Structure set_relationTheory
signature set_relationTheory =
sig
type thm = Thm.thm
(* Definitions *)
val acyclic_def : thm
val all_choices_def : thm
val antisym_def : thm
val chain_def : thm
val domain_def : thm
val fchains_def : thm
val finite_prefixes_def : thm
val get_min_def : thm
val irreflexive_def : thm
val linear_order_def : thm
val maximal_elements_def : thm
val minimal_elements_def : thm
val nth_min_curried_def : thm
val nth_min_tupled_primitive_def : thm
val num_order_def : thm
val partial_order_def : thm
val per_def : thm
val per_restrict_def : thm
val range_def : thm
val rcomp_def : thm
val reflexive_def : thm
val rrestrict_def : thm
val strict_def : thm
val strict_linear_order_def : thm
val tc_def : thm
val transitive_def : thm
val univ_reln_def : thm
val upper_bounds_def : thm
(* Theorems *)
val acyclic_bigunion : thm
val acyclic_irreflexive : thm
val acyclic_rrestrict : thm
val acyclic_subset : thm
val acyclic_union : thm
val all_choices_thm : thm
val countable_per : thm
val empty_linear_order : thm
val empty_strict_linear_order : thm
val extend_linear_order : thm
val finite_acyclic_has_maximal : thm
val finite_acyclic_has_maximal_path : thm
val finite_acyclic_has_minimal : thm
val finite_acyclic_has_minimal_path : thm
val finite_linear_order_has_maximal : thm
val finite_linear_order_has_minimal : thm
val finite_prefix_linear_order_has_unique_minimal : thm
val finite_prefix_po_has_minimal_path : thm
val finite_prefixes_comp : thm
val finite_prefixes_inj_image : thm
val finite_prefixes_range : thm
val finite_prefixes_subset : thm
val finite_prefixes_union : thm
val finite_strict_linear_order_has_maximal : thm
val finite_strict_linear_order_has_minimal : thm
val in_domain : thm
val in_range : thm
val in_rrestrict : thm
val linear_order : thm
val linear_order_dom_rng : thm
val linear_order_num_order : thm
val linear_order_of_countable_po : thm
val linear_order_restrict : thm
val linear_order_subset : thm
val maximal_TC : thm
val maximal_linear_order : thm
val maximal_union : thm
val minimal_TC : thm
val minimal_linear_order : thm
val minimal_linear_order_unique : thm
val minimal_union : thm
val nat_order_iso_thm : thm
val nth_min_def : thm
val nth_min_def_compute : thm
val nth_min_ind : thm
val num_order_finite_prefix : thm
val partial_order_dom_rng : thm
val partial_order_linear_order : thm
val partial_order_subset : thm
val per_delete : thm
val per_restrict_per : thm
val rextension : thm
val rrestrict_rrestrict : thm
val rrestrict_tc : thm
val rrestrict_union : thm
val rtc_ind_right : thm
val strict_linear_order : thm
val strict_linear_order_acyclic : thm
val strict_linear_order_dom_rng : thm
val strict_linear_order_restrict : thm
val strict_linear_order_union_acyclic : thm
val strict_partial_order : thm
val strict_partial_order_acyclic : thm
val strict_rrestrict : thm
val tc_cases : thm
val tc_cases_left : thm
val tc_cases_right : thm
val tc_domain_range : thm
val tc_empty : thm
val tc_implication : thm
val tc_ind : thm
val tc_ind_left : thm
val tc_ind_right : thm
val tc_rules : thm
val tc_strongind : thm
val tc_strongind_left : thm
val tc_strongind_right : thm
val tc_transitive : thm
val tc_union : thm
val transitive_tc : thm
val upper_bounds_lem : thm
val zorns_lemma : thm
val set_relation_grammars : type_grammar.grammar * term_grammar.grammar
(*
[pred_set] Parent theory of "set_relation"
[acyclic_def] Definition
|- ∀r. acyclic r ⇔ ∀x. (x,x) ∉ tc r
[all_choices_def] Definition
|- ∀xss.
