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
listlengths 0
101
| types
listlengths 0
101
| defs
listlengths 0
83
| output_key
stringclasses 1
value | input
stringlengths 37
79.6k
| output
stringlengths 24
16.1k
|
|---|---|---|---|---|---|---|---|---|---|---|
Matrix_LP/ComputeNumeral
|
ComputeNumeral.compute_numeral(18)
| null |
(True \<longrightarrow> ?P) = ?P
|
(True \<longrightarrow> x_1) = x_1
|
[] |
[] |
[] |
template
|
###lemma
(True \<longrightarrow> ?P) = ?P
###symbols
###defs
|
###output
(True \<longrightarrow> x_1) = x_1###end
|
Predicate_Compile_Examples/Predicate_Compile_Tests
|
Predicate_Compile_Tests.is_D_or_E_iE
| null |
pred.eval (is_D_or_E_i ?x) ?y \<Longrightarrow> (is_D_or_E ?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.is_D_or_E",
"Predicate_Compile_Tests.is_D_or_E_i",
"Predicate.pred.eval"
] |
[
"Predicate_Compile_Tests.char \\<Rightarrow> bool",
"Predicate_Compile_Tests.char \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive is_D_or_E\nwhere\n \"(x = D) \\<or> (x = E) ==> is_D_or_E x\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
template
|
###lemma
pred.eval (is_D_or_E_i ?x) ?y \<Longrightarrow> (is_D_or_E ?x \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Predicate_Compile_Tests.is_D_or_E :::: Predicate_Compile_Tests.char \<Rightarrow> bool
Predicate_Compile_Tests.is_D_or_E_i :::: Predicate_Compile_Tests.char \<Rightarrow> unit Predicate.pred
Predicate.pred.eval :::: 'a Predicate.pred \<Rightarrow> 'a \<Rightarrow> bool
###defs
inductive is_D_or_E
where
"(x = D) \<or> (x = E) ==> is_D_or_E x"
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
|
Map
|
Map.map_comp_None_iff
|
lemma map_comp_None_iff:
"((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) "
|
((?m1.0 \<circ>\<^sub>m ?m2.0) ?k = None) = (?m2.0 ?k = None \<or> (\<exists>k'. ?m2.0 ?k = Some k' \<and> ?m1.0 k' = None))
|
(?H1 x_1 x_2 x_3 = ?H2) = (x_2 x_3 = ?H2 \<or> (\<exists>y_0. x_2 x_3 = ?H3 y_0 \<and> x_1 y_0 = ?H2))
|
[
"Option.option.Some",
"Option.option.None",
"Map.map_comp"
] |
[
"'a \\<Rightarrow> 'a option",
"'a option",
"('a \\<Rightarrow> 'b option) \\<Rightarrow> ('c \\<Rightarrow> 'a option) \\<Rightarrow> 'c \\<Rightarrow> 'b option"
] |
[
"datatype 'a option =\n None\n | Some (the: 'a)",
"datatype 'a option =\n None\n | Some (the: 'a)",
"definition\n map_comp :: \"('b \\<rightharpoonup> 'c) \\<Rightarrow> ('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'c)\" (infixl \"\\<circ>\\<^sub>m\" 55) where\n \"f \\<circ>\\<^sub>m g = (\\<lambda>k. case g k of None \\<Rightarrow> None | Some v \\<Rightarrow> f v)\""
] |
template
|
###lemma
((?m1.0 \<circ>\<^sub>m ?m2.0) ?k = None) = (?m2.0 ?k = None \<or> (\<exists>k'. ?m2.0 ?k = Some k' \<and> ?m1.0 k' = None))
###symbols
Option.option.Some :::: 'a \<Rightarrow> 'a option
Option.option.None :::: 'a option
Map.map_comp :::: ('a \<Rightarrow> 'b option) \<Rightarrow> ('c \<Rightarrow> 'a option) \<Rightarrow> 'c \<Rightarrow> 'b option
###defs
datatype 'a option =
None
| Some (the: 'a)
datatype 'a option =
None
| Some (the: 'a)
definition
map_comp :: "('b \<rightharpoonup> 'c) \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'c)" (infixl "\<circ>\<^sub>m" 55) where
"f \<circ>\<^sub>m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
|
###output
(?H1 x_1 x_2 x_3 = ?H2) = (x_2 x_3 = ?H2 \<or> (\<exists>y_0. x_2 x_3 = ?H3 y_0 \<and> x_1 y_0 = ?H2))###end
|
Analysis/Weierstrass_Theorems
|
Weierstrass_Theorems.Stone_Weierstrass_real_polynomial_function
|
lemma Stone_Weierstrass_real_polynomial_function:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes "compact S" "continuous_on S f" "0 < e"
obtains g where "real_polynomial_function g" "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x - g x\<bar> < e"
|
compact ?S \<Longrightarrow> continuous_on ?S ?f \<Longrightarrow> 0 < ?e \<Longrightarrow> (\<And>g. real_polynomial_function g \<Longrightarrow> (\<And>x. x \<in> ?S \<Longrightarrow> \<bar> ?f x - g x\<bar> < ?e) \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
|
\<lbrakk> ?H1 x_1; ?H2 x_1 x_2; ?H3 < x_3; \<And>y_0. \<lbrakk> ?H4 y_0; \<And>y_1. y_1 \<in> x_1 \<Longrightarrow> ?H5 (?H6 (x_2 y_1) (y_0 y_1)) < x_3\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Groups.minus_class.minus",
"Groups.abs_class.abs",
"Weierstrass_Theorems.real_polynomial_function",
"Groups.zero_class.zero",
"Topological_Spaces.continuous_on",
"Topological_Spaces.topological_space_class.compact"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> real) \\<Rightarrow> bool",
"'a",
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool",
"'a set \\<Rightarrow> bool"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"class abs =\n fixes abs :: \"'a \\<Rightarrow> 'a\" (\"\\<bar>_\\<bar>\")",
"inductive real_polynomial_function :: \"('a::real_normed_vector \\<Rightarrow> real) \\<Rightarrow> bool\" where\n linear: \"bounded_linear f \\<Longrightarrow> real_polynomial_function f\"\n | const: \"real_polynomial_function (\\<lambda>x. c)\"\n | add: \"\\<lbrakk>real_polynomial_function f; real_polynomial_function g\\<rbrakk> \\<Longrightarrow> real_polynomial_function (\\<lambda>x. f x + g x)\"\n | mult: \"\\<lbrakk>real_polynomial_function f; real_polynomial_function g\\<rbrakk> \\<Longrightarrow> real_polynomial_function (\\<lambda>x. f x * g x)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"definition continuous_on :: \"'a set \\<Rightarrow> ('a::topological_space \\<Rightarrow> 'b::topological_space) \\<Rightarrow> bool\"\n where \"continuous_on s f \\<longleftrightarrow> (\\<forall>x\\<in>s. (f \\<longlongrightarrow> f x) (at x within s))\"",
"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
compact ?S \<Longrightarrow> continuous_on ?S ?f \<Longrightarrow> 0 < ?e \<Longrightarrow> (\<And>g. real_polynomial_function g \<Longrightarrow> (\<And>x. x \<in> ?S \<Longrightarrow> \<bar> ?f x - g x\<bar> < ?e) \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.abs_class.abs :::: 'a \<Rightarrow> 'a
Weierstrass_Theorems.real_polynomial_function :::: ('a \<Rightarrow> real) \<Rightarrow> bool
Groups.zero_class.zero :::: 'a
Topological_Spaces.continuous_on :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
Topological_Spaces.topological_space_class.compact :::: 'a set \<Rightarrow> bool
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
class abs =
fixes abs :: "'a \<Rightarrow> 'a" ("\<bar>_\<bar>")
inductive real_polynomial_function :: "('a::real_normed_vector \<Rightarrow> real) \<Rightarrow> bool" where
linear: "bounded_linear f \<Longrightarrow> real_polynomial_function f"
| const: "real_polynomial_function (\<lambda>x. c)"
| add: "\<lbrakk>real_polynomial_function f; real_polynomial_function g\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. f x + g x)"
| mult: "\<lbrakk>real_polynomial_function f; real_polynomial_function g\<rbrakk> \<Longrightarrow> real_polynomial_function (\<lambda>x. f x * g x)"
class zero =
fixes zero :: 'a ("0")
definition continuous_on :: "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
where "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
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
\<lbrakk> ?H1 x_1; ?H2 x_1 x_2; ?H3 < x_3; \<And>y_0. \<lbrakk> ?H4 y_0; \<And>y_1. y_1 \<in> x_1 \<Longrightarrow> ?H5 (?H6 (x_2 y_1) (y_0 y_1)) < x_3\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4###end
|
Analysis/Change_Of_Vars
|
Change_Of_Vars.measurable_shear_interval
| null |
?m \<noteq> ?n \<Longrightarrow> cbox ?a ?b \<noteq> {} \<Longrightarrow> 0 \<le> ?a $ ?n \<Longrightarrow> (\<lambda>x. \<chi>i. if i = ?m then x $ ?m + x $ ?n else x $ i) ` cbox ?a ?b \<in> lmeasurable
|
\<lbrakk>x_1 \<noteq> x_2; ?H1 x_3 x_4 \<noteq> ?H2; ?H3 \<le> ?H4 x_3 x_2\<rbrakk> \<Longrightarrow> ?H5 (\<lambda>y_0. ?H6 (\<lambda>y_1. if y_1 = x_1 then ?H7 (?H4 y_0 x_1) (?H4 y_0 x_2) else ?H4 y_0 y_1)) (?H1 x_3 x_4) \<in> ?H8
|
[
"Lebesgue_Measure.lmeasurable",
"Groups.plus_class.plus",
"Finite_Cartesian_Product.vec.vec_lambda",
"Set.image",
"Finite_Cartesian_Product.vec.vec_nth",
"Groups.zero_class.zero",
"Set.empty",
"Topology_Euclidean_Space.cbox"
] |
[
"'a set set",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('b, 'a) vec",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a",
"'a",
"'a set",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a set"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"definition \"vec x = (\\<chi> i. x)\"",
"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}\"",
"definition \"vec x = (\\<chi> i. x)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\""
] |
template
|
###lemma
?m \<noteq> ?n \<Longrightarrow> cbox ?a ?b \<noteq> {} \<Longrightarrow> 0 \<le> ?a $ ?n \<Longrightarrow> (\<lambda>x. \<chi>i. if i = ?m then x $ ?m + x $ ?n else x $ i) ` cbox ?a ?b \<in> lmeasurable
###symbols
Lebesgue_Measure.lmeasurable :::: 'a set set
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Finite_Cartesian_Product.vec.vec_lambda :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('b, 'a) vec
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Finite_Cartesian_Product.vec.vec_nth :::: ('a, 'b) vec \<Rightarrow> 'b \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Set.empty :::: 'a set
Topology_Euclidean_Space.cbox :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a set
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
definition "vec x = (\<chi> i. x)"
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
definition "vec x = (\<chi> i. x)"
class zero =
fixes zero :: 'a ("0")
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
|
###output
\<lbrakk>x_1 \<noteq> x_2; ?H1 x_3 x_4 \<noteq> ?H2; ?H3 \<le> ?H4 x_3 x_2\<rbrakk> \<Longrightarrow> ?H5 (\<lambda>y_0. ?H6 (\<lambda>y_1. if y_1 = x_1 then ?H7 (?H4 y_0 x_1) (?H4 y_0 x_2) else ?H4 y_0 y_1)) (?H1 x_3 x_4) \<in> ?H8###end
|
Analysis/Cartesian_Space
|
Cartesian_Space.matrix_vector_mul(1)
|
lemma matrix_vector_mul[simp]:
"Vector_Spaces.linear (*s) (*s) g \<Longrightarrow> (\<lambda>y. matrix g *v y) = g"
"linear f \<Longrightarrow> (\<lambda>x. matrix f *v x) = f"
"bounded_linear f \<Longrightarrow> (\<lambda>x. matrix f *v x) = f"
for f :: "real^'n \<Rightarrow> real ^'m"
|
Vector_Spaces.linear (*s) (*s) ?g \<Longrightarrow> (*v) (matrix ?g) = ?g
|
?H1 ?H2 ?H2 x_1 \<Longrightarrow> ?H3 (?H4 x_1) = x_1
|
[
"Finite_Cartesian_Product.matrix",
"Finite_Cartesian_Product.matrix_vector_mult",
"Finite_Cartesian_Product.vector_scalar_mult",
"Vector_Spaces.linear"
] |
[
"(('a, 'b) vec \\<Rightarrow> ('a, 'c) vec) \\<Rightarrow> (('a, 'b) vec, 'c) vec",
"(('a, 'b) vec, 'c) vec \\<Rightarrow> ('a, 'b) vec \\<Rightarrow> ('a, 'c) vec",
"'a \\<Rightarrow> ('a, 'b) vec \\<Rightarrow> ('a, 'b) vec",
"('a \\<Rightarrow> 'b \\<Rightarrow> 'b) \\<Rightarrow> ('a \\<Rightarrow> 'c \\<Rightarrow> 'c) \\<Rightarrow> ('b \\<Rightarrow> 'c) \\<Rightarrow> bool"
] |
[
"definition vector_scalar_mult:: \"'a::times \\<Rightarrow> 'a ^ 'n \\<Rightarrow> 'a ^ 'n\" (infixl \"*s\" 70)\n where \"c *s x = (\\<chi> i. c * (x$i))\""
] |
template
|
###lemma
Vector_Spaces.linear (*s) (*s) ?g \<Longrightarrow> (*v) (matrix ?g) = ?g
###symbols
Finite_Cartesian_Product.matrix :::: (('a, 'b) vec \<Rightarrow> ('a, 'c) vec) \<Rightarrow> (('a, 'b) vec, 'c) vec
Finite_Cartesian_Product.matrix_vector_mult :::: (('a, 'b) vec, 'c) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> ('a, 'c) vec
Finite_Cartesian_Product.vector_scalar_mult :::: 'a \<Rightarrow> ('a, 'b) vec \<Rightarrow> ('a, 'b) vec
Vector_Spaces.linear :::: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool
###defs
definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
where "c *s x = (\<chi> i. c * (x$i))"
|
###output
?H1 ?H2 ?H2 x_1 \<Longrightarrow> ?H3 (?H4 x_1) = x_1###end
|
Transitive_Closure
|
Transitive_Closure.rtrancl_reflcl_absorb
|
lemma rtrancl_reflcl_absorb[simp]: "(R\<^sup>*)\<^sup>= = R\<^sup>*"
|
(?R\<^sup>*)\<^sup>= = ?R\<^sup>*
|
?H1 (?H2 x_1) = ?H2 x_1
|
[
"Transitive_Closure.rtrancl",
"Transitive_Closure.reflcl"
] |
[
"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set",
"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set"
] |
[
"inductive_set rtrancl :: \"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set\" (\"(_\\<^sup>*)\" [1000] 999)\n for r :: \"('a \\<times> 'a) set\"\n where\n rtrancl_refl [intro!, Pure.intro!, simp]: \"(a, a) \\<in> r\\<^sup>*\"\n | rtrancl_into_rtrancl [Pure.intro]: \"(a, b) \\<in> r\\<^sup>* \\<Longrightarrow> (b, c) \\<in> r \\<Longrightarrow> (a, c) \\<in> r\\<^sup>*\"",
"abbreviation reflcl :: \"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set\" (\"(_\\<^sup>=)\" [1000] 999)\n where \"r\\<^sup>= \\<equiv> r \\<union> Id\""
] |
template
|
###lemma
(?R\<^sup>*)\<^sup>= = ?R\<^sup>*
###symbols
Transitive_Closure.rtrancl :::: ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set
Transitive_Closure.reflcl :::: ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set
###defs
inductive_set rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999)
for r :: "('a \<times> 'a) set"
where
rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) \<in> r\<^sup>*"
| rtrancl_into_rtrancl [Pure.intro]: "(a, b) \<in> r\<^sup>* \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>*"
abbreviation reflcl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_\<^sup>=)" [1000] 999)
where "r\<^sup>= \<equiv> r \<union> Id"
|
###output
?H1 (?H2 x_1) = ?H2 x_1###end
|
Proofs/Lambda/Standardization
|
Standardization.lemma4
|
lemma lemma4:
assumes r: "r \<rightarrow>\<^sub>s r'"
shows "r' \<rightarrow>\<^sub>\<beta> r'' \<Longrightarrow> r \<rightarrow>\<^sub>s r''"
|
?r \<rightarrow>\<^sub>s ?r' \<Longrightarrow> ?r' \<rightarrow>\<^sub>\<beta> ?r'' \<Longrightarrow> ?r \<rightarrow>\<^sub>s ?r''
|
\<lbrakk> ?H1 x_1 x_2; ?H2 x_2 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 x_3
|
[
"Lambda.beta",
"Standardization.sred"
] |
[
"dB \\<Rightarrow> dB \\<Rightarrow> bool",
"dB \\<Rightarrow> dB \\<Rightarrow> bool"
] |
[
"inductive beta :: \"[dB, dB] => bool\" (infixl \"\\<rightarrow>\\<^sub>\\<beta>\" 50)\n where\n beta [simp, intro!]: \"Abs s \\<degree> t \\<rightarrow>\\<^sub>\\<beta> s[t/0]\"\n | appL [simp, intro!]: \"s \\<rightarrow>\\<^sub>\\<beta> t ==> s \\<degree> u \\<rightarrow>\\<^sub>\\<beta> t \\<degree> u\"\n | appR [simp, intro!]: \"s \\<rightarrow>\\<^sub>\\<beta> t ==> u \\<degree> s \\<rightarrow>\\<^sub>\\<beta> u \\<degree> t\"\n | abs [simp, intro!]: \"s \\<rightarrow>\\<^sub>\\<beta> t ==> Abs s \\<rightarrow>\\<^sub>\\<beta> Abs t\""
] |
template
|
###lemma
?r \<rightarrow>\<^sub>s ?r' \<Longrightarrow> ?r' \<rightarrow>\<^sub>\<beta> ?r'' \<Longrightarrow> ?r \<rightarrow>\<^sub>s ?r''
###symbols
Lambda.beta :::: dB \<Rightarrow> dB \<Rightarrow> bool
Standardization.sred :::: dB \<Rightarrow> dB \<Rightarrow> bool
###defs
inductive beta :: "[dB, dB] => bool" (infixl "\<rightarrow>\<^sub>\<beta>" 50)
where
beta [simp, intro!]: "Abs s \<degree> t \<rightarrow>\<^sub>\<beta> s[t/0]"
| appL [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> s \<degree> u \<rightarrow>\<^sub>\<beta> t \<degree> u"
| appR [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> u \<degree> s \<rightarrow>\<^sub>\<beta> u \<degree> t"
| abs [simp, intro!]: "s \<rightarrow>\<^sub>\<beta> t ==> Abs s \<rightarrow>\<^sub>\<beta> Abs t"
|
###output
\<lbrakk> ?H1 x_1 x_2; ?H2 x_2 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 x_3###end
|
Real_Asymp/Multiseries_Expansion
|
Multiseries_Expansion.expands_to_add
|
lemma expands_to_add:
"basis_wf basis \<Longrightarrow> (f expands_to F) basis \<Longrightarrow> (g expands_to G) basis \<Longrightarrow>
((\<lambda>x. f x + g x) expands_to F + G) basis"
|
basis_wf ?basis \<Longrightarrow> (?f expands_to ?F) ?basis \<Longrightarrow> (?g expands_to ?G) ?basis \<Longrightarrow> ((\<lambda>x. ?f x + ?g x) expands_to ?F + ?G) ?basis
|
\<lbrakk> ?H1 x_1; ?H2 x_2 x_3 x_1; ?H2 x_4 x_5 x_1\<rbrakk> \<Longrightarrow> ?H2 (\<lambda>y_0. ?H3 (x_2 y_0) (x_4 y_0)) (?H3 x_3 x_5) x_1
|
[
"Groups.plus_class.plus",
"Multiseries_Expansion.expands_to",
"Multiseries_Expansion.basis_wf"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real \\<Rightarrow> real) \\<Rightarrow> 'a \\<Rightarrow> (real \\<Rightarrow> real) list \\<Rightarrow> bool",
"(real \\<Rightarrow> real) list \\<Rightarrow> bool"
] |
[
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"inductive expands_to :: \"(real \\<Rightarrow> real) \\<Rightarrow> 'a :: multiseries \\<Rightarrow> basis \\<Rightarrow> bool\" \n (infix \"(expands'_to)\" 50) where\n \"is_expansion F basis \\<Longrightarrow> eventually (\\<lambda>x. eval F x = f x) at_top \\<Longrightarrow> (f expands_to F) basis\"",
"definition basis_wf :: \"basis \\<Rightarrow> bool\" where\n \"basis_wf basis \\<longleftrightarrow> (\\<forall>f\\<in>set basis. filterlim f at_top at_top) \\<and> \n sorted_wrt (\\<lambda>f g. (\\<lambda>x. ln (g x)) \\<in> o(\\<lambda>x. ln (f x))) basis\""
] |
template
|
###lemma
basis_wf ?basis \<Longrightarrow> (?f expands_to ?F) ?basis \<Longrightarrow> (?g expands_to ?G) ?basis \<Longrightarrow> ((\<lambda>x. ?f x + ?g x) expands_to ?F + ?G) ?basis
###symbols
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Multiseries_Expansion.expands_to :::: (real \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool
Multiseries_Expansion.basis_wf :::: (real \<Rightarrow> real) list \<Rightarrow> bool
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
inductive expands_to :: "(real \<Rightarrow> real) \<Rightarrow> 'a :: multiseries \<Rightarrow> basis \<Rightarrow> bool"
(infix "(expands'_to)" 50) where
"is_expansion F basis \<Longrightarrow> eventually (\<lambda>x. eval F x = f x) at_top \<Longrightarrow> (f expands_to F) basis"
definition basis_wf :: "basis \<Rightarrow> bool" where
"basis_wf basis \<longleftrightarrow> (\<forall>f\<in>set basis. filterlim f at_top at_top) \<and>
sorted_wrt (\<lambda>f g. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (f x))) basis"
|
###output
\<lbrakk> ?H1 x_1; ?H2 x_2 x_3 x_1; ?H2 x_4 x_5 x_1\<rbrakk> \<Longrightarrow> ?H2 (\<lambda>y_0. ?H3 (x_2 y_0) (x_4 y_0)) (?H3 x_3 x_5) x_1###end
|
Data_Structures/Array_Braun
|
Array_Braun.list_add_lo
|
lemma list_add_lo: "braun t \<Longrightarrow> list (add_lo a t) = a # list t"
|
braun ?t \<Longrightarrow> list (add_lo ?a ?t) = ?a # list ?t
|
?H1 x_1 \<Longrightarrow> ?H2 (?H3 x_2 x_1) = ?H4 x_2 (?H2 x_1)
|
[
"List.list.Cons",
"Array_Braun.add_lo",
"Array_Braun.list",
"Braun_Tree.braun"
] |
[
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a \\<Rightarrow> 'a tree \\<Rightarrow> 'a tree",
"'a tree \\<Rightarrow> 'a list",
"'a tree \\<Rightarrow> bool"
