Theory "set_relation"

Parents pred_set

Signature

Constant	Type
RREFL_EXP	:α reln -> (α -> bool) -> α reln
RRUNIV	:(α -> bool) -> α reln
acyclic	:α set_relation$reln -> bool
all_choices	:((α -> bool) -> bool) -> (α -> bool) -> bool
antisym	:α set_relation$reln -> bool
chain	:(α -> bool) -> α set_relation$reln -> bool
domain	:(α # β -> bool) -> α -> bool
fchains	:α set_relation$reln -> (α -> bool) -> bool
finite_prefixes	:(α # β -> bool) -> (β -> bool) -> bool
get_min	:α set_relation$reln -> (α -> bool) # α set_relation$reln -> α option
irreflexive	:α set_relation$reln -> (α -> bool) -> bool
linear_order	:α set_relation$reln -> (α -> bool) -> bool
maximal_elements	:(α -> bool) -> α set_relation$reln -> α -> bool
minimal_elements	:(α -> bool) -> α set_relation$reln -> α -> bool
nth_min	:α set_relation$reln -> (α -> bool) # α set_relation$reln -> num -> α option
num_order	:(α -> num) -> (α -> bool) -> α set_relation$reln
partial_order	:α set_relation$reln -> (α -> bool) -> bool
per	:(α -> bool) -> ((α -> bool) -> bool) -> bool
per_restrict	:((α -> bool) -> bool) -> (α -> bool) reln
range	:(β # α -> bool) -> α -> bool
rcomp	:(α # γ -> bool) -> (γ # β -> bool) -> α # β -> bool
reflexive	:α set_relation$reln -> (α -> bool) -> bool
rel_to_reln	:(α -> β -> bool) -> α # β -> bool
reln_to_rel	:(α # β -> bool) -> α -> β -> bool
rrestrict	:α set_relation$reln -> (α -> bool) -> α set_relation$reln
strict	:α set_relation$reln -> α set_relation$reln
strict_linear_order	:α set_relation$reln -> (α -> bool) -> bool
tc	:α set_relation$reln -> α set_relation$reln
transitive	:α set_relation$reln -> bool
univ_reln	:(α -> bool) -> α set_relation$reln
upper_bounds	:(β -> bool) -> (β # α -> bool) -> α -> bool

Definitions

domain_def

⊢ ∀r. domain r = {x | ∃y. (x,y) ∈ r}

range_def

⊢ ∀r. range r = {y | ∃x. (x,y) ∈ r}

rrestrict_def

⊢ ∀r s. rrestrict r s = {(x,y) | (x,y) ∈ r ∧ x ∈ s ∧ y ∈ s}

rcomp_def

⊢ ∀r1 r2. r1 OO r2 = {(x,y) | ∃z. (x,z) ∈ r1 ∧ (z,y) ∈ r2}

strict_def

⊢ ∀r. strict r = {(x,y) | (x,y) ∈ r ∧ x ≠ y}

univ_reln_def

⊢ ∀xs. univ_reln xs = {(x1,x2) | x1 ∈ xs ∧ x2 ∈ xs}

finite_prefixes_def

⊢ ∀r s. finite_prefixes r s ⇔ ∀e. e ∈ s ⇒ FINITE {e' | (e',e) ∈ r}

tc_def

⊢ 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)

acyclic_def

⊢ ∀r. acyclic r ⇔ ∀x. (x,x) ∉ tc r

reflexive_def

⊢ ∀r s. reflexive r s ⇔ ∀x. x ∈ s ⇒ (x,x) ∈ r

irreflexive_def

⊢ ∀r s. irreflexive r s ⇔ ∀x. x ∈ s ⇒ (x,x) ∉ r

transitive_def

⊢ ∀r. transitive r ⇔ ∀x y z. (x,y) ∈ r ∧ (y,z) ∈ r ⇒ (x,z) ∈ r

antisym_def

⊢ ∀r. antisym r ⇔ ∀x y. (x,y) ∈ r ∧ (y,x) ∈ r ⇒ x = y

partial_order_def

⊢ ∀r s.
      partial_order r s ⇔
      domain r ⊆ s ∧ range r ⊆ s ∧ transitive r ∧ reflexive r s ∧ antisym r

linear_order_def

⊢ ∀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

strict_linear_order_def

⊢ ∀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

reln_to_rel_def

⊢ ∀r. reln_to_rel r = (λx y. (x,y) ∈ r)

rel_to_reln_def

⊢ ∀R. rel_to_reln R = {(x,y) | R x y}

RRUNIV_def

⊢ ∀s. RRUNIV s = (λx y. x ∈ s ∧ y ∈ s)

