theory_file stringclasses 672
values | lemma_name stringlengths 7 81 | lemma stringlengths 10 21.2k | symbols sequencelengths 1 152 | defs sequencelengths 0 110 | template stringlengths 7 11.1k | rediscoverable stringclasses 3
values |
|---|---|---|---|---|---|---|
SPARK/Examples/RIPEMD-160/F | Fun.surjI | (\<And>x. ?g (?f x) = x) \<Longrightarrow> ?g ` Orderings.top_class.top = Orderings.top_class.top | [
"(\\<Longrightarrow>)",
"Pure.all",
"(=)",
"(`)",
"Orderings.top_class.top",
"Orderings.top_class.top"
] | [] | PROP ?H0 (\<lambda>x. x_0 (x_1 x) = x) \<Longrightarrow> ?H1 x_0 ?H2 = ?H3 | False |
Computational_Algebra/Formal_Laurent_Series | Formal_Laurent_Series.fls_integral_X_power | Formal_Laurent_Series.fls_integral (Formal_Laurent_Series.fls_X ^ ?n) = Formal_Laurent_Series.fls_const (Fields.inverse_class.inverse (Nat.semiring_1_class.of_nat (Nat.Suc ?n))) * Formal_Laurent_Series.fls_X ^ Nat.Suc ?n | [
"(=)",
"Formal_Laurent_Series.fls_integral",
"(^)",
"Formal_Laurent_Series.fls_X",
"(*)",
"Formal_Laurent_Series.fls_const",
"Fields.inverse_class.inverse",
"Nat.semiring_1_class.of_nat",
"Nat.Suc",
"(^)",
"Formal_Laurent_Series.fls_X",
"Nat.Suc"
] | [
"definition fls_integral :: \"'a::{ring_1,inverse} fls \\<Rightarrow> 'a fls\"\n where \"fls_integral a = Abs_fls (\\<lambda>n. if n=0 then 0 else inverse (of_int n) * a$$(n - 1))\"",
"definition fls_const :: \"'a::zero \\<Rightarrow> 'a fls\"\n where \"fls_const c \\<equiv> Abs_fls (\\<lambda>n. if n = 0 then ... | ?H0 (?H1 ?H2 x_0) = ?H3 (?H4 (?H5 (?H6 (?H7 x_0)))) (?H1 ?H2 (?H7 x_0)) | Error |
Imperative_HOL/ex/List_Sublist | List_Sublist.nths_Nil' | List.nths ?xs ?inds = [] \<Longrightarrow> \<forall>i\<in>?inds. Nat.size_class.size ?xs \<le> i | [
"(\\<Longrightarrow>)",
"(=)",
"List.nths",
"[]",
"Set.Ball"
] | [
"function nths' :: \"nat \\<Rightarrow> nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\"\nwhere\n \"nths' n m [] = []\"\n| \"nths' n 0 xs = []\"\n| \"nths' 0 (Suc m) (x#xs) = (x#nths' 0 m xs)\"\n| \"nths' (Suc n) (Suc m) (x#xs) = nths' n m xs\""
] | ?H0 x_0 x_1 = ?H1 \<Longrightarrow> ?H2 x_1 (?H3 (?H4 x_0)) | False |
Analysis/Locally | Locally.locally_path_connected_real_interval(2) | Locally.locally_path_connected_space (Abstract_Topology.subtopology (Abstract_Topology.topology.topology Topological_Spaces.open_class.open) {?a<..<?b}) | [
"Locally.locally_path_connected_space",
"Abstract_Topology.subtopology",
"Abstract_Topology.topology.topology",
"Topological_Spaces.open_class.open",
"Set_Interval.ord_class.greaterThanLessThan"
] | [
"definition locally_path_connected_space\n where \"locally_path_connected_space X \\<equiv> neighbourhood_base_of (path_connectedin X) X\"",
"definition neighbourhood_base_at where\n \"neighbourhood_base_at x P X \\<equiv>\n \\<forall>W. openin X W \\<and> x \\<in> W\n \\<longrightarrow> (\\<ex... | ?H0 (?H1 (?H2 ?H3) (?H4 x_0 x_1)) | Error |
Number_Theory/Residue_Primitive_Roots | Residue_Primitive_Roots.prime_elements_with_ord(2) | 1 < ?n \<Longrightarrow> \<not> Finite_Set.card {x \<in> Totient.totatives ?n. Pocklington.ord ?n x = ?d} = 0 \<Longrightarrow> Factorial_Ring.normalization_semidom_class.prime ?n \<Longrightarrow> ?a \<in> Totient.totatives ?n \<Longrightarrow> Pocklington.ord ?n ?a = ?d \<Long... | [
"(\\<Longrightarrow>)",
"(<)",
"1",
"(\\<Longrightarrow>)",
"HOL.Not",
"(=)",
"Finite_Set.card",
"Set.Collect",
"0",
"(\\<Longrightarrow>)",
"Factorial_Ring.normalization_semidom_class.prime",
"(\\<Longrightarrow>)",
"(\\<in>)",
"Totient.totatives",
"(\\<Longrightarrow>)",
"(=)",
"Po... | [
"definition totatives :: \"nat \\<Rightarrow> nat set\" where\n \"totatives n = {k \\<in> {0<..n}. coprime k n}\"",
"inductive residue_primroot :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"n > 0 \\<Longrightarrow> coprime n g \\<Longrightarrow> ord n g = totient n \\<Longrightarrow> residue_primr... | \<lbrakk>?H0 ?H1 x_0; \<not> ?H2 (?H3 (\<lambda>x. ?H4 x (?H5 x_0) \<and> ?H6 x_0 x = x_1)) = ?H7; ?H8 x_0; ?H4 x_2 (?H5 x_0); ?H6 x_0 x_2 = x_1; \<not> x_2 = ?H1\<rbrakk> \<Longrightarrow> ?H3 (\<lambda>x. ?H4 x (?H9 x_0) \<and> ?H10 (?H11 x x_1) ?H1 x_0) = ?H12 (\<lambda>k. ?H13 (?H11 x_2 k) x_0) (?H9 x_1) | Error |
IMP/Compiler2 | Compiler2.acomp_exits | Compiler2.exits (Compiler.acomp ?a) = Set.insert (Nat.semiring_1_class.of_nat (Nat.size_class.size (Compiler.acomp ?a))) Orderings.bot_class.bot | [
"(=)",
"Compiler2.exits",
"Compiler.acomp",
"Set.insert",
"Nat.semiring_1_class.of_nat",
"Nat.size_class.size",
"Compiler.acomp",
"Orderings.bot_class.bot"
] | [
"definition exits :: \"instr list \\<Rightarrow> int set\" where\n\"exits P = succs P 0 - {0..< size P}\"",
"fun acomp :: \"aexp \\<Rightarrow> instr list\" where\n\"acomp (N n) = [LOADI n]\" |\n\"acomp (V x) = [LOAD x]\" |\n\"acomp (Plus a1 a2) = acomp a1 @ acomp a2 @ [ADD]\"",
"fun acomp :: \"aexp \\<Rightarr... | ?H0 (?H1 x_0) = ?H2 (?H3 (?H4 (?H1 x_0))) ?H5 | True |
Imperative_HOL/Array | Array.set | Array.set (Product_Type.prod.fst (Array.alloc ?ls ?h)) ?new_ls (Product_Type.prod.snd (Array.alloc ?ls ?h)) = Product_Type.prod.snd (Array.alloc ?new_ls ?h) | [
"(=)",
"Array.set",
"Product_Type.prod.fst",
"Array.alloc",
"Product_Type.prod.snd",
"Array.alloc",
"Product_Type.prod.snd",
"Array.alloc"
] | [
"definition set :: \"'a::heap array \\<Rightarrow> 'a list \\<Rightarrow> heap \\<Rightarrow> heap\" where\n \"set a x = arrays_update (\\<lambda>h. h(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))\"",
