diff options
Diffstat (limited to 'assignment8')
| -rw-r--r-- | assignment8/qs.prf | 1706 | ||||
| -rw-r--r-- | assignment8/qs.pvs | 150 |
2 files changed, 1856 insertions, 0 deletions
diff --git a/assignment8/qs.prf b/assignment8/qs.prf new file mode 100644 index 0000000..1467ace --- /dev/null +++ b/assignment8/qs.prf @@ -0,0 +1,1706 @@ +(my_list_induction + (less_wf 0 + (less_wf-1 nil 3667405436 + ("" (expand "well_founded?") + (("" (skolem 1 P) + (("" (flatten) + (("" (skolem -1 Y) + (("" (expand "less") + (("" (lemma naturalnumbers.wf_nat) + (("" (expand "well_founded?") + (("" + (inst -1 "(LAMBDA(n:nat):EXISTS(l:(P)):length(l)=n)") + (("" (split) + (("1" (skosimp*) + (("1" (typepred y!1) + (("1" (skolem -1 L) + (("1" (inst 1 L) + (("1" (skolem 1 G) + (("1" + (inst -2 "length(G)") + (("1" (assert) nil nil) + ("2" + (typepred G) + (("2" (inst 1 G) nil nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil) + ("2" (grind) + (("2" (inst 1 "length(Y)") + (("2" (inst 1 Y) nil nil)) nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil) + ((well_founded? const-decl "bool" orders nil) + (less const-decl "bool" my_list_induction nil) + (NOT const-decl "[bool -> bool]" booleans nil) + (P skolem-const-decl "pred[list[X]]" my_list_induction nil) + (G skolem-const-decl "(P)" my_list_induction nil) + (real_lt_is_strict_total_order name-judgement + "(strict_total_order?[real])" real_props nil) + (length def-decl "nat" list_props nil) + (= const-decl "[T, T -> boolean]" equalities nil) + (list type-decl nil list_adt nil) + (X formal-type-decl nil my_list_induction nil) + (pred type-eq-decl nil defined_types nil) + (nat nonempty-type-eq-decl nil naturalnumbers nil) + (>= const-decl "bool" reals nil) + (bool nonempty-type-eq-decl nil booleans nil) + (int nonempty-type-eq-decl nil integers nil) + (integer_pred const-decl "[rational -> boolean]" integers nil) + (rational nonempty-type-from-decl nil rationals nil) + (rational_pred const-decl "[real -> boolean]" rationals nil) + (real nonempty-type-from-decl nil reals nil) + (real_pred const-decl "[number_field -> boolean]" reals nil) + (number_field nonempty-type-from-decl nil number_fields nil) + (number_field_pred const-decl "[number -> boolean]" number_fields + nil) + (boolean nonempty-type-decl nil booleans nil) + (number nonempty-type-decl nil numbers nil) + (wf_nat formula-decl nil naturalnumbers nil)) + shostak)) + (list_length_induction 0 + (list_length_induction-1 nil 3667403384 + ("" (lemma "wf_induction[list[X],less].wf_induction") + (("1" (propax) nil nil) ("2" (rewrite less_wf) nil nil)) nil) + ((less_wf formula-decl nil my_list_induction nil) + (pred type-eq-decl nil defined_types nil) + (well_founded? const-decl "bool" orders nil) + (wf_induction formula-decl nil wf_induction nil) + (X formal-type-decl nil my_list_induction nil) + (list type-decl nil list_adt nil) + (boolean nonempty-type-decl nil booleans nil) + (bool nonempty-type-eq-decl nil booleans nil) + (less const-decl "bool" my_list_induction nil)) + shostak))) +(my_list_props + (occ_TCC1 0 + (occ_TCC1-1 nil 3667035231 ("" (termination-tcc) nil nil) + ((real_lt_is_strict_total_order 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(number_field_pred const-decl "[number -> boolean]" number_fields + nil) + (number_field nonempty-type-from-decl nil number_fields nil) + (real_pred const-decl "[number_field -> boolean]" reals nil) + (real nonempty-type-from-decl nil reals nil) + (rational_pred const-decl "[real -> boolean]" rationals nil) + (rational nonempty-type-from-decl nil rationals nil) + (integer_pred const-decl "[rational -> boolean]" integers nil) + (int nonempty-type-eq-decl nil integers nil) + (bool nonempty-type-eq-decl nil booleans nil) + (>= const-decl "bool" reals nil) + (nat nonempty-type-eq-decl nil naturalnumbers nil) + (< const-decl "bool" reals nil) + (NOT const-decl "[bool -> bool]" booleans nil) + (occ def-decl "nat" my_list_props nil) + (> const-decl "bool" reals nil) (lwb const-decl "bool" SPLIT nil) + (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) + (AND const-decl "[bool, bool -> bool]" booleans nil) + (PRED type-eq-decl nil defined_types nil) + (list type-decl nil list_adt nil)) + nil)) + (upb_perm 0 + (upb_perm-1 nil 3658132945 + ("" (skosimp*) + (("" (lemma occ_upb) + (("" (inst? -1) + (("" (assert) + (("" (expand "upb") + (("" (skosimp*) + (("" (expand "perm?") + (("" (lemma nth_occ) + (("" (inst? -1) + (("" (assert) + (("" (inst? -3) + (("" (lemma occ_nth) + (("" (inst -1 l!1 "nth(lp!1, i!1)") + (("" (assert) + (("" + (skosimp*) + (("" + (inst? -4) + (("" (assert) nil nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil) + ((occ_upb formula-decl nil QS_props nil) + (nth_occ formula-decl nil my_list_props nil) + (real_lt_is_strict_total_order name-judgement + "(strict_total_order?[real])" real_props nil) + (occ_nth formula-decl nil my_list_props nil) + (real_gt_is_strict_total_order name-judgement + "(strict_total_order?[real])" real_props nil) + (< const-decl "bool" reals nil) + (length def-decl "nat" list_props nil) + (below type-eq-decl nil nat_types nil) + (nth def-decl "T" list_props nil) + (perm? const-decl "bool" my_list_props nil) + (upb const-decl "bool" SPLIT nil) + (AND const-decl "[bool, bool -> bool]" booleans nil) + (PRED type-eq-decl nil defined_types nil) + (list type-decl nil list_adt nil) + (nat nonempty-type-eq-decl nil naturalnumbers nil) + (>= const-decl "bool" reals nil) + (bool nonempty-type-eq-decl nil booleans nil) + (int nonempty-type-eq-decl nil integers nil) + (integer_pred const-decl "[rational -> boolean]" integers nil) + (rational nonempty-type-from-decl nil rationals nil) + (rational_pred const-decl "[real -> boolean]" rationals nil) + (real nonempty-type-from-decl nil reals nil) + (real_pred const-decl "[number_field -> boolean]" reals nil) + (number_field nonempty-type-from-decl nil number_fields nil) + (number_field_pred const-decl "[number -> boolean]" number_fields + nil) + (boolean nonempty-type-decl nil booleans nil) + (number nonempty-type-decl nil numbers nil)) + shostak)) + (lwb_perm 0 + (lwb_perm-1 nil 3658138209 + ("" (skosimp*) + (("" (lemma occ_lwb) + (("" (inst? -1) + (("" (assert) + (("" (expand "lwb") + (("" (skosimp*) + (("" (expand "perm?") + (("" (lemma nth_occ) + (("" (inst? -1) + (("" (assert) + (("" (inst? -3) + (("" (lemma occ_nth) + (("" (inst -1 l!1 "nth(lp!1, i!1)") + (("" (assert) + (("" + (skosimp*) + (("" + (inst? -4) + (("" (assert) nil nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil)) + nil) + ((occ_lwb formula-decl nil QS_props nil) + (nth_occ formula-decl nil my_list_props nil) + (real_lt_is_strict_total_order name-judgement + "(strict_total_order?[real])" real_props nil) + (occ_nth formula-decl nil my_list_props nil) + (real_gt_is_strict_total_order name-judgement + "(strict_total_order?