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theory_file
stringclasses 970
values | lemma_name
stringlengths 6
108
| lemma_command
stringlengths 15
14.6k
⌀ | lemma_object
stringlengths 6
17.2k
| template
stringlengths 7
16.1k
| symbols
sequencelengths 0
101
| types
sequencelengths 0
101
| defs
sequencelengths 0
83
| output_key
stringclasses 1
value | input
stringlengths 37
79.6k
| output
stringlengths 24
16.1k
|
|---|---|---|---|---|---|---|---|---|---|---|
List
|
List.size_list_map
|
lemma size_list_map[simp]: "size_list f (map g xs) = size_list (f \<circ> g) xs"
|
size_list ?f (map ?g ?xs) = size_list (?f \<circ> ?g) ?xs
|
?H1 x_1 (?H2 x_2 x_3) = ?H1 (?H3 x_1 x_2) x_3
|
[
"Fun.comp",
"List.list.map",
"List.list.size_list"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'a) \\<Rightarrow> 'c \\<Rightarrow> 'b",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b list",
"('a \\<Rightarrow> nat) \\<Rightarrow> 'a list \\<Rightarrow> nat"
] |
[
"definition comp :: \"('b \\<Rightarrow> 'c) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'c\" (infixl \"\\<circ>\" 55)\n where \"f \\<circ> g = (\\<lambda>x. f (g x))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\""
] |
template
|
###lemma
size_list ?f (map ?g ?xs) = size_list (?f \<circ> ?g) ?xs
###symbols
Fun.comp :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'b
List.list.map :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list
List.list.size_list :::: ('a \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> nat
###defs
definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>" 55)
where "f \<circ> g = (\<lambda>x. f (g x))"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
|
###output
?H1 x_1 (?H2 x_2 x_3) = ?H1 (?H3 x_1 x_2) x_3###end
|
Predicate_Compile_Examples/Predicate_Compile_Tests
|
Predicate_Compile_Tests.foldlP_Foi_oB_i_i_oE
| null |
pred.eval (foldlP_Foi_oB_i_i_o ?x ?xa ?xb) ?xc \<Longrightarrow> (foldlP (\<lambda>xa xb xc. pred.eval (?x xb) (xa, xc)) ?xa ?xb ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 (\<lambda>y_0 y_1 y_2. ?H1 (x_1 y_1) (y_0, y_2)) x_2 x_3 x_4 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"Predicate_Compile_Tests.foldlP",
"Predicate_Compile_Tests.foldlP_Foi_oB_i_i_o",
"Predicate.pred.eval"
] |
[
"('a \\<Rightarrow> 'b \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> 'a \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<Rightarrow> bool",
"('a \\<Rightarrow> ('b \\<times> 'b) Predicate.pred) \\<Rightarrow> 'b \\<Rightarrow> 'a list \\<Rightarrow> 'b Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
template
|
###lemma
pred.eval (foldlP_Foi_oB_i_i_o ?x ?xa ?xb) ?xc \<Longrightarrow> (foldlP (\<lambda>xa xb xc. pred.eval (?x xb) (xa, xc)) ?xa ?xb ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Predicate_Compile_Tests.foldlP :::: ('a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> bool
Predicate_Compile_Tests.foldlP_Foi_oB_i_i_o :::: ('a \<Rightarrow> ('b \<times> 'b) Predicate.pred) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
###defs
datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a \<Rightarrow> bool")
|
###output
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 (\<lambda>y_0 y_1 y_2. ?H1 (x_1 y_1) (y_0, y_2)) x_2 x_3 x_4 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5###end
|
Analysis/Infinite_Products
|
Infinite_Products.convergent_prod_imp_convergent
| null |
convergent_prod ?f \<Longrightarrow> convergent (\<lambda>n. prod ?f {..n})
|
?H1 x_1 \<Longrightarrow> ?H2 (\<lambda>y_0. ?H3 x_1 (?H4 y_0))
|
[
"Set_Interval.ord_class.atMost",
"Groups_Big.comm_monoid_mult_class.prod",
"Topological_Spaces.topological_space_class.convergent",
"Infinite_Products.convergent_prod"
] |
[
"'a \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b",
"(nat \\<Rightarrow> 'a) \\<Rightarrow> bool",
"(nat \\<Rightarrow> 'a) \\<Rightarrow> bool"
] |
[
"class topological_space = \"open\" +\n assumes open_UNIV [simp, intro]: \"open UNIV\"\n assumes open_Int [intro]: \"open S \\<Longrightarrow> open T \\<Longrightarrow> open (S \\<inter> T)\"\n assumes open_Union [intro]: \"\\<forall>S\\<in>K. open S \\<Longrightarrow> open (\\<Union>K)\"\nbegin"
] |
template
|
###lemma
convergent_prod ?f \<Longrightarrow> convergent (\<lambda>n. prod ?f {..n})
###symbols
Set_Interval.ord_class.atMost :::: 'a \<Rightarrow> 'a set
Groups_Big.comm_monoid_mult_class.prod :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b
Topological_Spaces.topological_space_class.convergent :::: (nat \<Rightarrow> 'a) \<Rightarrow> bool
Infinite_Products.convergent_prod :::: (nat \<Rightarrow> 'a) \<Rightarrow> bool
###defs
class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
begin
|
###output
?H1 x_1 \<Longrightarrow> ?H2 (\<lambda>y_0. ?H3 x_1 (?H4 y_0))###end
|
Analysis/Generalised_Binomial_Theorem
|
Generalised_Binomial_Theorem.gen_binomial_complex
|
theorem gen_binomial_complex:
fixes z :: complex
assumes "norm z < 1"
shows "(\<lambda>n. (a gchoose n) * z^n) sums (1 + z) powr a"
|
cmod ?z < 1 \<Longrightarrow> (\<lambda>n. (?a gchoose n) * ?z ^ n) sums (1 + ?z) powr ?a
|
?H1 x_1 < ?H2 \<Longrightarrow> ?H3 (\<lambda>y_0. ?H4 (?H5 x_2 y_0) (?H6 x_1 y_0)) (?H7 (?H8 ?H2 x_1) x_2)
|
[
"Groups.plus_class.plus",
"Transcendental.powr",
"Power.power_class.power",
"Binomial.gbinomial",
"Groups.times_class.times",
"Series.sums",
"Groups.one_class.one",
"Complex.cmod"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(nat \\<Rightarrow> 'a) \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a",
"complex \\<Rightarrow> real"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition powr :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a::ln\" (infixr \"powr\" 80)\n \\<comment> \\<open>exponentation via ln and exp\\<close>\n where \"x powr a \\<equiv> if x = 0 then 0 else exp (a * ln x)\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"definition gbinomial :: \"'a::{semidom_divide,semiring_char_0} \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixl \"gchoose\" 65)\n where gbinomial_prod_rev: \"a gchoose k = prod (\\<lambda>i. a - of_nat i) {0..<k} div fact k\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"definition sums :: \"(nat \\<Rightarrow> 'a::{topological_space, comm_monoid_add}) \\<Rightarrow> 'a \\<Rightarrow> bool\"\n (infixr \"sums\" 80)\n where \"f sums s \\<longleftrightarrow> (\\<lambda>n. \\<Sum>i<n. f i) \\<longlonglongrightarrow> s\"",
"class one =\n fixes one :: 'a (\"1\")",
"abbreviation cmod :: \"complex \\<Rightarrow> real\"\n where \"cmod \\<equiv> norm\""
] |
template
|
###lemma
cmod ?z < 1 \<Longrightarrow> (\<lambda>n. (?a gchoose n) * ?z ^ n) sums (1 + ?z) powr ?a
###symbols
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Transcendental.powr :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Binomial.gbinomial :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Series.sums :::: (nat \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> bool
Groups.one_class.one :::: 'a
Complex.cmod :::: complex \<Rightarrow> real
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
definition powr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a::ln" (infixr "powr" 80)
\<comment> \<open>exponentation via ln and exp\<close>
where "x powr a \<equiv> if x = 0 then 0 else exp (a * ln x)"
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
definition gbinomial :: "'a::{semidom_divide,semiring_char_0} \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
where gbinomial_prod_rev: "a gchoose k = prod (\<lambda>i. a - of_nat i) {0..<k} div fact k"
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
definition sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
(infixr "sums" 80)
where "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> s"
class one =
fixes one :: 'a ("1")
abbreviation cmod :: "complex \<Rightarrow> real"
where "cmod \<equiv> norm"
|
###output
?H1 x_1 < ?H2 \<Longrightarrow> ?H3 (\<lambda>y_0. ?H4 (?H5 x_2 y_0) (?H6 x_1 y_0)) (?H7 (?H8 ?H2 x_1) x_2)###end
|
Analysis/Ball_Volume
|
Ball_Volume.emeasure_ball
| null |
0 \<le> ?r \<Longrightarrow> emeasure lborel (ball ?c ?r) = ennreal (unit_ball_vol (real DIM(?'a)) * ?r ^ DIM(?'a))
|
?H1 \<le> x_1 \<Longrightarrow> ?H2 ?H3 (?H4 x_2 x_1) = ?H5 (?H6 (?H7 (?H8 (?H9 ?H10))) (?H11 x_1 (?H9 ?H10)))
|
[
"Power.power_class.power",
"Euclidean_Space.euclidean_space_class.Basis",
"Finite_Set.card",
"Real.real",
"Ball_Volume.unit_ball_vol",
"Groups.times_class.times",
"Extended_Nonnegative_Real.ennreal",
"Elementary_Metric_Spaces.ball",
"Lebesgue_Measure.lborel",
"Sigma_Algebra.emeasure",
"Groups.zero_class.zero"
] |
[
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a set",
"'a set \\<Rightarrow> nat",
"nat \\<Rightarrow> real",
"real \\<Rightarrow> real",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"real \\<Rightarrow> ennreal",
"'a \\<Rightarrow> real \\<Rightarrow> 'a set",
"'a measure",
"'a measure \\<Rightarrow> 'a set \\<Rightarrow> ennreal",
"'a"
] |
[
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"class euclidean_space = real_inner +\n fixes Basis :: \"'a set\"\n assumes nonempty_Basis [simp]: \"Basis \\<noteq> {}\"\n assumes finite_Basis [simp]: \"finite Basis\"\n assumes inner_Basis:\n \"\\<lbrakk>u \\<in> Basis; v \\<in> Basis\\<rbrakk> \\<Longrightarrow> inner u v = (if u = v then 1 else 0)\"\n assumes euclidean_all_zero_iff:\n \"(\\<forall>u\\<in>Basis. inner x u = 0) \\<longleftrightarrow> (x = 0)\"",
"abbreviation real :: \"nat \\<Rightarrow> real\"\n where \"real \\<equiv> of_nat\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"typedef ennreal = \"{x :: ereal. 0 \\<le> x}\"\n morphisms enn2ereal e2ennreal'",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
0 \<le> ?r \<Longrightarrow> emeasure lborel (ball ?c ?r) = ennreal (unit_ball_vol (real DIM(?'a)) * ?r ^ DIM(?'a))
###symbols
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Euclidean_Space.euclidean_space_class.Basis :::: 'a set
Finite_Set.card :::: 'a set \<Rightarrow> nat
Real.real :::: nat \<Rightarrow> real
Ball_Volume.unit_ball_vol :::: real \<Rightarrow> real
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Extended_Nonnegative_Real.ennreal :::: real \<Rightarrow> ennreal
Elementary_Metric_Spaces.ball :::: 'a \<Rightarrow> real \<Rightarrow> 'a set
Lebesgue_Measure.lborel :::: 'a measure
Sigma_Algebra.emeasure :::: 'a measure \<Rightarrow> 'a set \<Rightarrow> ennreal
Groups.zero_class.zero :::: 'a
###defs
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
class euclidean_space = real_inner +
fixes Basis :: "'a set"
assumes nonempty_Basis [simp]: "Basis \<noteq> {}"
assumes finite_Basis [simp]: "finite Basis"
assumes inner_Basis:
"\<lbrakk>u \<in> Basis; v \<in> Basis\<rbrakk> \<Longrightarrow> inner u v = (if u = v then 1 else 0)"
assumes euclidean_all_zero_iff:
"(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> (x = 0)"
abbreviation real :: "nat \<Rightarrow> real"
where "real \<equiv> of_nat"
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
typedef ennreal = "{x :: ereal. 0 \<le> x}"
morphisms enn2ereal e2ennreal'
class zero =
fixes zero :: 'a ("0")
|
###output
?H1 \<le> x_1 \<Longrightarrow> ?H2 ?H3 (?H4 x_2 x_1) = ?H5 (?H6 (?H7 (?H8 (?H9 ?H10))) (?H11 x_1 (?H9 ?H10)))###end
|
Cardinals/Wellorder_Constructions
|
Wellorder_Constructions.omax_ordLess
|
lemma omax_ordLess:
assumes "finite R" and "R \<noteq> {}" and "\<forall> r \<in> R. r <o p"
shows "omax R <o p"
|
finite ?R \<Longrightarrow> ?R \<noteq> {} \<Longrightarrow> \<forall>r\<in> ?R. r <o ?p \<Longrightarrow> omax ?R <o ?p
|
\<lbrakk> ?H1 x_1; x_1 \<noteq> ?H2; \<forall>y_0\<in>x_1. ?H3 y_0 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1) x_2
|
[
"Wellorder_Constructions.omax",
"BNF_Wellorder_Constructions.ordLess2",
"Set.empty",
"Finite_Set.finite"
] |
[
"('a \\<times> 'a) set set \\<Rightarrow> ('a \\<times> 'a) set",
"('a \\<times> 'a) set \\<Rightarrow> ('b \\<times> 'b) set \\<Rightarrow> bool",
"'a set",
"'a set \\<Rightarrow> bool"
] |
[
"definition omax :: \"'a rel set \\<Rightarrow> 'a rel\"\n where\n \"omax R == SOME r. isOmax R r\"",
"abbreviation ordLess2 :: \"'a rel \\<Rightarrow> 'a' rel \\<Rightarrow> bool\" (infix \"<o\" 50)\n where \"r <o r' \\<equiv> (r,r') \\<in> ordLess\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] |
template
|
###lemma
finite ?R \<Longrightarrow> ?R \<noteq> {} \<Longrightarrow> \<forall>r\<in> ?R. r <o ?p \<Longrightarrow> omax ?R <o ?p
###symbols
Wellorder_Constructions.omax :::: ('a \<times> 'a) set set \<Rightarrow> ('a \<times> 'a) set
BNF_Wellorder_Constructions.ordLess2 :::: ('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> bool
Set.empty :::: 'a set
Finite_Set.finite :::: 'a set \<Rightarrow> bool
###defs
definition omax :: "'a rel set \<Rightarrow> 'a rel"
where
"omax R == SOME r. isOmax R r"
abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
where "r <o r' \<equiv> (r,r') \<in> ordLess"
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
|
###output
\<lbrakk> ?H1 x_1; x_1 \<noteq> ?H2; \<forall>y_0\<in>x_1. ?H3 y_0 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1) x_2###end
|
Deriv
|
Deriv.differentiable_inverse
|
lemma differentiable_inverse [simp, derivative_intros]:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
(\<lambda>x. inverse (f x)) differentiable (at x within s)"
|
?f differentiable at ?x within ?s \<Longrightarrow> ?f ?x \<noteq> (0:: ?'b) \<Longrightarrow> (\<lambda>x. inverse (?f x)) differentiable at ?x within ?s
|
\<lbrakk> ?H1 x_1 (?H2 x_2 x_3); x_1 x_2 \<noteq> ?H3\<rbrakk> \<Longrightarrow> ?H1 (\<lambda>y_0. ?H4 (x_1 y_0)) (?H2 x_2 x_3)
|
[
"Fields.inverse_class.inverse",
"Groups.zero_class.zero",
"Topological_Spaces.topological_space_class.at_within",
"Deriv.differentiable"
] |
[
"'a \\<Rightarrow> 'a",
"'a",
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a filter",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a filter \\<Rightarrow> bool"
] |
[
"class inverse = divide +\n fixes inverse :: \"'a \\<Rightarrow> 'a\"\nbegin",
"class zero =\n fixes zero :: 'a (\"0\")",
"class topological_space = \"open\" +\n assumes open_UNIV [simp, intro]: \"open UNIV\"\n assumes open_Int [intro]: \"open S \\<Longrightarrow> open T \\<Longrightarrow> open (S \\<inter> T)\"\n assumes open_Union [intro]: \"\\<forall>S\\<in>K. open S \\<Longrightarrow> open (\\<Union>K)\"\nbegin",
"definition differentiable :: \"('a::real_normed_vector \\<Rightarrow> 'b::real_normed_vector) \\<Rightarrow> 'a filter \\<Rightarrow> bool\"\n (infix \"differentiable\" 50)\n where \"f differentiable F \\<longleftrightarrow> (\\<exists>D. (f has_derivative D) F)\""
] |
template
|
###lemma
?f differentiable at ?x within ?s \<Longrightarrow> ?f ?x \<noteq> (0:: ?'b) \<Longrightarrow> (\<lambda>x. inverse (?f x)) differentiable at ?x within ?s
###symbols
Fields.inverse_class.inverse :::: 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Topological_Spaces.topological_space_class.at_within :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a filter
Deriv.differentiable :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool
###defs
class inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin
class zero =
fixes zero :: 'a ("0")
class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
begin
definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
(infix "differentiable" 50)
where "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
|
###output
\<lbrakk> ?H1 x_1 (?H2 x_2 x_3); x_1 x_2 \<noteq> ?H3\<rbrakk> \<Longrightarrow> ?H1 (\<lambda>y_0. ?H4 (x_1 y_0)) (?H2 x_2 x_3)###end
|
Combinatorics/Orbits
|
Orbits.funpow_neq_less_funpow_dist
|
lemma funpow_neq_less_funpow_dist:
assumes "y \<in> orbit f x" "m \<le> funpow_dist f x y" "n \<le> funpow_dist f x y" "m \<noteq> n"
shows "(f ^^ m) x \<noteq> (f ^^ n) x"
|
?y \<in> orbit ?f ?x \<Longrightarrow> ?m \<le> funpow_dist ?f ?x ?y \<Longrightarrow> ?n \<le> funpow_dist ?f ?x ?y \<Longrightarrow> ?m \<noteq> ?n \<Longrightarrow> (?f ^^ ?m) ?x \<noteq> (?f ^^ ?n) ?x
|
\<lbrakk>x_1 \<in> ?H1 x_2 x_3; x_4 \<le> ?H2 x_2 x_3 x_1; x_5 \<le> ?H2 x_2 x_3 x_1; x_4 \<noteq> x_5\<rbrakk> \<Longrightarrow> ?H3 x_2 x_4 x_3 \<noteq> ?H3 x_2 x_5 x_3
|
[
"Nat.compower",
"Orbits.funpow_dist",
"Orbits.orbit"
] |
[
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'a) \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> nat",
"('a \\<Rightarrow> 'a) \\<Rightarrow> 'a \\<Rightarrow> 'a set"
] |
[
"abbreviation compower :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^^\" 80)\n where \"f ^^ n \\<equiv> compow n f\"",
"definition funpow_dist :: \"('a \\<Rightarrow> 'a) \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> nat\" where\n \"funpow_dist f x y \\<equiv> LEAST n. (f ^^ n) x = y\"",
"inductive_set orbit :: \"('a \\<Rightarrow> 'a) \\<Rightarrow> 'a \\<Rightarrow> 'a set\" for f x where\n base: \"f x \\<in> orbit f x\" |\n step: \"y \\<in> orbit f x \\<Longrightarrow> f y \\<in> orbit f x\""
] |
template
|
###lemma
?y \<in> orbit ?f ?x \<Longrightarrow> ?m \<le> funpow_dist ?f ?x ?y \<Longrightarrow> ?n \<le> funpow_dist ?f ?x ?y \<Longrightarrow> ?m \<noteq> ?n \<Longrightarrow> (?f ^^ ?m) ?x \<noteq> (?f ^^ ?n) ?x
###symbols
Nat.compower :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Orbits.funpow_dist :::: ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> nat
Orbits.orbit :::: ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a set
###defs
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80)
where "f ^^ n \<equiv> compow n f"
definition funpow_dist :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> nat" where
"funpow_dist f x y \<equiv> LEAST n. (f ^^ n) x = y"
inductive_set orbit :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a set" for f x where
base: "f x \<in> orbit f x" |
step: "y \<in> orbit f x \<Longrightarrow> f y \<in> orbit f x"
|
###output
\<lbrakk>x_1 \<in> ?H1 x_2 x_3; x_4 \<le> ?H2 x_2 x_3 x_1; x_5 \<le> ?H2 x_2 x_3 x_1; x_4 \<noteq> x_5\<rbrakk> \<Longrightarrow> ?H3 x_2 x_4 x_3 \<noteq> ?H3 x_2 x_5 x_3###end
|
Probability/Probability
|
Probability_Mass_Function.rel_pmf_imp_rel_set
| null |
rel_pmf ?R ?p ?q \<Longrightarrow> rel_set ?R (set_pmf ?p) (set_pmf ?q)
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 x_1 (?H3 x_2) (?H3 x_3)
|
[
"Probability_Mass_Function.set_pmf",
"BNF_Def.rel_set",
"Probability_Mass_Function.rel_pmf"
] |
[
"'a pmf \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a set \\<Rightarrow> 'b set \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a pmf \\<Rightarrow> 'b pmf \\<Rightarrow> bool"
] |
[
"definition rel_set :: \"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a set \\<Rightarrow> 'b set \\<Rightarrow> bool\"\n where \"rel_set R = (\\<lambda>A B. (\\<forall>x\\<in>A. \\<exists>y\\<in>B. R x y) \\<and> (\\<forall>y\\<in>B. \\<exists>x\\<in>A. R x y))\"",
"inductive rel_pmf :: \"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a pmf \\<Rightarrow> 'b pmf \\<Rightarrow> bool\"\nfor R p q\nwhere\n \"\\<lbrakk> \\<And>x y. (x, y) \\<in> set_pmf pq \\<Longrightarrow> R x y;\n map_pmf fst pq = p; map_pmf snd pq = q \\<rbrakk>\n \\<Longrightarrow> rel_pmf R p q\""