] |
[
"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 [] = []\"",
"fun add_lo where\n \"add_lo x Leaf = Node Leaf x Leaf\" |\n \"add_lo x (Node l a r) = Node (add_lo a r) x l\"",
"fun list :: \"'a tree \\<Rightarrow> 'a list\" where\n \"list Leaf = []\" |\n \"list (Node l x r) = x # splice (list l) (list r)\"",
"fun braun :: \"'a tree \\<Rightarrow> bool\" where\n\"braun Leaf = True\" |\n\"braun (Node l x r) = ((size l = size r \\<or> size l = size r + 1) \\<and> braun l \\<and> braun r)\""
] |
template
|
###lemma
braun ?t \<Longrightarrow> list (add_lo ?a ?t) = ?a # list ?t
###symbols
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Array_Braun.add_lo :::: 'a \<Rightarrow> 'a tree \<Rightarrow> 'a tree
Array_Braun.list :::: 'a tree \<Rightarrow> 'a list
Braun_Tree.braun :::: 'a tree \<Rightarrow> bool
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
fun add_lo where
"add_lo x Leaf = Node Leaf x Leaf" |
"add_lo x (Node l a r) = Node (add_lo a r) x l"
fun list :: "'a tree \<Rightarrow> 'a list" where
"list Leaf = []" |
"list (Node l x r) = x # splice (list l) (list r)"
fun braun :: "'a tree \<Rightarrow> bool" where
"braun Leaf = True" |
"braun (Node l x r) = ((size l = size r \<or> size l = size r + 1) \<and> braun l \<and> braun r)"
|
###output
?H1 x_1 \<Longrightarrow> ?H2 (?H3 x_2 x_1) = ?H4 x_2 (?H2 x_1)###end
|
Analysis/Starlike
|
Starlike.rel_interior_sum_gen
|
lemma rel_interior_sum_gen:
fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
assumes "\<And>i. i\<in>I \<Longrightarrow> convex (S i)"
shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
|
(\<And>i. i \<in> ?I \<Longrightarrow> convex (?S i)) \<Longrightarrow> rel_interior (sum ?S ?I) = (\<Sum>i\<in> ?I. rel_interior (?S i))
|
(\<And>y_0. y_0 \<in> x_1 \<Longrightarrow> ?H1 (x_2 y_0)) \<Longrightarrow> ?H2 (?H3 x_2 x_1) = ?H3 (\<lambda>y_1. ?H2 (x_2 y_1)) x_1
|
[
"Groups_Big.comm_monoid_add_class.sum",
"Convex_Euclidean_Space.rel_interior",
"Convex.convex"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b",
"'a set \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> bool"
] |
[] |
template
|
###lemma
(\<And>i. i \<in> ?I \<Longrightarrow> convex (?S i)) \<Longrightarrow> rel_interior (sum ?S ?I) = (\<Sum>i\<in> ?I. rel_interior (?S i))
###symbols
Groups_Big.comm_monoid_add_class.sum :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b
Convex_Euclidean_Space.rel_interior :::: 'a set \<Rightarrow> 'a set
Convex.convex :::: 'a set \<Rightarrow> bool
###defs
|
###output
(\<And>y_0. y_0 \<in> x_1 \<Longrightarrow> ?H1 (x_2 y_0)) \<Longrightarrow> ?H2 (?H3 x_2 x_1) = ?H3 (\<lambda>y_1. ?H2 (x_2 y_1)) x_1###end
|
Analysis/Poly_Roots
|
Poly_Roots.polyfun_rootbound_card
| null |
\<exists>k\<le> ?n. ?c k \<noteq> (0:: ?'a) \<Longrightarrow> card {z. (\<Sum>i\<le> ?n. ?c i * z ^ i) = (0:: ?'a)} \<le> ?n
|
\<exists>y_0\<le>x_1. x_2 y_0 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (\<lambda>y_1. ?H4 (\<lambda>y_2. ?H5 (x_2 y_2) (?H6 y_1 y_2)) (?H7 x_1) = ?H1)) \<le> x_1
|
[
"Set_Interval.ord_class.atMost",
"Power.power_class.power",
"Groups.times_class.times",
"Groups_Big.comm_monoid_add_class.sum",
"Set.Collect",
"Finite_Set.card",
"Groups.zero_class.zero"
] |
[
"'a \\<Rightarrow> 'a set",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> nat",
"'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 times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
\<exists>k\<le> ?n. ?c k \<noteq> (0:: ?'a) \<Longrightarrow> card {z. (\<Sum>i\<le> ?n. ?c i * z ^ i) = (0:: ?'a)} \<le> ?n
###symbols
Set_Interval.ord_class.atMost :::: 'a \<Rightarrow> 'a set
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups_Big.comm_monoid_add_class.sum :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Finite_Set.card :::: 'a set \<Rightarrow> nat
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 times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class zero =
fixes zero :: 'a ("0")
|
###output
\<exists>y_0\<le>x_1. x_2 y_0 \<noteq> ?H1 \<Longrightarrow> ?H2 (?H3 (\<lambda>y_1. ?H4 (\<lambda>y_2. ?H5 (x_2 y_2) (?H6 y_1 y_2)) (?H7 x_1) = ?H1)) \<le> x_1###end
|
Nominal/Examples/Class1
|
Class1.interesting_subst1
|
lemma interesting_subst1:
assumes a: "x\<noteq>y" "x\<sharp>P" "y\<sharp>P"
shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<c>.Ax y c}{y:=<c>.P}"
|
?x \<noteq> ?y \<Longrightarrow> ?x \<sharp> ?P \<Longrightarrow> ?y \<sharp> ?P \<Longrightarrow> ?N{ ?y:=< ?c>. ?P}{ ?x:=< ?c>. ?P} = ?N{ ?x:=< ?c>.Ax ?y ?c}{ ?y:=< ?c>. ?P}
|
\<lbrakk>x_1 \<noteq> x_2; ?H1 x_1 x_3; ?H1 x_2 x_3\<rbrakk> \<Longrightarrow> ?H2 (?H2 x_4 x_2 x_5 x_3) x_1 x_5 x_3 = ?H2 (?H2 x_4 x_1 x_5 (?H3 x_2 x_5)) x_2 x_5 x_3
|
[
"Class1.trm.Ax",
"Class1.substn",
"Nominal.fresh"
] |
[
"name \\<Rightarrow> coname \\<Rightarrow> trm",
"trm \\<Rightarrow> name \\<Rightarrow> coname \\<Rightarrow> trm \\<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
?x \<noteq> ?y \<Longrightarrow> ?x \<sharp> ?P \<Longrightarrow> ?y \<sharp> ?P \<Longrightarrow> ?N{ ?y:=< ?c>. ?P}{ ?x:=< ?c>. ?P} = ?N{ ?x:=< ?c>.Ax ?y ?c}{ ?y:=< ?c>. ?P}
###symbols
Class1.trm.Ax :::: name \<Rightarrow> coname \<Rightarrow> trm
Class1.substn :::: trm \<Rightarrow> name \<Rightarrow> coname \<Rightarrow> trm \<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
\<lbrakk>x_1 \<noteq> x_2; ?H1 x_1 x_3; ?H1 x_2 x_3\<rbrakk> \<Longrightarrow> ?H2 (?H2 x_4 x_2 x_5 x_3) x_1 x_5 x_3 = ?H2 (?H2 x_4 x_1 x_5 (?H3 x_2 x_5)) x_2 x_5 x_3###end
|
HOLCF/Tr
|
Transfer.funpow_transfer
| null |
rel_fun (=) (rel_fun (rel_fun ?A ?A) (rel_fun ?A ?A)) compow compow
|
?H1 (=) (?H1 (?H1 x_1 x_1) (?H1 x_1 x_1)) ?H2 ?H2
|
[
"Nat.compow",
"BNF_Def.rel_fun"
] |
[
"nat \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('c \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'c) \\<Rightarrow> ('b \\<Rightarrow> 'd) \\<Rightarrow> bool"
] |
[
"consts compow :: \"nat \\<Rightarrow> 'a \\<Rightarrow> 'a\"",
"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))\""
] |
template
|
###lemma
rel_fun (=) (rel_fun (rel_fun ?A ?A) (rel_fun ?A ?A)) compow compow
###symbols
Nat.compow :::: nat \<Rightarrow> 'a \<Rightarrow> 'a
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
###defs
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
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))"
|
###output
?H1 (=) (?H1 (?H1 x_1 x_1) (?H1 x_1 x_1)) ?H2 ?H2###end
|
Predicate_Compile_Examples/Predicate_Compile_Tests
|
Predicate_Compile_Tests.maxP_intro_1
| null |
?f \<le> ?e \<Longrightarrow> maxP ?f ?e ?e
|
x_1 \<le> x_2 \<Longrightarrow> ?H1 x_1 x_2 x_2
|
[
"Predicate_Compile_Tests.maxP"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[] |
template
|
###lemma
?f \<le> ?e \<Longrightarrow> maxP ?f ?e ?e
###symbols
Predicate_Compile_Tests.maxP :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool
###defs
|
###output
x_1 \<le> x_2 \<Longrightarrow> ?H1 x_1 x_2 x_2###end
|
MicroJava/DFA/Kildall
|
Kildall.termination_lemma
|
lemma termination_lemma:
assumes semilat: "semilat (A, r, f)"
shows "\<lbrakk> ss \<in> list n A; \<forall>(q,t)\<in>set qs. q<n \<and> t\<in>A; p\<in>w \<rbrakk> \<Longrightarrow>
ss <[r] merges f qs ss \<or>
merges f qs ss = ss \<and> {q. \<exists>t. (q,t)\<in>set qs \<and> t +_f ss!q \<noteq> ss!q} Un (w-{p}) < w" (is "PROP ?P")
|
semilat (?A, ?r, ?f) \<Longrightarrow> ?ss \<in> list ?n ?A \<Longrightarrow> \<forall>(q, t)\<in>set ?qs. q < ?n \<and> t \<in> ?A \<Longrightarrow> ?p \<in> ?w \<Longrightarrow> ?ss <[ ?r] merges ?f ?qs ?ss \<or> merges ?f ?qs ?ss = ?ss \<and> {q. \<exists>t. (q, t) \<in> set ?qs \<and> t \<squnion>\<^bsub> ?f\<^esub> ?ss ! q \<noteq> ?ss ! q} \<union> (?w - { ?p}) \<subset> ?w
|
\<lbrakk> ?H1 (x_1, x_2, x_3); x_4 \<in> ?H2 x_5 x_1; Ball (?H3 x_6) (?H4 (\<lambda>y_0 y_1. y_0 < x_5 \<and> y_1 \<in> x_1)); x_7 \<in> x_8\<rbrakk> \<Longrightarrow> ?H5 x_4 x_2 (?H6 x_3 x_6 x_4) \<or> ?H6 x_3 x_6 x_4 = x_4 \<and> ?H7 (?H8 (?H9 (\<lambda>y_2. \<exists>y_3. (y_2, y_3) \<in> ?H3 x_6 \<and> ?H10 y_3 x_3 (?H11 x_4 y_2) \<noteq> ?H11 x_4 y_2)) (?H12 x_8 (?H13 x_7 ?H14))) x_8
|
[
"Set.empty",
"Set.insert",
"Groups.minus_class.minus",
"List.nth",
"Semilat.plussub",
"Set.Collect",
"Set.union",
"Set.subset",
"Kildall.merges",
"Listn.lesssublist_syntax",
"Product_Type.prod.case_prod",
"List.list.set",
"Listn.list",
"Semilat.semilat"
] |
[
"'a set",
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a list \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a \\<Rightarrow> ('a \\<Rightarrow> 'b \\<Rightarrow> 'c) \\<Rightarrow> 'b \\<Rightarrow> 'c",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"('a \\<Rightarrow> 'a \\<Rightarrow> 'a) \\<Rightarrow> (nat \\<times> 'a) list \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> 'c) \\<Rightarrow> 'a \\<times> 'b \\<Rightarrow> 'c",
"'a list \\<Rightarrow> 'a set",
"nat \\<Rightarrow> 'a set \\<Rightarrow> 'a list set",
"'a set \\<times> ('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<times> ('a \\<Rightarrow> 'a \\<Rightarrow> 'a) \\<Rightarrow> bool"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation union :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<union>\" 65)\n where \"union \\<equiv> sup\"",
"abbreviation subset :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset \\<equiv> less\"",
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\"",
"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
semilat (?A, ?r, ?f) \<Longrightarrow> ?ss \<in> list ?n ?A \<Longrightarrow> \<forall>(q, t)\<in>set ?qs. q < ?n \<and> t \<in> ?A \<Longrightarrow> ?p \<in> ?w \<Longrightarrow> ?ss <[ ?r] merges ?f ?qs ?ss \<or> merges ?f ?qs ?ss = ?ss \<and> {q. \<exists>t. (q, t) \<in> set ?qs \<and> t \<squnion>\<^bsub> ?f\<^esub> ?ss ! q \<noteq> ?ss ! q} \<union> (?w - { ?p}) \<subset> ?w
###symbols
Set.empty :::: 'a set
Set.insert :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a set
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
List.nth :::: 'a list \<Rightarrow> nat \<Rightarrow> 'a
Semilat.plussub :::: 'a \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Set.union :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
Set.subset :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Kildall.merges :::: ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<times> 'a) list \<Rightarrow> 'a list \<Rightarrow> 'a list
Listn.lesssublist_syntax :::: 'a list \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool
Product_Type.prod.case_prod :::: ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c
List.list.set :::: 'a list \<Rightarrow> 'a set
Listn.list :::: nat \<Rightarrow> 'a set \<Rightarrow> 'a list set
Semilat.semilat :::: 'a set \<times> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
primrec (nonexhaustive) nth :: "'a list => nat => 'a" (infixl "!" 100) where
nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
\<comment> \<open>Warning: simpset does not contain this definition, but separate
theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close>
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65)
where "union \<equiv> sup"
abbreviation subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset \<equiv> less"
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
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
\<lbrakk> ?H1 (x_1, x_2, x_3); x_4 \<in> ?H2 x_5 x_1; Ball (?H3 x_6) (?H4 (\<lambda>y_0 y_1. y_0 < x_5 \<and> y_1 \<in> x_1)); x_7 \<in> x_8\<rbrakk> \<Longrightarrow> ?H5 x_4 x_2 (?H6 x_3 x_6 x_4) \<or> ?H6 x_3 x_6 x_4 = x_4 \<and> ?H7 (?H8 (?H9 (\<lambda>y_2. \<exists>y_3. (y_2, y_3) \<in> ?H3 x_6 \<and> ?H10 y_3 x_3 (?H11 x_4 y_2) \<noteq> ?H11 x_4 y_2)) (?H12 x_8 (?H13 x_7 ?H14))) x_8###end
|
SPARK/Examples/RIPEMD-160/F
|
Factorial.fact_ge_Suc_0_nat
| null |
Suc 0 \<le> fact ?n
|
?H1 ?H2 \<le> ?H3 x_1
|
[
"Factorial.semiring_char_0_class.fact",
"Groups.zero_class.zero",
"Nat.Suc"
] |
[
"nat \\<Rightarrow> 'a",
"'a",
"nat \\<Rightarrow> nat"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\""
] |
template
|
###lemma
Suc 0 \<le> fact ?n
###symbols
Factorial.semiring_char_0_class.fact :::: nat \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Nat.Suc :::: nat \<Rightarrow> nat
###defs
class zero =
fixes zero :: 'a ("0")
definition Suc :: "nat \<Rightarrow> nat"
where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
|
###output
?H1 ?H2 \<le> ?H3 x_1###end
|
Analysis/Abstract_Topology
|
Abstract_Topology.continuous_map_image_closure_subset
|
lemma continuous_map_image_closure_subset:
assumes "continuous_map X Y f"
shows "f ` (X closure_of S) \<subseteq> Y closure_of f ` S"
|
continuous_map ?X ?Y ?f \<Longrightarrow> ?f ` (?X closure_of ?S) \<subseteq> ?Y closure_of ?f ` ?S
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_3 (?H4 x_1 x_4)) (?H4 x_2 (?H3 x_3 x_4))
|
[
"Abstract_Topology.closure_of",
"Set.image",
"Set.subset_eq",
"Abstract_Topology.continuous_map"
] |
[
"'a topology \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a topology \\<Rightarrow> 'b topology \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"definition closure_of :: \"'a topology \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixr \"closure'_of\" 80)\n where \"X closure_of S \\<equiv> {x \\<in> topspace X. \\<forall>T. x \\<in> T \\<and> openin X T \\<longrightarrow> (\\<exists>y \\<in> S. y \\<in> T)}\"",
"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 subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"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> ?f ` (?X closure_of ?S) \<subseteq> ?Y closure_of ?f ` ?S
###symbols
Abstract_Topology.closure_of :::: 'a topology \<Rightarrow> 'a set \<Rightarrow> 'a set
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Abstract_Topology.continuous_map :::: 'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition closure_of :: "'a topology \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixr "closure'_of" 80)
where "X closure_of S \<equiv> {x \<in> topspace X. \<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y \<in> S. y \<in> T)}"
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 subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
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
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_3 (?H4 x_1 x_4)) (?H4 x_2 (?H3 x_3 x_4))###end
|
MicroJava/DFA/Opt
|
Option.combine_options_simps(2)
| null |
combine_options ?f ?x None = ?x
|
?H1 x_1 x_2 ?H2 = 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 ?x None = ?x
###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 x_2 ?H2 = x_2###end
|
Analysis/Complex_Analysis_Basics
|
Complex_Analysis_Basics.analytic_on_compose
|
lemma analytic_on_compose:
assumes f: "f analytic_on S"
and g: "g analytic_on (f ` S)"
shows "(g \<circ> f) analytic_on S"
|
?f analytic_on ?S \<Longrightarrow> ?g analytic_on ?f ` ?S \<Longrightarrow> ?g \<circ> ?f analytic_on ?S
|
\<lbrakk> ?H1 x_1 x_2; ?H1 x_3 (?H2 x_1 x_2)\<rbrakk> \<Longrightarrow> ?H1 (?H3 x_3 x_1) x_2
|
[
"Fun.comp",
"Set.image",
"Complex_Analysis_Basics.analytic_on"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'a) \\<Rightarrow> 'c \\<Rightarrow> 'b",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"(complex \\<Rightarrow> complex) \\<Rightarrow> complex set \\<Rightarrow> bool"
] |
[
"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))\"",
"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}\""
] |
template
|
###lemma
?f analytic_on ?S \<Longrightarrow> ?g analytic_on ?f ` ?S \<Longrightarrow> ?g \<circ> ?f analytic_on ?S
###symbols
Fun.comp :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'b
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Complex_Analysis_Basics.analytic_on :::: (complex \<Rightarrow> complex) \<Rightarrow> complex set \<Rightarrow> bool
###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))"
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
|
###output
\<lbrakk> ?H1 x_1 x_2; ?H1 x_3 (?H2 x_1 x_2)\<rbrakk> \<Longrightarrow> ?H1 (?H3 x_3 x_1) x_2###end
|
Analysis/Cartesian_Euclidean_Space
|
Cartesian_Euclidean_Space.matrix_vector_mult_linear_continuous_on
| null |
continuous_on ?S ((*v) ?A)
|
?H1 x_1 (?H2 x_2)
|
[
"Finite_Cartesian_Product.matrix_vector_mult",
"Topological_Spaces.continuous_on"
] |
[
"(('a, 'b) vec, 'c) vec \\<Rightarrow> ('a, 'b) vec \\<Rightarrow> ('a, 'c) vec",
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"definition continuous_on :: \"'a set \\<Rightarrow> ('a::topological_space \\<Rightarrow> 'b::topological_space) \\<Rightarrow> bool\"\n where \"continuous_on s f \\<longleftrightarrow> (\\<forall>x\\<in>s. (f \\<longlongrightarrow> f x) (at x within s))\""
] |
template
|
###lemma
continuous_on ?S ((*v) ?A)
###symbols
Finite_Cartesian_Product.matrix_vector_mult :::: (('a, 'b) vec, 'c) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> ('a, 'c) vec
Topological_Spaces.continuous_on :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition continuous_on :: "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
where "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
|
###output
?H1 x_1 (?H2 x_2)###end
|
IMP/Hoare_Examples
|
Hoare_Examples.sum_via_bigstep
|
lemma sum_via_bigstep:
assumes "(''y'' ::= N 0;; wsum, s) \<Rightarrow> t"
shows "t ''y'' = sum (s ''x'')"
|
(''y'' ::= N 0;; wsum, ?s) \<Rightarrow> ?t \<Longrightarrow> ?t ''y'' = Hoare_Examples.sum (?s ''x'')
|
?H1 (?H2 (?H3 (?H4 (?H5 True False False True True True True False) ?H6) (?H7 ?H8)) ?H9, x_1) x_2 \<Longrightarrow> x_2 (?H4 (?H5 True False False True True True True False) ?H6) = ?H10 (x_1 (?H4 (?H5 False False False True True True True False) ?H6))
|
[
"Hoare_Examples.sum",
"Hoare_Examples.wsum",
"Groups.zero_class.zero",
"AExp.aexp.N",
"List.list.Nil",
"String.char.Char",
"List.list.Cons",
"Com.com.Assign",
"Com.com.Seq",
"Big_Step.big_step"
] |
[
"int \\<Rightarrow> int",
"com",
"'a",
"int \\<Rightarrow> aexp",
"'a list",
"bool \\<Rightarrow> bool \\<Rightarrow> bool \\<Rightarrow> bool \\<Rightarrow> bool \\<Rightarrow> bool \\<Rightarrow> bool \\<Rightarrow> bool \\<Rightarrow> char",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"char list \\<Rightarrow> aexp \\<Rightarrow> com",
"com \\<Rightarrow> com \\<Rightarrow> com",
"com \\<times> (char list \\<Rightarrow> int) \\<Rightarrow> (char list \\<Rightarrow> int) \\<Rightarrow> bool"
] |
[
"fun sum :: \"int \\<Rightarrow> int\" where\n\"sum i = (if i \\<le> 0 then 0 else sum (i - 1) + i)\"",
"abbreviation \"wsum ==\n WHILE Less (N 0) (V ''x'')\n DO (''y'' ::= Plus (V ''y'') (V ''x'');;\n ''x'' ::= Plus (V ''x'') (N (- 1)))\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"datatype aexp = N int | V vname | Plus aexp aexp",