RREFL_EXP_def

⊢ ∀R s. RREFL_EXP R s = R ∪ᵣ (λx y. x = y ∧ x ∉ s)

maximal_elements_def

⊢ ∀xs r.
      maximal_elements xs r =
      {x | x ∈ xs ∧ ∀x'. x' ∈ xs ∧ (x,x') ∈ r ⇒ x = x'}

minimal_elements_def

⊢ ∀xs r.
      minimal_elements xs r =
      {x | x ∈ xs ∧ ∀x'. x' ∈ xs ∧ (x',x) ∈ r ⇒ x = x'}

chain_def

⊢ ∀s r. chain s r ⇔ ∀x y. x ∈ s ∧ y ∈ s ⇒ (x,y) ∈ r ∨ (y,x) ∈ r

upper_bounds_def

⊢ ∀s r. upper_bounds s r = {x | x ∈ range r ∧ ∀y. y ∈ s ⇒ (y,x) ∈ r}

fchains_def

⊢ ∀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}

per_def

⊢ ∀xs xss.
      per xs xss ⇔
      BIGUNION xss ⊆ xs ∧ ∅ ∉ xss ∧
      ∀xs1 xs2. xs1 ∈ xss ∧ xs2 ∈ xss ∧ xs1 ≠ xs2 ⇒ DISJOINT xs1 xs2

per_restrict_def

⊢ ∀xss xs. per_restrict xss xs = {xs' ∩ xs | xs' ∈ xss} DELETE ∅

all_choices_def

⊢ ∀xss.
      all_choices xss =
      {IMAGE choice xss | choice | ∀xs. xs ∈ xss ⇒ choice xs ∈ xs}

num_order_def

⊢ ∀f s. num_order f s = {(x,y) | x ∈ s ∧ y ∈ s ∧ f x ≤ f y}

get_min_def

⊢ ∀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)

Theorems

reflexive_reln_to_rel_conv_UNIV

⊢ reflexive r 𝕌(:α) ⇔ reflexive (reln_to_rel r)

acyclic_bigunion

⊢ ∀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_SWAP

⊢ acyclic (IMAGE SWAP r) ⇔ acyclic r

acyclic_irreflexive

⊢ ∀r x. acyclic r ⇒ (x,x) ∉ r

acyclic_rrestrict

⊢ ∀r s. acyclic r ⇒ acyclic (rrestrict r s)

tc_mono

⊢ r ⊆ s ⇒ tc r ⊆ tc s

tc_idemp

⊢ tc (tc r) = tc r

subset_tc

⊢ r ⊆ tc r

tc_closure

⊢ r ⊆ tc s ⇒ tc r ⊆ tc s

tc_ind

⊢ ∀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

rextension

⊢ ∀s t. s = t ⇔ ∀x y. (x,y) ∈ s ⇔ (x,y) ∈ t

in_domain

⊢ ∀x r. x ∈ domain r ⇔ ∃y. (x,y) ∈ r

in_range

⊢ ∀y r. y ∈ range r ⇔ ∃x. (x,y) ∈ r

in_dom_rg

⊢ (x,y) ∈ r ⇒ x ∈ domain r ∧ y ∈ range r

domain_mono

⊢ r ⊆ r' ⇒ domain r ⊆ domain r'

range_mono

⊢ r ⊆ r' ⇒ range r ⊆ range r'

in_rrestrict

⊢ ∀x y r s. (x,y) ∈ rrestrict r s ⇔ (x,y) ∈ r ∧ x ∈ s ∧ y ∈ s

in_rrestrict_alt

⊢ x ∈ rrestrict r s ⇔ x ∈ r ∧ FST x ∈ s ∧ SND x ∈ s

rrestrict_SUBSET

⊢ rrestrict r s ⊆ r

rrestrict_union

⊢ ∀r1 r2 s. rrestrict (r1 ∪ r2) s = rrestrict r1 s ∪ rrestrict r2 s

rrestrict_rrestrict

⊢ ∀r x y. rrestrict (rrestrict r x) y = rrestrict r (x ∩ y)

