defs listlengths 0 26 | symbols listlengths 1 9 | lemma stringlengths 17 541 | theory_file stringlengths 43 62 | prediction stringlengths 32 658 | passes_generation bool 2
classes | passes_syntax bool 2
classes | passes_counterexample bool 1
class |
|---|---|---|---|---|---|---|---|
[
"record point =\n cx :: int\n cy :: int",
"record bw_point = point +\n black :: bool",
"typedef int' = \"UNIV::int set\"",
"definition n0 where \"n0 = Abs_int' 0\"",
"definition n1 where \"n1 = Abs_int' 1\"",
"definition n2 where \"n2 = Abs_int' 2\"",
"definition plus' where \"plus' n m = Abs_int' (R... | [
"HOL.Ex"
] | lemma "\<exists>f. map_option f (Some x) = Some (y + x)" | /pisa/Isabelle2022/src/HOL/SMT_Examples/SMT_Tests.thy | lemma
"\<exists>x. x = x"
"\<exists>x::int. x = x"
"\<exists>x::int. x = x + 1"
"\<exists>x::int. x = x + 1 + 1"
"\<exists>x::int. x = x + 1 + 1 + 1"
"\<exists>x::int. x = x + 1 + 1 + 1 + 1"
"\<exists>x::int. x = x + 1 + 1 + 1 + 1 + 1"
"\<exists>x::int. x = x + 1 + 1 + 1 + 1 + 1 + 1"
"\<exists>x::int.... | false | false | false |
[
"definition enum_word :: \\<open>'a word list\\<close>\n where \\<open>enum_word = map word_of_nat [0..<2 ^ LENGTH('a)]\\<close>",
"definition enum_all_word :: \\<open>('a word \\<Rightarrow> bool) \\<Rightarrow> bool\\<close>\n where \\<open>enum_all_word = Ball UNIV\\<close>",
"definition enum_ex_word :: \\... | [
""
] | lemma signed_numeral [simp]:
\<open>signed (numeral n :: 'b::len word) = of_int (signed_take_bit (LENGTH('b) - 1) (numeral n))\<close> | /pisa/Isabelle2022/src/HOL/Library/Word.thy | lemma [code]:
\<open>Word.the_int (Word.of_int x :: 'b::len word) = x\<close>
\<open>Word.the_int (Word.the_int x :: 'b::len word) = x\<close>
\<open>Word.the_int (Word.the_int x :: 'b::len word) = x\<close>
\<open>Word.the_int (Word.the_int x :: 'b::len word) = x\<close>
\<open>Word.the_int (Word.the_int x :... | false | false | false |
[
"definition subprob_algebra :: \"'a measure \\<Rightarrow> 'a measure measure\" where\n \"subprob_algebra K =\n (SUP A \\<in> sets K. vimage_algebra {M. subprob_space M \\<and> sets M = sets K} (\\<lambda>M. emeasure M A) borel)\"",
"definition return :: \"'a measure \\<Rightarrow> 'a \\<Rightarrow> 'a measur... | [
"Pure.imp",
"HOL.Trueprop",
"Set.member",
"Sigma_Algebra.measurable",
"HOL.eq",
"Sigma_Algebra.sets",
"Giry_Monad.subprob_algebra"
] | lemma measurable_join1:
"\<lbrakk> f \<in> measurable N K; sets M = sets (subprob_algebra N) \<rbrakk>
\<Longrightarrow> f \<in> measurable (join M) K" | /pisa/Isabelle2022/src/HOL/Probability/Giry_Monad.thy | lemma measurable_subprob_algebra:
"f \<in> measurable M (subprob_algebra N) \<Longrightarrow> sets M = sets (subprob_algebra N) \<Longrightarrow>
f \<in> measurable M N"{EOS} | true | false | false |
[] | [
"Hilbert_Choice.Eps"
] | lemma "P (Eps P)" | /pisa/Isabelle2022/src/HOL/Nitpick_Examples/Core_Nits.thy | lemma "Eps (\<lambda>x. x)"{EOS} | true | true | false |
[] | [
"HOL.All"
] | lemma "\<forall>x. f x y = f y x" | /pisa/Isabelle2022/src/HOL/Nitpick_Examples/Core_Nits.thy | lemma "\<forall>(x::'a::linordered_field) y z. x < y \<and> y < z \<longrightarrow> x < z"{EOS} | true | true | false |
[
"definition \"topspace T = \\<Union>{S. openin T S}\"",
"definition discrete_topology where \"discrete_topology U \\<equiv> topology (\\<lambda>S. S \\<subseteq> U)\"",
"definition derived_set_of :: \"'a topology \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"derived'_set'_of\" 80)\n where \"X derived... | [
"HOL.eq",
"Abstract_Topology.frontier_of",
"Set.inter",
"Abstract_Topology.closure_of",
"Set.union"
] | lemma frontier_of_Int:
"X frontier_of (S \<inter> T) =