all_choices xss =
{IMAGE choice xss | choice | ∀xs. xs ∈ xss ⇒ choice xs ∈ xs}
[antisym_def] Definition
|- ∀r. antisym r ⇔ ∀x y. (x,y) ∈ r ∧ (y,x) ∈ r ⇒ (x = y)
[chain_def] Definition
|- ∀s r. chain s r ⇔ ∀x y. x ∈ s ∧ y ∈ s ⇒ (x,y) ∈ r ∨ (y,x) ∈ r
[domain_def] Definition
|- ∀r. domain r = {x | ∃y. (x,y) ∈ r}
[fchains_def] Definition
|- ∀r.
fchains r =
{k |
chain k r ∧ k ≠ ∅ ∧
∀C.
chain C r ∧ C ⊆ k ∧ (upper_bounds C r DIFF C) ∩ k ≠ ∅ ⇒
CHOICE (upper_bounds C r DIFF C) ∈
minimal_elements ((upper_bounds C r DIFF C) ∩ k) r}
[finite_prefixes_def] Definition
|- ∀r s. finite_prefixes r s ⇔ ∀e. e ∈ s ⇒ FINITE {e' | (e',e) ∈ r}
[get_min_def] Definition
|- ∀r' s r.
get_min r' (s,r) =
(let mins = minimal_elements (minimal_elements s r) r'
in
if SING mins then SOME (CHOICE mins) else NONE)
[irreflexive_def] Definition
|- ∀r s. irreflexive r s ⇔ ∀x. x ∈ s ⇒ (x,x) ∉ r
[linear_order_def] Definition
|- ∀r s.
linear_order r s ⇔
domain r ⊆ s ∧ range r ⊆ s ∧ transitive r ∧ antisym r ∧
∀x y. x ∈ s ∧ y ∈ s ⇒ (x,y) ∈ r ∨ (y,x) ∈ r
[maximal_elements_def] Definition
|- ∀xs r.
maximal_elements xs r =
{x | x ∈ xs ∧ ∀x'. x' ∈ xs ∧ (x,x') ∈ r ⇒ (x = x')}
[minimal_elements_def] Definition
|- ∀xs r.
minimal_elements xs r =
{x | x ∈ xs ∧ ∀x'. x' ∈ xs ∧ (x',x) ∈ r ⇒ (x = x')}
[nth_min_curried_def] Definition
|- ∀x x1 x2. nth_min x x1 x2 = nth_min_tupled (x,x1,x2)
[nth_min_tupled_primitive_def] Definition
|- nth_min_tupled =
WFREC
(@R.
WF R ∧
∀n r s r' min.
(min = get_min r' (s,r)) ∧ min ≠ NONE ⇒
R (r',(s DELETE THE min,r),n) (r',(s,r),SUC n))
(λnth_min_tupled a.
case a of
(r',(s,r),0) => I (get_min r' (s,r))
| (r',(s,r),SUC n) =>
I
(let min = get_min r' (s,r)
in
if min = NONE then
NONE
else
nth_min_tupled (r',(s DELETE THE min,r),n)))
[num_order_def] Definition
|- ∀f s. num_order f s = {(x,y) | x ∈ s ∧ y ∈ s ∧ f x ≤ f y}
[partial_order_def] Definition
|- ∀r s.
partial_order r s ⇔
domain r ⊆ s ∧ range r ⊆ s ∧ transitive r ∧ reflexive r s ∧
antisym r
[per_def] Definition
|- ∀xs xss.