"definition alloc :: \"'a list \\<Rightarrow> heap \\<Rightarrow> 'a::heap array \\<times>... | ?H0 (?H1 (?H2 x_0 x_1)) x_2 (?H3 (?H2 x_0 x_1)) = ?H3 (?H2 x_2 x_1) | False |
Analysis/Henstock_Kurzweil_Integration | Henstock_Kurzweil_Integration.Sigma_Int_Paircomp1 | Lattices.inf_class.inf (Product_Type.Sigma ?A ?B) {(x, y). ?P x} = Product_Type.Sigma (Lattices.inf_class.inf ?A {x. ?P x}) ?B | [
"(=)",
"Lattices.inf_class.inf",
"Product_Type.Sigma",
"Set.Collect",
"Product_Type.prod.case_prod",
"Product_Type.Sigma",
"Lattices.inf_class.inf",
"Set.Collect"
] | [
"definition has_integral :: \"('n::euclidean_space \\<Rightarrow> 'b::real_normed_vector) \\<Rightarrow> 'b \\<Rightarrow> 'n set \\<Rightarrow> bool\"\n (infixr \"has'_integral\" 46)\n where \"(f has_integral I) s \\<longleftrightarrow>\n (if \\<exists>a b. s = cbox a b\n then ((\\<lambda>p. \\<Sum>(x,k)... | ?H0 (?H1 x_0 x_1) (?H2 (?H3 (\<lambda>x y. x_2 x))) = ?H1 (?H4 x_0 (?H5 x_2)) x_1 | False |
IMP/Com | Complete_Lattices.UNION_singleton_eq_range | (SUP x\<in>?A. Set.insert (?f x) Orderings.bot_class.bot) = ?f ` ?A | [
"(=)",
"Complete_Lattices.Sup_class.Sup",
"(`)",
"(`)"
] | [
"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)"
] | ?H0 (?H1 (\<lambda>x. ?H2 (x_0 x) ?H3) x_1) = ?H4 x_0 x_1 | False |
HOLCF/Tr | Transfer.transfer_raw(55) | Transfer.Rel (BNF_Def.rel_fun (BNF_Def.rel_fun ?Sa (BNF_Def.rel_fun ?Sc (=))) (BNF_Def.rel_fun (Tree.tree.rel_tree ?Sa) (BNF_Def.rel_fun (Tree.tree.rel_tree ?Sc) (=)))) Tree.tree.rel_tree Tree.tree.rel_tree | [
"Transfer.Rel",
"BNF_Def.rel_fun",
"BNF_Def.rel_fun",
"BNF_Def.rel_fun",
"(=)",
"BNF_Def.rel_fun",
"Tree.tree.rel_tree",
"BNF_Def.rel_fun",
"Tree.tree.rel_tree",
"(=)",
"Tree.tree.rel_tree",
"Tree.tree.rel_tree"
] | [
"type_synonym tr = \"bool lift\"",
"definition TT :: \"tr\"\n where \"TT = Def True\"",
"definition FF :: \"tr\"\n where \"FF = Def False\"",
"definition tr_case :: \"'a \\<rightarrow> 'a \\<rightarrow> tr \\<rightarrow> 'a\"\n where \"tr_case = (\\<Lambda> t e (Def b). if b then t else e)\"",
"definitio... | ?H0 (?H1 (?H2 x_0 (?H3 x_1 (=))) (?H4 (?H5 x_0) (?H6 (?H7 x_1) (=)))) ?H8 ?H9 | Error |
Library/Real_Mod | Real_Mod.rcong_0_right | [?x = ?y] (rmod 0) = (?x = ?y) | [
"(=)",
"Real_Mod.rcong",
"0",
"(=)"
] | [
"definition rcong :: \"real \\<Rightarrow> real \\<Rightarrow> real \\<Rightarrow> bool\" (\\<open>(1[_ = _] '(' rmod _'))\\<close>) where\n \"[x = y] (rmod m) \\<longleftrightarrow> x rmod m = y rmod m\"",
"definition rmod :: \"real \\<Rightarrow> real \\<Rightarrow> real\" (infixl \"rmod\" 70) where\n \"x rmo... | ?H0 x_0 x_1 ?H1 = (x_0 = x_1) | True |
Decision_Procs/Ferrack | Ferrack.finite_set_intervals | ?P ?x \<Longrightarrow> ?l \<le> ?x \<Longrightarrow> ?x \<le> ?u \<Longrightarrow> ?l \<in> ?S \<Longrightarrow> ?u \<in> ?S \<Longrightarrow> Finite_Set.finite ?S \<Longrightarrow> \<forall>x\<in>?S. ?l \<le> x \<Longrightarrow> \<forall>x\<in>?S. x \<le> ?u \<Longrightarrow>... | [
"(\\<Longrightarrow>)",
"(\\<Longrightarrow>)",
"(\\<le>)",
"(\\<Longrightarrow>)",
"(\\<le>)",
"(\\<Longrightarrow>)",
"(\\<in>)",
"(\\<Longrightarrow>)",
"(\\<in>)",
"(\\<Longrightarrow>)",
"Finite_Set.finite",
"(\\<Longrightarrow>)",
"Set.Ball",
"(\\<Longrightarrow>)",
"Set.Ball",
"... | [
"datatype (plugins del: size) num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num\n | Mul int num",
"primrec size_num :: \"num \\<Rightarrow> nat\"\nwhere\n \"size_num (C c) = 1\"\n| \"size_num (Bound n) = 1\"\n| \"size_num (Neg a) = 1 + size_num a\"\n| \"size_num (Add a b) = 1 + size_... | \<lbrakk>x_0 x_1; ?H0 x_2 x_1; ?H0 x_1 x_3; ?H1 x_2 x_4; ?H1 x_3 x_4; ?H2 x_4; ?H3 x_4 (?H0 x_2); ?H3 x_4 (\<lambda>x. ?H0 x x_3)\<rbrakk> \<Longrightarrow> ?H4 x_4 (\<lambda>a. ?H4 x_4 (\<lambda>b. (\<forall>y. ?H5 a y \<and> ?H5 y b \<longrightarrow> \<not> ?H1 y x_4) \<and> ?H0 a x_1 \<and> ?H0 x_1 b \<and> x_0 x_1)... | False |
SPARK/Examples/RIPEMD-160/F | Fields.divide_simps(11) | (?a = - (?b div ?c)) = (if \<not> ?c = (0::?'a) then ?a * ?c = - ?b else ?a = (0::?'a)) | [
"(=)",
"(=)",
"Groups.uminus_class.uminus",
"(div)",
"HOL.If",
"HOL.Not",
"(=)",
"0::?'a",
"(=)",
"(*)",
"Groups.uminus_class.uminus",
"(=)",
"0::?'a"
] | [] | (x_0 = ?H0 (?H1 x_1 x_2)) = (if \<not> x_2 = ?H2 then ?H3 x_0 x_2 = ?H0 x_1 else x_0 = ?H2) | Error |
Library/Countable_Set_Type | Countable_Set_Type.cDiff_cInt | ?A - Lattices.inf_class.inf ?B ?C = Lattices.sup_class.sup (?A - ?B) (?A - ?C) | [
"(=)",
"(-)",
"Lattices.inf_class.inf",
"Lattices.sup_class.sup",
"(-)",
"(-)"
] | [
"typedef 'a cset = \"{A :: 'a set. countable A}\" morphisms rcset acset",
"definition less_cset :: \"'a cset \\<Rightarrow> 'a cset \\<Rightarrow> bool\"\nwhere \"xs < ys \\<equiv> xs \\<le> ys \\<and> xs \\<noteq> (ys::'a cset)\""
] | ?H0 x_0 (?H1 x_1 x_2) = ?H2 (?H0 x_0 x_1) (?H0 x_0 x_2) | False |
Nonstandard_Analysis/StarDef | StarDef.star_of_min | StarDef.star_of (Orderings.ord_class.min ?x ?y) = Orderings.ord_class.min (StarDef.star_of ?x) (StarDef.star_of ?y) | [
"(=)",
"StarDef.star_of",
"Orderings.ord_class.min",
"Orderings.ord_class.min",
"StarDef.star_of",
"StarDef.star_of"
] | [
"definition star_of :: \"'a \\<Rightarrow> 'a star\"\n where \"star_of x \\<equiv> star_n (\\<lambda>n. x)\"",
"definition star_of :: \"'a \\<Rightarrow> 'a star\"\n where \"star_of x \\<equiv> star_n (\\<lambda>n. x)\"",
"definition star_of :: \"'a \\<Rightarrow> 'a star\"\n where \"star_of x \\<equiv> star... | ?H0 (?H1 x_0 x_1) = ?H2 (?H0 x_0) (?H0 x_1) | False |
