hcju/isabelleps huggingface dataset:
divides_primepow_nat: fixes p :: "nat"   and d :: "nat"   and k :: "nat" assumes "prime p" shows "(d dvd p ^ k) = (\<exists>i\<le>k. d = p ^ i)"
s_Pair_inv: fixes \<pi> :: "Syntax.term list"   and e\<^sub>1 :: "Syntax.term"   and e\<^sub>2 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Syntax.Pair e\<^sub>1 e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e\<^sub>1' :: "Syntax.term" where "e' = Syntax.Pair e\<^sub>1' e\<^sub>2" and "\<lless>\<pi>, e\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater>"   | e\<^sub>2'' :: "Syntax.term" where "e' = Syntax.Pair e\<^sub>1 e\<^sub>2''" and "value e\<^sub>1" and "\<lless>\<pi>, e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>2''\<ggreater>"
d_min: fixes C :: "'f" assumes "C \<in> N"   and "C < D" shows "I_of N \<Turnstile> {C}"
valid_rem_xs: shows "valid_unMultigraph (rem_unPath xs G)"
SKIP_starD: fixes s :: "char list \<Rightarrow> int"   and c :: "com"   and t :: "char list \<Rightarrow> int" assumes "(SKIP, s) \<rightarrow>* (c, t)" shows "c = SKIP"
Image_Collect_case_prod: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set" shows "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
assms: shows "z :\<^sub>h w = \<infinity>\<^sub>h"
span_is_submodule: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "LinearCombinations.submodule K (span S) V"
bounded_closure_image: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set" assumes "bounded (f ` closure S)" shows "bounded (f ` S)"
ring_hom_cringI: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "cring R"   and "cring S"   and "h \<in> ring_hom R S" shows "ring_hom_cring R S h"
sel_1: fixes x11 :: "nat"   and x12 :: "nat" shows "phil (picks x11 x12) = x11"
igSwapIGOp_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSwapGOp MOD" shows "igSwapIGOp (fromMOD MOD)"
not_all_ge: fixes d :: "'a"   and c :: "'a" assumes "arc_pos d" shows "\<exists>e\<ge>0::'a. arc_pos e \<and> \<not> d * e \<le> c"
basis_in_latticeI: fixes fs :: "int vec list"   and f :: "int vec" assumes "set fs \<subseteq> carrier_vec n"   and "f \<in> set fs" shows "f \<in> lattice_of fs"
zero_divisorE: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.mult_zero times zero"   and "mult_zero.zero_divisor times zero a" obtains b :: "'b" where "b \<noteq> zero" and "times a b = zero"
sin_int_2pin: fixes n :: "int" shows "sin (2 * pi * real_of_int n) = 0"
Special_axP_sf: fixes p :: "tm" shows "Sigma_fm (Special_axP p)"
anonymity_prefs_from_table: fixes xs :: "('agent \<times> 'alt set list) list"   and ys :: "('agent \<times> 'alt set list) list" assumes "prefs_from_table_wf agents alts xs"   and "prefs_from_table_wf agents alts ys"   and "mset (map snd xs) = mset (map snd ys)" shows "scf (prefs_from_table xs) = scf (prefs_from_table ys)"
no_overlaps_not_unefined: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and ft :: "('a, 'c) flow_entry_match list"   and p :: "'b" assumes "no_overlaps \<gamma> ft" shows "OF_priority_match \<gamma> ft p \<noteq> Undefined"
right_adjoint_simps_3: fixes f :: "'b" assumes "D.is_left_adjoint f" shows "trg\<^sub>D f\<^sup>*\<^sup>D = src\<^sub>D f"
disc_1: fixes x1 :: "'d" shows "is_Pure (Pure x1)"
rev_induct2: fixes xs :: "'b list"   and ys :: "'c list"   and P :: "'b list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (xs @ [x]) (ys @ [y])" shows "P xs ys"
proj_mult_scalar: fixes k :: "'k"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (p \<odot> q) = p * proj_poly k q"
some_GB_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "some_GB F = (SOME G. finite G \<and> is_Groebner_basis G \<and> pmdl G = pmdl F)"
rep_eq: shows "fls_nth 0 = (\<lambda>_. 0::'a)"
word_bw_comms_2: fixes x :: "'a word"   and y :: "'a word" shows "x OR y = y OR x"
sees: shows "P \<turnstile> C sees clinit, Static :  []\<rightarrow>Void = m' in C"
simps: fixes xs :: "'varSort"   and x :: "'var"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" shows "qGoodAbs (qAbs xs x X) = qGood X"