per xs xss ⇔
BIGUNION xss ⊆ xs ∧ ∅ ∉ xss ∧
∀xs1 xs2. xs1 ∈ xss ∧ xs2 ∈ xss ∧ xs1 ≠ xs2 ⇒ DISJOINT xs1 xs2
[per_restrict_def] Definition
|- ∀xss xs. per_restrict xss xs = {xs' ∩ xs | xs' ∈ xss} DELETE ∅
[range_def] Definition
|- ∀r. range r = {y | ∃x. (x,y) ∈ r}
[rcomp_def] Definition
|- ∀r1 r2. r1 OO r2 = {(x,y) | ∃z. (x,z) ∈ r1 ∧ (z,y) ∈ r2}
[reflexive_def] Definition
|- ∀r s. reflexive r s ⇔ ∀x. x ∈ s ⇒ (x,x) ∈ r
[rrestrict_def] Definition
|- ∀r s. rrestrict r s = {(x,y) | (x,y) ∈ r ∧ x ∈ s ∧ y ∈ s}
[strict_def] Definition
|- ∀r. strict r = {(x,y) | (x,y) ∈ r ∧ x ≠ y}
[strict_linear_order_def] Definition
|- ∀r s.
strict_linear_order r s ⇔
domain r ⊆ s ∧ range r ⊆ s ∧ transitive r ∧ (∀x. (x,x) ∉ r) ∧
∀x y. x ∈ s ∧ y ∈ s ∧ x ≠ y ⇒ (x,y) ∈ r ∨ (y,x) ∈ r
[tc_def] Definition
|- tc =
(λr a0.
∀tc'.
(∀a0.
(∃x y. (a0 = (x,y)) ∧ r (x,y)) ∨
(∃x y. (a0 = (x,y)) ∧ ∃z. tc' (x,z) ∧ tc' (z,y)) ⇒
tc' a0) ⇒
tc' a0)
[transitive_def] Definition
|- ∀r. transitive r ⇔ ∀x y z. (x,y) ∈ r ∧ (y,z) ∈ r ⇒ (x,z) ∈ r
[univ_reln_def] Definition
|- ∀xs. univ_reln xs = {(x1,x2) | x1 ∈ xs ∧ x2 ∈ xs}
[upper_bounds_def] Definition
|- ∀s r. upper_bounds s r = {x | x ∈ range r ∧ ∀y. y ∈ s ⇒ (y,x) ∈ r}
[acyclic_bigunion] Theorem
|- ∀rs.
(∀r r'.
r ∈ rs ∧ r' ∈ rs ∧ r ≠ r' ⇒
DISJOINT (domain r ∪ range r) (domain r' ∪ range r')) ∧
(∀r. r ∈ rs ⇒ acyclic r) ⇒
acyclic (BIGUNION rs)
[acyclic_irreflexive] Theorem
|- ∀r x. acyclic r ⇒ (x,x) ∉ r
[acyclic_rrestrict] Theorem
|- ∀r s. acyclic r ⇒ acyclic (rrestrict r s)
[acyclic_subset] Theorem
|- ∀r1 r2. acyclic r1 ∧ r2 ⊆ r1 ⇒ acyclic r2
[acyclic_union] Theorem
|- ∀r r'.
DISJOINT (domain r ∪ range r) (domain r' ∪ range r') ∧
acyclic r ∧ acyclic r' ⇒
acyclic (r ∪ r')
[all_choices_thm] Theorem
|- ∀x s y. x ∈ all_choices s ∧ y ∈ x ⇒ ∃z. z ∈ s ∧ y ∈ z
[countable_per] Theorem
|- ∀xs xss. countable xs ∧ per xs xss ⇒ countable xss
[empty_linear_order] Theorem
|- ∀r. linear_order r ∅ ⇔ (r = ∅)
[empty_strict_linear_order] Theorem
|- ∀r. strict_linear_order r ∅ ⇔ (r = ∅)
[extend_linear_order] Theorem
|- ∀r s x.
x ∉ s ∧ linear_order r s ⇒
linear_order (r ∪ {(y,x) | y | y ∈ s ∪ {x}}) (s ∪ {x})
[finite_acyclic_has_maximal] Theorem
|- ∀s.
FINITE s ⇒ s ≠ ∅ ⇒ ∀r. acyclic r ⇒ ∃x. x ∈ maximal_elements s r
[finite_acyclic_has_maximal_path] Theorem
|- ∀s r x.