] |
template
|
###lemma
rel_pmf ?R ?p ?q \<Longrightarrow> rel_set ?R (set_pmf ?p) (set_pmf ?q)
###symbols
Probability_Mass_Function.set_pmf :::: 'a pmf \<Rightarrow> 'a set
BNF_Def.rel_set :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool
Probability_Mass_Function.rel_pmf :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool
###defs
definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
where "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
inductive rel_pmf :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf \<Rightarrow> bool"
for R p q
where
"\<lbrakk> \<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y;
map_pmf fst pq = p; map_pmf snd pq = q \<rbrakk>
\<Longrightarrow> rel_pmf R p q"
|
###output
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 x_1 (?H3 x_2) (?H3 x_3)###end
|
MicroJava/DFA/Opt
|
Option.combine_options_simps(1)
| null |
combine_options ?f None ?y = ?y
|
?H1 x_1 ?H2 x_2 = x_2
|
[
"Option.option.None",
"Option.combine_options"
] |
[
"'a option",
"('a \\<Rightarrow> 'a \\<Rightarrow> 'a) \\<Rightarrow> 'a option \\<Rightarrow> 'a option \\<Rightarrow> 'a option"
] |
[
"datatype 'a option =\n None\n | Some (the: 'a)",
"definition combine_options :: \"('a \\<Rightarrow> 'a \\<Rightarrow> 'a) \\<Rightarrow> 'a option \\<Rightarrow> 'a option \\<Rightarrow> 'a option\"\n where \"combine_options f x y = \n (case x of None \\<Rightarrow> y | Some x \\<Rightarrow> (case y of None \\<Rightarrow> Some x | Some y \\<Rightarrow> Some (f x y)))\""
] |
template
|
###lemma
combine_options ?f None ?y = ?y
###symbols
Option.option.None :::: 'a option
Option.combine_options :::: ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a option \<Rightarrow> 'a option \<Rightarrow> 'a option
###defs
datatype 'a option =
None
| Some (the: 'a)
definition combine_options :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a option \<Rightarrow> 'a option \<Rightarrow> 'a option"
where "combine_options f x y =
(case x of None \<Rightarrow> y | Some x \<Rightarrow> (case y of None \<Rightarrow> Some x | Some y \<Rightarrow> Some (f x y)))"
|
###output
?H1 x_1 ?H2 x_2 = x_2###end
|
Metis_Examples/Big_O
|
Big_O.bigo_const_mult7
|
lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
|
?f \<in> O(?g) \<Longrightarrow> (\<lambda>x. ?c * ?f x) \<in> O(?g)
|
x_1 \<in> ?H1 x_2 \<Longrightarrow> (\<lambda>y_0. ?H2 x_3 (x_1 y_0)) \<in> ?H1 x_2
|
[
"Groups.times_class.times",
"Big_O.bigo"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('a \\<Rightarrow> 'b) set"
] |
[
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"definition bigo :: \"('a => 'b::linordered_idom) => ('a => 'b) set\" (\"(1O'(_'))\") where\n \"O(f::('a => 'b)) == {h. \\<exists>c. \\<forall>x. \\<bar>h x\\<bar> <= c * \\<bar>f x\\<bar>}\""
] |
template
|
###lemma
?f \<in> O(?g) \<Longrightarrow> (\<lambda>x. ?c * ?f x) \<in> O(?g)
###symbols
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Big_O.bigo :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
definition bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
"O(f::('a => 'b)) == {h. \<exists>c. \<forall>x. \<bar>h x\<bar> <= c * \<bar>f x\<bar>}"
|
###output
x_1 \<in> ?H1 x_2 \<Longrightarrow> (\<lambda>y_0. ?H2 x_3 (x_1 y_0)) \<in> ?H1 x_2###end
|
Auth/Smartcard/ShoupRubinBella
|
ShoupRubinBella.Inputs_B_Card_6
|
lemma Inputs_B_Card_6:
"\<lbrakk> Inputs B C \<lbrace>Agent A, Nonce Na\<rbrace> \<in> set evs; B \<noteq> Spy; evs \<in> srb \<rbrakk>
\<Longrightarrow> legalUse(C) \<and> C = (Card B) \<and> Gets B \<lbrace>Agent A, Nonce Na\<rbrace> \<in> set evs"
|
Inputs ?B ?C \<lbrace>Agent ?A, Nonce ?Na\<rbrace> \<in> set ?evs \<Longrightarrow> ?B \<noteq> Spy \<Longrightarrow> ?evs \<in> srb \<Longrightarrow> legalUse ?C \<and> ?C = Card ?B \<and> Gets ?B \<lbrace>Agent ?A, Nonce ?Na\<rbrace> \<in> set ?evs
|
\<lbrakk> ?H1 x_1 x_2 (?H2 (?H3 x_3) (?H4 x_4)) \<in> ?H5 x_5; x_1 \<noteq> ?H6; x_5 \<in> ?H7\<rbrakk> \<Longrightarrow> ?H8 x_2 \<and> x_2 = ?H9 x_1 \<and> ?H10 x_1 (?H2 (?H3 x_3) (?H4 x_4)) \<in> ?H5 x_5
|
[
"EventSC.event.Gets",
"EventSC.card.Card",
"Smartcard.legalUse",
"ShoupRubinBella.srb",
"Message.agent.Spy",
"List.list.set",
"Message.msg.Nonce",
"Message.msg.Agent",
"Message.msg.MPair",
"EventSC.event.Inputs"
] |
[
"agent \\<Rightarrow> msg \\<Rightarrow> event",
"agent \\<Rightarrow> card",
"card \\<Rightarrow> bool",
"event list set",
"agent",
"'a list \\<Rightarrow> 'a set",
"nat \\<Rightarrow> msg",
"agent \\<Rightarrow> msg",
"msg \\<Rightarrow> msg \\<Rightarrow> msg",
"agent \\<Rightarrow> card \\<Rightarrow> msg \\<Rightarrow> event"
] |
[
"definition legalUse :: \"card => bool\" (\"legalUse (_)\") where\n \"legalUse C == C \\<notin> stolen\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype\n msg = Agent agent \\<comment> \\<open>Agent names\\<close>\n | Number nat \\<comment> \\<open>Ordinary integers, timestamps, ...\\<close>\n | Nonce nat \\<comment> \\<open>Unguessable nonces\\<close>\n | Key key \\<comment> \\<open>Crypto keys\\<close>\n | Hash msg \\<comment> \\<open>Hashing\\<close>\n | MPair msg msg \\<comment> \\<open>Compound messages\\<close>\n | Crypt key msg \\<comment> \\<open>Encryption, public- or shared-key\\<close>",
"datatype\n msg = Agent agent \\<comment> \\<open>Agent names\\<close>\n | Number nat \\<comment> \\<open>Ordinary integers, timestamps, ...\\<close>\n | Nonce nat \\<comment> \\<open>Unguessable nonces\\<close>\n | Key key \\<comment> \\<open>Crypto keys\\<close>\n | Hash msg \\<comment> \\<open>Hashing\\<close>\n | MPair msg msg \\<comment> \\<open>Compound messages\\<close>\n | Crypt key msg \\<comment> \\<open>Encryption, public- or shared-key\\<close>",
"datatype\n msg = Agent agent \\<comment> \\<open>Agent names\\<close>\n | Number nat \\<comment> \\<open>Ordinary integers, timestamps, ...\\<close>\n | Nonce nat \\<comment> \\<open>Unguessable nonces\\<close>\n | Key key \\<comment> \\<open>Crypto keys\\<close>\n | Hash msg \\<comment> \\<open>Hashing\\<close>\n | MPair msg msg \\<comment> \\<open>Compound messages\\<close>\n | Crypt key msg \\<comment> \\<open>Encryption, public- or shared-key\\<close>"
] |
template
|
###lemma
Inputs ?B ?C \<lbrace>Agent ?A, Nonce ?Na\<rbrace> \<in> set ?evs \<Longrightarrow> ?B \<noteq> Spy \<Longrightarrow> ?evs \<in> srb \<Longrightarrow> legalUse ?C \<and> ?C = Card ?B \<and> Gets ?B \<lbrace>Agent ?A, Nonce ?Na\<rbrace> \<in> set ?evs
###symbols
EventSC.event.Gets :::: agent \<Rightarrow> msg \<Rightarrow> event
EventSC.card.Card :::: agent \<Rightarrow> card
Smartcard.legalUse :::: card \<Rightarrow> bool
ShoupRubinBella.srb :::: event list set
Message.agent.Spy :::: agent
List.list.set :::: 'a list \<Rightarrow> 'a set
Message.msg.Nonce :::: nat \<Rightarrow> msg
Message.msg.Agent :::: agent \<Rightarrow> msg
Message.msg.MPair :::: msg \<Rightarrow> msg \<Rightarrow> msg
EventSC.event.Inputs :::: agent \<Rightarrow> card \<Rightarrow> msg \<Rightarrow> event
###defs
definition legalUse :: "card => bool" ("legalUse (_)") where
"legalUse C == C \<notin> stolen"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
datatype
msg = Agent agent \<comment> \<open>Agent names\<close>
| Number nat \<comment> \<open>Ordinary integers, timestamps, ...\<close>
| Nonce nat \<comment> \<open>Unguessable nonces\<close>
| Key key \<comment> \<open>Crypto keys\<close>
| Hash msg \<comment> \<open>Hashing\<close>
| MPair msg msg \<comment> \<open>Compound messages\<close>
| Crypt key msg \<comment> \<open>Encryption, public- or shared-key\<close>
datatype
msg = Agent agent \<comment> \<open>Agent names\<close>
| Number nat \<comment> \<open>Ordinary integers, timestamps, ...\<close>
| Nonce nat \<comment> \<open>Unguessable nonces\<close>
| Key key \<comment> \<open>Crypto keys\<close>
| Hash msg \<comment> \<open>Hashing\<close>
| MPair msg msg \<comment> \<open>Compound messages\<close>
| Crypt key msg \<comment> \<open>Encryption, public- or shared-key\<close>
datatype
msg = Agent agent \<comment> \<open>Agent names\<close>
| Number nat \<comment> \<open>Ordinary integers, timestamps, ...\<close>
| Nonce nat \<comment> \<open>Unguessable nonces\<close>
| Key key \<comment> \<open>Crypto keys\<close>
| Hash msg \<comment> \<open>Hashing\<close>
| MPair msg msg \<comment> \<open>Compound messages\<close>
| Crypt key msg \<comment> \<open>Encryption, public- or shared-key\<close>
|
###output
\<lbrakk> ?H1 x_1 x_2 (?H2 (?H3 x_3) (?H4 x_4)) \<in> ?H5 x_5; x_1 \<noteq> ?H6; x_5 \<in> ?H7\<rbrakk> \<Longrightarrow> ?H8 x_2 \<and> x_2 = ?H9 x_1 \<and> ?H10 x_1 (?H2 (?H3 x_3) (?H4 x_4)) \<in> ?H5 x_5###end
|
HOLCF/FOCUS/Stream_adm
|
Stream_adm.infinite_chain_adm_lemma
|
lemma infinite_chain_adm_lemma:
"\<lbrakk>Porder.chain Y; \<forall>i. P (Y i);
\<And>Y. \<lbrakk>Porder.chain Y; \<forall>i. P (Y i); \<not> finite_chain Y\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk>
\<Longrightarrow> P (\<Squnion>i. Y i)"
|
chain ?Y \<Longrightarrow> \<forall>i. ?P (?Y i) \<Longrightarrow> (\<And>Y. chain Y \<Longrightarrow> \<forall>i. ?P (Y i) \<Longrightarrow> \<not> finite_chain Y \<Longrightarrow> ?P (\<Squnion>i. Y i)) \<Longrightarrow> ?P (\<Squnion>i. ?Y i)
|
\<lbrakk> ?H1 x_1; \<forall>y_0. x_2 (x_1 y_0); \<And>y_1. \<lbrakk> ?H1 y_1; \<forall>y_2. x_2 (y_1 y_2); \<not> ?H2 y_1\<rbrakk> \<Longrightarrow> x_2 (?H3 y_1)\<rbrakk> \<Longrightarrow> x_2 (?H3 x_1)
|
[
"Porder.po_class.Lub",
"Porder.po_class.finite_chain",
"Porder.po_class.chain"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b",
"(nat \\<Rightarrow> 'a) \\<Rightarrow> bool",
"(nat \\<Rightarrow> 'a) \\<Rightarrow> bool"
] |
[
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin",
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin",
"class po = below +\n assumes below_refl [iff]: \"x \\<sqsubseteq> x\"\n assumes below_trans: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> z \\<Longrightarrow> x \\<sqsubseteq> z\"\n assumes below_antisym: \"x \\<sqsubseteq> y \\<Longrightarrow> y \\<sqsubseteq> x \\<Longrightarrow> x = y\"\nbegin"
] |
template
|
###lemma
chain ?Y \<Longrightarrow> \<forall>i. ?P (?Y i) \<Longrightarrow> (\<And>Y. chain Y \<Longrightarrow> \<forall>i. ?P (Y i) \<Longrightarrow> \<not> finite_chain Y \<Longrightarrow> ?P (\<Squnion>i. Y i)) \<Longrightarrow> ?P (\<Squnion>i. ?Y i)
###symbols
Porder.po_class.Lub :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'b
Porder.po_class.finite_chain :::: (nat \<Rightarrow> 'a) \<Rightarrow> bool
Porder.po_class.chain :::: (nat \<Rightarrow> 'a) \<Rightarrow> bool
###defs
class po = below +
assumes below_refl [iff]: "x \<sqsubseteq> x"
assumes below_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
assumes below_antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
begin
class po = below +
assumes below_refl [iff]: "x \<sqsubseteq> x"
assumes below_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
assumes below_antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
begin
class po = below +
assumes below_refl [iff]: "x \<sqsubseteq> x"
assumes below_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
assumes below_antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
begin
|
###output
\<lbrakk> ?H1 x_1; \<forall>y_0. x_2 (x_1 y_0); \<And>y_1. \<lbrakk> ?H1 y_1; \<forall>y_2. x_2 (y_1 y_2); \<not> ?H2 y_1\<rbrakk> \<Longrightarrow> x_2 (?H3 y_1)\<rbrakk> \<Longrightarrow> x_2 (?H3 x_1)###end
|
Library/Word
|
Word.word_rec_id_eq
|
lemma word_rec_id_eq: "(\<And>m. m < n \<Longrightarrow> f m = id) \<Longrightarrow> word_rec z f n = z"
|
(\<And>m. m < ?n \<Longrightarrow> ?f m = id) \<Longrightarrow> word_rec ?z ?f ?n = ?z
|
(\<And>y_0. y_0 < x_1 \<Longrightarrow> x_2 y_0 = ?H1) \<Longrightarrow> ?H2 x_3 x_2 x_1 = x_3
|
[
"Word.word_rec",
"Fun.id"
] |
[
"'a \\<Rightarrow> ('b word \\<Rightarrow> 'a \\<Rightarrow> 'a) \\<Rightarrow> 'b word \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a"
] |
[
"definition word_rec :: \"'a \\<Rightarrow> ('b::len word \\<Rightarrow> 'a \\<Rightarrow> 'a) \\<Rightarrow> 'b word \\<Rightarrow> 'a\"\n where \"word_rec forZero forSuc n = rec_nat forZero (forSuc \\<circ> of_nat) (unat n)\"",
"definition id :: \"'a \\<Rightarrow> 'a\"\n where \"id = (\\<lambda>x. x)\""
] |
template
|
###lemma
(\<And>m. m < ?n \<Longrightarrow> ?f m = id) \<Longrightarrow> word_rec ?z ?f ?n = ?z
###symbols
Word.word_rec :::: 'a \<Rightarrow> ('b word \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b word \<Rightarrow> 'a
Fun.id :::: 'a \<Rightarrow> 'a
###defs
definition word_rec :: "'a \<Rightarrow> ('b::len word \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b word \<Rightarrow> 'a"
where "word_rec forZero forSuc n = rec_nat forZero (forSuc \<circ> of_nat) (unat n)"
definition id :: "'a \<Rightarrow> 'a"
where "id = (\<lambda>x. x)"
|
###output
(\<And>y_0. y_0 < x_1 \<Longrightarrow> x_2 y_0 = ?H1) \<Longrightarrow> ?H2 x_3 x_2 x_1 = x_3###end
|
Nonstandard_Analysis/NSA
|
NSA.hypreal_gt_isUb
|
lemma hypreal_gt_isUb: "isUb R S x \<Longrightarrow> x < y \<Longrightarrow> y \<in> R \<Longrightarrow> isUb R S y"
for x y :: hypreal
|
isUb ?R ?S ?x \<Longrightarrow> ?x < ?y \<Longrightarrow> ?y \<in> ?R \<Longrightarrow> isUb ?R ?S ?y
|
\<lbrakk> ?H1 x_1 x_2 x_3; x_3 < x_4; x_4 \<in> x_1\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 x_4
|
[
"Lub_Glb.isUb"
] |
[
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition isUb :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a::ord \\<Rightarrow> bool\"\n where \"isUb R S x = (S *<= x \\<and> x \\<in> R)\""
] |
template
|
###lemma
isUb ?R ?S ?x \<Longrightarrow> ?x < ?y \<Longrightarrow> ?y \<in> ?R \<Longrightarrow> isUb ?R ?S ?y
###symbols
Lub_Glb.isUb :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool
###defs
definition isUb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "isUb R S x = (S *<= x \<and> x \<in> R)"
|
###output
\<lbrakk> ?H1 x_1 x_2 x_3; x_3 < x_4; x_4 \<in> x_1\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 x_4###end
|
Computational_Algebra/Normalized_Fraction
|
Normalized_Fraction.normalize_quot_aux(4)
|
lemma normalize_quot_aux:
fixes a b
assumes "b \<noteq> 0"
defines "d \<equiv> gcd a b * unit_factor b"
shows "a = fst (normalize_quot (a,b)) * d" "b = snd (normalize_quot (a,b)) * d"
"d dvd a" "d dvd b" "d \<noteq> 0"
|
?b \<noteq> (0:: ?'a) \<Longrightarrow> gcd ?a ?b * unit_factor ?b dvd ?b
|
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (?H4 x_2 x_1) (?H5 x_1)) x_1
|
[
"Rings.unit_factor_class.unit_factor",
"GCD.gcd_class.gcd",
"Groups.times_class.times",
"Rings.dvd_class.dvd",
"Groups.zero_class.zero"
] |
[
"'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a"
] |
[
"class unit_factor =\n fixes unit_factor :: \"'a \\<Rightarrow> 'a\"",
"class gcd = zero + one + dvd +\n fixes gcd :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\"\n and lcm :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"definition dvd :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" (infix \"dvd\" 50)\n where \"b dvd a \\<longleftrightarrow> (\\<exists>k. a = b * k)\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
?b \<noteq> (0:: ?'a) \<Longrightarrow> gcd ?a ?b * unit_factor ?b dvd ?b
###symbols
Rings.unit_factor_class.unit_factor :::: 'a \<Rightarrow> 'a
GCD.gcd_class.gcd :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Rings.dvd_class.dvd :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
Groups.zero_class.zero :::: 'a
###defs
class unit_factor =
fixes unit_factor :: "'a \<Rightarrow> 'a"
class gcd = zero + one + dvd +
fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
class zero =
fixes zero :: 'a ("0")
|
###output
x_1 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (?H4 x_2 x_1) (?H5 x_1)) x_1###end
|
Record
|
Record.update_accessor_congruence_foldE
|
lemma update_accessor_congruence_foldE:
assumes uac: "iso_tuple_update_accessor_cong_assist upd ac"
and r: "r = r'" and v: "ac r' = v'"
and f: "\<And>v. v' = v \<Longrightarrow> f v = f' v"
shows "upd f r = upd f' r'"
|
Record.iso_tuple_update_accessor_cong_assist ?upd ?ac \<Longrightarrow> ?r = ?r' \<Longrightarrow> ?ac ?r' = ?v' \<Longrightarrow> (\<And>v. ?v' = v \<Longrightarrow> ?f v = ?f' v) \<Longrightarrow> ?upd ?f ?r = ?upd ?f' ?r'
|
\<lbrakk> ?H1 x_1 x_2; x_3 = x_4; x_2 x_4 = x_5; \<And>y_0. x_5 = y_0 \<Longrightarrow> x_6 y_0 = x_7 y_0\<rbrakk> \<Longrightarrow> x_1 x_6 x_3 = x_1 x_7 x_4
|
[
"Record.iso_tuple_update_accessor_cong_assist"
] |
[
"(('a \\<Rightarrow> 'a) \\<Rightarrow> 'b \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a) \\<Rightarrow> bool"
] |
[
"definition\n iso_tuple_update_accessor_cong_assist ::\n \"(('b \\<Rightarrow> 'b) \\<Rightarrow> ('a \\<Rightarrow> 'a)) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool\" where\n \"iso_tuple_update_accessor_cong_assist upd ac \\<longleftrightarrow>\n (\\<forall>f v. upd (\\<lambda>x. f (ac v)) v = upd f v) \\<and> (\\<forall>v. upd id v = v)\""
] |
template
|
###lemma
Record.iso_tuple_update_accessor_cong_assist ?upd ?ac \<Longrightarrow> ?r = ?r' \<Longrightarrow> ?ac ?r' = ?v' \<Longrightarrow> (\<And>v. ?v' = v \<Longrightarrow> ?f v = ?f' v) \<Longrightarrow> ?upd ?f ?r = ?upd ?f' ?r'
###symbols
Record.iso_tuple_update_accessor_cong_assist :::: (('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool
###defs
definition
iso_tuple_update_accessor_cong_assist ::
"(('b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'a)) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
"iso_tuple_update_accessor_cong_assist upd ac \<longleftrightarrow>
(\<forall>f v. upd (\<lambda>x. f (ac v)) v = upd f v) \<and> (\<forall>v. upd id v = v)"
|
###output
\<lbrakk> ?H1 x_1 x_2; x_3 = x_4; x_2 x_4 = x_5; \<And>y_0. x_5 = y_0 \<Longrightarrow> x_6 y_0 = x_7 y_0\<rbrakk> \<Longrightarrow> x_1 x_6 x_3 = x_1 x_7 x_4###end
|
Analysis/Tagged_Division
|
Tagged_Division.tagged_division_split_right_inj
|
lemma tagged_division_split_right_inj:
assumes d: "d tagged_division_of i"
and tags: "(x1, K1) \<in> d" "(x2, K2) \<in> d"
and "K1 \<noteq> K2"
and eq: "K1 \<inter> {x. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}"
shows "interior (K1 \<inter> {x. x\<bullet>k \<ge> c}) = {}"
|
?d tagged_division_of ?i \<Longrightarrow> (?x1.0, ?K1.0) \<in> ?d \<Longrightarrow> (?x2.0, ?K2.0) \<in> ?d \<Longrightarrow> ?K1.0 \<noteq> ?K2.0 \<Longrightarrow> ?K1.0 \<inter> {x. ?c \<le> x \<bullet> ?k} = ?K2.0 \<inter> {x. ?c \<le> x \<bullet> ?k} \<Longrightarrow> interior (?K1.0 \<inter> {x. ?c \<le> x \<bullet> ?k}) = {}
|
\<lbrakk> ?H1 x_1 x_2; (x_3, x_4) \<in> x_1; (x_5, x_6) \<in> x_1; x_4 \<noteq> x_6; ?H2 x_4 (?H3 (\<lambda>y_0. x_7 \<le> ?H4 y_0 x_8)) = ?H2 x_6 (?H3 (\<lambda>y_1. x_7 \<le> ?H4 y_1 x_8))\<rbrakk> \<Longrightarrow> ?H5 (?H2 x_4 (?H3 (\<lambda>y_2. x_7 \<le> ?H4 y_2 x_8))) = ?H6
|
[
"Set.empty",
"Elementary_Topology.interior",
"Inner_Product.real_inner_class.inner",
"Set.Collect",
"Set.inter",
"Tagged_Division.tagged_division_of"
] |
[
"'a set",
"'a set \\<Rightarrow> 'a set",
