Structure util_probTheory

signature util_probTheory =
sig
  type thm = Thm.thm
  
  (*  Definitions  *)
    val lg_def : thm
    val logr_def : thm
    val minimal_def : thm
    val pair_def : thm
    val powr_def : thm
    val prod_sets_def : thm
  
  (*  Theorems  *)
    val EQ_T_IMP : thm
    val GBIGUNION_IMAGE : thm
    val HALF_CANCEL : thm
    val HALF_LT_1 : thm
    val HALF_POS : thm
    val INCREASING_SEQ : thm
    val INF_CLOSE : thm
    val INF_DEF_ALT : thm
    val INF_GREATER : thm
    val INF_LE : thm
    val IN_PAIR : thm
    val IN_PROD_SETS : thm
    val IN_o : thm
    val K_PARTIAL : thm
    val LE_INF : thm
    val LE_SUC : thm
    val LT_SUC : thm
    val MAX_LE_X : thm
    val MINIMAL_EQ : thm
    val MINIMAL_EQ_IMP : thm
    val MINIMAL_EXISTS : thm
    val MINIMAL_EXISTS0 : thm
    val MINIMAL_EXISTS_IMP : thm
    val MINIMAL_SUC : thm
    val MINIMAL_SUC_IMP : thm
    val NUM_2D_BIJ : thm
    val NUM_2D_BIJ_BIG_SQUARE : thm
    val NUM_2D_BIJ_INV : thm
    val NUM_2D_BIJ_NZ : thm
    val NUM_2D_BIJ_NZ_ALT : thm
    val NUM_2D_BIJ_NZ_ALT2 : thm
    val NUM_2D_BIJ_NZ_ALT2_INV : thm
    val NUM_2D_BIJ_NZ_ALT_INV : thm
    val NUM_2D_BIJ_NZ_INV : thm
    val NUM_2D_BIJ_SMALL_SQUARE : thm
    val ONE_MINUS_HALF : thm
    val PAIRED_BETA_THM : thm
    val PAIR_UNIV : thm
    val POS_SUMMABLE : thm
    val POW_HALF_MONO : thm
    val POW_HALF_POS : thm
    val POW_HALF_SER : thm
    val POW_HALF_SMALL : thm
    val PREIMAGE_REAL_COMPL1 : thm
    val PREIMAGE_REAL_COMPL2 : thm
    val PREIMAGE_REAL_COMPL3 : thm
    val PREIMAGE_REAL_COMPL4 : thm
    val REAL_LE_LT_MUL : thm
    val REAL_LT_LE_MUL : thm
    val REAL_MUL_IDEMPOT : thm
    val REAL_SUP_LE_X : thm
    val REAL_X_LE_SUP : thm
    val SEQ_SANDWICH : thm
    val SER_POS : thm
    val SER_POS_COMPARE : thm
    val SER_POS_MONO : thm
    val SUMINF_2D : thm
    val SUMINF_ADD : thm
    val SUMINF_POS : thm
    val SUMMABLE_LE : thm
    val SUMS_EQ : thm
    val SUMS_ZERO : thm
    val SUM_CONST_R : thm
    val SUM_LT : thm
    val SUM_PICK : thm
    val TRANSFORM_2D_NUM : thm
    val TRIANGLE_2D_NUM : thm
    val X_HALF_HALF : thm
    val X_LE_MAX : thm
    val finite_enumeration_of_sets_has_max_non_empty : thm
    val lg_1 : thm
    val lg_2 : thm
    val lg_inv : thm
    val lg_mul : thm
    val lg_nonzero : thm
    val lg_pow : thm
    val logr_1 : thm
    val logr_div : thm
    val logr_inv : thm
    val logr_mul : thm
    val neg_lg : thm
    val neg_logr : thm
  
  val util_prob_grammars : type_grammar.grammar * term_grammar.grammar
(*
   [real_sigma] Parent theory of "util_prob"
   