[real])" real_props nil) + (< const-decl "bool" reals nil) + (length def-decl "nat" list_props nil) + (below type-eq-decl nil nat_types nil) + (nth def-decl "T" list_props nil) + (perm? const-decl "bool" my_list_props nil) + (lwb const-decl "bool" SPLIT nil) + (AND const-decl "[bool, bool -> bool]" booleans nil) + (PRED type-eq-decl nil defined_types nil) + (list type-decl nil list_adt nil) + (nat nonempty-type-eq-decl nil naturalnumbers nil) + (>= const-decl "bool" reals nil) + (bool nonempty-type-eq-decl nil booleans nil) + (int nonempty-type-eq-decl nil integers nil) + (integer_pred const-decl "[rational -> boolean]" integers nil) + (rational nonempty-type-from-decl nil rationals nil) + (rational_pred const-decl "[real -> boolean]" rationals nil) + (real nonempty-type-from-decl nil reals nil) + (real_pred const-decl "[number_field -> boolean]" reals nil) + (number_field nonempty-type-from-decl nil number_fields nil) + (number_field_pred const-decl "[number -> boolean]" number_fields + nil) + (boolean nonempty-type-decl nil booleans nil) + (number nonempty-type-decl nil numbers nil)) + nil)) + (qs_sorted 0 + (qs_sorted-1 nil 3658128963 + ("" (lemma list_length_induction) (("" (postpone) nil nil)) nil) + ((list type-decl nil list_adt nil) + (PRED type-eq-decl nil defined_types nil) + (AND const-decl "[bool, bool -> bool]" booleans nil) + (pred type-eq-decl nil defined_types nil) + (sorted? const-decl "bool" SORTED nil) + (qs def-decl "list[nat]" QS nil) + (split_lwb_upb formula-decl nil SPLIT nil) + (cdr adt-accessor-decl "[(cons?) -> list]" list_adt nil) + (car adt-accessor-decl "[(cons?) -> T]" list_adt nil) + (cons? adt-recognizer-decl "[list -> boolean]" list_adt nil) + (split_length formula-decl nil SPLIT nil) + (sorted_append formula-decl nil SORTED nil) + (lwb_perm formula-decl nil QS_props nil) + (qs_perm formula-decl nil QS_props nil) + (perm_commutative formula-decl nil my_list_props nil) + (nth def-decl "T" list_props nil) + (real_lt_is_strict_total_order name-judgement + "(strict_total_order?[real])" real_props nil) + (posint_plus_nnint_is_posint application-judgement "posint" + integers nil) + (real_le_is_total_order name-judgement "(total_order?[real])" + real_props nil) + (numfield nonempty-type-eq-decl nil number_fields nil) + (- const-decl "[numfield, numfield -> numfield]" number_fields nil) + (int_minus_int_is_int application-judgement "int" integers nil) + (real_ge_is_total_order name-judgement "(total_order?[real])" + real_props nil) + (lwb const-decl "bool" SPLIT nil) + (upb_perm formula-decl nil QS_props nil) + (less const-decl "bool" my_list_induction nil) + (split def-decl "[list[nat], list[nat]]" SPLIT nil) + (cons adt-constructor-decl "[[T, list] -> (cons?)]" list_adt nil) + (nnint_plus_nnint_is_nnint application-judgement "nonneg_int" + integers nil) + (length def-decl "nat" list_props nil) + (list_length_induction formula-decl nil my_list_induction nil) + (number nonempty-type-decl nil numbers nil) + (boolean nonempty-type-decl nil booleans nil) + (number_field_pred const-decl "[number -> boolean]" number_fields + nil) + (number_field nonempty-type-from-decl nil number_fields nil) + (real_pred const-decl "[number_field -> boolean]" reals nil) + (real nonempty-type-from-decl nil reals nil) + (rational_pred const-decl "[real -> boolean]" rationals nil) + (rational nonempty-type-from-decl nil rationals nil) + (integer_pred const-decl "[rational -> boolean]" integers nil) + (int nonempty-type-eq-decl nil integers nil) + (bool nonempty-type-eq-decl nil booleans nil) + (>= const-decl "bool" reals nil) + (nat nonempty-type-eq-decl nil naturalnumbers nil)) + shostak))) + diff --git a/assignment8/qs.pvs b/assignment8/qs.pvs new file mode 100644 index 0000000..086321a --- /dev/null +++ b/assignment8/qs.pvs @@ -0,0 +1,150 @@ +my_list_induction[X:TYPE] : THEORY +BEGIN + less(l1,l2: list[X]) : bool = length(l1) < length(l2) + + less_wf: LEMMA well_founded?