HOLCF/Domain | Domain_Aux.sel_app_rules(14) | \<not> ?x = \<bottom> \<Longrightarrow> \<not> ?y = \<bottom> \<Longrightarrow> \<not> (:?x, ?y:) = \<bottom> | [
"(\\<Longrightarrow>)",
"HOL.Not",
"(=)",
"\\<bottom>",
"(\\<Longrightarrow>)",
"HOL.Not",
"(=)",
"\\<bottom>",
"HOL.Not",
"(=)",
"Cfun.cfun.Rep_cfun",
"Cfun.cfun.Rep_cfun",
"Sprod.spair",
"\\<bottom>"
] | [
"definition spair :: \"'a \\<rightarrow> 'b \\<rightarrow> ('a ** 'b)\"\n where \"spair = (\\<Lambda> a b. Abs_sprod (seq\\<cdot>b\\<cdot>a, seq\\<cdot>a\\<cdot>b))\"",
"definition defl_set :: \"'a::bifinite defl \\<Rightarrow> 'a set\"\nwhere \"defl_set A = {x. cast\\<cdot>A\\<cdot>x = x}\"",
"definition isod... | \<lbrakk>\<not> x_0 = ?H0; \<not> x_1 = ?H1\<rbrakk> \<Longrightarrow> \<not> ?H2 (?H3 ?H4 x_0) x_1 = ?H5 | Error |
Analysis/Function_Topology | Function_Topology.quotient_map_product_projection | ?i \<in> ?I \<Longrightarrow> Abstract_Topology.quotient_map (Function_Topology.product_topology ?X ?I) (?X ?i) (\<lambda>x. x ?i) = (Function_Topology.product_topology ?X ?I = Abstract_Topology.discrete_topology Orderings.bot_class.bot \<longrightarrow> ?X ?i = Abstract_Topology... | [
"(\\<Longrightarrow>)",
"(\\<in>)",
"(=)",
"Abstract_Topology.quotient_map",
"Function_Topology.product_topology",
"(\\<longrightarrow>)",
"(=)",
"Function_Topology.product_topology",
"Abstract_Topology.discrete_topology",
"Orderings.bot_class.bot",
"(=)",
"Abstract_Topology.discrete_topology"... | [
"definition quotient_map where\n \"quotient_map X X' f \\<longleftrightarrow>\n f ` (topspace X) = topspace X' \\<and>\n (\\<forall>U. U \\<subseteq> topspace X' \\<longrightarrow> (openin X {x. x \\<in> topspace X \\<and> f x \\<in> U} \\<longleftrightarrow> openin X' U))\"",
"definition discrete_t... | ?H0 x_0 x_1 \<Longrightarrow> ?H1 (?H2 x_2 x_1) (x_2 x_0) (\<lambda>x. x x_0) = (?H2 x_2 x_1 = ?H3 ?H4 \<longrightarrow> x_2 x_0 = ?H5 ?H6) | Error |
Algebra/Order | Orderings.order_le_less | (?x \<le> ?y) = (?x < ?y \<or> ?x = ?y) | [
"(=)",
"(\\<le>)",
"(\\<or>)",
"(<)",
"(=)"
] | [
"record 'a gorder = \"'a eq_object\" +\n le :: \"['a, 'a] => bool\" (infixl \"\\<sqsubseteq>\\<index>\" 50)",
"definition\n lless :: \"[_, 'a, 'a] => bool\" (infixl \"\\<sqsubset>\\<index>\" 50)\n where \"x \\<sqsubset>\\<^bsub>L\\<^esub> y \\<longleftrightarrow> x \\<sqsubseteq>\\<^bsub>L\\<^esub> y \\<and> x... | ?H0 x_0 x_1 = (?H1 x_0 x_1 \<or> x_0 = x_1) | False |
Quotient_Examples/Quotient_Message | Quotient_Message.discrim_Decrypt | Quotient_Message.discrim (Quotient_Message.Decrypt ?K ?X) = Quotient_Message.discrim ?X - 2 | [
"(=)",
"Quotient_Message.discrim",
"Quotient_Message.Decrypt",
"(-)",
"Quotient_Message.discrim",
"Num.numeral_class.numeral",
"Num.num.Bit0",
"Num.num.One"
] | [
"datatype\n freemsg = NONCE nat\n | MPAIR freemsg freemsg\n | CRYPT nat freemsg\n | DECRYPT nat freemsg",
"inductive\n msgrel::\"freemsg \\<Rightarrow> freemsg \\<Rightarrow> bool\" (infixl \"\\<sim>\" 50)\nwhere\n CD: \"CRYPT K (DECRYPT K X) \\<sim> X\"\n| DC: \"DECRYPT K (CRYPT... | ?H0 (?H1 x_0 x_1) = ?H2 (?H0 x_1) (?H3 (?H4 ?H5)) | True |
HOLCF/IOA/Pred | Predicate.the_eqI | (THE x. Predicate.pred.eval ?P x) = ?x \<Longrightarrow> Predicate.the ?P = ?x | [
"(\\<Longrightarrow>)",
"(=)",
"HOL.The",
"(=)",
"Predicate.the"
] | [
"type_synonym 'a predicate = \"'a \\<Rightarrow> bool\"",
"definition satisfies :: \"'a \\<Rightarrow> 'a predicate \\<Rightarrow> bool\" (\"_ \\<Turnstile> _\" [100, 9] 8)\n where \"(s \\<Turnstile> P) \\<longleftrightarrow> P s\"",
"definition valid :: \"'a predicate \\<Rightarrow> bool\" (\"\\<TTurnstile>... | (THE x. ?H0 x_0 x) = x_1 \<Longrightarrow> ?H1 x_0 = x_1 | False |
Bali/DeclConcepts | DeclConcepts.is_static_method_simp | DeclConcepts.has_static_class.is_static (DeclConcepts.method ?sig ?m) = DeclConcepts.has_static_class.is_static ?m | [
"(=)",
"DeclConcepts.has_static_class.is_static",
"DeclConcepts.method",
"DeclConcepts.has_static_class.is_static"
] | [
"definition is_public :: \"prog \\<Rightarrow> qtname \\<Rightarrow> bool\" where\n\"is_public G qn = (case class G qn of\n None \\<Rightarrow> (case iface G qn of\n None \\<Rightarrow> False\n | Some i \\<Righta... | ?H0 (?H1 x_0 x_1) = ?H2 x_1 | True |
Library/RBT_Mapping | RBT_Mapping.lookup_Mapping | Mapping.lookup (RBT_Mapping.Mapping ?t) = RBT.lookup ?t | [
"(=)",
"Mapping.lookup",
"RBT_Mapping.Mapping",
"RBT.lookup"
] | [] | ?H0 (?H1 x_0) = ?H2 x_0 | False |
MicroJava/BV/Typing_Framework_JVM | Typing_Framework_JVM.special_ex_swap_lemma | (\<exists>X. (\<exists>n. X = ?A n \<and> ?P n) \<and> ?Q X) = (\<exists>n. ?Q (?A n) \<and> ?P n) | [
"(=)",
"HOL.Ex",
"HOL.Ex"
] | [
"definition exec :: \"jvm_prog \\<Rightarrow> nat \\<Rightarrow> ty \\<Rightarrow> exception_table \\<Rightarrow> instr list \\<Rightarrow> JVMType.state step_type\" where\n \"exec G maxs rT et bs == \n err_step (size bs) (\\<lambda>pc. app (bs!pc) G maxs rT pc et) (\\<lambda>pc. eff (bs!pc) G pc et)\"",
"defin... | (\<exists>X. (\<exists>n. X = x_0 n \<and> x_1 n) \<and> x_2 X) = (\<exists>n. x_2 (x_0 n) \<and> x_1 n) | False |
Analysis/Henstock_Kurzweil_Integration | Henstock_Kurzweil_Integration.uminus_interval_vector | Groups.uminus_class.uminus ` Topology_Euclidean_Space.cbox ?a ?b = Topology_Euclidean_Space.cbox (- ?b) (- ?a) | [
"(=)",
"(`)",
"Groups.uminus_class.uminus",
"Topology_Euclidean_Space.cbox",
"Topology_Euclidean_Space.cbox",
"Groups.uminus_class.uminus",
