Theory BNF_Least_Fixpoint
section ‹Least Fixpoint (Datatype) Operation on Bounded Natural Functors›
theory BNF_Least_Fixpoint
imports BNF_Fixpoint_Base
keywords
"datatype" :: thy_defn and
"datatype_compat" :: thy_defn
begin
lemma subset_emptyI: "(⋀x. x ∈ A ⟹ False) ⟹ A ⊆ {}"
by blast
lemma image_Collect_subsetI: "(⋀x. P x ⟹ f x ∈ B) ⟹ f ` {x. P x} ⊆ B"
by blast
lemma Collect_restrict: "{x. x ∈ X ∧ P x} ⊆ X"
by auto
lemma prop_restrict: "⟦x ∈ Z; Z ⊆ {x. x ∈ X ∧ P x}⟧ ⟹ P x"
by auto
lemma underS_I: "⟦i ≠ j; (i, j) ∈ R⟧ ⟹ i ∈ underS R j"
unfolding underS_def by simp
lemma underS_E: "i ∈ underS R j ⟹ i ≠ j ∧ (i, j) ∈ R"
unfolding underS_def by simp
lemma underS_Field: "i ∈ underS R j ⟹ i ∈ Field R"
unfolding underS_def Field_def by auto
lemma ex_bij_betw: "|A| ≤o (r :: 'b rel) ⟹ ∃f B :: 'b set. bij_betw f B A"
by (subst (asm) internalize_card_of_ordLeq) (auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric])
lemma bij_betwI':
"⟦⋀x y. ⟦x ∈ X; y ∈ X⟧ ⟹ (f x = f y) = (x = y);
⋀x. x ∈ X ⟹ f x ∈ Y;
⋀y. y ∈ Y ⟹ ∃x ∈ X. y = f x⟧ ⟹ bij_betw f X Y"
unfolding bij_betw_def inj_on_def by blast
lemma surj_fun_eq:
assumes surj_on: "f ` X = UNIV" and eq_on: "∀x ∈ X. (g1 ∘ f) x = (g2 ∘ f) x"
shows "g1 = g2"
proof (rule ext)
fix y
from surj_on obtain x where "x ∈ X" and "y = f x" by blast
thus "g1 y = g2 y" using eq_on by simp
qed
lemma Card_order_wo_rel: "Card_order r ⟹ wo_rel r"
unfolding wo_rel_def card_order_on_def by blast
lemma Cinfinite_limit: "⟦x ∈ Field r; Cinfinite r⟧ ⟹ ∃y ∈ Field r. x ≠ y ∧ (x, y) ∈ r"
unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
lemma Card_order_trans:
"⟦Card_order r; x ≠ y; (x, y) ∈ r; y ≠ z; (y, z) ∈ r⟧ ⟹ x ≠ z ∧ (x, z) ∈ r"
unfolding card_order_on_def well_order_on_def linear_order_on_def
partial_order_on_def preorder_on_def trans_def antisym_def by blast
lemma Cinfinite_limit2:
assumes x1: "x1 ∈ Field r" and x2: "x2 ∈ Field r" and r: "Cinfinite r"
shows "∃y ∈ Field r. (x1 ≠ y ∧ (x1, y) ∈ r) ∧ (x2 ≠ y ∧ (x2, y) ∈ r)"
proof -
from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
unfolding card_order_on_def well_order_on_def linear_order_on_def
partial_order_on_def preorder_on_def by auto
obtain y1 where y1: "y1 ∈ Field r" "x1 ≠ y1" "(x1, y1) ∈ r"
using Cinfinite_limit[OF x1 r] by blast
obtain y2 where y2: "y2 ∈ Field r" "x2 ≠ y2" "(x2, y2) ∈ r"
using Cinfinite_limit[OF x2 r] by blast
show ?thesis
proof (cases "y1 = y2")
case True with y1 y2 show ?thesis by blast
next
case False
with y1(1) y2(1) total have "(y1, y2) ∈ r ∨ (y2, y1) ∈ r"
unfolding total_on_def by auto
thus ?thesis
proof
assume *: "(y1, y2) ∈ r"
with trans y1(3) have "(x1, y2) ∈ r" unfolding trans_def by blast
with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
next
assume *: "(y2, y1) ∈ r"
with trans y2(3) have "(x2, y1) ∈ r" unfolding trans_def by blast
with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
qed
qed
qed
lemma Cinfinite_limit_finite:
"⟦finite X; X ⊆ Field r; Cinfinite r⟧ ⟹ ∃y ∈ Field r. ∀x ∈ X. (x ≠ y ∧ (x, y) ∈ r)"
proof (induct X rule: finite_induct)
case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
next
case (insert x X)
then obtain y where y: "y ∈ Field r" "∀x ∈ X. (x ≠ y ∧ (x, y) ∈ r)" by blast
then obtain z where z: "z ∈ Field r" "x ≠ z ∧ (x, z) ∈ r" "y ≠ z ∧ (y, z) ∈ r"
