|
(real_continuity
(IMP_metric_continuity_TCC1 0
(IMP_metric_continuity_TCC1-1 nil 3429890484
("" (expand "metric?")
(("" (expand "metric_zero?")
(("" (expand "metric_symmetric?")
(("" (expand "metric_triangle?") (("" (grind) nil nil)) nil))
nil))
nil))
nil)
((real_minus_real_is_real application-judgement "real" reals nil)
(metric_zero? const-decl "bool" metric_def nil)
(metric_triangle? const-decl "bool" metric_def nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(minus_real_is_real application-judgement "real" reals nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props 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)
(number nonempty-type-decl nil numbers nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil)
(metric_symmetric? const-decl "bool" metric_def nil)
(metric? const-decl "bool" metric_def nil))
nil))
(sum_continuous 0
(sum_continuous-1 nil 3429893655
("" (skosimp)
(("" (typepred "g2!1")
(("" (typepred "g1!1")
(("" (expand "metric_continuous?")
(("" (skosimp)
(("" (inst - "x!1")
(("" (inst - "x!1")
(("" (expand "metric_continuous_at?")
(("" (skosimp)
(("" (inst - "epsilon!1/2")
(("" (inst - "epsilon!1/2")
(("" (skosimp*)
(("" (inst + "min(delta!1,delta!2)")
(("" (skosimp)
((""
(inst - "x!2")
((""
(inst - "x!2")
((""
(expand "member")
((""
(expand "ball")
((""
(expand "+")
((""
(assert)
((""
(lemma
"triangle"
("x"
"g1!1(x!1) - g1!1(x!2)"
"y"
"g2!1(x!1) - g2!1(x!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)
((metric_continuous type-eq-decl nil metric_continuity nil)
(metric_continuous? const-decl "bool" metric_continuity nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(d formal-const-decl "metric" real_continuity nil)
(metric nonempty-type-eq-decl nil metric_def nil)
(metric? const-decl "bool" metric_def nil)
(nnreal type-eq-decl nil real_types nil)
(>= const-decl "bool" reals 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)
(number nonempty-type-decl nil numbers nil)
(T formal-type-decl nil real_continuity nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil)
(metric_continuous_at? const-decl "bool" metric_continuity nil)
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil)
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
(nznum nonempty-type-eq-decl nil number_fields nil)
(/= const-decl "boolean" notequal nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(real_minus_real_is_real application-judgement "real" reals nil)
(ball const-decl "set[T]" metric_space_def nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
real_types nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(triangle formula-decl nil real_props nil)
(+ const-decl "[T -> real]" real_fun_ops "reals/")
(real_plus_real_is_real application-judgement "real" reals nil)
(member const-decl "bool" sets nil)
(<= const-decl "bool" reals nil)
(min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
(posreal_min application-judgement
"{z: posreal | z <= x AND z <= y}" real_defs nil))
nil))
(opp_continuous 0
(opp_continuous-1 nil 3429893655
("" (skosimp)
(("" (typepred "g!1")
(("" (expand "metric_continuous?")
(("" (skosimp)
(("" (inst - "x!1")
(("" (expand "metric_continuous_at?")
