast
then show ?thesis
unfolding f_def
by (rule THE_default_none)
qed
definition in_rel_def[simp]:
"in_rel R x y \ (x, y) \ R"
lemma wf_in_rel:
"wf R \ wfp (in_rel R)"
by (simp add: wfp_def)
ML_file
‹Tools/
Function/function_core.ML
›
ML_file
‹Tools/
Function/mutual.ML
›
ML_file
‹Tools/
Function/pattern_split.ML
›
ML_file
‹Tools/
Function/relation.ML
›
ML_file
‹Tools/
Function/function_elims.ML
›
method_setup relation =
‹
Args.
term >> (fn t => fn ctxt => SIMPLE_METHOD
' (Function_Relation.relation_infer_tac ctxt t))
› "prove termination using a user-specified wellfounded relation"
ML_file
‹Tools/
Function/
function.ML
›
ML_file
‹Tools/
Function/pat_completeness.ML
›
method_setup pat_completeness =
‹
Scan.succeed (SIMPLE_METHOD
' o Pat_Completeness.pat_completeness_tac)
› "prove completeness of (co)datatype patterns"
ML_file
‹Tools/
Function/
fun.ML
›
ML_file
‹Tools/
Function/induction_schema.ML
›
method_setup induction_schema =
‹
Scan.succeed (CONTEXT_TACTIC oo Induction_Schema.induction_schema_tac)
› "prove an induction principle"
subsection ‹Measure functions
›
inductive is_measure ::
"('a \ nat) \ bool"
where is_measure_trivial:
"is_measure f"
named_theorems measure_function
"rules that guide the heuristic generation of measure functions"
ML_file
‹Tools/
Function/measure_functions.ML
›
lemma measure_size[measure_function]:
"is_measure size"
by (rule is_measure_trivial)
lemma measure_fst[measure_function]:
"is_measure f \ is_measure (\p. f (fst p))"
by (rule is_measure_trivial)
lemma measure_snd[measure_function]:
"is_measure f \ is_measure (\p. f (snd p))"
by (rule is_measure_trivial)
ML_file
‹Tools/
Function/lexicographic_order.ML
›
method_setup lexicographic_order =
‹
Method.sections clasimp_modifiers >>
(K (SIMPLE_METHOD o Lexicographic_Order.lexicographic_order_tac false))
› "termination prover for lexicographic orderings"
subsection ‹Congruence rules
›
lemma let_cong [fundef_cong]:
"M = N \ (\x. x = N \ f x = g x) \ Let M f = Let N g"
unfolding Let_def
by blast
lemmas [fundef_cong] =
if_cong image_cong
bex_cong ball_cong imp_cong map_option_cong Option.bind_cong
lemma split_cong [fundef_cong]:
"(\x y. (x, y) = q \ f x y = g x y) \ p = q \ case_prod f p = case_prod g q"
by (auto simp: split_def)
lemma comp_cong [fundef_cong]:
"f (g x) = f' (g' x') \ (f \ g) x = (f' \ g') x'"
by (simp only: o_apply)
subsection ‹Simp rules
for termination proofs
›
declare
trans_less_add1[termination_simp]
trans_less_add2[termination_simp]
trans_le_add1[termination_simp]
trans_le_add2[termination_simp]
less_imp_le_nat[termination_simp]
le_imp_less_Suc[termination_simp]
lemma size_prod_simp[termination_simp]:
"size_prod f g p = f (fst p) + g (snd p) + Suc 0"
by (induct p) auto
subsection ‹Decomposition
›
lemma less_by_empty:
"A = {} \ A \ B"
and union_comp_emptyL:
"A O C = {} \ B O C = {} \ (A \ B) O C = {}"
and union_comp_emptyR:
"A O B = {} \ A O C = {} \ A O (B \ C) = {}"
and wf_no_loop:
"R O R = {} \ wf R"
by (auto simp add: wf_comp_self [of R])
subsection ‹Reduction pairs
›
definition "reduction_pair P \ wf (fst P) \ fst P O snd P \ fst P"
lemma reduction_pairI[intro]:
"wf R \ R O S \ R \ reduction_pair (R, S)"
by (auto simp: reduction_pair_def)
lemma reduction_pair_lemma:
assumes rp:
"reduction_pair P"
assumes "R \ fst P"
assumes "S \ snd P"
assumes "wf S"
shows "wf (R \ S)"
proof -
from rp
‹S
⊆ snd P
› have "wf (fst P)" "fst P O S \ fst P"
unfolding reduction_pair_def
by auto
with ‹wf S
› have "wf (fst P \ S)"
by (auto intro: wf_union_compatible)
moreover from ‹R
⊆ fst P
› have "R \ S \ fst P \ S" by auto
ultimately show ?thesis
by (rule wf_subset)
qed
definition "rp_inv_image = (\(R,S) f. (inv_image R f, inv_image S f))"
lemma rp_inv_image_rp:
"reduction_pair P \ reduction_pair (rp_inv_image P f)"
unfolding reduction_pair_def rp_inv_image_def split_def
by force
subsection ‹Concrete orders
for SCNP
termination proofs
›
definition "pair_less = less_than <*lex*> less_than"
definition "pair_leq = pair_less\<^sup>="
definition "max_strict = max_ext pair_less"
definition "max_weak = max_ext pair_leq \ {({}, {})}"
definition "min_strict = min_ext pair_less"