"'a \\<Rightarrow> 'a \\<Rightarrow> real",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"('a \\<times> 'a set) set \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +\n fixes inner :: \"'a \\<Rightarrow> 'a \\<Rightarrow> real\"\n assumes inner_commute: \"inner x y = inner y x\"\n and inner_add_left: \"inner (x + y) z = inner x z + inner y z\"\n and inner_scaleR_left [simp]: \"inner (scaleR r x) y = r * (inner x y)\"\n and inner_ge_zero [simp]: \"0 \\<le> inner x x\"\n and inner_eq_zero_iff [simp]: \"inner x x = 0 \\<longleftrightarrow> x = 0\"\n and norm_eq_sqrt_inner: \"norm x = sqrt (inner x x)\"\nbegin",
"abbreviation inter :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<inter>\" 70)\n where \"(\\<inter>) \\<equiv> inf\""
] |
template
|
###lemma
?d tagged_division_of ?i \<Longrightarrow> (?x1.0, ?K1.0) \<in> ?d \<Longrightarrow> (?x2.0, ?K2.0) \<in> ?d \<Longrightarrow> ?K1.0 \<noteq> ?K2.0 \<Longrightarrow> ?K1.0 \<inter> {x. ?c \<le> x \<bullet> ?k} = ?K2.0 \<inter> {x. ?c \<le> x \<bullet> ?k} \<Longrightarrow> interior (?K1.0 \<inter> {x. ?c \<le> x \<bullet> ?k}) = {}
###symbols
Set.empty :::: 'a set
Elementary_Topology.interior :::: 'a set \<Rightarrow> 'a set
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Set.inter :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
Tagged_Division.tagged_division_of :::: ('a \<times> 'a set) set \<Rightarrow> 'a set \<Rightarrow> bool
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
assumes inner_commute: "inner x y = inner y x"
and inner_add_left: "inner (x + y) z = inner x z + inner y z"
and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)"
and inner_ge_zero [simp]: "0 \<le> inner x x"
and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> x = 0"
and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)"
begin
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<inter>" 70)
where "(\<inter>) \<equiv> inf"
|
###output
\<lbrakk> ?H1 x_1 x_2; (x_3, x_4) \<in> x_1; (x_5, x_6) \<in> x_1; x_4 \<noteq> x_6; ?H2 x_4 (?H3 (\<lambda>y_0. x_7 \<le> ?H4 y_0 x_8)) = ?H2 x_6 (?H3 (\<lambda>y_1. x_7 \<le> ?H4 y_1 x_8))\<rbrakk> \<Longrightarrow> ?H5 (?H2 x_4 (?H3 (\<lambda>y_2. x_7 \<le> ?H4 y_2 x_8))) = ?H6###end
|
Examples/Gauss_Numbers
|
Gauss_Numbers.Im_i_times
|
lemma Im_i_times [simp]:
\<open>Im (\<i> * z) = Re z\<close>
|
Im (\<i> * ?z) = Re ?z
|
?H1 (?H2 ?H3 x_1) = ?H4 x_1
|
[
"Gauss_Numbers.gauss.Re",
"Gauss_Numbers.imaginary_unit",
"Groups.times_class.times",
"Gauss_Numbers.gauss.Im"
] |
[
"gauss \\<Rightarrow> int",
"gauss",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"gauss \\<Rightarrow> int"
] |
[
"codatatype gauss = Gauss (Re: int) (Im: int)",
"primcorec imaginary_unit :: gauss (\\<open>\\<i>\\<close>)\n where\n \\<open>Re \\<i> = 0\\<close>\n | \\<open>Im \\<i> = 1\\<close>",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"codatatype gauss = Gauss (Re: int) (Im: int)"
] |
template
|
###lemma
Im (\<i> * ?z) = Re ?z
###symbols
Gauss_Numbers.gauss.Re :::: gauss \<Rightarrow> int
Gauss_Numbers.imaginary_unit :::: gauss
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Gauss_Numbers.gauss.Im :::: gauss \<Rightarrow> int
###defs
codatatype gauss = Gauss (Re: int) (Im: int)
primcorec imaginary_unit :: gauss (\<open>\<i>\<close>)
where
\<open>Re \<i> = 0\<close>
| \<open>Im \<i> = 1\<close>
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
codatatype gauss = Gauss (Re: int) (Im: int)
|
###output
?H1 (?H2 ?H3 x_1) = ?H4 x_1###end
|
Library/Complete_Partial_Order2
|
Complete_Partial_Order2.mcont_call
|
lemma mcont_call [cont_intro, simp]:
"mcont (fun_lub lub) (fun_ord ord) lub ord (\<lambda>f. f t)"
|
mcont (fun_lub ?lub) (fun_ord ?ord) ?lub ?ord (\<lambda>f. f ?t)
|
?H1 (?H2 x_1) (?H3 x_2) x_1 x_2 (\<lambda>y_0. y_0 x_3)
|
[
"Partial_Function.fun_ord",
"Partial_Function.fun_lub",
"Complete_Partial_Order2.mcont"
] |
[
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('c \\<Rightarrow> 'a) \\<Rightarrow> ('c \\<Rightarrow> 'b) \\<Rightarrow> bool",
"('a set \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'a) set \\<Rightarrow> 'c \\<Rightarrow> 'b",
"('a set \\<Rightarrow> 'a) \\<Rightarrow> ('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('b set \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"definition \"fun_ord ord f g \\<longleftrightarrow> (\\<forall>x. ord (f x) (g x))\"",
"definition \"fun_lub L A = (\\<lambda>x. L {y. \\<exists>f\\<in>A. y = f x})\"",
"definition mcont :: \"('a set \\<Rightarrow> 'a) \\<Rightarrow> ('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('b set \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\nwhere\n \"mcont luba orda lubb ordb f \\<longleftrightarrow>\n monotone orda ordb f \\<and> cont luba orda lubb ordb f\""
] |
template
|
###lemma
mcont (fun_lub ?lub) (fun_ord ?ord) ?lub ?ord (\<lambda>f. f ?t)
###symbols
Partial_Function.fun_ord :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> bool
Partial_Function.fun_lub :::: ('a set \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'a) set \<Rightarrow> 'c \<Rightarrow> 'b
Complete_Partial_Order2.mcont :::: ('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b set \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition "fun_ord ord f g \<longleftrightarrow> (\<forall>x. ord (f x) (g x))"
definition "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
definition mcont :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b set \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
where
"mcont luba orda lubb ordb f \<longleftrightarrow>
monotone orda ordb f \<and> cont luba orda lubb ordb f"
|
###output
?H1 (?H2 x_1) (?H3 x_2) x_1 x_2 (\<lambda>y_0. y_0 x_3)###end
|
Bali/Trans
|
Transfer.transfer_raw(29)
| null |
bi_total ?A \<Longrightarrow> bi_unique ?A \<Longrightarrow> Transfer.Rel (rel_fun (rel_fun (rel_fun ?A (=)) (=)) (=)) is_filter is_filter
|
\<lbrakk> ?H1 x_1; ?H2 x_1\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H4 (?H4 x_1 (=)) (=)) (=)) ?H5 ?H5
|
[
"Filter.is_filter",
"BNF_Def.rel_fun",
"Transfer.Rel",
"Transfer.bi_unique",
"Transfer.bi_total"
] |
[
"(('a \\<Rightarrow> bool) \\<Rightarrow> bool) \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('c \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'c) \\<Rightarrow> ('b \\<Rightarrow> 'd) \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a \\<Rightarrow> 'b \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> bool"
] |
[
"definition\n rel_fun :: \"('a \\<Rightarrow> 'c \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'd) \\<Rightarrow> bool\"\nwhere\n \"rel_fun A B = (\\<lambda>f g. \\<forall>x y. A x y \\<longrightarrow> B (f x) (g y))\"",
"definition Rel :: \"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a \\<Rightarrow> 'b \\<Rightarrow> bool\"\n where \"Rel r \\<equiv> r\"",
"definition bi_unique :: \"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"bi_unique R \\<longleftrightarrow>\n (\\<forall>x y z. R x y \\<longrightarrow> R x z \\<longrightarrow> y = z) \\<and>\n (\\<forall>x y z. R x z \\<longrightarrow> R y z \\<longrightarrow> x = y)\"",
"definition bi_total :: \"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"bi_total R \\<longleftrightarrow> (\\<forall>x. \\<exists>y. R x y) \\<and> (\\<forall>y. \\<exists>x. R x y)\""
] |
template
|
###lemma
bi_total ?A \<Longrightarrow> bi_unique ?A \<Longrightarrow> Transfer.Rel (rel_fun (rel_fun (rel_fun ?A (=)) (=)) (=)) is_filter is_filter
###symbols
Filter.is_filter :::: (('a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> bool
BNF_Def.rel_fun :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> bool
Transfer.Rel :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool
Transfer.bi_unique :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool
Transfer.bi_total :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool
###defs
definition
rel_fun :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"
where
"rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
where "Rel r \<equiv> r"
definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "bi_unique R \<longleftrightarrow>
(\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
(\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
|
###output
\<lbrakk> ?H1 x_1; ?H2 x_1\<rbrakk> \<Longrightarrow> ?H3 (?H4 (?H4 (?H4 x_1 (=)) (=)) (=)) ?H5 ?H5###end
|
Library/FuncSet
|
FuncSet.Pi_Int
|
lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
|
Pi ?I ?E \<inter> Pi ?I ?F = (\<Pi> i\<in> ?I. ?E i \<inter> ?F i)
|
?H1 (?H2 x_1 x_2) (?H2 x_1 x_3) = ?H2 x_1 (\<lambda>y_0. ?H1 (x_2 y_0) (x_3 y_0))
|
[
"FuncSet.Pi",
"Set.inter"
] |
[
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b set) \\<Rightarrow> ('a \\<Rightarrow> 'b) set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set"
] |
[
"definition Pi :: \"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b set) \\<Rightarrow> ('a \\<Rightarrow> 'b) set\"\n where \"Pi A B = {f. \\<forall>x. x \\<in> A \\<longrightarrow> f x \\<in> B x}\"",
"abbreviation inter :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<inter>\" 70)\n where \"(\\<inter>) \\<equiv> inf\""
] |
template
|
###lemma
Pi ?I ?E \<inter> Pi ?I ?F = (\<Pi> i\<in> ?I. ?E i \<inter> ?F i)
###symbols
FuncSet.Pi :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set
Set.inter :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
###defs
definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<inter>" 70)
where "(\<inter>) \<equiv> inf"
|
###output
?H1 (?H2 x_1 x_2) (?H2 x_1 x_3) = ?H2 x_1 (\<lambda>y_0. ?H1 (x_2 y_0) (x_3 y_0))###end
|
Analysis/Tagged_Division
|
Tagged_Division.additive_tagged_division_1
|
lemma additive_tagged_division_1:
fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
assumes "a \<le> b"
and "p tagged_division_of {a..b}"
shows "sum (\<lambda>(x,K). f(Sup K) - f(Inf K)) p = f b - f a"
|
?a \<le> ?b \<Longrightarrow> ?p tagged_division_of { ?a.. ?b} \<Longrightarrow> (\<Sum>(x, K)\<in> ?p. ?f (Sup K) - ?f (Inf K)) = ?f ?b - ?f ?a
|
\<lbrakk>x_1 \<le> x_2; ?H1 x_3 (?H2 x_1 x_2)\<rbrakk> \<Longrightarrow> ?H3 (?H4 (\<lambda>y_0 y_1. ?H5 (x_4 (?H6 y_1)) (x_4 (?H7 y_1)))) x_3 = ?H5 (x_4 x_2) (x_4 x_1)
|
[
"Complete_Lattices.Inf_class.Inf",
"Complete_Lattices.Sup_class.Sup",
"Groups.minus_class.minus",
"Product_Type.prod.case_prod",
"Groups_Big.comm_monoid_add_class.sum",
"Set_Interval.ord_class.atLeastAtMost",
"Tagged_Division.tagged_division_of"
] |
[
"'a set \\<Rightarrow> 'a",
"'a set \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b \\<Rightarrow> 'c) \\<Rightarrow> 'a \\<times> 'b \\<Rightarrow> 'c",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a set",
"('a \\<times> 'a set) set \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"class Inf =\n fixes Inf :: \"'a set \\<Rightarrow> 'a\" (\"\\<Sqinter> _\" [900] 900)",
"class Sup =\n fixes Sup :: \"'a set \\<Rightarrow> 'a\" (\"\\<Squnion> _\" [900] 900)",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\""
] |
template
|
###lemma
?a \<le> ?b \<Longrightarrow> ?p tagged_division_of { ?a.. ?b} \<Longrightarrow> (\<Sum>(x, K)\<in> ?p. ?f (Sup K) - ?f (Inf K)) = ?f ?b - ?f ?a
###symbols
Complete_Lattices.Inf_class.Inf :::: 'a set \<Rightarrow> 'a
Complete_Lattices.Sup_class.Sup :::: 'a set \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Product_Type.prod.case_prod :::: ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c
Groups_Big.comm_monoid_add_class.sum :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b
Set_Interval.ord_class.atLeastAtMost :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a set
Tagged_Division.tagged_division_of :::: ('a \<times> 'a set) set \<Rightarrow> 'a set \<Rightarrow> bool
###defs
class Inf =
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter> _" [900] 900)
class Sup =
fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion> _" [900] 900)
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
|
###output
\<lbrakk>x_1 \<le> x_2; ?H1 x_3 (?H2 x_1 x_2)\<rbrakk> \<Longrightarrow> ?H3 (?H4 (\<lambda>y_0 y_1. ?H5 (x_4 (?H6 y_1)) (x_4 (?H7 y_1)))) x_3 = ?H5 (x_4 x_2) (x_4 x_1)###end
|
Analysis/Abstract_Euclidean_Space
|
Abstract_Euclidean_Space.Hausdorff_Euclidean_space
|
lemma Hausdorff_Euclidean_space:
"Hausdorff_space (Euclidean_space n)"
|
Hausdorff_space (Euclidean_space ?n)
|
?H1 (?H2 x_1)
|
[
"Abstract_Euclidean_Space.Euclidean_space",
"T1_Spaces.Hausdorff_space"
] |
[
"nat \\<Rightarrow> (nat \\<Rightarrow> real) topology",
"'a topology \\<Rightarrow> bool"
] |
[
"definition Euclidean_space :: \"nat \\<Rightarrow> (nat \\<Rightarrow> real) topology\"\n where \"Euclidean_space n \\<equiv> subtopology (powertop_real UNIV) {x. \\<forall>i\\<ge>n. x i = 0}\"",
"definition Hausdorff_space\n where\n \"Hausdorff_space X \\<equiv>\n \\<forall>x y. x \\<in> topspace X \\<and> y \\<in> topspace X \\<and> (x \\<noteq> y)\n \\<longrightarrow> (\\<exists>U V. openin X U \\<and> openin X V \\<and> x \\<in> U \\<and> y \\<in> V \\<and> disjnt U V)\""
] |
template
|
###lemma
Hausdorff_space (Euclidean_space ?n)
###symbols
Abstract_Euclidean_Space.Euclidean_space :::: nat \<Rightarrow> (nat \<Rightarrow> real) topology
T1_Spaces.Hausdorff_space :::: 'a topology \<Rightarrow> bool
###defs
definition Euclidean_space :: "nat \<Rightarrow> (nat \<Rightarrow> real) topology"
where "Euclidean_space n \<equiv> subtopology (powertop_real UNIV) {x. \<forall>i\<ge>n. x i = 0}"
definition Hausdorff_space
where
"Hausdorff_space X \<equiv>
\<forall>x y. x \<in> topspace X \<and> y \<in> topspace X \<and> (x \<noteq> y)
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> x \<in> U \<and> y \<in> V \<and> disjnt U V)"
|
###output
?H1 (?H2 x_1)###end
|
Nonstandard_Analysis/HyperNat
|
HyperNat.of_hypnat_add
|
lemma of_hypnat_add [simp]: "\<And>m n. of_hypnat (m + n) = of_hypnat m + of_hypnat n"
|
of_hypnat (?m + ?n) = of_hypnat ?m + of_hypnat ?n
|
?H1 (?H2 x_1 x_2) = ?H2 (?H1 x_1) (?H1 x_2)
|
[
"Groups.plus_class.plus",
"HyperNat.of_hypnat"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"nat star \\<Rightarrow> 'a star"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition of_hypnat :: \"hypnat \\<Rightarrow> 'a::semiring_1_cancel star\"\n where of_hypnat_def [transfer_unfold]: \"of_hypnat = *f* of_nat\""
] |
template
|
###lemma
of_hypnat (?m + ?n) = of_hypnat ?m + of_hypnat ?n
###symbols
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
HyperNat.of_hypnat :::: nat star \<Rightarrow> 'a star
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
definition of_hypnat :: "hypnat \<Rightarrow> 'a::semiring_1_cancel star"
where of_hypnat_def [transfer_unfold]: "of_hypnat = *f* of_nat"
|
###output
?H1 (?H2 x_1 x_2) = ?H2 (?H1 x_1) (?H1 x_2)###end
|
Topological_Spaces
|
Topological_Spaces.tendsto_compose_filtermap
|
lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)"
|
((?g \<circ> ?f) \<longlongrightarrow> ?T) ?F = (?g \<longlongrightarrow> ?T) (filtermap ?f ?F)
|
?H1 (?H2 x_1 x_2) x_3 x_4 = ?H1 x_1 x_3 (?H3 x_2 x_4)
|
[
"Filter.filtermap",
"Fun.comp",
"Topological_Spaces.topological_space_class.tendsto"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a filter \\<Rightarrow> 'b filter",
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'a) \\<Rightarrow> 'c \\<Rightarrow> 'b",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b \\<Rightarrow> 'a filter \\<Rightarrow> bool"
] |
[
"definition filtermap :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a filter \\<Rightarrow> 'b filter\"\n where \"filtermap f F = Abs_filter (\\<lambda>P. eventually (\\<lambda>x. P (f x)) F)\"",
"definition comp :: \"('b \\<Rightarrow> 'c) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'c\" (infixl \"\\<circ>\" 55)\n where \"f \\<circ> g = (\\<lambda>x. f (g x))\"",
"class topological_space = \"open\" +\n assumes open_UNIV [simp, intro]: \"open UNIV\"\n assumes open_Int [intro]: \"open S \\<Longrightarrow> open T \\<Longrightarrow> open (S \\<inter> T)\"\n assumes open_Union [intro]: \"\\<forall>S\\<in>K. open S \\<Longrightarrow> open (\\<Union>K)\"\nbegin"
] |
template
|
###lemma
((?g \<circ> ?f) \<longlongrightarrow> ?T) ?F = (?g \<longlongrightarrow> ?T) (filtermap ?f ?F)
###symbols
Filter.filtermap :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter
Fun.comp :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'b
Topological_Spaces.topological_space_class.tendsto :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool
###defs
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>" 55)
where "f \<circ> g = (\<lambda>x. f (g x))"
class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
begin
|
###output
?H1 (?H2 x_1 x_2) x_3 x_4 = ?H1 x_1 x_3 (?H3 x_2 x_4)###end
|
Analysis/Complex_Transcendental
|
Complex_Transcendental.Arg2pi_inverse_eq_0
|
lemma Arg2pi_inverse_eq_0: "Arg2pi(inverse z) = 0 \<longleftrightarrow> Arg2pi z = 0"
|
(Arg2pi (inverse ?z) = 0) = (Arg2pi ?z = 0)
|
(?H1 (?H2 x_1) = ?H3) = (?H1 x_1 = ?H3)
|
[
"Groups.zero_class.zero",
"Fields.inverse_class.inverse",
"Complex_Transcendental.Arg2pi"
] |
[
"'a",
"'a \\<Rightarrow> 'a",
"complex \\<Rightarrow> real"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"class inverse = divide +\n fixes inverse :: \"'a \\<Rightarrow> 'a\"\nbegin"
] |
template
|
###lemma
(Arg2pi (inverse ?z) = 0) = (Arg2pi ?z = 0)
###symbols
Groups.zero_class.zero :::: 'a
Fields.inverse_class.inverse :::: 'a \<Rightarrow> 'a
Complex_Transcendental.Arg2pi :::: complex \<Rightarrow> real
###defs
class zero =
fixes zero :: 'a ("0")
class inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin
|
###output
(?H1 (?H2 x_1) = ?H3) = (?H1 x_1 = ?H3)###end
|
Number_Theory/Cong
|
Cong.cong_0_1_nat
|
lemma cong_0_1_nat: "[0 = 1] (mod n) \<longleftrightarrow> n = 1"
for n :: nat
|
[0 = 1] (mod ?n) = (?n = 1)
|
?H1 ?H2 ?H3 x_1 = (x_1 = ?H3)
|
[
"Groups.one_class.one",
"Groups.zero_class.zero",
"Cong.unique_euclidean_semiring_class.cong"
] |
[
"'a",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
[0 = 1] (mod ?n) = (?n = 1)
###symbols
Groups.one_class.one :::: 'a
Groups.zero_class.zero :::: 'a
Cong.unique_euclidean_semiring_class.cong :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool
###defs
class one =
fixes one :: 'a ("1")
class zero =
fixes zero :: 'a ("0")
|
###output
?H1 ?H2 ?H3 x_1 = (x_1 = ?H3)###end
|
SET_Protocol/Merchant_Registration
|
Merchant_Registration.Spy_see_private_Key
|
lemma Spy_see_private_Key [simp]:
"evs \<in> set_mr
==> (Key(invKey (publicKey b A)) \<in> parts(knows Spy evs)) = (A \<in> bad)"
|
?evs \<in> set_mr \<Longrightarrow> (Key (invKey (publicKey ?b ?A)) \<in> parts (knows Spy ?evs)) = (?A \<in> bad)
|
x_1 \<in> ?H1 \<Longrightarrow> (?H2 (?H3 (?H4 x_2 x_3)) \<in> ?H5 (?H6 ?H7 x_1)) = (x_3 \<in> ?H8)
|
[
"Event_SET.bad",
"Message_SET.agent.Spy",
"Event_SET.knows",
"Message_SET.parts",
"Public_SET.publicKey",
"Message_SET.invKey",
"Message_SET.msg.Key",
"Merchant_Registration.set_mr"
] |
[
"agent set",
"agent",
"agent \\<Rightarrow> event list \\<Rightarrow> msg set",
"msg set \\<Rightarrow> msg set",
"bool \\<Rightarrow> agent \\<Rightarrow> nat",
"nat \\<Rightarrow> nat",
"nat \\<Rightarrow> msg",