"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 char =\n Char (digit0: bool) (digit1: bool) (digit2: bool) (digit3: bool)\n (digit4: bool) (digit5: bool) (digit6: bool) (digit7: bool)",
"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 com = SKIP \n | Assign vname aexp (\"_ ::= _\" [1000, 61] 61)\n | Seq com com (\"_;;/ _\" [60, 61] 60)\n | If bexp com com (\"(IF _/ THEN _/ ELSE _)\" [0, 0, 61] 61)\n | While bexp com (\"(WHILE _/ DO _)\" [0, 61] 61)",
"datatype\n com = SKIP \n | Assign vname aexp (\"_ ::= _\" [1000, 61] 61)\n | Seq com com (\"_;;/ _\" [60, 61] 60)\n | If bexp com com (\"(IF _/ THEN _/ ELSE _)\" [0, 0, 61] 61)\n | While bexp com (\"(WHILE _/ DO _)\" [0, 61] 61)",
"inductive\n big_step :: \"com \\<times> state \\<Rightarrow> state \\<Rightarrow> bool\" (infix \"\\<Rightarrow>\" 55)\nwhere\nSkip: \"(SKIP,s) \\<Rightarrow> s\" |\nAssign: \"(x ::= a,s) \\<Rightarrow> s(x := aval a s)\" |\nSeq: \"\\<lbrakk> (c\\<^sub>1,s\\<^sub>1) \\<Rightarrow> s\\<^sub>2; (c\\<^sub>2,s\\<^sub>2) \\<Rightarrow> s\\<^sub>3 \\<rbrakk> \\<Longrightarrow> (c\\<^sub>1;;c\\<^sub>2, s\\<^sub>1) \\<Rightarrow> s\\<^sub>3\" |\nIfTrue: \"\\<lbrakk> bval b s; (c\\<^sub>1,s) \\<Rightarrow> t \\<rbrakk> \\<Longrightarrow> (IF b THEN c\\<^sub>1 ELSE c\\<^sub>2, s) \\<Rightarrow> t\" |\nIfFalse: \"\\<lbrakk> \\<not>bval b s; (c\\<^sub>2,s) \\<Rightarrow> t \\<rbrakk> \\<Longrightarrow> (IF b THEN c\\<^sub>1 ELSE c\\<^sub>2, s) \\<Rightarrow> t\" |\nWhileFalse: \"\\<not>bval b s \\<Longrightarrow> (WHILE b DO c,s) \\<Rightarrow> s\" |\nWhileTrue:\n\"\\<lbrakk> bval b s\\<^sub>1; (c,s\\<^sub>1) \\<Rightarrow> s\\<^sub>2; (WHILE b DO c, s\\<^sub>2) \\<Rightarrow> s\\<^sub>3 \\<rbrakk> \n\\<Longrightarrow> (WHILE b DO c, s\\<^sub>1) \\<Rightarrow> s\\<^sub>3\""
] |
template
|
###lemma
(''y'' ::= N 0;; wsum, ?s) \<Rightarrow> ?t \<Longrightarrow> ?t ''y'' = Hoare_Examples.sum (?s ''x'')
###symbols
Hoare_Examples.sum :::: int \<Rightarrow> int
Hoare_Examples.wsum :::: com
Groups.zero_class.zero :::: 'a
AExp.aexp.N :::: int \<Rightarrow> aexp
List.list.Nil :::: 'a list
String.char.Char :::: bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> char
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Com.com.Assign :::: char list \<Rightarrow> aexp \<Rightarrow> com
Com.com.Seq :::: com \<Rightarrow> com \<Rightarrow> com
Big_Step.big_step :::: com \<times> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool
###defs
fun sum :: "int \<Rightarrow> int" where
"sum i = (if i \<le> 0 then 0 else sum (i - 1) + i)"
abbreviation "wsum ==
WHILE Less (N 0) (V ''x'')
DO (''y'' ::= Plus (V ''y'') (V ''x'');;
''x'' ::= Plus (V ''x'') (N (- 1)))"
class zero =
fixes zero :: 'a ("0")
datatype aexp = N int | V vname | Plus aexp aexp
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 char =
Char (digit0: bool) (digit1: bool) (digit2: bool) (digit3: bool)
(digit4: bool) (digit5: bool) (digit6: bool) (digit7: bool)
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
com = SKIP
| Assign vname aexp ("_ ::= _" [1000, 61] 61)
| Seq com com ("_;;/ _" [60, 61] 60)
| If bexp com com ("(IF _/ THEN _/ ELSE _)" [0, 0, 61] 61)
| While bexp com ("(WHILE _/ DO _)" [0, 61] 61)
datatype
com = SKIP
| Assign vname aexp ("_ ::= _" [1000, 61] 61)
| Seq com com ("_;;/ _" [60, 61] 60)
| If bexp com com ("(IF _/ THEN _/ ELSE _)" [0, 0, 61] 61)
| While bexp com ("(WHILE _/ DO _)" [0, 61] 61)
inductive
big_step :: "com \<times> state \<Rightarrow> state \<Rightarrow> bool" (infix "\<Rightarrow>" 55)
where
Skip: "(SKIP,s) \<Rightarrow> s" |
Assign: "(x ::= a,s) \<Rightarrow> s(x := aval a s)" |
Seq: "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> s\<^sub>2; (c\<^sub>2,s\<^sub>2) \<Rightarrow> s\<^sub>3 \<rbrakk> \<Longrightarrow> (c\<^sub>1;;c\<^sub>2, s\<^sub>1) \<Rightarrow> s\<^sub>3" |
IfTrue: "\<lbrakk> bval b s; (c\<^sub>1,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t" |
IfFalse: "\<lbrakk> \<not>bval b s; (c\<^sub>2,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t" |
WhileFalse: "\<not>bval b s \<Longrightarrow> (WHILE b DO c,s) \<Rightarrow> s" |
WhileTrue:
"\<lbrakk> bval b s\<^sub>1; (c,s\<^sub>1) \<Rightarrow> s\<^sub>2; (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3 \<rbrakk>
\<Longrightarrow> (WHILE b DO c, s\<^sub>1) \<Rightarrow> s\<^sub>3"
|
###output
?H1 (?H2 (?H3 (?H4 (?H5 True False False True True True True False) ?H6) (?H7 ?H8)) ?H9, x_1) x_2 \<Longrightarrow> x_2 (?H4 (?H5 True False False True True True True False) ?H6) = ?H10 (x_1 (?H4 (?H5 False False False True True True True False) ?H6))###end
|
Algebra/Group
|
Group.units_of_carrier
|
lemma units_of_carrier: "carrier (units_of G) = Units G"
|
carrier (units_of ?G) = Units ?G
|
?H1 (?H2 x_1) = ?H3 x_1
|
[
"Group.Units",
"Group.units_of",
"Congruence.partial_object.carrier"
] |
[
"('a, 'b) monoid_scheme \\<Rightarrow> 'a set",
"('a, 'b) monoid_scheme \\<Rightarrow> 'a monoid",
"('a, 'b) partial_object_scheme \\<Rightarrow> 'a set"
] |
[
"definition\n Units :: \"_ => 'a set\"\n \\<comment> \\<open>The set of invertible elements\\<close>\n where \"Units G = {y. y \\<in> carrier G \\<and> (\\<exists>x \\<in> carrier G. x \\<otimes>\\<^bsub>G\\<^esub> y = \\<one>\\<^bsub>G\\<^esub> \\<and> y \\<otimes>\\<^bsub>G\\<^esub> x = \\<one>\\<^bsub>G\\<^esub>)}\"",
"definition units_of :: \"('a, 'b) monoid_scheme \\<Rightarrow> 'a monoid\"\n where \"units_of G =\n \\<lparr>carrier = Units G, Group.monoid.mult = Group.monoid.mult G, one = one G\\<rparr>\""
] |
template
|
###lemma
carrier (units_of ?G) = Units ?G
###symbols
Group.Units :::: ('a, 'b) monoid_scheme \<Rightarrow> 'a set
Group.units_of :::: ('a, 'b) monoid_scheme \<Rightarrow> 'a monoid
Congruence.partial_object.carrier :::: ('a, 'b) partial_object_scheme \<Rightarrow> 'a set
###defs
definition
Units :: "_ => 'a set"
\<comment> \<open>The set of invertible elements\<close>
where "Units G = {y. y \<in> carrier G \<and> (\<exists>x \<in> carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)}"
definition units_of :: "('a, 'b) monoid_scheme \<Rightarrow> 'a monoid"
where "units_of G =
\<lparr>carrier = Units G, Group.monoid.mult = Group.monoid.mult G, one = one G\<rparr>"
|
###output
?H1 (?H2 x_1) = ?H3 x_1###end
|
UNITY/Comp/Alloc
|
Alloc.rename_Client_Bounded
|
lemma rename_Client_Bounded: "i \<in> I
==> rename sysOfClient (plam x: I. rename client_map Client) \<in>
UNIV guarantees
Always {s. \<forall>elt \<in> set ((ask o sub i o client) s). elt \<le> NbT}"
|
?i \<in> ?I \<Longrightarrow> rename sysOfClient (plam x: ?I. rename client_map Client) \<in> UNIV guarantees Always {s. \<forall>elt\<in>set ((ask \<circ> sub ?i \<circ> client) s). elt \<le> NbT}
|
x_1 \<in> x_2 \<Longrightarrow> ?H1 ?H2 (?H3 x_2 (\<lambda>y_0. ?H1 ?H4 ?H5)) \<in> ?H6 ?H7 (?H8 (?H9 (\<lambda>y_1. \<forall>y_2\<in> ?H10 (?H11 (?H11 ?H12 (?H13 x_1)) ?H14 y_1). y_2 \<le> ?H15)))
|
[
"AllocBase.NbT",
"Alloc.systemState.client",
"Lift_prog.sub",
"Alloc.clientState.ask",
"Fun.comp",
"List.list.set",
"Set.Collect",
"Constrains.Always",
"Set.UNIV",
"Guar.guar",
"Alloc.Client",
"Alloc.client_map",
"PPROD.PLam",
"Alloc.sysOfClient",
"Rename.rename"
] |
[
"nat",
"('a, 'b) systemState_scheme \\<Rightarrow> nat \\<Rightarrow> clientState",
"'a \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'b",
"'a clientState_scheme \\<Rightarrow> nat list",
"('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'a) \\<Rightarrow> 'c \\<Rightarrow> 'b",
"'a list \\<Rightarrow> 'a set",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a program set",
"'a set",
"'a program set \\<Rightarrow> 'a program set \\<Rightarrow> 'a program set",
"'a clientState_d program",
"'a clientState_d \\<Rightarrow> clientState \\<times> 'a",
"nat set \\<Rightarrow> (nat \\<Rightarrow> ('a \\<times> (nat \\<Rightarrow> 'a) \\<times> 'b) program) \\<Rightarrow> ((nat \\<Rightarrow> 'a) \\<times> 'b) program",
"(nat \\<Rightarrow> clientState) \\<times> 'a allocState_d \\<Rightarrow> 'a systemState",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a program \\<Rightarrow> 'b program"
] |
[
"definition sub :: \"['a, 'a=>'b] => 'b\" where\n \"sub == %i f. f i\"",
"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 [] = []\"",
"definition Always :: \"'a set => 'a program set\" where\n \"Always A == {F. Init F \\<subseteq> A} \\<inter> Stable A\"",
"abbreviation UNIV :: \"'a set\"\n where \"UNIV \\<equiv> top\"",
"definition guar :: \"['a program set, 'a program set] => 'a program set\" (infixl \"guarantees\" 55) where\n (*higher than membership, lower than Co*)\n \"X guarantees Y == {F. \\<forall>G. F ok G --> F\\<squnion>G \\<in> X --> F\\<squnion>G \\<in> Y}\"",
"definition PLam :: \"[nat set, nat => ('b * ((nat=>'b) * 'c)) program]\n => ((nat=>'b) * 'c) program\" where\n \"PLam I F == \\<Squnion>i \\<in> I. lift i (F i)\"",
"definition rename :: \"['a => 'b, 'a program] => 'b program\" where\n \"rename h == extend (%(x,u::unit). h x)\""
] |
template
|
###lemma
?i \<in> ?I \<Longrightarrow> rename sysOfClient (plam x: ?I. rename client_map Client) \<in> UNIV guarantees Always {s. \<forall>elt\<in>set ((ask \<circ> sub ?i \<circ> client) s). elt \<le> NbT}
###symbols
AllocBase.NbT :::: nat
Alloc.systemState.client :::: ('a, 'b) systemState_scheme \<Rightarrow> nat \<Rightarrow> clientState
Lift_prog.sub :::: 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b
Alloc.clientState.ask :::: 'a clientState_scheme \<Rightarrow> nat list
Fun.comp :::: ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'b
List.list.set :::: 'a list \<Rightarrow> 'a set
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
Constrains.Always :::: 'a set \<Rightarrow> 'a program set
Set.UNIV :::: 'a set
Guar.guar :::: 'a program set \<Rightarrow> 'a program set \<Rightarrow> 'a program set
Alloc.Client :::: 'a clientState_d program
Alloc.client_map :::: 'a clientState_d \<Rightarrow> clientState \<times> 'a
PPROD.PLam :::: nat set \<Rightarrow> (nat \<Rightarrow> ('a \<times> (nat \<Rightarrow> 'a) \<times> 'b) program) \<Rightarrow> ((nat \<Rightarrow> 'a) \<times> 'b) program
Alloc.sysOfClient :::: (nat \<Rightarrow> clientState) \<times> 'a allocState_d \<Rightarrow> 'a systemState
Rename.rename :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a program \<Rightarrow> 'b program
###defs
definition sub :: "['a, 'a=>'b] => 'b" where
"sub == %i f. f i"
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 [] = []"
definition Always :: "'a set => 'a program set" where
"Always A == {F. Init F \<subseteq> A} \<inter> Stable A"
abbreviation UNIV :: "'a set"
where "UNIV \<equiv> top"
definition guar :: "['a program set, 'a program set] => 'a program set" (infixl "guarantees" 55) where
(*higher than membership, lower than Co*)
"X guarantees Y == {F. \<forall>G. F ok G --> F\<squnion>G \<in> X --> F\<squnion>G \<in> Y}"
definition PLam :: "[nat set, nat => ('b * ((nat=>'b) * 'c)) program]
=> ((nat=>'b) * 'c) program" where
"PLam I F == \<Squnion>i \<in> I. lift i (F i)"
definition rename :: "['a => 'b, 'a program] => 'b program" where
"rename h == extend (%(x,u::unit). h x)"
|
###output
x_1 \<in> x_2 \<Longrightarrow> ?H1 ?H2 (?H3 x_2 (\<lambda>y_0. ?H1 ?H4 ?H5)) \<in> ?H6 ?H7 (?H8 (?H9 (\<lambda>y_1. \<forall>y_2\<in> ?H10 (?H11 (?H11 ?H12 (?H13 x_1)) ?H14 y_1). y_2 \<le> ?H15)))###end
|
Analysis/Arcwise_Connected
|
Arcwise_Connected.segment_to_point_exists
|
lemma segment_to_point_exists:
fixes S :: "'a :: euclidean_space set"
assumes "closed S" "S \<noteq> {}"
obtains b where "b \<in> S" "open_segment a b \<inter> S = {}"
|
closed ?S \<Longrightarrow> ?S \<noteq> {} \<Longrightarrow> (\<And>b. b \<in> ?S \<Longrightarrow> open_segment ?a b \<inter> ?S = {} \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
|
\<lbrakk> ?H1 x_1; x_1 \<noteq> ?H2; \<And>y_0. \<lbrakk>y_0 \<in> x_1; ?H3 (?H4 x_2 y_0) x_1 = ?H2\<rbrakk> \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
|
[
"Line_Segment.open_segment",
"Set.inter",
"Set.empty",
"Topological_Spaces.topological_space_class.closed"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a set",
"'a set \\<Rightarrow> bool"
] |
[
"abbreviation inter :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<inter>\" 70)\n where \"(\\<inter>) \\<equiv> inf\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"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
closed ?S \<Longrightarrow> ?S \<noteq> {} \<Longrightarrow> (\<And>b. b \<in> ?S \<Longrightarrow> open_segment ?a b \<inter> ?S = {} \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
###symbols
Line_Segment.open_segment :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a set
Set.inter :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
Set.empty :::: 'a set
Topological_Spaces.topological_space_class.closed :::: 'a set \<Rightarrow> bool
###defs
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<inter>" 70)
where "(\<inter>) \<equiv> inf"
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
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
\<lbrakk> ?H1 x_1; x_1 \<noteq> ?H2; \<And>y_0. \<lbrakk>y_0 \<in> x_1; ?H3 (?H4 x_2 y_0) x_1 = ?H2\<rbrakk> \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3###end
|
Hoare_Parallel/OG_Tactics
|
OG_Tactics.AnnBasic_assertions
|
lemma AnnBasic_assertions:
"\<lbrakk>interfree_aux(None, r, Some a); interfree_aux(None, q, Some a)\<rbrakk> \<Longrightarrow>
interfree_aux(Some (AnnBasic r f), q, Some a)"
|
interfree_aux (None, ?r, Some ?a) \<Longrightarrow> interfree_aux (None, ?q, Some ?a) \<Longrightarrow> interfree_aux (Some (AnnBasic ?r ?f), ?q, Some ?a)
|
\<lbrakk> ?H1 (?H2, x_1, ?H3 x_2); ?H1 (?H2, x_3, ?H3 x_2)\<rbrakk> \<Longrightarrow> ?H1 (?H3 (?H4 x_1 x_4), x_3, ?H3 x_2)
|
[
"OG_Com.ann_com.AnnBasic",
"Option.option.Some",
"Option.option.None",
"OG_Hoare.interfree_aux"
] |
[
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'a) \\<Rightarrow> 'a ann_com",
"'a \\<Rightarrow> 'a option",
"'a option",
"'a ann_com option \\<times> 'a set \\<times> 'a ann_com option \\<Rightarrow> bool"
] |
[
"datatype 'a ann_com =\n AnnBasic \"('a assn)\" \"('a \\<Rightarrow> 'a)\"\n | AnnSeq \"('a ann_com)\" \"('a ann_com)\"\n | AnnCond1 \"('a assn)\" \"('a bexp)\" \"('a ann_com)\" \"('a ann_com)\"\n | AnnCond2 \"('a assn)\" \"('a bexp)\" \"('a ann_com)\"\n | AnnWhile \"('a assn)\" \"('a bexp)\" \"('a assn)\" \"('a ann_com)\"\n | AnnAwait \"('a assn)\" \"('a bexp)\" \"('a com)\"\nand 'a com =\n Parallel \"('a ann_com option \\<times> 'a assn) list\"\n | Basic \"('a \\<Rightarrow> 'a)\"\n | Seq \"('a com)\" \"('a com)\"\n | Cond \"('a bexp)\" \"('a com)\" \"('a com)\"\n | While \"('a bexp)\" \"('a assn)\" \"('a com)\"",
"datatype 'a option =\n None\n | Some (the: 'a)",
"datatype 'a option =\n None\n | Some (the: 'a)",
"definition interfree_aux :: \"('a ann_com_op \\<times> 'a assn \\<times> 'a ann_com_op) \\<Rightarrow> bool\" where\n \"interfree_aux \\<equiv> \\<lambda>(co, q, co'). co'= None \\<or>\n (\\<forall>(r,a) \\<in> atomics (the co'). \\<parallel>= (q \\<inter> r) a q \\<and>\n (co = None \\<or> (\\<forall>p \\<in> assertions (the co). \\<parallel>= (p \\<inter> r) a p)))\""
] |
template
|
###lemma
interfree_aux (None, ?r, Some ?a) \<Longrightarrow> interfree_aux (None, ?q, Some ?a) \<Longrightarrow> interfree_aux (Some (AnnBasic ?r ?f), ?q, Some ?a)
###symbols
OG_Com.ann_com.AnnBasic :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a ann_com
Option.option.Some :::: 'a \<Rightarrow> 'a option
Option.option.None :::: 'a option
OG_Hoare.interfree_aux :::: 'a ann_com option \<times> 'a set \<times> 'a ann_com option \<Rightarrow> bool
###defs
datatype 'a ann_com =
AnnBasic "('a assn)" "('a \<Rightarrow> 'a)"
| AnnSeq "('a ann_com)" "('a ann_com)"
| AnnCond1 "('a assn)" "('a bexp)" "('a ann_com)" "('a ann_com)"
| AnnCond2 "('a assn)" "('a bexp)" "('a ann_com)"
| AnnWhile "('a assn)" "('a bexp)" "('a assn)" "('a ann_com)"
| AnnAwait "('a assn)" "('a bexp)" "('a com)"
and 'a com =
Parallel "('a ann_com option \<times> 'a assn) list"
| Basic "('a \<Rightarrow> 'a)"
| Seq "('a com)" "('a com)"
| Cond "('a bexp)" "('a com)" "('a com)"
| While "('a bexp)" "('a assn)" "('a com)"
datatype 'a option =
None
| Some (the: 'a)
datatype 'a option =
None
| Some (the: 'a)
definition interfree_aux :: "('a ann_com_op \<times> 'a assn \<times> 'a ann_com_op) \<Rightarrow> bool" where
"interfree_aux \<equiv> \<lambda>(co, q, co'). co'= None \<or>
(\<forall>(r,a) \<in> atomics (the co'). \<parallel>= (q \<inter> r) a q \<and>
(co = None \<or> (\<forall>p \<in> assertions (the co). \<parallel>= (p \<inter> r) a p)))"
|
###output
\<lbrakk> ?H1 (?H2, x_1, ?H3 x_2); ?H1 (?H2, x_3, ?H3 x_2)\<rbrakk> \<Longrightarrow> ?H1 (?H3 (?H4 x_1 x_4), x_3, ?H3 x_2)###end
|
List
|
List.filter_transfer
|
lemma filter_transfer [transfer_rule]:
"((A ===> (=)) ===> list_all2 A ===> list_all2 A) filter filter"
|
rel_fun (rel_fun ?A (=)) (rel_fun (list_all2 ?A) (list_all2 ?A)) filter filter
|
?H1 (?H1 x_1 (=)) (?H1 (?H2 x_1) (?H2 x_1)) ?H3 ?H3
|
[
"List.filter",
"List.list.list_all2",
"BNF_Def.rel_fun"
] |
[
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('c \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'c) \\<Rightarrow> ('b \\<Rightarrow> 'd) \\<Rightarrow> bool"
] |
[
"primrec filter:: \"('a \\<Rightarrow> bool) \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"filter P [] = []\" |\n\"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)\"",
"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 [] = []\"",
"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))\""
] |
template
|
###lemma
rel_fun (rel_fun ?A (=)) (rel_fun (list_all2 ?A) (list_all2 ?A)) filter filter
###symbols
List.filter :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list
List.list.list_all2 :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<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
###defs
primrec filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"filter P [] = []" |
"filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
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))"
|
###output
?H1 (?H1 x_1 (=)) (?H1 (?H2 x_1) (?H2 x_1)) ?H3 ?H3###end
|
Auth/Guard/P1
|
P1.chain_guard_Nonce_neq
|
lemma chain_guard_Nonce_neq [intro]: "n \<noteq> ofr
\<Longrightarrow> chain B ofr A' L C \<in> guard n {priK A}"
|
?n \<noteq> ?ofr \<Longrightarrow> chain ?B ?ofr ?A' ?L ?C \<in> guard ?n {priEK ?A}
|
x_1 \<noteq> x_2 \<Longrightarrow> ?H1 x_3 x_2 x_4 x_5 x_6 \<in> ?H2 x_1 (?H3 (?H4 x_7) ?H5)
|
[
"Set.empty",
"Public.priEK",
"Set.insert",
"Guard.guard",
"P1.chain"
] |
[
"'a set",
"agent \\<Rightarrow> nat",
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"nat \\<Rightarrow> nat set \\<Rightarrow> msg set",