FINITE s ∧ acyclic r ∧ x ∈ s ∧ x ∉ maximal_elements s r ⇒
∃y. y ∈ maximal_elements s r ∧ (x,y) ∈ tc r
[finite_acyclic_has_minimal] Theorem
|- ∀s.
FINITE s ⇒ s ≠ ∅ ⇒ ∀r. acyclic r ⇒ ∃x. x ∈ minimal_elements s r
[finite_acyclic_has_minimal_path] Theorem
|- ∀s r x.
FINITE s ∧ acyclic r ∧ x ∈ s ∧ x ∉ minimal_elements s r ⇒
∃y. y ∈ minimal_elements s r ∧ (y,x) ∈ tc r
[finite_linear_order_has_maximal] Theorem
|- ∀s r.
FINITE s ∧ linear_order r s ∧ s ≠ ∅ ⇒
∃x. x ∈ maximal_elements s r
[finite_linear_order_has_minimal] Theorem
|- ∀s r.
FINITE s ∧ linear_order r s ∧ s ≠ ∅ ⇒
∃x. x ∈ minimal_elements s r
[finite_prefix_linear_order_has_unique_minimal] Theorem
|- ∀r s s'.
linear_order r s ∧ finite_prefixes r s ∧ x ∈ s' ∧ s' ⊆ s ⇒
SING (minimal_elements s' r)
[finite_prefix_po_has_minimal_path] Theorem
|- ∀r s x s'.
partial_order r s ∧ finite_prefixes r s ∧
x ∉ minimal_elements s' r ∧ x ∈ s' ∧ s' ⊆ s ⇒
∃x'. x' ∈ minimal_elements s' r ∧ (x',x) ∈ r
[finite_prefixes_comp] Theorem
|- ∀r1 r2 s1 s2.
finite_prefixes r1 s1 ∧ finite_prefixes r2 s2 ∧
{x | ∃y. y ∈ s2 ∧ (x,y) ∈ r2} ⊆ s1 ⇒
finite_prefixes (r1 OO r2) s2
[finite_prefixes_inj_image] Theorem
|- ∀f r s.
(∀x y. (f x = f y) ⇒ (x = y)) ∧ finite_prefixes r s ⇒
finite_prefixes {(f x,f y) | (x,y) ∈ r} (IMAGE f s)
[finite_prefixes_range] Theorem
|- ∀r s t.
finite_prefixes r s ∧ DISJOINT t (range r) ⇒
finite_prefixes r (s ∪ t)
[finite_prefixes_subset] Theorem
|- ∀r s s'.
finite_prefixes r s ∧ s' ⊆ s ⇒
finite_prefixes r s' ∧ finite_prefixes (rrestrict r s') s'
[finite_prefixes_union] Theorem
|- ∀r1 r2 s1 s2.
finite_prefixes r1 s1 ∧ finite_prefixes r2 s2 ⇒
finite_prefixes (r1 ∪ r2) (s1 ∩ s2)
[finite_strict_linear_order_has_maximal] Theorem
|- ∀s r.
FINITE s ∧ strict_linear_order r s ∧ s ≠ ∅ ⇒
∃x. x ∈ maximal_elements s r
[finite_strict_linear_order_has_minimal] Theorem
|- ∀s r.
FINITE s ∧ strict_linear_order r s ∧ s ≠ ∅ ⇒
∃x. x ∈ minimal_elements s r
[in_domain] Theorem
|- ∀x r. x ∈ domain r ⇔ ∃y. (x,y) ∈ r
[in_range] Theorem
|- ∀y r. y ∈ range r ⇔ ∃x. (x,y) ∈ r
[in_rrestrict] Theorem
|- ∀x y r s. (x,y) ∈ rrestrict r s ⇔ (x,y) ∈ r ∧ x ∈ s ∧ y ∈ s
[linear_order] Theorem
|- ∀r s.