"event list set"
] |
[
"consts bad :: \"agent set\"",
"datatype\n agent = CA nat | Cardholder nat | Merchant nat | PG nat | Spy",
"primrec knows :: \"[agent, event list] \\<Rightarrow> msg set\"\nwhere\n knows_Nil:\n \"knows A [] = initState A\"\n| knows_Cons:\n \"knows A (ev # evs) =\n (if A = Spy then\n (case ev of\n Says A' B X \\<Rightarrow> insert X (knows Spy evs)\n | Gets A' X \\<Rightarrow> knows Spy evs\n | Notes A' X \\<Rightarrow>\n if A' \\<in> bad then insert X (knows Spy evs) else knows Spy evs)\n else\n (case ev of\n Says A' B X \\<Rightarrow>\n if A'=A then insert X (knows A evs) else knows A evs\n | Gets A' X \\<Rightarrow>\n if A'=A then insert X (knows A evs) else knows A evs\n | Notes A' X \\<Rightarrow>\n if A'=A then insert X (knows A evs) else knows A evs))\"",
"inductive_set\n parts :: \"msg set \\<Rightarrow> msg set\"\n for H :: \"msg set\"\n where\n Inj [intro]: \"X \\<in> H ==> X \\<in> parts H\"\n | Fst: \"\\<lbrace>X,Y\\<rbrace> \\<in> parts H ==> X \\<in> parts H\"\n | Snd: \"\\<lbrace>X,Y\\<rbrace> \\<in> parts H ==> Y \\<in> parts H\"\n | Body: \"Crypt K X \\<in> parts H ==> X \\<in> parts H\"",
"consts\n publicKey :: \"[bool, agent] \\<Rightarrow> key\"\n \\<comment> \\<open>the boolean is TRUE if a signing key\\<close>",
"datatype\n msg = Agent agent \\<comment> \\<open>Agent names\\<close>\n | Number nat \\<comment> \\<open>Ordinary integers, timestamps, ...\\<close>\n | Nonce nat \\<comment> \\<open>Unguessable nonces\\<close>\n | Pan nat \\<comment> \\<open>Unguessable Primary Account Numbers (??)\\<close>\n | Key key \\<comment> \\<open>Crypto keys\\<close>\n | Hash msg \\<comment> \\<open>Hashing\\<close>\n | MPair msg msg \\<comment> \\<open>Compound messages\\<close>\n | Crypt key msg \\<comment> \\<open>Encryption, public- or shared-key\\<close>",
"inductive_set\n set_mr :: \"event list set\"\nwhere\n\n Nil: \\<comment> \\<open>Initial trace is empty\\<close>\n \"[] \\<in> set_mr\"\n\n\n| Fake: \\<comment> \\<open>The spy MAY say anything he CAN say.\\<close>\n \"[| evsf \\<in> set_mr; X \\<in> synth (analz (knows Spy evsf)) |]\n ==> Says Spy B X # evsf \\<in> set_mr\"\n \n\n| Reception: \\<comment> \\<open>If A sends a message X to B, then B might receive it\\<close>\n \"[| evsr \\<in> set_mr; Says A B X \\<in> set evsr |]\n ==> Gets B X # evsr \\<in> set_mr\"\n\n\n| SET_MR1: \\<comment> \\<open>RegFormReq: M requires a registration form to a CA\\<close>\n \"[| evs1 \\<in> set_mr; M = Merchant k; Nonce NM1 \\<notin> used evs1 |]\n ==> Says M (CA i) \\<lbrace>Agent M, Nonce NM1\\<rbrace> # evs1 \\<in> set_mr\"\n\n\n| SET_MR2: \\<comment> \\<open>RegFormRes: CA replies with the registration form and the \n certificates for her keys\\<close>\n \"[| evs2 \\<in> set_mr; Nonce NCA \\<notin> used evs2;\n Gets (CA i) \\<lbrace>Agent M, Nonce NM1\\<rbrace> \\<in> set evs2 |]\n ==> Says (CA i) M \\<lbrace>sign (priSK (CA i)) \\<lbrace>Agent M, Nonce NM1, Nonce NCA\\<rbrace>,\n cert (CA i) (pubEK (CA i)) onlyEnc (priSK RCA),\n cert (CA i) (pubSK (CA i)) onlySig (priSK RCA) \\<rbrace>\n # evs2 \\<in> set_mr\"\n\n| SET_MR3:\n \\<comment> \\<open>CertReq: M submits the key pair to be certified. The Notes\n event allows KM1 to be lost if M is compromised. Piero remarks\n that the agent mentioned inside the signature is not verified to\n correspond to M. As in CR, each Merchant has fixed key pairs. M\n is only optionally required to send NCA back, so M doesn't do so\n in the model\\<close>\n \"[| evs3 \\<in> set_mr; M = Merchant k; Nonce NM2 \\<notin> used evs3;\n Key KM1 \\<notin> used evs3; KM1 \\<in> symKeys;\n Gets M \\<lbrace>sign (invKey SKi) \\<lbrace>Agent X, Nonce NM1, Nonce NCA\\<rbrace>,\n cert (CA i) EKi onlyEnc (priSK RCA),\n cert (CA i) SKi onlySig (priSK RCA) \\<rbrace>\n \\<in> set evs3;\n Says M (CA i) \\<lbrace>Agent M, Nonce NM1\\<rbrace> \\<in> set evs3 |]\n ==> Says M (CA i)\n \\<lbrace>Crypt KM1 (sign (priSK M) \\<lbrace>Agent M, Nonce NM2,\n Key (pubSK M), Key (pubEK M)\\<rbrace>),\n Crypt EKi (Key KM1)\\<rbrace>\n # Notes M \\<lbrace>Key KM1, Agent (CA i)\\<rbrace>\n # evs3 \\<in> set_mr\"\n\n| SET_MR4:\n \\<comment> \\<open>CertRes: CA issues the certificates for merSK and merEK,\n while checking never to have certified the m even\n separately. NOTE: In Cardholder Registration the\n corresponding rule (6) doesn't use the \"sign\" primitive. \"The\n CertRes shall be signed but not encrypted if the EE is a Merchant\n or Payment Gateway.\"-- Programmer's Guide, page 191.\\<close>\n \"[| evs4 \\<in> set_mr; M = Merchant k;\n merSK \\<notin> symKeys; merEK \\<notin> symKeys;\n Notes (CA i) (Key merSK) \\<notin> set evs4;\n Notes (CA i) (Key merEK) \\<notin> set evs4;\n Gets (CA i) \\<lbrace>Crypt KM1 (sign (invKey merSK)\n \\<lbrace>Agent M, Nonce NM2, Key merSK, Key merEK\\<rbrace>),\n Crypt (pubEK (CA i)) (Key KM1) \\<rbrace>\n \\<in> set evs4 |]\n ==> Says (CA i) M \\<lbrace>sign (priSK(CA i)) \\<lbrace>Agent M, Nonce NM2, Agent(CA i)\\<rbrace>,\n cert M merSK onlySig (priSK (CA i)),\n cert M merEK onlyEnc (priSK (CA i)),\n cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\\<rbrace>\n # Notes (CA i) (Key merSK)\n # Notes (CA i) (Key merEK)\n # evs4 \\<in> set_mr\""
] |
template
|
###lemma
?evs \<in> set_mr \<Longrightarrow> (Key (invKey (publicKey ?b ?A)) \<in> parts (knows Spy ?evs)) = (?A \<in> bad)
###symbols
Event_SET.bad :::: agent set
Message_SET.agent.Spy :::: agent
Event_SET.knows :::: agent \<Rightarrow> event list \<Rightarrow> msg set
Message_SET.parts :::: msg set \<Rightarrow> msg set
Public_SET.publicKey :::: bool \<Rightarrow> agent \<Rightarrow> nat
Message_SET.invKey :::: nat \<Rightarrow> nat
Message_SET.msg.Key :::: nat \<Rightarrow> msg
Merchant_Registration.set_mr :::: event list set
###defs
consts bad :: "agent set"
datatype
agent = CA nat | Cardholder nat | Merchant nat | PG nat | Spy
primrec knows :: "[agent, event list] \<Rightarrow> msg set"
where
knows_Nil:
"knows A [] = initState A"
| knows_Cons:
"knows A (ev # evs) =
(if A = Spy then
(case ev of
Says A' B X \<Rightarrow> insert X (knows Spy evs)
| Gets A' X \<Rightarrow> knows Spy evs
| Notes A' X \<Rightarrow>
if A' \<in> bad then insert X (knows Spy evs) else knows Spy evs)
else
(case ev of
Says A' B X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs
| Gets A' X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs
| Notes A' X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs))"
inductive_set
parts :: "msg set \<Rightarrow> msg set"
for H :: "msg set"
where
Inj [intro]: "X \<in> H ==> X \<in> parts H"
| Fst: "\<lbrace>X,Y\<rbrace> \<in> parts H ==> X \<in> parts H"
| Snd: "\<lbrace>X,Y\<rbrace> \<in> parts H ==> Y \<in> parts H"
| Body: "Crypt K X \<in> parts H ==> X \<in> parts H"
consts
publicKey :: "[bool, agent] \<Rightarrow> key"
\<comment> \<open>the boolean is TRUE if a signing key\<close>
datatype
msg = Agent agent \<comment> \<open>Agent names\<close>
| Number nat \<comment> \<open>Ordinary integers, timestamps, ...\<close>
| Nonce nat \<comment> \<open>Unguessable nonces\<close>
| Pan nat \<comment> \<open>Unguessable Primary Account Numbers (??)\<close>
| Key key \<comment> \<open>Crypto keys\<close>
| Hash msg \<comment> \<open>Hashing\<close>
| MPair msg msg \<comment> \<open>Compound messages\<close>
| Crypt key msg \<comment> \<open>Encryption, public- or shared-key\<close>
inductive_set
set_mr :: "event list set"
where
Nil: \<comment> \<open>Initial trace is empty\<close>
"[] \<in> set_mr"
| Fake: \<comment> \<open>The spy MAY say anything he CAN say.\<close>
"[| evsf \<in> set_mr; X \<in> synth (analz (knows Spy evsf)) |]
==> Says Spy B X # evsf \<in> set_mr"
| Reception: \<comment> \<open>If A sends a message X to B, then B might receive it\<close>
"[| evsr \<in> set_mr; Says A B X \<in> set evsr |]
==> Gets B X # evsr \<in> set_mr"
| SET_MR1: \<comment> \<open>RegFormReq: M requires a registration form to a CA\<close>
"[| evs1 \<in> set_mr; M = Merchant k; Nonce NM1 \<notin> used evs1 |]
==> Says M (CA i) \<lbrace>Agent M, Nonce NM1\<rbrace> # evs1 \<in> set_mr"
| SET_MR2: \<comment> \<open>RegFormRes: CA replies with the registration form and the
certificates for her keys\<close>
"[| evs2 \<in> set_mr; Nonce NCA \<notin> used evs2;
Gets (CA i) \<lbrace>Agent M, Nonce NM1\<rbrace> \<in> set evs2 |]
==> Says (CA i) M \<lbrace>sign (priSK (CA i)) \<lbrace>Agent M, Nonce NM1, Nonce NCA\<rbrace>,
cert (CA i) (pubEK (CA i)) onlyEnc (priSK RCA),
cert (CA i) (pubSK (CA i)) onlySig (priSK RCA) \<rbrace>
# evs2 \<in> set_mr"
| SET_MR3:
\<comment> \<open>CertReq: M submits the key pair to be certified. The Notes
event allows KM1 to be lost if M is compromised. Piero remarks
that the agent mentioned inside the signature is not verified to
correspond to M. As in CR, each Merchant has fixed key pairs. M
is only optionally required to send NCA back, so M doesn't do so
in the model\<close>
"[| evs3 \<in> set_mr; M = Merchant k; Nonce NM2 \<notin> used evs3;
Key KM1 \<notin> used evs3; KM1 \<in> symKeys;
Gets M \<lbrace>sign (invKey SKi) \<lbrace>Agent X, Nonce NM1, Nonce NCA\<rbrace>,
cert (CA i) EKi onlyEnc (priSK RCA),
cert (CA i) SKi onlySig (priSK RCA) \<rbrace>
\<in> set evs3;
Says M (CA i) \<lbrace>Agent M, Nonce NM1\<rbrace> \<in> set evs3 |]
==> Says M (CA i)
\<lbrace>Crypt KM1 (sign (priSK M) \<lbrace>Agent M, Nonce NM2,
Key (pubSK M), Key (pubEK M)\<rbrace>),
Crypt EKi (Key KM1)\<rbrace>
# Notes M \<lbrace>Key KM1, Agent (CA i)\<rbrace>
# evs3 \<in> set_mr"
| SET_MR4:
\<comment> \<open>CertRes: CA issues the certificates for merSK and merEK,
while checking never to have certified the m even
separately. NOTE: In Cardholder Registration the
corresponding rule (6) doesn't use the "sign" primitive. "The
CertRes shall be signed but not encrypted if the EE is a Merchant
or Payment Gateway."-- Programmer's Guide, page 191.\<close>
"[| evs4 \<in> set_mr; M = Merchant k;
merSK \<notin> symKeys; merEK \<notin> symKeys;
Notes (CA i) (Key merSK) \<notin> set evs4;
Notes (CA i) (Key merEK) \<notin> set evs4;
Gets (CA i) \<lbrace>Crypt KM1 (sign (invKey merSK)
\<lbrace>Agent M, Nonce NM2, Key merSK, Key merEK\<rbrace>),
Crypt (pubEK (CA i)) (Key KM1) \<rbrace>
\<in> set evs4 |]
==> Says (CA i) M \<lbrace>sign (priSK(CA i)) \<lbrace>Agent M, Nonce NM2, Agent(CA i)\<rbrace>,
cert M merSK onlySig (priSK (CA i)),
cert M merEK onlyEnc (priSK (CA i)),
cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace>
# Notes (CA i) (Key merSK)
# Notes (CA i) (Key merEK)
# evs4 \<in> set_mr"
|
###output
x_1 \<in> ?H1 \<Longrightarrow> (?H2 (?H3 (?H4 x_2 x_3)) \<in> ?H5 (?H6 ?H7 x_1)) = (x_3 \<in> ?H8)###end
|
Analysis/Derivative
|
Derivative.has_derivative_within_alt
|
lemma has_derivative_within_alt:
"(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
(\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x))"
|
(?f has_derivative ?f') (at ?x within ?s) = (bounded_linear ?f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>y\<in> ?s. norm (y - ?x) < d \<longrightarrow> norm (?f y - ?f ?x - ?f' (y - ?x)) \<le> e * norm (y - ?x)))
|
?H1 x_1 x_2 (?H2 x_3 x_4) = (?H3 x_2 \<and> (\<forall>y_0> ?H4. \<exists>y_1> ?H4. \<forall>y_2\<in>x_4. ?H5 (?H6 y_2 x_3) < y_1 \<longrightarrow> ?H5 (?H6 (?H6 (x_1 y_2) (x_1 x_3)) (x_2 (?H6 y_2 x_3))) \<le> ?H7 y_0 (?H5 (?H6 y_2 x_3))))
|
[
"Groups.times_class.times",
"Groups.minus_class.minus",
"Real_Vector_Spaces.norm_class.norm",
"Groups.zero_class.zero",
"Real_Vector_Spaces.bounded_linear",
"Topological_Spaces.topological_space_class.at_within",
"Deriv.has_derivative"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> real",
"'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a filter",
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'a filter \\<Rightarrow> bool"
] |
[
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"class norm =\n fixes norm :: \"'a \\<Rightarrow> real\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"class topological_space = \"open\" +\n assumes open_UNIV [simp, intro]: \"open UNIV\"\n assumes open_Int [intro]: \"open S \\<Longrightarrow> open T \\<Longrightarrow> open (S \\<inter> T)\"\n assumes open_Union [intro]: \"\\<forall>S\\<in>K. open S \\<Longrightarrow> open (\\<Union>K)\"\nbegin",
"definition has_derivative :: \"('a::real_normed_vector \\<Rightarrow> 'b::real_normed_vector) \\<Rightarrow>\n ('a \\<Rightarrow> 'b) \\<Rightarrow> 'a filter \\<Rightarrow> bool\" (infix \"(has'_derivative)\" 50)\n where \"(f has_derivative f') F \\<longleftrightarrow>\n bounded_linear f' \\<and>\n ((\\<lambda>y. ((f y - f (Lim F (\\<lambda>x. x))) - f' (y - Lim F (\\<lambda>x. x))) /\\<^sub>R norm (y - Lim F (\\<lambda>x. x))) \\<longlongrightarrow> 0) F\""
] |
template
|
###lemma
(?f has_derivative ?f') (at ?x within ?s) = (bounded_linear ?f' \<and> (\<forall>e>0. \<exists>d>0. \<forall>y\<in> ?s. norm (y - ?x) < d \<longrightarrow> norm (?f y - ?f ?x - ?f' (y - ?x)) \<le> e * norm (y - ?x)))
###symbols
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Real_Vector_Spaces.norm_class.norm :::: 'a \<Rightarrow> real
Groups.zero_class.zero :::: 'a
Real_Vector_Spaces.bounded_linear :::: ('a \<Rightarrow> 'b) \<Rightarrow> bool
Topological_Spaces.topological_space_class.at_within :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a filter
Deriv.has_derivative :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
class norm =
fixes norm :: "'a \<Rightarrow> real"
class zero =
fixes zero :: 'a ("0")
class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
begin
definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow>
('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool" (infix "(has'_derivative)" 50)
where "(f has_derivative f') F \<longleftrightarrow>
bounded_linear f' \<and>
((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> 0) F"
|
###output
?H1 x_1 x_2 (?H2 x_3 x_4) = (?H3 x_2 \<and> (\<forall>y_0> ?H4. \<exists>y_1> ?H4. \<forall>y_2\<in>x_4. ?H5 (?H6 y_2 x_3) < y_1 \<longrightarrow> ?H5 (?H6 (?H6 (x_1 y_2) (x_1 x_3)) (x_2 (?H6 y_2 x_3))) \<le> ?H7 y_0 (?H5 (?H6 y_2 x_3))))###end
|
Cardinals/Order_Relation_More
|
Order_Relation_More.aboveS_notIn
|
lemma aboveS_notIn: "a \<notin> aboveS r a"
|
?a \<notin> aboveS ?r ?a
|
?H1 x_1 (?H2 x_2 x_1)
|
[
"Order_Relation.aboveS",
"Set.not_member"
] |
[
"('a \\<times> 'a) set \\<Rightarrow> 'a \\<Rightarrow> 'a set",
"'a \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"definition aboveS :: \"'a rel \\<Rightarrow> 'a \\<Rightarrow> 'a set\"\n where \"aboveS r a \\<equiv> {b. b \\<noteq> a \\<and> (a, b) \\<in> r}\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>"
] |
template
|
###lemma
?a \<notin> aboveS ?r ?a
###symbols
Order_Relation.aboveS :::: ('a \<times> 'a) set \<Rightarrow> 'a \<Rightarrow> 'a set
Set.not_member :::: 'a \<Rightarrow> 'a set \<Rightarrow> bool
###defs
definition aboveS :: "'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
where "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a, b) \<in> r}"
abbreviation not_member
where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close>
|
###output
?H1 x_1 (?H2 x_2 x_1)###end
|
Auth/Event
|
Event.analz_impI
| null |
(?Y \<notin> analz (knows Spy ?evs) \<Longrightarrow> ?Q) \<Longrightarrow> ?Y \<notin> analz (knows Spy ?evs) \<longrightarrow> ?Q
|
(?H1 x_1 (?H2 (?H3 ?H4 x_2)) \<Longrightarrow> x_3) \<Longrightarrow> ?H1 x_1 (?H2 (?H3 ?H4 x_2)) \<longrightarrow> x_3
|
[
"Message.agent.Spy",
"Event.knows",
"Message.analz",
"Set.not_member"
] |
[
"agent",
"agent \\<Rightarrow> event list \\<Rightarrow> msg set",
"msg set \\<Rightarrow> msg set",
"'a \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"primrec knows :: \"agent \\<Rightarrow> event list \\<Rightarrow> msg set\"\nwhere\n knows_Nil: \"knows A [] = initState A\"\n| knows_Cons:\n \"knows A (ev # evs) =\n (if A = Spy then \n (case ev of\n Says A' B X \\<Rightarrow> insert X (knows Spy evs)\n | Gets A' X \\<Rightarrow> knows Spy evs\n | Notes A' X \\<Rightarrow> \n if A' \\<in> bad then insert X (knows Spy evs) else knows Spy evs)\n else\n (case ev of\n Says A' B X \\<Rightarrow> \n if A'=A then insert X (knows A evs) else knows A evs\n | Gets A' X \\<Rightarrow> \n if A'=A then insert X (knows A evs) else knows A evs\n | Notes A' X \\<Rightarrow> \n if A'=A then insert X (knows A evs) else knows A evs))\"",
"inductive_set\n analz :: \"msg set \\<Rightarrow> msg set\"\n for H :: \"msg set\"\n where\n Inj [intro,simp]: \"X \\<in> H \\<Longrightarrow> X \\<in> analz H\"\n | Fst: \"\\<lbrace>X,Y\\<rbrace> \\<in> analz H \\<Longrightarrow> X \\<in> analz H\"\n | Snd: \"\\<lbrace>X,Y\\<rbrace> \\<in> analz H \\<Longrightarrow> Y \\<in> analz H\"\n | Decrypt [dest]: \n \"\\<lbrakk>Crypt K X \\<in> analz H; Key(invKey K) \\<in> analz H\\<rbrakk> \\<Longrightarrow> X \\<in> analz H\"",
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>"
] |
template
|
###lemma
(?Y \<notin> analz (knows Spy ?evs) \<Longrightarrow> ?Q) \<Longrightarrow> ?Y \<notin> analz (knows Spy ?evs) \<longrightarrow> ?Q
###symbols
Message.agent.Spy :::: agent
Event.knows :::: agent \<Rightarrow> event list \<Rightarrow> msg set
Message.analz :::: msg set \<Rightarrow> msg set
Set.not_member :::: 'a \<Rightarrow> 'a set \<Rightarrow> bool
###defs
primrec knows :: "agent \<Rightarrow> event list \<Rightarrow> msg set"
where
knows_Nil: "knows A [] = initState A"
| knows_Cons:
"knows A (ev # evs) =
(if A = Spy then
(case ev of
Says A' B X \<Rightarrow> insert X (knows Spy evs)
| Gets A' X \<Rightarrow> knows Spy evs
| Notes A' X \<Rightarrow>
if A' \<in> bad then insert X (knows Spy evs) else knows Spy evs)
else
(case ev of
Says A' B X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs
| Gets A' X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs
| Notes A' X \<Rightarrow>
if A'=A then insert X (knows A evs) else knows A evs))"
inductive_set
analz :: "msg set \<Rightarrow> msg set"
for H :: "msg set"
where
Inj [intro,simp]: "X \<in> H \<Longrightarrow> X \<in> analz H"
| Fst: "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> X \<in> analz H"
| Snd: "\<lbrace>X,Y\<rbrace> \<in> analz H \<Longrightarrow> Y \<in> analz H"
| Decrypt [dest]:
"\<lbrakk>Crypt K X \<in> analz H; Key(invKey K) \<in> analz H\<rbrakk> \<Longrightarrow> X \<in> analz H"
abbreviation not_member
where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close>
|
###output
(?H1 x_1 (?H2 (?H3 ?H4 x_2)) \<Longrightarrow> x_3) \<Longrightarrow> ?H1 x_1 (?H2 (?H3 ?H4 x_2)) \<longrightarrow> x_3###end
|
Nonstandard_Analysis/HyperNat
|
HyperNat.hypnat_of_nat_less_whn
|
lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn"
|
hypnat_of_nat ?n < whn
|
?H1 x_1 < ?H2
|
[
"HyperNat.whn",
"HyperNat.hypnat_of_nat"
] |
[
"nat star",
"nat \\<Rightarrow> nat star"
] |
[
"definition whn :: hypnat\n where hypnat_omega_def: \"whn = star_n (\\<lambda>n::nat. n)\"",
"abbreviation hypnat_of_nat :: \"nat \\<Rightarrow> nat star\"\n where \"hypnat_of_nat \\<equiv> star_of\""
] |
template
|
###lemma
hypnat_of_nat ?n < whn
###symbols
HyperNat.whn :::: nat star
HyperNat.hypnat_of_nat :::: nat \<Rightarrow> nat star
