Structure realaxTheory
signature realaxTheory =
sig
type thm = Thm.thm
(* Definitions *)
val hreal_of_real : thm
val hreal_of_treal : thm
val real_0 : thm
val real_1 : thm
val real_ABS_def : thm
val real_REP_def : thm
val real_TY_DEF : thm
val real_add : thm
val real_bijections : thm
val real_inv : thm
val real_lt : thm
val real_mul : thm
val real_neg : thm
val real_of_hreal : thm
val treal_0 : thm
val treal_1 : thm
val treal_add : thm
val treal_eq : thm
val treal_inv : thm
val treal_lt : thm
val treal_mul : thm
val treal_neg : thm
val treal_of_hreal : thm
(* Theorems *)
val HREAL_EQ_ADDL : thm
val HREAL_EQ_ADDR : thm
val HREAL_EQ_LADD : thm
val HREAL_LT_ADD2 : thm
val HREAL_LT_ADDL : thm
val HREAL_LT_ADDR : thm
val HREAL_LT_GT : thm
val HREAL_LT_LADD : thm
val HREAL_LT_NE : thm
val HREAL_LT_REFL : thm
val HREAL_RDISTRIB : thm
val REAL_10 : thm
val REAL_ADD_ASSOC : thm
val REAL_ADD_LID : thm
val REAL_ADD_LINV : thm
val REAL_ADD_SYM : thm
val REAL_INV_0 : thm
val REAL_ISO_EQ : thm
val REAL_LDISTRIB : thm
val REAL_LT_IADD : thm
val REAL_LT_MUL : thm
val REAL_LT_REFL : thm
val REAL_LT_TOTAL : thm
val REAL_LT_TRANS : thm
val REAL_MUL_ASSOC : thm
val REAL_MUL_LID : thm
val REAL_MUL_LINV : thm
val REAL_MUL_SYM : thm
val REAL_POS : thm
val REAL_SUP_ALLPOS : thm
val SUP_ALLPOS_LEMMA1 : thm
val SUP_ALLPOS_LEMMA2 : thm
val SUP_ALLPOS_LEMMA3 : thm
val SUP_ALLPOS_LEMMA4 : thm
val TREAL_10 : thm
val TREAL_ADD_ASSOC : thm
val TREAL_ADD_LID : thm
val TREAL_ADD_LINV : thm
val TREAL_ADD_SYM : thm
val TREAL_ADD_WELLDEF : thm
val TREAL_ADD_WELLDEFR : thm
val TREAL_BIJ : thm
val TREAL_BIJ_WELLDEF : thm
val TREAL_EQ_AP : thm
val TREAL_EQ_EQUIV : thm
val TREAL_EQ_REFL : thm
val TREAL_EQ_SYM : thm
val TREAL_EQ_TRANS : thm
val TREAL_INV_0 : thm
val TREAL_INV_WELLDEF : thm
val TREAL_ISO : thm
val TREAL_LDISTRIB : thm
val TREAL_LT_ADD : thm
val TREAL_LT_MUL : thm
val TREAL_LT_REFL : thm
val TREAL_LT_TOTAL : thm
val TREAL_LT_TRANS : thm
val TREAL_LT_WELLDEF : thm
val TREAL_LT_WELLDEFL : thm
val TREAL_LT_WELLDEFR : thm
val TREAL_MUL_ASSOC : thm
val TREAL_MUL_LID : thm
val TREAL_MUL_LINV : thm
val TREAL_MUL_SYM : thm
val TREAL_MUL_WELLDEF : thm
val TREAL_MUL_WELLDEFR : thm
val TREAL_NEG_WELLDEF : thm
val real_ABS_REP_CLASS : thm
val real_QUOTIENT : thm
val realax_grammars : type_grammar.grammar * term_grammar.grammar
(*
[hreal] Parent theory of "realax"
[hreal_of_real] Definition
⊢ ∀T1. hreal_of_real T1 = hreal_of_treal (real_REP T1)
[hreal_of_treal] Definition
⊢ ∀x y. hreal_of_treal (x,y) = @d. x = y hreal_add d
[real_0] Definition
⊢ real_0 = real_ABS treal_0
[real_1] Definition
⊢ real_1 = real_ABS treal_1
[real_ABS_def] Definition
⊢ ∀r. real_ABS r = real_ABS_CLASS ($treal_eq r)
[real_REP_def] Definition
⊢ ∀a. real_REP a = $@ (real_REP_CLASS a)
[real_TY_DEF] Definition
⊢ ∃rep.