domain_rrestrict_SUBSET

⊢ domain (rrestrict r s) ⊆ s

range_rrestrict_SUBSET

⊢ range (rrestrict r s) ⊆ s

strict_rrestrict

⊢ ∀r s. strict (rrestrict r s) = rrestrict (strict r) s

finite_prefixes_subset_s

⊢ ∀r s s'. finite_prefixes r s ∧ s' ⊆ s ⇒ finite_prefixes r s'

finite_prefixes_subset_r

⊢ ∀r r' s. finite_prefixes r s ∧ r' ⊆ r ⇒ finite_prefixes r' s

finite_prefixes_subset_rs

⊢ ∀r s r' s'. finite_prefixes r s ⇒ r' ⊆ r ⇒ s' ⊆ s ⇒ finite_prefixes r' s'

finite_prefixes_subset

⊢ ∀r s s'.
      finite_prefixes r s ∧ s' ⊆ s ⇒
      finite_prefixes r s' ∧ finite_prefixes (rrestrict r s') s'

finite_prefixes_union

⊢ ∀r1 r2 s1 s2.
      finite_prefixes r1 s1 ∧ finite_prefixes r2 s2 ⇒
      finite_prefixes (r1 ∪ r2) (s1 ∩ s2)

finite_prefixes_comp

⊢ ∀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

⊢ ∀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

⊢ ∀r s t.
      finite_prefixes r s ∧ DISJOINT t (range r) ⇒ finite_prefixes r (s ∪ t)

tc_rules

⊢ ∀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_cases

⊢ ∀r x y. (x,y) ∈ tc r ⇔ (x,y) ∈ r ∨ ∃z. (x,z) ∈ tc r ∧ (z,y) ∈ tc r

tc_strongind

⊢ ∀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_cases_right

⊢ ∀r x y. (x,y) ∈ tc r ⇔ (x,y) ∈ r ∨ ∃z. (x,z) ∈ tc r ∧ (z,y) ∈ r

tc_cases_left

⊢ ∀r x y. (x,y) ∈ tc r ⇔ (x,y) ∈ r ∨ ∃z. (x,z) ∈ r ∧ (z,y) ∈ tc r

tc_ind_left

⊢ ∀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_strongind_left

⊢ ∀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_ind_right

⊢ ∀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

rtc_ind_right

⊢ ∀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

tc_strongind_right

⊢ ∀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_union

⊢ ∀x y. (x,y) ∈ tc r1 ⇒ ∀r2. (x,y) ∈ tc (r1 ∪ r2)

tc_implication

⊢ ∀r1 r2.
      (∀x y. (x,y) ∈ r1 ⇒ (x,y) ∈ r2) ⇒ ∀x y. (x,y) ∈ tc r1 ⇒ (x,y) ∈ tc r2

tc_empty

⊢ ∀x y. (x,y) ∉ tc ∅

tc_empty_eqn

⊢ tc ∅ = ∅

tc_domain_range

⊢ ∀x y. (x,y) ∈ tc r ⇒ x ∈ domain r ∧ y ∈ range r

rrestrict_tc

⊢ ∀e e'. (e,e') ∈ tc (rrestrict r x) ⇒ (e,e') ∈ tc r

tc_SWAP

⊢ tc (IMAGE SWAP r) = IMAGE SWAP (tc r)

acyclic_subset

⊢ ∀r1 r2. acyclic r1 ∧ r2 ⊆ r1 ⇒ acyclic r2

acyclic_union

⊢ ∀r r'.
      DISJOINT (domain r ∪ range r) (domain r' ∪ range r') ∧ acyclic r ∧
      acyclic r' ⇒
      acyclic (r ∪ r')

transitive_tc

⊢ ∀r. transitive r ⇒ tc r = r

tc_transitive

⊢ ∀r. transitive (tc r)

antisym_subset

⊢ antisym t ⇒ s ⊆ t ⇒ antisym s

partial_order_dom_rng

⊢ ∀r s x y. (x,y) ∈ r ∧ partial_order r s ⇒ x ∈ s ∧ y ∈ s

partial_order_subset

⊢ ∀r s s'. partial_order r s ∧ s' ⊆ s ⇒ partial_order (rrestrict r s') s'

strict_partial_order

⊢ ∀r s.
      partial_order r s ⇒
      domain (strict r) ⊆ s ∧ range (strict r) ⊆ s ∧ transitive (strict r) ∧
      antisym (strict r)

strict_partial_order_acyclic

⊢ ∀r s. partial_order r s ⇒ acyclic (strict r)