"Groups.uminus_class.uminus"
] | [
"definition has_integral :: \"('n::euclidean_space \\<Rightarrow> 'b::real_normed_vector) \\<Rightarrow> 'b \\<Rightarrow> 'n set \\<Rightarrow> bool\"\n (infixr \"has'_integral\" 46)\n where \"(f has_integral I) s \\<longleftrightarrow>\n (if \\<exists>a b. s = cbox a b\n then ((\\<lambda>p. \\<Sum>(x,k)... | ?H0 ?H1 (?H2 x_0 x_1) = ?H2 (?H1 x_1) (?H1 x_0) | False |
Nominal/Examples/Pattern | Pattern.trm_btrm_perm_append(2) | (?pi1.0 @ ?pi2.0) \<bullet> ?btrm_Rep = ?pi1.0 \<bullet> ?pi2.0 \<bullet> ?btrm_Rep | [
"(=)",
"(\\<bullet>)",
"(@)",
"(\\<bullet>)",
"(\\<bullet>)"
] | [
"datatype pat =\n PVar name ty\n | PTuple pat pat (\"(1'\\<langle>\\<langle>_,/ _'\\<rangle>\\<rangle>)\")",
"primrec pat_perm\nwhere\n \"pat_perm pi (PVar x ty) = PVar (pi \\<bullet> x) (pi \\<bullet> ty)\"\n| \"pat_perm pi \\<langle>\\<langle>p, q\\<rangle>\\<rangle> = \\<langle>\\<langle>pat_perm pi p, p... | ?H0 (?H1 x_0 x_1) x_2 = ?H0 x_0 (?H0 x_1 x_2) | Error |
MicroJava/BV/Effect | Effect.appNew | Effect.app (JVMInstructions.instr.New ?C) ?G ?maxs ?rT ?pc ?et (Option.option.Some ?s) = (\<exists>ST LT. ?s = (ST, LT) \<and> Decl.is_class ?G ?C \<and> Nat.size_class.size ST < ?maxs \<and> (\<forall>x\<in>List.list.set (Effect.match ?G Type.xcpt.OutOfMemory ?pc ?e... | [
"(=)",
"Effect.app",
"JVMInstructions.instr.New",
"Option.option.Some",
"HOL.Ex"
] | [
"type_synonym succ_type = \"(p_count \\<times> state_type option) list\"",
"primrec succs :: \"instr \\<Rightarrow> p_count \\<Rightarrow> p_count list\" where\n \"succs (Load idx) pc = [pc+1]\"\n| \"succs (Store idx) pc = [pc+1]\"\n| \"succs (LitPush v) pc = [pc+1]\"\n| \"succs (Getfield F... | ?H0 (?H1 x_0) x_1 x_2 x_3 x_4 x_5 (?H2 x_6) = (\<exists>ST LT. x_6 = ?H3 ST LT \<and> ?H4 x_1 x_0 \<and> ?H5 (?H6 ST) x_2 \<and> ?H7 (?H8 (?H9 x_1 ?H10 x_4 x_5)) (?H4 x_1)) | Error |
Auth/Message | Message.analz_insert_Crypt_subset | Message.analz (Set.insert (Message.msg.Crypt ?K ?X) ?H) \<le> Set.insert (Message.msg.Crypt ?K ?X) (Message.analz (Set.insert ?X ?H)) | [
"(\\<le>)",
"Message.analz",
"Set.insert",
"Message.msg.Crypt",
"Set.insert",
"Message.msg.Crypt",
"Message.analz",
"Set.insert"
] | [
"type_synonym\n key = nat",
"definition symKeys :: \"key set\" where\n \"symKeys == {K. invKey K = K}\"",
"datatype \\<comment> \\<open>We allow any number of friendly agents\\<close>\n agent = Server | Friend nat | Spy",
"datatype\n msg = Agent agent \\<comment> \\<open>Agent names\\<close>\n ... | ?H0 (?H1 (?H2 (?H3 x_0 x_1) x_2)) (?H2 (?H3 x_0 x_1) (?H1 (?H2 x_1 x_2))) | True |
Induct/Com | Complete_Lattices.Sup1_I | ?P \<in> ?A \<Longrightarrow> ?P ?a \<Longrightarrow> Complete_Lattices.Sup_class.Sup ?A ?a | [
"(\\<Longrightarrow>)",
"(\\<in>)",
"(\\<Longrightarrow>)",
"Complete_Lattices.Sup_class.Sup"
] | [
"typedecl loc",
"type_synonym state = \"loc => nat\"",
"datatype\n exp = N nat\n | X loc\n | Op \"nat => nat => nat\" exp exp\n | valOf com exp (\"VALOF _ RESULTIS _\" 60)\nand\n com = SKIP\n | Assign loc exp (infixl \":=\" 60)\n | Semi com com (\"_;;_\" [... | \<lbrakk>?H0 x_0 x_1; x_0 x_2\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 | False |
Algebra/Group | Groups.field_split_simps(28) | - (?a div ?z) + ?b = (if ?z = (0::?'a) then ?b else (- ?a + ?b * ?z) div ?z) | [
"(=)",
"(+)",
"Groups.uminus_class.uminus",
"(div)",
"HOL.If",
"(=)",
"0::?'a",
"(div)",
"(+)",
"Groups.uminus_class.uminus",
"(*)"
] | [
"record 'a monoid = \"'a partial_object\" +\n mult :: \"['a, 'a] \\<Rightarrow> 'a\" (infixl \"\\<otimes>\\<index>\" 70)\n one :: 'a (\"\\<one>\\<index>\")",
"definition\n m_inv :: \"('a, 'b) monoid_scheme => 'a => 'a\" (\"inv\\<index> _\" [81] 80)\n where \"inv\\<^bsub>G\\<^esub> x = (THE y. y \\<in>... | ?H0 (?H1 (?H2 x_0 x_1)) x_2 = (if x_1 = ?H3 then x_2 else ?H2 (?H0 (?H1 x_0) (?H4 x_2 x_1)) x_1) | Error |
Nominal/Examples/Crary | Crary.fresh_star_eqvt(1) | ?pi \<bullet> ?a \<sharp>* ?x = (?pi \<bullet> ?a) \<sharp>* (?pi \<bullet> ?x) | [
"(=)",
"(\\<bullet>)",
"Nominal.fresh_star",
"Nominal.fresh_star",
"(\\<bullet>)",
"(\\<bullet>)"
] | [
"type_synonym Ctxt = \"(name\\<times>ty) list\"",
"type_synonym Subst = \"(name\\<times>trm) list\"",
"fun\n lookup :: \"Subst \\<Rightarrow> name \\<Rightarrow> trm\" \nwhere\n \"lookup [] x = Var x\"\n| \"lookup ((y,T)#\\<theta>) x = (if x=y then T else lookup \\<theta> x)\"",
"inductive\n whr_... | ?H0 x_0 (?H1 x_1 x_2) = ?H1 (?H2 x_0 x_1) (?H3 x_0 x_2) | Error |
Decision_Procs/Rat_Pair | Rat_Pair.Nneg_normN | Rat_Pair.isnormNum ?x \<Longrightarrow> Rat_Pair.isnormNum (~\<^sub>N ?x) | [
"(\\<Longrightarrow>)",
"Rat_Pair.isnormNum",
"Rat_Pair.isnormNum",
"~\\<^sub>N"
] | [
"definition isnormNum :: \"Num \\<Rightarrow> bool\"\n where \"isnormNum = (\\<lambda>(a, b). if a = 0 then b = 0 else b > 0 \\<and> gcd a b = 1)\"",
"definition isnormNum :: \"Num \\<Rightarrow> bool\"\n where \"isnormNum = (\\<lambda>(a, b). if a = 0 then b = 0 else b > 0 \\<and> gcd a b = 1)\"",
"type_syno... | ?H0 x_0 \<Longrightarrow> ?H0 (?H1 x_0) | True |
Library/Extended_Nonnegative_Real | Extended_Nonnegative_Real.sup_const_add_ennreal | Lattices.sup_class.sup (?c + ?a) (?c + ?b) = ?c + Lattices.sup_class.sup ?a ?b | [
"(=)",
"Lattices.sup_class.sup",
"(+)",
"(+)",
"(+)",
"Lattices.sup_class.sup"
] | [
"typedef ennreal = \"{x :: ereal. 0 \\<le> x}\"\n morphisms enn2ereal e2ennreal'",
"definition \"e2ennreal x = e2ennreal' (max 0 x)\"",
