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|
(BG_PUSH
; array_length
(FORALL (t i v) (EQ (array_length (store t i v)) (array_length t)))
; booleans
(DISTINCT |@true| |@false|)
; exchange
(FORALL (t1 t2 i j)
(IFF (EQ (exchange t1 t2 i j) |@true|)
(AND (EQ (array_length t1) (array_length t2))
(EQ (select t1 i) (select t2 j))
(EQ (select t1 j) (select t2 i))
(FORALL (k) (IMPLIES (AND (NEQ k i) (NEQ k j))
(EQ (select t1 k) (select t2 k)))))))
(FORALL (t1 t2 i j)
(IMPLIES (EQ (exchange t1 t2 i j) |@true|)
(EQ (array_length t1) (array_length t2))))
; permut
(FORALL (t) (EQ (permut t t) |@true|))
(FORALL (t1 t2) (IMPLIES (EQ (permut t1 t2) |@true|)
(EQ (permut t2 t1) |@true|)))
(FORALL (t1 t2 t3) (IMPLIES (AND (EQ (permut t1 t2) |@true|)
(EQ (permut t2 t3) |@true|))
(EQ (permut t1 t3) |@true|)))
(FORALL
(t i j)
(EQ (permut t (store (store t i (select t j)) j (select t i))) |@true|))
(FORALL (t1 t2)
(IMPLIES (EQ (permut t1 t2) |@true|)
(EQ (array_length t1) (array_length t2))))
; sub_permut
(FORALL (t g d) (EQ (sub_permut g d t t) |@true|))
(FORALL (t1 t2 g d) (IMPLIES (EQ (sub_permut g d t1 t2) |@true|)
(EQ (sub_permut g d t2 t1) |@true|)))
(FORALL (t1 t2 t3 g d) (IMPLIES (AND (EQ (sub_permut g d t1 t2) |@true|)
(EQ (sub_permut g d t2 t3) |@true|))
(EQ (sub_permut g d t1 t3) |@true|)))
(FORALL
(t g d i j)
(IMPLIES (AND (<= g i) (<= i d) (<= g j) (<= j d))
(EQ (sub_permut g d t
(store (store t i (select t j)) j (select t i)))
|@true|)))
(FORALL
(t1 t2 g d i j)
(IMPLIES (AND (EQ (exchange t1 t2 i j) |@true|)
(<= g i) (<= i d) (<= g j) (<= j d))
(EQ (sub_permut g d t1 t2) |@true|)))
(FORALL (t1 t2 i j)
(IMPLIES (EQ (sub_permut i j t1 t2) |@true|)
(EQ (permut t1 t2) |@true|)))
(FORALL (t1 t2 i j)
(IMPLIES (EQ (sub_permut i j t1 t2) |@true|)
(EQ (array_length t1) (array_length t2))))
; sorted_array
(FORALL
(t i j)
(IFF (EQ (sorted_array t i j) |@true|)
(IMPLIES (<= i j)
(FORALL (k) (IMPLIES (AND (<= i k) (< k j))
(<= (select t k) (select t (+ k 1))))))))
)
(BG_PUSH
;; Why axiom Maximize_def
(FORALL (t)
(FORALL (n)
(FORALL (x)
(FORALL (k)
(IFF (EQ (Maximize t n x k) |@true|)
(FORALL (i) (IMPLIES (AND (<= k i) (<= i n)) (<= (select t k) x)))))))))
;; DO NOT EDIT BELOW THIS LINE
(BG_PUSH (NEQ |@true| |@false|))
(BG_PUSH
;; Why axiom bool_and_def
(FORALL (a b)
(IFF (EQ (bool_and a b) |@true|) (AND (EQ a |@true|) (EQ b |@true|)))))
(BG_PUSH
;; Why axiom bool_or_def
(FORALL (a b)
(IFF (EQ (bool_or a b) |@true|) (OR (EQ a |@true|) (EQ b |@true|)))))
(BG_PUSH
;; Why axiom bool_xor_def
(FORALL (a b) (IFF (EQ (bool_xor a b) |@true|) (NEQ a b))))
(BG_PUSH
;; Why axiom bool_not_def
(FORALL (a) (IFF (EQ (bool_not a) |@true|) (EQ a |@false|))))