using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
show ?case
apply (intro bexI ballI)
apply (erule insertE)
apply hypsubst
apply (rule z(2))
using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
apply blast
apply (rule z(1))
done
qed
lemma insert_subsetI: "⟦x ∈ A; X ⊆ A⟧ ⟹ insert x X ⊆ A"
by auto
lemmas well_order_induct_imp = wo_rel.well_order_induct[of r "λx. x ∈ Field r ⟶ P x" for r P]
lemma meta_spec2:
assumes "(⋀x y. PROP P x y)"
shows "PROP P x y"
by (rule assms)
lemma nchotomy_relcomppE:
assumes "⋀y. ∃x. y = f x" "(r OO s) a c" "⋀b. r a (f b) ⟹ s (f b) c ⟹ P"
shows P
proof (rule relcompp.cases[OF assms(2)], hypsubst)
fix b assume "r a b" "s b c"
moreover from assms(1) obtain b' where "b = f b'" by blast
ultimately show P by (blast intro: assms(3))
qed
lemma predicate2D_vimage2p: "⟦R ≤ vimage2p f g S; R x y⟧ ⟹ S (f x) (g y)"
unfolding vimage2p_def by auto
lemma ssubst_Pair_rhs: "⟦(r, s) ∈ R; s' = s⟧ ⟹ (r, s') ∈ R"
by (rule ssubst)
lemma all_mem_range1:
"(⋀y. y ∈ range f ⟹ P y) ≡ (⋀x. P (f x)) "
by (rule equal_intr_rule) fast+
lemma all_mem_range2:
"(⋀fa y. fa ∈ range f ⟹ y ∈ range fa ⟹ P y) ≡ (⋀x xa. P (f x xa))"
by (rule equal_intr_rule) fast+
lemma all_mem_range3:
"(⋀fa fb y. fa ∈ range f ⟹ fb ∈ range fa ⟹ y ∈ range fb ⟹ P y) ≡ (⋀x xa xb. P (f x xa xb))"
by (rule equal_intr_rule) fast+
lemma all_mem_range4:
"(⋀fa fb fc y. fa ∈ range f ⟹ fb ∈ range fa ⟹ fc ∈ range fb ⟹ y ∈ range fc ⟹ P y) ≡
(⋀x xa xb xc. P (f x xa xb xc))"
by (rule equal_intr_rule) fast+
lemma all_mem_range5:
"(⋀fa fb fc fd y. fa ∈ range f ⟹ fb ∈ range fa ⟹ fc ∈ range fb ⟹ fd ∈ range fc ⟹
y ∈ range fd ⟹ P y) ≡
(⋀x xa xb xc xd. P (f x xa xb xc xd))"
by (rule equal_intr_rule) fast+
lemma all_mem_range6:
"(⋀fa fb fc fd fe ff y. fa ∈ range f ⟹ fb ∈ range fa ⟹ fc ∈ range fb ⟹ fd ∈ range fc ⟹
fe ∈ range fd ⟹ ff ∈ range fe ⟹ y ∈ range ff ⟹ P y) ≡
(⋀x xa xb xc xd xe xf. P (f x xa xb xc xd xe xf))"
by (rule equal_intr_rule) (fastforce, fast)
lemma all_mem_range7:
"(⋀fa fb fc fd fe ff fg y. fa ∈ range f ⟹ fb ∈ range fa ⟹ fc ∈ range fb ⟹ fd ∈ range fc ⟹
fe ∈ range fd ⟹ ff ∈ range fe ⟹ fg ∈ range ff ⟹ y ∈ range fg ⟹ P y) ≡
(⋀x xa xb xc xd xe xf xg. P (f x xa xb xc xd xe xf xg))"
by (rule equal_intr_rule) (fastforce, fast)
lemma all_mem_range8:
"(⋀fa fb fc fd fe ff fg fh y. fa ∈ range f ⟹ fb ∈ range fa ⟹ fc ∈ range fb ⟹ fd ∈ range fc ⟹
fe ∈ range fd ⟹ ff ∈ range fe ⟹ fg ∈ range ff ⟹ fh ∈ range fg ⟹ y ∈ range fh ⟹ P y) ≡
(⋀x xa xb xc xd xe xf xg xh. P (f x xa xb xc xd xe xf xg xh))"
by (rule equal_intr_rule) (fastforce, fast)
lemmas all_mem_range = all_mem_range1 all_mem_range2 all_mem_range3 all_mem_range4 all_mem_range5
all_mem_range6 all_mem_range7 all_mem_range8
lemma pred_fun_True_id: "NO_MATCH id p ⟹ pred_fun (λx. True) p f = pred_fun (λx. True) id (p ∘ f)"
unfolding fun.pred_map unfolding comp_def id_def ..
ML_file ‹Tools/BNF/bnf_lfp_util.ML›
ML_file ‹Tools/BNF/bnf_lfp_tactics.ML›
ML_file ‹Tools/BNF/bnf_lfp.ML›
ML_file ‹Tools/BNF/bnf_lfp_compat.ML›
ML_file ‹Tools/BNF/bnf_lfp_rec_sugar_more.ML›
ML_file ‹Tools/BNF/bnf_lfp_size.ML›
ML_file ‹Tools/datatype_simprocs.ML›
simproc_setup datatype_no_proper_subterm
("(x :: 'a :: size) = y") = ‹K Datatype_Simprocs.no_proper_subterm_simproc›
end