linear_order_subset

⊢ ∀r s s'. linear_order r s ∧ s' ⊆ s ⇒ linear_order (rrestrict r s') s'

partial_order_linear_order

⊢ ∀r s. linear_order r s ⇒ partial_order r s

strict_linear_order_dom_rng

⊢ ∀r s x y. (x,y) ∈ r ∧ strict_linear_order r s ⇒ x ∈ s ∧ y ∈ s

linear_order_dom_rng

⊢ ∀r s x y. (x,y) ∈ r ∧ linear_order r s ⇒ x ∈ s ∧ y ∈ s

RREFL_EXP_RSUBSET

⊢ R ⊆ᵣ RREFL_EXP R s

RREFL_EXP_UNIV

⊢ RREFL_EXP R 𝕌(:α) = R

REL_RESTRICT_UNIV

⊢ REL_RESTRICT R 𝕌(:α) = R

in_rel_to_reln

⊢ xy ∈ rel_to_reln R ⇔ R (FST xy) (SND xy)

reln_to_rel_app

⊢ reln_to_rel r x y ⇔ (x,y) ∈ r

rel_to_reln_IS_UNCURRY

⊢ rel_to_reln = UNCURRY

reln_to_rel_IS_CURRY

⊢ reln_to_rel = CURRY

rel_to_reln_inv

⊢ reln_to_rel (rel_to_reln R) = R

reln_to_rel_inv

⊢ rel_to_reln (reln_to_rel r) = r

reln_to_rel_11

⊢ reln_to_rel r1 = reln_to_rel r2 ⇔ r1 = r2

rel_to_reln_11

⊢ rel_to_reln R1 = rel_to_reln R2 ⇔ R1 = R2

rel_to_reln_swap

⊢ r = rel_to_reln R ⇔ reln_to_rel r = R

domain_to_rel_conv

⊢ domain r = RDOM (reln_to_rel r)

range_to_rel_conv

⊢ range r = RRANGE (reln_to_rel r)

strict_to_rel_conv

⊢ strict r = rel_to_reln (STRORD (reln_to_rel r))

rrestrict_to_rel_conv

⊢ rrestrict r s = rel_to_reln (REL_RESTRICT (reln_to_rel r) s)

rcomp_to_rel_conv

⊢ r1 OO r2 = rel_to_reln (reln_to_rel r2 ∘ᵣ reln_to_rel r1)

univ_reln_to_rel_conv

⊢ univ_reln s = rel_to_reln (RRUNIV s)

tc_to_rel_conv

⊢ tc r = rel_to_reln (reln_to_rel r)⁺

acyclic_reln_to_rel_conv

⊢ acyclic r ⇔ irreflexive (reln_to_rel r)⁺

irreflexive_reln_to_rel_conv

⊢ irreflexive r s ⇔ irreflexive (REL_RESTRICT (reln_to_rel r) s)

irreflexive_reln_to_rel_conv_UNIV

⊢ irreflexive r 𝕌(:α) ⇔ irreflexive (reln_to_rel r)

reflexive_reln_to_rel_conv

⊢ transitive r ⇔ transitive (reln_to_rel r)

antisym_reln_to_rel_conv

⊢ antisym r ⇔ antisymmetric (reln_to_rel r)

partial_order_reln_to_rel_conv

⊢ partial_order r s ⇔
  reln_to_rel r ⊆ᵣ RRUNIV s ∧ WeakOrder (RREFL_EXP (reln_to_rel r) s)

partial_order_reln_to_rel_conv_UNIV

⊢ partial_order r 𝕌(:α) ⇔ WeakOrder (reln_to_rel r)

linear_order_reln_to_rel_conv_UNIV

⊢ linear_order r 𝕌(:α) ⇔ WeakLinearOrder (reln_to_rel r)

strict_linear_order_reln_to_rel_conv_UNIV

⊢ strict_linear_order r 𝕌(:α) ⇔ StrongLinearOrder (reln_to_rel r)

