|
(results_commutation
(Local_Comu_and_Noeth 0
(Local_Comu_and_Noeth-1 nil 3374068475
("" (skeep)
(("" (lemma "noetherian_induction")
(("" (inst? -1 ("R" "union(R1, R2)"))
(("1"
(inst -1
"(LAMBDA (a: T): (FORALL (b, c: T): RTC(R1)(a, b) AND RTC(R2)(a, c) IMPLIES (EXISTS (d: T): RTC(R2)(b, d) AND RTC(R1)(c, d))))")
(("1" (split)
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nil))
nil))
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nil)
("2" (hide 2)
(("2" (skeep)
(("2" (skeep)
(("2" (expand "RTC" (-2 -3))
(("2" (expand "IUnion" (-2 -3))
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(("2" (skolem * "j")
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(inst 1 "c")
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(expand* "RTC" "IUnion")
(("1"
(split)
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(expand "iterate" -1)
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(replaces -1)
(("1" (inst 1 "j") nil nil))
nil))
nil)
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(inst 1 "0")
(("2"
(expand "iterate" 1)
(("2" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (case-replace "j = 0" :hide? t)
(("1"
(hide-all-but (-2 -3 2))
(("1"
(inst 1 "b")
(("1"
(expand* "RTC" "IUnion")
(("1"
(split)
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(inst 1 "0")
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(expand "iterate" 1)
(("1" (propax) nil nil))
nil))
nil)
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(expand "iterate" -2)
(("2"
(replaces -2)
(("2" (inst 1 "i") nil nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(lemma iterate_add_one)
(("2"
(copy -1)
(("2"
(inst? -1 ("R" "R1" "n" "i - 1"))
(("1"
(inst? -2 ("R" "R2" "n" "j - 1"))
(("1"
(decompose-equality)
(("1"
(decompose-equality)
(("1"
(inst -2 "(x, b)")
(("1"
(inst -1 "(x, c)")
(("1"
(replaces -1)
(("1"
(replaces -1)
(("1"
(expand "o")
(("1"
(skolem * "x1")
(("1"
(skolem * "x2")
(("1"
(flatten)
(("1"
(lemma
"iterate_RTC")
(("1"
(copy -1)
(("1"
(inst
-1
"R1"
"i - 1")
(("1"
(inst
-2
"R2"
"j - 1")
(("1"
(expand*
"subset?"
"member")
(("1"
(inst
-1
"(x1, b)")
(("1"
(inst
-2
"(x2, c)")
(("1"
(assert)
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(expand
"locally_commute?")
(("1"
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"x"
"x1"
"x2")
(("1"
(assert)
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(skolem
*
"u")
(("1"
(flatten)
(("1"
(hide
(-5
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(("1"
(copy
-3)
(("1"
(inst
-4
"x1")
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(split)
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(inst
-1
"b"
"u")
(("1"
(assert)
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*
"v")
(("1"
(flatten)
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(typepred
"RTC(R1)")
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(expand
"reflexive_transitive?")
(("1"
(flatten)
(("1"
(hide
-1)
(("1"
(expand
"transitive?")
(("1"
(inst
-1
"x2"
"u"
"v")
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(assert)
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(-10
-3))
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"x2")
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(inst
-1
"v"
"c")
(("1"
(assert)
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(hide-all-but
(-1
-3
3))
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(skolem
*
"w")
(("1"
(flatten)
(("1"
(inst
1
"w")
(("1"
(typepred
"RTC(R2)")
(("1"
(expand
"reflexive_transitive?")
(("1"
(flatten)
(("1"
(hide
-1)
(("1"
(expand
"transitive?")
(("1"
(inst
-1
"b"
"v"
"w")
(("1"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but
(-6
1))
(("2"
(case
"subset?(R2, union(R1, R2))")
(("1"
(lemma
"R_subset_TC")
(("1"
(inst
-1
"union(R1, R2)")
(("1"
(expand*
"subset?"
"member")
(("1"
(inst
-1
"(x, x2)")
(("1"
(assert)
(("1"
(inst
-1
"(x, x2)")
(("1"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but
1)
(("2"
(expand*
"subset?"