strict_linear_order r s ⇒ linear_order (r ∪ {(x,x) | x ∈ s}) s
[linear_order_dom_rng] Theorem
|- ∀r s x y. (x,y) ∈ r ∧ linear_order r s ⇒ x ∈ s ∧ y ∈ s
[linear_order_num_order] Theorem
|- ∀f s t. INJ f s t ⇒ linear_order (num_order f s) s
[linear_order_of_countable_po] Theorem
|- ∀r s.
countable s ∧ partial_order r s ∧ finite_prefixes r s ⇒
∃r'. linear_order r' s ∧ finite_prefixes r' s ∧ r ⊆ r'
[linear_order_restrict] Theorem
|- ∀s r s'. linear_order r s ⇒ linear_order (rrestrict r s') (s ∩ s')
[linear_order_subset] Theorem
|- ∀r s s'.
linear_order r s ∧ s' ⊆ s ⇒ linear_order (rrestrict r s') s'
[maximal_TC] Theorem
|- ∀s r.
domain r ⊆ s ∧ range r ⊆ s ⇒
(maximal_elements s (tc r) = maximal_elements s r)
[maximal_linear_order] Theorem
|- ∀s r x y.
y ∈ s ∧ linear_order r s ∧ x ∈ maximal_elements s r ⇒ (y,x) ∈ r
[maximal_union] Theorem
|- ∀e s r1 r2.
e ∈ maximal_elements s (r1 ∪ r2) ⇒
e ∈ maximal_elements s r1 ∧ e ∈ maximal_elements s r2
[minimal_TC] Theorem
|- ∀s r.
domain r ⊆ s ∧ range r ⊆ s ⇒
(minimal_elements s (tc r) = minimal_elements s r)
[minimal_linear_order] Theorem
|- ∀s r x y.
y ∈ s ∧ linear_order r s ∧ x ∈ minimal_elements s r ⇒ (x,y) ∈ r
[minimal_linear_order_unique] Theorem
|- ∀r s s' x y.
linear_order r s ∧ x ∈ minimal_elements s' r ∧
y ∈ minimal_elements s' r ∧ s' ⊆ s ⇒
(x = y)
[minimal_union] Theorem
|- ∀e s r1 r2.
e ∈ minimal_elements s (r1 ∪ r2) ⇒
e ∈ minimal_elements s r1 ∧ e ∈ minimal_elements s r2
[nat_order_iso_thm] Theorem
|- ∀f s.
(∀n m. (f m = f n) ∧ f m ≠ NONE ⇒ (m = n)) ∧
(∀x. x ∈ s ⇒ ∃m. f m = SOME x) ∧
(∀m x. (f m = SOME x) ⇒ x ∈ s) ⇒
linear_order
{(x,y) | ∃m n. m ≤ n ∧ (f m = SOME x) ∧ (f n = SOME y)} s ∧
finite_prefixes
{(x,y) | ∃m n. m ≤ n ∧ (f m = SOME x) ∧ (f n = SOME y)} s
[nth_min_def] Theorem
|- (∀s r' r. nth_min r' (s,r) 0 = get_min r' (s,r)) ∧
∀s r' r n.
nth_min r' (s,r) (SUC n) =
(let min = get_min r' (s,r)
in
if min = NONE then
NONE
else
nth_min r' (s DELETE THE min,r) n)
[nth_min_def_compute] Theorem
|- (∀s r' r. nth_min r' (s,r) 0 = get_min r' (s,r)) ∧
(∀s r' r n.
nth_min r' (s,r) (NUMERAL (BIT1 n)) =
(let min = get_min r' (s,r)
in
if min = NONE then
NONE
else
nth_min r' (s DELETE THE min,r) (NUMERAL (BIT1 n) − 1))) ∧
∀s r' r n.
nth_min r' (s,r) (NUMERAL (BIT2 n)) =
(let min = get_min r' (s,r)
in
if min = NONE then
NONE
else
nth_min r' (s DELETE THE min,r) (NUMERAL (BIT1 n)))
[nth_min_ind] Theorem
|- ∀P.