"definition infinity_ennreal :: ennreal\nwhere\n [simp]: \"\\<infinity> = (top::ennreal)\"",
"definition divide_ennreal :: \"ennreal \\<Rightarrow> ennreal \\<Rightarrow> e... | ?H0 (?H1 x_0 x_1) (?H1 x_0 x_2) = ?H1 x_0 (?H0 x_1 x_2) | Error |
Matrix_LP/ComputeNumeral | ComputeNumeral.compute_div_mod(178) | (0::?'a) < Num.numeral_class.numeral ?n | [
"(<)",
"0::?'a",
"Num.numeral_class.numeral"
] | [
"fun natfac :: \"nat \\<Rightarrow> nat\"\n where \"natfac n = (if n = 0 then 1 else n * (natfac (n - 1)))\""
] | ?H0 ?H1 (?H2 x_0) | Error |
UNITY/Simple/Lift | Lifting.relator_distr_raw(19) | Lifting.NEG (?R2 OO ?R'2) ?C1 \<Longrightarrow> Lifting.POS (?S2 OO ?S'2) ?D1 \<Longrightarrow> Lifting.POS (BNF_Def.rel_fun ?R2 ?S2 OO BNF_Def.rel_fun ?R'2 ?S'2) (BNF_Def.rel_fun ?C1 ?D1) | [
"(\\<Longrightarrow>)",
"Lifting.NEG",
"(OO)",
"(\\<Longrightarrow>)",
"Lifting.POS",
"(OO)",
"Lifting.POS",
"(OO)",
"BNF_Def.rel_fun",
"BNF_Def.rel_fun",
"BNF_Def.rel_fun"
] | [
"record state =\n floor :: \"int\" \\<comment> \\<open>current position of the lift\\<close>\n \"open\" :: \"bool\" \\<comment> \\<open>whether the door is opened at floor\\<close>\n stop :: \"bool\" \\<comment> \\<open>whether the lift is stopped at floor\\<close>\n req :: \"in... | \<lbrakk>?H0 (?H1 x_0 x_1) x_2; ?H2 (?H3 x_3 x_4) x_5\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H6 x_0 x_3) (?H7 x_1 x_4)) (?H8 x_2 x_5) | Error |
MicroJava/BV/Correct | Correct.approx_loc_Err | Correct.approx_loc ?G ?hp (List.replicate ?n ?v) (List.replicate ?n Err.err.Err) | [
"Correct.approx_loc",
"List.replicate",
"List.replicate",
"Err.err.Err"
] | [
"definition approx_val :: \"[jvm_prog,aheap,val,ty err] \\<Rightarrow> bool\" where\n \"approx_val G h v any == case any of Err \\<Rightarrow> True | OK T \\<Rightarrow> G,h\\<turnstile>v::\\<preceq>T\"",
"definition approx_loc :: \"[jvm_prog,aheap,val list,locvars_type] \\<Rightarrow> bool\" where\n \"approx_l... | ?H0 x_0 x_1 (?H1 x_2 x_3) (?H2 x_2 ?H3) | Error |
Library/List_Lexorder | List_Lexorder.not_less_Nil | \<not> ?x < [] | [
"HOL.Not",
"(<)",
"[]"
] | [
"definition\n list_less_def: \"xs < ys \\<longleftrightarrow> (xs, ys) \\<in> lexord {(u, v). u < v}\"",
"definition\n list_le_def: \"(xs :: _ list) \\<le> ys \\<longleftrightarrow> xs < ys \\<or> xs = ys\"",
"definition \"(inf :: 'a list \\<Rightarrow> _) = min\"",
"definition \"(sup :: 'a list \\<Rightarr... | \<not> ?H0 x_0 ?H1 | False |
Data_Structures/Tree23_Set | Tree23_Set.height'_node33 | 0 < Tree23.height_class.height ?m \<Longrightarrow> Tree23_Set.hD (Tree23_Set.node33 ?l ?a ?m ?b ?r) = Orderings.ord_class.max (Tree23.height_class.height ?l) (Orderings.ord_class.max (Tree23.height_class.height ?m) (Tree23_Set.hD ?r)) + 1 | [
"(\\<Longrightarrow>)",
"(<)",
"0",
"Tree23.height_class.height",
"(=)",
"Tree23_Set.hD",
"Tree23_Set.node33",
"(+)",
"Orderings.ord_class.max",
"Tree23.height_class.height",
"Orderings.ord_class.max",
"Tree23.height_class.height",
"Tree23_Set.hD",
"1"
] | [
"fun hD :: \"'a upD \\<Rightarrow> nat\" where\n\"hD (TD t) = height t\" |\n\"hD (UF t) = height t + 1\"",
"fun node33 :: \"'a tree23 \\<Rightarrow> 'a \\<Rightarrow> 'a tree23 \\<Rightarrow> 'a \\<Rightarrow> 'a upD \\<Rightarrow> 'a upD\" where\n\"node33 t1 a t2 b (TD t3) = TD(Node3 t1 a t2 b t3)\" |\n\"node33 ... | ?H0 ?H1 (?H2 x_0) \<Longrightarrow> ?H3 (?H4 x_1 x_2 x_0 x_3 x_4) = ?H5 (?H6 (?H2 x_1) (?H6 (?H2 x_0) (?H3 x_4))) ?H7 | Error |
Computational_Algebra/Formal_Laurent_Series | Formal_Laurent_Series.fls_X_inv_times_conv_shift(1) | Formal_Laurent_Series.fls_X_inv * ?f = Formal_Laurent_Series.fls_shift 1 ?f | [
"(=)",
"(*)",
"Formal_Laurent_Series.fls_X_inv",
"Formal_Laurent_Series.fls_shift",
"1"
] | [
"definition fls_shift :: \"int \\<Rightarrow> ('a::zero) fls \\<Rightarrow> 'a fls\"\n where \"fls_shift n f \\<equiv> Abs_fls (\\<lambda>k. f $$ (k+n))\"\n\\<comment> \\<open>Since the index set is unbounded in both directions, we can shift in either direction.\\<close>",
"typedef (overloaded) 'a fls = \"{f::in... | ?H0 ?H1 x_0 = ?H2 ?H3 x_0 | Error |
SPARK/Examples/Liseq/Longest_Increasing_Subsequence | Longest_Increasing_Subsequence.iseq_singleton | ?i < ?l \<Longrightarrow> Set.insert ?i Orderings.bot_class.bot \<in> Longest_Increasing_Subsequence.iseq ?xs ?l | [
"(\\<Longrightarrow>)",
"(<)",
"(\\<in>)",
"Set.insert",
"Orderings.bot_class.bot",
"Longest_Increasing_Subsequence.iseq"
] | [
"definition iseq :: \"(nat \\<Rightarrow> 'a::linorder) \\<Rightarrow> nat \\<Rightarrow> nat set set\" where\n \"iseq xs l = {is. (\\<forall>i\\<in>is. i < l) \\<and>\n (\\<forall>i\\<in>is. \\<forall>j\\<in>is. i \\<le> j \\<longrightarrow> xs i \\<le> xs j)}\"",
"definition liseq :: \"(nat \\<Rightarrow> ... | ?H0 x_0 x_1 \<Longrightarrow> ?H1 (?H2 x_0 ?H3) (?H4 x_2 x_1) | Error |
Library/Order_Continuity | Order_Continuity.cclfp_lowerbound | Fun.monotone_on Orderings.top_class.top (\<le>) (\<le>) ?f \<Longrightarrow> ?f ?A \<le> ?A \<Longrightarrow> Order_Continuity.countable_complete_lattice_class.cclfp ?f \<le> ?A | [
"(\\<Longrightarrow>)",
"Fun.monotone_on",
"Orderings.top_class.top",
"(\\<le>)",
"(\\<le>)",
"(\\<Longrightarrow>)",
"(\\<le>)",
"(\\<le>)",
"Order_Continuity.countable_complete_lattice_class.cclfp"
] | [
"definition\n sup_continuous :: \"('a::countable_complete_lattice \\<Rightarrow> 'b::countable_complete_lattice) \\<Rightarrow> bool\"\nwhere\n \"sup_continuous F \\<longleftrightarrow> (\\<forall>M::nat \\<Rightarrow> 'a. mono M \\<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))\"",