TYPE_DEFINITION (λc. ∃r. r treal_eq r ∧ (c = $treal_eq r)) rep
[real_add] Definition
⊢ ∀T1 T2. T1 + T2 = real_ABS (real_REP T1 treal_add real_REP T2)
[real_bijections] Definition
⊢ (∀a. real_ABS_CLASS (real_REP_CLASS a) = a) ∧
∀r. (λc. ∃r. r treal_eq r ∧ (c = $treal_eq r)) r ⇔
(real_REP_CLASS (real_ABS_CLASS r) = r)
[real_inv] Definition
⊢ ∀T1. T1⁻¹ = real_ABS (treal_inv (real_REP T1))
[real_lt] Definition
⊢ ∀T1 T2. T1 < T2 ⇔ real_REP T1 treal_lt real_REP T2
[real_mul] Definition
⊢ ∀T1 T2. T1 * T2 = real_ABS (real_REP T1 treal_mul real_REP T2)
[real_neg] Definition
⊢ ∀T1. -T1 = real_ABS (treal_neg (real_REP T1))
[real_of_hreal] Definition
⊢ ∀T1. real_of_hreal T1 = real_ABS (treal_of_hreal T1)
[treal_0] Definition
⊢ treal_0 = (hreal_1,hreal_1)
[treal_1] Definition
⊢ treal_1 = (hreal_1 hreal_add hreal_1,hreal_1)
[treal_add] Definition
⊢ ∀x1 y1 x2 y2.
(x1,y1) treal_add (x2,y2) = (x1 hreal_add x2,y1 hreal_add y2)
[treal_eq] Definition
⊢ ∀x1 y1 x2 y2.
(x1,y1) treal_eq (x2,y2) ⇔ (x1 hreal_add y2 = x2 hreal_add y1)
[treal_inv] Definition
⊢ ∀x y.
treal_inv (x,y) =
if x = y then treal_0
else if y hreal_lt x then
(hreal_inv (x hreal_sub y) hreal_add hreal_1,hreal_1)
else (hreal_1,hreal_inv (y hreal_sub x) hreal_add hreal_1)
[treal_lt] Definition
⊢ ∀x1 y1 x2 y2.
(x1,y1) treal_lt (x2,y2) ⇔
x1 hreal_add y2 hreal_lt x2 hreal_add y1
[treal_mul] Definition
⊢ ∀x1 y1 x2 y2.
(x1,y1) treal_mul (x2,y2) =
(x1 hreal_mul x2 hreal_add y1 hreal_mul y2,
x1 hreal_mul y2 hreal_add y1 hreal_mul x2)
[treal_neg] Definition
⊢ ∀x y. treal_neg (x,y) = (y,x)
[treal_of_hreal] Definition
⊢ ∀x. treal_of_hreal x = (x hreal_add hreal_1,hreal_1)
[HREAL_EQ_ADDL] Theorem
⊢ ∀x y. x ≠ x hreal_add y
[HREAL_EQ_ADDR] Theorem
⊢ ∀x y. x hreal_add y ≠ x
[HREAL_EQ_LADD] Theorem
⊢ ∀x y z. (x hreal_add y = x hreal_add z) ⇔ (y = z)
[HREAL_LT_ADD2] Theorem
⊢ ∀x1 x2 y1 y2.
x1 hreal_lt y1 ∧ x2 hreal_lt y2 ⇒
x1 hreal_add x2 hreal_lt y1 hreal_add y2
[HREAL_LT_ADDL] Theorem
⊢ ∀x y. x hreal_lt x hreal_add y
[HREAL_LT_ADDR] Theorem
⊢ ∀x y. ¬(x hreal_add y hreal_lt x)
[HREAL_LT_GT] Theorem
⊢ ∀x y. x hreal_lt y ⇒ ¬(y hreal_lt x)
[HREAL_LT_LADD] Theorem
⊢ ∀x y z. x hreal_add y hreal_lt x hreal_add z ⇔ y hreal_lt z
[HREAL_LT_NE] Theorem
⊢ ∀x y. x hreal_lt y ⇒ x ≠ y
[HREAL_LT_REFL] Theorem
⊢ ∀x. ¬(x hreal_lt x)
[HREAL_RDISTRIB] Theorem
⊢ ∀x y z.