X closure_of (S \<inter> T) \<inter> (X frontier_of S \<union> X frontier_of T)" | /pisa/Isabelle2022/src/HOL/Analysis/Abstract_Topology.thy | lemma
shows quotient_map_frontier_of: "X quotient_map_to Y f \<Longrightarrow> X frontier_of S = Y closure_of (X frontier_of S \<inter> S)"
and quotient_map_closure_of: "X quotient_map_to Y f \<Longrightarrow> X closure_of S = Y closure_of (X frontier_of S \<inter> S)"
and quotient_map_frontier_of_closure_of:
... | false | false | false |
[
"definition \"einterval a b = {x. a < ereal x \\<and> ereal x < b}\""
] | [
"Interval_Integral.interval_lebesgue_integrable",
"HOL.eq",
"Interval_Integral.interval_lebesgue_integral",
"Groups.plus_class.plus"
] | lemma interval_lebesgue_integral_add [intro, simp]:
fixes M a b f
assumes "interval_lebesgue_integrable M a b f" "interval_lebesgue_integrable M a b g"
shows "interval_lebesgue_integrable M a b (\<lambda>x. f x + g x)" and
"interval_lebesgue_integral M a b (\<lambda>x. f x + g x) =
interval_lebesgue_integr... | /pisa/Isabelle2022/src/HOL/Analysis/Interval_Integral.thy | lemma interval_integral_add:
fixes f g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes "interval_lebesgue_integrable M A f" "interval_lebesgue_integrable M A g"
shows "interval_lebesgue_integral M A (\<lambda>x. f x + g x) =
interval_lebesgue_integral M A f + interval_lebesgue_integral M... | true | false | false |
[] | [
"Topological_Spaces.continuous",
"Topological_Spaces.topological_space_class.at_within",
"Transcendental.exp"
] | lemma continuous_within_exp:
fixes z::"'a::{real_normed_field,banach}"
shows "continuous (at z within s) exp" | /pisa/Isabelle2022/src/HOL/Analysis/Complex_Transcendental.thy | lemma continuous_within_exp [continuous_intros]: "continuous (at x within s) exp"{EOS} | true | false | false |
[
"definition prime_nat :: \"nat \\<Rightarrow> bool\" where\n \"prime_nat p = (1 < p \\<and> (\\<forall>m. m dvd p --> m = 1 \\<or> m = p))\"",
"fun dec_10 :: \"int \\<Rightarrow> int\" where\n \"dec_10 n = (if n < 10 then n else dec_10 (n - 10))\"",
"definition Pred :: \"'a \\<Rightarrow> bool\" where\n \"Pr... | [
"Orderings.ord_class.less_eq",
"Complete_Lattices.Sup_class.Sup",
"Set.Collect"
] | lemma
assumes "Sup {a | i::bool. True} \<le> Sup {b | i::bool. True}"
and "Sup {b | i::bool. True} \<le> Sup {a | i::bool. True}"
shows "Sup {a | i::bool. True} \<le> Sup {a | i::bool. True}" | /pisa/Isabelle2022/src/HOL/SMT_Examples/SMT_Examples.thy | lemma_10: "(SUP x\<in>{x. P x}. f x) \<le> (SUP x\<in>{x. P x}. g x)"{EOS} | true | false | false |
[
"definition affine_parallel :: \"'a::real_vector set \\<Rightarrow> 'a::real_vector set \\<Rightarrow> bool\"\n where \"affine_parallel S T \\<longleftrightarrow> (\\<exists>a. T = (\\<lambda>x. a + x) ` S)\""
] | [
"Set.not_member",
"HOL.eq",
"Hull.hull",
"Affine.affine",
"Set.insert",
"Set.Collect"
] | lemma affine_hull_insert_span:
assumes "a \<notin> S"
shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x. x \<in> S}}" | /pisa/Isabelle2022/src/HOL/Analysis/Affine.thy | lemma affine_hull_insert_eq:
assumes "a \<notin> S"
shows "affine hull (insert a S) = {x. \<exists>b. x = a + b \<and> b \<in> S}"
(is "?lhs =?rhs"){EOS} | true | false | false |
[
"definition path_component_of\n where \"path_component_of X x y \\<equiv> \\<exists>g. pathin X g \\<and> g 0 = x \\<and> g 1 = y\"",
"definition path_components_of :: \"'a topology \\<Rightarrow> 'a set set\"\n where \"path_components_of X \\<equiv> path_component_of_set X ` topspace X\""
] | [
"Abstract_Topology_2.pathin",
"Abstract_Topology_2.path_connectedin",
"Set.image",
"Set_Interval.ord_class.atLeastAtMost",
"Groups.zero_class.zero",
"Groups.one_class.one"
] | lemma path_connectedin_path_image:
assumes "pathin X g" shows "path_connectedin X (g ` ({0..1}))" | /pisa/Isabelle2022/src/HOL/Analysis/Path_Connected.thy | lemma path_connectedin_interval:
assumes "pathin X g"
shows "path_connectedin X (g ` {0..1})"{EOS} | true | false | false |
[
"definition has_sum :: \\<open>('a \\<Rightarrow> 'b :: {comm_monoid_add, topological_space}) \\<Rightarrow> 'a set \\<Rightarrow> 'b \\<Rightarrow> bool\\<close> where\n \\<open>has_sum f A x \\<longleftrightarrow> (sum f \\<longlongrightarrow> x) (finite_subsets_at_top A)\\<close>",
"definition summable_on :: ... | [
"Infinite_Sum.summable_on",
"HOL.eq",
"Infinite_Sum.infsum",
"Complex.complex.Im"
] | lemma infsum_Im:
assumes "f summable_on M"
shows "infsum (\<lambda>x. Im (f x)) M = Im (infsum f M)" | /pisa/Isabelle2022/src/HOL/Analysis/Infinite_Sum.thy | lemma infsum_Im:
assumes "f summable_on A"
shows "infsum (\<lambda>x. Im (f x)) A = Im (infsum f A)"{EOS} | true | false | false |
[
"definition eqpoll :: \"'a set \\<Rightarrow> 'b set \\<Rightarrow> bool\" (infixl \"\\<approx>\" 50)\n where \"eqpoll A B \\<equiv> \\<exists>f. bij_betw f A B\"",
"definition lepoll :: \"'a set \\<Rightarrow> 'b set \\<Rightarrow> bool\" (infixl \"\\<lesssim>\" 50)\n where \"lepoll A B \\<equiv> \\<exists>f. ... | [
"Equipollence.lepoll",
"Equipollence.eqpoll"
] | lemma lepoll_antisym:
assumes "A \<lesssim> B" "B \<lesssim> A" shows "A \<approx> B" | /pisa/Isabelle2022/src/HOL/Library/Equipollence.thy | lemma lepoll_refl [iff]: "A \<lesssim> A"
and lepoll_sym [sym]: "A \<lesssim> B \<longleftrightarrow> B \<lesssim> A"
and lepoll_trans [trans]: "A \<lesssim> B \<Longrightarrow> B \<lesssim> C \<Longrightarrow> A \<lesssim> C"
and lepoll_antisym [intro?]: "A \<lesssim> B \<Longrightarrow> B \<lesssim> A \<L... | false | false | false |
[
"definition prefix :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> bool\"\n where \"prefix xs ys \\<longleftrightarrow> (\\<exists>zs. ys = xs @ zs)\"",
"definition strict_prefix :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> bool\"\n where \"strict_prefix xs ys \\<longleftrightarrow> prefix xs ys \\<and... | [
"HOL.not_equal",
"Set.empty",
"Set.not_member",
"List.list.Nil",
"Set.Ball",
"HOL.eq",
"Sublist.Longest_common_prefix",
"List.list.Cons",
"Set.Collect"
] | lemma Longest_common_prefix_eq_Cons: assumes "L \<noteq> {}" "[] \<notin> L" "\<forall>xs\<in>L. hd xs = x"
shows "Longest_common_prefix L = x # Longest_common_prefix {ys. x#ys \<in> L}" | /pisa/Isabelle2022/src/HOL/Library/Sublist.thy | lemma Longest_common_prefix_Cons:
assumes "L \<noteq> {}" and "L \<noteq> []" and "[] \<notin> L" and "\<forall>xs \<in> L. prefix ps xs"
shows "Longest_common_prefix (L - {ps}) = ps"
(is "?L =?R"){EOS} | true | false | false |
[] | [
"Set.set",
"HOL.iff",
"Retracts.ANR",
"HOL.All",
"HOL.eq",
"Pure.dummy_pattern"
] | lemma ANR_eq_absolute_neighbourhood_extensor:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<longleftrightarrow>
(\<forall>f :: 'a * real \<Rightarrow> 'a.
\<forall>U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow>
closedin (top_of_set U) T \<long... | /pisa/Isabelle2022/src/HOL/Analysis/Retracts.thy | lemma ANR_eq_eq:
fixes S :: "'a::euclidean_space set"
shows "ANR S \<longleftrightarrow> (\<forall>x. x \<in> S \<longrightarrow> (\<forall>y. y \<in> S \<longrightarrow> x = y))"
(is "?lhs =?rhs"){EOS} | true | false | false |
[] | [
"Orderings.ord_class.less_eq",
"Relation.Field",
"local.ofilter",
"local.UnderS"
] | lemma ofilter_UnderS[simp]:
assumes "A \<le> Field r"
shows "ofilter(UnderS A)" | /pisa/Isabelle2022/src/HOL/Cardinals/Wellorder_Relation.thy | lemma OF_UnderS_le: "Field ofilter \<le> UnderS ofilter"{EOS} | true | false | false |
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