   [lg_def]  Definition
      
      ⊢ ∀x. lg x = logr 2 x
   
   [logr_def]  Definition
      
      ⊢ ∀a x. logr a x = ln x / ln a
   
   [minimal_def]  Definition
      
      ⊢ ∀p. minimal p = @n. p n ∧ ∀m. m < n ⇒ ¬p m
   
   [pair_def]  Definition
      
      ⊢ ∀X Y. pair X Y = (λ(x,y). x ∈ X ∧ y ∈ Y)
   
   [powr_def]  Definition
      
      ⊢ ∀x a. x powr a = exp (a * ln x)
   
   [prod_sets_def]  Definition
      
      ⊢ ∀a b. prod_sets a b = {s × t | s ∈ a ∧ t ∈ b}
   
   [EQ_T_IMP]  Theorem
      
      ⊢ ∀x. x ⇔ T ⇒ x
   
   [GBIGUNION_IMAGE]  Theorem
      
      ⊢ ∀s p n. {s | ∃n. p s n} = BIGUNION (IMAGE (λn. {s | p s n}) 𝕌(:γ))
   
   [HALF_CANCEL]  Theorem
      
      ⊢ 2 * (1 / 2) = 1
   
   [HALF_LT_1]  Theorem
      
      ⊢ 1 / 2 < 1
   
   [HALF_POS]  Theorem
      
      ⊢ 0 < 1 / 2
   
   [INCREASING_SEQ]  Theorem
      
      ⊢ ∀f l.
            (∀n. f n ≤ f (SUC n)) ∧ (∀n. f n ≤ l) ∧
            (∀e. 0 < e ⇒ ∃n. l < f n + e) ⇒
            f --> l
   
   [INF_CLOSE]  Theorem
      
      ⊢ ∀p e. (∃x. x ∈ p) ∧ 0 < e ⇒ ∃x. x ∈ p ∧ x < inf p + e
   
   [INF_DEF_ALT]  Theorem
      
      ⊢ ∀p. inf p = -sup (λr. -r ∈ p)
   
   [INF_GREATER]  Theorem
      
      ⊢ ∀p z. (∃x. x ∈ p) ∧ inf p < z ⇒ ∃x. x ∈ p ∧ x < z
   
   [INF_LE]  Theorem
      
      ⊢ ∀p r. (∃z. ∀x. x ∈ p ⇒ z ≤ x) ∧ (∃x. x ∈ p ∧ x ≤ r) ⇒ inf p ≤ r
   
   [IN_PAIR]  Theorem
      
      ⊢ ∀x X Y. x ∈ pair X Y ⇔ FST x ∈ X ∧ SND x ∈ Y
   
   [IN_PROD_SETS]  Theorem
      
      ⊢ ∀s a b. s ∈ prod_sets a b ⇔ ∃t u. (s = t × u) ∧ t ∈ a ∧ u ∈ b
   
   [IN_o]  Theorem
      
      ⊢ ∀x f s. x ∈ s ∘ f ⇔ f x ∈ s
   
   [K_PARTIAL]  Theorem
      
      ⊢ ∀x. K x = (λz. x)
   
   [LE_INF]  Theorem
      
      ⊢ ∀p r. (∃x. x ∈ p) ∧ (∀x. x ∈ p ⇒ r ≤ x) ⇒ r ≤ inf p
   
   [LE_SUC]  Theorem
      
      ⊢ ∀a b. a ≤ SUC b ⇔ a ≤ b ∨ (a = SUC b)
   
   [LT_SUC]  Theorem
      
      ⊢ ∀a b. a < SUC b ⇔ a < b ∨ (a = b)
   
   [MAX_LE_X]  Theorem
      
      ⊢ ∀m n k. MAX m n ≤ k ⇔ m ≤ k ∧ n ≤ k
   
   [MINIMAL_EQ]  Theorem
      
      ⊢ ∀p m. p m ∧ (m = minimal p) ⇔ p m ∧ ∀n. n < m ⇒ ¬p n
   
   [MINIMAL_EQ_IMP]  Theorem
      
      ⊢ ∀m p. p m ∧ (∀n. n < m ⇒ ¬p n) ⇒ (m = minimal p)
   
