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harmonic_polynomials.prf
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harmonic_polynomials.prf
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(harmonic_polynomials
(harmonic_poly_real_TCC1 0
(harmonic_poly_real_TCC1-1 nil 3261618989
("" (skeep)
(("" (rewrite "odd_iff_not_even") (("" (rewrite "even_div2") nil nil))
nil))
nil)
((odd_iff_not_even formula-decl nil naturalnumbers 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)
(even_div2 formula-decl nil naturalnumbers nil))
shostak
(harmonic_poly_real subtype "(number_fields./(harmonic_polynomials.i, 2))"
"{i: integers.int | booleans.TRUE}")))
(harmonic_poly_real_TCC2 0
(harmonic_poly_real_TCC2-2 "" 3790088995
("" (skosimp*)
(("" (lemma "even_div2") (("" (inst?) (("" (assert) nil nil)) nil)) nil))
nil)
nil shostak
(harmonic_poly_real subtype "harmonic_polynomials.i"
"{i: naturalnumbers.nat | reals.<=(i, harmonic_polynomials.pn)}"))
(harmonic_poly_real_TCC2-1 nil 3261619116
("" (skosimp*)
(("" (lemma "even_div2")
(("" (inst?)
(("" (assert)
(("" (lemma "even_iff_not_odd")
(("" (inst?) (("" (assert) nil nil)) nil)) nil))
nil))
nil))
nil))
nil)
((even_div2 formula-decl nil naturalnumbers nil)
(real_le_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props 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
(harmonic_poly_real subtype "harmonic_polynomials.i"
"{i: naturalnumbers.nat | reals.<=(i, harmonic_polynomials.pn)}")))
(harmonic_poly_real_TCC3 0
(harmonic_poly_real_TCC3-2 "" 3790088995
("" (skosimp*)
(("" (lemma "even_div2") (("" (inst?) (("" (assert) nil nil)) nil)) nil))
nil)
nil shostak
(harmonic_poly_real subtype "harmonic_polynomials.pn"
"{i: integers.int | booleans.OR(harmonic_polynomials.x /= 0, reals.>=(i, 0))}"))
(harmonic_poly_real_TCC3-1 nil 3261619130
("" (skosimp*)
(("" (lemma "even_div2")
(("" (inst?)
(("" (assert)
(("" (lemma "odd_iff_not_even")
(("" (inst?) (("" (assert) nil nil)) nil)) nil))
nil))
nil))
nil))
nil)
((even_div2 formula-decl nil naturalnumbers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props 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
(harmonic_poly_real subtype "harmonic_polynomials.pn"
"{i: integers.int | booleans.OR(harmonic_polynomials.x /= 0, reals.>=(i, 0))}")))
(harmonic_poly_real_TCC4 0
(harmonic_poly_real_TCC4-1 nil 3261619140 ("" (grind) nil nil)
((odd? const-decl "bool" integers nil)
(/= const-decl "boolean" notequal nil)
(even_times_int_is_even application-judgement "even_int" integers nil))
shostak
(harmonic_poly_real subtype "harmonic_polynomials.pn"
"{i: integers.int | booleans.OR(harmonic_polynomials.y /= 0, reals.>=(i, 0))}")))
(harmonic_poly_real_TCC5 0
(harmonic_poly_real_TCC5-1 nil 3261619149 ("" (grind) nil nil)
((boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(number nonempty-type-decl nil numbers 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)
(> const-decl "bool" 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)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(posnat nonempty-type-eq-decl nil integers nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(real_ge_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(odd? const-decl "bool" integers nil)
