- HAS_SIZE_def
-
|- ∀s n. s HAS_SIZE n ⇔ FINITE s ∧ (CARD s = n)
- finite_image_TY_DEF
-
|- ∃rep. TYPE_DEFINITION (λx. (x = ARB) ∨ FINITE 𝕌(:α)) rep
- finite_image_tybij
-
|- (∀a. mk_finite_image (dest_finite_image a) = a) ∧
∀r.
(λx. (x = ARB) ∨ FINITE 𝕌(:α)) r ⇔
(dest_finite_image (mk_finite_image r) = r)
- dimindex_def
-
|- dimindex (:α) = if FINITE 𝕌(:α) then CARD 𝕌(:α) else 1
- finite_index_def
-
|- finite_index = @f. ∀x. ∃!n. n < dimindex (:α) ∧ (f n = x)
- cart_TY_DEF
-
|- ∃rep. TYPE_DEFINITION (λf. T) rep
- cart_tybij
-
|- (∀a. mk_cart (dest_cart a) = a) ∧
∀r. (λf. T) r ⇔ (dest_cart (mk_cart r) = r)
- fcp_index
-
|- ∀x i. x ' i = dest_cart x (finite_index i)
- fcp_case_def
-
|- ∀h f. fcp_CASE (mk_cart h) f = f h
- FCP
-
|- $FCP = (λg. @f. ∀i. i < dimindex (:β) ⇒ (f ' i = g i))
- bit0_TY_DEF
-
|- ∃rep.
TYPE_DEFINITION
(λa0.
∀'bit0' .
(∀a0.
(∃a. a0 = (λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM)) a) ∨
(∃a.
a0 =
(λa. ind_type$CONSTR (SUC 0) a (λn. ind_type$BOTTOM)) a) ⇒
'bit0' a0) ⇒
'bit0' a0) rep
- bit0_case_def
-
|- (∀a f f1. bit0_CASE (BIT0A a) f f1 = f a) ∧
∀a f f1. bit0_CASE (BIT0B a) f f1 = f1 a
- bit0_size_def
-
|- (∀f a. bit0_size f (BIT0A a) = 1 + f a) ∧
∀f a. bit0_size f (BIT0B a) = 1 + f a
- bit1_TY_DEF
-
|- ∃rep.
TYPE_DEFINITION
(λa0.
∀'bit1' .
(∀a0.
(∃a. a0 = (λa. ind_type$CONSTR 0 a (λn. ind_type$BOTTOM)) a) ∨
(∃a.
a0 =
(λa. ind_type$CONSTR (SUC 0) a (λn. ind_type$BOTTOM)) a) ∨
(a0 =
ind_type$CONSTR (SUC (SUC 0)) ARB (λn. ind_type$BOTTOM)) ⇒
'bit1' a0) ⇒
'bit1' a0) rep
- bit1_case_def
-
|- (∀a f f1 v. bit1_CASE (BIT1A a) f f1 v = f a) ∧
(∀a f f1 v. bit1_CASE (BIT1B a) f f1 v = f1 a) ∧
∀f f1 v. bit1_CASE BIT1C f f1 v = v
- bit1_size_def
-
|- (∀f a. bit1_size f (BIT1A a) = 1 + f a) ∧
(∀f a. bit1_size f (BIT1B a) = 1 + f a) ∧ ∀f. bit1_size f BIT1C = 0
- FCP_UPDATE_def
-
|- ∀a b. a :+ b = (λm. FCP c. if a = c then b else m ' c)
- FCP_HD_def
-
|- ∀v. FCP_HD v = v ' 0
- FCP_TL_def
-
|- ∀v. FCP_TL v = FCP i. v ' (SUC i)
- FCP_CONS_def
-
|- ∀h v. FCP_CONS h v = (0 :+ h) (FCP i. v ' (PRE i))
- FCP_MAP_def
-
|- ∀f v. FCP_MAP f v = FCP i. f (v ' i)
- FCP_EXISTS_def
-
|- ∀P v. FCP_EXISTS P v ⇔ ∃i. i < dimindex (:α) ∧ P (v ' i)
- FCP_EVERY_def
-
|- ∀P v. FCP_EVERY P v ⇔ ∀i. dimindex (:α) ≤ i ∨ P (v ' i)
- FCP_CONCAT_def
-
|- ∀a b.