"agent \\<Rightarrow> nat \\<Rightarrow> agent \\<Rightarrow> msg \\<Rightarrow> agent \\<Rightarrow> msg"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"abbreviation\n (*BEWARE!! priEK, priSK DON'T WORK with inj, range, image, etc.*)\n priEK :: \"agent \\<Rightarrow> key\" where\n \"priEK A == privateKey Encryption A\"",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"inductive_set\n guard :: \"nat \\<Rightarrow> key set \\<Rightarrow> msg set\"\n for n :: nat and Ks :: \"key set\"\nwhere\n No_Nonce [intro]: \"Nonce n \\<notin> parts {X} \\<Longrightarrow> X \\<in> guard n Ks\"\n| Guard_Nonce [intro]: \"invKey K \\<in> Ks \\<Longrightarrow> Crypt K X \\<in> guard n Ks\"\n| Crypt [intro]: \"X \\<in> guard n Ks \\<Longrightarrow> Crypt K X \\<in> guard n Ks\"\n| Pair [intro]: \"\\<lbrakk>X \\<in> guard n Ks; Y \\<in> guard n Ks\\<rbrakk> \\<Longrightarrow> \\<lbrace>X,Y\\<rbrace> \\<in> guard n Ks\""
] |
template
|
###lemma
?n \<noteq> ?ofr \<Longrightarrow> chain ?B ?ofr ?A' ?L ?C \<in> guard ?n {priEK ?A}
###symbols
Set.empty :::: 'a set
Public.priEK :::: agent \<Rightarrow> nat
Set.insert :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a set
Guard.guard :::: nat \<Rightarrow> nat set \<Rightarrow> msg set
P1.chain :::: agent \<Rightarrow> nat \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> agent \<Rightarrow> msg
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
abbreviation
(*BEWARE!! priEK, priSK DON'T WORK with inj, range, image, etc.*)
priEK :: "agent \<Rightarrow> key" where
"priEK A == privateKey Encryption A"
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
inductive_set
guard :: "nat \<Rightarrow> key set \<Rightarrow> msg set"
for n :: nat and Ks :: "key set"
where
No_Nonce [intro]: "Nonce n \<notin> parts {X} \<Longrightarrow> X \<in> guard n Ks"
| Guard_Nonce [intro]: "invKey K \<in> Ks \<Longrightarrow> Crypt K X \<in> guard n Ks"
| Crypt [intro]: "X \<in> guard n Ks \<Longrightarrow> Crypt K X \<in> guard n Ks"
| Pair [intro]: "\<lbrakk>X \<in> guard n Ks; Y \<in> guard n Ks\<rbrakk> \<Longrightarrow> \<lbrace>X,Y\<rbrace> \<in> guard n Ks"
|
###output
x_1 \<noteq> x_2 \<Longrightarrow> ?H1 x_3 x_2 x_4 x_5 x_6 \<in> ?H2 x_1 (?H3 (?H4 x_7) ?H5)###end
|
Analysis/Line_Segment
|
Line_Segment.dist_midpoint(4)
|
lemma dist_midpoint:
fixes a b :: "'a::real_normed_vector" shows
"dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
"dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
"dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
"dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
|
dist (midpoint ?a ?b) ?b = dist ?a ?b / 2
|
?H1 (?H2 x_1 x_2) x_2 = ?H3 (?H1 x_1 x_2) (?H4 (?H5 ?H6))
|
[
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Fields.inverse_class.inverse_divide",
"Line_Segment.midpoint",
"Real_Vector_Spaces.dist_class.dist"
] |
[
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] |
[
"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 inverse = divide +\n fixes inverse :: \"'a \\<Rightarrow> 'a\"\nbegin",
"class dist =\n fixes dist :: \"'a \\<Rightarrow> 'a \\<Rightarrow> real\""
] |
template
|
###lemma
dist (midpoint ?a ?b) ?b = dist ?a ?b / 2
###symbols
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Fields.inverse_class.inverse_divide :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Line_Segment.midpoint :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Real_Vector_Spaces.dist_class.dist :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
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 inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin
class dist =
fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
|
###output
?H1 (?H2 x_1 x_2) x_2 = ?H3 (?H1 x_1 x_2) (?H4 (?H5 ?H6))###end
|
Transcendental
|
Transcendental.sinh_real_pos_iff
|
lemma sinh_real_pos_iff [simp]: "sinh (x :: real) > 0 \<longleftrightarrow> x > 0"
|
(0 < sinh ?x) = (0 < ?x)
|
(?H1 < ?H2 x_1) = (?H1 < x_1)
|
[
"Transcendental.sinh",
"Groups.zero_class.zero"
] |
[
"'a \\<Rightarrow> 'a",
"'a"
] |
[
"definition sinh :: \"'a :: {banach, real_normed_algebra_1} \\<Rightarrow> 'a\" where\n \"sinh x = (exp x - exp (-x)) /\\<^sub>R 2\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
(0 < sinh ?x) = (0 < ?x)
###symbols
Transcendental.sinh :::: 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
###defs
definition sinh :: "'a :: {banach, real_normed_algebra_1} \<Rightarrow> 'a" where
"sinh x = (exp x - exp (-x)) /\<^sub>R 2"
class zero =
fixes zero :: 'a ("0")
|
###output
(?H1 < ?H2 x_1) = (?H1 < x_1)###end
|
Transfer
|
Transfer.transfer_raw(225)
| null |
Transfer.Rel (rel_fun (rel_fun ?S1a (rel_fun ?S1c (=))) (rel_fun (rel_fun ?S2a (rel_fun ?S2c (=))) (rel_fun (rel_sum ?S1a ?S2a) (rel_fun (rel_sum ?S1c ?S2c) (=))))) rel_sum rel_sum
|
?H1 (?H2 (?H2 x_1 (?H2 x_2 (=))) (?H2 (?H2 x_3 (?H2 x_4 (=))) (?H2 (?H3 x_1 x_3) (?H2 (?H3 x_2 x_4) (=))))) ?H3 ?H3
|
[
"BNF_Def.rel_sum",
"BNF_Def.rel_fun",
"Transfer.Rel"
] |
[
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('c \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> 'a + 'c \\<Rightarrow> 'b + 'd \\<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"
] |
[
"inductive\n rel_sum :: \"('a \\<Rightarrow> 'c \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> 'a + 'b \\<Rightarrow> 'c + 'd \\<Rightarrow> bool\" for R1 R2\nwhere\n \"R1 a c \\<Longrightarrow> rel_sum R1 R2 (Inl a) (Inl c)\"\n| \"R2 b d \\<Longrightarrow> rel_sum R1 R2 (Inr b) (Inr d)\"",
"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\""
] |
template
|
###lemma
Transfer.Rel (rel_fun (rel_fun ?S1a (rel_fun ?S1c (=))) (rel_fun (rel_fun ?S2a (rel_fun ?S2c (=))) (rel_fun (rel_sum ?S1a ?S2a) (rel_fun (rel_sum ?S1c ?S2c) (=))))) rel_sum rel_sum
###symbols
BNF_Def.rel_sum :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'c \<Rightarrow> 'b + 'd \<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
###defs
inductive
rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool" for R1 R2
where
"R1 a c \<Longrightarrow> rel_sum R1 R2 (Inl a) (Inl c)"
| "R2 b d \<Longrightarrow> rel_sum R1 R2 (Inr b) (Inr d)"
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"
|
###output
?H1 (?H2 (?H2 x_1 (?H2 x_2 (=))) (?H2 (?H2 x_3 (?H2 x_4 (=))) (?H2 (?H3 x_1 x_3) (?H2 (?H3 x_2 x_4) (=))))) ?H3 ?H3###end
|
Library/Sublist
|
Sublist.Cons_parallelI2
|
lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
|
?a = ?b \<Longrightarrow> ?as \<parallel> ?bs \<Longrightarrow> ?a # ?as \<parallel> ?b # ?bs
|
\<lbrakk>x_1 = x_2; ?H1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_1 x_3) (?H2 x_2 x_4)
|
[
"List.list.Cons",
"Sublist.parallel"
] |
[
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> bool"
] |
[
"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 [] = []\"",
"definition parallel :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> bool\" (infixl \"\\<parallel>\" 50)\n where \"(xs \\<parallel> ys) = (\\<not> prefix xs ys \\<and> \\<not> prefix ys xs)\""
] |
template
|
###lemma
?a = ?b \<Longrightarrow> ?as \<parallel> ?bs \<Longrightarrow> ?a # ?as \<parallel> ?b # ?bs
###symbols
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Sublist.parallel :::: 'a list \<Rightarrow> 'a list \<Rightarrow> bool
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
definition parallel :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "\<parallel>" 50)
where "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)"
|
###output
\<lbrakk>x_1 = x_2; ?H1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_1 x_3) (?H2 x_2 x_4)###end
|
Library/Word
|
Word.unat_plus_if'
|
lemma unat_plus_if':
\<open>unat (a + b) =
(if unat a + unat b < 2 ^ LENGTH('a)
then unat a + unat b
else unat a + unat b - 2 ^ LENGTH('a))\<close> for a b :: \<open>'a::len word\<close>
|
unat (?a + ?b) = (if unat ?a + unat ?b < 2 ^ LENGTH(?'a) then unat ?a + unat ?b else unat ?a + unat ?b - 2 ^ LENGTH(?'a))
|
?H1 (?H2 x_1 x_2) = (if ?H2 (?H1 x_1) (?H1 x_2) < ?H3 (?H4 (?H5 ?H6)) (?H7 TYPE(?'a)) then ?H2 (?H1 x_1) (?H1 x_2) else ?H8 (?H2 (?H1 x_1) (?H1 x_2)) (?H3 (?H4 (?H5 ?H6)) (?H7 TYPE(?'a))))
|
[
"Groups.minus_class.minus",
"Type_Length.len0_class.len_of",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Power.power_class.power",
"Groups.plus_class.plus",
"Word.unat"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a itself \\<Rightarrow> nat",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a word \\<Rightarrow> nat"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"class len0 =\n fixes len_of :: \"'a itself \\<Rightarrow> nat\"",
"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\"",
"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 plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"abbreviation unat :: \\<open>'a::len word \\<Rightarrow> nat\\<close>\n where \\<open>unat \\<equiv> unsigned\\<close>"
] |
template
|
###lemma
unat (?a + ?b) = (if unat ?a + unat ?b < 2 ^ LENGTH(?'a) then unat ?a + unat ?b else unat ?a + unat ?b - 2 ^ LENGTH(?'a))
###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Type_Length.len0_class.len_of :::: 'a itself \<Rightarrow> nat
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Word.unat :::: 'a word \<Rightarrow> nat
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
class len0 =
fixes len_of :: "'a itself \<Rightarrow> nat"
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"
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
abbreviation unat :: \<open>'a::len word \<Rightarrow> nat\<close>
where \<open>unat \<equiv> unsigned\<close>
|
###output
?H1 (?H2 x_1 x_2) = (if ?H2 (?H1 x_1) (?H1 x_2) < ?H3 (?H4 (?H5 ?H6)) (?H7 TYPE(?'a)) then ?H2 (?H1 x_1) (?H1 x_2) else ?H8 (?H2 (?H1 x_1) (?H1 x_2)) (?H3 (?H4 (?H5 ?H6)) (?H7 TYPE(?'a))))###end
|
Probability/SPMF
|
SPMF.map_spmf_mono
|
lemma map_spmf_mono [partial_function_mono]: "mono_spmf B \<Longrightarrow> mono_spmf (\<lambda>g. map_spmf f (B g))"
|
monotone spmf.le_fun (ord_spmf (=)) ?B \<Longrightarrow> monotone spmf.le_fun (ord_spmf (=)) (\<lambda>g. map_spmf ?f (?B g))
|
?H1 ?H2 (?H3 (=)) x_1 \<Longrightarrow> ?H1 ?H2 (?H3 (=)) (\<lambda>y_0. ?H4 x_2 (x_1 y_0))
|
[
"SPMF.map_spmf",
"SPMF.ord_spmf",
"SPMF.spmf.le_fun",
"Fun.monotone"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a spmf \\<Rightarrow> 'b spmf",
"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> 'a spmf \\<Rightarrow> 'a spmf \\<Rightarrow> bool",
"('b \\<Rightarrow> 'a spmf) \\<Rightarrow> ('b \\<Rightarrow> 'a spmf) \\<Rightarrow> bool",
"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"abbreviation map_spmf :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a spmf \\<Rightarrow> 'b spmf\"\n where \"map_spmf f \\<equiv> map_pmf (map_option f)\"",
"abbreviation ord_spmf :: \"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> 'a spmf \\<Rightarrow> 'a spmf \\<Rightarrow> bool\"\n where \"ord_spmf ord \\<equiv> rel_pmf (ord_option ord)\"",
"abbreviation spmf :: \"'a spmf \\<Rightarrow> 'a \\<Rightarrow> real\"\n where \"spmf p x \\<equiv> pmf p (Some x)\"",
"abbreviation monotone :: \"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"monotone \\<equiv> monotone_on UNIV\""
] |
template
|
###lemma
monotone spmf.le_fun (ord_spmf (=)) ?B \<Longrightarrow> monotone spmf.le_fun (ord_spmf (=)) (\<lambda>g. map_spmf ?f (?B g))
###symbols
SPMF.map_spmf :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a spmf \<Rightarrow> 'b spmf
SPMF.ord_spmf :::: ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a spmf \<Rightarrow> 'a spmf \<Rightarrow> bool
SPMF.spmf.le_fun :::: ('b \<Rightarrow> 'a spmf) \<Rightarrow> ('b \<Rightarrow> 'a spmf) \<Rightarrow> bool
Fun.monotone :::: ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
abbreviation map_spmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a spmf \<Rightarrow> 'b spmf"
where "map_spmf f \<equiv> map_pmf (map_option f)"
abbreviation ord_spmf :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a spmf \<Rightarrow> 'a spmf \<Rightarrow> bool"
where "ord_spmf ord \<equiv> rel_pmf (ord_option ord)"
abbreviation spmf :: "'a spmf \<Rightarrow> 'a \<Rightarrow> real"
where "spmf p x \<equiv> pmf p (Some x)"
abbreviation monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
where "monotone \<equiv> monotone_on UNIV"
|
###output
?H1 ?H2 (?H3 (=)) x_1 \<Longrightarrow> ?H1 ?H2 (?H3 (=)) (\<lambda>y_0. ?H4 x_2 (x_1 y_0))###end
|
ex/Tree23
|
Tree23.bal_del0
|
lemma bal_del0: "bal t \<Longrightarrow> bal (del0 k t)"
|
bal ?t \<Longrightarrow> bal (del0 ?k ?t)
|
?H1 x_1 \<Longrightarrow> ?H1 (?H2 x_2 x_1)
|
[
"Tree23.del0",
"Tree23.bal"
] |
[
"int \\<Rightarrow> 'a tree23 \\<Rightarrow> 'a tree23",
"'a tree23 \\<Rightarrow> bool"
] |
[
"definition del0 :: \"key \\<Rightarrow> 'a tree23 \\<Rightarrow> 'a tree23\" where\n\"del0 k t = (case del (Some k) t of None \\<Rightarrow> t | Some(_,(_,t')) \\<Rightarrow> t')\"",
"fun bal :: \"'a tree23 \\<Rightarrow> bool\" where\n\"bal Empty = True\" |\n\"bal (Branch2 l _ r) = (bal l & bal r & height l = height r)\" |\n\"bal (Branch3 l _ m _ r) = (bal l & bal m & bal r & height l = height m & height m = height r)\""
] |
template
|
###lemma
bal ?t \<Longrightarrow> bal (del0 ?k ?t)
###symbols
Tree23.del0 :::: int \<Rightarrow> 'a tree23 \<Rightarrow> 'a tree23
Tree23.bal :::: 'a tree23 \<Rightarrow> bool
###defs
definition del0 :: "key \<Rightarrow> 'a tree23 \<Rightarrow> 'a tree23" where
"del0 k t = (case del (Some k) t of None \<Rightarrow> t | Some(_,(_,t')) \<Rightarrow> t')"
fun bal :: "'a tree23 \<Rightarrow> bool" where
"bal Empty = True" |
"bal (Branch2 l _ r) = (bal l & bal r & height l = height r)" |
"bal (Branch3 l _ m _ r) = (bal l & bal m & bal r & height l = height m & height m = height r)"
|
###output
?H1 x_1 \<Longrightarrow> ?H1 (?H2 x_2 x_1)###end
|
Number_Theory/Pocklington
|
Pocklington.pocklington_lemma
|
lemma pocklington_lemma:
fixes p :: nat
assumes n: "n \<ge> 2" and nqr: "n - 1 = q * r"
and an: "[a^ (n - 1) = 1] (mod n)"
and aq: "\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a ^ ((n - 1) div p) - 1) n"
and pp: "prime p" and pn: "p dvd n"
shows "[p = 1] (mod q)"
|
2 \<le> ?n \<Longrightarrow> ?n - 1 = ?q * ?r \<Longrightarrow> [ ?a ^ (?n - 1) = 1] (mod ?n) \<Longrightarrow> \<forall>p. prime p \<and> p dvd ?q \<longrightarrow> coprime (?a ^ ((?n - 1) div p) - 1) ?n \<Longrightarrow> prime ?p \<Longrightarrow> ?p dvd ?n \<Longrightarrow> [ ?p = 1] (mod ?q)
|
\<lbrakk> ?H1 (?H2 ?H3) \<le> x_1; ?H4 x_1 ?H5 = ?H6 x_2 x_3; ?H7 (?H8 x_4 (?H4 x_1 ?H5)) ?H5 x_1; \<forall>y_0. ?H9 y_0 \<and> ?H10 y_0 x_2 \<longrightarrow> ?H11 (?H4 (?H8 x_4 (?H12 (?H4 x_1 ?H5) y_0)) ?H5) x_1; ?H9 x_5; ?H10 x_5 x_1\<rbrakk> \<Longrightarrow> ?H7 x_5 ?H5 x_2
|
[
"Rings.divide_class.divide",
"Rings.algebraic_semidom_class.coprime",
"Rings.dvd_class.dvd",
"Factorial_Ring.normalization_semidom_class.prime",
"Power.power_class.power",
"Cong.unique_euclidean_semiring_class.cong",
"Groups.times_class.times",
"Groups.one_class.one",
"Groups.minus_class.minus",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> bool",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a"
] |
[
"class divide =\n fixes divide :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"div\" 70)",
"class algebraic_semidom = semidom_divide\nbegin",
"definition dvd :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" (infix \"dvd\" 50)\n where \"b dvd a \\<longleftrightarrow> (\\<exists>k. a = b * k)\"",
"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 times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"class one =\n fixes one :: 'a (\"1\")",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"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\""
] |
template
|
###lemma
2 \<le> ?n \<Longrightarrow> ?n - 1 = ?q * ?r \<Longrightarrow> [ ?a ^ (?n - 1) = 1] (mod ?n) \<Longrightarrow> \<forall>p. prime p \<and> p dvd ?q \<longrightarrow> coprime (?a ^ ((?n - 1) div p) - 1) ?n \<Longrightarrow> prime ?p \<Longrightarrow> ?p dvd ?n \<Longrightarrow> [ ?p = 1] (mod ?q)
###symbols
Rings.divide_class.divide :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Rings.algebraic_semidom_class.coprime :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
Rings.dvd_class.dvd :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
Factorial_Ring.normalization_semidom_class.prime :::: 'a \<Rightarrow> bool
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
Cong.unique_euclidean_semiring_class.cong :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.one_class.one :::: 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
###defs
class divide =
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
class algebraic_semidom = semidom_divide
begin
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class one =
fixes one :: 'a ("1")
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
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"
|
###output
\<lbrakk> ?H1 (?H2 ?H3) \<le> x_1; ?H4 x_1 ?H5 = ?H6 x_2 x_3; ?H7 (?H8 x_4 (?H4 x_1 ?H5)) ?H5 x_1; \<forall>y_0. ?H9 y_0 \<and> ?H10 y_0 x_2 \<longrightarrow> ?H11 (?H4 (?H8 x_4 (?H12 (?H4 x_1 ?H5) y_0)) ?H5) x_1; ?H9 x_5; ?H10 x_5 x_1\<rbrakk> \<Longrightarrow> ?H7 x_5 ?H5 x_2###end
|
HOLCF/Tutorial/New_Domain
|
New_Domain.ltree_induct
|
lemma ltree_induct:
fixes P :: "'a ltree \<Rightarrow> bool"
assumes adm: "adm P"
assumes bot: "P \<bottom>"
assumes Leaf: "\<And>x. P (Leaf\<cdot>x)"
assumes Branch: "\<And>f l. \<forall>x. P (f\<cdot>x) \<Longrightarrow> P (Branch\<cdot>(llist_map\<cdot>f\<cdot>l))"
shows "P x"
|
adm ?P \<Longrightarrow> ?P \<bottom> \<Longrightarrow> (\<And>x. ?P (Leaf\<cdot>x)) \<Longrightarrow> (\<And>f l. \<forall>x. ?P (f\<cdot>x) \<Longrightarrow> ?P (Branch\<cdot>(llist_map\<cdot>f\<cdot>l))) \<Longrightarrow> ?P ?x
|
\<lbrakk> ?H1 x_1; x_1 ?H2; \<And>y_0. x_1 (?H3 ?H4 y_0); \<And>y_1 y_2. \<forall>y_3. x_1 (?H3 y_1 y_3) \<Longrightarrow> x_1 (?H3 ?H5 (?H3 (?H3 ?H6 y_1) y_2))\<rbrakk> \<Longrightarrow> x_1 x_2
|
[
"New_Domain.llist_map",
"New_Domain.ltree.Branch",
"New_Domain.ltree.Leaf",
"Cfun.cfun.Rep_cfun",
"Pcpo.pcpo_class.bottom",
"Adm.adm"
] |
[
"('a \\<rightarrow> 'a) \\<rightarrow> 'a llist \\<rightarrow> 'a llist",
"'a ltree llist \\<rightarrow> 'a ltree",
"'a \\<rightarrow> 'a ltree",
"('a \\<rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'b",
"'a",
"('a \\<Rightarrow> bool) \\<Rightarrow> bool"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition adm :: \"('a::cpo \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"adm P \\<longleftrightarrow> (\\<forall>Y. chain Y \\<longrightarrow> (\\<forall>i. P (Y i)) \\<longrightarrow> P (\\<Squnion>i. Y i))\""
] |
template
|
###lemma