(/= const-decl "boolean" notequal nil)
(even_times_int_is_even application-judgement "even_int" integers nil))
shostak
(harmonic_poly_real subtype
"((number_fields.-)(harmonic_polynomials.pn, harmonic_polynomials.i))"
"{i: integers.int | booleans.OR(harmonic_polynomials.x /= 0, reals.>=(i, 0))}")))
(harmonic_poly_real_TCC6 0
(harmonic_poly_real_TCC6-1 nil 3261619159 ("" (grind) nil nil)
((odd? const-decl "bool" integers nil)
(/= const-decl "boolean" notequal nil)
(even_times_int_is_even application-judgement "even_int" integers nil))
shostak
(harmonic_poly_real subtype "harmonic_polynomials.i"
"{i: integers.int | booleans.OR(harmonic_polynomials.y /= 0, reals.>=(i, 0))}")))
(harmonic_poly_imag_TCC1 0
(harmonic_poly_imag_TCC1-1 nil 3261619175
("" (skosimp*)
(("" (lemma "odd_div2" ("x" "i!1"))
(("" (rewrite "even_or_odd") (("" (assert) nil nil)) nil)) nil))
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)
(odd_div2 formula-decl nil naturalnumbers nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(even_or_odd formula-decl nil naturalnumbers nil))
shostak
(harmonic_poly_imag subtype
"(number_fields./(((number_fields.-)(harmonic_polynomials.i, 1)), 2))"
"{i: integers.int | booleans.TRUE}")))
(harmonic_poly_imag_TCC2 0
(harmonic_poly_imag_TCC2-1 nil 3261619184 ("" (grind) nil nil)
((boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(number nonempty-type-decl nil numbers 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)
(> const-decl "bool" 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)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(posnat nonempty-type-eq-decl nil integers nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(real_le_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(real_ge_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(even? const-decl "bool" integers nil))
shostak
(harmonic_poly_imag subtype "harmonic_polynomials.i"
"{i: naturalnumbers.nat | reals.<=(i, harmonic_polynomials.pn)}")))
(harmonic_poly_imag_TCC3 0
(harmonic_poly_imag_TCC3-1 nil 3261619194 ("" (grind) nil nil)
((even? const-decl "bool" integers nil)
(/= const-decl "boolean" notequal nil))
shostak
(harmonic_poly_imag subtype "harmonic_polynomials.pn"
"{i: integers.int | booleans.OR(harmonic_polynomials.y /= 0, reals.>=(i, 0))}")))
(harmonic_poly_imag_TCC4 0
(harmonic_poly_imag_TCC4-1 nil 3261619204 ("" (grind) nil nil)
((boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(number nonempty-type-decl nil numbers 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)
(> const-decl "bool" 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)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(posnat nonempty-type-eq-decl nil integers nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(real_ge_is_total_order name-judgement "(total_order?[real])" real_props
nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(even? const-decl "bool" integers nil)
(/= const-decl "boolean" notequal nil))
shostak
(harmonic_poly_imag subtype
"((number_fields.-)(harmonic_polynomials.pn, harmonic_polynomials.i))"
"{i: integers.int | booleans.OR(harmonic_polynomials.x /= 0, reals.>=(i, 0))}")))
(harmonic_poly_imag_TCC5 0