(BG_PUSH
;; Why axiom ite_true
(FORALL (x y) (EQ (ite |@true| x y) x)))
(BG_PUSH
;; Why axiom ite_false
(FORALL (x y) (EQ (ite |@false| x y) y)))
(BG_PUSH
;; Why axiom lt_int_bool_axiom
(FORALL (x y) (IFF (EQ (lt_int_bool x y) |@true|) (< x y))))
(BG_PUSH
;; Why axiom le_int_bool_axiom
(FORALL (x y) (IFF (EQ (le_int_bool x y) |@true|) (<= x y))))
(BG_PUSH
;; Why axiom gt_int_bool_axiom
(FORALL (x y) (IFF (EQ (gt_int_bool x y) |@true|) (> x y))))
(BG_PUSH
;; Why axiom ge_int_bool_axiom
(FORALL (x y) (IFF (EQ (ge_int_bool x y) |@true|) (>= x y))))
(BG_PUSH
;; Why axiom eq_int_bool_axiom
(FORALL (x y) (IFF (EQ (eq_int_bool x y) |@true|) (EQ x y))))
(BG_PUSH
;; Why axiom neq_int_bool_axiom
(FORALL (x y) (IFF (EQ (neq_int_bool x y) |@true|) (NEQ x y))))
(BG_PUSH
;; Why axiom lt_real_bool_axiom
(FORALL (x y)
(IFF (EQ (lt_real_bool x y) |@true|) (EQ (lt_real x y) |@true|))))
(BG_PUSH
;; Why axiom le_real_bool_axiom
(FORALL (x y)
(IFF (EQ (le_real_bool x y) |@true|) (EQ (le_real x y) |@true|))))
(BG_PUSH
;; Why axiom gt_real_bool_axiom
(FORALL (x y)
(IFF (EQ (gt_real_bool x y) |@true|) (EQ (gt_real x y) |@true|))))
(BG_PUSH
;; Why axiom ge_real_bool_axiom
(FORALL (x y)
(IFF (EQ (ge_real_bool x y) |@true|) (EQ (ge_real x y) |@true|))))
(BG_PUSH
;; Why axiom eq_real_bool_axiom
(FORALL (x y) (IFF (EQ (eq_real_bool x y) |@true|) (EQ x y))))
(BG_PUSH
;; Why axiom neq_real_bool_axiom
(FORALL (x y) (IFF (EQ (neq_real_bool x y) |@true|) (NEQ x y))))
(DEFPRED (zwf_zero a b) (AND (<= 0 b) (< a b)))
(BG_PUSH
;; Why axiom access_update
(FORALL (a i v) (EQ (access (update a i v) i) v)))
(BG_PUSH
;; Why axiom access_update_neq
(FORALL (a i j v)
(IMPLIES (NEQ i j) (EQ (access (update a i v) j) (access a j))))
(FORALL (i j)
(IMPLIES (NEQ i j)
(FORALL (a v) (EQ (access (update a i v) j) (access a j))))))
(DEFPRED (sorted_array t i j)
(FORALL (k1 k2)
(IMPLIES (AND (AND (<= i k1) (<= k1 k2)) (<= k2 j))
(<= (access t k1) (access t k2)))))
(DEFPRED (exchange a1 a2 i j)
(AND (EQ (array_length a1) (array_length a2))
(AND (EQ (access a1 i) (access a2 j))
(AND (EQ (access a2 i) (access a1 j))
(FORALL (k)
(IMPLIES (AND (NEQ k i) (NEQ k j)) (EQ (access a1 k) (access a2 k))))))))
(BG_PUSH
;; Why axiom permut_refl
(FORALL (t l u) (EQ (permut t t l u) |@true|)))
(BG_PUSH
;; Why axiom permut_sym
(FORALL (t1 t2 l u)
(IMPLIES (EQ (permut t1 t2 l u) |@true|) (EQ (permut t2 t1 l u) |@true|))))
(BG_PUSH
;; Why axiom permut_trans
(FORALL (t1 t2 t3 l u)
(IMPLIES (EQ (permut t1 t2 l u) |@true|)
(IMPLIES (EQ (permut t2 t3 l u) |@true|) (EQ (permut t1 t3 l u) |@true|))))
(FORALL (t1 t2 l u)
(IMPLIES (EQ (permut t1 t2 l u) |@true|)
(FORALL (t3)