###defs
definition whn :: hypnat
where hypnat_omega_def: "whn = star_n (\<lambda>n::nat. n)"
abbreviation hypnat_of_nat :: "nat \<Rightarrow> nat star"
where "hypnat_of_nat \<equiv> star_of"
|
###output
?H1 x_1 < ?H2###end
|
Predicate_Compile_Examples/Predicate_Compile_Tests
|
Predicate_Compile_Tests.one_or_two''_iE
| null |
pred.eval (one_or_two''_i ?x) ?y \<Longrightarrow> (one_or_two'' ?x \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1) x_2; ?H3 x_1 \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
|
[
"Predicate_Compile_Tests.one_or_two''",
"Predicate_Compile_Tests.one_or_two''_i",
"Predicate.pred.eval"
] |
[
"nat \\<Rightarrow> bool",
"nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition one_or_two'':\n \"one_or_two'' == (%x. x = 1 \\<or> x = (2::nat))\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
template
|
###lemma
pred.eval (one_or_two''_i ?x) ?y \<Longrightarrow> (one_or_two'' ?x \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Predicate_Compile_Tests.one_or_two'' :::: nat \<Rightarrow> bool
Predicate_Compile_Tests.one_or_two''_i :::: nat \<Rightarrow> unit Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
###defs
definition one_or_two'':
"one_or_two'' == (%x. x = 1 \<or> x = (2::nat))"
datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a \<Rightarrow> bool")
|
###output
\<lbrakk> ?H1 (?H2 x_1) x_2; ?H3 x_1 \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3###end
|
Analysis/Affine
|
Affine.aff_dim_affine_independent
|
lemma aff_dim_affine_independent:
fixes B :: "'n::euclidean_space set"
assumes "\<not> affine_dependent B"
shows "of_nat (card B) = aff_dim B + 1"
|
\<not> affine_dependent ?B \<Longrightarrow> int (card ?B) = aff_dim ?B + 1
|
\<not> ?H1 x_1 \<Longrightarrow> ?H2 (?H3 x_1) = ?H4 (?H5 x_1) ?H6
|
[
"Groups.one_class.one",
"Affine.aff_dim",
"Groups.plus_class.plus",
"Finite_Set.card",
"Int.int",
"Affine.affine_dependent"
] |
[
"'a",
"'a set \\<Rightarrow> int",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a set \\<Rightarrow> nat",
"nat \\<Rightarrow> int",
"'a set \\<Rightarrow> bool"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"abbreviation int :: \"nat \\<Rightarrow> int\"\n where \"int \\<equiv> of_nat\""
] |
template
|
###lemma
\<not> affine_dependent ?B \<Longrightarrow> int (card ?B) = aff_dim ?B + 1
###symbols
Groups.one_class.one :::: 'a
Affine.aff_dim :::: 'a set \<Rightarrow> int
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Finite_Set.card :::: 'a set \<Rightarrow> nat
Int.int :::: nat \<Rightarrow> int
Affine.affine_dependent :::: 'a set \<Rightarrow> bool
###defs
class one =
fixes one :: 'a ("1")
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
abbreviation int :: "nat \<Rightarrow> int"
where "int \<equiv> of_nat"
|
###output
\<not> ?H1 x_1 \<Longrightarrow> ?H2 (?H3 x_1) = ?H4 (?H5 x_1) ?H6###end
|
Predicate
|
Predicate.eval_map
|
lemma eval_map [simp]:
"eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
|
pred.eval (Predicate.map ?f ?P) = (SUP x\<in>{x. pred.eval ?P x}. (=) (?f x))
|
?H1 (?H2 x_1 x_2) = ?H3 (?H4 (\<lambda>y_0. (=) (x_1 y_0)) (?H5 (?H1 x_2)))
|
[
"Set.Collect",
"Set.image",
"Complete_Lattices.Sup_class.Sup",
"Predicate.map",
"Predicate.pred.eval"
] |
[
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a Predicate.pred \\<Rightarrow> 'b Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition image :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set\" (infixr \"`\" 90)\n where \"f ` A = {y. \\<exists>x\\<in>A. y = f x}\"",
"class Sup =\n fixes Sup :: \"'a set \\<Rightarrow> 'a\" (\"\\<Squnion> _\" [900] 900)",
"definition map :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a pred \\<Rightarrow> 'b pred\" where\n \"map f P = P \\<bind> (single \\<circ> f)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
template
|
###lemma
pred.eval (Predicate.map ?f ?P) = (SUP x\<in>{x. pred.eval ?P x}. (=) (?f x))
###symbols
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Complete_Lattices.Sup_class.Sup :::: 'a set \<Rightarrow> 'a
Predicate.map :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a Predicate.pred \<Rightarrow> 'b Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
###defs
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
class Sup =
fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion> _" [900] 900)
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
"map f P = P \<bind> (single \<circ> f)"
datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a \<Rightarrow> bool")
|
###output
?H1 (?H2 x_1 x_2) = ?H3 (?H4 (\<lambda>y_0. (=) (x_1 y_0)) (?H5 (?H1 x_2)))###end
|
Algebra/Group
|
Groups.field_simps(50)
| null |
((1:: ?'a) / ?a) ^ ?n = (1:: ?'a) / ?a ^ ?n
|
?H1 (?H2 ?H3 x_1) x_2 = ?H2 ?H3 (?H1 x_1 x_2)
|
[
"Groups.one_class.one",
"Fields.inverse_class.inverse_divide",
"Power.power_class.power"
] |
[
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"class inverse = divide +\n fixes inverse :: \"'a \\<Rightarrow> 'a\"\nbegin",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\""
] |
template
|
###lemma
((1:: ?'a) / ?a) ^ ?n = (1:: ?'a) / ?a ^ ?n
###symbols
Groups.one_class.one :::: 'a
Fields.inverse_class.inverse_divide :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
###defs
class one =
fixes one :: 'a ("1")
class inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
|
###output
?H1 (?H2 ?H3 x_1) x_2 = ?H2 ?H3 (?H1 x_1 x_2)###end
|
Relation
|
Relation.Range_Int_subset
|
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
|
Range (?A \<inter> ?B) \<subseteq> Range ?A \<inter> Range ?B
|
?H1 (?H2 (?H3 x_1 x_2)) (?H3 (?H2 x_1) (?H2 x_2))
|
[
"Set.inter",
"Relation.Range",
"Set.subset_eq"
] |
[
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"('a \\<times> 'b) set \\<Rightarrow> 'b set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"abbreviation inter :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<inter>\" 70)\n where \"(\\<inter>) \\<equiv> inf\"",
"inductive_set Range :: \"('a \\<times> 'b) set \\<Rightarrow> 'b set\" for r :: \"('a \\<times> 'b) set\"\n where RangeI [intro]: \"(a, b) \\<in> r \\<Longrightarrow> b \\<in> Range r\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
template
|
###lemma
Range (?A \<inter> ?B) \<subseteq> Range ?A \<inter> Range ?B
###symbols
Set.inter :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
Relation.Range :::: ('a \<times> 'b) set \<Rightarrow> 'b set
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
###defs
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<inter>" 70)
where "(\<inter>) \<equiv> inf"
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" for r :: "('a \<times> 'b) set"
where RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
?H1 (?H2 (?H3 x_1 x_2)) (?H3 (?H2 x_1) (?H2 x_2))###end
|
Library/Stream
|
Stream.smap_smap2
|
lemma smap_smap2[simp]:
"smap f (smap2 g s1 s2) = smap2 (\<lambda>x y. f (g x y)) s1 s2"
|
smap ?f (smap2 ?g ?s1.0 ?s2.0) = smap2 (\<lambda>x y. ?f (?g x y)) ?s1.0 ?s2.0
|
?H1 x_1 (?H2 x_2 x_3 x_4) = ?H2 (\<lambda>y_0 y_1. x_1 (x_2 y_0 y_1)) x_3 x_4
|
[
"Stream.smap2",
"Stream.stream.smap"
] |
[
"('a \\<Rightarrow> 'b \\<Rightarrow> 'c) \\<Rightarrow> 'a stream \\<Rightarrow> 'b stream \\<Rightarrow> 'c stream",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a stream \\<Rightarrow> 'b stream"
] |
[
"primcorec smap2 where\n \"shd (smap2 f s1 s2) = f (shd s1) (shd s2)\"\n| \"stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)\"",
"codatatype (sset: 'a) stream =\n SCons (shd: 'a) (stl: \"'a stream\") (infixr \\<open>##\\<close> 65)\nfor\n map: smap\n rel: stream_all2"
] |
template
|
###lemma
smap ?f (smap2 ?g ?s1.0 ?s2.0) = smap2 (\<lambda>x y. ?f (?g x y)) ?s1.0 ?s2.0
###symbols
Stream.smap2 :::: ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a stream \<Rightarrow> 'b stream \<Rightarrow> 'c stream
Stream.stream.smap :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a stream \<Rightarrow> 'b stream
###defs
primcorec smap2 where
"shd (smap2 f s1 s2) = f (shd s1) (shd s2)"
| "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)"
codatatype (sset: 'a) stream =
SCons (shd: 'a) (stl: "'a stream") (infixr \<open>##\<close> 65)
for
map: smap
rel: stream_all2
|
###output
?H1 x_1 (?H2 x_2 x_3 x_4) = ?H2 (\<lambda>y_0 y_1. x_1 (x_2 y_0 y_1)) x_3 x_4###end
|
Complete_Lattices
|
Complete_Lattices.UN_insert_distrib
|
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
|
?u \<in> ?A \<Longrightarrow> (\<Union>x\<in> ?A. insert ?a (?B x)) = insert ?a (\<Union> (?B ` ?A))
|
x_1 \<in> x_2 \<Longrightarrow> ?H1 (?H2 (\<lambda>y_0. ?H3 x_3 (x_4 y_0)) x_2) = ?H3 x_3 (?H1 (?H2 x_4 x_2))
|
[
"Set.insert",
"Set.image",
"Complete_Lattices.Union"
] |
[
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set set \\<Rightarrow> 'a set"
] |
[
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"definition image :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set\" (infixr \"`\" 90)\n where \"f ` A = {y. \\<exists>x\\<in>A. y = f x}\"",
"abbreviation Union :: \"'a set set \\<Rightarrow> 'a set\" (\"\\<Union>\")\n where \"\\<Union>S \\<equiv> \\<Squnion>S\""
] |
template
|
###lemma
?u \<in> ?A \<Longrightarrow> (\<Union>x\<in> ?A. insert ?a (?B x)) = insert ?a (\<Union> (?B ` ?A))
###symbols
Set.insert :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a set
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Complete_Lattices.Union :::: 'a set set \<Rightarrow> 'a set
###defs
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
abbreviation Union :: "'a set set \<Rightarrow> 'a set" ("\<Union>")
where "\<Union>S \<equiv> \<Squnion>S"
|
###output
x_1 \<in> x_2 \<Longrightarrow> ?H1 (?H2 (\<lambda>y_0. ?H3 x_3 (x_4 y_0)) x_2) = ?H3 x_3 (?H1 (?H2 x_4 x_2))###end
|
Library/Omega_Words_Fun
|
Omega_Words_Fun.build_split
|
lemma build_split[intro]: "w = w 0 ## suffix 1 w"
|
?w = ?w 0 ## suffix 1 ?w
|
x_1 = ?H1 (x_1 ?H2) (?H3 ?H4 x_1)
|
[
"Groups.one_class.one",
"Omega_Words_Fun.suffix",
"Groups.zero_class.zero",
"Omega_Words_Fun.build"
] |
[
"'a",
"nat \\<Rightarrow> (nat \\<Rightarrow> 'a) \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a",
"'a \\<Rightarrow> (nat \\<Rightarrow> 'a) \\<Rightarrow> nat \\<Rightarrow> 'a"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"definition suffix :: \"[nat, 'a word] \\<Rightarrow> 'a word\"\n where \"suffix k x \\<equiv> \\<lambda>n. x (k+n)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec build :: \"'a \\<Rightarrow> 'a word \\<Rightarrow> 'a word\" (infixr \\<open>##\\<close> 65)\n where \"(a ## w) 0 = a\" | \"(a ## w) (Suc i) = w i\""
] |
template
|
###lemma
?w = ?w 0 ## suffix 1 ?w
###symbols
Groups.one_class.one :::: 'a
Omega_Words_Fun.suffix :::: nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Omega_Words_Fun.build :::: 'a \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a
###defs
class one =
fixes one :: 'a ("1")
definition suffix :: "[nat, 'a word] \<Rightarrow> 'a word"
where "suffix k x \<equiv> \<lambda>n. x (k+n)"
class zero =
fixes zero :: 'a ("0")
primrec build :: "'a \<Rightarrow> 'a word \<Rightarrow> 'a word" (infixr \<open>##\<close> 65)
where "(a ## w) 0 = a" | "(a ## w) (Suc i) = w i"
|
###output
x_1 = ?H1 (x_1 ?H2) (?H3 ?H4 x_1)###end
|
ex/Sqrt_Script
|
Sqrt_Script.prime_not_square
|
lemma prime_not_square:
"prime (p::nat) \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))"
|
prime ?p \<Longrightarrow> 0 < ?k \<Longrightarrow> ?m * ?m \<noteq> ?p * (?k * ?k)
|
\<lbrakk> ?H1 x_1; ?H2 < x_2\<rbrakk> \<Longrightarrow> ?H3 x_3 x_3 \<noteq> ?H3 x_1 (?H3 x_2 x_2)
|
[
"Groups.times_class.times",
"Groups.zero_class.zero",
"Factorial_Ring.normalization_semidom_class.prime"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a",
"'a \\<Rightarrow> bool"
] |
[
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
prime ?p \<Longrightarrow> 0 < ?k \<Longrightarrow> ?m * ?m \<noteq> ?p * (?k * ?k)
###symbols
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Factorial_Ring.normalization_semidom_class.prime :::: 'a \<Rightarrow> bool
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class zero =
fixes zero :: 'a ("0")
|
###output
\<lbrakk> ?H1 x_1; ?H2 < x_2\<rbrakk> \<Longrightarrow> ?H3 x_3 x_3 \<noteq> ?H3 x_1 (?H3 x_2 x_2)###end
|
Library/Extended_Real
|
Extended_Real.ereal_tendsto_simps(2)
| null |
((?f \<circ> real_of_ereal) \<longlongrightarrow> ?y) (at_right (ereal ?x)) = (?f \<longlongrightarrow> ?y) (at_right ?x)
|
?H1 (?H2 x_1 ?H3) x_2 (?H4 (?H5 x_3)) = ?H1 x_1 x_2 (?H4 x_3)
|
[
"Extended_Real.ereal.ereal",
"Topological_Spaces.order_topology_class.at_right",
"Extended_Real.real_of_ereal",
"Fun.comp",
"Topological_Spaces.topological_space_class.tendsto"
] |
[
"real \\<Rightarrow> ereal",
"'a \\<Rightarrow> 'a filter",
"ereal \\<Rightarrow> real",
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'a) \\<Rightarrow> 'c \\<Rightarrow> 'b",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b \\<Rightarrow> 'a filter \\<Rightarrow> bool"
] |
[
"datatype ereal = ereal real | PInfty | MInfty",
"class order_topology = order + \"open\" +\n assumes open_generated_order: \"open = generate_topology (range (\\<lambda>a. {..< a}) \\<union> range (\\<lambda>a. {a <..}))\"\nbegin",
"function real_of_ereal :: \"ereal \\<Rightarrow> real\" where\n \"real_of_ereal (ereal r) = r\"\n| \"real_of_ereal \\<infinity> = 0\"\n| \"real_of_ereal (-\\<infinity>) = 0\"",
"definition comp :: \"('b \\<Rightarrow> 'c) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'c\" (infixl \"\\<circ>\" 55)\n where \"f \\<circ> g = (\\<lambda>x. f (g x))\"",
"class topological_space = \"open\" +\n assumes open_UNIV [simp, intro]: \"open UNIV\"\n assumes open_Int [intro]: \"open S \\<Longrightarrow> open T \\<Longrightarrow> open (S \\<inter> T)\"\n assumes open_Union [intro]: \"\\<forall>S\\<in>K. open S \\<Longrightarrow> open (\\<Union>K)\"\nbegin"
] |
template
|
###lemma
((?f \<circ> real_of_ereal) \<longlongrightarrow> ?y) (at_right (ereal ?x)) = (?f \<longlongrightarrow> ?y) (at_right ?x)
###symbols
Extended_Real.ereal.ereal :::: real \<Rightarrow> ereal
Topological_Spaces.order_topology_class.at_right :::: 'a \<Rightarrow> 'a filter
Extended_Real.real_of_ereal :::: ereal \<Rightarrow> real
Fun.comp :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'b
Topological_Spaces.topological_space_class.tendsto :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool
###defs
datatype ereal = ereal real | PInfty | MInfty
class order_topology = order + "open" +
assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
begin
function real_of_ereal :: "ereal \<Rightarrow> real" where
"real_of_ereal (ereal r) = r"
| "real_of_ereal \<infinity> = 0"
| "real_of_ereal (-\<infinity>) = 0"
definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>" 55)
where "f \<circ> g = (\<lambda>x. f (g x))"
class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
begin
|
###output
?H1 (?H2 x_1 ?H3) x_2 (?H4 (?H5 x_3)) = ?H1 x_1 x_2 (?H4 x_3)###end
|
Analysis/Homotopy
|
Homotopy.nullhomotopic_into_contractible_space
|
lemma nullhomotopic_into_contractible_space:
assumes f: "continuous_map X Y f" and Y: "contractible_space Y"
obtains c where "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
|
continuous_map ?X ?Y ?f \<Longrightarrow> contractible_space ?Y \<Longrightarrow> (\<And>c. homotopic_with (\<lambda>h. True) ?X ?Y ?f (\<lambda>x. c) \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_2; \<And>y_0. ?H3 (\<lambda>y_1. True) x_1 x_2 x_3 (\<lambda>y_2. y_0) \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Homotopy.homotopic_with",
"Homotopy.contractible_space",
"Abstract_Topology.continuous_map"
] |
[
"(('a \\<Rightarrow> 'b) \\<Rightarrow> bool) \\<Rightarrow> 'a topology \\<Rightarrow> 'b topology \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool",
"'a topology \\<Rightarrow> bool",
"'a topology \\<Rightarrow> 'b topology \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"definition contractible_space where\n \"contractible_space X \\<equiv> \\<exists>a. homotopic_with (\\<lambda>x. True) X X id (\\<lambda>x. a)\"",
"definition continuous_map where\n \"continuous_map X Y f \\<equiv>\n f \\<in> topspace X \\<rightarrow> topspace Y \\<and>\n (\\<forall>U. openin Y U \\<longrightarrow> openin X {x \\<in> topspace X. f x \\<in> U})\""
] |
template
|
###lemma
continuous_map ?X ?Y ?f \<Longrightarrow> contractible_space ?Y \<Longrightarrow> (\<And>c. homotopic_with (\<lambda>h. True) ?X ?Y ?f (\<lambda>x. c) \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
###symbols
Homotopy.homotopic_with :::: (('a \<Rightarrow> 'b) \<Rightarrow> bool) \<Rightarrow> 'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
Homotopy.contractible_space :::: 'a topology \<Rightarrow> bool
Abstract_Topology.continuous_map :::: 'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition contractible_space where
"contractible_space X \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
definition continuous_map where
"continuous_map X Y f \<equiv>
f \<in> topspace X \<rightarrow> topspace Y \<and>
(\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U})"
|
###output
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_2; \<And>y_0. ?H3 (\<lambda>y_1. True) x_1 x_2 x_3 (\<lambda>y_2. y_0) \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4###end
|
Analysis/Further_Topology
|
Further_Topology.Jordan_Brouwer_frontier
| null |
?S homeomorphic sphere ?a ?r \<Longrightarrow> ?T \<in> components (- ?S) \<Longrightarrow> 2 \<le> DIM(?'a) \<Longrightarrow> frontier ?T = ?S
|
\<lbrakk> ?H1 x_1 (?H2 x_2 x_3); x_4 \<in> ?H3 (?H4 x_1); ?H5 (?H6 ?H7) \<le> ?H8 ?H9\<rbrakk> \<Longrightarrow> ?H10 x_4 = x_1
|
[
"Elementary_Topology.frontier",
"Euclidean_Space.euclidean_space_class.Basis",
"Finite_Set.card",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.uminus_class.uminus",
"Connected.components",
"Elementary_Metric_Spaces.sphere",
"Elementary_Topology.homeomorphic"
] |
[
"'a set \\<Rightarrow> 'a set",
"'a set",
"'a set \\<Rightarrow> nat",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a",
"'a set \\<Rightarrow> 'a set set",
"'a \\<Rightarrow> real \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'b set \\<Rightarrow> bool"
] |
[
"class euclidean_space = real_inner +\n fixes Basis :: \"'a set\"\n assumes nonempty_Basis [simp]: \"Basis \\<noteq> {}\"\n assumes finite_Basis [simp]: \"finite Basis\"\n assumes inner_Basis:\n \"\\<lbrakk>u \\<in> Basis; v \\<in> Basis\\<rbrakk> \\<Longrightarrow> inner u v = (if u = v then 1 else 0)\"\n assumes euclidean_all_zero_iff:\n \"(\\<forall>u\\<in>Basis. inner x u = 0) \\<longleftrightarrow> (x = 0)\"",
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\"- _\" [81] 80)"
] |
template
|
###lemma
?S homeomorphic sphere ?a ?r \<Longrightarrow> ?T \<in> components (- ?S) \<Longrightarrow> 2 \<le> DIM(?'a) \<Longrightarrow> frontier ?T = ?S
###symbols
Elementary_Topology.frontier :::: 'a set \<Rightarrow> 'a set
Euclidean_Space.euclidean_space_class.Basis :::: 'a set
Finite_Set.card :::: 'a set \<Rightarrow> nat
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
