Structure sumTheory

signature sumTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val INL_DEF : thm
    val INR_DEF : thm
    val ISL : thm
    val ISR : thm
    val IS_SUM_REP : thm
    val OUTL : thm
    val OUTR : thm
    val SUM_ALL_def : thm
    val SUM_MAP_def : thm
    val sum_ISO_DEF : thm
    val sum_TY_DEF : thm
    val sum_case_def : thm
  
  (*  Theorems  *)
    val EXISTS_SUM : thm
    val FORALL_SUM : thm
    val INL : thm
    val INL_11 : thm
    val INR : thm
    val INR_11 : thm
    val INR_INL_11 : thm
    val INR_neq_INL : thm
    val ISL_OR_ISR : thm
    val NOT_ISL_ISR : thm
    val NOT_ISR_ISL : thm
    val SUM_ALL_CONG : thm
    val SUM_ALL_MONO : thm
    val SUM_MAP : thm
    val SUM_MAP_CASE : thm
    val SUM_MAP_I : thm
    val SUM_MAP_o : thm
    val cond_sum_expand : thm
    val datatype_sum : thm
    val sum_Axiom : thm
    val sum_CASES : thm
    val sum_INDUCT : thm
    val sum_axiom : thm
    val sum_case_cong : thm
    val sum_distinct : thm
    val sum_distinct1 : thm
  
  val sum_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [pair] Parent theory of "sum"
   
   [INL_DEF]  Definition
      
      ⊢ ∀e. INL e = ABS_sum (λb x y. x = e ∧ b)
   
   [INR_DEF]  Definition
      
      ⊢ ∀e. INR e = ABS_sum (λb x y. y = e ∧ ¬b)
   
   [ISL]  Definition
      
      ⊢ (∀x. ISL (INL x)) ∧ ∀y. ¬ISL (INR y)
   
   [ISR]  Definition
      
      ⊢ (∀x. ISR (INR x)) ∧ ∀y. ¬ISR (INL y)
   
   [IS_SUM_REP]  Definition
      
      ⊢ ∀f. IS_SUM_REP f ⇔
            ∃v1 v2. f = (λb x y. x = v1 ∧ b) ∨ f = (λb x y. y = v2 ∧ ¬b)
   
   [OUTL]  Definition
      
      ⊢ ∀x. OUTL (INL x) = x
   
   [OUTR]  Definition
      
      ⊢ ∀x. OUTR (INR x) = x
   
   [SUM_ALL_def]  Definition
      
      ⊢ (∀P Q x. SUM_ALL P Q (INL x) ⇔ P x) ∧
        ∀P Q y. SUM_ALL P Q (INR y) ⇔ Q y
   
   [SUM_MAP_def]  Definition
      
      ⊢ (∀f g a. SUM_MAP f g (INL a) = INL (f a)) ∧
        ∀f g b. SUM_MAP f g (INR b) = INR (g b)
   
   [sum_ISO_DEF]  Definition
      
      ⊢ (∀a. ABS_sum (REP_sum a) = a) ∧
        ∀r. IS_SUM_REP r ⇔ REP_sum (ABS_sum r) = r
   
   [sum_TY_DEF]  Definition
      
      ⊢ ∃rep. TYPE_DEFINITION IS_SUM_REP rep
   
   [sum_case_def]  Definition
      
      ⊢ (∀x f f1. sum_CASE (INL x) f f1 = f x) ∧
        ∀y f f1. sum_CASE (INR y) f f1 = f1 y
   
   [EXISTS_SUM]  Theorem
      
      ⊢ ∀P. (∃s. P s) ⇔ (∃x. P (INL x)) ∨ ∃y. P (INR y)
   
   [FORALL_SUM]  Theorem
      
      ⊢ (∀s. P s) ⇔ (∀x. P (INL x)) ∧ ∀y. P (INR y)
   