(("" (skosimp)
(("" (inst - "epsilon!1")
(("" (skosimp)
(("" (inst + "delta!1")
(("" (skosimp)
(("" (inst - "x!2") (("" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((metric_continuous type-eq-decl nil metric_continuity nil)
(metric_continuous? const-decl "bool" metric_continuity nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(d formal-const-decl "metric" real_continuity nil)
(metric nonempty-type-eq-decl nil metric_def nil)
(metric? const-decl "bool" metric_def nil)
(nnreal type-eq-decl nil real_types nil)
(>= const-decl "bool" reals 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)
(number nonempty-type-decl nil numbers nil)
(T formal-type-decl nil real_continuity nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil)
(metric_continuous_at? const-decl "bool" metric_continuity nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(minus_real_is_real application-judgement "real" reals nil)
(ball_is_metric_open application-judgement "metric_open"
real_continuity nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(ball_is_metric_open application-judgement "metric_open"
real_continuity nil)
(ball const-decl "set[T]" metric_space_def nil)
(member const-decl "bool" sets nil)
(- const-decl "[T -> real]" real_fun_ops "reals/")
(real_minus_real_is_real application-judgement "real" reals nil))
nil))
(diff_continuous 0
(diff_continuous-1 nil 3429893655
("" (skosimp)
(("" (lemma "opp_continuous" ("g" "g2!1"))
(("" (lemma "sum_continuous" ("g1" "g1!1" "g2" "-g2!1"))
(("" (expand "+")
(("" (expand "-") (("" (assert) nil nil)) nil)) nil))
nil))
nil))
nil)
((metric_continuous type-eq-decl nil metric_continuity nil)
(metric_continuous? const-decl "bool" metric_continuity nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(d formal-const-decl "metric" real_continuity nil)
(metric nonempty-type-eq-decl nil metric_def nil)
(metric? const-decl "bool" metric_def nil)
(nnreal type-eq-decl nil real_types nil)
(>= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans 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)
(T formal-type-decl nil real_continuity nil)
(opp_continuous judgement-tcc nil real_continuity nil)
(+ const-decl "[T -> real]" real_fun_ops "reals/")
(minus_real_is_real application-judgement "real" reals nil)
(real_plus_real_is_real application-judgement "real" reals nil)
(- const-decl "[T -> real]" real_fun_ops "reals/")
(real_minus_real_is_real application-judgement "real" reals nil)
(opp_continuous application-judgement "metric_continuous"
real_continuity nil)
(sum_continuous judgement-tcc nil real_continuity nil)
(- const-decl "[T -> real]" real_fun_ops "reals/"))
nil))
(scal_continuous 0
(scal_continuous-1 nil 3429893655
("" (skosimp)
(("" (typepred "g!1")
(("" (expand "*")
(("" (expand "metric_continuous?")
(("" (skosimp)
(("" (inst - "x!1")
(("" (expand "metric_continuous_at?")
(("" (skosimp)
(("" (case-replace "c!1=0")
(("1" (inst + "1")
(("1" (expand "member")
(("1" (expand "ball")
(("1" (skosimp)
(("1" (expand "abs")
(("1" (assert) nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (expand "member")
(("2" (expand "ball")
(("2"
(lemma "posreal_div_posreal_is_posreal"
("px" "epsilon!1" "py" "abs(c!1)"))
(("1" (inst - "epsilon!1 / abs(c!1)")
(("1" (skosimp)
(("1"
(inst + "delta!1")
(("1"
(skosimp)
(("1"
(inst - "x!2")
(("1"
(assert)
(("1"
(rewrite "div_mult_pos_lt2" -2)
(("1"
(rewrite
"abs_mult"
-2
:dir
rl)
(("1" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (assert) nil nil))
nil)
("2" (hide 3 -1) (("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((metric_continuous type-eq-decl nil metric_continuity nil)
(metric_continuous? const-decl "bool" metric_continuity nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(d formal-const-decl "metric" real_continuity nil)
(metric nonempty-type-eq-decl nil metric_def nil)
(metric? const-decl "bool" metric_def nil)
(nnreal type-eq-decl nil real_types nil)
(>= const-decl "bool" reals 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)
(number nonempty-type-decl nil numbers nil)
(T formal-type-decl nil real_continuity nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil)
(posreal_div_posreal_is_posreal judgement-tcc nil real_types nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(abs_mult formula-decl nil real_props nil)
(nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
real_types nil)
(div_mult_pos_lt2 formula-decl nil real_props nil)
(c!1 skolem-const-decl "real" real_continuity nil)
(epsilon!1 skolem-const-decl "posreal" real_continuity nil)
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
(nznum nonempty-type-eq-decl nil number_fields nil)
(/= const-decl "boolean" notequal nil)
(nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
real_types nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(real_minus_real_is_real application-judgement "real" reals nil)
(ball const-decl "set[T]" metric_space_def nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(member const-decl "bool" sets nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(metric_continuous_at? const-decl "bool" metric_continuity nil)
(* const-decl "[T -> real]" real_fun_ops "reals/")
(real_times_real_is_real application-judgement "real" reals nil))
nil))
(abs_continuous 0
(abs_continuous-1 nil 3429890484
("" (skosimp)
(("" (typepred "g!1")
(("" (expand "metric_continuous?")