"definition\n inf_conti... | \<lbrakk>?H0 ?H1 ?H2 ?H2 x_0; ?H2 (x_0 x_1) x_1\<rbrakk> \<Longrightarrow> ?H2 (?H3 x_0) x_1 | False |
Analysis/Bochner_Integration | Bochner_Integration.has_bochner_integral_diff | Bochner_Integration.has_bochner_integral ?M ?f ?x \<Longrightarrow> Bochner_Integration.has_bochner_integral ?M ?g ?y \<Longrightarrow> Bochner_Integration.has_bochner_integral ?M (\<lambda>x. ?f x - ?g x) (?x - ?y) | [
"(\\<Longrightarrow>)",
"Bochner_Integration.has_bochner_integral",
"(\\<Longrightarrow>)",
"Bochner_Integration.has_bochner_integral",
"Bochner_Integration.has_bochner_integral",
"(-)"
] | [
"inductive has_bochner_integral :: \"'a measure \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'b::{real_normed_vector, second_countable_topology} \\<Rightarrow> bool\"\n for M f x where\n \"f \\<in> borel_measurable M \\<Longrightarrow>\n (\\<And>i. simple_bochner_integrable M (s i)) \\<Longrightarrow>\... | \<lbrakk>?H0 x_0 x_1 x_2; ?H0 x_0 x_3 x_4\<rbrakk> \<Longrightarrow> ?H0 x_0 (\<lambda>x. ?H1 (x_1 x) (x_3 x)) (?H1 x_2 x_4) | False |
HOLCF/Adm | Adm.admD2 | Adm.adm (\<lambda>x. \<not> ?P x) \<Longrightarrow> Porder.po_class.chain ?Y \<Longrightarrow> ?P (Porder.po_class.lub (?Y ` Orderings.top_class.top)) \<Longrightarrow> \<exists>i. ?P (?Y i) | [
"(\\<Longrightarrow>)",
"Adm.adm",
"(\\<Longrightarrow>)",
"Porder.po_class.chain",
"(\\<Longrightarrow>)",
"Porder.po_class.lub",
"(`)",
"Orderings.top_class.top",
"HOL.Ex"
] | [
"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))\"",
"definition adm :: \"('a::cpo \\<Rightarrow> bool) \\<Rightarrow> bool\"\n where \"adm P \... | \<lbrakk>?H0 (\<lambda>x. \<not> x_0 x); ?H1 x_1; x_0 (?H2 (?H3 x_1 ?H4))\<rbrakk> \<Longrightarrow> \<exists>i. x_0 (x_1 i) | False |
Computational_Algebra/Formal_Power_Series | Formal_Power_Series.fps_div_zero | 0 div ?g = 0 | [
"(=)",
"(div)",
"0",
"0"
] | [
"typedef 'a fps = \"{f :: nat \\<Rightarrow> 'a. True}\"\n morphisms fps_nth Abs_fps",
"definition fps_zero_def: \"0 = Abs_fps (\\<lambda>n. 0)\"",
"definition fps_one_def: \"1 = Abs_fps (\\<lambda>n. if n = 0 then 1 else 0)\"",
"definition fps_plus_def: \"(+) = (\\<lambda>f g. Abs_fps (\\<lambda>n. f $ n + ... | ?H0 ?H1 x_0 = ?H1 | False |
Datatype_Examples/Regex_ACIDZ | Regex_ACIDZ.elim_zeros_distribute_idem | Regex_ACIDZ.elim_zeros (Regex_ACIDZ.distribute ?s (Regex_ACIDZ.elim_zeros ?r)) = Regex_ACIDZ.elim_zeros (Regex_ACIDZ.distribute ?s ?r) | [
"(=)",
"Regex_ACIDZ.elim_zeros",
"Regex_ACIDZ.distribute",
"Regex_ACIDZ.elim_zeros",
"Regex_ACIDZ.elim_zeros",
"Regex_ACIDZ.distribute"
] | [
"fun elim_zeros where\n \"elim_zeros (Alt r s) =\n (let r' = elim_zeros r; s' = elim_zeros s in\n if r' = Zero then s' else if s' = Zero then r' else Alt r' s')\"\n| \"elim_zeros (Conc r s) = (let r' = elim_zeros r in if r' = Zero then Zero else Conc r' s)\"\n| \"elim_zeros r = r\"",
"fun distribute where... | ?H0 (?H1 x_0 (?H0 x_1)) = ?H0 (?H1 x_0 x_1) | False |
Auth/Smartcard/ShoupRubin | ShoupRubin.Outpts_A_Card_4 | EventSC.event.Outpts ?C ?A \<lbrace>Message.msg.Nonce ?Na, Message.msg.Crypt (Smartcard.crdK (EventSC.card.Card ?A)) (Message.msg.Nonce ?Na)\<rbrace> \<in> List.list.set ?evs \<Longrightarrow> ?evs \<in> ShoupRubin.sr \<Longrightarrow> legalUse ?C \<and> ?C = EventSC.ca... | [
"(\\<Longrightarrow>)",
"(\\<in>)",
"EventSC.event.Outpts",
"Message.msg.MPair",
"Message.msg.Nonce",
"Message.msg.Crypt",
"Smartcard.crdK",
"EventSC.card.Card",
"Message.msg.Nonce",
"List.list.set",
"(\\<Longrightarrow>)",
"(\\<in>)",
"ShoupRubin.sr",
"(\\<and>)",
"Smartcard.legalUse",
... | [
"definition legalUse :: \"card => bool\" (\"legalUse (_)\") where\n \"legalUse C == C \\<notin> stolen\"",
"definition Unique :: \"[event, event list] => bool\" (\"Unique _ on _\") where\n \"Unique ev on evs == \n ev \\<notin> set (tl (dropWhile (% z. z \\<noteq> ev) evs))\""
] | \<lbrakk>?H0 (?H1 x_0 x_1 (?H2 (?H3 x_2) (?H4 (?H5 (?H6 x_1)) (?H3 x_2)))) (?H7 x_3); ?H8 x_3 ?H9\<rbrakk> \<Longrightarrow> ?H10 x_0 \<and> x_0 = ?H6 x_1 \<and> ?H0 (?H11 x_1 (?H6 x_1) (?H12 x_1)) (?H7 x_3) | Error |
SPARK/Examples/RIPEMD-160/F | Fun.inj_on_imageI2 | Fun.inj_on (?f' \<circ> ?f) ?A \<Longrightarrow> Fun.inj_on ?f ?A | [
"(\\<Longrightarrow>)",
"Fun.inj_on",
"(\\<circ>)",
"Fun.inj_on"
] | [] | ?H0 (?H1 x_0 x_1) x_2 \<Longrightarrow> ?H2 x_1 x_2 | False |
Analysis/Cross3 | Cross3.cross_components(3) | Cross3.cross3 ?x ?y $ 3 = ?x $ 1 * ?y $ 2 - ?y $ 1 * ?x $ 2 | [
"(=)",
"($)",
"Cross3.cross3",
"Num.numeral_class.numeral",
"Num.num.Bit1",
"Num.num.One",
"(-)",
"(*)",
"($)",
"1",
"($)",
"Num.numeral_class.numeral",
"Num.num.Bit0",
"Num.num.One",
"(*)",
"($)",
"1",
"($)",
"Num.numeral_class.numeral",
"Num.num.Bit0",
"Num.num.One"
] | [
"definition\\<^marker>\\<open>tag important\\<close> cross3 :: \"[real^3, real^3] \\<Rightarrow> real^3\" (infixr \"\\<times>\" 80)\n where \"a \\<times> b \\<equiv>\n vector [a$2 * b$3 - a$3 * b$2,\n a$3 * b$1 - a$1 * b$3,\n a$1 * b$2 - a$2 * b$1]\""
] | ?H0 (?H1 x_0 x_1) (?H2 (?H3 ?H4)) = ?H5 (?H6 (?H0 x_0 ?H7) (?H0 x_1 (?H2 (?H8 ?H4)))) (?H6 (?H0 x_1 ?H7) (?H0 x_0 (?H2 (?H8 ?H4)))) | Error |
Library/FSet | FSet.fsingleton_finsert_inj_eq' | (FSet.finsert ?a ?A = FSet.finsert ?b Orderings.bot_class.bot) = (?a = ?b \<and> ?A \<le> FSet.finsert ?b Orderings.bot_class.bot) | [
"(=)",
"(=)",
"FSet.finsert",
"FSet.finsert",
"Orderings.bot_class.bot",
"(\\<and>)",
"(=)",
"(\\<le>)",
"FSet.finsert",