(x hreal_add y) hreal_mul z =
x hreal_mul z hreal_add y hreal_mul z
[REAL_10] Theorem
⊢ real_1 ≠ real_0
[REAL_ADD_ASSOC] Theorem
⊢ ∀x y z. x + (y + z) = x + y + z
[REAL_ADD_LID] Theorem
⊢ ∀x. real_0 + x = x
[REAL_ADD_LINV] Theorem
⊢ ∀x. -x + x = real_0
[REAL_ADD_SYM] Theorem
⊢ ∀x y. x + y = y + x
[REAL_INV_0] Theorem
⊢ real_0⁻¹ = real_0
[REAL_ISO_EQ] Theorem
⊢ ∀h i. h hreal_lt i ⇔ real_of_hreal h < real_of_hreal i
[REAL_LDISTRIB] Theorem
⊢ ∀x y z. x * (y + z) = x * y + x * z
[REAL_LT_IADD] Theorem
⊢ ∀x y z. y < z ⇒ x + y < x + z
[REAL_LT_MUL] Theorem
⊢ ∀x y. real_0 < x ∧ real_0 < y ⇒ real_0 < x * y
[REAL_LT_REFL] Theorem
⊢ ∀x. ¬(x < x)
[REAL_LT_TOTAL] Theorem
⊢ ∀x y. (x = y) ∨ x < y ∨ y < x
[REAL_LT_TRANS] Theorem
⊢ ∀x y z. x < y ∧ y < z ⇒ x < z
[REAL_MUL_ASSOC] Theorem
⊢ ∀x y z. x * (y * z) = x * y * z
[REAL_MUL_LID] Theorem
⊢ ∀x. real_1 * x = x
[REAL_MUL_LINV] Theorem
⊢ ∀x. x ≠ real_0 ⇒ (x⁻¹ * x = real_1)
[REAL_MUL_SYM] Theorem
⊢ ∀x y. x * y = y * x
[REAL_POS] Theorem
⊢ ∀X. real_0 < real_of_hreal X
[REAL_SUP_ALLPOS] Theorem
⊢ ∀P. (∀x. P x ⇒ real_0 < x) ∧ (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x < z) ⇒
∃s. ∀y. (∃x. P x ∧ y < x) ⇔ y < s
[SUP_ALLPOS_LEMMA1] Theorem
⊢ ∀P y.
(∀x. P x ⇒ real_0 < x) ⇒
((∃x. P x ∧ y < x) ⇔
∃X. P (real_of_hreal X) ∧ y < real_of_hreal X)
[SUP_ALLPOS_LEMMA2] Theorem
⊢ ∀P X. P (real_of_hreal X) ⇔ (λh. P (real_of_hreal h)) X
[SUP_ALLPOS_LEMMA3] Theorem
⊢ ∀P. (∀x. P x ⇒ real_0 < x) ∧ (∃x. P x) ∧ (∃z. ∀x. P x ⇒ x < z) ⇒
(∃X. (λh. P (real_of_hreal h)) X) ∧
∃Y. ∀X. (λh. P (real_of_hreal h)) X ⇒ X hreal_lt Y
[SUP_ALLPOS_LEMMA4] Theorem
⊢ ∀y. ¬(real_0 < y) ⇒ ∀x. y < real_of_hreal x
[TREAL_10] Theorem
⊢ ¬(treal_1 treal_eq treal_0)
[TREAL_ADD_ASSOC] Theorem
⊢ ∀x y z. x treal_add (y treal_add z) = x treal_add y treal_add z
[TREAL_ADD_LID] Theorem
⊢ ∀x. treal_0 treal_add x treal_eq x
[TREAL_ADD_LINV] Theorem
⊢ ∀x. treal_neg x treal_add x treal_eq treal_0
[TREAL_ADD_SYM] Theorem
⊢ ∀x y. x treal_add y = y treal_add x
[TREAL_ADD_WELLDEF] Theorem
⊢ ∀x1 x2 y1 y2.