   [MINIMAL_EXISTS]  Theorem
      
      ⊢ ∀P. (∃n. P n) ⇔ P (minimal P) ∧ ∀n. n < minimal P ⇒ ¬P n
   
   [MINIMAL_EXISTS0]  Theorem
      
      ⊢ (∃n. P n) ⇔ ∃n. P n ∧ ∀m. m < n ⇒ ¬P m
   
   [MINIMAL_EXISTS_IMP]  Theorem
      
      ⊢ ∀P. (∃n. P n) ⇒ ∃m. P m ∧ ∀n. n < m ⇒ ¬P n
   
   [MINIMAL_SUC]  Theorem
      
      ⊢ ∀n p.
            (SUC n = minimal p) ∧ p (SUC n) ⇔
            ¬p 0 ∧ (n = minimal (p ∘ SUC)) ∧ p (SUC n)
   
   [MINIMAL_SUC_IMP]  Theorem
      
      ⊢ ∀n p.
            p (SUC n) ∧ ¬p 0 ∧ (n = minimal (p ∘ SUC)) ⇒
            (SUC n = minimal p)
   
   [NUM_2D_BIJ]  Theorem
      
      ⊢ ∃f. BIJ f (𝕌(:num) × 𝕌(:num)) 𝕌(:num)
   
   [NUM_2D_BIJ_BIG_SQUARE]  Theorem
      
      ⊢ ∀f N.
            BIJ f 𝕌(:num) (𝕌(:num) × 𝕌(:num)) ⇒
            ∃k. IMAGE f (count N) ⊆ count k × count k
   
   [NUM_2D_BIJ_INV]  Theorem
      
      ⊢ ∃f. BIJ f 𝕌(:num) (𝕌(:num) × 𝕌(:num))
   
   [NUM_2D_BIJ_NZ]  Theorem
      
      ⊢ ∃f. BIJ f (𝕌(:num) × (𝕌(:num) DIFF {0})) 𝕌(:num)
   
   [NUM_2D_BIJ_NZ_ALT]  Theorem
      
      ⊢ ∃f. BIJ f (𝕌(:num) × 𝕌(:num)) (𝕌(:num) DIFF {0})
   
   [NUM_2D_BIJ_NZ_ALT2]  Theorem
      
      ⊢ ∃f. BIJ f ((𝕌(:num) DIFF {0}) × (𝕌(:num) DIFF {0})) 𝕌(:num)
   
   [NUM_2D_BIJ_NZ_ALT2_INV]  Theorem
      
      ⊢ ∃f. BIJ f 𝕌(:num) ((𝕌(:num) DIFF {0}) × (𝕌(:num) DIFF {0}))
   
   [NUM_2D_BIJ_NZ_ALT_INV]  Theorem
      
      ⊢ ∃f. BIJ f (𝕌(:num) DIFF {0}) (𝕌(:num) × 𝕌(:num))
   
   [NUM_2D_BIJ_NZ_INV]  Theorem
      
      ⊢ ∃f. BIJ f 𝕌(:num) (𝕌(:num) × (𝕌(:num) DIFF {0}))
   
   [NUM_2D_BIJ_SMALL_SQUARE]  Theorem
      
      ⊢ ∀f k.
            BIJ f 𝕌(:num) (𝕌(:num) × 𝕌(:num)) ⇒
            ∃N. count k × count k ⊆ IMAGE f (count N)
   
   [ONE_MINUS_HALF]  Theorem
      
      ⊢ 1 − 1 / 2 = 1 / 2
   
   [PAIRED_BETA_THM]  Theorem
      
      ⊢ ∀f z. UNCURRY f z = f (FST z) (SND z)
   
   [PAIR_UNIV]  Theorem
      
      ⊢ pair 𝕌(:α) 𝕌(:β) = 𝕌(:α # β)
   
