Structure posetTheory

signature posetTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val bottom_def : thm
    val carrier_def : thm
    val chain_def : thm
    val complete_def : thm
    val continuous_def : thm
    val down_continuous_def : thm
    val function_def : thm
    val gfp_def : thm
    val glb_def : thm
    val lfp_def : thm
    val lub_def : thm
    val monotonic_def : thm
    val pointwise_lift_def : thm
    val poset_def : thm
    val relation_def : thm
    val top_def : thm
    val up_continuous_def : thm
  
  (*  Theorems  *)
    val complete_bottom : thm
    val complete_down : thm
    val complete_pointwise : thm
    val complete_top : thm
    val complete_up : thm
    val gfp_unique : thm
    val glb_pred : thm
    val knaster_tarski : thm
    val knaster_tarski_gfp : thm
    val knaster_tarski_lfp : thm
    val lfp_unique : thm
    val lub_pred : thm
    val poset_antisym : thm
    val poset_nonempty : thm
    val poset_refl : thm
    val poset_trans : thm
  
  val poset_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [pair] Parent theory of "poset"
   
   [bottom_def]  Definition
      
      ⊢ ∀s r x. bottom (s,r) x ⇔ s x ∧ ∀y. s y ⇒ r x y
   
   [carrier_def]  Definition
      
      ⊢ ∀s r. carrier (s,r) = s
   
   [chain_def]  Definition
      
      ⊢ ∀s r c. chain (s,r) c ⇔ ∀x y. s x ∧ s y ∧ c x ∧ c y ⇒ r x y ∨ r y x
   
   [complete_def]  Definition
      
      ⊢ ∀p. complete p ⇔ ∀c. (∃x. lub p c x) ∧ ∃x. glb p c x
   
   [continuous_def]  Definition
      
      ⊢ ∀p f. continuous p f ⇔ up_continuous p f ∧ down_continuous p f
   
   [down_continuous_def]  Definition
      
      ⊢ ∀s r f.
            down_continuous (s,r) f ⇔
            ∀c x.
                chain (s,r) c ∧ glb (s,r) c x ⇒
                glb (s,r) (λy. ∃z. (s z ∧ c z) ∧ y = f z) (f x)
   
   [function_def]  Definition
      
      ⊢ ∀a b f. function a b f ⇔ ∀x. a x ⇒ b (f x)
   
   [gfp_def]  Definition
      
      ⊢ ∀s r f x.
            gfp (s,r) f x ⇔ s x ∧ f x = x ∧ ∀y. s y ∧ r y (f y) ⇒ r y x
   
   [glb_def]  Definition
      
      ⊢ ∀s r p x.
            glb (s,r) p x ⇔
            s x ∧ (∀y. s y ∧ p y ⇒ r x y) ∧
            ∀z. s z ∧ (∀y. s y ∧ p y ⇒ r z y) ⇒ r z x
   
   [lfp_def]  Definition
      
      ⊢ ∀s r f x.
            lfp (s,r) f x ⇔ s x ∧ f x = x ∧ ∀y. s y ∧ r (f y) y ⇒ r x y
   
   [lub_def]  Definition
      
      ⊢ ∀s r p x.
            lub (s,r) p x ⇔
            s x ∧ (∀y. s y ∧ p y ⇒ r y x) ∧
            ∀z. s z ∧ (∀y. s y ∧ p y ⇒ r y z) ⇒ r x z
   
   [monotonic_def]  Definition
      
      ⊢ ∀s r f. monotonic (s,r) f ⇔ ∀x y. s x ∧ s y ∧ r x y ⇒ r (f x) (f y)
   
   [pointwise_lift_def]  Definition
      
      ⊢ ∀t s r.
            pointwise_lift t (s,r) =
            (function t s,(λf g. ∀x. t x ⇒ r (f x) (g x)))
   
   [poset_def]  Definition
      
      ⊢ ∀s r.
            poset (s,r) ⇔
            (∃x. s x) ∧ (∀x. s x ⇒ r x x) ∧
            (∀x y. s x ∧ s y ∧ r x y ∧ r y x ⇒ x = y) ∧
            ∀x y z. s x ∧ s y ∧ s z ∧ r x y ∧ r y z ⇒ r x z
   
   [relation_def]  Definition
      
      ⊢ ∀s r. relation (s,r) = r
   
   [top_def]  Definition
      
      ⊢ ∀s r x. top (s,r) x ⇔ s x ∧ ∀y. s y ⇒ r y x
   
   [up_continuous_def]  Definition
      
      ⊢ ∀s r f.
            up_continuous (s,r) f ⇔
            ∀c x.
                chain (s,r) c ∧ lub (s,r) c x ⇒
                lub (s,r) (λy. ∃z. (s z ∧ c z) ∧ y = f z) (f x)
   
   [complete_bottom]  Theorem
      
      ⊢ ∀p. poset p ∧ complete p ⇒ ∃x. bottom p x
   
   [complete_down]  Theorem
      
      ⊢ ∀p c. complete p ⇒ ∃x. glb p c x
   
   [complete_pointwise]  Theorem
      
      ⊢ ∀p t. complete p ⇒ complete (pointwise_lift t p)
   
   [complete_top]  Theorem
      
      ⊢ ∀p. poset p ∧ complete p ⇒ ∃x. top p x
   
   [complete_up]  Theorem
      
      ⊢ ∀p c. complete p ⇒ ∃x. lub p c x
   
   [gfp_unique]  Theorem
      
      ⊢ ∀p f x x'. poset p ∧ gfp p f x ∧ gfp p f x' ⇒ x = x'
   
   [glb_pred]  Theorem
      
      ⊢ ∀s r p x. glb (s,r) (λj. s j ∧ p j) x ⇔ glb (s,r) p x
   
   [knaster_tarski]  Theorem
      
      ⊢ ∀p f.
            poset p ∧ complete p ∧ function (carrier p) (carrier p) f ∧
            monotonic p f ⇒
            (∃x. lfp p f x) ∧ ∃x. gfp p f x
   
   [knaster_tarski_gfp]  Theorem
      
      ⊢ ∀p f.
            poset p ∧ complete p ∧ function (carrier p) (carrier p) f ∧
            monotonic p f ⇒
            ∃x. gfp p f x
   
   [knaster_tarski_lfp]  Theorem
      
      ⊢ ∀p f.
            poset p ∧ complete p ∧ function (carrier p) (carrier p) f ∧
            monotonic p f ⇒
            ∃x. lfp p f x
   
   [lfp_unique]  Theorem
      
      ⊢ ∀p f x x'. poset p ∧ lfp p f x ∧ lfp p f x' ⇒ x = x'
   
   [lub_pred]  Theorem
      
      ⊢ ∀s r p x. lub (s,r) (λj. s j ∧ p j) x ⇔ lub (s,r) p x
   
   [poset_antisym]  Theorem
      
      ⊢ ∀s r x y. poset (s,r) ∧ s x ∧ s y ∧ r x y ∧ r y x ⇒ x = y
   
   [poset_nonempty]  Theorem
      
      ⊢ ∀s r. poset (s,r) ⇒ ∃x. s x
   
   [poset_refl]  Theorem
      
      ⊢ ∀s r x. poset (s,r) ∧ s x ⇒ r x x
   
   [poset_trans]  Theorem
      
      ⊢ ∀s r x y z. poset (s,r) ∧ s x ∧ s y ∧ s z ∧ r x y ∧ r y z ⇒ r x z
   
   
*)
end

HOL 4, Kananaskis-13