x1 treal_eq x2 ∧ y1 treal_eq y2 ⇒
x1 treal_add y1 treal_eq x2 treal_add y2
[TREAL_ADD_WELLDEFR] Theorem
⊢ ∀x1 x2 y. x1 treal_eq x2 ⇒ x1 treal_add y treal_eq x2 treal_add y
[TREAL_BIJ] Theorem
⊢ (∀h. hreal_of_treal (treal_of_hreal h) = h) ∧
∀r. treal_0 treal_lt r ⇔
treal_of_hreal (hreal_of_treal r) treal_eq r
[TREAL_BIJ_WELLDEF] Theorem
⊢ ∀h i. h treal_eq i ⇒ (hreal_of_treal h = hreal_of_treal i)
[TREAL_EQ_AP] Theorem
⊢ ∀p q. (p = q) ⇒ p treal_eq q
[TREAL_EQ_EQUIV] Theorem
⊢ ∀p q. p treal_eq q ⇔ ($treal_eq p = $treal_eq q)
[TREAL_EQ_REFL] Theorem
⊢ ∀x. x treal_eq x
[TREAL_EQ_SYM] Theorem
⊢ ∀x y. x treal_eq y ⇔ y treal_eq x
[TREAL_EQ_TRANS] Theorem
⊢ ∀x y z. x treal_eq y ∧ y treal_eq z ⇒ x treal_eq z
[TREAL_INV_0] Theorem
⊢ treal_inv treal_0 treal_eq treal_0
[TREAL_INV_WELLDEF] Theorem
⊢ ∀x1 x2. x1 treal_eq x2 ⇒ treal_inv x1 treal_eq treal_inv x2
[TREAL_ISO] Theorem
⊢ ∀h i. h hreal_lt i ⇒ treal_of_hreal h treal_lt treal_of_hreal i
[TREAL_LDISTRIB] Theorem
⊢ ∀x y z.
x treal_mul (y treal_add z) =
x treal_mul y treal_add x treal_mul z
[TREAL_LT_ADD] Theorem
⊢ ∀x y z. y treal_lt z ⇒ x treal_add y treal_lt x treal_add z
[TREAL_LT_MUL] Theorem
⊢ ∀x y.
treal_0 treal_lt x ∧ treal_0 treal_lt y ⇒
treal_0 treal_lt x treal_mul y
[TREAL_LT_REFL] Theorem
⊢ ∀x. ¬(x treal_lt x)
[TREAL_LT_TOTAL] Theorem
⊢ ∀x y. x treal_eq y ∨ x treal_lt y ∨ y treal_lt x
[TREAL_LT_TRANS] Theorem
⊢ ∀x y z. x treal_lt y ∧ y treal_lt z ⇒ x treal_lt z
[TREAL_LT_WELLDEF] Theorem
⊢ ∀x1 x2 y1 y2.
x1 treal_eq x2 ∧ y1 treal_eq y2 ⇒
(x1 treal_lt y1 ⇔ x2 treal_lt y2)
[TREAL_LT_WELLDEFL] Theorem
⊢ ∀x y1 y2. y1 treal_eq y2 ⇒ (x treal_lt y1 ⇔ x treal_lt y2)
[TREAL_LT_WELLDEFR] Theorem
⊢ ∀x1 x2 y. x1 treal_eq x2 ⇒ (x1 treal_lt y ⇔ x2 treal_lt y)
[TREAL_MUL_ASSOC] Theorem
⊢ ∀x y z. x treal_mul (y treal_mul z) = x treal_mul y treal_mul z
[TREAL_MUL_LID] Theorem
⊢ ∀x. treal_1 treal_mul x treal_eq x
[TREAL_MUL_LINV] Theorem
⊢ ∀x. ¬(x treal_eq treal_0) ⇒
treal_inv x treal_mul x treal_eq treal_1
[TREAL_MUL_SYM] Theorem
⊢ ∀x y. x treal_mul y = y treal_mul x
[TREAL_MUL_WELLDEF] Theorem
⊢ ∀x1 x2 y1 y2.
x1 treal_eq x2 ∧ y1 treal_eq y2 ⇒
x1 treal_mul y1 treal_eq x2 treal_mul y2
[TREAL_MUL_WELLDEFR] Theorem
⊢ ∀x1 x2 y. x1 treal_eq x2 ⇒ x1 treal_mul y treal_eq x2 treal_mul y
[TREAL_NEG_WELLDEF] Theorem
⊢ ∀x1 x2. x1 treal_eq x2 ⇒ treal_neg x1 treal_eq treal_neg x2
[real_ABS_REP_CLASS] Theorem
⊢ (∀a. real_ABS_CLASS (real_REP_CLASS a) = a) ∧
∀c. (∃r. r treal_eq r ∧ (c = $treal_eq r)) ⇔
(real_REP_CLASS (real_ABS_CLASS c) = c)
[real_QUOTIENT] Theorem
⊢ QUOTIENT $treal_eq real_ABS real_REP
*)
end
HOL 4, Kananaskis-14