(("" (expand "metric_continuous_at?")
(("" (skosimp)
(("" (skosimp)
(("" (inst - "x!1")
(("" (inst - "epsilon!1")
(("" (skosimp)
(("" (inst + "delta!1")
(("" (expand "member")
(("" (skosimp)
(("" (inst - "x!2")
(("" (assert)
((""
(hide -2)
((""
(expand "ball")
((""
(expand "abs")
((""
(expand "abs")
(("" (grind) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((metric_continuous type-eq-decl nil metric_continuity nil)
(metric_continuous? const-decl "bool" metric_continuity nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(d formal-const-decl "metric" real_continuity nil)
(metric nonempty-type-eq-decl nil metric_def nil)
(metric? const-decl "bool" metric_def nil)
(nnreal type-eq-decl nil real_types nil)
(>= const-decl "bool" reals 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)
(number nonempty-type-decl nil numbers nil)
(T formal-type-decl nil real_continuity nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil)
(metric_continuous_at? const-decl "bool" metric_continuity nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(ball_is_metric_open application-judgement "metric_open"
real_continuity nil)
(ball_is_metric_open application-judgement "metric_open"
real_continuity nil)
(real_minus_real_is_real application-judgement "real" reals nil)
(ball const-decl "set[T]" metric_space_def nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(minus_real_is_real application-judgement "real" reals nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(abs const-decl "[T -> nonneg_real]" real_fun_ops "reals/")
(member const-decl "bool" sets nil))
nil))
(expt_real_continuous_TCC1 0
(expt_real_continuous_TCC1-1 nil 3433761206
("" (subtype-tcc) nil nil)
((boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(T formal-type-decl nil real_continuity 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)
(nnreal type-eq-decl nil real_types nil)
(metric? const-decl "bool" metric_def nil)
(metric nonempty-type-eq-decl nil metric_def nil)
(d formal-const-decl "metric" real_continuity nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(metric_continuous? const-decl "bool" metric_continuity nil)
(metric_continuous type-eq-decl nil metric_continuity nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(metric_continuous_at? const-decl "bool" metric_continuity nil)
(member const-decl "bool" sets nil)
(ball const-decl "set[T]" metric_space_def nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(real_minus_real_is_real application-judgement "real" reals nil))
nil))
(expt_real_continuous 0
(expt_real_continuous-1 nil 3429893655
(""
(case "forall (a,eps:posreal,x:nnreal): exists (delta:posreal): forall (y:nnreal): abs(x-y) < delta => abs(x^a-y^a) < eps")
(("1" (skosimp)
(("1" (expand "metric_continuous?")
(("1" (expand "^" 1)
(("1" (expand "metric_continuous_at?")
(("1" (expand "member")
(("1" (expand "ball")
(("1" (skosimp*)
(("1" (typepred "a!1")
(("1" (typepred "h!1")
(("1" (inst -5 "a!1" "epsilon!1" "h!1(x!1)")
(("1" (skosimp)
(("1" (expand "metric_continuous?")
(("1" (expand "metric_continuous_at?")