adm ?P \<Longrightarrow> ?P \<bottom> \<Longrightarrow> (\<And>x. ?P (Leaf\<cdot>x)) \<Longrightarrow> (\<And>f l. \<forall>x. ?P (f\<cdot>x) \<Longrightarrow> ?P (Branch\<cdot>(llist_map\<cdot>f\<cdot>l))) \<Longrightarrow> ?P ?x
###symbols
New_Domain.llist_map :::: ('a \<rightarrow> 'a) \<rightarrow> 'a llist \<rightarrow> 'a llist
New_Domain.ltree.Branch :::: 'a ltree llist \<rightarrow> 'a ltree
New_Domain.ltree.Leaf :::: 'a \<rightarrow> 'a ltree
Cfun.cfun.Rep_cfun :::: ('a \<rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b
Pcpo.pcpo_class.bottom :::: 'a
Adm.adm :::: ('a \<Rightarrow> bool) \<Rightarrow> bool
###defs
class pcpo = cpo +
assumes least: "\<exists>x. \<forall>y. x \<sqsubseteq> y"
begin
definition adm :: "('a::cpo \<Rightarrow> bool) \<Rightarrow> bool"
where "adm P \<longleftrightarrow> (\<forall>Y. chain Y \<longrightarrow> (\<forall>i. P (Y i)) \<longrightarrow> P (\<Squnion>i. Y i))"
|
###output
\<lbrakk> ?H1 x_1; x_1 ?H2; \<And>y_0. x_1 (?H3 ?H4 y_0); \<And>y_1 y_2. \<forall>y_3. x_1 (?H3 y_1 y_3) \<Longrightarrow> x_1 (?H3 ?H5 (?H3 (?H3 ?H6 y_1) y_2))\<rbrakk> \<Longrightarrow> x_1 x_2###end
|
GCD
|
GCD.dvd_lcm_I1_int
|
lemma dvd_lcm_I1_int [simp]: "i dvd m \<Longrightarrow> i dvd lcm m n"
for i m n :: int
|
?i dvd ?m \<Longrightarrow> ?i dvd lcm ?m ?n
|
?H1 x_1 x_2 \<Longrightarrow> ?H1 x_1 (?H2 x_2 x_3)
|
[
"GCD.gcd_class.lcm",
"Rings.dvd_class.dvd"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"class gcd = zero + one + dvd +\n fixes gcd :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\"\n and lcm :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\"",
"definition dvd :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" (infix \"dvd\" 50)\n where \"b dvd a \\<longleftrightarrow> (\\<exists>k. a = b * k)\""
] |
template
|
###lemma
?i dvd ?m \<Longrightarrow> ?i dvd lcm ?m ?n
###symbols
GCD.gcd_class.lcm :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Rings.dvd_class.dvd :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
###defs
class gcd = zero + one + dvd +
fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
|
###output
?H1 x_1 x_2 \<Longrightarrow> ?H1 x_1 (?H2 x_2 x_3)###end
|
HOLCF/Lift
|
Lift.DefE
|
lemma DefE: "Def x = \<bottom> \<Longrightarrow> R"
|
Def ?x = \<bottom> \<Longrightarrow> ?R
|
?H1 x_1 = ?H2 \<Longrightarrow> x_2
|
[
"Pcpo.pcpo_class.bottom",
"Lift.Def"
] |
[
"'a",
"'a \\<Rightarrow> 'a lift"
] |
[
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"definition\n Def :: \"'a \\<Rightarrow> 'a lift\" where\n \"Def x = Abs_lift (up\\<cdot>(Discr x))\""
] |
template
|
###lemma
Def ?x = \<bottom> \<Longrightarrow> ?R
###symbols
Pcpo.pcpo_class.bottom :::: 'a
Lift.Def :::: 'a \<Rightarrow> 'a lift
###defs
class pcpo = cpo +
assumes least: "\<exists>x. \<forall>y. x \<sqsubseteq> y"
begin
definition
Def :: "'a \<Rightarrow> 'a lift" where
"Def x = Abs_lift (up\<cdot>(Discr x))"
|
###output
?H1 x_1 = ?H2 \<Longrightarrow> x_2###end
|
Analysis/Infinite_Set_Sum
|
Infinite_Set_Sum.abs_summable_on_diff
|
lemma abs_summable_on_diff [intro]:
assumes "f abs_summable_on A" and "g abs_summable_on A"
shows "(\<lambda>x. f x - g x) abs_summable_on A"
|
?f abs_summable_on ?A \<Longrightarrow> ?g abs_summable_on ?A \<Longrightarrow> (\<lambda>x. ?f x - ?g x) abs_summable_on ?A
|
\<lbrakk> ?H1 x_1 x_2; ?H1 x_3 x_2\<rbrakk> \<Longrightarrow> ?H1 (\<lambda>y_0. ?H2 (x_1 y_0) (x_3 y_0)) x_2
|
[
"Groups.minus_class.minus",
"Infinite_Set_Sum.abs_summable_on"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)"
] |
template
|
###lemma
?f abs_summable_on ?A \<Longrightarrow> ?g abs_summable_on ?A \<Longrightarrow> (\<lambda>x. ?f x - ?g x) abs_summable_on ?A
###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Infinite_Set_Sum.abs_summable_on :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
|
###output
\<lbrakk> ?H1 x_1 x_2; ?H1 x_3 x_2\<rbrakk> \<Longrightarrow> ?H1 (\<lambda>y_0. ?H2 (x_1 y_0) (x_3 y_0)) x_2###end
|
Analysis/Arcwise_Connected
|
Arcwise_Connected.dense_accessible_frontier_points_connected
|
lemma dense_accessible_frontier_points_connected:
fixes S :: "'a::{complete_space,real_normed_vector} set"
assumes "open S" "connected S" "x \<in> S" "V \<noteq> {}"
and ope: "openin (top_of_set (frontier S)) V"
obtains g where "arc g" "g ` {0..<1} \<subseteq> S" "pathstart g = x" "pathfinish g \<in> V"
|
open ?S \<Longrightarrow> connected ?S \<Longrightarrow> ?x \<in> ?S \<Longrightarrow> ?V \<noteq> {} \<Longrightarrow> openin (top_of_set (frontier ?S)) ?V \<Longrightarrow> (\<And>g. arc g \<Longrightarrow> g ` {0..<1} \<subseteq> ?S \<Longrightarrow> pathstart g = ?x \<Longrightarrow> pathfinish g \<in> ?V \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
|
\<lbrakk> ?H1 x_1; ?H2 x_1; x_2 \<in> x_1; x_3 \<noteq> ?H3; ?H4 (?H5 (?H6 x_1)) x_3; \<And>y_0. \<lbrakk> ?H7 y_0; ?H8 (?H9 y_0 (?H10 ?H11 ?H12)) x_1; ?H13 y_0 = x_2; ?H14 y_0 \<in> x_3\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Path_Connected.pathfinish",
"Path_Connected.pathstart",
"Groups.one_class.one",
"Groups.zero_class.zero",
"Set_Interval.ord_class.atLeastLessThan",
"Set.image",
"Set.subset_eq",
"Path_Connected.arc",
"Elementary_Topology.frontier",
"Abstract_Topology.top_of_set",
"Abstract_Topology.topology.openin",
"Set.empty",
"Topological_Spaces.topological_space_class.connected",
"Topological_Spaces.open_class.open"
] |
[
"(real \\<Rightarrow> 'a) \\<Rightarrow> 'a",
"(real \\<Rightarrow> 'a) \\<Rightarrow> 'a",
"'a",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a set",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"(real \\<Rightarrow> 'a) \\<Rightarrow> bool",
"'a set \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a topology",
"'a topology \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a set",
"'a set \\<Rightarrow> bool",
"'a set \\<Rightarrow> bool"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"class zero =\n fixes zero :: 'a (\"0\")",
"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 subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"abbreviation top_of_set :: \"'a::topological_space set \\<Rightarrow> 'a topology\"\n where \"top_of_set \\<equiv> subtopology (topology open)\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"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",
"class \"open\" =\n fixes \"open\" :: \"'a set \\<Rightarrow> bool\""
] |
template
|
###lemma
open ?S \<Longrightarrow> connected ?S \<Longrightarrow> ?x \<in> ?S \<Longrightarrow> ?V \<noteq> {} \<Longrightarrow> openin (top_of_set (frontier ?S)) ?V \<Longrightarrow> (\<And>g. arc g \<Longrightarrow> g ` {0..<1} \<subseteq> ?S \<Longrightarrow> pathstart g = ?x \<Longrightarrow> pathfinish g \<in> ?V \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
###symbols
Path_Connected.pathfinish :::: (real \<Rightarrow> 'a) \<Rightarrow> 'a
Path_Connected.pathstart :::: (real \<Rightarrow> 'a) \<Rightarrow> 'a
Groups.one_class.one :::: 'a
Groups.zero_class.zero :::: 'a
Set_Interval.ord_class.atLeastLessThan :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a set
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Path_Connected.arc :::: (real \<Rightarrow> 'a) \<Rightarrow> bool
Elementary_Topology.frontier :::: 'a set \<Rightarrow> 'a set
Abstract_Topology.top_of_set :::: 'a set \<Rightarrow> 'a topology
Abstract_Topology.topology.openin :::: 'a topology \<Rightarrow> 'a set \<Rightarrow> bool
Set.empty :::: 'a set
Topological_Spaces.topological_space_class.connected :::: 'a set \<Rightarrow> bool
Topological_Spaces.open_class.open :::: 'a set \<Rightarrow> bool
###defs
class one =
fixes one :: 'a ("1")
class zero =
fixes zero :: 'a ("0")
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 subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
abbreviation top_of_set :: "'a::topological_space set \<Rightarrow> 'a topology"
where "top_of_set \<equiv> subtopology (topology open)"
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
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
class "open" =
fixes "open" :: "'a set \<Rightarrow> bool"
|
###output
\<lbrakk> ?H1 x_1; ?H2 x_1; x_2 \<in> x_1; x_3 \<noteq> ?H3; ?H4 (?H5 (?H6 x_1)) x_3; \<And>y_0. \<lbrakk> ?H7 y_0; ?H8 (?H9 y_0 (?H10 ?H11 ?H12)) x_1; ?H13 y_0 = x_2; ?H14 y_0 \<in> x_3\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4###end
|
Library/Finite_Map
|
Finite_Map.fmimage_restrict_fset
|
lemma fmimage_restrict_fset[simp]: "fmimage (fmrestrict_fset B m) A = fmimage m (A |\<inter>| B)"
|
fmimage (fmrestrict_fset ?B ?m) ?A = fmimage ?m (?A |\<inter>| ?B)
|
?H1 (?H2 x_1 x_2) x_3 = ?H1 x_2 (?H3 x_3 x_1)
|
[
"FSet.finter",
"Finite_Map.fmrestrict_fset",
"Finite_Map.fmimage"
] |
[
"'a fset \\<Rightarrow> 'a fset \\<Rightarrow> 'a fset",
"'a fset \\<Rightarrow> ('a, 'b) fmap \\<Rightarrow> ('a, 'b) fmap",
"('a, 'b) fmap \\<Rightarrow> 'a fset \\<Rightarrow> 'b fset"
] |
[
"abbreviation finter :: \"'a fset \\<Rightarrow> 'a fset \\<Rightarrow> 'a fset\" (infixl \"|\\<inter>|\" 65) where \"xs |\\<inter>| ys \\<equiv> inf xs ys\""
] |
template
|
###lemma
fmimage (fmrestrict_fset ?B ?m) ?A = fmimage ?m (?A |\<inter>| ?B)
###symbols
FSet.finter :::: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset
Finite_Map.fmrestrict_fset :::: 'a fset \<Rightarrow> ('a, 'b) fmap \<Rightarrow> ('a, 'b) fmap
Finite_Map.fmimage :::: ('a, 'b) fmap \<Rightarrow> 'a fset \<Rightarrow> 'b fset
###defs
abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys"
|
###output
?H1 (?H2 x_1 x_2) x_3 = ?H1 x_2 (?H3 x_3 x_1)###end
|
Bali/Decl
|
Decl.memberid_pair_simp1
|
lemma memberid_pair_simp1: "memberid p = memberid (snd p)"
|
memberid ?p = memberid (snd ?p)
|
?H1 x_1 = ?H1 (?H2 x_1)
|
[
"Product_Type.prod.snd",
"Decl.has_memberid_class.memberid"
] |
[
"'a \\<times> 'b \\<Rightarrow> 'b",
"'a \\<Rightarrow> memberid"
] |
[
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\"",
"class has_memberid =\n fixes memberid :: \"'a \\<Rightarrow> memberid\""
] |
template
|
###lemma
memberid ?p = memberid (snd ?p)
###symbols
Product_Type.prod.snd :::: 'a \<times> 'b \<Rightarrow> 'b
Decl.has_memberid_class.memberid :::: 'a \<Rightarrow> memberid
###defs
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
class has_memberid =
fixes memberid :: "'a \<Rightarrow> memberid"
|
###output
?H1 x_1 = ?H1 (?H2 x_1)###end
|
Analysis/Fashoda_Theorem
|
Fashoda_Theorem.fashoda_interlace
| null |
path ?f \<Longrightarrow> path ?g \<Longrightarrow> path_image ?f \<subseteq> cbox ?a ?b \<Longrightarrow> path_image ?g \<subseteq> cbox ?a ?b \<Longrightarrow> pathstart ?f $ 2 = ?a $ 2 \<Longrightarrow> pathfinish ?f $ 2 = ?a $ 2 \<Longrightarrow> pathstart ?g $ 2 = ?a $ 2 \<Longrightarrow> pathfinish ?g $ 2 = ?a $ 2 \<Longrightarrow> pathstart ?f $ 1 < pathstart ?g $ 1 \<Longrightarrow> pathstart ?g $ 1 < pathfinish ?f $ 1 \<Longrightarrow> pathfinish ?f $ 1 < pathfinish ?g $ 1 \<Longrightarrow> (\<And>z. z \<in> path_image ?f \<Longrightarrow> z \<in> path_image ?g \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
|
\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 (?H3 x_1) (?H4 x_3 x_4); ?H2 (?H3 x_2) (?H4 x_3 x_4); ?H5 (?H6 x_1) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H10 x_1) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H6 x_2) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H10 x_2) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H6 x_1) ?H11 < ?H5 (?H6 x_2) ?H11; ?H5 (?H6 x_2) ?H11 < ?H5 (?H10 x_1) ?H11; ?H5 (?H10 x_1) ?H11 < ?H5 (?H10 x_2) ?H11; \<And>y_0. \<lbrakk>y_0 \<in> ?H3 x_1; y_0 \<in> ?H3 x_2\<rbrakk> \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"Groups.one_class.one",
"Path_Connected.pathfinish",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Path_Connected.pathstart",
"Finite_Cartesian_Product.vec.vec_nth",
"Topology_Euclidean_Space.cbox",
"Path_Connected.path_image",
"Set.subset_eq",
"Path_Connected.path"
] |
[
"'a",
"(real \\<Rightarrow> 'a) \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"(real \\<Rightarrow> 'a) \\<Rightarrow> 'a",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a set",
"(real \\<Rightarrow> 'a) \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"(real \\<Rightarrow> 'a) \\<Rightarrow> bool"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"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\"",
"definition \"vec x = (\\<chi> i. x)\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
template
|
###lemma
path ?f \<Longrightarrow> path ?g \<Longrightarrow> path_image ?f \<subseteq> cbox ?a ?b \<Longrightarrow> path_image ?g \<subseteq> cbox ?a ?b \<Longrightarrow> pathstart ?f $ 2 = ?a $ 2 \<Longrightarrow> pathfinish ?f $ 2 = ?a $ 2 \<Longrightarrow> pathstart ?g $ 2 = ?a $ 2 \<Longrightarrow> pathfinish ?g $ 2 = ?a $ 2 \<Longrightarrow> pathstart ?f $ 1 < pathstart ?g $ 1 \<Longrightarrow> pathstart ?g $ 1 < pathfinish ?f $ 1 \<Longrightarrow> pathfinish ?f $ 1 < pathfinish ?g $ 1 \<Longrightarrow> (\<And>z. z \<in> path_image ?f \<Longrightarrow> z \<in> path_image ?g \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
###symbols
Groups.one_class.one :::: 'a
Path_Connected.pathfinish :::: (real \<Rightarrow> 'a) \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Path_Connected.pathstart :::: (real \<Rightarrow> 'a) \<Rightarrow> 'a
Finite_Cartesian_Product.vec.vec_nth :::: ('a, 'b) vec \<Rightarrow> 'b \<Rightarrow> 'a
Topology_Euclidean_Space.cbox :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a set
Path_Connected.path_image :::: (real \<Rightarrow> 'a) \<Rightarrow> 'a set
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Path_Connected.path :::: (real \<Rightarrow> 'a) \<Rightarrow> bool
###defs
class one =
fixes one :: 'a ("1")
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"
definition "vec x = (\<chi> i. x)"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 (?H3 x_1) (?H4 x_3 x_4); ?H2 (?H3 x_2) (?H4 x_3 x_4); ?H5 (?H6 x_1) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H10 x_1) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H6 x_2) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H10 x_2) (?H7 (?H8 ?H9)) = ?H5 x_3 (?H7 (?H8 ?H9)); ?H5 (?H6 x_1) ?H11 < ?H5 (?H6 x_2) ?H11; ?H5 (?H6 x_2) ?H11 < ?H5 (?H10 x_1) ?H11; ?H5 (?H10 x_1) ?H11 < ?H5 (?H10 x_2) ?H11; \<And>y_0. \<lbrakk>y_0 \<in> ?H3 x_1; y_0 \<in> ?H3 x_2\<rbrakk> \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5###end
|
Hoare_Parallel/OG_Tran
|
OG_Tran.L3_5v_lemma5
|
lemma L3_5v_lemma5 [rule_format]:
"\<forall>s. (fwhile b c k, s) -P*\<rightarrow> (Parallel Ts, t) \<longrightarrow> All_None Ts \<longrightarrow>
(While b i c, s) -P*\<rightarrow> (Parallel Ts,t)"
|
(fwhile ?b ?c ?k, ?s) -P*\<rightarrow> (Parallel ?Ts, ?t) \<Longrightarrow> All_None ?Ts \<Longrightarrow> (While ?b ?i ?c, ?s) -P*\<rightarrow> (Parallel ?Ts, ?t)
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3, x_4) (?H3 x_5, x_6); ?H4 x_5\<rbrakk> \<Longrightarrow> ?H1 (?H5 x_1 x_7 x_2, x_4) (?H3 x_5, x_6)
|
[
"OG_Com.com.While",
"OG_Tran.All_None",
"OG_Com.com.Parallel",
"OG_Tran.fwhile",
"OG_Tran.transitions"
] |
[
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a com \\<Rightarrow> 'a com",
"('a ann_com option \\<times> 'a set) list \\<Rightarrow> bool",
"('a ann_com option \\<times> 'a set) list \\<Rightarrow> 'a com",
"'a set \\<Rightarrow> 'a com \\<Rightarrow> nat \\<Rightarrow> 'a com",
"'a com \\<times> 'a \\<Rightarrow> 'a com \\<times> 'a \\<Rightarrow> bool"
] |
[
"definition All_None :: \"'a ann_triple_op list \\<Rightarrow> bool\" where\n \"All_None Ts \\<equiv> \\<forall>(c, q) \\<in> set Ts. c = None\"",
"primrec fwhile :: \"'a bexp \\<Rightarrow> 'a com \\<Rightarrow> nat \\<Rightarrow> 'a com\" where\n \"fwhile b c 0 = \\<Omega>\"\n | \"fwhile b c (Suc n) = Cond b (Seq c (fwhile b c n)) (Basic id)\""
] |
template
|
###lemma
(fwhile ?b ?c ?k, ?s) -P*\<rightarrow> (Parallel ?Ts, ?t) \<Longrightarrow> All_None ?Ts \<Longrightarrow> (While ?b ?i ?c, ?s) -P*\<rightarrow> (Parallel ?Ts, ?t)
###symbols
OG_Com.com.While :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a com \<Rightarrow> 'a com
OG_Tran.All_None :::: ('a ann_com option \<times> 'a set) list \<Rightarrow> bool
OG_Com.com.Parallel :::: ('a ann_com option \<times> 'a set) list \<Rightarrow> 'a com
OG_Tran.fwhile :::: 'a set \<Rightarrow> 'a com \<Rightarrow> nat \<Rightarrow> 'a com
OG_Tran.transitions :::: 'a com \<times> 'a \<Rightarrow> 'a com \<times> 'a \<Rightarrow> bool
###defs
definition All_None :: "'a ann_triple_op list \<Rightarrow> bool" where
"All_None Ts \<equiv> \<forall>(c, q) \<in> set Ts. c = None"
primrec fwhile :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> nat \<Rightarrow> 'a com" where
"fwhile b c 0 = \<Omega>"
| "fwhile b c (Suc n) = Cond b (Seq c (fwhile b c n)) (Basic id)"
|
###output
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3, x_4) (?H3 x_5, x_6); ?H4 x_5\<rbrakk> \<Longrightarrow> ?H1 (?H5 x_1 x_7 x_2, x_4) (?H3 x_5, x_6)###end
|
Nominal/Nominal
|
Nominal.pt_bij2
|
lemma pt_bij2:
fixes pi :: "'x prm"
and x :: "'a"
and y :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
and a: "x = (rev pi)\<bullet>y"
shows "(pi\<bullet>x)=y"
|
pt TYPE(?'a) TYPE(?'x) \<Longrightarrow> at TYPE(?'x) \<Longrightarrow> ?x = rev ?pi \<bullet> ?y \<Longrightarrow> ?pi \<bullet> ?x = ?y
|
\<lbrakk> ?H1 TYPE(?'a) TYPE(?'x); ?H2 TYPE(?'x); x_1 = ?H3 (?H4 x_2) x_3\<rbrakk> \<Longrightarrow> ?H3 x_2 x_1 = x_3
|
[
"List.rev",
"Nominal.perm",
"Nominal.at",
"Nominal.pt"
] |
[
"'a list \\<Rightarrow> 'a list",
"('a \\<times> 'a) list \\<Rightarrow> 'b \\<Rightarrow> 'b",
"'a itself \\<Rightarrow> bool",
"'a itself \\<Rightarrow> 'b itself \\<Rightarrow> bool"
] |
[
"primrec rev :: \"'a list \\<Rightarrow> 'a list\" where\n\"rev [] = []\" |\n\"rev (x # xs) = rev xs @ [x]\"",
"consts \n perm :: \"'x prm \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixr \\<open>\\<bullet>\\<close> 80)\n swap :: \"('x \\<times> 'x) \\<Rightarrow> 'x \\<Rightarrow> 'x\"",
"definition\n \"at TYPE('x) \\<equiv> \n (\\<forall>(x::'x). ([]::'x prm)\\<bullet>x = x) \\<and>\n (\\<forall>(a::'x) (b::'x) (pi::'x prm) (x::'x). ((a,b)#(pi::'x prm))\\<bullet>x = swap (a,b) (pi\\<bullet>x)) \\<and> \n (\\<forall>(a::'x) (b::'x) (c::'x). swap (a,b) c = (if a=c then b else (if b=c then a else c))) \\<and> \n (infinite (UNIV::'x set))\"",
"definition\n \"pt TYPE('a) TYPE('x) \\<equiv> \n (\\<forall>(x::'a). ([]::'x prm)\\<bullet>x = x) \\<and> \n (\\<forall>(pi1::'x prm) (pi2::'x prm) (x::'a). (pi1@pi2)\\<bullet>x = pi1\\<bullet>(pi2\\<bullet>x)) \\<and> \n (\\<forall>(pi1::'x prm) (pi2::'x prm) (x::'a). pi1 \\<triangleq> pi2 \\<longrightarrow> pi1\\<bullet>x = pi2\\<bullet>x)\""
] |
template
|
###lemma
pt TYPE(?'a) TYPE(?'x) \<Longrightarrow> at TYPE(?'x) \<Longrightarrow> ?x = rev ?pi \<bullet> ?y \<Longrightarrow> ?pi \<bullet> ?x = ?y