definition "min_weak = min_ext pair_leq \ {({}, {})}"
lemma wf_pair_less[simp]:
"wf pair_less"
by (auto simp: pair_less_def)
lemma total_pair_less [iff]:
"total_on A pair_less" and trans_pair_less [iff]:
"trans pair_less"
by (auto simp: total_on_def pair_less_def)
text ‹Introduction rules
for ‹pair_less
›/
‹pair_leq
››
lemma pair_leqI1:
"a < b \ ((a, s), (b, t)) \ pair_leq"
and pair_leqI2:
"a \ b \ s \ t \ ((a, s), (b, t)) \ pair_leq"
and pair_lessI1:
"a < b \ ((a, s), (b, t)) \ pair_less"
and pair_lessI2:
"a \ b \ s < t \ ((a, s), (b, t)) \ pair_less"
by (auto simp: pair_leq_def pair_less_def)
lemma pair_less_iff1 [simp]:
"((x,y), (x,z)) \ pair_less \ y
by (simp add: pair_less_def)
text ‹Introduction rules for max›
lemma smax_emptyI: "finite Y \ Y \ {} \ ({}, Y) \ max_strict"
and smax_insertI:
"y \ Y \ (x, y) \ pair_less \ (X, Y) \ max_strict \ (insert x X, Y) \ max_strict"
and wmax_emptyI: "finite X \ ({}, X) \ max_weak"
and wmax_insertI:
"y \ YS \ (x, y) \ pair_leq \ (XS, YS) \ max_weak \ (insert x XS, YS) \ max_weak"
by (auto simp: max_strict_def max_weak_def elim!: max_ext.cases)
text ‹Introduction rules for min›
lemma smin_emptyI: "X \ {} \ (X, {}) \ min_strict"
and smin_insertI:
"x \ XS \ (x, y) \ pair_less \ (XS, YS) \ min_strict \ (XS, insert y YS) \ min_strict"
and wmin_emptyI: "(X, {}) \ min_weak"
and wmin_insertI:
"x \ XS \ (x, y) \ pair_leq \ (XS, YS) \ min_weak \ (XS, insert y YS) \ min_weak"
by (auto simp: min_strict_def min_weak_def min_ext_def)
text ‹Reduction Pairs.›
lemma max_ext_compat:
assumes "R O S \ R"
shows "max_ext R O (max_ext S \ {({}, {})}) \ max_ext R"
proof -
have "\X Y Z. (X, Y) \ max_ext R \ (Y, Z) \ max_ext S \ (X, Z) \ max_ext R"
proof -
fix X Y Z
assume "(X,Y)\max_ext R"
"(Y, Z)\max_ext S"
then have *: "finite X" "finite Y" "finite Z" "Y\{}" "Z\{}"
"(\x. x\X \ \y\Y. (x, y)\R)"
"(\y. y\Y \ \z\Z. (y, z)\S)"
by (auto elim: max_ext.cases)
moreover have "\x. x\X \ \z\Z. (x, z)\R"
proof -
fix x
assume "x\X"
then obtain y where 1: "y\Y" "(x, y)\R"
using * by auto
then obtain z where "z\Z" "(y, z)\S"
using * by auto
then show "\z\Z. (x, z)\R"
using assms 1 by (auto elim: max_ext.cases)
qed
ultimately show "(X,Z)\max_ext R"
by auto
qed
then show "max_ext R O (max_ext S \ {({}, {})}) \ max_ext R"
by auto
qed
lemma max_rpair_set: "reduction_pair (max_strict, max_weak)"
unfolding max_strict_def max_weak_def
apply (intro reduction_pairI max_ext_wf)
apply simp
apply (rule max_ext_compat)
apply (auto simp: pair_less_def pair_leq_def)
done
lemma min_ext_compat:
assumes "R O S \ R"
shows "min_ext R O (min_ext S \ {({},{})}) \ min_ext R"
proof -
have "\X Y Z z. \y\Y. \x\X. (x, y) \ R \ \z\Z. \y\Y. (y, z) \ S
==> z ∈ Z ==> ∃x∈X. (x, z) ∈ R"
proof -
fix X Y Z z
assume *: "\y\Y. \x\X. (x, y) \ R"
"\z\Z. \y\Y. (y, z) \ S"
"z\Z"
then obtain y' where 1: "y'∈Y" "(y', z) \ S"
by auto
then obtain x' where 2: "x'∈X" "(x', y') ∈ R"
using * by auto
show "\x\X. (x, z) \ R"
using 1 2 assms by auto
qed
then show ?thesis
using assms by (auto simp: min_ext_def)
qed
lemma min_rpair_set: "reduction_pair (min_strict, min_weak)"
unfolding min_strict_def min_weak_def
apply (intro reduction_pairI min_ext_wf)
apply simp
apply (rule min_ext_compat)
apply (auto simp: pair_less_def pair_leq_def)
done
subsection ‹Yet more induction principles on the natural numbers›
lemma nat_descend_induct [case_names base descend]:
fixes P :: "nat \ bool"
assumes H1: "\k. k > n \ P k"
assumes H2: "\k. k \ n \ (\i. i > k \ P i) \ P k"
shows "P m"
using assms by induction_schema (force intro!: wf_measure [of "\k. Suc n - k"])+
lemma induct_nat_012[case_names 0 1 ge2]:
"P 0 \ P (Suc 0) \ (\n. P n \ P (Suc n) \ P (Suc (Suc n))) \ P n"
proof (induction_schema, pat_completeness)
show "wf (Wellfounded.measure id)"
by simp
qed auto
subsection ‹Tool setup›
ML_file ‹Tools/Function/termination.ML›
ML_file ‹Tools/Function/scnp_solve.ML›
ML_file ‹Tools/Function/scnp_reconstruct.ML›
ML_file ‹Tools/Function/fun_cases.ML›
ML_val 🍋 ‹setup inactive›
‹
Context.theory_map (Function_Common.set_termination_prover
(K (ScnpReconstruct.decomp_scnp_tac [ScnpSolve.MAX, ScnpSolve.MIN, ScnpSolve.MS])))
›
end