(("1"
(inst - "x!1")
(("1"
(inst - "delta!1")
(("1"
(skosimp)
(("1"
(expand "member")
(("1"
(expand "ball")
(("1"
(inst + "delta!2")
(("1"
(skosimp)
(("1"
(inst - "x!2")
(("1"
(inst -5 "h!1(x!2)")
(("1" (assert) nil nil)
("2"
(inst - "x!2")
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (inst - "x!1") nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide 2)
(("2" (skosimp)
(("2" (case-replace "x!1=0")
(("1" (rewrite "real_expt_0x")
(("1"
(case "EXISTS (delta: posreal):
FORALL (y: nnreal): y < delta => y ^ a!1 < eps!1")
(("1" (skosimp)
(("1" (inst + "delta!1")
(("1" (skosimp)
(("1" (inst - "y!1")
(("1" (expand "abs")
(("1" (assert)
(("1" (case-replace "y!1=0")
(("1" (assert)
(("1"
(rewrite "real_expt_0x")
(("1" (assert) nil nil))
nil))
nil)
("2" (assert)
(("2"
(assert)
(("2"
(lemma
"real_expt_pos"
("px" "y!1" "a" "a!1"))
(("2" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide 2)
(("2" (inst + "eps!1^(1/a!1)")
(("1" (skosimp)
(("1"
(lemma "real_expt_strict_increasing"
("x" "y!1" "y" "eps!1 ^ (1 / a!1)" "pa" "a!1"))
(("1" (assert)
(("1" (rewrite "real_expt_times" -1 :dir rl)
(("1" (assert)
(("1" (rewrite "real_expt_x1") nil nil))
nil))
nil))
nil))
nil))
nil)
("2" (rewrite "real_expt_pos") nil nil))
nil))
nil))
nil))
nil)
("2" (case "x!1>0")
(("1" (hide 1)
(("1"
(case "EXISTS (delta: posreal):
FORALL (y: nnreal): y>x!1& y-x!1<delta => y^a!1-x!1^a!1<eps!1")
(("1"
(case "EXISTS (delta: posreal):
FORALL (y: nnreal): y<x!1& x!1-y<delta => x!1^a!1-y^a!1<eps!1")
(("1" (skosimp*)
(("1" (inst + "min(delta!1,delta!2)")
(("1" (skosimp)
(("1" (lemma "trich_lt" ("x" "x!1" "y" "y!1"))
(("1" (split -1)
(("1" (inst -3 "y!1")
(("1"
(assert)
(("1"
(lemma
"real_expt_strict_increasing"
("x" "x!1" "y" "y!1" "pa" "a!1"))
(("1"
(expand "abs")
(("1" (assert) nil nil))
nil))
nil))
nil))
nil)
("2" (replace -1)
(("2"
(assert)
(("2"
(expand "abs")
(("2" (assert) nil nil))
nil))
nil))
nil)
("3" (inst - "y!1")
(("3"
(assert)
(("3"
(lemma
"real_expt_strict_increasing"
("y" "x!1" "x" "y!1" "pa" "a!1"))
(("3"
(expand "abs")
(("3" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide -1 2)
(("2"
(lemma "trich_lt" ("x" "x!1^a!1" "y" "eps!1"))
(("2" (split -1)
(("1" (inst + "1")
(("1" (skosimp)
(("1"
(lemma "real_expt_pos"
("px" "y!1" "a" "a!1"))
(("1" (assert) nil nil)
("2" (assert) nil nil))
nil))
nil))
nil)
("2" (replace -1)
(("2" (inst + "x!1/2")
(("1" (skosimp)
(("1"
(lemma
"real_expt_pos"
("px" "y!1" "a" "a!1"))
(("1" (assert) nil nil)
("2" (assert) nil nil))
nil))
nil)
("2" (assert) nil nil))
nil))
nil)
("3" (inst + "x!1-(x!1^a!1-eps!1)^(1/a!1)")
(("1" (skosimp)
(("1"
(lemma "real_expt_strict_increasing"
("x"
"(x!1 ^ a!1 - eps!1) ^ (1 / a!1)"
"y"
"y!1"
"pa"
"a!1"))
(("1"
(assert)
(("1"
(rewrite
"real_expt_times"
-1
:dir
rl)
(("1"
(rewrite "real_expt_x1")
(("1" (assert) nil nil))
nil))
nil))
nil)
("2" (assert) nil nil))
nil))
nil)
("2"
(lemma "real_expt_strict_increasing"
("x" "x!1 ^ a!1 - eps!1" "y" "x!1 ^ a!1"
"pa" "1/a!1"))
(("1" (assert)
(("1"
(rewrite "real_expt_times" -1 :dir rl)
(("1"
(rewrite "real_expt_x1")
(("1" (assert) nil nil))
nil))
nil))
nil)
("2" (assert) nil nil))
nil)
("3" (flatten) (("3" (assert) nil nil)) nil)
("4" (assert) nil nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide 2)
(("2" (inst + "(eps!1+x!1^a!1)^(1/a!1)-x!1")
(("1" (skosimp)
(("1"
(lemma "real_expt_strict_increasing"
("x" "y!1" "y"
"(eps!1 + x!1 ^ a!1) ^ (1 / a!1)" "pa"
"a!1"))
(("1" (assert)
(("1" (rewrite "real_expt_times" -1 :dir rl)
(("1" (rewrite "real_expt_x1" -1)
(("1" (assert) nil nil)) nil))
nil))
nil))
nil))
nil)
("2"
(lemma "real_expt_strict_increasing"
("x" "x!1^a!1" "y" "x!1 ^ a!1 + eps!1" "pa"
"1/a!1"))
(("2" (assert)