(IMPLIES (EQ (permut t2 t3 l u) |@true|) (EQ (permut t1 t3 l u) |@true|))))))
(BG_PUSH
;; Why axiom permut_exchange
(FORALL (a1 a2 l u i j)
(IMPLIES (AND (<= l i) (<= i u))
(IMPLIES (AND (<= l j) (<= j u))
(IMPLIES (exchange a1 a2 i j) (EQ (permut a1 a2 l u) |@true|)))))
(FORALL (l u i)
(IMPLIES (AND (<= l i) (<= i u))
(FORALL (j)
(IMPLIES (AND (<= l j) (<= j u))
(FORALL (a1 a2)
(IMPLIES (exchange a1 a2 i j) (EQ (permut a1 a2 l u) |@true|))))))))
(BG_PUSH
;; Why axiom exchange_upd
(FORALL (a i j)
(exchange a (update (update a i (access a j)) j (access a i)) i j)))
(BG_PUSH
;; Why axiom permut_weakening
(FORALL (a1 a2 l1 r1 l2 r2)
(IMPLIES (AND (AND (<= l1 l2) (<= l2 r2)) (<= r2 r1))
(IMPLIES (EQ (permut a1 a2 l2 r2) |@true|)
(EQ (permut a1 a2 l1 r1) |@true|))))
(FORALL (l1 r1 l2 r2)
(IMPLIES (AND (AND (<= l1 l2) (<= l2 r2)) (<= r2 r1))
(FORALL (a2 a1)
(IMPLIES (EQ (permut a1 a2 l2 r2) |@true|)
(EQ (permut a1 a2 l1 r1) |@true|))))))
(BG_PUSH
;; Why axiom permut_eq
(FORALL (a1 a2 l u)
(IMPLIES (<= l u)
(IMPLIES (EQ (permut a1 a2 l u) |@true|)
(FORALL (i) (IMPLIES (OR (< i l) (< u i)) (EQ (access a2 i) (access a1 i)))))))
(FORALL (l u)
(IMPLIES (<= l u)
(FORALL (a2 a1)
(IMPLIES (EQ (permut a1 a2 l u) |@true|)
(FORALL (i) (IMPLIES (OR (< i l) (< u i)) (EQ (access a2 i) (access a1 i)))))))))
(DEFPRED (permutation a1 a2)
(EQ (permut a1 a2 0 (- (array_length a1) 1)) |@true|))
(BG_PUSH
;; Why axiom array_length_update
(FORALL (a i v) (EQ (array_length (update a i v)) (array_length a))))
(BG_PUSH
;; Why axiom permut_array_length
(FORALL (a1 a2 l u)
(IMPLIES (EQ (permut a1 a2 l u) |@true|)
(EQ (array_length a1) (array_length a2)))))
;; swap_po_1, File "maximumsort.mlw", line 10, characters 5-14
(FORALL (i)
(FORALL (j)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 i) (< i (array_length t)))
(AND (<= 0 j) (< j (array_length t))))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (result)
(IMPLIES (EQ result (access t i))
(IMPLIES (AND (<= 0 j) (< j (array_length t)))
(FORALL (result0)
(IMPLIES (EQ result0 (access t j))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (t0)
(IMPLIES (EQ t0 (update t i result0)) (AND (<= 0 j) (< j (array_length t0))))))))))))))))
;; swap_po_2, File "maximumsort.mlw", line 12, characters 4-22
(FORALL (i)
(FORALL (j)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 i) (< i (array_length t)))
(AND (<= 0 j) (< j (array_length t))))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (result)
(IMPLIES (EQ result (access t i))
(IMPLIES (AND (<= 0 j) (< j (array_length t)))
(FORALL (result0)
(IMPLIES (EQ result0 (access t j))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (t0)