"member")
(("2"
(skolem
*
"a")
(("2"
(flatten)
(("2"
(expand*
"union"
"member")
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but
(-4
1))
(("2"
(case
"subset?(R1, union(R1, R2))")
(("1"
(lemma
"R_subset_TC")
(("1"
(inst
-1
"union(R1, R2)")
(("1"
(expand*
"subset?"
"member")
(("1"
(inst
-1
"(x, x1)")
(("1"
(assert)
(("1"
(inst
-1
"(x, x1)")
(("1"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but
1)
(("2"
(expand*
"subset?"
"member")
(("2"
(skolem
*
"a")
(("2"
(flatten)
(("2"
(expand*
"union"
"member")
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but
(1
2))
(("2"
(typepred
"j")
(("2"
(assert)
nil
nil))
nil))
nil))
nil)
("2"
(hide-all-but
(1
3))
(("2"
(typepred
"i")
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but (1 2))
(("2"
(typepred "j")
(("2" (assert) nil nil))
nil))
nil)
("3"
(hide-all-but (1 2))
(("3"
(typepred "j")
(("3" (assert) nil nil))
nil))
nil))
nil)
("2"
(hide-all-but (1 3))
(("2"
(typepred "i")
(("2" (assert) nil nil))
nil))
nil)
("3"
(hide-all-but (1 3))
(("3"
(typepred "i")
(("3" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (propax) nil nil))
nil))
nil))
nil)
((T formal-type-decl nil results_commutation nil)
(noetherian_induction formula-decl nil noetherian nil)
(RTC const-decl "reflexive_transitive" relations_closure nil)
(reflexive_transitive type-eq-decl nil relations_closure nil)
(reflexive_transitive? const-decl "bool" relations_closure nil)
(pred type-eq-decl nil defined_types nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(IUnion const-decl "set[T]" indexed_sets nil)
(j skolem-const-decl "nat" results_commutation nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(transitive? const-decl "bool" relations nil)
(R_subset_TC formula-decl nil relations_closure nil)
(locally_commute? const-decl "bool" ars_terminology nil)
(subset? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(iterate_RTC formula-decl nil relations_closure nil)
(odd_minus_odd_is_even application-judgement "even_int" integers
nil)
(int_plus_int_is_int application-judgement "int" integers nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(O const-decl "bool" relation_props nil)
(i skolem-const-decl "nat" results_commutation nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil 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)
(iterate_add_one formula-decl nil relation_iterate "orders/")
(iterate def-decl "pred[[T, T]]" relation_iterate "orders/")
(number nonempty-type-decl nil numbers nil)
(= const-decl "[T, T -> boolean]" equalities 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)
(>= const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(commute? const-decl "bool" ars_terminology nil)
(noetherian type-eq-decl nil noetherian nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(PRED type-eq-decl nil defined_types nil)
(noetherian? const-decl "bool" noetherian nil)
(set type-eq-decl nil sets nil) (union const-decl "set" sets nil)
(R1 skolem-const-decl "PRED[[T, T]]" results_commutation nil)
(R2 skolem-const-decl "PRED[[T, T]]" results_commutation nil))
shostak))
(commute_and_iterate_one 0
(commute_and_iterate_one-1 nil 3374066489
("" (induct "n")
(("1" (skeep)
(("1" (inst 1 "z")
(("1" (split)
(("1" (expand* "RTC" "IUnion")
(("1" (inst 1 "0")
(("1" (expand "iterate") (("1" (propax) nil nil)) nil))
nil))
nil)
("2" (expand "iterate")
(("2" (expand* "RC" "union" "member")
(("2" (assert) nil nil)) nil))
nil))
nil))
nil))
nil)
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(("2" (skeep)
(("2" (rewrite "iterate_add")
(("2" (expand "o")
(("2" (skolem * "y1")
(("2" (flatten)
(("2" (rewrite "iterate_1")
(("2" (inst -1 "R1" "R2" "x" "y1" "z")
(("2" (assert)
(("2" (skolem * "z1")
(("2" (flatten)
(("2" (hide -4)
(("2" (expand "RC" -2)
(("2"
(expand* "union" "member")
(("2"
(split)
(("1"
(expand "strong_commute?")