   [POS_SUMMABLE]  Theorem
      
      ⊢ ∀f. (∀n. 0 ≤ f n) ∧ (∃x. ∀n. sum (0,n) f ≤ x) ⇒ summable f
   
   [POW_HALF_MONO]  Theorem
      
      ⊢ ∀m n. m ≤ n ⇒ (1 / 2) pow n ≤ (1 / 2) pow m
   
   [POW_HALF_POS]  Theorem
      
      ⊢ ∀n. 0 < (1 / 2) pow n
   
   [POW_HALF_SER]  Theorem
      
      ⊢ (λn. (1 / 2) pow (n + 1)) sums 1
   
   [POW_HALF_SMALL]  Theorem
      
      ⊢ ∀e. 0 < e ⇒ ∃n. (1 / 2) pow n < e
   
   [PREIMAGE_REAL_COMPL1]  Theorem
      
      ⊢ ∀c. COMPL {x | c < x} = {x | x ≤ c}
   
   [PREIMAGE_REAL_COMPL2]  Theorem
      
      ⊢ ∀c. COMPL {x | c ≤ x} = {x | x < c}
   
   [PREIMAGE_REAL_COMPL3]  Theorem
      
      ⊢ ∀c. COMPL {x | x ≤ c} = {x | c < x}
   
   [PREIMAGE_REAL_COMPL4]  Theorem
      
      ⊢ ∀c. COMPL {x | x < c} = {x | c ≤ x}
   
   [REAL_LE_LT_MUL]  Theorem
      
      ⊢ ∀x y. 0 ≤ x ∧ 0 < y ⇒ 0 ≤ x * y
   
   [REAL_LT_LE_MUL]  Theorem
      
      ⊢ ∀x y. 0 < x ∧ 0 ≤ y ⇒ 0 ≤ x * y
   
   [REAL_MUL_IDEMPOT]  Theorem
      
      ⊢ ∀r. (r * r = r) ⇔ (r = 0) ∨ (r = 1)
   
   [REAL_SUP_LE_X]  Theorem
      
      ⊢ ∀P x. (∃r. P r) ∧ (∀r. P r ⇒ r ≤ x) ⇒ sup P ≤ x
   
   [REAL_X_LE_SUP]  Theorem
      
      ⊢ ∀P x.
            (∃r. P r) ∧ (∃z. ∀r. P r ⇒ r ≤ z) ∧ (∃r. P r ∧ x ≤ r) ⇒
            x ≤ sup P
   
   [SEQ_SANDWICH]  Theorem
      
      ⊢ ∀f g h l. f --> l ∧ h --> l ∧ (∀n. f n ≤ g n ∧ g n ≤ h n) ⇒ g --> l
   
   [SER_POS]  Theorem
      
      ⊢ ∀f. summable f ∧ (∀n. 0 ≤ f n) ⇒ 0 ≤ suminf f
   
   [SER_POS_COMPARE]  Theorem
      
      ⊢ ∀f g.
            (∀n. 0 ≤ f n) ∧ summable g ∧ (∀n. f n ≤ g n) ⇒
            summable f ∧ suminf f ≤ suminf g
   
   [SER_POS_MONO]  Theorem
      
      ⊢ ∀f. (∀n. 0 ≤ f n) ⇒ mono (λn. sum (0,n) f)
   
   [SUMINF_2D]  Theorem
      
      ⊢ ∀f g h.
            (∀m n. 0 ≤ f m n) ∧ (∀n. f n sums g n) ∧ summable g ∧
            BIJ h 𝕌(:num) (𝕌(:num) × 𝕌(:num)) ⇒
            UNCURRY f ∘ h sums suminf g
   
   [SUMINF_ADD]  Theorem
      
      ⊢ ∀f g.
            summable f ∧ summable g ⇒
            summable (λn. f n + g n) ∧
            (suminf f + suminf g = suminf (λn. f n + g n))
   