"Orderings.bot_class.bot"
] | [
"typedef 'a fset = \"{A :: 'a set. finite A}\" morphisms fset Abs_fset",
"definition less_fset :: \"'a fset \\<Rightarrow> 'a fset \\<Rightarrow> bool\" where \"xs < ys \\<equiv> xs \\<le> ys \\<and> xs \\<noteq> (ys::'a fset)\"",
"definition \"HOL.equal A B \\<longleftrightarrow> A |\\<subseteq>| B \\<and> B ... | (?H0 x_0 x_1 = ?H0 x_2 ?H1) = (x_0 = x_2 \<and> ?H2 x_1 (?H0 x_2 ?H1)) | False |
Library/Extended_Real | Extended_Real.sup_continuous_ereal_of_enat | Order_Continuity.sup_continuous ?f \<Longrightarrow> Order_Continuity.sup_continuous (\<lambda>x. Extended_Real.ereal_of_enat (?f x)) | [
"(\\<Longrightarrow>)",
"Order_Continuity.sup_continuous",
"Order_Continuity.sup_continuous"
] | [
"definition\n sup_continuous :: \"('a::countable_complete_lattice \\<Rightarrow> 'b::countable_complete_lattice) \\<Rightarrow> bool\"\nwhere\n \"sup_continuous F \\<longleftrightarrow> (\\<forall>M::nat \\<Rightarrow> 'a. mono M \\<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))\"",
"definition\n sup_conti... | ?H0 x_0 \<Longrightarrow> ?H1 (\<lambda>x. ?H2 (x_0 x)) | False |
ex/Meson_Test | Meson_Test.TOP001_2 | (\<forall>Vf U. ?element_of_set (U, ?union_of_members Vf) \<longrightarrow> ?element_of_set (U, ?f1.0 (Vf, U))) \<and> (\<forall>U Vf. ?element_of_set (U, ?union_of_members Vf) \<longrightarrow> ?element_of_collection (?f1.0 (Vf, U), Vf)) \<and> (\<forall>U Uu1 Vf. ... | [
"(\\<longrightarrow>)",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)",
"HOL.All",
"(\\<and>)... | [] | (\<forall>Vf U. x_0 (?H0 U (x_1 Vf)) \<longrightarrow> x_0 (?H0 U (x_2 (?H0 Vf U)))) \<and> (\<forall>U Vf. x_0 (?H0 U (x_1 Vf)) \<longrightarrow> x_3 (?H0 (x_2 (?H0 Vf U)) Vf)) \<and> (\<forall>U Uu1 Vf. x_0 (?H0 U Uu1) \<and> x_3 (?H0 Uu1 Vf) \<longrightarrow> x_0 (?H0 U (x_1 Vf))) \<and> (\<forall>Vf X. x_4 (?H0 X V... | False |
Unix/Unix | Unix.eval(1) | Unix.access ?root ?path ?uid ?perms = (case Nested_Environment.lookup ?root ?path of Option.option.None \<Rightarrow> Option.option.None | Option.option.Some file \<Rightarrow> if ?uid = 0 \<or> ?uid = Unix.att.owner (Unix.attributes file) \<or> ?perms \<le> Un... | [
"(=)",
"Unix.access",
"Option.option.case_option",
"Option.option.None",
"Nested_Environment.lookup"
] | [
"primrec lookup :: \"('a, 'b, 'c) env \\<Rightarrow> 'c list \\<Rightarrow> ('a, 'b, 'c) env option\"\n and lookup_option :: \"('a, 'b, 'c) env option \\<Rightarrow> 'c list \\<Rightarrow> ('a, 'b, 'c) env option\"\nwhere\n \"lookup (Val a) xs = (if xs = [] then Some (Val a) else None)\"\n | \"lookup (Env b es... | ?H0 x_0 x_1 x_2 x_3 = ?H1 ?H2 (\<lambda>file. if x_2 = ?H3 \<or> x_2 = ?H4 (?H5 file) \<or> ?H6 x_3 (?H7 (?H5 file)) then ?H8 file else ?H2) (?H9 x_0 x_1) | Error |
ex/Tree23 | Tree23.ord'_add0 | Tree23.ord' ?k1.0 ?t ?k2.0 \<Longrightarrow> Tree23.opt_less ?k1.0 (Option.option.Some ?k) \<Longrightarrow> Tree23.opt_less (Option.option.Some ?k) ?k2.0 \<Longrightarrow> Tree23.ord' ?k1.0 (Tree23.add0 ?k ?y ?t) ?k2.0 | [
"(\\<Longrightarrow>)",
"Tree23.ord'",
"(\\<Longrightarrow>)",
"Tree23.opt_less",
"Option.option.Some",
"(\\<Longrightarrow>)",
"Tree23.opt_less",
"Option.option.Some",
"Tree23.ord'",
"Tree23.add0"
] | [
"primrec ord' :: \"key option \\<Rightarrow> 'a tree23 \\<Rightarrow> key option \\<Rightarrow> bool\" where\n\"ord' x Empty y = opt_less x y\" |\n\"ord' x (Branch2 l p r) y = (ord' x l (Some (fst p)) & ord' (Some (fst p)) r y)\" |\n\"ord' x (Branch3 l p m q r) y = (ord' x l (Some (fst p)) & ord' (Some (fst p)) m (... | \<lbrakk>?H0 x_0 x_1 x_2; ?H1 x_0 (?H2 x_3); ?H1 (?H2 x_3) x_2\<rbrakk> \<Longrightarrow> ?H0 x_0 (?H3 x_3 x_4 x_1) x_2 | False |
Analysis/Complex_Transcendental | Complex_Transcendental.Arg2pi_lt_2pi | Complex_Transcendental.Arg2pi ?z < 2 * Transcendental.pi | [
"(<)",
"Complex_Transcendental.Arg2pi",
"(*)",
"Num.numeral_class.numeral",
"Num.num.Bit0",
"Num.num.One",
"Transcendental.pi"
] | [
"definition\\<^marker>\\<open>tag important\\<close> \"moebius a b c d \\<equiv> (\\<lambda>z. (a*z+b) / (c*z+d :: 'a :: field))\"",
"definition\\<^marker>\\<open>tag important\\<close> is_Arg :: \"[complex,real] \\<Rightarrow> bool\"\n where \"is_Arg z r \\<equiv> z = of_real(norm z) * exp(\\<i> * of_real r)\""... | ?H0 (?H1 x_0) (?H2 (?H3 (?H4 ?H5)) ?H6) | Error |
ex/Conditional_Parametricity_Examples | Conditional_Parametricity_Examples.t4_thm | BNF_Def.rel_fun (BNF_Def.rel_fun (List.list.list_all2 ?A) (BNF_Def.rel_fun (List.list.list_all2 ?A) (List.list.list_all2 ?A))) (BNF_Def.rel_fun (List.list.list_all2 ?A) (List.list.list_all2 ?A)) Conditional_Parametricity_Examples.t4 Conditional_Parametricity_Examples.t4 | [
"BNF_Def.rel_fun",
"BNF_Def.rel_fun",
"List.list.list_all2",
"BNF_Def.rel_fun",
"List.list.list_all2",
"List.list.list_all2",
"BNF_Def.rel_fun",
"List.list.list_all2",
"List.list.list_all2",
"Conditional_Parametricity_Examples.t4",
"Conditional_Parametricity_Examples.t4"
] | [] | ?H0 (?H1 (?H2 x_0) (?H3 (?H2 x_0) (?H2 x_0))) (?H3 (?H2 x_0) (?H2 x_0)) ?H4 ?H5 | False |
HOLCF/Sprod | Sprod.sprodE | (?p = \<bottom> \<Longrightarrow> ?thesis) \<Longrightarrow> (\<And>x y. ?p = (:x, y:) \<Longrightarrow> \<not> x = \<bottom> \<Longrightarrow> \<not> y = \<bottom> \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis | [
"(\\<Longrightarrow>)",
"(\\<Longrightarrow>)",
"(=)",
"\\<bottom>",
"(\\<Longrightarrow>)",
"Pure.all"
] | [
"definition sfst :: \"('a ** 'b) \\<rightarrow> 'a\"\n where \"sfst = (\\<Lambda> p. fst (Rep_sprod p))\"",