(IMPLIES (EQ t0 (update t i result0))
(IMPLIES (AND (<= 0 j) (< j (array_length t0)))
(FORALL (t1) (IMPLIES (EQ t1 (update t0 j result)) (exchange t1 t i j)))))))))))))))))
(DEFPRED (Maximize t n x k)
(FORALL (i) (IMPLIES (AND (<= k i) (<= i n)) (<= (access t k) x))))
;; maximum_po_1, File "maximumsort.mlw", line 31, characters 4-50
(FORALL (n)
(FORALL (k)
(FORALL (i)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 k) (<= k i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) k))))
(IMPLIES (EQ k 0)
(AND (AND (<= 0 i) (<= i n)) (Maximize t n (access t i) 0))))))))
;; maximum_po_2, File "maximumsort.mlw", line 28, characters 10-15
(FORALL (n)
(FORALL (k)
(FORALL (i)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 k) (<= k i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) k))))
(IMPLIES (NEQ k 0) (AND (<= 0 (- k 1)) (< (- k 1) (array_length t)))))))))
;; maximum_po_3, File "maximumsort.mlw", line 28, characters 16-20
(FORALL (n)
(FORALL (k)
(FORALL (i)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 k) (<= k i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) k))))
(IMPLIES (NEQ k 0)
(IMPLIES (AND (<= 0 (- k 1)) (< (- k 1) (array_length t)))
(FORALL (result)
(IMPLIES (EQ result (access t (- k 1)))
(AND (<= 0 i) (< i (array_length t))))))))))))
;; maximum_po_4, File "maximumsort.mlw", line 29, characters 9-26
(FORALL (n)
(FORALL (k)
(FORALL (i)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 k) (<= k i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) k))))
(IMPLIES (NEQ k 0)
(IMPLIES (AND (<= 0 (- k 1)) (< (- k 1) (array_length t)))
(FORALL (result)
(IMPLIES (EQ result (access t (- k 1)))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (result0)
(IMPLIES (EQ result0 (access t i))
(IMPLIES (> result result0) (AND (<= 0 k) (< (- k 1) k)))))))))))))))
;; maximum_po_5, File "maximumsort.mlw", line 29, characters 9-26
(FORALL (n)
(FORALL (k)
(FORALL (i)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 k) (<= k i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) k))))
(IMPLIES (NEQ k 0)
(IMPLIES (AND (<= 0 (- k 1)) (< (- k 1) (array_length t)))
(FORALL (result)
(IMPLIES (EQ result (access t (- k 1)))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (result0)
(IMPLIES (EQ result0 (access t i))
(IMPLIES (> result result0)
(IMPLIES (AND (<= 0 k) (< (- k 1) k))
(AND (AND (<= 0 (- k 1)) (<= (- k 1) (- k 1)))
(AND (<= (- k 1) n)
(AND (< n (array_length t)) (Maximize t n (access t (- k 1)) (- k 1)))))))))))))))))))
;; maximum_po_6, File "maximumsort.mlw", line 30, characters 9-25
(FORALL (n)
(FORALL (k)
(FORALL (i)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 k) (<= k i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) k))))