(("2" (rewrite "real_expt_times" -1 :dir rl)
(("2" (rewrite "real_expt_x1")
(("2" (assert) nil nil)) nil))
nil))
nil))
nil)
("3" (flatten) (("3" (assert) nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (assert) nil nil))
nil))
nil))
nil))
nil))
nil)
((= const-decl "[T, T -> boolean]" equalities nil)
(real_times_real_is_real application-judgement "real" reals nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(minus_real_is_real application-judgement "real" reals nil)
(real_expt_pos formula-decl nil real_expt "power/")
(mult_divides2 application-judgement "(divides(m))" divides nil)
(mult_divides1 application-judgement "(divides(n))" divides nil)
(int_times_even_is_even application-judgement "even_int" integers
nil)
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil)
(eps!1 skolem-const-decl "posreal" real_continuity nil)
(/= const-decl "boolean" notequal nil)
(nznum nonempty-type-eq-decl nil number_fields nil)
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
(a!1 skolem-const-decl "posreal" real_continuity nil)
(real_expt_strict_increasing formula-decl nil real_expt "power/")
(real_expt_times formula-decl nil real_expt "power/")
(posreal_times_posreal_is_posreal application-judgement "posreal"
real_types nil)
(real_expt_x1 formula-decl nil real_expt "power/")
(real_expt_0x formula-decl nil real_expt "power/")
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(posreal_plus_nnreal_is_posreal application-judgement "posreal"
real_types nil)
(nnreal_plus_posreal_is_posreal application-judgement "posreal"
real_types nil)
(posreal_min application-judgement
"{z: posreal | z <= x AND z <= y}" real_defs nil)
(min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
(<= const-decl "bool" reals nil)
(trich_lt formula-decl nil real_props nil)
(x!1 skolem-const-decl "nnreal" real_continuity nil)
(nnreal_div_posreal_is_nnreal application-judgement "nnreal"
real_types nil)
(^ const-decl "[T -> nnreal]" real_fun_power "power/")
(member const-decl "bool" sets nil)
(T formal-type-decl nil real_continuity nil)
(metric? const-decl "bool" metric_def nil)
(metric nonempty-type-eq-decl nil metric_def nil)
(d formal-const-decl "metric" real_continuity nil)
(metric_continuous type-eq-decl nil metric_continuity nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(x!2 skolem-const-decl "T" real_continuity nil)
(x!1 skolem-const-decl "T" real_continuity nil)
(h!1 skolem-const-decl "{g | FORALL x: g(x) >= 0}" real_continuity
nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(ball const-decl "set[T]" metric_space_def nil)
(metric_continuous_at? const-decl "bool" metric_continuity nil)
(metric_continuous? const-decl "bool" metric_continuity nil)
(real_minus_real_is_real application-judgement "real" reals 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)
(bool nonempty-type-eq-decl nil booleans nil)
(>= const-decl "bool" reals nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(nnreal type-eq-decl nil real_types nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(< const-decl "bool" reals nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(^ const-decl "nnreal" real_expt "power/"))
nil))
(expt_nat_continuous 0
(expt_nat_continuous-1 nil 3429907898
("" (skosimp)
(("" (case-replace "n!1=0")
(("1" (expand "expt")
(("1" (expand "expt")
(("1" (lemma "const_continuous" ("a" "1"))
(("1" (rewrite "metric_continuous_def")
(("1" (expand "const_fun") (("1" (propax) nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (case "n!1>0")
(("1" (hide 1)
(("1" (typepred "g!1")
(("1"
(case "FORALL (epsilon:posreal,x:real): EXISTS (delta:posreal): FORALL (y:real): abs(x-y) abs(expt(x,n!1)-expt(y,n!1)))
(("1" (expand "metric_continuous?")