   [INL]  Theorem
      
      ⊢ ∀x. ISL x ⇒ INL (OUTL x) = x
   
   [INL_11]  Theorem
      
      ⊢ INL x = INL y ⇔ x = y
   
   [INR]  Theorem
      
      ⊢ ∀x. ISR x ⇒ INR (OUTR x) = x
   
   [INR_11]  Theorem
      
      ⊢ INR x = INR y ⇔ x = y
   
   [INR_INL_11]  Theorem
      
      ⊢ (∀y x. INL x = INL y ⇔ x = y) ∧ ∀y x. INR x = INR y ⇔ x = y
   
   [INR_neq_INL]  Theorem
      
      ⊢ ∀v1 v2. INR v2 ≠ INL v1
   
   [ISL_OR_ISR]  Theorem
      
      ⊢ ∀x. ISL x ∨ ISR x
   
   [NOT_ISL_ISR]  Theorem
      
      ⊢ ∀x. ¬ISL x ⇔ ISR x
   
   [NOT_ISR_ISL]  Theorem
      
      ⊢ ∀x. ¬ISR x ⇔ ISL x
   
   [SUM_ALL_CONG]  Theorem
      
      ⊢ ∀s s' P P' Q Q'.
          s = s' ∧ (∀a. s' = INL a ⇒ (P a ⇔ P' a)) ∧
          (∀b. s' = INR b ⇒ (Q b ⇔ Q' b)) ⇒
          (SUM_ALL P Q s ⇔ SUM_ALL P' Q' s')
   
   [SUM_ALL_MONO]  Theorem
      
      ⊢ (∀x. P x ⇒ P' x) ∧ (∀y. Q y ⇒ Q' y) ⇒
        SUM_ALL P Q s ⇒
        SUM_ALL P' Q' s
   
   [SUM_MAP]  Theorem
      
      ⊢ ∀f g z.
          SUM_MAP f g z =
          if ISL z then INL (f (OUTL z)) else INR (g (OUTR z))
   
   [SUM_MAP_CASE]  Theorem
      
      ⊢ ∀f g z. SUM_MAP f g z = sum_CASE z (INL ∘ f) (INR ∘ g)
   
   [SUM_MAP_I]  Theorem
      
      ⊢ SUM_MAP I I = I
   
   [SUM_MAP_o]  Theorem
      
      ⊢ SUM_MAP f g ∘ SUM_MAP h k = SUM_MAP (f ∘ h) (g ∘ k)
   
   [cond_sum_expand]  Theorem
      
      ⊢ (∀x y z. (if P then INR x else INL y) = INR z ⇔ P ∧ z = x) ∧
        (∀x y z. (if P then INR x else INL y) = INL z ⇔ ¬P ∧ z = y) ∧
        (∀x y z. (if P then INL x else INR y) = INL z ⇔ P ∧ z = x) ∧
        ∀x y z. (if P then INL x else INR y) = INR z ⇔ ¬P ∧ z = y
   
   [datatype_sum]  Theorem
      
      ⊢ DATATYPE (sum INL INR)
   
   [sum_Axiom]  Theorem
      
      ⊢ ∀f g. ∃h. (∀x. h (INL x) = f x) ∧ ∀y. h (INR y) = g y
   
   [sum_CASES]  Theorem
      
      ⊢ ∀ss. (∃x. ss = INL x) ∨ ∃y. ss = INR y
   
   [sum_INDUCT]  Theorem
      
      ⊢ ∀P. (∀x. P (INL x)) ∧ (∀y. P (INR y)) ⇒ ∀s. P s
   
   [sum_axiom]  Theorem
      
      ⊢ ∀f g. ∃!h. h ∘ INL = f ∧ h ∘ INR = g
   
   [sum_case_cong]  Theorem
      
      ⊢ ∀M M' f f1.
          M = M' ∧ (∀x. M' = INL x ⇒ f x = f' x) ∧
          (∀y. M' = INR y ⇒ f1 y = f1' y) ⇒
          sum_CASE M f f1 = sum_CASE M' f' f1'
   
   [sum_distinct]  Theorem
      
      ⊢ ∀x y. INL x ≠ INR y
   
   [sum_distinct1]  Theorem
      
      ⊢ ∀x y. INR y ≠ INL x
   
   
*)
end

HOL 4, Kananaskis-14