Connected.components :::: 'a set \<Rightarrow> 'a set set
Elementary_Metric_Spaces.sphere :::: 'a \<Rightarrow> real \<Rightarrow> 'a set
Elementary_Topology.homeomorphic :::: 'a set \<Rightarrow> 'b set \<Rightarrow> bool
###defs
class euclidean_space = real_inner +
fixes Basis :: "'a set"
assumes nonempty_Basis [simp]: "Basis \<noteq> {}"
assumes finite_Basis [simp]: "finite Basis"
assumes inner_Basis:
"\<lbrakk>u \<in> Basis; v \<in> Basis\<rbrakk> \<Longrightarrow> inner u v = (if u = v then 1 else 0)"
assumes euclidean_all_zero_iff:
"(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> (x = 0)"
datatype num = One | Bit0 num | Bit1 num
datatype num = One | Bit0 num | Bit1 num
primrec numeral :: "num \<Rightarrow> 'a"
where
numeral_One: "numeral One = 1"
| numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n"
| numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
|
###output
\<lbrakk> ?H1 x_1 (?H2 x_2 x_3); x_4 \<in> ?H3 (?H4 x_1); ?H5 (?H6 ?H7) \<le> ?H8 ?H9\<rbrakk> \<Longrightarrow> ?H10 x_4 = x_1###end
|
Predicate_Compile_Examples/Predicate_Compile_Tests
|
Predicate_Compile_Tests.map_prods_hoaux_PPiii_PoiiI
| null |
map_prods_hoaux ?x (?xa, ?xb) \<Longrightarrow> pred.eval (map_prods_hoaux_PPiii_Poii ?x ?xb) ?xa
|
?H1 x_1 (x_2, x_3) \<Longrightarrow> ?H2 (?H3 x_1 x_3) x_2
|
[
"Predicate_Compile_Tests.map_prods_hoaux_PPiii_Poii",
"Predicate.pred.eval",
"Predicate_Compile_Tests.map_prods_hoaux"
] |
[
"('a \\<times> 'b) \\<times> 'c \\<Rightarrow> 'b \\<times> 'c \\<Rightarrow> 'a Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"('a \\<times> 'b) \\<times> 'c \\<Rightarrow> 'a \\<times> 'b \\<times> 'c \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
template
|
###lemma
map_prods_hoaux ?x (?xa, ?xb) \<Longrightarrow> pred.eval (map_prods_hoaux_PPiii_Poii ?x ?xb) ?xa
###symbols
Predicate_Compile_Tests.map_prods_hoaux_PPiii_Poii :::: ('a \<times> 'b) \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
Predicate_Compile_Tests.map_prods_hoaux :::: ('a \<times> 'b) \<times> 'c \<Rightarrow> 'a \<times> 'b \<times> 'c \<Rightarrow> bool
###defs
datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a \<Rightarrow> bool")
|
###output
?H1 x_1 (x_2, x_3) \<Longrightarrow> ?H2 (?H3 x_1 x_3) x_2###end
|
Predicate_Compile_Examples/Predicate_Compile_Tests
|
Predicate_Compile_Tests.partition_FiBB_o_i_iE
| null |
pred.eval (partition_FiBB_o_i_i ?x ?xb ?xc) ?xa \<Longrightarrow> (Predicate_Compile_Tests.partition (\<lambda>xa. pred.eval (?x xa) ()) ?xa ?xb ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 (\<lambda>y_0. ?H1 (x_1 y_0) ?H4) x_4 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"Product_Type.Unity",
"Predicate_Compile_Tests.partition",
"Predicate_Compile_Tests.partition_FiBB_o_i_i",
"Predicate.pred.eval"
] |
[
"unit",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"('a \\<Rightarrow> unit Predicate.pred) \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"inductive partition :: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool\"\n for f where\n \"partition f [] [] []\"\n | \"f x \\<Longrightarrow> partition f xs ys zs \\<Longrightarrow> partition f (x # xs) (x # ys) zs\"\n | \"\\<not> f x \\<Longrightarrow> partition f xs ys zs \\<Longrightarrow> partition f (x # xs) ys (x # zs)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
template
|
###lemma
pred.eval (partition_FiBB_o_i_i ?x ?xb ?xc) ?xa \<Longrightarrow> (Predicate_Compile_Tests.partition (\<lambda>xa. pred.eval (?x xa) ()) ?xa ?xb ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Product_Type.Unity :::: unit
Predicate_Compile_Tests.partition :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool
Predicate_Compile_Tests.partition_FiBB_o_i_i :::: ('a \<Rightarrow> unit Predicate.pred) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
###defs
definition Unity :: unit ("'(')")
where "() = Abs_unit True"
inductive partition :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"
for f where
"partition f [] [] []"
| "f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) (x # ys) zs"
| "\<not> f x \<Longrightarrow> partition f xs ys zs \<Longrightarrow> partition f (x # xs) ys (x # zs)"
datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a \<Rightarrow> bool")
|
###output
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 (\<lambda>y_0. ?H1 (x_1 y_0) ?H4) x_4 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5###end
|
Set
|
Set.DiffE
|
lemma DiffE [elim!]: "c \<in> A - B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P"
|
?c \<in> ?A - ?B \<Longrightarrow> (?c \<in> ?A \<Longrightarrow> ?c \<notin> ?B \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk>x_1 \<in> ?H1 x_2 x_3; \<lbrakk>x_1 \<in> x_2; ?H2 x_1 x_3\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Set.not_member",
"Groups.minus_class.minus"
] |
[
"'a \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a"
] |
[
"abbreviation not_member\n where \"not_member x A \\<equiv> \\<not> (x \\<in> A)\" \\<comment> \\<open>non-membership\\<close>",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)"
] |
template
|
###lemma
?c \<in> ?A - ?B \<Longrightarrow> (?c \<in> ?A \<Longrightarrow> ?c \<notin> ?B \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Set.not_member :::: 'a \<Rightarrow> 'a set \<Rightarrow> bool
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
###defs
abbreviation not_member
where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close>
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
|
###output
\<lbrakk>x_1 \<in> ?H1 x_2 x_3; \<lbrakk>x_1 \<in> x_2; ?H2 x_1 x_3\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4###end
|
Nominal/Examples/Class1
|
Class1.crename_fresh
|
lemma crename_fresh:
assumes a: "a\<sharp>M"
shows "M[a\<turnstile>c>b] = M"
|
?a \<sharp> ?M \<Longrightarrow> ?M[ ?a\<turnstile>c> ?b] = ?M
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 x_2 x_1 x_3 = x_2
|
[
"Class1.crename",
"Nominal.fresh"
] |
[
"trm \\<Rightarrow> coname \\<Rightarrow> coname \\<Rightarrow> trm",
"'a \\<Rightarrow> 'b \\<Rightarrow> bool"
] |
[
"definition fresh :: \"'x \\<Rightarrow> 'a \\<Rightarrow> bool\" (\\<open>_ \\<sharp> _\\<close> [80,80] 80) where\n \"a \\<sharp> x \\<longleftrightarrow> a \\<notin> supp x\""
] |
template
|
###lemma
?a \<sharp> ?M \<Longrightarrow> ?M[ ?a\<turnstile>c> ?b] = ?M
###symbols
Class1.crename :::: trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm
Nominal.fresh :::: 'a \<Rightarrow> 'b \<Rightarrow> bool
###defs
definition fresh :: "'x \<Rightarrow> 'a \<Rightarrow> bool" (\<open>_ \<sharp> _\<close> [80,80] 80) where
"a \<sharp> x \<longleftrightarrow> a \<notin> supp x"
|
###output
?H1 x_1 x_2 \<Longrightarrow> ?H2 x_2 x_1 x_3 = x_2###end
|
Limits
|
Limits.BseqD
|
lemma BseqD: "Bseq X \<Longrightarrow> \<exists>K. 0 < K \<and> (\<forall>n. norm (X n) \<le> K)"
|
Bseq ?X \<Longrightarrow> \<exists>K>0. \<forall>n. norm (?X n) \<le> K
|
?H1 x_1 \<Longrightarrow> \<exists>y_0> ?H2. \<forall>y_1. ?H3 (x_1 y_1) \<le> y_0
|
[
"Real_Vector_Spaces.norm_class.norm",
"Groups.zero_class.zero",
"Limits.Bseq"
] |
[
"'a \\<Rightarrow> real",
"'a",
"(nat \\<Rightarrow> 'a) \\<Rightarrow> bool"
] |
[
"class norm =\n fixes norm :: \"'a \\<Rightarrow> real\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"abbreviation Bseq :: \"(nat \\<Rightarrow> 'a::metric_space) \\<Rightarrow> bool\"\n where \"Bseq X \\<equiv> Bfun X sequentially\""
] |
template
|
###lemma
Bseq ?X \<Longrightarrow> \<exists>K>0. \<forall>n. norm (?X n) \<le> K
###symbols
Real_Vector_Spaces.norm_class.norm :::: 'a \<Rightarrow> real
Groups.zero_class.zero :::: 'a
Limits.Bseq :::: (nat \<Rightarrow> 'a) \<Rightarrow> bool
###defs
class norm =
fixes norm :: "'a \<Rightarrow> real"
class zero =
fixes zero :: 'a ("0")
abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool"
where "Bseq X \<equiv> Bfun X sequentially"
|
###output
?H1 x_1 \<Longrightarrow> \<exists>y_0> ?H2. \<forall>y_1. ?H3 (x_1 y_1) \<le> y_0###end
|
GCD
|
GCD.lcm_cases_int
|
lemma lcm_cases_int:
fixes x y :: int
assumes "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm x y)"
and "x \<ge> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm x (- y))"
and "x \<le> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> P (lcm (- x) y)"
and "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> P (lcm (- x) (- y))"
shows "P (lcm x y)"
|
(0 \<le> ?x \<Longrightarrow> 0 \<le> ?y \<Longrightarrow> ?P (lcm ?x ?y)) \<Longrightarrow> (0 \<le> ?x \<Longrightarrow> ?y \<le> 0 \<Longrightarrow> ?P (lcm ?x (- ?y))) \<Longrightarrow> (?x \<le> 0 \<Longrightarrow> 0 \<le> ?y \<Longrightarrow> ?P (lcm (- ?x) ?y)) \<Longrightarrow> (?x \<le> 0 \<Longrightarrow> ?y \<le> 0 \<Longrightarrow> ?P (lcm (- ?x) (- ?y))) \<Longrightarrow> ?P (lcm ?x ?y)
|
\<lbrakk>\<lbrakk> ?H1 \<le> x_1; ?H1 \<le> x_2\<rbrakk> \<Longrightarrow> x_3 (?H2 x_1 x_2); \<lbrakk> ?H1 \<le> x_1; x_2 \<le> ?H1\<rbrakk> \<Longrightarrow> x_3 (?H2 x_1 (?H3 x_2)); \<lbrakk>x_1 \<le> ?H1; ?H1 \<le> x_2\<rbrakk> \<Longrightarrow> x_3 (?H2 (?H3 x_1) x_2); \<lbrakk>x_1 \<le> ?H1; x_2 \<le> ?H1\<rbrakk> \<Longrightarrow> x_3 (?H2 (?H3 x_1) (?H3 x_2))\<rbrakk> \<Longrightarrow> x_3 (?H2 x_1 x_2)
|
[
"Groups.uminus_class.uminus",
"GCD.gcd_class.lcm",
"Groups.zero_class.zero"
] |
[
"'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a"
] |
[
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\"- _\" [81] 80)",
"class gcd = zero + one + dvd +\n fixes gcd :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\"\n and lcm :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
(0 \<le> ?x \<Longrightarrow> 0 \<le> ?y \<Longrightarrow> ?P (lcm ?x ?y)) \<Longrightarrow> (0 \<le> ?x \<Longrightarrow> ?y \<le> 0 \<Longrightarrow> ?P (lcm ?x (- ?y))) \<Longrightarrow> (?x \<le> 0 \<Longrightarrow> 0 \<le> ?y \<Longrightarrow> ?P (lcm (- ?x) ?y)) \<Longrightarrow> (?x \<le> 0 \<Longrightarrow> ?y \<le> 0 \<Longrightarrow> ?P (lcm (- ?x) (- ?y))) \<Longrightarrow> ?P (lcm ?x ?y)
###symbols
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
GCD.gcd_class.lcm :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
###defs
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
class gcd = zero + one + dvd +
fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
class zero =
fixes zero :: 'a ("0")
|
###output
\<lbrakk>\<lbrakk> ?H1 \<le> x_1; ?H1 \<le> x_2\<rbrakk> \<Longrightarrow> x_3 (?H2 x_1 x_2); \<lbrakk> ?H1 \<le> x_1; x_2 \<le> ?H1\<rbrakk> \<Longrightarrow> x_3 (?H2 x_1 (?H3 x_2)); \<lbrakk>x_1 \<le> ?H1; ?H1 \<le> x_2\<rbrakk> \<Longrightarrow> x_3 (?H2 (?H3 x_1) x_2); \<lbrakk>x_1 \<le> ?H1; x_2 \<le> ?H1\<rbrakk> \<Longrightarrow> x_3 (?H2 (?H3 x_1) (?H3 x_2))\<rbrakk> \<Longrightarrow> x_3 (?H2 x_1 x_2)###end
|
Combinatorics/Permutations
|
Permutations.permutes_inv_eq
|
lemma permutes_inv_eq:
\<open>inv p y = x \<longleftrightarrow> p x = y\<close>
|
?p permutes ?S \<Longrightarrow> (inv ?p ?y = ?x) = (?p ?x = ?y)
|
?H1 x_1 x_2 \<Longrightarrow> (?H2 x_1 x_3 = x_4) = (x_1 x_4 = x_3)
|
[
"Hilbert_Choice.inv",
"Permutations.permutes"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'a) \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"abbreviation inv :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b \\<Rightarrow> 'a)\" where\n\"inv \\<equiv> inv_into UNIV\"",
"definition permutes :: \\<open>('a \\<Rightarrow> 'a) \\<Rightarrow> 'a set \\<Rightarrow> bool\\<close> (infixr \\<open>permutes\\<close> 41)\n where \\<open>p permutes S \\<longleftrightarrow> (\\<forall>x. x \\<notin> S \\<longrightarrow> p x = x) \\<and> (\\<forall>y. \\<exists>!x. p x = y)\\<close>"
] |
template
|
###lemma
?p permutes ?S \<Longrightarrow> (inv ?p ?y = ?x) = (?p ?x = ?y)
###symbols
Hilbert_Choice.inv :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a
Permutations.permutes :::: ('a \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> bool
###defs
abbreviation inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
"inv \<equiv> inv_into UNIV"
definition permutes :: \<open>('a \<Rightarrow> 'a) \<Rightarrow> 'a set \<Rightarrow> bool\<close> (infixr \<open>permutes\<close> 41)
where \<open>p permutes S \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)\<close>
|
###output
?H1 x_1 x_2 \<Longrightarrow> (?H2 x_1 x_3 = x_4) = (x_1 x_4 = x_3)###end
|
Fun
|
Fun.strict_mono_on_imp_inj_on
|
lemma strict_mono_on_imp_inj_on:
assumes "strict_mono_on A (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder))"
shows "inj_on f A"
|
strict_mono_on ?A ?f \<Longrightarrow> inj_on ?f ?A
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 x_2 x_1
|
[
"Fun.inj_on",
"Fun.ord_class.strict_mono_on"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"definition inj_on :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> bool\" \\<comment> \\<open>injective\\<close>\n where \"inj_on f A \\<longleftrightarrow> (\\<forall>x\\<in>A. \\<forall>y\\<in>A. f x = f y \\<longrightarrow> x = y)\""
] |
template
|
###lemma
strict_mono_on ?A ?f \<Longrightarrow> inj_on ?f ?A
###symbols
Fun.inj_on :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool
Fun.ord_class.strict_mono_on :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> \<open>injective\<close>
where "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
|
###output
?H1 x_1 x_2 \<Longrightarrow> ?H2 x_2 x_1###end
|
SPARK/Examples/RIPEMD-160/F
|
Filter.prod_filter_assoc
| null |
(?F \<times>\<^sub>F ?G) \<times>\<^sub>F ?H = filtermap (\<lambda>(x, y, z). ((x, y), z)) (?F \<times>\<^sub>F ?G \<times>\<^sub>F ?H)
|
?H1 (?H1 x_1 x_2) x_3 = ?H2 (?H3 (\<lambda>y_0. ?H3 (\<lambda>y_1. Pair (y_0, y_1)))) (?H1 x_1 (?H1 x_2 x_3))
|
[
"Product_Type.prod.case_prod",
"Filter.filtermap",
"Filter.prod_filter"
] |
[
"('a \\<Rightarrow> 'b \\<Rightarrow> 'c) \\<Rightarrow> 'a \\<times> 'b \\<Rightarrow> 'c",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a filter \\<Rightarrow> 'b filter",
"'a filter \\<Rightarrow> 'b filter \\<Rightarrow> ('a \\<times> 'b) filter"
] |
[
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\"",
"definition filtermap :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a filter \\<Rightarrow> 'b filter\"\n where \"filtermap f F = Abs_filter (\\<lambda>P. eventually (\\<lambda>x. P (f x)) F)\"",
"definition prod_filter :: \"'a filter \\<Rightarrow> 'b filter \\<Rightarrow> ('a \\<times> 'b) filter\" (infixr \"\\<times>\\<^sub>F\" 80) where\n \"prod_filter F G =\n (\\<Sqinter>(P, Q)\\<in>{(P, Q). eventually P F \\<and> eventually Q G}. principal {(x, y). P x \\<and> Q y})\""
] |
template
|
###lemma
(?F \<times>\<^sub>F ?G) \<times>\<^sub>F ?H = filtermap (\<lambda>(x, y, z). ((x, y), z)) (?F \<times>\<^sub>F ?G \<times>\<^sub>F ?H)
###symbols
Product_Type.prod.case_prod :::: ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c
Filter.filtermap :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter
Filter.prod_filter :::: 'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter
###defs
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where
"prod_filter F G =
(\<Sqinter>(P, Q)\<in>{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})"
|
###output
?H1 (?H1 x_1 x_2) x_3 = ?H2 (?H3 (\<lambda>y_0. ?H3 (\<lambda>y_1. Pair (y_0, y_1)))) (?H1 x_1 (?H1 x_2 x_3))###end
|
IMPP/Com
|
Complete_Lattices.UN_simps(7)
| null |
(\<Union>x\<in> ?C. ?A - ?B x) = ?A - \<Inter> (?B ` ?C)
|
?H1 (?H2 (\<lambda>y_0. ?H3 x_1 (x_2 y_0)) x_3) = ?H3 x_1 (?H4 (?H2 x_2 x_3))
|
[
"Complete_Lattices.Inter",
"Groups.minus_class.minus",
"Set.image",
"Complete_Lattices.Union"
] |
[
"'a set set \\<Rightarrow> 'a set",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set set \\<Rightarrow> 'a set"
] |
[
"abbreviation Inter :: \"'a set set \\<Rightarrow> 'a set\" (\"\\<Inter>\")\n where \"\\<Inter>S \\<equiv> \\<Sqinter>S\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"definition image :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set\" (infixr \"`\" 90)\n where \"f ` A = {y. \\<exists>x\\<in>A. y = f x}\"",
"abbreviation Union :: \"'a set set \\<Rightarrow> 'a set\" (\"\\<Union>\")\n where \"\\<Union>S \\<equiv> \\<Squnion>S\""
] |
template
|
###lemma
(\<Union>x\<in> ?C. ?A - ?B x) = ?A - \<Inter> (?B ` ?C)
###symbols
Complete_Lattices.Inter :::: 'a set set \<Rightarrow> 'a set
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Complete_Lattices.Union :::: 'a set set \<Rightarrow> 'a set
###defs
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" ("\<Inter>")
where "\<Inter>S \<equiv> \<Sqinter>S"
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
abbreviation Union :: "'a set set \<Rightarrow> 'a set" ("\<Union>")
where "\<Union>S \<equiv> \<Squnion>S"
|
###output
?H1 (?H2 (\<lambda>y_0. ?H3 x_1 (x_2 y_0)) x_3) = ?H3 x_1 (?H4 (?H2 x_2 x_3))###end
|
Bali/WellType
|
WellType.wt_elim_cases(17)
| null |
?E, ?dt\<Turnstile>In1l (Body ?D ?blk)\<Colon> ?T \<Longrightarrow> (\<And>T. ?T = Inl T \<Longrightarrow> is_class (prg ?E) ?D \<Longrightarrow> ?E, ?dt\<Turnstile> ?blk\<Colon>\<surd> \<Longrightarrow> lcl ?E Result = Some T \<Longrightarrow> is_type (prg ?E) T \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 x_1 x_2 (?H2 (?H3 x_3 x_4)) x_5; \<And>y_0. \<lbrakk>x_5 = ?H4 y_0; ?H5 (?H6 x_1) x_3; ?H7 x_1 x_2 x_4; ?H8 x_1 ?H9 = ?H10 y_0; ?H11 (?H6 x_1) y_0\<rbrakk> \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"Decl.is_type",
"Option.option.Some",
"Name.Result",
"WellType.env.lcl",
"WellType.wt_stmt",
"WellType.env.prg",
"Decl.is_class",
"Sum_Type.Inl",
"Term.expr.Body",
"Basis.In1l",
"WellType.wt"
] |
[
"prog \\<Rightarrow> ty \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a option",
"lname",
"'a env_scheme \\<Rightarrow> lenv",
"env \\<Rightarrow> (loc \\<Rightarrow> ty option) \\<Rightarrow> stmt \\<Rightarrow> bool",
"'a env_scheme \\<Rightarrow> prog",
"prog \\<Rightarrow> qtname \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a + 'b",
"qtname \\<Rightarrow> stmt \\<Rightarrow> expr",
"'a \\<Rightarrow> ('a + 'b, 'c, 'd) sum3",
"env \\<Rightarrow> (loc \\<Rightarrow> ty option) \\<Rightarrow> Term.term \\<Rightarrow> tys \\<Rightarrow> bool"
] |
[
"primrec is_type :: \"prog \\<Rightarrow> ty \\<Rightarrow> bool\"\n and isrtype :: \"prog \\<Rightarrow> ref_ty \\<Rightarrow> bool\"\nwhere\n \"is_type G (PrimT pt) = True\"\n| \"is_type G (RefT rt) = isrtype G rt\"\n| \"isrtype G (NullT) = True\"\n| \"isrtype G (IfaceT tn) = is_iface G tn\"\n| \"isrtype G (ClassT tn) = is_class G tn\"\n| \"isrtype G (ArrayT T ) = is_type G T\"",
"datatype 'a option =\n None\n | Some (the: 'a)",
"abbreviation Result :: lname\n where \"Result == EName Res\"",
"record env = \n prg:: \"prog\" \\<comment> \\<open>program\\<close>\n cls:: \"qtname\" \\<comment> \\<open>current package and class name\\<close>\n lcl:: \"lenv\" \\<comment> \\<open>local environment\\<close>",
"record env = \n prg:: \"prog\" \\<comment> \\<open>program\\<close>\n cls:: \"qtname\" \\<comment> \\<open>current package and class name\\<close>\n lcl:: \"lenv\" \\<comment> \\<open>local environment\\<close>",
"abbreviation\n is_class :: \"prog \\<Rightarrow> qtname \\<Rightarrow> bool\"\n where \"is_class G C == class G C \\<noteq> None\"",