###symbols
List.rev :::: 'a list \<Rightarrow> 'a list
Nominal.perm :::: ('a \<times> 'a) list \<Rightarrow> 'b \<Rightarrow> 'b
Nominal.at :::: 'a itself \<Rightarrow> bool
Nominal.pt :::: 'a itself \<Rightarrow> 'b itself \<Rightarrow> bool
###defs
primrec rev :: "'a list \<Rightarrow> 'a list" where
"rev [] = []" |
"rev (x # xs) = rev xs @ [x]"
consts
perm :: "'x prm \<Rightarrow> 'a \<Rightarrow> 'a" (infixr \<open>\<bullet>\<close> 80)
swap :: "('x \<times> 'x) \<Rightarrow> 'x \<Rightarrow> 'x"
definition
"at TYPE('x) \<equiv>
(\<forall>(x::'x). ([]::'x prm)\<bullet>x = x) \<and>
(\<forall>(a::'x) (b::'x) (pi::'x prm) (x::'x). ((a,b)#(pi::'x prm))\<bullet>x = swap (a,b) (pi\<bullet>x)) \<and>
(\<forall>(a::'x) (b::'x) (c::'x). swap (a,b) c = (if a=c then b else (if b=c then a else c))) \<and>
(infinite (UNIV::'x set))"
definition
"pt TYPE('a) TYPE('x) \<equiv>
(\<forall>(x::'a). ([]::'x prm)\<bullet>x = x) \<and>
(\<forall>(pi1::'x prm) (pi2::'x prm) (x::'a). (pi1@pi2)\<bullet>x = pi1\<bullet>(pi2\<bullet>x)) \<and>
(\<forall>(pi1::'x prm) (pi2::'x prm) (x::'a). pi1 \<triangleq> pi2 \<longrightarrow> pi1\<bullet>x = pi2\<bullet>x)"
|
###output
\<lbrakk> ?H1 TYPE(?'a) TYPE(?'x); ?H2 TYPE(?'x); x_1 = ?H3 (?H4 x_2) x_3\<rbrakk> \<Longrightarrow> ?H3 x_2 x_1 = x_3###end
|
Bali/Trans
|
Transfer.right_total_Domainp_transfer
| null |
right_total ?B \<Longrightarrow> rel_fun (rel_fun ?A (rel_fun ?B (=))) (rel_fun ?A (=)) (\<lambda>T x. \<exists>y\<in>Collect (Domainp ?B). T x y) Domainp
|
?H1 x_1 \<Longrightarrow> ?H2 (?H2 x_2 (?H2 x_1 (=))) (?H2 x_2 (=)) (\<lambda>y_0 y_1. \<exists>y_2\<in> ?H3 (?H4 x_1). y_0 y_1 y_2) ?H4
|
[
"Relation.Domainp",
"Set.Collect",
"BNF_Def.rel_fun",
"Transfer.right_total"
] |
[
"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> 'a \\<Rightarrow> bool",
"('a \\<Rightarrow> bool) \\<Rightarrow> 'a set",
"('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> 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 right_total :: \"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"right_total R \\<longleftrightarrow> (\\<forall>y. \\<exists>x. R x y)\""
] |
template
|
###lemma
right_total ?B \<Longrightarrow> rel_fun (rel_fun ?A (rel_fun ?B (=))) (rel_fun ?A (=)) (\<lambda>T x. \<exists>y\<in>Collect (Domainp ?B). T x y) Domainp
###symbols
Relation.Domainp :::: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool
Set.Collect :::: ('a \<Rightarrow> bool) \<Rightarrow> 'a set
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.right_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 right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
|
###output
?H1 x_1 \<Longrightarrow> ?H2 (?H2 x_2 (?H2 x_1 (=))) (?H2 x_2 (=)) (\<lambda>y_0 y_1. \<exists>y_2\<in> ?H3 (?H4 x_1). y_0 y_1 y_2) ?H4###end
|
Bali/Trans
|
Transitive_Closure.tranclD
| null |
(?x, ?y) \<in> ?R\<^sup>+ \<Longrightarrow> \<exists>z. (?x, z) \<in> ?R \<and> (z, ?y) \<in> ?R\<^sup>*
|
(x_1, x_2) \<in> ?H1 x_3 \<Longrightarrow> \<exists>y_0. (x_1, y_0) \<in> x_3 \<and> (y_0, x_2) \<in> ?H2 x_3
|
[
"Transitive_Closure.rtrancl",
"Transitive_Closure.trancl"
] |
[
"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set",
"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set"
] |
[
"inductive_set rtrancl :: \"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set\" (\"(_\\<^sup>*)\" [1000] 999)\n for r :: \"('a \\<times> 'a) set\"\n where\n rtrancl_refl [intro!, Pure.intro!, simp]: \"(a, a) \\<in> r\\<^sup>*\"\n | rtrancl_into_rtrancl [Pure.intro]: \"(a, b) \\<in> r\\<^sup>* \\<Longrightarrow> (b, c) \\<in> r \\<Longrightarrow> (a, c) \\<in> r\\<^sup>*\"",
"inductive_set trancl :: \"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set\" (\"(_\\<^sup>+)\" [1000] 999)\n for r :: \"('a \\<times> 'a) set\"\n where\n r_into_trancl [intro, Pure.intro]: \"(a, b) \\<in> r \\<Longrightarrow> (a, b) \\<in> r\\<^sup>+\"\n | trancl_into_trancl [Pure.intro]: \"(a, b) \\<in> r\\<^sup>+ \\<Longrightarrow> (b, c) \\<in> r \\<Longrightarrow> (a, c) \\<in> r\\<^sup>+\""
] |
template
|
###lemma
(?x, ?y) \<in> ?R\<^sup>+ \<Longrightarrow> \<exists>z. (?x, z) \<in> ?R \<and> (z, ?y) \<in> ?R\<^sup>*
###symbols
Transitive_Closure.rtrancl :::: ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set
Transitive_Closure.trancl :::: ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set
###defs
inductive_set rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999)
for r :: "('a \<times> 'a) set"
where
rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) \<in> r\<^sup>*"
| rtrancl_into_rtrancl [Pure.intro]: "(a, b) \<in> r\<^sup>* \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>*"
inductive_set trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999)
for r :: "('a \<times> 'a) set"
where
r_into_trancl [intro, Pure.intro]: "(a, b) \<in> r \<Longrightarrow> (a, b) \<in> r\<^sup>+"
| trancl_into_trancl [Pure.intro]: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
|
###output
(x_1, x_2) \<in> ?H1 x_3 \<Longrightarrow> \<exists>y_0. (x_1, y_0) \<in> x_3 \<and> (y_0, x_2) \<in> ?H2 x_3###end
|
Library/FSet
|
FSet.fcard_funion_fsubset
|
lemma fcard_funion_fsubset:
"B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
|
?B |\<subseteq>| ?A \<Longrightarrow> fcard (?A |-| ?B) = fcard ?A - fcard ?B
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_2 x_1) = ?H4 (?H2 x_2) (?H2 x_1)
|
[
"Groups.minus_class.minus",
"FSet.fminus",
"FSet.fcard",
"FSet.fsubset_eq"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a fset \\<Rightarrow> 'a fset \\<Rightarrow> 'a fset",
"'a fset \\<Rightarrow> nat",
"'a fset \\<Rightarrow> 'a fset \\<Rightarrow> bool"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"abbreviation fminus :: \"'a fset \\<Rightarrow> 'a fset \\<Rightarrow> 'a fset\" (infixl \"|-|\" 65) where \"xs |-| ys \\<equiv> minus xs ys\"",
"abbreviation fsubset_eq :: \"'a fset \\<Rightarrow> 'a fset \\<Rightarrow> bool\" (infix \"|\\<subseteq>|\" 50) where \"xs |\\<subseteq>| ys \\<equiv> xs \\<le> ys\""
] |
template
|
###lemma
?B |\<subseteq>| ?A \<Longrightarrow> fcard (?A |-| ?B) = fcard ?A - fcard ?B
###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
FSet.fminus :::: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset
FSet.fcard :::: 'a fset \<Rightarrow> nat
FSet.fsubset_eq :::: 'a fset \<Rightarrow> 'a fset \<Rightarrow> bool
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys"
abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
|
###output
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_2 x_1) = ?H4 (?H2 x_2) (?H2 x_1)###end
|
Nominal/Examples/Class2
|
Class2.NEG_elim(2)
|
lemma NEG_elim:
shows "M \<in> \<parallel><B>\<parallel> \<Longrightarrow> M \<in> NEGc B (\<parallel>(B)\<parallel>)"
and "N \<in> \<parallel>(B)\<parallel> \<Longrightarrow> N \<in> NEGn B (\<parallel><B>\<parallel>)"
|
?N \<in> \<parallel>(?B)\<parallel> \<Longrightarrow> ?N \<in> NEGn ?B (\<parallel>< ?B>\<parallel>)
|
x_1 \<in> ?H1 x_2 \<Longrightarrow> x_1 \<in> ?H2 x_2 (?H3 x_2)
|
[
"Class2.CANDc",
"Class2.NEGn",
"Class2.CANDn"
] |
[
"ty \\<Rightarrow> ctrm set",
"ty \\<Rightarrow> ctrm set \\<Rightarrow> ntrm set",
"ty \\<Rightarrow> ntrm set"
] |
[] |
template
|
###lemma
?N \<in> \<parallel>(?B)\<parallel> \<Longrightarrow> ?N \<in> NEGn ?B (\<parallel>< ?B>\<parallel>)
###symbols
Class2.CANDc :::: ty \<Rightarrow> ctrm set
Class2.NEGn :::: ty \<Rightarrow> ctrm set \<Rightarrow> ntrm set
Class2.CANDn :::: ty \<Rightarrow> ntrm set
###defs
|
###output
x_1 \<in> ?H1 x_2 \<Longrightarrow> x_1 \<in> ?H2 x_2 (?H3 x_2)###end
|
UNITY/Lift_prog
|
Lift_prog.bij_drop_map
|
lemma bij_drop_map [iff]: "bij (drop_map i)"
|
bij (drop_map ?i)
|
?H1 (?H2 x_1)
|
[
"Lift_prog.drop_map",
"Fun.bij"
] |
[
"nat \\<Rightarrow> (nat \\<Rightarrow> 'a) \\<times> 'b \\<Rightarrow> 'a \\<times> (nat \\<Rightarrow> 'a) \\<times> 'b",
"('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"definition drop_map :: \"[nat, (nat=>'b) * 'c] => 'b * ((nat=>'b) * 'c)\" where\n \"drop_map i == %(g, uu). (g i, (delete_map i g, uu))\"",
"abbreviation bij :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"bij f \\<equiv> bij_betw f UNIV UNIV\""
] |
template
|
###lemma
bij (drop_map ?i)
###symbols
Lift_prog.drop_map :::: nat \<Rightarrow> (nat \<Rightarrow> 'a) \<times> 'b \<Rightarrow> 'a \<times> (nat \<Rightarrow> 'a) \<times> 'b
Fun.bij :::: ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition drop_map :: "[nat, (nat=>'b) * 'c] => 'b * ((nat=>'b) * 'c)" where
"drop_map i == %(g, uu). (g i, (delete_map i g, uu))"
abbreviation bij :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"
where "bij f \<equiv> bij_betw f UNIV UNIV"
|
###output
?H1 (?H2 x_1)###end
|
Library/Complete_Partial_Order2
|
Complete_Partial_Order2.cont_intro(3)
| null |
monotone ?orda ?ordb ?F \<Longrightarrow> monotone ?orda ?ordb ?G \<Longrightarrow> monotone ?orda ?ordb (\<lambda>f. if ?c then ?F f else ?G f)
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H1 x_1 x_2 x_4\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 (\<lambda>y_0. if x_5 then x_3 y_0 else x_4 y_0)
|
[
"Fun.monotone"
] |
[
"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] |
[
"abbreviation monotone :: \"('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool\"\n where \"monotone \\<equiv> monotone_on UNIV\""
] |
template
|
###lemma
monotone ?orda ?ordb ?F \<Longrightarrow> monotone ?orda ?ordb ?G \<Longrightarrow> monotone ?orda ?ordb (\<lambda>f. if ?c then ?F f else ?G f)
###symbols
Fun.monotone :::: ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
abbreviation monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
where "monotone \<equiv> monotone_on UNIV"
|
###output
\<lbrakk> ?H1 x_1 x_2 x_3; ?H1 x_1 x_2 x_4\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 (\<lambda>y_0. if x_5 then x_3 y_0 else x_4 y_0)###end
|
Computational_Algebra/Polynomial
|
Polynomial.pseudo_divmod_eq_div_mod
|
lemma pseudo_divmod_eq_div_mod:
\<open>pseudo_divmod f g = (f div g, f mod g)\<close> if \<open>lead_coeff g = 1\<close>
|
lead_coeff ?g = (1:: ?'a) \<Longrightarrow> pseudo_divmod ?f ?g = (?f div ?g, ?f mod ?g)
|
?H1 x_1 = ?H2 \<Longrightarrow> ?H3 x_2 x_1 = (?H4 x_2 x_1, ?H5 x_2 x_1)
|
[
"Rings.modulo_class.modulo",
"Rings.divide_class.divide",
"Polynomial.pseudo_divmod",
"Groups.one_class.one",
"Polynomial.lead_coeff"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a poly \\<Rightarrow> 'a poly \\<Rightarrow> 'a poly \\<times> 'a poly",
"'a",
"'a poly \\<Rightarrow> 'a"
] |
[
"class modulo = dvd + divide +\n fixes modulo :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"mod\" 70)",
"class divide =\n fixes divide :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"div\" 70)",
"definition pseudo_divmod :: \"'a :: comm_ring_1 poly \\<Rightarrow> 'a poly \\<Rightarrow> 'a poly \\<times> 'a poly\"\n where \"pseudo_divmod p q \\<equiv>\n if q = 0 then (0, p)\n else\n pseudo_divmod_main (coeff q (degree q)) 0 p q (degree p)\n (1 + length (coeffs p) - length (coeffs q))\"",
"class one =\n fixes one :: 'a (\"1\")",
"abbreviation lead_coeff:: \"'a::zero poly \\<Rightarrow> 'a\"\n where \"lead_coeff p \\<equiv> coeff p (degree p)\""
] |
template
|
###lemma
lead_coeff ?g = (1:: ?'a) \<Longrightarrow> pseudo_divmod ?f ?g = (?f div ?g, ?f mod ?g)
###symbols
Rings.modulo_class.modulo :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Rings.divide_class.divide :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Polynomial.pseudo_divmod :::: 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly
Groups.one_class.one :::: 'a
Polynomial.lead_coeff :::: 'a poly \<Rightarrow> 'a
###defs
class modulo = dvd + divide +
fixes modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
class divide =
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
definition pseudo_divmod :: "'a :: comm_ring_1 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
where "pseudo_divmod p q \<equiv>
if q = 0 then (0, p)
else
pseudo_divmod_main (coeff q (degree q)) 0 p q (degree p)
(1 + length (coeffs p) - length (coeffs q))"
class one =
fixes one :: 'a ("1")
abbreviation lead_coeff:: "'a::zero poly \<Rightarrow> 'a"
where "lead_coeff p \<equiv> coeff p (degree p)"
|
###output
?H1 x_1 = ?H2 \<Longrightarrow> ?H3 x_2 x_1 = (?H4 x_2 x_1, ?H5 x_2 x_1)###end
|
Analysis/Abstract_Topological_Spaces
|
Abstract_Topological_Spaces.quasi_eq_connected_component_of
|
lemma quasi_eq_connected_component_of:
"finite(connected_components_of X) \<or>
finite(quasi_components_of X) \<or>
locally_connected_space X \<or>
compact_space X \<and> (Hausdorff_space X \<or> regular_space X \<or> normal_space X)
\<Longrightarrow> quasi_component_of X x = connected_component_of X x"
|
finite (connected_components_of ?X) \<or> finite (quasi_components_of ?X) \<or> locally_connected_space ?X \<or> compact_space ?X \<and> (Hausdorff_space ?X \<or> regular_space ?X \<or> normal_space ?X) \<Longrightarrow> quasi_component_of ?X ?x = connected_component_of ?X ?x
|
?H1 (?H2 x_1) \<or> ?H1 (?H3 x_1) \<or> ?H4 x_1 \<or> ?H5 x_1 \<and> (?H6 x_1 \<or> ?H7 x_1 \<or> ?H8 x_1) \<Longrightarrow> ?H9 x_1 x_2 = ?H10 x_1 x_2
|
[
"Abstract_Topology_2.connected_component_of",
"Abstract_Topological_Spaces.quasi_component_of",
"Abstract_Topological_Spaces.normal_space",
"Abstract_Topological_Spaces.regular_space",
"T1_Spaces.Hausdorff_space",
"Abstract_Topology.compact_space",
"Locally.locally_connected_space",
"Abstract_Topological_Spaces.quasi_components_of",
"Abstract_Topology_2.connected_components_of",
"Finite_Set.finite"
] |
[
"'a topology \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a topology \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a topology \\<Rightarrow> bool",
"'a topology \\<Rightarrow> bool",
"'a topology \\<Rightarrow> bool",
"'a topology \\<Rightarrow> bool",
"'a topology \\<Rightarrow> bool",
"'a topology \\<Rightarrow> 'a set set",
"'a topology \\<Rightarrow> 'a set set",
"'a set \\<Rightarrow> bool"
] |
[
"definition connected_component_of :: \"'a topology \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\n where \"connected_component_of X x y \\<equiv>\n \\<exists>T. connectedin X T \\<and> x \\<in> T \\<and> y \\<in> T\"",
"definition quasi_component_of :: \"'a topology \\<Rightarrow> 'a \\<Rightarrow> 'a \\<Rightarrow> bool\"\n where\n \"quasi_component_of X x y \\<equiv>\n x \\<in> topspace X \\<and> y \\<in> topspace X \\<and>\n (\\<forall>T. closedin X T \\<and> openin X T \\<longrightarrow> (x \\<in> T \\<longleftrightarrow> y \\<in> T))\"",
"definition normal_space \n where \"normal_space X \\<equiv>\n \\<forall>S T. closedin X S \\<and> closedin X T \\<and> disjnt S T \n \\<longrightarrow> (\\<exists>U V. openin X U \\<and> openin X V \\<and> S \\<subseteq> U \\<and> T \\<subseteq> V \\<and> disjnt U V)\"",
"definition regular_space \n where \"regular_space X \\<equiv>\n \\<forall>C a. closedin X C \\<and> a \\<in> topspace X - C\n \\<longrightarrow> (\\<exists>U V. openin X U \\<and> openin X V \\<and> a \\<in> U \\<and> C \\<subseteq> V \\<and> disjnt U V)\"",
"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)\"",
"definition compact_space where\n \"compact_space X \\<equiv> compactin X (topspace X)\"",
"definition locally_connected_space \n where \"locally_connected_space X \\<equiv> neighbourhood_base_of (connectedin X) X\"",
"definition quasi_components_of :: \"'a topology \\<Rightarrow> ('a set) set\"\n where\n \"quasi_components_of X = quasi_component_of_set X ` topspace X\"",
"definition connected_components_of :: \"'a topology \\<Rightarrow> ('a set) set\"\n where \"connected_components_of X \\<equiv> connected_component_of_set X ` topspace X\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] |
template
|
###lemma
finite (connected_components_of ?X) \<or> finite (quasi_components_of ?X) \<or> locally_connected_space ?X \<or> compact_space ?X \<and> (Hausdorff_space ?X \<or> regular_space ?X \<or> normal_space ?X) \<Longrightarrow> quasi_component_of ?X ?x = connected_component_of ?X ?x
###symbols
Abstract_Topology_2.connected_component_of :::: 'a topology \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool
Abstract_Topological_Spaces.quasi_component_of :::: 'a topology \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool
Abstract_Topological_Spaces.normal_space :::: 'a topology \<Rightarrow> bool
Abstract_Topological_Spaces.regular_space :::: 'a topology \<Rightarrow> bool
T1_Spaces.Hausdorff_space :::: 'a topology \<Rightarrow> bool
Abstract_Topology.compact_space :::: 'a topology \<Rightarrow> bool
Locally.locally_connected_space :::: 'a topology \<Rightarrow> bool
Abstract_Topological_Spaces.quasi_components_of :::: 'a topology \<Rightarrow> 'a set set
Abstract_Topology_2.connected_components_of :::: 'a topology \<Rightarrow> 'a set set
Finite_Set.finite :::: 'a set \<Rightarrow> bool
###defs
definition connected_component_of :: "'a topology \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
where "connected_component_of X x y \<equiv>
\<exists>T. connectedin X T \<and> x \<in> T \<and> y \<in> T"
definition quasi_component_of :: "'a topology \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
where
"quasi_component_of X x y \<equiv>
x \<in> topspace X \<and> y \<in> topspace X \<and>
(\<forall>T. closedin X T \<and> openin X T \<longrightarrow> (x \<in> T \<longleftrightarrow> y \<in> T))"
definition normal_space
where "normal_space X \<equiv>
\<forall>S T. closedin X S \<and> closedin X T \<and> disjnt S T
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V)"
definition regular_space
where "regular_space X \<equiv>
\<forall>C a. closedin X C \<and> a \<in> topspace X - C
\<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> a \<in> U \<and> C \<subseteq> V \<and> disjnt U V)"
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)"
definition compact_space where
"compact_space X \<equiv> compactin X (topspace X)"
definition locally_connected_space
where "locally_connected_space X \<equiv> neighbourhood_base_of (connectedin X) X"
definition quasi_components_of :: "'a topology \<Rightarrow> ('a set) set"
where
"quasi_components_of X = quasi_component_of_set X ` topspace X"
definition connected_components_of :: "'a topology \<Rightarrow> ('a set) set"
where "connected_components_of X \<equiv> connected_component_of_set X ` topspace X"
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
|
###output
?H1 (?H2 x_1) \<or> ?H1 (?H3 x_1) \<or> ?H4 x_1 \<or> ?H5 x_1 \<and> (?H6 x_1 \<or> ?H7 x_1 \<or> ?H8 x_1) \<Longrightarrow> ?H9 x_1 x_2 = ?H10 x_1 x_2###end
|
HOLCF/IOA/CompoScheds
|
CompoScheds.mkex2_unfold
|
lemma mkex2_unfold:
"mkex2 A B =
(LAM sch exA exB.