reln_rel_conv_thms

⊢ ((xy ∈ rel_to_reln R ⇔ R (FST xy) (SND xy)) ∧
   (reln_to_rel r x y ⇔ (x,y) ∈ r) ∧ reln_to_rel (rel_to_reln R) = R ∧
   rel_to_reln (reln_to_rel r) = r ∧
   (reln_to_rel r1 = reln_to_rel r2 ⇔ r1 = r2) ∧
   (rel_to_reln R1 = rel_to_reln R2 ⇔ R1 = R2)) ∧ RREFL_EXP R 𝕌(:α) = R ∧
  REL_RESTRICT R 𝕌(:α) = R ∧ domain r = RDOM (reln_to_rel r) ∧
  range r = RRANGE (reln_to_rel r) ∧
  strict r = rel_to_reln (STRORD (reln_to_rel r)) ∧
  rrestrict r s = rel_to_reln (REL_RESTRICT (reln_to_rel r) s) ∧
  r1 OO r2 = rel_to_reln (reln_to_rel r2 ∘ᵣ reln_to_rel r1) ∧
  univ_reln s = rel_to_reln (RRUNIV s) ∧ tc r = rel_to_reln (reln_to_rel r)⁺ ∧
  (acyclic r ⇔ irreflexive (reln_to_rel r)⁺) ∧
  (irreflexive r s ⇔ irreflexive (REL_RESTRICT (reln_to_rel r) s)) ∧
  (reflexive r s ⇔ reflexive (RREFL_EXP (reln_to_rel r) s)) ∧
  (transitive r ⇔ transitive (reln_to_rel r)) ∧
  (antisym r ⇔ antisymmetric (reln_to_rel r)) ∧
  (partial_order r 𝕌(:α) ⇔ WeakOrder (reln_to_rel r)) ∧
  (linear_order r 𝕌(:α) ⇔ WeakLinearOrder (reln_to_rel r)) ∧
  (strict_linear_order r 𝕌(:α) ⇔ StrongLinearOrder (reln_to_rel r))

acyclic_WF

⊢ FINITE s ∧ acyclic r ∧ domain r ⊆ s ∧ range r ⊆ s ⇒ WF (reln_to_rel r)

minimal_elements_subset

⊢ minimal_elements s lo ⊆ s

minimal_elements_SWAP

⊢ minimal_elements xs (IMAGE SWAP r) = maximal_elements xs r

maximal_union

⊢ ∀e s r1 r2.
      e ∈ maximal_elements s (r1 ∪ r2) ⇒
      e ∈ maximal_elements s r1 ∧ e ∈ maximal_elements s r2

minimal_union

⊢ ∀e s r1 r2.
      e ∈ minimal_elements s (r1 ∪ r2) ⇒
      e ∈ minimal_elements s r1 ∧ e ∈ minimal_elements s r2

minimal_elements_mono

⊢ r ⊆ r' ⇒ minimal_elements xs r' ⊆ minimal_elements xs r

minimal_elements_rrestrict

⊢ minimal_elements xs (rrestrict r xs) = minimal_elements xs r

WF_has_minimal_path

⊢ WF (reln_to_rel r) ⇒
  x ∈ s ⇒
  ∃y. y ∈ minimal_elements s r ∧ ((y,x) ∈ tc r ∨ y = x)

finite_acyclic_has_maximal

⊢ ∀s. FINITE s ⇒ s ≠ ∅ ⇒ ∀r. acyclic r ⇒ ∃x. x ∈ maximal_elements s r

finite_acyclic_has_minimal

⊢ ∀s. FINITE s ⇒ s ≠ ∅ ⇒ ∀r. acyclic r ⇒ ∃x. x ∈ minimal_elements s r

maximal_TC

⊢ ∀s r.
      domain r ⊆ s ∧ range r ⊆ s ⇒
      maximal_elements s (tc r) = maximal_elements s r

minimal_TC

⊢ ∀s r.
      domain r ⊆ s ∧ range r ⊆ s ⇒
      minimal_elements s (tc r) = minimal_elements s r

finite_acyclic_has_minimal_path

⊢ ∀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_acyclic_has_maximal_path

⊢ ∀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_prefix_po_has_minimal_path

⊢ ∀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

empty_strict_linear_order

⊢ ∀r. strict_linear_order r ∅ ⇔ r = ∅

empty_linear_order

⊢ ∀r. linear_order r ∅ ⇔ r = ∅

linear_order_restrict

⊢ ∀s r s'. linear_order r s ⇒ linear_order (rrestrict r s') (s ∩ s')

strict_linear_order_restrict

⊢ ∀s r s'.
      strict_linear_order r s ⇒ strict_linear_order (rrestrict r s') (s ∩ s')