"definition ssnd :: \"('a ** 'b) \\<rightarrow> 'b\"\n where \"ssnd = (\\<Lambda> p. snd (Rep_sprod p))\"",
"definition spair :: \"'a \\<rightarrow> 'b \\<rightarrow> ('a ** 'b)\"\n where \"spair = (\\... | \<lbrakk>x_0 = ?H0 \<Longrightarrow> x_1; PROP ?H1 (\<lambda>x. PROP ?H2 (\<lambda>y. (\<lbrakk>x_0 = ?H3 (?H4 ?H5 x) y; \<not> x = ?H6; \<not> y = ?H7\<rbrakk> \<Longrightarrow> x_1)))\<rbrakk> \<Longrightarrow> x_1 | False |
Data_Structures/Trie_Fun | Trie_Fun.set_insert | Trie_Fun.set (Trie_Fun.insert ?xs ?t) = Lattices.sup_class.sup (Trie_Fun.set ?t) (Set.insert ?xs Orderings.bot_class.bot) | [
"(=)",
"Trie_Fun.set",
"Trie_Fun.insert",
"Lattices.sup_class.sup",
"Trie_Fun.set",
"Set.insert",
"Orderings.bot_class.bot"
] | [
"definition set :: \"'a trie \\<Rightarrow> 'a list set\" where\n[simp]: \"set t = {xs. isin t xs}\"",
"fun insert :: \"'a list \\<Rightarrow> 'a trie \\<Rightarrow> 'a trie\" where\n\"insert [] (Nd b m) = Nd True m\" |\n\"insert (x#xs) (Nd b m) =\n (let s = (case m x of None \\<Rightarrow> empty | Some t \\<Ri... | ?H0 (?H1 x_0 x_1) = ?H2 (?H0 x_1) (?H3 x_0 ?H4) | False |
Bali/Decl | Decl.type_is_iface | Decl.is_type ?G (Type.ty.RefT (Type.ref_ty.IfaceT ?I)) \<Longrightarrow> \<not> Map.map_of (Decl.prog.ifaces ?G) ?I = Option.option.None | [
"(\\<Longrightarrow>)",
"Decl.is_type",
"Type.ty.RefT",
"Type.ref_ty.IfaceT",
"HOL.Not",
"(=)",
"Map.map_of",
"Decl.prog.ifaces",
"Option.option.None"
] | [
"primrec is_type :: \"prog \\<Rightarrow> ty \\<Rightarrow> bool\"\n and isrtype :: \"prog \\<Rightarrow> ref_ty \\<Rightarrow> bool\"\nwhere\n \"is_type G (PrimT pt) = True\"\n| \"is_type G (RefT rt) = isrtype G rt\"\n| \"isrtype G (NullT) = True\"\n| \"isrtype G (IfaceT tn) = is_iface G tn\"\n| \"isrtype G (... | ?H0 x_0 (?H1 (?H2 x_1)) \<Longrightarrow> \<not> ?H3 (?H4 x_0) x_1 = ?H5 | Error |
HOLCF/Library/List_Cpo | List_Cpo.list_chain_cases | Porder.po_class.chain ?S \<Longrightarrow> (?S = (\<lambda>i. []) \<Longrightarrow> ?thesis) \<Longrightarrow> (\<And>A B. Porder.po_class.chain A \<Longrightarrow> Porder.po_class.chain B \<Longrightarrow> ?S = (\<lambda>i. A i # B i) \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis | [
"(\\<Longrightarrow>)",
"Porder.po_class.chain",
"(\\<Longrightarrow>)",
"(\\<Longrightarrow>)",
"(=)",
"(\\<Longrightarrow>)",
"Pure.all"
] | [
"definition\n \"xs \\<sqsubseteq> ys \\<longleftrightarrow> list_all2 (\\<sqsubseteq>) xs ys\"",
"definition\n match_Nil :: \"'a::cpo list \\<rightarrow> 'b match \\<rightarrow> 'b match\"\nwhere\n \"match_Nil = (\\<Lambda> xs k. case xs of [] \\<Rightarrow> k | y # ys \\<Rightarrow> Fixrec.fail)\"",
"defini... | \<lbrakk>?H0 x_0; x_0 = (\<lambda>i. ?H1) \<Longrightarrow> x_1; PROP ?H2 (\<lambda>A. PROP ?H3 (\<lambda>B. (\<lbrakk>?H4 A; ?H0 B; x_0 = (\<lambda>i. ?H5 (A i) (B i))\<rbrakk> \<Longrightarrow> x_1)))\<rbrakk> \<Longrightarrow> x_1 | False |
MicroJava/DFA/Product | Product_Type.SigmaD1 | (?a, ?b) \<in> Product_Type.Sigma ?A ?B \<Longrightarrow> ?a \<in> ?A | [
"(\\<Longrightarrow>)",
"(\\<in>)",
"Product_Type.Pair",
"Product_Type.Sigma",
"(\\<in>)"
] | [
"definition le :: \"'a ord \\<Rightarrow> 'b ord \\<Rightarrow> ('a * 'b) ord\" where\n\"le rA rB == %(a,b) (a',b'). a <=_rA a' & b <=_rB b'\"",
"definition sup :: \"'a ebinop \\<Rightarrow> 'b ebinop \\<Rightarrow> ('a * 'b)ebinop\" where\n\"sup f g == %(a1,b1)(a2,b2). Err.sup Pair (a1 +_f a2) (b1 +_g b2)\"",
... | ?H0 (?H1 x_0 x_1) (?H2 x_2 x_3) \<Longrightarrow> ?H3 x_0 x_2 | False |
Analysis/Nonnegative_Lebesgue_Integration | Nonnegative_Lebesgue_Integration.simple_function_point_measure | Nonnegative_Lebesgue_Integration.simple_function (Nonnegative_Lebesgue_Integration.point_measure ?A ?f) ?g = Finite_Set.finite (?g ` ?A) | [
"(=)",
"Nonnegative_Lebesgue_Integration.simple_function",
"Nonnegative_Lebesgue_Integration.point_measure",
"Finite_Set.finite",
"(`)"
] | [
"definition\\<^marker>\\<open>tag important\\<close> \"simple_function M g \\<longleftrightarrow>\n finite (g ` space M) \\<and>\n (\\<forall>x \\<in> g ` space M. g -` {x} \\<inter> space M \\<in> sets M)\"",
"definition\\<^marker>\\<open>tag important\\<close> simple_integral :: \"'a measure \\<Rightarrow... | ?H0 (?H1 x_0 x_1) x_2 = ?H2 (?H3 x_2 x_0) | False |
Auth/Smartcard/ShoupRubin | ShoupRubin.Outpts_B_Card_unique_key | EventSC.event.Outpts (EventSC.card.Card ?B) ?B \<lbrace>Message.msg.Nonce ?Nb, Message.msg.Key ?SK, ?Cert1.0, ?Cert2.0\<rbrace> \<in> List.list.set ?evs \<Longrightarrow> EventSC.event.Outpts (EventSC.card.Card ?B') ?B' \<lbrace>Message.msg.Nonce ?Nb', Message.msg.Key ?SK, ?Cert1', ?Cert2'\<rb... | [
"(\\<Longrightarrow>)",
"(\\<in>)",
"EventSC.event.Outpts",
"EventSC.card.Card",
"Message.msg.MPair",
"Message.msg.Nonce",
"Message.msg.MPair",
"Message.msg.Key",
"Message.msg.MPair",
"List.list.set",
"(\\<Longrightarrow>)",
"(\\<in>)",
"EventSC.event.Outpts",
"EventSC.card.Card",
"Messa... | [
"definition Unique :: \"[event, event list] => bool\" (\"Unique _ on _\") where\n \"Unique ev on evs == \n ev \\<notin> set (tl (dropWhile (% z. z \\<noteq> ev) evs))\""
] | \<lbrakk>?H0 (?H1 (?H2 x_0) x_0 (?H3 (?H4 x_1) (?H3 (?H5 x_2) (?H3 x_3 x_4)))) (?H6 x_5); ?H0 (?H1 (?H2 x_6) x_6 (?H3 (?H4 x_7) (?H3 (?H5 x_2) (?H3 x_8 x_9)))) (?H6 x_5); ?H7 x_5 ?H8\<rbrakk> \<Longrightarrow> x_0 = x_6 \<and> x_1 = x_7 \<and> x_3 = x_8 \<and> x_4 = x_9 | Error |
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