(harmonic_poly_imag_TCC5-1 nil 3261620046 ("" (grind) nil nil)
((even? const-decl "bool" integers nil)
(/= const-decl "boolean" notequal nil))
shostak
(harmonic_poly_imag subtype "harmonic_polynomials.i"
"{i: integers.int | booleans.OR(harmonic_polynomials.y /= 0, reals.>=(i, 0))}")))
(harmonic_polynomial_real_n1 0
(harmonic_polynomial_real_n1-2 "" 3790088995
("" (skolem 1 ("x" "y"))
(("" (expand "harmonic_poly_real")
(("" (expand "sigma")
(("" (expand "sigma")
(("" (rewrite "expt_x0")
(("" (rewrite "expt_x1")
(("" (rewrite "C_0")
(("" (case "odd?(0)")
(("1" (expand "odd?") (("1" (propax) nil nil)) nil)
("2" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
nil shostak)
(harmonic_polynomial_real_n1-1 nil 3261619224
("" (skolem 1 ("x" "y"))
(("" (expand "harmonic_poly_real")
(("" (expand "sigma")
(("" (expand "sigma")
(("" (rewrite "expt_x0")
(("" (rewrite "expt_x1")
(("" (rewrite "C_0")
(("" (case "odd?(0)")
(("1" (expand "odd?") (("1" (propax) nil nil)) nil)
("2" (assert)
(("2" (expand "sigma") (("2" (propax) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((harmonic_poly_real const-decl "real" harmonic_polynomials nil)
(expt_x1 formula-decl nil exponentiation nil)
(posint_times_posint_is_posint application-judgement "posint" integers
nil)
(odd? const-decl "bool" integers nil)
(even_times_int_is_even application-judgement "even_int" integers nil)
(real_plus_real_is_real application-judgement "real" reals nil)
(real_times_real_is_real application-judgement "real" reals nil)
(sigma_0_neg formula-decl nil sigma_nat nil)
(C_0 formula-decl nil binomial 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)
(rat_exp application-judgement "rat" exponentiation nil)
(nzint_times_nzint_is_nzint application-judgement "nzint" integers nil)
(nzreal_exp application-judgement "nzreal" exponentiation nil)
(int_exp application-judgement "int" exponentiation nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields 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)
(expt_x0 formula-decl nil exponentiation nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(sigma def-decl "real" sigma nil))
shostak))
(harmonic_polynomial_imag_n1 0
(harmonic_polynomial_imag_n1-2 "" 3790088996
("" (skolem 1 ("x" "y"))
(("" (expand "harmonic_poly_imag")
(("" (expand "sigma")
(("" (expand "sigma")
(("" (rewrite "expt_x1")
(("" (rewrite "expt_x0")
(("" (rewrite "C_n")
(("" (case "even?(1)")
(("1" (expand "even?") (("1" (propax) nil nil)) nil)
("2" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
nil shostak)
(harmonic_polynomial_imag_n1-1 nil 3261619351
("" (skolem 1 ("x" "y"))
(("" (expand "harmonic_poly_imag")
(("" (expand "sigma")
(("" (expand "sigma")
(("" (rewrite "expt_x1")
(("" (rewrite "expt_x0")
(("" (rewrite "C_n")
(("" (case "even?(1)")
(("1" (expand "even?") (("1" (propax) nil nil)) nil)
("2" (assert)
(("2" (expand "sigma") (("2" (propax) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((int_minus_int_is_int application-judgement "int" integers nil)
(harmonic_poly_imag const-decl "real" harmonic_polynomials nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(expt_x0 formula-decl nil exponentiation nil)
(even? const-decl "bool" integers nil)
(real_plus_real_is_real application-judgement "real" reals nil)