(∀r' s r. P r' (s,r) 0) ∧
(∀r' s r n.
(∀min.
(min = get_min r' (s,r)) ∧ min ≠ NONE ⇒
P r' (s DELETE THE min,r) n) ⇒
P r' (s,r) (SUC n)) ⇒
∀v v1 v2 v3. P v (v1,v2) v3
[num_order_finite_prefix] Theorem
|- ∀f s t. INJ f s t ⇒ finite_prefixes (num_order f s) s
[partial_order_dom_rng] Theorem
|- ∀r s x y. (x,y) ∈ r ∧ partial_order r s ⇒ x ∈ s ∧ y ∈ s
[partial_order_linear_order] Theorem
|- ∀r s. linear_order r s ⇒ partial_order r s
[partial_order_subset] Theorem
|- ∀r s s'.
partial_order r s ∧ s' ⊆ s ⇒ partial_order (rrestrict r s') s'
[per_delete] Theorem
|- ∀xs xss e.
per xs xss ⇒
per (xs DELETE e)
{es | es ∈ IMAGE (λes. es DELETE e) xss ∧ es ≠ ∅}
[per_restrict_per] Theorem
|- ∀r s s'. per s r ⇒ per s' (per_restrict r s')
[rextension] Theorem
|- ∀s t. (s = t) ⇔ ∀x y. (x,y) ∈ s ⇔ (x,y) ∈ t
[rrestrict_rrestrict] Theorem
|- ∀r x y. rrestrict (rrestrict r x) y = rrestrict r (x ∩ y)
[rrestrict_tc] Theorem
|- ∀e e'. (e,e') ∈ tc (rrestrict r x) ⇒ (e,e') ∈ tc r
[rrestrict_union] Theorem
|- ∀r1 r2 s. rrestrict (r1 ∪ r2) s = rrestrict r1 s ∪ rrestrict r2 s
[rtc_ind_right] Theorem
|- ∀r tc'.
(∀x. x ∈ domain r ∨ x ∈ range r ⇒ tc' x x) ∧
(∀x y. (∃z. tc' x z ∧ (z,y) ∈ r) ⇒ tc' x y) ⇒
∀x y. (x,y) ∈ tc r ⇒ tc' x y
[strict_linear_order] Theorem
|- ∀r s. linear_order r s ⇒ strict_linear_order (strict r) s
[strict_linear_order_acyclic] Theorem
|- ∀r s. strict_linear_order r s ⇒ acyclic r
[strict_linear_order_dom_rng] Theorem
|- ∀r s x y. (x,y) ∈ r ∧ strict_linear_order r s ⇒ x ∈ s ∧ y ∈ s
[strict_linear_order_restrict] Theorem
|- ∀s r s'.
strict_linear_order r s ⇒
strict_linear_order (rrestrict r s') (s ∩ s')
[strict_linear_order_union_acyclic] Theorem
|- ∀r1 r2 s.
strict_linear_order r1 s ∧ domain r2 ∪ range r2 ⊆ s ⇒
(acyclic (r1 ∪ r2) ⇔ r2 ⊆ r1)
[strict_partial_order] Theorem
|- ∀r s.
partial_order r s ⇒
domain (strict r) ⊆ s ∧ range (strict r) ⊆ s ∧
transitive (strict r) ∧ antisym (strict r)
[strict_partial_order_acyclic] Theorem
|- ∀r s. partial_order r s ⇒ acyclic (strict r)
[strict_rrestrict] Theorem
|- ∀r s. strict (rrestrict r s) = rrestrict (strict r) s
[tc_cases] Theorem
|- ∀r x y. (x,y) ∈ tc r ⇔ (x,y) ∈ r ∨ ∃z. (x,z) ∈ tc r ∧ (z,y) ∈ tc r
[tc_cases_left] Theorem
|- ∀r x y. (x,y) ∈ tc r ⇔ (x,y) ∈ r ∨ ∃z. (x,z) ∈ r ∧ (z,y) ∈ tc r
[tc_cases_right] Theorem
|- ∀r x y. (x,y) ∈ tc r ⇔ (x,y) ∈ r ∨ ∃z. (x,z) ∈ tc r ∧ (z,y) ∈ r
[tc_domain_range] Theorem
|- ∀x y. (x,y) ∈ tc r ⇒ x ∈ domain r ∧ y ∈ range r
[tc_empty] Theorem
|- ∀x y. (x,y) ∉ tc ∅
[tc_implication] Theorem
|- ∀r1 r2.