   [SUMINF_POS]  Theorem
      
      ⊢ ∀f. (∀n. 0 ≤ f n) ∧ summable f ⇒ 0 ≤ suminf f
   
   [SUMMABLE_LE]  Theorem
      
      ⊢ ∀f x. summable f ∧ (∀n. sum (0,n) f ≤ x) ⇒ suminf f ≤ x
   
   [SUMS_EQ]  Theorem
      
      ⊢ ∀f x. f sums x ⇔ summable f ∧ (suminf f = x)
   
   [SUMS_ZERO]  Theorem
      
      ⊢ K 0 sums 0
   
   [SUM_CONST_R]  Theorem
      
      ⊢ ∀n r. sum (0,n) (K r) = &n * r
   
   [SUM_LT]  Theorem
      
      ⊢ ∀f g m n.
            (∀r. m ≤ r ∧ r < n + m ⇒ f r < g r) ∧ 0 < n ⇒
            sum (m,n) f < sum (m,n) g
   
   [SUM_PICK]  Theorem
      
      ⊢ ∀n k x.
            sum (0,n) (λm. if m = k then x else 0) = if k < n then x else 0
   
   [TRANSFORM_2D_NUM]  Theorem
      
      ⊢ ∀P. (∀m n. P m n ⇒ P n m) ∧ (∀m n. P m (m + n)) ⇒ ∀m n. P m n
   
   [TRIANGLE_2D_NUM]  Theorem
      
      ⊢ ∀P. (∀d n. P n (d + n)) ⇒ ∀m n. m ≤ n ⇒ P m n
   
   [X_HALF_HALF]  Theorem
      
      ⊢ ∀x. 1 / 2 * x + 1 / 2 * x = x
   
   [X_LE_MAX]  Theorem
      
      ⊢ ∀m n k. k ≤ MAX m n ⇔ k ≤ m ∨ k ≤ n
   
   [finite_enumeration_of_sets_has_max_non_empty]  Theorem
      
      ⊢ ∀f s.
            FINITE s ∧ (∀x. f x ∈ s) ∧ (∀m n. m ≠ n ⇒ DISJOINT (f m) (f n)) ⇒
            ∃N. ∀n. n ≥ N ⇒ (f n = ∅)
   
   [lg_1]  Theorem
      
      ⊢ lg 1 = 0
   
   [lg_2]  Theorem
      
      ⊢ lg 2 = 1
   
   [lg_inv]  Theorem
      
      ⊢ ∀x. 0 < x ⇒ (lg x⁻¹ = -lg x)
   
   [lg_mul]  Theorem
      
      ⊢ ∀x y. 0 < x ∧ 0 < y ⇒ (lg (x * y) = lg x + lg y)
   
   [lg_nonzero]  Theorem
      
      ⊢ ∀x. x ≠ 0 ∧ 0 ≤ x ⇒ (lg x ≠ 0 ⇔ x ≠ 1)
   
   [lg_pow]  Theorem
      
      ⊢ ∀n. lg (2 pow n) = &n
   
   [logr_1]  Theorem
      
      ⊢ ∀b. logr b 1 = 0
   
   [logr_div]  Theorem
      
      ⊢ ∀b x y. 0 < x ∧ 0 < y ⇒ (logr b (x / y) = logr b x − logr b y)
   
   [logr_inv]  Theorem
      
      ⊢ ∀b x. 0 < x ⇒ (logr b x⁻¹ = -logr b x)
   
   [logr_mul]  Theorem
      
      ⊢ ∀b x y. 0 < x ∧ 0 < y ⇒ (logr b (x * y) = logr b x + logr b y)
   
   [neg_lg]  Theorem
      
      ⊢ ∀x. 0 < x ⇒ (-lg x = lg x⁻¹)
   
   [neg_logr]  Theorem
      
      ⊢ ∀b x. 0 < x ⇒ (-logr b x = logr b x⁻¹)
   
   
*)
end

HOL 4, Kananaskis-11