(real_times_real_is_real application-judgement "real" reals nil)
(sigma_0_neg formula-decl nil sigma_nat nil)
(posint_times_posint_is_posint application-judgement "posint" integers
nil)
(C_n formula-decl nil binomial 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)
(rat_exp application-judgement "rat" exponentiation nil)
(nzint_times_nzint_is_nzint application-judgement "nzint" integers nil)
(nzreal_exp application-judgement "nzreal" exponentiation nil)
(int_exp application-judgement "int" exponentiation 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)
(expt_x1 formula-decl nil exponentiation nil)
(sigma def-decl "real" sigma nil))
shostak))
(harmonic_polynomial_real_rec 0
(harmonic_polynomial_real_rec-1 nil 3261620367
("" (skolem 1 ("n" "x" "y"))
(("" (expand "harmonic_poly_real")
(("" (expand "harmonic_poly_imag")
((""
(name "F1" "LAMBDA (i: nat):
IF i > 1 + n OR odd?(i) THEN 0
ELSE C(1 + n, i) * (-1) ^ (i / 2) *
(IF i = 0 THEN x ^ (1 + n)
ELSIF i = 1 + n THEN y ^ (1 + n)
ELSE (x ^ (1 - i + n)) * y ^ i
ENDIF)
ENDIF")
(("1" (replace -1)
(("1"
(lemma "sigma_scal[nat]"
("low" "0" "high" "n" "a" "x" "F" " LAMBDA (i: nat):
IF i > n OR odd?(i) THEN 0
ELSE C(n, i) * (-1) ^ (i / 2) *
(IF i = 0 THEN x ^ n
ELSIF i = n THEN y ^ n
ELSE x ^ (n - i) * y ^ i
ENDIF)
ENDIF"))
(("1" (replace -1 1 rl)
(("1" (hide -1)
(("1"
(name "F2" "LAMBDA (i_1: nat):
x *
(LAMBDA (i: nat):
IF i > n OR odd?(i) THEN 0
ELSE C(n, i) * (-1) ^ (i / 2) *
(IF i = 0 THEN x ^ n
ELSIF i = n THEN y ^ n
ELSE x ^ (n - i) * y ^ i
ENDIF)
ENDIF)
(i_1)")
(("1" (replace -1)
(("1"
(name "F3" "LAMBDA (i: nat):
IF i > n OR even?(i) THEN 0
ELSE C(n, i) * (-1) ^ ((i - 1) / 2) *
(IF i = n THEN y ^ n ELSE x ^ (n - i) * y ^ i ENDIF)
ENDIF")
(("1" (replace -1)
(("1"
(lemma "sigma_scal[nat]"
("low" "0" "high" "n" "F" "F3" "a" "y"))
(("1" (replace -1 1 rl)
(("1"
(case "sigma(0, n, F2) = sigma(0, n+1, F2)")
(("1" (replace -1 1)
(("1"
(case "FORALL (n:nat): sigma(0, n, LAMBDA (i: nat): y * F3(i)) = sigma(0,n+1, LAMBDA (i:nat): IF i = 0 THEN 0 ELSE y*F3(i-1) ENDIF)")
(("1" (inst - "n")
(("1"
(replace -1 1)
(("1"
(lemma
"sigma_minus[nat]"
("low"
"0"
"high"
"n+1"
"F"
"F2"
"G"
"LAMBDA (i: nat): IF i = 0 THEN 0 ELSE y * F3(i - 1) ENDIF"))
(("1"
(replace -1 1)
(("1"
(hide -1 -2 -3 -4)
(("1"
(lemma
"sigma_eq[nat]"
("low"
"0"
"high"
"n+1"
"F"
"F1"
"G"
"LAMBDA (i_1: nat):
F2(i_1) -
(LAMBDA (i: nat): IF i = 0 THEN 0 ELSE y * F3(i - 1) ENDIF)
(i_1)"))
(("1"
(split -1)
(("1" (assert) nil nil)
("2"
(hide 2)
(("2"
(skosimp*)
(("2"
(typepred "n!1")
(("2"
(expand "F1")
(("2"
(hide -5 -4 -3)
(("2"
(expand "F2")
(("2"
(expand "F3")
(("2"
(case
"n!1=0")
(("1"
(replace
-1)
(("1"
(rewrite
"C_0")
(("1"
(rewrite
"C_0")
(("1"
(rewrite
"expt_x0")
(("1"
(case
"odd?(0)")
(("1"
(expand
"odd?")
(("1"
(propax)
nil
nil))
nil)
("2"
(assert)
(("2"
(expand
"^")
(("2"
(expand
"expt"
2
1)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(assert)
(("2"
(case
"n!1= 1+n")
(("1"
(assert)
(("1"
(replace
-1)
(("1"
(rewrite
"C_n")
(("1"
(rewrite
"C_n")
(("1"
(case
"even?(n)")
(("1"
(case
"odd?(n+1)")
(("1"
(assert)
nil
nil)
("2"
(hide-all-but
(-1
1))
(("2"
(expand
"even?")
(("2"
(skosimp*)
(("2"
(replace
-1)
(("2"
(expand
"odd?")