"definition Inl :: \"'a \\<Rightarrow> 'a + 'b\"\n where \"Inl = Abs_sum \\<circ> Inl_Rep\"",
"abbreviation In1l :: \"'al \\<Rightarrow> ('al + 'ar, 'b, 'c) sum3\"\n where \"In1l e \\<equiv> In1 (Inl e)\"",
"inductive wt :: \"env \\<Rightarrow> dyn_ty \\<Rightarrow> [term,tys] \\<Rightarrow> bool\" (\"_,_\\<Turnstile>_\\<Colon>_\" [51,51,51,51] 50)\n and wt_stmt :: \"env \\<Rightarrow> dyn_ty \\<Rightarrow> stmt \\<Rightarrow> bool\" (\"_,_\\<Turnstile>_\\<Colon>\\<surd>\" [51,51,51] 50)\n and ty_expr :: \"env \\<Rightarrow> dyn_ty \\<Rightarrow> [expr ,ty ] \\<Rightarrow> bool\" (\"_,_\\<Turnstile>_\\<Colon>-_\" [51,51,51,51] 50)\n and ty_var :: \"env \\<Rightarrow> dyn_ty \\<Rightarrow> [var ,ty ] \\<Rightarrow> bool\" (\"_,_\\<Turnstile>_\\<Colon>=_\" [51,51,51,51] 50)\n and ty_exprs :: \"env \\<Rightarrow> dyn_ty \\<Rightarrow> [expr list, ty list] \\<Rightarrow> bool\"\n (\"_,_\\<Turnstile>_\\<Colon>\\<doteq>_\" [51,51,51,51] 50)\nwhere\n\n \"E,dt\\<Turnstile>s\\<Colon>\\<surd> \\<equiv> E,dt\\<Turnstile>In1r s\\<Colon>Inl (PrimT Void)\"\n| \"E,dt\\<Turnstile>e\\<Colon>-T \\<equiv> E,dt\\<Turnstile>In1l e\\<Colon>Inl T\"\n| \"E,dt\\<Turnstile>e\\<Colon>=T \\<equiv> E,dt\\<Turnstile>In2 e\\<Colon>Inl T\"\n| \"E,dt\\<Turnstile>e\\<Colon>\\<doteq>T \\<equiv> E,dt\\<Turnstile>In3 e\\<Colon>Inr T\"\n\n\\<comment> \\<open>well-typed statements\\<close>\n\n| Skip: \"E,dt\\<Turnstile>Skip\\<Colon>\\<surd>\"\n\n| Expr: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-T\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Expr e\\<Colon>\\<surd>\"\n \\<comment> \\<open>cf. 14.6\\<close>\n| Lab: \"E,dt\\<Turnstile>c\\<Colon>\\<surd> \\<Longrightarrow> \n E,dt\\<Turnstile>l\\<bullet> c\\<Colon>\\<surd>\" \n\n| Comp: \"\\<lbrakk>E,dt\\<Turnstile>c1\\<Colon>\\<surd>; \n E,dt\\<Turnstile>c2\\<Colon>\\<surd>\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>c1;; c2\\<Colon>\\<surd>\"\n\n \\<comment> \\<open>cf. 14.8\\<close>\n| If: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-PrimT Boolean;\n E,dt\\<Turnstile>c1\\<Colon>\\<surd>;\n E,dt\\<Turnstile>c2\\<Colon>\\<surd>\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>If(e) c1 Else c2\\<Colon>\\<surd>\"\n\n \\<comment> \\<open>cf. 14.10\\<close>\n| Loop: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-PrimT Boolean;\n E,dt\\<Turnstile>c\\<Colon>\\<surd>\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>l\\<bullet> While(e) c\\<Colon>\\<surd>\"\n \\<comment> \\<open>cf. 14.13, 14.15, 14.16\\<close>\n| Jmp: \"E,dt\\<Turnstile>Jmp jump\\<Colon>\\<surd>\"\n\n \\<comment> \\<open>cf. 14.16\\<close>\n| Throw: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-Class tn;\n prg E\\<turnstile>tn\\<preceq>\\<^sub>C SXcpt Throwable\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Throw e\\<Colon>\\<surd>\"\n \\<comment> \\<open>cf. 14.18\\<close>\n| Try: \"\\<lbrakk>E,dt\\<Turnstile>c1\\<Colon>\\<surd>; prg E\\<turnstile>tn\\<preceq>\\<^sub>C SXcpt Throwable;\n lcl E (VName vn)=None; E \\<lparr>lcl := (lcl E)(VName vn\\<mapsto>Class tn)\\<rparr>,dt\\<Turnstile>c2\\<Colon>\\<surd>\\<rbrakk>\n \\<Longrightarrow>\n E,dt\\<Turnstile>Try c1 Catch(tn vn) c2\\<Colon>\\<surd>\"\n\n \\<comment> \\<open>cf. 14.18\\<close>\n| Fin: \"\\<lbrakk>E,dt\\<Turnstile>c1\\<Colon>\\<surd>; E,dt\\<Turnstile>c2\\<Colon>\\<surd>\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>c1 Finally c2\\<Colon>\\<surd>\"\n\n| Init: \"\\<lbrakk>is_class (prg E) C\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Init C\\<Colon>\\<surd>\"\n \\<comment> \\<open>\\<^term>\\<open>Init\\<close> is created on the fly during evaluation (see Eval.thy). \n The class isn't necessarily accessible from the points \\<^term>\\<open>Init\\<close> \n is called. Therefor we only demand \\<^term>\\<open>is_class\\<close> and not \n \\<^term>\\<open>is_acc_class\\<close> here.\\<close>\n\n\\<comment> \\<open>well-typed expressions\\<close>\n\n \\<comment> \\<open>cf. 15.8\\<close>\n| NewC: \"\\<lbrakk>is_acc_class (prg E) (pkg E) C\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>NewC C\\<Colon>-Class C\"\n \\<comment> \\<open>cf. 15.9\\<close>\n| NewA: \"\\<lbrakk>is_acc_type (prg E) (pkg E) T;\n E,dt\\<Turnstile>i\\<Colon>-PrimT Integer\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>New T[i]\\<Colon>-T.[]\"\n\n \\<comment> \\<open>cf. 15.15\\<close>\n| Cast: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-T; is_acc_type (prg E) (pkg E) T';\n prg E\\<turnstile>T\\<preceq>? T'\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Cast T' e\\<Colon>-T'\"\n\n \\<comment> \\<open>cf. 15.19.2\\<close>\n| Inst: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-RefT T; is_acc_type (prg E) (pkg E) (RefT T');\n prg E\\<turnstile>RefT T\\<preceq>? RefT T'\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>e InstOf T'\\<Colon>-PrimT Boolean\"\n\n \\<comment> \\<open>cf. 15.7.1\\<close>\n| Lit: \"\\<lbrakk>typeof dt x = Some T\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Lit x\\<Colon>-T\"\n\n| UnOp: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-Te; wt_unop unop Te; T=PrimT (unop_type unop)\\<rbrakk> \n \\<Longrightarrow>\n E,dt\\<Turnstile>UnOp unop e\\<Colon>-T\"\n \n| BinOp: \"\\<lbrakk>E,dt\\<Turnstile>e1\\<Colon>-T1; E,dt\\<Turnstile>e2\\<Colon>-T2; wt_binop (prg E) binop T1 T2; \n T=PrimT (binop_type binop)\\<rbrakk> \n \\<Longrightarrow>\n E,dt\\<Turnstile>BinOp binop e1 e2\\<Colon>-T\"\n \n \\<comment> \\<open>cf. 15.10.2, 15.11.1\\<close>\n| Super: \"\\<lbrakk>lcl E This = Some (Class C); C \\<noteq> Object;\n class (prg E) C = Some c\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Super\\<Colon>-Class (super c)\"\n\n \\<comment> \\<open>cf. 15.13.1, 15.10.1, 15.12\\<close>\n| Acc: \"\\<lbrakk>E,dt\\<Turnstile>va\\<Colon>=T\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Acc va\\<Colon>-T\"\n\n \\<comment> \\<open>cf. 15.25, 15.25.1\\<close>\n| Ass: \"\\<lbrakk>E,dt\\<Turnstile>va\\<Colon>=T; va \\<noteq> LVar This;\n E,dt\\<Turnstile>v \\<Colon>-T';\n prg E\\<turnstile>T'\\<preceq>T\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>va:=v\\<Colon>-T'\"\n\n \\<comment> \\<open>cf. 15.24\\<close>\n| Cond: \"\\<lbrakk>E,dt\\<Turnstile>e0\\<Colon>-PrimT Boolean;\n E,dt\\<Turnstile>e1\\<Colon>-T1; E,dt\\<Turnstile>e2\\<Colon>-T2;\n prg E\\<turnstile>T1\\<preceq>T2 \\<and> T = T2 \\<or> prg E\\<turnstile>T2\\<preceq>T1 \\<and> T = T1\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>e0 ? e1 : e2\\<Colon>-T\"\n\n \\<comment> \\<open>cf. 15.11.1, 15.11.2, 15.11.3\\<close>\n| Call: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-RefT statT;\n E,dt\\<Turnstile>ps\\<Colon>\\<doteq>pTs;\n max_spec (prg E) (cls E) statT \\<lparr>name=mn,parTs=pTs\\<rparr> \n = {((statDeclT,m),pTs')}\n \\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>{cls E,statT,invmode m e}e\\<cdot>mn({pTs'}ps)\\<Colon>-(resTy m)\"\n\n| Methd: \"\\<lbrakk>is_class (prg E) C;\n methd (prg E) C sig = Some m;\n E,dt\\<Turnstile>Body (declclass m) (stmt (mbody (mthd m)))\\<Colon>-T\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Methd C sig\\<Colon>-T\"\n \\<comment> \\<open>The class \\<^term>\\<open>C\\<close> is the dynamic class of the method call \n (cf. Eval.thy). \n It hasn't got to be directly accessible from the current package \n \\<^term>\\<open>(pkg E)\\<close>. \n Only the static class must be accessible (enshured indirectly by \n \\<^term>\\<open>Call\\<close>). \n Note that l is just a dummy value. It is only used in the smallstep \n semantics. To proof typesafety directly for the smallstep semantics \n we would have to assume conformance of l here!\\<close>\n\n| Body: \"\\<lbrakk>is_class (prg E) D;\n E,dt\\<Turnstile>blk\\<Colon>\\<surd>;\n (lcl E) Result = Some T;\n is_type (prg E) T\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>Body D blk\\<Colon>-T\"\n\\<comment> \\<open>The class \\<^term>\\<open>D\\<close> implementing the method must not directly be \n accessible from the current package \\<^term>\\<open>(pkg E)\\<close>, but can also \n be indirectly accessible due to inheritance (enshured in \\<^term>\\<open>Call\\<close>)\n The result type hasn't got to be accessible in Java! (If it is not \n accessible you can only assign it to Object).\n For dummy value l see rule \\<^term>\\<open>Methd\\<close>.\\<close>\n\n\\<comment> \\<open>well-typed variables\\<close>\n\n \\<comment> \\<open>cf. 15.13.1\\<close>\n| LVar: \"\\<lbrakk>lcl E vn = Some T; is_acc_type (prg E) (pkg E) T\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>LVar vn\\<Colon>=T\"\n \\<comment> \\<open>cf. 15.10.1\\<close>\n| FVar: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-Class C; \n accfield (prg E) (cls E) C fn = Some (statDeclC,f)\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>{cls E,statDeclC,is_static f}e..fn\\<Colon>=(type f)\"\n \\<comment> \\<open>cf. 15.12\\<close>\n| AVar: \"\\<lbrakk>E,dt\\<Turnstile>e\\<Colon>-T.[]; \n E,dt\\<Turnstile>i\\<Colon>-PrimT Integer\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>e.[i]\\<Colon>=T\"\n\n\n\\<comment> \\<open>well-typed expression lists\\<close>\n\n \\<comment> \\<open>cf. 15.11.???\\<close>\n| Nil: \"E,dt\\<Turnstile>[]\\<Colon>\\<doteq>[]\"\n\n \\<comment> \\<open>cf. 15.11.???\\<close>\n| Cons: \"\\<lbrakk>E,dt\\<Turnstile>e \\<Colon>-T;\n E,dt\\<Turnstile>es\\<Colon>\\<doteq>Ts\\<rbrakk> \\<Longrightarrow>\n E,dt\\<Turnstile>e#es\\<Colon>\\<doteq>T#Ts\""
] |
template
|
###lemma
?E, ?dt\<Turnstile>In1l (Body ?D ?blk)\<Colon> ?T \<Longrightarrow> (\<And>T. ?T = Inl T \<Longrightarrow> is_class (prg ?E) ?D \<Longrightarrow> ?E, ?dt\<Turnstile> ?blk\<Colon>\<surd> \<Longrightarrow> lcl ?E Result = Some T \<Longrightarrow> is_type (prg ?E) T \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Decl.is_type :::: prog \<Rightarrow> ty \<Rightarrow> bool
Option.option.Some :::: 'a \<Rightarrow> 'a option
Name.Result :::: lname
WellType.env.lcl :::: 'a env_scheme \<Rightarrow> lenv
WellType.wt_stmt :::: env \<Rightarrow> (loc \<Rightarrow> ty option) \<Rightarrow> stmt \<Rightarrow> bool
WellType.env.prg :::: 'a env_scheme \<Rightarrow> prog
Decl.is_class :::: prog \<Rightarrow> qtname \<Rightarrow> bool
Sum_Type.Inl :::: 'a \<Rightarrow> 'a + 'b
Term.expr.Body :::: qtname \<Rightarrow> stmt \<Rightarrow> expr
Basis.In1l :::: 'a \<Rightarrow> ('a + 'b, 'c, 'd) sum3
WellType.wt :::: env \<Rightarrow> (loc \<Rightarrow> ty option) \<Rightarrow> Term.term \<Rightarrow> tys \<Rightarrow> bool
###defs
primrec is_type :: "prog \<Rightarrow> ty \<Rightarrow> bool"
and isrtype :: "prog \<Rightarrow> ref_ty \<Rightarrow> bool"
where
"is_type G (PrimT pt) = True"
| "is_type G (RefT rt) = isrtype G rt"
| "isrtype G (NullT) = True"
| "isrtype G (IfaceT tn) = is_iface G tn"
| "isrtype G (ClassT tn) = is_class G tn"
| "isrtype G (ArrayT T ) = is_type G T"
datatype 'a option =
None
| Some (the: 'a)
abbreviation Result :: lname
where "Result == EName Res"
record env =
prg:: "prog" \<comment> \<open>program\<close>
cls:: "qtname" \<comment> \<open>current package and class name\<close>
lcl:: "lenv" \<comment> \<open>local environment\<close>
record env =
prg:: "prog" \<comment> \<open>program\<close>
cls:: "qtname" \<comment> \<open>current package and class name\<close>
lcl:: "lenv" \<comment> \<open>local environment\<close>
abbreviation
is_class :: "prog \<Rightarrow> qtname \<Rightarrow> bool"
where "is_class G C == class G C \<noteq> None"
definition Inl :: "'a \<Rightarrow> 'a + 'b"
where "Inl = Abs_sum \<circ> Inl_Rep"
abbreviation In1l :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
where "In1l e \<equiv> In1 (Inl e)"
inductive wt :: "env \<Rightarrow> dyn_ty \<Rightarrow> [term,tys] \<Rightarrow> bool" ("_,_\<Turnstile>_\<Colon>_" [51,51,51,51] 50)
and wt_stmt :: "env \<Rightarrow> dyn_ty \<Rightarrow> stmt \<Rightarrow> bool" ("_,_\<Turnstile>_\<Colon>\<surd>" [51,51,51] 50)
and ty_expr :: "env \<Rightarrow> dyn_ty \<Rightarrow> [expr ,ty ] \<Rightarrow> bool" ("_,_\<Turnstile>_\<Colon>-_" [51,51,51,51] 50)
and ty_var :: "env \<Rightarrow> dyn_ty \<Rightarrow> [var ,ty ] \<Rightarrow> bool" ("_,_\<Turnstile>_\<Colon>=_" [51,51,51,51] 50)
and ty_exprs :: "env \<Rightarrow> dyn_ty \<Rightarrow> [expr list, ty list] \<Rightarrow> bool"
("_,_\<Turnstile>_\<Colon>\<doteq>_" [51,51,51,51] 50)
where
"E,dt\<Turnstile>s\<Colon>\<surd> \<equiv> E,dt\<Turnstile>In1r s\<Colon>Inl (PrimT Void)"
| "E,dt\<Turnstile>e\<Colon>-T \<equiv> E,dt\<Turnstile>In1l e\<Colon>Inl T"
| "E,dt\<Turnstile>e\<Colon>=T \<equiv> E,dt\<Turnstile>In2 e\<Colon>Inl T"
| "E,dt\<Turnstile>e\<Colon>\<doteq>T \<equiv> E,dt\<Turnstile>In3 e\<Colon>Inr T"
\<comment> \<open>well-typed statements\<close>
| Skip: "E,dt\<Turnstile>Skip\<Colon>\<surd>"
| Expr: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-T\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Expr e\<Colon>\<surd>"
\<comment> \<open>cf. 14.6\<close>
| Lab: "E,dt\<Turnstile>c\<Colon>\<surd> \<Longrightarrow>
E,dt\<Turnstile>l\<bullet> c\<Colon>\<surd>"
| Comp: "\<lbrakk>E,dt\<Turnstile>c1\<Colon>\<surd>;
E,dt\<Turnstile>c2\<Colon>\<surd>\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>c1;; c2\<Colon>\<surd>"
\<comment> \<open>cf. 14.8\<close>
| If: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-PrimT Boolean;
E,dt\<Turnstile>c1\<Colon>\<surd>;
E,dt\<Turnstile>c2\<Colon>\<surd>\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>If(e) c1 Else c2\<Colon>\<surd>"
\<comment> \<open>cf. 14.10\<close>
| Loop: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-PrimT Boolean;
E,dt\<Turnstile>c\<Colon>\<surd>\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>l\<bullet> While(e) c\<Colon>\<surd>"
\<comment> \<open>cf. 14.13, 14.15, 14.16\<close>
| Jmp: "E,dt\<Turnstile>Jmp jump\<Colon>\<surd>"
\<comment> \<open>cf. 14.16\<close>
| Throw: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-Class tn;
prg E\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Throw e\<Colon>\<surd>"
\<comment> \<open>cf. 14.18\<close>
| Try: "\<lbrakk>E,dt\<Turnstile>c1\<Colon>\<surd>; prg E\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable;
lcl E (VName vn)=None; E \<lparr>lcl := (lcl E)(VName vn\<mapsto>Class tn)\<rparr>,dt\<Turnstile>c2\<Colon>\<surd>\<rbrakk>
\<Longrightarrow>
E,dt\<Turnstile>Try c1 Catch(tn vn) c2\<Colon>\<surd>"
\<comment> \<open>cf. 14.18\<close>
| Fin: "\<lbrakk>E,dt\<Turnstile>c1\<Colon>\<surd>; E,dt\<Turnstile>c2\<Colon>\<surd>\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>c1 Finally c2\<Colon>\<surd>"
| Init: "\<lbrakk>is_class (prg E) C\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Init C\<Colon>\<surd>"
\<comment> \<open>\<^term>\<open>Init\<close> is created on the fly during evaluation (see Eval.thy).
The class isn't necessarily accessible from the points \<^term>\<open>Init\<close>
is called. Therefor we only demand \<^term>\<open>is_class\<close> and not
\<^term>\<open>is_acc_class\<close> here.\<close>
\<comment> \<open>well-typed expressions\<close>
\<comment> \<open>cf. 15.8\<close>
| NewC: "\<lbrakk>is_acc_class (prg E) (pkg E) C\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>NewC C\<Colon>-Class C"
\<comment> \<open>cf. 15.9\<close>
| NewA: "\<lbrakk>is_acc_type (prg E) (pkg E) T;
E,dt\<Turnstile>i\<Colon>-PrimT Integer\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>New T[i]\<Colon>-T.[]"
\<comment> \<open>cf. 15.15\<close>
| Cast: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-T; is_acc_type (prg E) (pkg E) T';
prg E\<turnstile>T\<preceq>? T'\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Cast T' e\<Colon>-T'"
\<comment> \<open>cf. 15.19.2\<close>
| Inst: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-RefT T; is_acc_type (prg E) (pkg E) (RefT T');
prg E\<turnstile>RefT T\<preceq>? RefT T'\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>e InstOf T'\<Colon>-PrimT Boolean"
\<comment> \<open>cf. 15.7.1\<close>
| Lit: "\<lbrakk>typeof dt x = Some T\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Lit x\<Colon>-T"
| UnOp: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-Te; wt_unop unop Te; T=PrimT (unop_type unop)\<rbrakk>
\<Longrightarrow>
E,dt\<Turnstile>UnOp unop e\<Colon>-T"
| BinOp: "\<lbrakk>E,dt\<Turnstile>e1\<Colon>-T1; E,dt\<Turnstile>e2\<Colon>-T2; wt_binop (prg E) binop T1 T2;
T=PrimT (binop_type binop)\<rbrakk>
\<Longrightarrow>
E,dt\<Turnstile>BinOp binop e1 e2\<Colon>-T"
\<comment> \<open>cf. 15.10.2, 15.11.1\<close>
| Super: "\<lbrakk>lcl E This = Some (Class C); C \<noteq> Object;
class (prg E) C = Some c\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Super\<Colon>-Class (super c)"
\<comment> \<open>cf. 15.13.1, 15.10.1, 15.12\<close>
| Acc: "\<lbrakk>E,dt\<Turnstile>va\<Colon>=T\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Acc va\<Colon>-T"
\<comment> \<open>cf. 15.25, 15.25.1\<close>
| Ass: "\<lbrakk>E,dt\<Turnstile>va\<Colon>=T; va \<noteq> LVar This;
E,dt\<Turnstile>v \<Colon>-T';
prg E\<turnstile>T'\<preceq>T\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>va:=v\<Colon>-T'"
\<comment> \<open>cf. 15.24\<close>
| Cond: "\<lbrakk>E,dt\<Turnstile>e0\<Colon>-PrimT Boolean;
E,dt\<Turnstile>e1\<Colon>-T1; E,dt\<Turnstile>e2\<Colon>-T2;
prg E\<turnstile>T1\<preceq>T2 \<and> T = T2 \<or> prg E\<turnstile>T2\<preceq>T1 \<and> T = T1\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>e0 ? e1 : e2\<Colon>-T"
\<comment> \<open>cf. 15.11.1, 15.11.2, 15.11.3\<close>
| Call: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-RefT statT;
E,dt\<Turnstile>ps\<Colon>\<doteq>pTs;
max_spec (prg E) (cls E) statT \<lparr>name=mn,parTs=pTs\<rparr>
= {((statDeclT,m),pTs')}
\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>{cls E,statT,invmode m e}e\<cdot>mn({pTs'}ps)\<Colon>-(resTy m)"
| Methd: "\<lbrakk>is_class (prg E) C;
methd (prg E) C sig = Some m;
E,dt\<Turnstile>Body (declclass m) (stmt (mbody (mthd m)))\<Colon>-T\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Methd C sig\<Colon>-T"
\<comment> \<open>The class \<^term>\<open>C\<close> is the dynamic class of the method call
(cf. Eval.thy).
It hasn't got to be directly accessible from the current package
\<^term>\<open>(pkg E)\<close>.
Only the static class must be accessible (enshured indirectly by
\<^term>\<open>Call\<close>).
Note that l is just a dummy value. It is only used in the smallstep
semantics. To proof typesafety directly for the smallstep semantics
we would have to assume conformance of l here!\<close>
| Body: "\<lbrakk>is_class (prg E) D;
E,dt\<Turnstile>blk\<Colon>\<surd>;
(lcl E) Result = Some T;
is_type (prg E) T\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>Body D blk\<Colon>-T"
\<comment> \<open>The class \<^term>\<open>D\<close> implementing the method must not directly be
accessible from the current package \<^term>\<open>(pkg E)\<close>, but can also
be indirectly accessible due to inheritance (enshured in \<^term>\<open>Call\<close>)
The result type hasn't got to be accessible in Java! (If it is not
accessible you can only assign it to Object).