(\<lambda>s t.
case sch of
nil \<Rightarrow> nil
| x ## xs \<Rightarrow>
(case x of
UU \<Rightarrow> UU
| Def y \<Rightarrow>
(if y \<in> act A then
(if y \<in> act B then
(case HD \<cdot> exA of
UU \<Rightarrow> UU
| Def a \<Rightarrow>
(case HD \<cdot> exB of
UU \<Rightarrow> UU
| Def b \<Rightarrow>
(y, (snd a, snd b)) \<leadsto>
(mkex2 A B \<cdot> xs \<cdot> (TL \<cdot> exA) \<cdot> (TL \<cdot> exB)) (snd a) (snd b)))
else
(case HD \<cdot> exA of
UU \<Rightarrow> UU
| Def a \<Rightarrow> (y, (snd a, t)) \<leadsto> (mkex2 A B \<cdot> xs \<cdot> (TL \<cdot> exA) \<cdot> exB) (snd a) t))
else
(if y \<in> act B then
(case HD \<cdot> exB of
UU \<Rightarrow> UU
| Def b \<Rightarrow> (y, (s, snd b)) \<leadsto> (mkex2 A B \<cdot> xs \<cdot> exA \<cdot> (TL \<cdot> exB)) s (snd b))
else UU)))))"
|
mkex2 ?A ?B = (\<Lambda> sch exA exB. (\<lambda>s t. case sch of nil \<Rightarrow> nil | x ## xs \<Rightarrow> case x of \<bottom> \<Rightarrow> \<bottom> | Def y \<Rightarrow> if y \<in> act ?A then if y \<in> act ?B then case HD\<cdot>exA of \<bottom> \<Rightarrow> \<bottom> | Def a \<Rightarrow> case HD\<cdot>exB of \<bottom> \<Rightarrow> \<bottom> | Def b \<Rightarrow> (y, snd a, snd b)\<leadsto>(mkex2 ?A ?B\<cdot>xs\<cdot>(TL\<cdot>exA)\<cdot>(TL\<cdot>exB)) (snd a) (snd b) else case HD\<cdot>exA of \<bottom> \<Rightarrow> \<bottom> | Def a \<Rightarrow> (y, snd a, t)\<leadsto>(mkex2 ?A ?B\<cdot>xs\<cdot>(TL\<cdot>exA)\<cdot>exB) (snd a) t else if y \<in> act ?B then case HD\<cdot>exB of \<bottom> \<Rightarrow> \<bottom> | Def b \<Rightarrow> (y, s, snd b)\<leadsto>(mkex2 ?A ?B\<cdot>xs\<cdot>exA\<cdot>(TL\<cdot>exB)) s (snd b) else \<bottom>))
|
?H1 x_1 x_2 = ?H2 (\<lambda>y_0. ?H2 (\<lambda>y_1. ?H2 (\<lambda>y_2 y_3 y_4. ?H3 (?H3 (?H3 ?H4 ?H5) (?H2 (\<lambda>y_5. ?H2 (\<lambda>y_6. ?H6 ?H7 (\<lambda>y_7. if y_7 \<in> ?H8 x_1 then if y_7 \<in> ?H8 x_2 then ?H6 ?H7 (\<lambda>y_8. ?H6 ?H7 (\<lambda>y_9. ?H9 (y_7, ?H10 y_8, ?H10 y_9) (?H3 (?H3 (?H3 (?H1 x_1 x_2) y_6) (?H3 ?H11 y_1)) (?H3 ?H11 y_2) (?H10 y_8) (?H10 y_9))) (?H3 ?H12 y_2)) (?H3 ?H12 y_1) else ?H6 ?H7 (\<lambda>y_10. ?H9 (y_7, ?H10 y_10, y_4) (?H3 (?H3 (?H3 (?H1 x_1 x_2) y_6) (?H3 ?H11 y_1)) y_2 (?H10 y_10) y_4)) (?H3 ?H12 y_1) else if y_7 \<in> ?H8 x_2 then ?H6 ?H7 (\<lambda>y_11. ?H9 (y_7, y_3, ?H10 y_11) (?H3 (?H3 (?H3 (?H1 x_1 x_2) y_6) y_1) (?H3 ?H11 y_2) y_3 (?H10 y_11))) (?H3 ?H12 y_2) else ?H7) y_5)))) y_0)))
|
[
"Seq.seq.HD",
"Seq.seq.TL",
"Product_Type.prod.snd",
"Sequence.Consq_syn",
"Automata.act",
"Pcpo.pcpo_class.bottom",
"Lift.lift.case_lift",
"Seq.seq.nil",
"Seq.seq.seq_case",
"Cfun.cfun.Rep_cfun",
"Cfun.cfun.Abs_cfun",
"CompoScheds.mkex2"
] |
[
"'a seq \\<rightarrow> 'a",
"'a seq \\<rightarrow> 'a seq",
"'a \\<times> 'b \\<Rightarrow> 'b",
"'a \\<Rightarrow> 'a lift seq \\<Rightarrow> 'a lift seq",
"('a set \\<times> 'a set \\<times> 'a set) \\<times> 'b set \\<times> ('b \\<times> 'a \\<times> 'b) set \\<times> 'a set set \\<times> 'a set set \\<Rightarrow> 'a set",
"'a",
"'a \\<Rightarrow> ('b \\<Rightarrow> 'a) \\<Rightarrow> 'b lift \\<Rightarrow> 'a",
"'a seq",
"'a \\<rightarrow> ('b \\<rightarrow> 'b seq \\<rightarrow> 'a) \\<rightarrow> 'b seq \\<rightarrow> 'a",
"('a \\<rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'b",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<rightarrow> 'b",
"('a set \\<times> 'a set \\<times> 'a set) \\<times> 'b set \\<times> ('b \\<times> 'a \\<times> 'b) set \\<times> 'a set set \\<times> 'a set set \\<Rightarrow> ('a set \\<times> 'a set \\<times> 'a set) \\<times> 'c set \\<times> ('c \\<times> 'a \\<times> 'c) set \\<times> 'a set set \\<times> 'a set set \\<Rightarrow> 'a lift seq \\<rightarrow> ('a \\<times> 'b) lift seq \\<rightarrow> ('a \\<times> 'c) lift seq \\<rightarrow> 'b \\<Rightarrow> 'c \\<Rightarrow> ('a \\<times> 'b \\<times> 'c) lift seq"
] |
[
"datatype 'a seq = Empty | Seq 'a \"'a seq\"",
"datatype 'a seq = Empty | Seq 'a \"'a seq\"",
"definition \"prod = {f. \\<exists>a b. f = Pair_Rep (a::'a) (b::'b)}\"",
"class pcpo = cpo +\n assumes least: \"\\<exists>x. \\<forall>y. x \\<sqsubseteq> y\"\nbegin",
"datatype 'a seq = Empty | Seq 'a \"'a seq\"",
"datatype 'a seq = Empty | Seq 'a \"'a seq\""
] |
template
|
###lemma
mkex2 ?A ?B = (\<Lambda> sch exA exB. (\<lambda>s t. case sch of nil \<Rightarrow> nil | x ## xs \<Rightarrow> case x of \<bottom> \<Rightarrow> \<bottom> | Def y \<Rightarrow> if y \<in> act ?A then if y \<in> act ?B then case HD\<cdot>exA of \<bottom> \<Rightarrow> \<bottom> | Def a \<Rightarrow> case HD\<cdot>exB of \<bottom> \<Rightarrow> \<bottom> | Def b \<Rightarrow> (y, snd a, snd b)\<leadsto>(mkex2 ?A ?B\<cdot>xs\<cdot>(TL\<cdot>exA)\<cdot>(TL\<cdot>exB)) (snd a) (snd b) else case HD\<cdot>exA of \<bottom> \<Rightarrow> \<bottom> | Def a \<Rightarrow> (y, snd a, t)\<leadsto>(mkex2 ?A ?B\<cdot>xs\<cdot>(TL\<cdot>exA)\<cdot>exB) (snd a) t else if y \<in> act ?B then case HD\<cdot>exB of \<bottom> \<Rightarrow> \<bottom> | Def b \<Rightarrow> (y, s, snd b)\<leadsto>(mkex2 ?A ?B\<cdot>xs\<cdot>exA\<cdot>(TL\<cdot>exB)) s (snd b) else \<bottom>))
###symbols
Seq.seq.HD :::: 'a seq \<rightarrow> 'a
Seq.seq.TL :::: 'a seq \<rightarrow> 'a seq
Product_Type.prod.snd :::: 'a \<times> 'b \<Rightarrow> 'b
Sequence.Consq_syn :::: 'a \<Rightarrow> 'a lift seq \<Rightarrow> 'a lift seq
Automata.act :::: ('a set \<times> 'a set \<times> 'a set) \<times> 'b set \<times> ('b \<times> 'a \<times> 'b) set \<times> 'a set set \<times> 'a set set \<Rightarrow> 'a set
Pcpo.pcpo_class.bottom :::: 'a
Lift.lift.case_lift :::: 'a \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b lift \<Rightarrow> 'a
Seq.seq.nil :::: 'a seq
Seq.seq.seq_case :::: 'a \<rightarrow> ('b \<rightarrow> 'b seq \<rightarrow> 'a) \<rightarrow> 'b seq \<rightarrow> 'a
Cfun.cfun.Rep_cfun :::: ('a \<rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b
Cfun.cfun.Abs_cfun :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<rightarrow> 'b
CompoScheds.mkex2 :::: ('a set \<times> 'a set \<times> 'a set) \<times> 'b set \<times> ('b \<times> 'a \<times> 'b) set \<times> 'a set set \<times> 'a set set \<Rightarrow> ('a set \<times> 'a set \<times> 'a set) \<times> 'c set \<times> ('c \<times> 'a \<times> 'c) set \<times> 'a set set \<times> 'a set set \<Rightarrow> 'a lift seq \<rightarrow> ('a \<times> 'b) lift seq \<rightarrow> ('a \<times> 'c) lift seq \<rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('a \<times> 'b \<times> 'c) lift seq
###defs
datatype 'a seq = Empty | Seq 'a "'a seq"
datatype 'a seq = Empty | Seq 'a "'a seq"
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
class pcpo = cpo +
assumes least: "\<exists>x. \<forall>y. x \<sqsubseteq> y"
begin
datatype 'a seq = Empty | Seq 'a "'a seq"
datatype 'a seq = Empty | Seq 'a "'a seq"
|
###output
?H1 x_1 x_2 = ?H2 (\<lambda>y_0. ?H2 (\<lambda>y_1. ?H2 (\<lambda>y_2 y_3 y_4. ?H3 (?H3 (?H3 ?H4 ?H5) (?H2 (\<lambda>y_5. ?H2 (\<lambda>y_6. ?H6 ?H7 (\<lambda>y_7. if y_7 \<in> ?H8 x_1 then if y_7 \<in> ?H8 x_2 then ?H6 ?H7 (\<lambda>y_8. ?H6 ?H7 (\<lambda>y_9. ?H9 (y_7, ?H10 y_8, ?H10 y_9) (?H3 (?H3 (?H3 (?H1 x_1 x_2) y_6) (?H3 ?H11 y_1)) (?H3 ?H11 y_2) (?H10 y_8) (?H10 y_9))) (?H3 ?H12 y_2)) (?H3 ?H12 y_1) else ?H6 ?H7 (\<lambda>y_10. ?H9 (y_7, ?H10 y_10, y_4) (?H3 (?H3 (?H3 (?H1 x_1 x_2) y_6) (?H3 ?H11 y_1)) y_2 (?H10 y_10) y_4)) (?H3 ?H12 y_1) else if y_7 \<in> ?H8 x_2 then ?H6 ?H7 (\<lambda>y_11. ?H9 (y_7, y_3, ?H10 y_11) (?H3 (?H3 (?H3 (?H1 x_1 x_2) y_6) y_1) (?H3 ?H11 y_2) y_3 (?H10 y_11))) (?H3 ?H12 y_2) else ?H7) y_5)))) y_0)))###end
|
Library/Disjoint_Sets
|
Disjoint_Sets.disjoint_family_on_mono
|
lemma disjoint_family_on_mono:
"A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A"
|
?A \<subseteq> ?B \<Longrightarrow> disjoint_family_on ?f ?B \<Longrightarrow> disjoint_family_on ?f ?A
|
\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_2\<rbrakk> \<Longrightarrow> ?H2 x_3 x_1
|
[
"Disjoint_Sets.disjoint_family_on",
"Set.subset_eq"
] |
[
"('a \\<Rightarrow> 'b set) \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool"
] |
[
"definition disjoint_family_on :: \"('i \\<Rightarrow> 'a set) \\<Rightarrow> 'i set \\<Rightarrow> bool\" where\n \"disjoint_family_on A S \\<longleftrightarrow> (\\<forall>m\\<in>S. \\<forall>n\\<in>S. m \\<noteq> n \\<longrightarrow> A m \\<inter> A n = {})\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
template
|
###lemma
?A \<subseteq> ?B \<Longrightarrow> disjoint_family_on ?f ?B \<Longrightarrow> disjoint_family_on ?f ?A
###symbols
Disjoint_Sets.disjoint_family_on :::: ('a \<Rightarrow> 'b set) \<Rightarrow> 'a set \<Rightarrow> bool
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
###defs
definition disjoint_family_on :: "('i \<Rightarrow> 'a set) \<Rightarrow> 'i set \<Rightarrow> bool" where
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_2\<rbrakk> \<Longrightarrow> ?H2 x_3 x_1###end
|
Library/Extended_Real
|
Extended_Real.ereal_power_numeral
|
lemma ereal_power_numeral[simp]:
"(numeral num :: ereal) ^ n = ereal (numeral num ^ n)"
|
numeral ?num ^ ?n = ereal (numeral ?num ^ ?n)
|
?H1 (?H2 x_1) x_2 = ?H3 (?H1 (?H2 x_1) x_2)
|
[
"Extended_Real.ereal.ereal",
"Num.numeral_class.numeral",
"Power.power_class.power"
] |
[
"real \\<Rightarrow> ereal",
"num \\<Rightarrow> 'a",
"'a \\<Rightarrow> nat \\<Rightarrow> 'a"
] |
[
"datatype ereal = ereal real | PInfty | MInfty",
"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\"",
"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
numeral ?num ^ ?n = ereal (numeral ?num ^ ?n)
###symbols
Extended_Real.ereal.ereal :::: real \<Rightarrow> ereal
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Power.power_class.power :::: 'a \<Rightarrow> nat \<Rightarrow> 'a
###defs
datatype ereal = ereal real | PInfty | MInfty
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"
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 x_1) x_2 = ?H3 (?H1 (?H2 x_1) x_2)###end
|
Fun
|
Fun.bij_betw_imp_inj_on
|
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
|
bij_betw ?f ?A ?B \<Longrightarrow> inj_on ?f ?A
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 x_1 x_2
|
[
"Fun.inj_on",
"Fun.bij_betw"
] |
[
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> bool",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set \\<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)\"",
"definition bij_betw :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set \\<Rightarrow> bool\" \\<comment> \\<open>bijective\\<close>\n where \"bij_betw f A B \\<longleftrightarrow> inj_on f A \\<and> f ` A = B\""
] |
template
|
###lemma
bij_betw ?f ?A ?B \<Longrightarrow> inj_on ?f ?A
###symbols
Fun.inj_on :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool
Fun.bij_betw :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<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)"
definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" \<comment> \<open>bijective\<close>
where "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
|
###output
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 x_1 x_2###end
|
TLA/Action
|
Action.actionD
|
lemma actionD [dest]: "\<turnstile> A \<Longrightarrow> (s,t) \<Turnstile> A"
|
\<turnstile> ?A \<Longrightarrow> ?A (?s, ?t)
|
?H1 x_1 \<Longrightarrow> x_1 (x_2, x_3)
|
[
"Intensional.Valid"
] |
[
"('a \\<Rightarrow> bool) \\<Rightarrow> bool"
] |
[
"definition Valid :: \"('w::world) form \\<Rightarrow> bool\"\n where \"Valid A \\<equiv> \\<forall>w. A w\""
] |
template
|
###lemma
\<turnstile> ?A \<Longrightarrow> ?A (?s, ?t)
###symbols
Intensional.Valid :::: ('a \<Rightarrow> bool) \<Rightarrow> bool
###defs
definition Valid :: "('w::world) form \<Rightarrow> bool"
where "Valid A \<equiv> \<forall>w. A w"
|
###output
?H1 x_1 \<Longrightarrow> x_1 (x_2, x_3)###end
|
Analysis/Sigma_Algebra
|
Sigma_Algebra.sigma_sets_UNION
|
lemma sigma_sets_UNION:
"countable B \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A) \<Longrightarrow> \<Union> B \<in> sigma_sets X A"
|
countable ?B \<Longrightarrow> (\<And>b. b \<in> ?B \<Longrightarrow> b \<in> sigma_sets ?X ?A) \<Longrightarrow> \<Union> ?B \<in> sigma_sets ?X ?A
|
\<lbrakk> ?H1 x_1; \<And>y_0. y_0 \<in> x_1 \<Longrightarrow> y_0 \<in> ?H2 x_2 x_3\<rbrakk> \<Longrightarrow> ?H3 x_1 \<in> ?H2 x_2 x_3
|
[
"Complete_Lattices.Union",
"Sigma_Algebra.sigma_sets",
"Countable_Set.countable"
] |
[
"'a set set \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set set \\<Rightarrow> 'a set set",
"'a set \\<Rightarrow> bool"
] |
[
"abbreviation Union :: \"'a set set \\<Rightarrow> 'a set\" (\"\\<Union>\")\n where \"\\<Union>S \\<equiv> \\<Squnion>S\"",
"definition countable :: \"'a set \\<Rightarrow> bool\" where\n \"countable S \\<longleftrightarrow> (\\<exists>f::'a \\<Rightarrow> nat. inj_on f S)\""
] |
template
|
###lemma
countable ?B \<Longrightarrow> (\<And>b. b \<in> ?B \<Longrightarrow> b \<in> sigma_sets ?X ?A) \<Longrightarrow> \<Union> ?B \<in> sigma_sets ?X ?A
###symbols
Complete_Lattices.Union :::: 'a set set \<Rightarrow> 'a set
Sigma_Algebra.sigma_sets :::: 'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set
Countable_Set.countable :::: 'a set \<Rightarrow> bool
###defs
abbreviation Union :: "'a set set \<Rightarrow> 'a set" ("\<Union>")
where "\<Union>S \<equiv> \<Squnion>S"
definition countable :: "'a set \<Rightarrow> bool" where
"countable S \<longleftrightarrow> (\<exists>f::'a \<Rightarrow> nat. inj_on f S)"
|
###output
\<lbrakk> ?H1 x_1; \<And>y_0. y_0 \<in> x_1 \<Longrightarrow> y_0 \<in> ?H2 x_2 x_3\<rbrakk> \<Longrightarrow> ?H3 x_1 \<in> ?H2 x_2 x_3###end
|
Relation
|
Relation.transp_onI
|
lemma transp_onI:
"(\<And>x y z. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> z \<in> A \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z) \<Longrightarrow> transp_on A R"
|
(\<And>x y z. x \<in> ?A \<Longrightarrow> y \<in> ?A \<Longrightarrow> z \<in> ?A \<Longrightarrow> ?R x y \<Longrightarrow> ?R y z \<Longrightarrow> ?R x z) \<Longrightarrow> transp_on ?A ?R
|
(\<And>y_0 y_1 y_2. \<lbrakk>y_0 \<in> x_1; y_1 \<in> x_1; y_2 \<in> x_1; x_2 y_0 y_1; x_2 y_1 y_2\<rbrakk> \<Longrightarrow> x_2 y_0 y_2) \<Longrightarrow> ?H1 x_1 x_2
|
[
"Relation.transp_on"
] |
[
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> bool"
] |
[
"definition transp_on :: \"'a set \\<Rightarrow> ('a \\<Rightarrow> 'a \\<Rightarrow> bool) \\<Rightarrow> bool\" where\n \"transp_on A R \\<longleftrightarrow> (\\<forall>x \\<in> A. \\<forall>y \\<in> A. \\<forall>z \\<in> A. R x y \\<longrightarrow> R y z \\<longrightarrow> R x z)\""
] |
template
|
###lemma
(\<And>x y z. x \<in> ?A \<Longrightarrow> y \<in> ?A \<Longrightarrow> z \<in> ?A \<Longrightarrow> ?R x y \<Longrightarrow> ?R y z \<Longrightarrow> ?R x z) \<Longrightarrow> transp_on ?A ?R
###symbols
Relation.transp_on :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool
###defs
definition transp_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
"transp_on A R \<longleftrightarrow> (\<forall>x \<in> A. \<forall>y \<in> A. \<forall>z \<in> A. R x y \<longrightarrow> R y z \<longrightarrow> R x z)"
|
###output
(\<And>y_0 y_1 y_2. \<lbrakk>y_0 \<in> x_1; y_1 \<in> x_1; y_2 \<in> x_1; x_2 y_0 y_1; x_2 y_1 y_2\<rbrakk> \<Longrightarrow> x_2 y_0 y_2) \<Longrightarrow> ?H1 x_1 x_2###end
|
Analysis/Affine
|
Affine.affine_dependent_explicit
| null |
affine_dependent ?p = (\<exists>S U. finite S \<and> S \<subseteq> ?p \<and> sum U S = 0 \<and> (\<exists>v\<in>S. U v \<noteq> 0) \<and> (\<Sum>v\<in>S. U v *\<^sub>R v) = (0:: ?'a))
|
?H1 x_1 = (\<exists>y_0 y_1. ?H2 y_0 \<and> ?H3 y_0 x_1 \<and> ?H4 y_1 y_0 = ?H5 \<and> (\<exists>y_2\<in>y_0. y_1 y_2 \<noteq> ?H5) \<and> ?H4 (\<lambda>y_3. ?H6 (y_1 y_3) y_3) y_0 = ?H5)
|
[
"Real_Vector_Spaces.scaleR_class.scaleR",
"Groups.zero_class.zero",
"Groups_Big.comm_monoid_add_class.sum",
"Set.subset_eq",
"Finite_Set.finite",
"Affine.affine_dependent"
] |
[
"real \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a set \\<Rightarrow> bool",
"'a set \\<Rightarrow> bool"
] |
[
"class scaleR =\n fixes scaleR :: \"real \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixr \"*\\<^sub>R\" 75)\nbegin",
"class zero =\n fixes zero :: 'a (\"0\")",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] |
template
|
###lemma
affine_dependent ?p = (\<exists>S U. finite S \<and> S \<subseteq> ?p \<and> sum U S = 0 \<and> (\<exists>v\<in>S. U v \<noteq> 0) \<and> (\<Sum>v\<in>S. U v *\<^sub>R v) = (0:: ?'a))
###symbols
Real_Vector_Spaces.scaleR_class.scaleR :::: real \<Rightarrow> 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Groups_Big.comm_monoid_add_class.sum :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Finite_Set.finite :::: 'a set \<Rightarrow> bool
Affine.affine_dependent :::: 'a set \<Rightarrow> bool
###defs
class scaleR =
fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
begin
class zero =
fixes zero :: 'a ("0")
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
|
###output
?H1 x_1 = (\<exists>y_0 y_1. ?H2 y_0 \<and> ?H3 y_0 x_1 \<and> ?H4 y_1 y_0 = ?H5 \<and> (\<exists>y_2\<in>y_0. y_1 y_2 \<noteq> ?H5) \<and> ?H4 (\<lambda>y_3. ?H6 (y_1 y_3) y_3) y_0 = ?H5)###end
|
UNITY/Guar
|
Guar.uv1
|
lemma uv1:
assumes "uv_prop X"
and "finite GG"
and "GG \<subseteq> X"
and "OK GG (%G. G)"
shows "(\<Squnion>G \<in> GG. G) \<in> X"
|
uv_prop ?X \<Longrightarrow> finite ?GG \<Longrightarrow> ?GG \<subseteq> ?X \<Longrightarrow> OK ?GG (\<lambda>G. G) \<Longrightarrow> (\<Squnion>G\<in> ?GG. G) \<in> ?X
|
\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 x_1; ?H4 x_2 (\<lambda>y_0. y_0)\<rbrakk> \<Longrightarrow> ?H5 x_2 (\<lambda>y_1. y_1) \<in> x_1
|
[
"Union.JOIN",
"Union.OK",
"Set.subset_eq",
"Finite_Set.finite",
"Guar.uv_prop"
] |
[
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b program) \\<Rightarrow> 'b program",
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b program) \\<Rightarrow> bool",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool",
"'a set \\<Rightarrow> bool",
"'a program set \\<Rightarrow> bool"
] |
[
"definition\n JOIN :: \"['a set, 'a => 'b program] => 'b program\"\n where \"JOIN I F = mk_program (\\<Inter>i \\<in> I. Init (F i), \\<Union>i \\<in> I. Acts (F i),\n \\<Inter>i \\<in> I. AllowedActs (F i))\"",
"definition\n OK :: \"['a set, 'a => 'b program] => bool\"\n where \"OK I F = (\\<forall>i \\<in> I. \\<forall>j \\<in> I-{i}. Acts (F i) \\<subseteq> AllowedActs (F j))\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin",
"definition uv_prop :: \"'a program set => bool\" where\n \"uv_prop X == SKIP \\<in> X & (\\<forall>F G. F ok G --> F \\<in> X & G \\<in> X --> (F\\<squnion>G) \\<in> X)\""
] |
template
|
###lemma
uv_prop ?X \<Longrightarrow> finite ?GG \<Longrightarrow> ?GG \<subseteq> ?X \<Longrightarrow> OK ?GG (\<lambda>G. G) \<Longrightarrow> (\<Squnion>G\<in> ?GG. G) \<in> ?X
###symbols
Union.JOIN :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'b program) \<Rightarrow> 'b program
Union.OK :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'b program) \<Rightarrow> bool
Set.subset_eq :::: 'a set \<Rightarrow> 'a set \<Rightarrow> bool
Finite_Set.finite :::: 'a set \<Rightarrow> bool