(("2"
(inst
+
"j!1")
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(assert)
(("2"
(case
"NOT (odd?(1+n))")
(("1"
(assert)
(("1"
(rewrite
"even_or_odd")
(("1"
(hide
1)
(("1"
(expand
"odd?")
(("1"
(skosimp*)
(("1"
(replace
-1)
(("1"
(lemma
"div_cancel1"
("x"
"j!1"
"n0z"
"2"))
(("1"
(replace
-1)
(("1"
(lemma
"div_cancel1"
("x"
"j!1+1"
"n0z"
"2"))
(("1"
(replace
-1)
(("1"
(replace
-3
2
rl)
(("1"
(replace
-3
*
rl)
(("1"
(typepred
"n")
(("1"
(expand
"^"
2)
(("1"
(assert)
(("1"
(expand
"expt"
2
1)
(("1"
(expand
"expt"
2
2)
(("1"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but
(-1
1))
(("2"
(expand
"odd?")
(("2"
(skosimp*)
(("2"
(replace
-1)
(("2"
(expand
"even?")
(("2"
(inst
+
"j!1")
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(assert)
(("2"
(case
"odd?(n!1)")
(("1"
(case
"even?(n!1-1)")
(("1"
(assert)
nil
nil)
("2"
(hide-all-but
(1
-1))
(("2"
(expand
"odd?")
(("2"
(skosimp*)
(("2"
(replace
-1)
(("2"
(expand
"even?")
(("2"
(inst
+
"j!1")
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(case
"NOT (even?(n!1-1))")
(("1"
(assert)
(("1"
(rewrite
"even_or_odd")
(("1"
(hide
1)
(("1"
(expand
"odd?")
(("1"
(skosimp*)
(("1"
(lemma
"div_cancel1"
("x"
"1+j!1"
"n0z"
"2"))
(("1"
(lemma
"div_cancel1"
("x"
"j!1"
"n0z"
"2"))
(("1"
(case
"(n!1 - 2) / 2 = j!1")
(("1"
(replace
-1)
(("1"
(case
"(n!1) / 2 = j!1+1")
(("1"
(replace
-1)
(("1"
(case
"n!1=n")
(("1"
(replace
-1)
(("1"
(rewrite
"C_n")
(("1"
(rewrite
"C_n_1")
(("1"
(rewrite
"C_n_1")
(("1"
(lemma
"expt_plus"
("n0x"
"-1"
"i"
"1"
"j"
"j!1"))
(("1"
(rewrite
"expt_x1"
-1)
(("1"
(replace
-1)
(("1"
(name-replace
"K1"
"(-1)^j!1")
(("1"
(rewrite
"expt_x1"
3)
(("1"
(expand
"^"
3)
(("1"
(expand
"expt"
3
1)
(("1"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(assert)
(("2"
(lemma
"C_n_plus_1"
("n"
"n"
"k"
"n!1"))
(("2"
(replace
-1
4)
(("2"
(lemma
"expt_plus"
("n0x"
"-1"
"i"
"1"
"j"
"j!1"))
(("2"
(rewrite
"expt_x1"
-1)
(("2"
(replace
-1
4)
(("2"
(name-replace
"K10"
"(-1)^j!1")
(("2"
(name-replace
"K11"
"C(n,n!1-1)")
(("2"
(name-replace
"K12"
"C(n,n!1)")
(("2"
(expand
"^"
4)
(("2"
(expand
"expt"
4
7)
(("2"
(expand
"expt"
4
6)
(("2"
(expand
"expt"
4
4)
(("2"
(expand
"expt"
4
3)
(("2"
(expand
"expt"
4
2)
(("2"
(expand
"expt"
4
1)
(("2"
(assert)
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
-3))
(("2"
(grind)
nil
nil))
nil))
nil))
nil)
("2"
(hide-all-but
(1
-1
-3))
(("2"
(grind)
nil
nil))
nil))
nil))
nil))
nil))