(less) % :-) + + list_length_induction :LEMMA % :-) + (FORALL (p:pred[list[X]]): + (FORALL (g:list[X]): + (FORALL (s:list[X]): + less(s,g) IMPLIES p(s)) + IMPLIES p(g)) + IMPLIES (FORALL (l:list[X]): p(l))) + +END my_list_induction + +my_list_props[X:TYPE] : THEORY +BEGIN + occ(x:X,l:list[X]) : RECURSIVE nat = + CASES l + OF null : 0 , + cons(y,l) : IF x=y THEN occ(x,l)+1 ELSE occ(x,l) ENDIF + ENDCASES + MEASURE length(l) + + occ_nth: LEMMA + FORALL(l:list[X]): FORALL(x:X): occ(x,l) > 0 IMPLIES + EXISTS(i:nat): i<length(l) AND nth(l,i) = x + + nth_occ: LEMMA + FORALL(l:list[X]): FORALL(i:nat): i<length(l) IMPLIES occ(nth(l,i),l) > 0 + + perm?( l1,l2 : list[X]) : bool = FORALL(x:X): occ(x,l1) = occ(x,l2) + + perm_null: LEMMA % :-) + FORALL(l:list[X]): perm?(null, l) IMPLIES l = null + + perm_commutative: LEMMA % :-) + FORALL(l1,l2: list[X]):perm?(l1,l2) IMPLIES perm?(l2,l1) + + append_occ: LEMMA % :-) + FORALL( l1,l2:list[X] ): FORALL(x:X): occ(x,append(l1,l2)) = occ(x,l1)+occ(x,l2) + + length_append: LEMMA % :-) + FORALL( l1,l2:list[X] ):length(append(l1,l2)) = length(l1) + length(l2) + + nth_append: LEMMA % :-( to be done + FORALL( l1,l2:list[X] ): FORALL(i:nat): i < length(l1)+length(l2) IMPLIES + nth(append(l1,l2),i) = IF i < length(l1) THEN nth(l1,i) ELSE nth(l2,i-length(l1)) ENDIF + +END my_list_props + + +SPLIT : THEORY +BEGIN + IMPORTING my_list_props + + split (m: nat, l : list[nat]) : RECURSIVE [list[nat], list[nat]] = + CASES l OF + null : (null, null), + cons(h,t) : LET + (l,r) = split (m, t) + IN + IF h < m + THEN (cons(h,l),r) + ELSE (l,cons(h,r)) + ENDIF + ENDCASES + MEASURE (length(l)) + + split_length: LEMMA % :-) + FORALL( m:nat, l:list[nat] ): LET s = split(m,l) IN length(l) = length(s`1) + length(s`2) + + split_occ: LEMMA % :-) + FORALL( n: nat, l:list[nat] ): LET s = split(n, l) IN FORALL(x:nat):occ(x,l) = occ(x,s`1)+occ(x,s`2) + + upb( g:nat, l:list[nat] ): bool = + FORALL(i:nat): i < length(l) IMPLIES nth(l,i) < g + + lwb( s:nat, l:list[nat] ): bool = + FORALL(i:nat): i < length(l) IMPLIES NOT nth(l,i) < s + + split_lwb_upb: LEMMA % :-( to be done + FORALL( m:nat, l:list[nat] ): LET s = split(m,l) IN upb(m,s`1) AND lwb(m,s`2) + +END SPLIT + +QS : THEORY +BEGIN + IMPORTING SPLIT + qs (l : list[nat]) : RECURSIVE list[nat] = + CASES l OF + null : null, + cons(h,t) : LET + (l,r) = split (h, t) + IN + append(qs(l),cons(h,qs(r))) + ENDCASES + MEASURE (length(l)) + +END QS + +SORTED: THEORY +BEGIN + IMPORTING my_list_props[nat] + IMPORTING SPLIT + + sorted?(l:list[nat]) : bool = + FORALL (i,j: nat): i < j AND j < length(l) IMPLIES nth(l,i) <= nth(l,j) + + sorted_append: LEMMA % :-( to be done + FORALL( m:nat, l1,l2:list[nat] ): + sorted?(l1) AND sorted?(l2) and upb(m,l1) AND lwb(m,l2) + IMPLIES sorted?(append(l1,l2)) +END SORTED + +QS_props : THEORY +BEGIN + IMPORTING QS, SORTED + IMPORTING my_list_props[nat], my_list_induction[nat] + + qs_perm: LEMMA % :-( to be done + FORALL(l:list[nat]):perm?(qs(l),l) + + occ_upb: LEMMA % :-) + FORALL( n: nat, l:list[nat] ): upb(n,l) IMPLIES + (FORALL(x:nat):occ(x,l) > 0 IMPLIES x < n) + + occ_lwb: LEMMA % :-) + FORALL( n: nat, l:list[nat] ): lwb(n,l) IMPLIES + (FORALL(x:nat):occ(x,l) > 0 IMPLIES NOT x < n) + + upb_perm: LEMMA % :-) + FORALL(l, lp: list[nat]): perm?(l,lp) IMPLIES + FORALL(n:nat): upb(n,l) IMPLIES upb(n,lp) + + lwb_perm: LEMMA % :-) + FORALL(l, lp: list[nat]): perm?(l,lp) IMPLIES + FORALL(n:nat): lwb(n,l) IMPLIES lwb(n,lp) + + qs_sorted :LEMMA % :-( to be done + FORALL (l:list[nat]): sorted?(qs(l)) + +% tl : VAR list[nat] + +% test : LEMMA +% qs ((:4,3,6,2,5:)) = tl + + +END QS_props |