(IMPLIES (NEQ k 0)
(IMPLIES (AND (<= 0 (- k 1)) (< (- k 1) (array_length t)))
(FORALL (result)
(IMPLIES (EQ result (access t (- k 1)))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (result0)
(IMPLIES (EQ result0 (access t i))
(IMPLIES (<= result result0) (AND (<= 0 k) (< (- k 1) k)))))))))))))))
;; maximum_po_7, File "maximumsort.mlw", line 30, characters 9-25
(FORALL (n)
(FORALL (k)
(FORALL (i)
(FORALL (t)
(IMPLIES (AND (AND (<= 0 k) (<= k i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) k))))
(IMPLIES (NEQ k 0)
(IMPLIES (AND (<= 0 (- k 1)) (< (- k 1) (array_length t)))
(FORALL (result)
(IMPLIES (EQ result (access t (- k 1)))
(IMPLIES (AND (<= 0 i) (< i (array_length t)))
(FORALL (result0)
(IMPLIES (EQ result0 (access t i))
(IMPLIES (<= result result0)
(IMPLIES (AND (<= 0 k) (< (- k 1) k))
(AND (AND (<= 0 (- k 1)) (<= (- k 1) i))
(AND (<= i n)
(AND (< n (array_length t)) (Maximize t n (access t i) (- k 1)))))))))))))))))))
;; maxisort_po_1, File "maximumsort.mlw", line 39, characters 18-198
(FORALL (t)
(IMPLIES (<= 0 (array_length t))
(FORALL (result)
(IMPLIES (EQ result (array_length t))
(AND (AND (<= 0 (+ (- result 1) 1)) (<= (+ (- result 1) 1) (array_length t)))
(AND (sorted_array t (+ (- result 1) 1) (- (array_length t) 1))
(AND (permutation t t)
(IMPLIES (< (+ (- result 1) 1) (array_length t))
(Maximize t (- result 1) (access t (+ (- result 1) 1)) 0)))))))))
;; maxisort_po_2, File "maximumsort.mlw", line 44, characters 16-34
(FORALL (t)
(IMPLIES (<= 0 (array_length t))
(FORALL (result)
(IMPLIES (EQ result (array_length t))
(FORALL (i)
(FORALL (t0)
(IMPLIES (AND (AND (<= 0 (+ i 1)) (<= (+ i 1) (array_length t0)))
(AND (sorted_array t0 (+ i 1) (- (array_length t0) 1))
(AND (permutation t0 t)
(IMPLIES (< (+ i 1) (array_length t0))
(Maximize t0 i (access t0 (+ i 1)) 0)))))
(IMPLIES (>= i 0)
(AND (AND (<= 0 i) (<= i i))
(AND (<= i i) (AND (< i (array_length t0)) (Maximize t0 i (access t0 i) i))))))))))))
;; maxisort_po_3, File "maximumsort.mlw", line 45, characters 8-19
(FORALL (t)
(IMPLIES (<= 0 (array_length t))
(FORALL (result)
(IMPLIES (EQ result (array_length t))
(FORALL (i)
(FORALL (t0)
(IMPLIES (AND (AND (<= 0 (+ i 1)) (<= (+ i 1) (array_length t0)))
(AND (sorted_array t0 (+ i 1) (- (array_length t0) 1))
(AND (permutation t0 t)
(IMPLIES (< (+ i 1) (array_length t0))
(Maximize t0 i (access t0 (+ i 1)) 0)))))
(IMPLIES (>= i 0)
(IMPLIES (AND (AND (<= 0 i) (<= i i))
(AND (<= i i)
(AND (< i (array_length t0)) (Maximize t0 i (access t0 i) i))))
(FORALL (result0)
(IMPLIES (AND (AND (<= 0 result0) (<= result0 i))
(Maximize t0 i (access t0 result0) 0))