(("1"
(inst -3 "y1" "y" "z1")
(("1"
(assert)
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(skolem * "z2")
(("1"
(flatten)
(("1"
(inst 1 "z2")
(("1"
(split)
(("1"
(hide-all-but
(-2 -4 1))
(("1"
(typepred
"RTC(R1)")
(("1"
(expand
"reflexive_transitive?")
(("1"
(flatten)
(("1"
(hide -1)
(("1"
(expand
"transitive?")
(("1"
(inst
-1
"z"
"z1"
"z2")
(("1"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(propax)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(inst 1 "y")
(("2"
(split)
(("1"
(replaces -1)
(("1"
(hide (-2 -4))
(("1"
(lemma "R_subset_RTC")
(("1"
(inst?)
(("1"
(expand*
"subset?"
"member")
(("1"
(inst -1 "(z1, y)")
(("1"
(assert)
(("1"
(hide -3)
(("1"
(typepred
"RTC(R1)")
(("1"
(expand
"reflexive_transitive?")
(("1"
(flatten)
(("1"
(hide -1)
(("1"
(expand
"transitive?")
(("1"
(inst
-1
"z"
"z1"
"y")
(("1"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but 1)
(("2"
(expand*
"RC"
"union"
"member")
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((iterate_add formula-decl nil relation_iterate "orders/")
(iterate_1 formula-decl nil relation_iterate "orders/")
(transitive? const-decl "bool" relations nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(subset? const-decl "bool" sets nil)
(R_subset_RTC formula-decl nil relations_closure nil)
(O const-decl "bool" relation_props nil)
(IUnion const-decl "set[T]" indexed_sets nil)
(union const-decl "set" sets nil)
(member const-decl "bool" sets nil)
(nat_induction formula-decl nil naturalnumbers nil)
(RC const-decl "reflexive" relations_closure nil)
(reflexive type-eq-decl nil relations_closure nil)
(reflexive? const-decl "bool" relations nil)
(RTC const-decl "reflexive_transitive" relations_closure nil)
(reflexive_transitive type-eq-decl nil relations_closure nil)
(reflexive_transitive? const-decl "bool" relations_closure nil)
(iterate def-decl "pred[[T, T]]" relation_iterate "orders/")
(strong_commute? const-decl "bool" ars_terminology nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(PRED type-eq-decl nil defined_types nil)
(T formal-type-decl nil results_commutation 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))
shostak))
(commute_and_iterate_two 0
(commute_and_iterate_two-1 nil 3374066577
("" (induct "m")
(("1" (skeep)
(("1" (inst 1 "y")
(("1" (split)
(("1" (expand* "RTC" "IUnion")
(("1" (inst 1 "0")
(("1" (expand "iterate") (("1" (propax) nil nil)) nil))
nil))
nil)
("2" (expand "iterate" -3)
(("2" (expand* "RTC" "IUnion")
(("2" (inst 1 "n") (("2" (assert) nil nil)) nil)) nil))
nil))
nil))
nil))
nil)
("2" (skeep)
(("2" (skeep)
(("2" (rewrite "iterate_add")
(("2" (expand "o")
(("2" (skolem * "z1")
(("2" (flatten)
(("2" (inst -1 "R1" "R2" "x" "y" "z1" "n")
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(("2" (skolem * "r1")
(("2" (flatten)
(("2" (rewrite "iterate_1")
(("2" (expand "RTC" -2)
(("2" (expand "IUnion" -2)
(("2"
(skolem * "i")
(("2"
(lemma "commute_and_iterate_one")
(("2"
(inst
-1
"R1"
"R2"
"z1"
"r1"
"z"
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(("2"
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(skolem * "r2")
(("2"
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(inst 1 "r2")
(("2"
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(hide-all-but
(-2 -3 1))
(("2"
(case
"subset?(RC(R2), RTC(R2))")
(("1"
(expand*
"subset?"
"member")
(("1"
(inst
-1
"(r1, r2)")
(("1"
(assert)
(("1"
(typepred
"RTC(R2)")
(("1"
(expand
"reflexive_transitive?")
(("1"
(flatten)
(("1"
(hide
(-1
-3))
(("1"
(expand
"transitive?")
(("1"
(inst
-1
"y"
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