For dummy value l see rule \<^term>\<open>Methd\<close>.\<close>
\<comment> \<open>well-typed variables\<close>
\<comment> \<open>cf. 15.13.1\<close>
| LVar: "\<lbrakk>lcl E vn = Some T; is_acc_type (prg E) (pkg E) T\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>LVar vn\<Colon>=T"
\<comment> \<open>cf. 15.10.1\<close>
| FVar: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-Class C;
accfield (prg E) (cls E) C fn = Some (statDeclC,f)\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>{cls E,statDeclC,is_static f}e..fn\<Colon>=(type f)"
\<comment> \<open>cf. 15.12\<close>
| AVar: "\<lbrakk>E,dt\<Turnstile>e\<Colon>-T.[];
E,dt\<Turnstile>i\<Colon>-PrimT Integer\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>e.[i]\<Colon>=T"
\<comment> \<open>well-typed expression lists\<close>
\<comment> \<open>cf. 15.11.???\<close>
| Nil: "E,dt\<Turnstile>[]\<Colon>\<doteq>[]"
\<comment> \<open>cf. 15.11.???\<close>
| Cons: "\<lbrakk>E,dt\<Turnstile>e \<Colon>-T;
E,dt\<Turnstile>es\<Colon>\<doteq>Ts\<rbrakk> \<Longrightarrow>
E,dt\<Turnstile>e#es\<Colon>\<doteq>T#Ts"
|
###output
\<lbrakk> ?H1 x_1 x_2 (?H2 (?H3 x_3 x_4)) x_5; \<And>y_0. \<lbrakk>x_5 = ?H4 y_0; ?H5 (?H6 x_1) x_3; ?H7 x_1 x_2 x_4; ?H8 x_1 ?H9 = ?H10 y_0; ?H11 (?H6 x_1) y_0\<rbrakk> \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6###end
|
Predicate_Compile_Examples/Predicate_Compile_Tests
|
Predicate_Compile_Tests.predicate_where_argument_is_condition_and_value_oE
| null |
pred.eval predicate_where_argument_is_condition_and_value_o ?x \<Longrightarrow> (predicate_where_argument_is_condition_and_value ?x \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 ?H2 x_1; ?H3 x_1 \<Longrightarrow> x_2\<rbrakk> \<Longrightarrow> x_2
|
[
"Predicate_Compile_Tests.predicate_where_argument_is_condition_and_value",
"Predicate_Compile_Tests.predicate_where_argument_is_condition_and_value_o",
"Predicate.pred.eval"
] |
[
"bool \\<Rightarrow> bool",
"bool Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive predicate_where_argument_is_condition_and_value :: \"bool \\<Rightarrow> bool\"\nwhere\n \"predicate_where_argument_is_condition_and_value ck \\<Longrightarrow> ck\n \\<Longrightarrow> predicate_where_argument_is_condition_and_value ck\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
template
|
###lemma
pred.eval predicate_where_argument_is_condition_and_value_o ?x \<Longrightarrow> (predicate_where_argument_is_condition_and_value ?x \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Predicate_Compile_Tests.predicate_where_argument_is_condition_and_value :::: bool \<Rightarrow> bool
Predicate_Compile_Tests.predicate_where_argument_is_condition_and_value_o :::: bool Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
###defs
inductive predicate_where_argument_is_condition_and_value :: "bool \<Rightarrow> bool"
where
"predicate_where_argument_is_condition_and_value ck \<Longrightarrow> ck
\<Longrightarrow> predicate_where_argument_is_condition_and_value ck"
datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: "'a \<Rightarrow> bool")
|
###output
\<lbrakk> ?H1 ?H2 x_1; ?H3 x_1 \<Longrightarrow> x_2\<rbrakk> \<Longrightarrow> x_2###end
|
Limits
|
Limits.filterlim_at_top_add_at_top
|
lemma filterlim_at_top_add_at_top:
assumes f: "LIM x F. f x :> at_top"
and g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x + g x :: real) :> at_top"
|
filterlim ?f at_top ?F \<Longrightarrow> filterlim ?g at_top ?F \<Longrightarrow> LIM x ?F. ?f x + ?g x :> at_top
|
\<lbrakk> ?H1 x_1 ?H2 x_2; ?H1 x_3 ?H2 x_2\<rbrakk> \<Longrightarrow> ?H1 (\<lambda>y_2. ?H3 (x_1 y_2) (x_3 y_2)) ?H2 x_2
|
[
"Groups.plus_class.plus",
"Filter.at_top",
"Filter.filterlim"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a filter",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b filter \\<Rightarrow> 'a filter \\<Rightarrow> bool"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition at_top :: \"('a::order) filter\"\n where \"at_top = (\\<Sqinter>k. principal {k ..})\"",
"definition filterlim :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b filter \\<Rightarrow> 'a filter \\<Rightarrow> bool\" where\n \"filterlim f F2 F1 \\<longleftrightarrow> filtermap f F1 \\<le> F2\""
] |
template
|
###lemma
filterlim ?f at_top ?F \<Longrightarrow> filterlim ?g at_top ?F \<Longrightarrow> LIM x ?F. ?f x + ?g x :> at_top
###symbols
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Filter.at_top :::: 'a filter
Filter.filterlim :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
definition at_top :: "('a::order) filter"
where "at_top = (\<Sqinter>k. principal {k ..})"
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
|
###output
\<lbrakk> ?H1 x_1 ?H2 x_2; ?H1 x_3 ?H2 x_2\<rbrakk> \<Longrightarrow> ?H1 (\<lambda>y_2. ?H3 (x_1 y_2) (x_3 y_2)) ?H2 x_2###end
|
Complex
|
Complex.sum_roots_unity
|
lemma sum_roots_unity:
assumes "n > 1"
shows "\<Sum>{z::complex. z ^ n = 1} = 0"
|
1 < ?n \<Longrightarrow> \<Sum> {z. z ^ ?n = 1} = 0
|
?H1 < x_1 \<Longrightarrow> ?H2 (?H3 (\<lambda>y_0. ?H4 y_0 x_1 = ?H1)) = ?H5
|
[
"Groups.zero_class.zero",
"Power.power_class.power",
"Set.Collect",
"Groups_Big.comm_monoid_add_class.Sum",
"Groups.one_class.one"
] |
[
"'a",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a",
"'a"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"class one =\n fixes one :: 'a (\"1\")"
] |
template
|
###lemma
1 < ?n \<Longrightarrow> \<Sum> {z. z ^ ?n = 1} = 0
###symbols
Groups.zero_class.zero :::: 'a
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Groups_Big.comm_monoid_add_class.Sum :::: 'a set \<Rightarrow> 'a
Groups.one_class.one :::: 'a
###defs
class zero =
fixes zero :: 'a ("0")
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
class one =
fixes one :: 'a ("1")
|
###output
?H1 < x_1 \<Longrightarrow> ?H2 (?H3 (\<lambda>y_0. ?H4 y_0 x_1 = ?H1)) = ?H5###end
|
Analysis/Brouwer_Fixpoint
|
Brouwer_Fixpoint.swap_apply2
| null |
(?f \<circ> Transposition.transpose ?a ?b) ?b = ?f ?a
|
?H1 x_1 (?H2 x_2 x_3) x_3 = x_1 x_2
|
[
"Transposition.transpose",
"Fun.comp"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'a) \\<Rightarrow> 'c \\<Rightarrow> 'b"
] |
[
"definition transpose :: \\<open>'a \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close>\n where \\<open>transpose a b c = (if c = a then b else if c = b then a else c)\\<close>",
"definition comp :: \"('b \\<Rightarrow> 'c) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'c\" (infixl \"\\<circ>\" 55)\n where \"f \\<circ> g = (\\<lambda>x. f (g x))\""
] |
template
|
###lemma
(?f \<circ> Transposition.transpose ?a ?b) ?b = ?f ?a
###symbols
Transposition.transpose :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a
Fun.comp :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'b
###defs
definition transpose :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>
where \<open>transpose a b c = (if c = a then b else if c = b then a else c)\<close>
definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>" 55)
where "f \<circ> g = (\<lambda>x. f (g x))"
|
###output
?H1 x_1 (?H2 x_2 x_3) x_3 = x_1 x_2###end
|
Transcendental
|
Transcendental.arccos_ubound
|
lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi"
|
- 1 \<le> ?y \<Longrightarrow> ?y \<le> 1 \<Longrightarrow> arccos ?y \<le> pi
|
\<lbrakk> ?H1 ?H2 \<le> x_1; x_1 \<le> ?H2\<rbrakk> \<Longrightarrow> ?H3 x_1 \<le> ?H4
|
[
"Transcendental.pi",
"Transcendental.arccos",
"Groups.one_class.one",
"Groups.uminus_class.uminus"
] |
[
"real",
"real \\<Rightarrow> real",
"'a",
"'a \\<Rightarrow> 'a"
] |
[
"definition pi :: real\n where \"pi = 2 * (THE x. 0 \\<le> x \\<and> x \\<le> 2 \\<and> cos x = 0)\"",
"definition arccos :: \"real \\<Rightarrow> real\"\n where \"arccos y = (THE x. 0 \\<le> x \\<and> x \\<le> pi \\<and> cos x = y)\"",
"class one =\n fixes one :: 'a (\"1\")",
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\"- _\" [81] 80)"
] |
template
|
###lemma
- 1 \<le> ?y \<Longrightarrow> ?y \<le> 1 \<Longrightarrow> arccos ?y \<le> pi
###symbols
Transcendental.pi :::: real
Transcendental.arccos :::: real \<Rightarrow> real
Groups.one_class.one :::: 'a
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
###defs
definition pi :: real
where "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
definition arccos :: "real \<Rightarrow> real"
where "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)"
class one =
fixes one :: 'a ("1")
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
|
###output
\<lbrakk> ?H1 ?H2 \<le> x_1; x_1 \<le> ?H2\<rbrakk> \<Longrightarrow> ?H3 x_1 \<le> ?H4###end
|
HOLCF/IOA/CompoTraces
|
CompoTraces.subst_lemma1
|
lemma subst_lemma1: "f \<sqsubseteq> g x \<Longrightarrow> x = h x \<Longrightarrow> f \<sqsubseteq> g (h x)"
|
?f \<sqsubseteq> ?g ?x \<Longrightarrow> ?x = ?h ?x \<Longrightarrow> ?f \<sqsubseteq> ?g (?h ?x)
|
\<lbrakk> ?H1 x_1 (x_2 x_3); x_3 = x_4 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 (x_2 (x_4 x_3))
|
[
"Porder.below_class.below"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"class below =\n fixes below :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\nbegin"
] |
template
|
###lemma
?f \<sqsubseteq> ?g ?x \<Longrightarrow> ?x = ?h ?x \<Longrightarrow> ?f \<sqsubseteq> ?g (?h ?x)
###symbols
Porder.below_class.below :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
###defs
class below =
fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
begin
|
###output
\<lbrakk> ?H1 x_1 (x_2 x_3); x_3 = x_4 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 (x_2 (x_4 x_3))###end
|
Real_Asymp/Multiseries_Expansion_Bounds
|
Multiseries_Expansion_Bounds.abs_powr_nat_le
|
lemma abs_powr_nat_le: "abs (powr_nat x y) \<le> powr_nat (abs x) y"
|
\<bar>powr_nat ?x ?y\<bar> \<le> powr_nat \<bar> ?x\<bar> ?y
|
?H1 (?H2 x_1 x_2) \<le> ?H2 (?H1 x_1) x_2
|
[
"Multiseries_Expansion.powr_nat",
"Groups.abs_class.abs"
] |
[
"real \\<Rightarrow> real \\<Rightarrow> real",
"'a \\<Rightarrow> 'a"
] |
[
"definition powr_nat :: \"real \\<Rightarrow> real \\<Rightarrow> real\" where \n \"powr_nat x y = \n (if y = 0 then 1\n else if x < 0 then cos (pi * y) * (-x) powr y else x powr y)\"",
"class abs =\n fixes abs :: \"'a \\<Rightarrow> 'a\" (\"\\<bar>_\\<bar>\")"
] |
template
|
###lemma
\<bar>powr_nat ?x ?y\<bar> \<le> powr_nat \<bar> ?x\<bar> ?y
###symbols
Multiseries_Expansion.powr_nat :::: real \<Rightarrow> real \<Rightarrow> real
Groups.abs_class.abs :::: 'a \<Rightarrow> 'a
###defs
definition powr_nat :: "real \<Rightarrow> real \<Rightarrow> real" where
"powr_nat x y =
(if y = 0 then 1
else if x < 0 then cos (pi * y) * (-x) powr y else x powr y)"
class abs =
fixes abs :: "'a \<Rightarrow> 'a" ("\<bar>_\<bar>")
|
###output
?H1 (?H2 x_1 x_2) \<le> ?H2 (?H1 x_1) x_2###end
|
Imperative_HOL/Array
|
Array.noteq_sym
|
lemma noteq_sym: "a =!!= b \<Longrightarrow> b =!!= a"
and unequal [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
|
?a =!!= ?b \<Longrightarrow> ?b =!!= ?a
|
?H1 x_1 x_2 \<Longrightarrow> ?H1 x_2 x_1
|
[
"Array.noteq"
] |
[
"'a array \\<Rightarrow> 'b array \\<Rightarrow> bool"
] |
[
"definition noteq :: \"'a::heap array \\<Rightarrow> 'b::heap array \\<Rightarrow> bool\" (infix \"=!!=\" 70) where\n \"r =!!= s \\<longleftrightarrow> TYPEREP('a) \\<noteq> TYPEREP('b) \\<or> addr_of_array r \\<noteq> addr_of_array s\""
] |
template
|
###lemma
?a =!!= ?b \<Longrightarrow> ?b =!!= ?a
###symbols
Array.noteq :::: 'a array \<Rightarrow> 'b array \<Rightarrow> bool
###defs
definition noteq :: "'a::heap array \<Rightarrow> 'b::heap array \<Rightarrow> bool" (infix "=!!=" 70) where
"r =!!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s"
|
###output
?H1 x_1 x_2 \<Longrightarrow> ?H1 x_2 x_1###end
|
Nominal/Examples/Class3
|
Class3.crename_NotL
|
lemma crename_NotL:
assumes a: "R[a\<turnstile>c>b] = NotL <c>.N y" "c\<sharp>(R,a,b)"
shows "\<exists>N'. (R = NotL <c>.N' y) \<and> N'[a\<turnstile>c>b] = N"
|
?R[ ?a\<turnstile>c> ?b] = NotL < ?c>. ?N ?y \<Longrightarrow> ?c \<sharp> (?R, ?a, ?b) \<Longrightarrow> \<exists>N'. ?R = NotL < ?c>.N' ?y \<and> N'[ ?a\<turnstile>c> ?b] = ?N
|
\<lbrakk> ?H1 x_1 x_2 x_3 = ?H2 x_4 x_5 x_6; ?H3 x_4 (x_1, x_2, x_3)\<rbrakk> \<Longrightarrow> \<exists>y_0. x_1 = ?H2 x_4 y_0 x_6 \<and> ?H1 y_0 x_2 x_3 = x_5
|
[
"Nominal.fresh",
"Class1.trm.NotL",
"Class1.crename"
] |
[
"'a \\<Rightarrow> 'b \\<Rightarrow> bool",
"coname \\<Rightarrow> trm \\<Rightarrow> name \\<Rightarrow> trm",
"trm \\<Rightarrow> coname \\<Rightarrow> coname \\<Rightarrow> trm"
] |
[
"definition fresh :: \"'x \\<Rightarrow> 'a \\<Rightarrow> bool\" (\\<open>_ \\<sharp> _\\<close> [80,80] 80) where\n \"a \\<sharp> x \\<longleftrightarrow> a \\<notin> supp x\""
] |
template
|
###lemma
?R[ ?a\<turnstile>c> ?b] = NotL < ?c>. ?N ?y \<Longrightarrow> ?c \<sharp> (?R, ?a, ?b) \<Longrightarrow> \<exists>N'. ?R = NotL < ?c>.N' ?y \<and> N'[ ?a\<turnstile>c> ?b] = ?N
###symbols
Nominal.fresh :::: 'a \<Rightarrow> 'b \<Rightarrow> bool
Class1.trm.NotL :::: coname \<Rightarrow> trm \<Rightarrow> name \<Rightarrow> trm
Class1.crename :::: trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm
###defs
definition fresh :: "'x \<Rightarrow> 'a \<Rightarrow> bool" (\<open>_ \<sharp> _\<close> [80,80] 80) where
"a \<sharp> x \<longleftrightarrow> a \<notin> supp x"
|
###output
\<lbrakk> ?H1 x_1 x_2 x_3 = ?H2 x_4 x_5 x_6; ?H3 x_4 (x_1, x_2, x_3)\<rbrakk> \<Longrightarrow> \<exists>y_0. x_1 = ?H2 x_4 y_0 x_6 \<and> ?H1 y_0 x_2 x_3 = x_5###end
|
Groups
|
Groups.algebra_simps(22)
| null |
(?a - ?b) * ?c = ?a * ?c - ?b * ?c
|
?H1 (?H2 x_1 x_2) x_3 = ?H2 (?H1 x_1 x_3) (?H1 x_2 x_3)
|
[
"Groups.minus_class.minus",
"Groups.times_class.times"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)"
] |
template
|
###lemma
(?a - ?b) * ?c = ?a * ?c - ?b * ?c
###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
|
###output
?H1 (?H2 x_1 x_2) x_3 = ?H2 (?H1 x_1 x_3) (?H1 x_2 x_3)###end
|
Bit_Operations
|
Bit_Operations.and_int_code(1)
|
lemma and_int_code [code]:
fixes i j :: int shows
\<open>0 AND j = 0\<close>
\<open>i AND 0 = 0\<close>
\<open>Int.Pos n AND Int.Pos m = (case and_num n m of None \<Rightarrow> 0 | Some n' \<Rightarrow> Int.Pos n')\<close>
\<open>Int.Neg n AND Int.Neg m = NOT (Num.sub n num.One OR Num.sub m num.One)\<close>
\<open>Int.Pos n AND Int.Neg num.One = Int.Pos n\<close>
\<open>Int.Pos n AND Int.Neg (num.Bit0 m) = Num.sub (or_not_num_neg (Num.BitM m) n) num.One\<close>
\<open>Int.Pos n AND Int.Neg (num.Bit1 m) = Num.sub (or_not_num_neg (num.Bit0 m) n) num.One\<close>
\<open>Int.Neg num.One AND Int.Pos m = Int.Pos m\<close>
\<open>Int.Neg (num.Bit0 n) AND Int.Pos m = Num.sub (or_not_num_neg (Num.BitM n) m) num.One\<close>
\<open>Int.Neg (num.Bit1 n) AND Int.Pos m = Num.sub (or_not_num_neg (num.Bit0 n) m) num.One\<close>
|
and 0 ?j = 0
|
?H1 ?H2 x_1 = ?H2
|
[
"Groups.zero_class.zero",
"Bit_Operations.semiring_bit_operations_class.and"
] |
[
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"class semiring_bit_operations = semiring_bits +\n fixes \"and\" :: \\<open>'a \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close> (infixr \\<open>AND\\<close> 64)\n and or :: \\<open>'a \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close> (infixr \\<open>OR\\<close> 59)\n and xor :: \\<open>'a \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close> (infixr \\<open>XOR\\<close> 59)\n and mask :: \\<open>nat \\<Rightarrow> 'a\\<close>\n and set_bit :: \\<open>nat \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close>\n and unset_bit :: \\<open>nat \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close>\n and flip_bit :: \\<open>nat \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close>\n and push_bit :: \\<open>nat \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close>\n and drop_bit :: \\<open>nat \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close>\n and take_bit :: \\<open>nat \\<Rightarrow> 'a \\<Rightarrow> 'a\\<close>\n assumes and_rec: \\<open>a AND b = of_bool (odd a \\<and> odd b) + 2 * ((a div 2) AND (b div 2))\\<close>\n and or_rec: \\<open>a OR b = of_bool (odd a \\<or> odd b) + 2 * ((a div 2) OR (b div 2))\\<close>\n and xor_rec: \\<open>a XOR b = of_bool (odd a \\<noteq> odd b) + 2 * ((a div 2) XOR (b div 2))\\<close>\n and mask_eq_exp_minus_1: \\<open>mask n = 2 ^ n - 1\\<close>\n and set_bit_eq_or: \\<open>set_bit n a = a OR push_bit n 1\\<close>\n and unset_bit_eq_or_xor: \\<open>unset_bit n a = (a OR push_bit n 1) XOR push_bit n 1\\<close>\n and flip_bit_eq_xor: \\<open>flip_bit n a = a XOR push_bit n 1\\<close>\n and push_bit_eq_mult: \\<open>push_bit n a = a * 2 ^ n\\<close>\n and drop_bit_eq_div: \\<open>drop_bit n a = a div 2 ^ n\\<close>\n and take_bit_eq_mod: \\<open>take_bit n a = a mod 2 ^ n\\<close>\nbegin"
] |
template
|
###lemma
and 0 ?j = 0
###symbols
Groups.zero_class.zero :::: 'a
Bit_Operations.semiring_bit_operations_class.and :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
###defs
class zero =
fixes zero :: 'a ("0")
class semiring_bit_operations = semiring_bits +
fixes "and" :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>AND\<close> 64)
and or :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>OR\<close> 59)
and xor :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>XOR\<close> 59)
and mask :: \<open>nat \<Rightarrow> 'a\<close>
and set_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
and unset_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
and flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
and push_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
and drop_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
and take_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close>
assumes and_rec: \<open>a AND b = of_bool (odd a \<and> odd b) + 2 * ((a div 2) AND (b div 2))\<close>
and or_rec: \<open>a OR b = of_bool (odd a \<or> odd b) + 2 * ((a div 2) OR (b div 2))\<close>
and xor_rec: \<open>a XOR b = of_bool (odd a \<noteq> odd b) + 2 * ((a div 2) XOR (b div 2))\<close>
and mask_eq_exp_minus_1: \<open>mask n = 2 ^ n - 1\<close>
and set_bit_eq_or: \<open>set_bit n a = a OR push_bit n 1\<close>
and unset_bit_eq_or_xor: \<open>unset_bit n a = (a OR push_bit n 1) XOR push_bit n 1\<close>
and flip_bit_eq_xor: \<open>flip_bit n a = a XOR push_bit n 1\<close>
and push_bit_eq_mult: \<open>push_bit n a = a * 2 ^ n\<close>
and drop_bit_eq_div: \<open>drop_bit n a = a div 2 ^ n\<close>
and take_bit_eq_mod: \<open>take_bit n a = a mod 2 ^ n\<close>
begin
|
###output
?H1 ?H2 x_1 = ?H2###end
|
Set
|
Set.emptyE
|
lemma emptyE [elim!]: "a \<in> {} \<Longrightarrow> P"
|
?a \<in> {} \<Longrightarrow> ?P
|
x_1 \<in> ?H1 \<Longrightarrow> x_2
|
[
"Set.empty"
] |
[
"'a set"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\""
] |
template
|
###lemma
?a \<in> {} \<Longrightarrow> ?P
###symbols
Set.empty :::: 'a set
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
|
###output
x_1 \<in> ?H1 \<Longrightarrow> x_2###end
|
Analysis/Extended_Real_Limits
|
Extended_Real_Limits.ereal_limsup_add_mono
|
lemma ereal_limsup_add_mono:
fixes u v::"nat \<Rightarrow> ereal"
shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
|
limsup (\<lambda>n. ?u n + ?v n) \<le> limsup ?u + limsup ?v
|
?H1 (\<lambda>y_0. ?H2 (x_1 y_0) (x_2 y_0)) \<le> ?H2 (?H1 x_1) (?H1 x_2)
|
[
"Groups.plus_class.plus",
"Liminf_Limsup.limsup"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(nat \\<Rightarrow> 'a) \\<Rightarrow> 'a"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"abbreviation \"limsup \\<equiv> Limsup sequentially\""
] |
template
|
###lemma
limsup (\<lambda>n. ?u n + ?v n) \<le> limsup ?u + limsup ?v
###symbols
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Liminf_Limsup.limsup :::: (nat \<Rightarrow> 'a) \<Rightarrow> 'a
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
abbreviation "limsup \<equiv> Limsup sequentially"
|
###output
?H1 (\<lambda>y_0. ?H2 (x_1 y_0) (x_2 y_0)) \<le> ?H2 (?H1 x_1) (?H1 x_2)###end
|
Analysis/Starlike
|
Starlike.aff_dim_halfspace_ge
|
lemma aff_dim_halfspace_ge:
fixes a :: "'a::euclidean_space"
shows "aff_dim {x. a \<bullet> x \<ge> r} =
(if a = 0 \<and> r > 0 then -1 else DIM('a))"
|
aff_dim {x. ?r \<le> ?a \<bullet> x} = (if ?a = (0:: ?'a) \<and> 0 < ?r then - 1 else int DIM(?'a))
|
?H1 (?H2 (\<lambda>y_0. x_1 \<le> ?H3 x_2 y_0)) = (if x_2 = ?H4 \<and> ?H4 < x_1 then ?H5 ?H6 else ?H7 (?H8 ?H9))
|
[
"Euclidean_Space.euclidean_space_class.Basis",
"Finite_Set.card",
"Int.int",
"Groups.one_class.one",
"Groups.uminus_class.uminus",
"Groups.zero_class.zero",
"Inner_Product.real_inner_class.inner",
"Set.Collect",
"Affine.aff_dim"
] |
[
"'a set",
"'a set \\<Rightarrow> nat",
"nat \\<Rightarrow> int",
"'a",
"'a \\<Rightarrow> 'a",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> real",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> int"
] |
[
"class euclidean_space = real_inner +\n fixes Basis :: \"'a set\"\n assumes nonempty_Basis [simp]: \"Basis \\<noteq> {}\"\n assumes finite_Basis [simp]: \"finite Basis\"\n assumes inner_Basis:\n \"\\<lbrakk>u \\<in> Basis; v \\<in> Basis\\<rbrakk> \\<Longrightarrow> inner u v = (if u = v then 1 else 0)\"\n assumes euclidean_all_zero_iff:\n \"(\\<forall>u\\<in>Basis. inner x u = 0) \\<longleftrightarrow> (x = 0)\"",
"abbreviation int :: \"nat \\<Rightarrow> int\"\n where \"int \\<equiv> of_nat\"",
"class one =\n fixes one :: 'a (\"1\")",
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\"- _\" [81] 80)",
"class zero =\n fixes zero :: 'a (\"0\")",
"class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +\n fixes inner :: \"'a \\<Rightarrow> 'a \\<Rightarrow> real\"\n assumes inner_commute: \"inner x y = inner y x\"\n and inner_add_left: \"inner (x + y) z = inner x z + inner y z\"\n and inner_scaleR_left [simp]: \"inner (scaleR r x) y = r * (inner x y)\"\n and inner_ge_zero [simp]: \"0 \\<le> inner x x\"\n and inner_eq_zero_iff [simp]: \"inner x x = 0 \\<longleftrightarrow> x = 0\"\n and norm_eq_sqrt_inner: \"norm x = sqrt (inner x x)\"\nbegin"
] |
template
|
###lemma
aff_dim {x. ?r \<le> ?a \<bullet> x} = (if ?a = (0:: ?'a) \<and> 0 < ?r then - 1 else int DIM(?'a))
###symbols
Euclidean_Space.euclidean_space_class.Basis :::: 'a set
Finite_Set.card :::: 'a set \<Rightarrow> nat
Int.int :::: nat \<Rightarrow> int
Groups.one_class.one :::: 'a
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Affine.aff_dim :::: 'a set \<Rightarrow> int
###defs
class euclidean_space = real_inner +
fixes Basis :: "'a set"
assumes nonempty_Basis [simp]: "Basis \<noteq> {}"
assumes finite_Basis [simp]: "finite Basis"
assumes inner_Basis:
"\<lbrakk>u \<in> Basis; v \<in> Basis\<rbrakk> \<Longrightarrow> inner u v = (if u = v then 1 else 0)"
assumes euclidean_all_zero_iff:
"(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> (x = 0)"
abbreviation int :: "nat \<Rightarrow> int"
where "int \<equiv> of_nat"
class one =
fixes one :: 'a ("1")
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
class zero =
fixes zero :: 'a ("0")
class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
assumes inner_commute: "inner x y = inner y x"
and inner_add_left: "inner (x + y) z = inner x z + inner y z"
and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)"
and inner_ge_zero [simp]: "0 \<le> inner x x"
and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> x = 0"
and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)"
begin
|
###output
?H1 (?H2 (\<lambda>y_0. x_1 \<le> ?H3 x_2 y_0)) = (if x_2 = ?H4 \<and> ?H4 < x_1 then ?H5 ?H6 else ?H7 (?H8 ?H9))###end
|
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