(AND (AND (<= 0 i) (< i (array_length t0)))
(AND (<= 0 result0) (< result0 (array_length t0)))))))))))))))
;; maxisort_po_4, File "maximumsort.mlw", line 39, characters 18-198
(FORALL (t)
(IMPLIES (<= 0 (array_length t))
(FORALL (result)
(IMPLIES (EQ result (array_length t))
(FORALL (i)
(FORALL (t0)
(IMPLIES (AND (AND (<= 0 (+ i 1)) (<= (+ i 1) (array_length t0)))
(AND (sorted_array t0 (+ i 1) (- (array_length t0) 1))
(AND (permutation t0 t)
(IMPLIES (< (+ i 1) (array_length t0))
(Maximize t0 i (access t0 (+ i 1)) 0)))))
(IMPLIES (>= i 0)
(IMPLIES (AND (AND (<= 0 i) (<= i i))
(AND (<= i i)
(AND (< i (array_length t0)) (Maximize t0 i (access t0 i) i))))
(FORALL (result0)
(IMPLIES (AND (AND (<= 0 result0) (<= result0 i))
(Maximize t0 i (access t0 result0) 0))
(IMPLIES (AND (AND (<= 0 i) (< i (array_length t0)))
(AND (<= 0 result0) (< result0 (array_length t0))))
(FORALL (t1)
(IMPLIES (exchange t1 t0 i result0)
(FORALL (i0)
(IMPLIES (EQ i0 (- i 1))
(AND (AND (<= 0 (+ i0 1)) (<= (+ i0 1) (array_length t1)))
(AND (sorted_array t1 (+ i0 1) (- (array_length t1) 1))
(AND (permutation t1 t)
(IMPLIES (< (+ i0 1) (array_length t1))
(Maximize t1 i0 (access t1 (+ i0 1)) 0)))))))))))))))))))))
;; maxisort_po_5, File "maximumsort.mlw", line 43, characters 16-19
(FORALL (t)
(IMPLIES (<= 0 (array_length t))
(FORALL (result)
(IMPLIES (EQ result (array_length t))
(FORALL (i)
(FORALL (t0)
(IMPLIES (AND (AND (<= 0 (+ i 1)) (<= (+ i 1) (array_length t0)))
(AND (sorted_array t0 (+ i 1) (- (array_length t0) 1))
(AND (permutation t0 t)
(IMPLIES (< (+ i 1) (array_length t0))
(Maximize t0 i (access t0 (+ i 1)) 0)))))
(IMPLIES (>= i 0)
(IMPLIES (AND (AND (<= 0 i) (<= i i))
(AND (<= i i)
(AND (< i (array_length t0)) (Maximize t0 i (access t0 i) i))))
(FORALL (result0)
(IMPLIES (AND (AND (<= 0 result0) (<= result0 i))
(Maximize t0 i (access t0 result0) 0))
(IMPLIES (AND (AND (<= 0 i) (< i (array_length t0)))
(AND (<= 0 result0) (< result0 (array_length t0))))
(FORALL (t1)
(IMPLIES (exchange t1 t0 i result0)
(FORALL (i0)
(IMPLIES (EQ i0 (- i 1)) (AND (<= 0 (+ i 1)) (< (+ i0 1) (+ i 1)))))))))))))))))))
;; maxisort_po_6, File "maximumsort.mlw", line 48, characters 4-61
(FORALL (t)
(IMPLIES (<= 0 (array_length t))
(FORALL (result)
(IMPLIES (EQ result (array_length t))
(FORALL (i)
(FORALL (t0)
(IMPLIES (AND (AND (<= 0 (+ i 1)) (<= (+ i 1) (array_length t0)))
(AND (sorted_array t0 (+ i 1) (- (array_length t0) 1))
(AND (permutation t0 t)
(IMPLIES (< (+ i 1) (array_length t0))
(Maximize t0 i (access t0 (+ i 1)) 0)))))
(IMPLIES (< i 0)
(AND (sorted_array t0 0 (- (array_length t0) 1)) (permutation t0 t))))))))))
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