linear_order_dom_rg

⊢ linear_order lo X ⇒ domain lo ∪ range lo = X

linear_order_refl

⊢ linear_order lo X ⇒ x ∈ X ⇒ (x,x) ∈ lo

linear_order_in_set

⊢ linear_order lo X ⇒ (x,y) ∈ lo ⇒ x ∈ X ∧ y ∈ X

IN_MIN_LO

⊢ x ∈ X ⇒ linear_order lo X ⇒ y ∈ minimal_elements X lo ⇒ (y,x) ∈ lo

extend_linear_order

⊢ ∀r s x.
      x ∉ s ∧ linear_order r s ⇒
      linear_order (r ∪ {(y,x) | y | y ∈ s ∪ {x}}) (s ∪ {x})

strict_linear_order_acyclic

⊢ ∀r s. strict_linear_order r s ⇒ acyclic r

strict_linear_order_union_acyclic

⊢ ∀r1 r2 s.
      strict_linear_order r1 s ∧ domain r2 ∪ range r2 ⊆ s ⇒
      (acyclic (r1 ∪ r2) ⇔ r2 ⊆ r1)

strict_linear_order

⊢ ∀r s. linear_order r s ⇒ strict_linear_order (strict r) s

linear_order

⊢ ∀r s. strict_linear_order r s ⇒ linear_order (r ∪ {(x,x) | x ∈ s}) s

finite_strict_linear_order_has_maximal

⊢ ∀s r.
      FINITE s ∧ strict_linear_order r s ∧ s ≠ ∅ ⇒
      ∃x. x ∈ maximal_elements s r

finite_strict_linear_order_has_minimal

⊢ ∀s r.
      FINITE s ∧ strict_linear_order r s ∧ s ≠ ∅ ⇒
      ∃x. x ∈ minimal_elements s r

finite_linear_order_has_maximal

⊢ ∀s r. FINITE s ∧ linear_order r s ∧ s ≠ ∅ ⇒ ∃x. x ∈ maximal_elements s r

finite_linear_order_has_minimal

⊢ ∀s r. FINITE s ∧ linear_order r s ∧ s ≠ ∅ ⇒ ∃x. x ∈ minimal_elements s r

maximal_linear_order

⊢ ∀s r x y. y ∈ s ∧ linear_order r s ∧ x ∈ maximal_elements s r ⇒ (y,x) ∈ r

minimal_linear_order

⊢ ∀s r x y. y ∈ s ∧ linear_order r s ∧ x ∈ minimal_elements s r ⇒ (x,y) ∈ r

minimal_linear_order_unique

⊢ ∀r s s' x y.
      linear_order r s ∧ x ∈ minimal_elements s' r ∧
      y ∈ minimal_elements s' r ∧ s' ⊆ s ⇒
      x = y

finite_prefix_linear_order_has_unique_minimal

⊢ ∀r s s'.
      linear_order r s ∧ finite_prefixes r s ∧ x ∈ s' ∧ s' ⊆ s ⇒
      SING (minimal_elements s' r)

nat_order_iso_thm

⊢ ∀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

upper_bounds_lem

⊢ ∀r s x1 x2.
      transitive r ∧ x1 ∈ upper_bounds s r ∧ (x1,x2) ∈ r ⇒
      x2 ∈ upper_bounds s r

zorns_lemma

⊢ ∀r s.
      s ≠ ∅ ∧ partial_order r s ∧ (∀t. chain t r ⇒ upper_bounds t r ≠ ∅) ⇒
      ∃x. x ∈ maximal_elements s r

per_delete

⊢ ∀xs xss e.
      per xs xss ⇒
      per (xs DELETE e) {es | es ∈ IMAGE (λes. es DELETE e) xss ∧ es ≠ ∅}

per_restrict_per

⊢ ∀r s s'. per s r ⇒ per s' (per_restrict r s')

countable_per

⊢ ∀xs xss. countable xs ∧ per xs xss ⇒ countable xss

all_choices_thm

⊢ ∀x s y. x ∈ all_choices s ∧ y ∈ x ⇒ ∃z. z ∈ s ∧ y ∈ z

linear_order_num_order

⊢ ∀f s t. INJ f s t ⇒ linear_order (num_order f s) s

num_order_finite_prefix

⊢ ∀f s t. INJ f s t ⇒ finite_prefixes (num_order f s) s

nth_min_ind

⊢ ∀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

nth_min_def

⊢ (∀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

⊢ (∀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)))

linear_order_of_countable_po

⊢ ∀r s.
      countable s ∧ partial_order r s ∧ finite_prefixes r s ⇒
      ∃r'. linear_order r' s ∧ finite_prefixes r' s ∧ r ⊆ r'