(∀x y. (x,y) ∈ r1 ⇒ (x,y) ∈ r2) ⇒
∀x y. (x,y) ∈ tc r1 ⇒ (x,y) ∈ tc r2
[tc_ind] Theorem
|- ∀r tc'.
(∀x y. (x,y) ∈ r ⇒ tc' x y) ∧
(∀x y. (∃z. tc' x z ∧ tc' z y) ⇒ tc' x y) ⇒
∀x y. (x,y) ∈ tc r ⇒ tc' x y
[tc_ind_left] Theorem
|- ∀r tc'.
(∀x y. (x,y) ∈ r ⇒ tc' x y) ∧
(∀x y. (∃z. (x,z) ∈ r ∧ tc' z y) ⇒ tc' x y) ⇒
∀x y. (x,y) ∈ tc r ⇒ tc' x y
[tc_ind_right] Theorem
|- ∀r tc'.
(∀x y. (x,y) ∈ r ⇒ tc' x y) ∧
(∀x y. (∃z. tc' x z ∧ (z,y) ∈ r) ⇒ tc' x y) ⇒
∀x y. (x,y) ∈ tc r ⇒ tc' x y
[tc_rules] Theorem
|- ∀r.
(∀x y. (x,y) ∈ r ⇒ (x,y) ∈ tc r) ∧
∀x y. (∃z. (x,z) ∈ tc r ∧ (z,y) ∈ tc r) ⇒ (x,y) ∈ tc r
[tc_strongind] Theorem
|- ∀r tc'.
(∀x y. (x,y) ∈ r ⇒ tc' x y) ∧
(∀x y.
(∃z. (x,z) ∈ tc r ∧ tc' x z ∧ (z,y) ∈ tc r ∧ tc' z y) ⇒
tc' x y) ⇒
∀x y. (x,y) ∈ tc r ⇒ tc' x y
[tc_strongind_left] Theorem
|- ∀r tc'.
(∀x y. (x,y) ∈ r ⇒ tc' x y) ∧
(∀x y. (∃z. (x,z) ∈ r ∧ (z,y) ∈ tc r ∧ tc' z y) ⇒ tc' x y) ⇒
∀x y. (x,y) ∈ tc r ⇒ tc' x y
[tc_strongind_right] Theorem
|- ∀r tc'.
(∀x y. (x,y) ∈ r ⇒ tc' x y) ∧
(∀x y. (∃z. (x,z) ∈ tc r ∧ tc' x z ∧ (z,y) ∈ r) ⇒ tc' x y) ⇒
∀x y. (x,y) ∈ tc r ⇒ tc' x y
[tc_transitive] Theorem
|- ∀r. transitive (tc r)
[tc_union] Theorem
|- ∀x y. (x,y) ∈ tc r1 ⇒ ∀r2. (x,y) ∈ tc (r1 ∪ r2)
[transitive_tc] Theorem
|- ∀r. transitive r ⇒ (tc r = r)
[upper_bounds_lem] Theorem
|- ∀r s x1 x2.
transitive r ∧ x1 ∈ upper_bounds s r ∧ (x1,x2) ∈ r ⇒
x2 ∈ upper_bounds s r
[zorns_lemma] Theorem
|- ∀r s.
s ≠ ∅ ∧ partial_order r s ∧
(∀t. chain t r ⇒ upper_bounds t r ≠ ∅) ⇒
∃x. x ∈ maximal_elements s r
*)
end
HOL 4, Kananaskis-8