(("1" (skosimp)
(("1" (inst - "x!1")
(("1" (expand "metric_continuous_at?")
(("1" (expand "ball")
(("1" (expand "member")
(("1" (skosimp)
(("1" (inst - "epsilon!1" "g!1(x!1)")
(("1"
(skosimp)
(("1"
(inst -2 "delta!1")
(("1"
(skosimp)
(("1"
(inst + "delta!2")
(("1"
(skosimp)
(("1"
(inst -2 "x!2")
(("1"
(assert)
(("1"
(inst - "g!1(x!2)")
(("1"
(assert)
(("1"
(expand "expt" 1)
(("1"
(propax)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide -1 2)
(("2" (skosimp)
(("2"
(case "forall (n:nat,x:nnreal): EXISTS (delta: posreal):
FORALL (y: nnreal):
abs(x - y) < delta =>
abs(expt(x, n) - expt(y, n)) < epsilon!1")
(("1"
(case "forall (n:nat,x:negreal): EXISTS (delta: posreal):
FORALL (y: negreal):
abs(x - y) < delta =>
abs(expt(x, n) - expt(y, n)) < epsilon!1")
(("1" (case "x!1>=0")
(("1" (hide -2)
(("1" (inst - "n!1" "x!1")
(("1" (skosimp)
(("1"
(expand ">=" -1)
(("1"
(expand "<=" -1)
(("1"
(split -1)
(("1"
(inst + "min(x!1,delta!1)")
(("1"
(skosimp)
(("1"
(inst - "y!1")
(("1" (assert) nil nil)
("2"
(hide 2)
(("2" (grind) nil nil))
nil))
nil))
nil)
("2"
(hide -2)
(("2" (grind) nil nil))
nil))
nil)
("2"
(replace -1 * rl)
(("2"
(expand "expt" -2 1)
(("2"
(expand "expt" 1 1)
(("2"
(assert)
(("2"
(inst + "delta!1")
(("2"
(skosimp)
(("2"
(inst - "abs(y!1)")
(("2"
(split -2)
(("1"
(rewrite
"expt_of_abs")
(("1"
(name-replace
"YN"
"expt(y!1, n!1)")
(("1"
(grind)
nil
nil))
nil))
nil)
("2"
(hide 2)
(("2"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide -2)
(("2" (inst - "n!1" "x!1")
(("1" (skosimp)
(("1"
(inst + "min(delta!1,-x!1)")
(("1"
(skosimp)
(("1"
(inst - "y!1")
(("1"
(split -1)
(("1" (propax) nil nil)
("2"
(hide 3)
(("2" (grind) nil nil))
nil))
nil)
("2"
(hide 3)
(("2" (grind) nil nil))
nil))
nil))
nil)
("2"
(hide -1)
(("2" (grind) nil nil))
nil))
nil))
nil)
("2" (assert) nil nil))
nil))
nil))
nil)
("2" (hide 2)
(("2" (hide -2)
(("2" (skosimp)
(("2" (lemma "odd_or_even_int" ("x" "n!2"))
(("2"
(split -1)
(("1"
(case
"forall (x:real): expt(-x,n!2)=-expt(x,n!2)")
(("1"
(inst - "n!2" "-x!2")
(("1"
(skosimp)
(("1"
(inst + "min(delta!1,-x!2)")
(("1"
(skosimp)
(("1"
(inst -4 "-y!1")
(("1"
(split -4)
(("1"
(inst-cp - "x!2")
(("1"
(inst - "y!1")
(("1"
(replace -2)
(("1"
(replace -3)
(("1"
(hide
-2
-3
-4
-5)
(("1"
(name-replace
"XN"
"expt(x!2, n!2)")
(("1"
(name-replace
"YN"
"expt(y!1, n!2)")
(("1"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but (-3 1))
(("2" (grind) nil nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but 1)
(("2" (grind) nil nil))
nil))
nil))
nil))
nil)
("2"
(hide -2 2)
(("2"
(case
"forall (j:nat,x:real): expt(-x, 1+2*j) = -expt(x, 1+2*j)")
(("1"
(expand "odd?")