FCP_CONCAT a b =
FCP i. if i < dimindex (:γ) then b ' i else a ' (i − dimindex (:γ))
- FCP_ZIP_def
-
|- ∀a b. FCP_ZIP a b = FCP i. (a ' i,b ' i)
- V2L_def
-
|- ∀v. V2L v = GENLIST ($' v) (dimindex (:β))
- L2V_def
-
|- ∀L. L2V L = FCP i. EL i L
- FCP_FOLD_def
-
|- ∀f i v. FCP_FOLD f i v = FOLDL f i (V2L v)
- NOT_FINITE_IMP_dimindex_1
-
|- INFINITE 𝕌(:α) ⇒ (dimindex (:α) = 1)
- DIMINDEX_GE_1
-
|- 1 ≤ dimindex (:α)
- fcp_Axiom
-
|- ∀f. ∃g. ∀h. g (mk_cart h) = f h
- fcp_ind
-
|- ∀P. (∀f. P (mk_cart f)) ⇒ ∀a. P a
- CART_EQ
-
|- ∀x y. (x = y) ⇔ ∀i. i < dimindex (:β) ⇒ (x ' i = y ' i)
- FCP_BETA
-
|- ∀i. i < dimindex (:β) ⇒ ($FCP g ' i = g i)
- FCP_UNIQUE
-
|- ∀f g. (∀i. i < dimindex (:β) ⇒ (f ' i = g i)) ⇔ ($FCP g = f)
- FCP_ETA
-
|- ∀g. (FCP i. g ' i) = g
- card_dimindex
-
|- FINITE 𝕌(:α) ⇒ (CARD 𝕌(:α) = dimindex (:α))
- index_sum
-
|- dimindex (:α + β) =
if FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) then dimindex (:α) + dimindex (:β) else 1
- finite_sum
-
|- FINITE 𝕌(:α + β) ⇔ FINITE 𝕌(:α) ∧ FINITE 𝕌(:β)
- datatype_bit0
-
|- DATATYPE (bit0 BIT0A BIT0B)
- bit0_11
-
|- (∀a a'. (BIT0A a = BIT0A a') ⇔ (a = a')) ∧
∀a a'. (BIT0B a = BIT0B a') ⇔ (a = a')
- bit0_distinct
-
|- ∀a' a. BIT0A a ≠ BIT0B a'
- bit0_case_cong
-
|- ∀M M' f f1.
(M = M') ∧ (∀a. (M' = BIT0A a) ⇒ (f a = f' a)) ∧
(∀a. (M' = BIT0B a) ⇒ (f1 a = f1' a)) ⇒
(bit0_CASE M f f1 = bit0_CASE M' f' f1')
- bit0_nchotomy
-
|- ∀bb. (∃a. bb = BIT0A a) ∨ ∃a. bb = BIT0B a
- bit0_Axiom
-
|- ∀f0 f1. ∃fn. (∀a. fn (BIT0A a) = f0 a) ∧ ∀a. fn (BIT0B a) = f1 a
- bit0_induction
-
|- ∀P. (∀a. P (BIT0A a)) ∧ (∀a. P (BIT0B a)) ⇒ ∀b. P b
- index_bit0
-
|- dimindex (:α bit0) = if FINITE 𝕌(:α) then 2 * dimindex (:α) else 1
- finite_bit0
-
|- FINITE 𝕌(:α bit0) ⇔ FINITE 𝕌(:α)
- datatype_bit1
-
|- DATATYPE (bit1 BIT1A BIT1B BIT1C)
- bit1_11
-
|- (∀a a'. (BIT1A a = BIT1A a') ⇔ (a = a')) ∧
∀a a'. (BIT1B a = BIT1B a') ⇔ (a = a')
- bit1_distinct
-
|- (∀a' a. BIT1A a ≠ BIT1B a') ∧ (∀a. BIT1A a ≠ BIT1C) ∧ ∀a. BIT1B a ≠ BIT1C
- bit1_case_cong
-
|- ∀M M' f f1 v.