Guar.uv_prop :::: 'a program set \<Rightarrow> bool
###defs
definition
JOIN :: "['a set, 'a => 'b program] => 'b program"
where "JOIN I F = mk_program (\<Inter>i \<in> I. Init (F i), \<Union>i \<in> I. Acts (F i),
\<Inter>i \<in> I. AllowedActs (F i))"
definition
OK :: "['a set, 'a => 'b program] => bool"
where "OK I F = (\<forall>i \<in> I. \<forall>j \<in> I-{i}. Acts (F i) \<subseteq> AllowedActs (F j))"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
definition uv_prop :: "'a program set => bool" where
"uv_prop X == SKIP \<in> X & (\<forall>F G. F ok G --> F \<in> X & G \<in> X --> (F\<squnion>G) \<in> X)"
|
###output
\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 x_1; ?H4 x_2 (\<lambda>y_0. y_0)\<rbrakk> \<Longrightarrow> ?H5 x_2 (\<lambda>y_1. y_1) \<in> x_1###end
|
Library/Stream
|
Stream.sset_range
|
lemma sset_range: "sset s = range (snth s)"
|
sset ?s = range ((!!) ?s)
|
?H1 x_1 = ?H2 (?H3 x_1)
|
[
"Stream.snth",
"Set.range",
"Stream.stream.sset"
] |
[
"'a stream \\<Rightarrow> nat \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b set",
"'a stream \\<Rightarrow> 'a set"
] |
[
"primrec snth :: \"'a stream \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixl \\<open>!!\\<close> 100) where\n \"s !! 0 = shd s\"\n| \"s !! Suc n = stl s !! n\"",
"abbreviation range :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b set\" \\<comment> \\<open>of function\\<close>\n where \"range f \\<equiv> f ` UNIV\"",
"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
sset ?s = range ((!!) ?s)
###symbols
Stream.snth :::: 'a stream \<Rightarrow> nat \<Rightarrow> 'a
Set.range :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'b set
Stream.stream.sset :::: 'a stream \<Rightarrow> 'a set
###defs
primrec snth :: "'a stream \<Rightarrow> nat \<Rightarrow> 'a" (infixl \<open>!!\<close> 100) where
"s !! 0 = shd s"
| "s !! Suc n = stl s !! n"
abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set" \<comment> \<open>of function\<close>
where "range f \<equiv> f ` UNIV"
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 (?H3 x_1)###end
|
Bit_Operations
|
Bit_Operations.AND_upper1''
|
lemma AND_upper1'' [simp]: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
\<open>y AND x < z\<close> if \<open>0 \<le> y\<close> \<open>y < z\<close> for x y z :: int
|
0 \<le> ?y \<Longrightarrow> ?y < ?z \<Longrightarrow> and ?y ?x < ?z
|
\<lbrakk> ?H1 \<le> x_1; x_1 < x_2\<rbrakk> \<Longrightarrow> ?H2 x_1 x_3 < x_2
|
[
"Bit_Operations.semiring_bit_operations_class.and",
"Groups.zero_class.zero"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a"
] |
[
"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",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
0 \<le> ?y \<Longrightarrow> ?y < ?z \<Longrightarrow> and ?y ?x < ?z
###symbols
Bit_Operations.semiring_bit_operations_class.and :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
###defs
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
class zero =
fixes zero :: 'a ("0")
|
###output
\<lbrakk> ?H1 \<le> x_1; x_1 < x_2\<rbrakk> \<Longrightarrow> ?H2 x_1 x_3 < x_2###end
|
Hoare_Parallel/OG_Tactics
|
OG_Tactics.my_simp_list(61)
| null |
atom_com (While ?b ?i ?c) = atom_com ?c
|
?H1 (?H2 x_1 x_2 x_3) = ?H1 x_3
|
[
"OG_Com.com.While",
"OG_Com.atom_com"
] |
[
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a com \\<Rightarrow> 'a com",
"'a com \\<Rightarrow> bool"
] |
[
"primrec atom_com :: \"'a com \\<Rightarrow> bool\" where\n \"atom_com (Parallel Ts) = False\"\n| \"atom_com (Basic f) = True\"\n| \"atom_com (Seq c1 c2) = (atom_com c1 \\<and> atom_com c2)\"\n| \"atom_com (Cond b c1 c2) = (atom_com c1 \\<and> atom_com c2)\"\n| \"atom_com (While b i c) = atom_com c\""
] |
template
|
###lemma
atom_com (While ?b ?i ?c) = atom_com ?c
###symbols
OG_Com.com.While :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a com \<Rightarrow> 'a com
OG_Com.atom_com :::: 'a com \<Rightarrow> bool
###defs
primrec atom_com :: "'a com \<Rightarrow> bool" where
"atom_com (Parallel Ts) = False"
| "atom_com (Basic f) = True"
| "atom_com (Seq c1 c2) = (atom_com c1 \<and> atom_com c2)"
| "atom_com (Cond b c1 c2) = (atom_com c1 \<and> atom_com c2)"
| "atom_com (While b i c) = atom_com c"
|
###output
?H1 (?H2 x_1 x_2 x_3) = ?H1 x_3###end
|
Decision_Procs/Cooper
|
Cooper.conj_qf
|
lemma conj_qf: "qfree p \<Longrightarrow> qfree q \<Longrightarrow> qfree (conj p q)"
|
qfree ?p \<Longrightarrow> qfree ?q \<Longrightarrow> qfree (Cooper.conj ?p ?q)
|
\<lbrakk> ?H1 x_1; ?H1 x_2\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_1 x_2)
|
[
"Cooper.conj",
"Cooper.qfree"
] |
[
"fm \\<Rightarrow> fm \\<Rightarrow> fm",
"fm \\<Rightarrow> bool"
] |
[
"definition conj :: \"fm \\<Rightarrow> fm \\<Rightarrow> fm\"\n where \"conj p q =\n (if p = F \\<or> q = F then F\n else if p = T then q\n else if q = T then p\n else And p q)\"",
"fun qfree :: \"fm \\<Rightarrow> bool\" \\<comment> \\<open>Quantifier freeness\\<close>\n where\n \"qfree (E p) \\<longleftrightarrow> False\"\n | \"qfree (A p) \\<longleftrightarrow> False\"\n | \"qfree (Not p) \\<longleftrightarrow> qfree p\"\n | \"qfree (And p q) \\<longleftrightarrow> qfree p \\<and> qfree q\"\n | \"qfree (Or p q) \\<longleftrightarrow> qfree p \\<and> qfree q\"\n | \"qfree (Imp p q) \\<longleftrightarrow> qfree p \\<and> qfree q\"\n | \"qfree (Iff p q) \\<longleftrightarrow> qfree p \\<and> qfree q\"\n | \"qfree p \\<longleftrightarrow> True\""
] |
template
|
###lemma
qfree ?p \<Longrightarrow> qfree ?q \<Longrightarrow> qfree (Cooper.conj ?p ?q)
###symbols
Cooper.conj :::: fm \<Rightarrow> fm \<Rightarrow> fm
Cooper.qfree :::: fm \<Rightarrow> bool
###defs
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
where "conj p q =
(if p = F \<or> q = F then F
else if p = T then q
else if q = T then p
else And p q)"
fun qfree :: "fm \<Rightarrow> bool" \<comment> \<open>Quantifier freeness\<close>
where
"qfree (E p) \<longleftrightarrow> False"
| "qfree (A p) \<longleftrightarrow> False"
| "qfree (Not p) \<longleftrightarrow> qfree p"
| "qfree (And p q) \<longleftrightarrow> qfree p \<and> qfree q"
| "qfree (Or p q) \<longleftrightarrow> qfree p \<and> qfree q"
| "qfree (Imp p q) \<longleftrightarrow> qfree p \<and> qfree q"
| "qfree (Iff p q) \<longleftrightarrow> qfree p \<and> qfree q"
| "qfree p \<longleftrightarrow> True"
|
###output
\<lbrakk> ?H1 x_1; ?H1 x_2\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_1 x_2)###end
|
Bali/DeclConcepts
|
DeclConcepts.is_static_fieldm_simp
|
lemma is_static_fieldm_simp[simp]: "is_static (fieldm n f) = is_static f"
|
is_static (fieldm ?n ?f) = is_static ?f
|
?H1 (?H2 x_1 x_2) = ?H1 x_2
|
[
"DeclConcepts.fieldm",
"DeclConcepts.has_static_class.is_static"
] |
[
"vname \\<Rightarrow> qtname \\<times> Decl.field \\<Rightarrow> qtname \\<times> memberdecl",
"'a \\<Rightarrow> bool"
] |
[
"definition\n fieldm :: \"vname \\<Rightarrow> (qtname \\<times> field) \\<Rightarrow> (qtname \\<times> memberdecl)\"\n where \"fieldm n f = (declclass f, fdecl (n, fld f))\"",
"class has_static =\n fixes is_static :: \"'a \\<Rightarrow> bool\""
] |
template
|
###lemma
is_static (fieldm ?n ?f) = is_static ?f
###symbols
DeclConcepts.fieldm :::: vname \<Rightarrow> qtname \<times> Decl.field \<Rightarrow> qtname \<times> memberdecl
DeclConcepts.has_static_class.is_static :::: 'a \<Rightarrow> bool
###defs
definition
fieldm :: "vname \<Rightarrow> (qtname \<times> field) \<Rightarrow> (qtname \<times> memberdecl)"
where "fieldm n f = (declclass f, fdecl (n, fld f))"
class has_static =
fixes is_static :: "'a \<Rightarrow> bool"
|
###output
?H1 (?H2 x_1 x_2) = ?H1 x_2###end
|
Set
|
Set.UnE
|
lemma UnE [elim!]: "c \<in> A \<union> B \<Longrightarrow> (c \<in> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
|
?c \<in> ?A \<union> ?B \<Longrightarrow> (?c \<in> ?A \<Longrightarrow> ?P) \<Longrightarrow> (?c \<in> ?B \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk>x_1 \<in> ?H1 x_2 x_3; x_1 \<in> x_2 \<Longrightarrow> x_4; x_1 \<in> x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Set.union"
] |
[
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set"
] |
[
"abbreviation union :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<union>\" 65)\n where \"union \\<equiv> sup\""
] |
template
|
###lemma
?c \<in> ?A \<union> ?B \<Longrightarrow> (?c \<in> ?A \<Longrightarrow> ?P) \<Longrightarrow> (?c \<in> ?B \<Longrightarrow> ?P) \<Longrightarrow> ?P
###symbols
Set.union :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
###defs
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65)
where "union \<equiv> sup"
|
###output
\<lbrakk>x_1 \<in> ?H1 x_2 x_3; x_1 \<in> x_2 \<Longrightarrow> x_4; x_1 \<in> x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4###end
|
UNITY/Transformers
|
Transformers.wens_single_finite_Suc
|
lemma wens_single_finite_Suc:
"single_valued act
==> wens_single_finite act B (Suc k) =
wens_single_finite act B k \<union> wp act (wens_single_finite act B k)"
|
single_valued ?act \<Longrightarrow> wens_single_finite ?act ?B (Suc ?k) = wens_single_finite ?act ?B ?k \<union> wp ?act (wens_single_finite ?act ?B ?k)
|
?H1 x_1 \<Longrightarrow> ?H2 x_1 x_2 (?H3 x_3) = ?H4 (?H2 x_1 x_2 x_3) (?H5 x_1 (?H2 x_1 x_2 x_3))
|
[
"Transformers.wp",
"Set.union",
"Nat.Suc",
"Transformers.wens_single_finite",
"Relation.single_valued"
] |
[
"('a \\<times> 'a) set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"nat \\<Rightarrow> nat",
"('a \\<times> 'a) set \\<Rightarrow> 'a set \\<Rightarrow> nat \\<Rightarrow> 'a set",
"('a \\<times> 'b) set \\<Rightarrow> bool"
] |
[
"definition wp :: \"[('a*'a) set, 'a set] => 'a set\" where \n \\<comment> \\<open>Dijkstra's weakest-precondition operator (for an individual command)\\<close>\n \"wp act B == - (act\\<inverse> `` (-B))\"",
"abbreviation union :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<union>\" 65)\n where \"union \\<equiv> sup\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"definition wens_single_finite :: \"[('a*'a) set, 'a set, nat] => 'a set\" where \n \"wens_single_finite act B k == \\<Union>i \\<in> atMost k. (wp act ^^ i) B\"",
"definition single_valued :: \"('a \\<times> 'b) set \\<Rightarrow> bool\"\n where \"single_valued r \\<longleftrightarrow> (\\<forall>x y. (x, y) \\<in> r \\<longrightarrow> (\\<forall>z. (x, z) \\<in> r \\<longrightarrow> y = z))\""
] |
template
|
###lemma
single_valued ?act \<Longrightarrow> wens_single_finite ?act ?B (Suc ?k) = wens_single_finite ?act ?B ?k \<union> wp ?act (wens_single_finite ?act ?B ?k)
###symbols
Transformers.wp :::: ('a \<times> 'a) set \<Rightarrow> 'a set \<Rightarrow> 'a set
Set.union :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
Nat.Suc :::: nat \<Rightarrow> nat
Transformers.wens_single_finite :::: ('a \<times> 'a) set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set
Relation.single_valued :::: ('a \<times> 'b) set \<Rightarrow> bool
###defs
definition wp :: "[('a*'a) set, 'a set] => 'a set" where
\<comment> \<open>Dijkstra's weakest-precondition operator (for an individual command)\<close>
"wp act B == - (act\<inverse> `` (-B))"
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65)
where "union \<equiv> sup"
definition Suc :: "nat \<Rightarrow> nat"
where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
definition wens_single_finite :: "[('a*'a) set, 'a set, nat] => 'a set" where
"wens_single_finite act B k == \<Union>i \<in> atMost k. (wp act ^^ i) B"
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
where "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
|
###output
?H1 x_1 \<Longrightarrow> ?H2 x_1 x_2 (?H3 x_3) = ?H4 (?H2 x_1 x_2 x_3) (?H5 x_1 (?H2 x_1 x_2 x_3))###end
|
Analysis/Elementary_Normed_Spaces
|
Elementary_Normed_Spaces.interior_translation_subtract
|
lemma interior_translation_subtract:
"interior ((\<lambda>x. x - a) ` S) = (\<lambda>x. x - a) ` interior S" for S :: "'a::real_normed_vector set"
|
interior ((\<lambda>x. x - ?a) ` ?S) = (\<lambda>x. x - ?a) ` interior ?S
|
?H1 (?H2 (\<lambda>y_0. ?H3 y_0 x_1) x_2) = ?H2 (\<lambda>y_1. ?H3 y_1 x_1) (?H1 x_2)
|
[
"Groups.minus_class.minus",
"Set.image",
"Elementary_Topology.interior"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set",
"'a set \\<Rightarrow> 'a set"
] |
[
"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}\""
] |
template
|
###lemma
interior ((\<lambda>x. x - ?a) ` ?S) = (\<lambda>x. x - ?a) ` interior ?S
###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Set.image :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set
Elementary_Topology.interior :::: 'a set \<Rightarrow> 'a set
###defs
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}"
|
###output
?H1 (?H2 (\<lambda>y_0. ?H3 y_0 x_1) x_2) = ?H2 (\<lambda>y_1. ?H3 y_1 x_1) (?H1 x_2)###end
|
Proofs/Extraction/Euclid
|
Euclidean_Algorithm.prime_int_iff
| null |
prime ?p = (0 < ?p \<and> prime_elem ?p)
|
?H1 x_1 = (?H2 < x_1 \<and> ?H3 x_1)
|
[
"Factorial_Ring.comm_semiring_1_class.prime_elem",
"Groups.zero_class.zero",
"Factorial_Ring.normalization_semidom_class.prime"
] |
[
"'a \\<Rightarrow> bool",
"'a",
"'a \\<Rightarrow> bool"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
prime ?p = (0 < ?p \<and> prime_elem ?p)
###symbols
Factorial_Ring.comm_semiring_1_class.prime_elem :::: 'a \<Rightarrow> bool
Groups.zero_class.zero :::: 'a
Factorial_Ring.normalization_semidom_class.prime :::: 'a \<Rightarrow> bool
###defs
class zero =
fixes zero :: 'a ("0")
|
###output
?H1 x_1 = (?H2 < x_1 \<and> ?H3 x_1)###end
|
Real
|
Real.floor_divide_real_eq_div
|
lemma floor_divide_real_eq_div:
assumes "0 \<le> b"
shows "\<lfloor>a / real_of_int b\<rfloor> = \<lfloor>a\<rfloor> div b"
|
0 \<le> ?b \<Longrightarrow> \<lfloor> ?a / real_of_int ?b\<rfloor> = \<lfloor> ?a\<rfloor> div ?b
|
?H1 \<le> x_1 \<Longrightarrow> ?H2 (?H3 x_2 (?H4 x_1)) = ?H5 (?H2 x_2) x_1
|
[
"Rings.divide_class.divide",
"Real.real_of_int",
"Fields.inverse_class.inverse_divide",
"Archimedean_Field.floor_ceiling_class.floor",
"Groups.zero_class.zero"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"int \\<Rightarrow> real",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> int",
"'a"
] |
[
"class divide =\n fixes divide :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"div\" 70)",
"abbreviation real_of_int :: \"int \\<Rightarrow> real\"\n where \"real_of_int \\<equiv> of_int\"",
"class inverse = divide +\n fixes inverse :: \"'a \\<Rightarrow> 'a\"\nbegin",
"class floor_ceiling = archimedean_field +\n fixes floor :: \"'a \\<Rightarrow> int\" (\"\\<lfloor>_\\<rfloor>\")\n assumes floor_correct: \"of_int \\<lfloor>x\\<rfloor> \\<le> x \\<and> x < of_int (\\<lfloor>x\\<rfloor> + 1)\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
template
|
###lemma
0 \<le> ?b \<Longrightarrow> \<lfloor> ?a / real_of_int ?b\<rfloor> = \<lfloor> ?a\<rfloor> div ?b
###symbols
Rings.divide_class.divide :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Real.real_of_int :::: int \<Rightarrow> real
Fields.inverse_class.inverse_divide :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Archimedean_Field.floor_ceiling_class.floor :::: 'a \<Rightarrow> int
Groups.zero_class.zero :::: 'a
###defs
class divide =
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
abbreviation real_of_int :: "int \<Rightarrow> real"
where "real_of_int \<equiv> of_int"
class inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin
class floor_ceiling = archimedean_field +
fixes floor :: "'a \<Rightarrow> int" ("\<lfloor>_\<rfloor>")
assumes floor_correct: "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)"
class zero =
fixes zero :: 'a ("0")
|
###output
?H1 \<le> x_1 \<Longrightarrow> ?H2 (?H3 x_2 (?H4 x_1)) = ?H5 (?H2 x_2) x_1###end
|
Complete_Lattices
|
Complete_Lattices.Inter_Un_subset
|
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
|
\<Inter> ?A \<union> \<Inter> ?B \<subseteq> \<Inter> (?A \<inter> ?B)
|
?H1 (?H2 (?H3 x_1) (?H3 x_2)) (?H3 (?H4 x_1 x_2))
|
[
"Set.inter",
"Complete_Lattices.Inter",
"Set.union",
"Set.subset_eq"
] |
[
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a set set \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a 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\"",
"abbreviation Inter :: \"'a set set \\<Rightarrow> 'a set\" (\"\\<Inter>\")\n where \"\\<Inter>S \\<equiv> \\<Sqinter>S\"",
"abbreviation union :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<union>\" 65)\n where \"union \\<equiv> sup\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
template
|
###lemma
\<Inter> ?A \<union> \<Inter> ?B \<subseteq> \<Inter> (?A \<inter> ?B)
###symbols
Set.inter :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set
Complete_Lattices.Inter :::: 'a set set \<Rightarrow> 'a set
Set.union :::: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a 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"
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" ("\<Inter>")
where "\<Inter>S \<equiv> \<Sqinter>S"
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65)
where "union \<equiv> sup"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
?H1 (?H2 (?H3 x_1) (?H3 x_2)) (?H3 (?H4 x_1 x_2))###end
|
Bali/TypeRel
|
TypeRel.subclseq_trans
|
lemma subclseq_trans: "\<lbrakk>G\<turnstile>A \<preceq>\<^sub>C B; G\<turnstile>B \<preceq>\<^sub>C C\<rbrakk> \<Longrightarrow> G\<turnstile>A \<preceq>\<^sub>C C"
|
?G\<turnstile> ?A\<preceq>\<^sub>C ?B \<Longrightarrow> ?G\<turnstile> ?B\<preceq>\<^sub>C ?C \<Longrightarrow> ?G\<turnstile> ?A\<preceq>\<^sub>C ?C
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H1 x_1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 x_4
|
[
"Decl.subclseq_syntax"
] |
[
"prog \\<Rightarrow> qtname \\<Rightarrow> qtname \\<Rightarrow> bool"
] |
[
"abbreviation\n subclseq_syntax :: \"prog => [qtname, qtname] => bool\" (\"_\\<turnstile>_\\<preceq>\\<^sub>C _\" [71,71,71] 70) \n where \"G\\<turnstile>C \\<preceq>\\<^sub>C D == (C,D) \\<in>(subcls1 G)\\<^sup>*\""
] |
template
|
###lemma
?G\<turnstile> ?A\<preceq>\<^sub>C ?B \<Longrightarrow> ?G\<turnstile> ?B\<preceq>\<^sub>C ?C \<Longrightarrow> ?G\<turnstile> ?A\<preceq>\<^sub>C ?C
###symbols
Decl.subclseq_syntax :::: prog \<Rightarrow> qtname \<Rightarrow> qtname \<Rightarrow> bool
###defs
abbreviation
subclseq_syntax :: "prog => [qtname, qtname] => bool" ("_\<turnstile>_\<preceq>\<^sub>C _" [71,71,71] 70)
where "G\<turnstile>C \<preceq>\<^sub>C D == (C,D) \<in>(subcls1 G)\<^sup>*"
|
###output
\<lbrakk> ?H1 x_1 x_2 x_3; ?H1 x_1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 x_4###end
|
Fun_Def_Base
|
Fun_Def_Base.termination_simp(12)
| null |
(?m = ?n) = (nat_of_natural ?m = nat_of_natural ?n)
|
(x_1 = x_2) = (?H1 x_1 = ?H1 x_2)
|
[
"Code_Numeral.natural.nat_of_natural"
] |
[
"natural \\<Rightarrow> nat"
] |
[
"typedef natural = \"UNIV :: nat set\"\n morphisms nat_of_natural natural_of_nat"
] |
template
|
###lemma
(?m = ?n) = (nat_of_natural ?m = nat_of_natural ?n)
###symbols
Code_Numeral.natural.nat_of_natural :::: natural \<Rightarrow> nat
###defs
typedef natural = "UNIV :: nat set"
morphisms nat_of_natural natural_of_nat
|
###output
(x_1 = x_2) = (?H1 x_1 = ?H1 x_2)###end
|
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