(("1"
(skosimp*)
(("1"
(replace -2)
(("1"
(inst - "j!1" "x!3")
(("1" (assert) nil nil))
nil))
nil))
nil))
nil)
("2"
(hide -1 2)
(("2"
(induct "j")
(("1"
(expand "expt")
(("1"
(expand "expt")
(("1" (propax) nil nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(expand "expt" 1)
(("2"
(expand "expt" 1)
(("2"
(inst - "x!3")
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(case
"forall (x:real): expt(-x,n!2)=expt(x,n!2)")
(("1"
(typepred "x!2")
(("1"
(inst - "n!2" "-x!2")
(("1"
(skosimp)
(("1"
(inst + "min(delta!1,-x!2)")
(("1"
(skosimp)
(("1"
(inst -6 "-y!1")
(("1"
(split -6)
(("1"
(inst-cp - "x!2")
(("1"
(inst - "y!1")
(("1"
(replace -4)
(("1"
(replace -5)
(("1"
(propax)
nil
nil))
nil))
nil))
nil))
nil)
("2"
(hide -3 2)
(("2"
(grind)
nil
nil))
nil))
nil))
nil))
nil)
("2"
(hide -3 -4 -5)
(("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but (-1 1))
(("2"
(case
"forall (x:real,j:nat): expt(-x,2*j) = expt(x,2*j)")
(("1"
(expand "even?")
(("1"
(skosimp)
(("1"
(skosimp)
(("1"
(replace -2)
(("1"
(inst - "x!3" "j!1")
(("1"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide-all-but 1)
(("2"
(induct "j")
(("1"
(skosimp)
(("1"
(expand "expt")
(("1" (propax) nil nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(expand "expt" 1)
(("2"
(expand "expt" 1)
(("2"
(inst - "x!3")
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (hide 2)
(("2" (skosimp)
(("2" (hide -1)
(("2" (case-replace "n!2=0")
(("1" (expand "expt")
(("1"
(expand "abs")
(("1" (propax) nil nil))
nil))
nil)
("2" (case "n!2>0")
(("1"
(hide 1)
(("1"
(case
"EXISTS (delta: posreal):
FORALL (y: nnreal):
x!2 <= y &y < x!2+delta =>
expt(y, n!2) < expt(x!2, n!2)+epsilon!1")
(("1"
(case
"EXISTS (delta: posreal):
FORALL (y: nnreal):
x!2 - delta < y & y <= x!2 =>
expt(x!2, n!2) - epsilon!1 < expt(y, n!2)")
(("1"
(skosimp*)
(("1"
(inst + "min(delta!1,delta!2)")
(("1"
(skosimp)
(("1"
(inst - "y!1")
(("1"
(inst - "y!1")
(("1"
(case-replace "x!2=0")
(("1"
(expand "expt" 1 1)
(("1"
(expand
"expt"
-2
1)
(("1"
(expand
"expt"
-3
2)
(("1"
(assert)
(("1"
(name-replace
"YN"
"expt(y!1, n!2)")
(("1"
(grind)
nil
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(case "x!2>0")
(("1"
(hide 1)
(("1"
(case-replace
"y!1=0")
(("1"
(expand
"expt"
1
2)
(("1"
(expand
"expt"
-3
2)
(("1"
(assert)
(("1"
(name-replace
"XN"
"expt(x!2, n!2)")
(("1"
(grind)
nil
nil))
nil))
nil))
nil))
nil)
("2"
(case "y!1>0")
(("1"
(hide 1)
(("1"
--> --------------------
--> maximum size reached
--> --------------------
¤ Dauer der Verarbeitung: 0.52 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