(M = M') ∧ (∀a. (M' = BIT1A a) ⇒ (f a = f' a)) ∧
(∀a. (M' = BIT1B a) ⇒ (f1 a = f1' a)) ∧ ((M' = BIT1C) ⇒ (v = v')) ⇒
(bit1_CASE M f f1 v = bit1_CASE M' f' f1' v')
- bit1_nchotomy
-
|- ∀bb. (∃a. bb = BIT1A a) ∨ (∃a. bb = BIT1B a) ∨ (bb = BIT1C)
- bit1_Axiom
-
|- ∀f0 f1 f2.
∃fn.
(∀a. fn (BIT1A a) = f0 a) ∧ (∀a. fn (BIT1B a) = f1 a) ∧ (fn BIT1C = f2)
- bit1_induction
-
|- ∀P. (∀a. P (BIT1A a)) ∧ (∀a. P (BIT1B a)) ∧ P BIT1C ⇒ ∀b. P b
- index_bit1
-
|- dimindex (:α bit1) = if FINITE 𝕌(:α) then 2 * dimindex (:α) + 1 else 1
- finite_bit1
-
|- FINITE 𝕌(:α bit1) ⇔ FINITE 𝕌(:α)
- index_one
-
|- dimindex (:unit) = 1
- finite_one
-
|- FINITE 𝕌(:unit)
- FCP_UPDATE_COMMUTES
-
|- ∀m a b c d. a ≠ b ⇒ ((a :+ c) ((b :+ d) m) = (b :+ d) ((a :+ c) m))
- FCP_UPDATE_EQ
-
|- ∀m a b c. (a :+ c) ((a :+ b) m) = (a :+ c) m
- FCP_UPDATE_IMP_ID
-
|- ∀m a v. (m ' a = v) ⇒ ((a :+ v) m = m)
- APPLY_FCP_UPDATE_ID
-
|- ∀m a. (a :+ m ' a) m = m
- FCP_APPLY_UPDATE_THM
-
|- ∀m a w b.
(a :+ w) m ' b =
if b < dimindex (:β) then if a = b then w else m ' b
else FAIL $' index out of range ((a :+ w) m) b
- LENGTH_V2L
-
|- ∀v. LENGTH (V2L v) = dimindex (:β)
- EL_V2L
-
|- ∀i v. i < dimindex (:β) ⇒ (EL i (V2L v) = v ' i)
- FCP_MAP
-
|- ∀f v. FCP_MAP f v = L2V (MAP f (V2L v))
- FCP_TL
-
|- ∀v.
1 < dimindex (:β) ∧ (dimindex (:γ) = dimindex (:β) − 1) ⇒
(FCP_TL v = L2V (TL (V2L v)))
- FCP_EXISTS
-
|- ∀P v. FCP_EXISTS P v ⇔ EXISTS P (V2L v)
- FCP_EVERY
-
|- ∀P v. FCP_EVERY P v ⇔ EVERY P (V2L v)
- FCP_HD
-
|- ∀v. FCP_HD v = HD (V2L v)
- FCP_CONS
-
|- ∀a v. FCP_CONS a v = L2V (a::V2L v)
- V2L_L2V
-
|- ∀x. (dimindex (:β) = LENGTH x) ⇒ (V2L (L2V x) = x)
- NULL_V2L
-
|- ∀v. ¬NULL (V2L v)
- READ_TL
-
|- ∀i a. i < dimindex (:β) ⇒ (FCP_TL a ' i = a ' (SUC i))
- READ_L2V
-
|- ∀i a. i < dimindex (:β) ⇒ (L2V a ' i = EL i a)
- index_comp
-
|- ∀f n.
$FCP f ' n =
if n < dimindex (:β) then f n else FAIL $' FCP out of bounds ($FCP f) n
- fcp_subst_comp
-
|- ∀a b f. (x :+ y) ($FCP f) = FCP c. if x = c then y else f c