comm_ring_hom_axioms: shows "comm_ring_hom of_rat"
combine_options_cases: fixes x :: "'b option"   and P :: "'b option \<Rightarrow> 'c option \<Rightarrow> bool"   and y :: "'c option" assumes "x = None \<Longrightarrow> P x y"   and "y = None \<Longrightarrow> P x y"   and "\<And>a b. \<lbrakk>x = Some a; y = Some b\<rbrakk> \<Longrightarrow> P x y" shows "P x y"
polynomial_of_mset_uminus: fixes za :: "(char list multiset \<times> int) multiset" shows "polynomial_of_mset {#case x of (a, b) \<Rightarrow> (a, - b). x \<in># za#} = - polynomial_of_mset za"
z_def: shows "z \<equiv> add_rat_1 r1 y"
Mp_ident_iff: fixes f :: "int poly" shows "(Mp f = f) = (\<forall>n. poly.coeff f n \<in> {0..<p})"
sel_2: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "ltl (greedy_conforming_path p \<sigma> \<sigma>' v0) = (if deadend v0 then LNil else if v0 \<in> VV p then greedy_conforming_path p \<sigma> \<sigma>' (\<sigma> v0) else greedy_conforming_path p \<sigma> \<sigma>' (\<sigma>' v0))"
exec_move_ALength: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e\<bullet>length) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
cfWritePreservesLastBlockInvariant: fixes cfile1 :: "CFile"   and byteIndex :: "nat"   and "value" :: "byte"   and cfile2 :: "CFile" assumes "reachabilityInvariant cfile1"   and "cfWrite cfile1 byteIndex value = Some cfile2" shows "lastBlockInvariant cfile2"
finite_basis_exists: assumes "fin_dim" shows "\<exists>\<beta>. finite \<beta> \<and> basis \<beta>"
sc_sound: fixes P :: "'b \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "sound (\<lambda>s. c * P s)"
GroupHom_idhom: shows "GroupHom R (id \<down> R)"
maximally_connected_def: fixes H :: "('a, 'b) graph"   and G :: "('a, 'b) graph" shows "maximally_connected H G \<equiv> \<forall>v\<in>nodes G. \<forall>v'\<in>nodes G. nodes_connected G v v' \<longrightarrow> nodes_connected H v v'"
Cons: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "r x y"   and "listrelp r xs ys" shows "listrelp r (x # xs) (y # ys)"
simps: fixes data :: "'a"   and subtrees :: "'a rose_tree list" shows "children (Tree (data, subtrees)) = subtrees"
weave_append1: fixes xs :: "'a list"   and ys :: "'a list"   and A :: "nat set"   and x :: "'a" assumes "length xs + length ys \<in> A"   and "length xs = card {a \<in> A. a < length xs + length ys}" shows "weave A (xs @ [x]) ys = weave A xs ys @ [x]"
partition_axioms: shows "Set_Theory.partition (Sigma I \<FF>) rel.Partition"
vector_space_prod_axioms: shows "vector_space_prod (*\<^sub>R) (*\<^sub>R)"
equiv_subseteq_in_sym: fixes r :: "('a \<times> 'a) set"   and X :: "'a set"   and x :: "'a"   and y :: "'a"   and Y :: "'a set" assumes "r `` X \<subseteq> X"   and "(x, y) \<in> r"   and "y \<in> X"   and "equiv Y r"   and "X \<subseteq> Y" shows "x \<in> X"
traces_list_all: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and P :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool" assumes "c \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> c'"   and "P c'"   and "\<And>c c'. \<lbrakk>c \<Rightarrow> c'; P c'\<rbrakk> \<Longrightarrow> P c" shows "(\<forall>x\<in>set T. P x) \<and> P c"
hyps_for_simp: fixes h :: "('form, 'var) out_port"   and n :: "('form, 'rule) graph_node"   and p :: "('form, 'var) antecedent" shows "(h |\<in>| hyps_for n p) = (hyps n h = Some p)"
simps_16: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) = Some (if bval b s then (c\<^sub>1, s) else (c\<^sub>2, s))"
diff_funI: fixes k :: "enat"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> manifold_eucl.carrier \<Longrightarrow> \<exists>c1\<in>manifold_eucl.atlas k. x \<in> domain c1 \<and> k-smooth_on (codomain c1) (f \<circ> inv_chart c1)" shows "diff_fun k charts_eucl f"
domain_atlas_subset_carrier: fixes c :: "('a, 'b) chart" assumes "c \<in> sub.atlas" shows "domain c \<subseteq> sub.carrier"
from_hma\<^sub>m_def: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>m a = Matrix.mat CARD('f) CARD('e) (\<lambda>(i, j). a $h from_nat i $h from_nat j)"
hline: fixes i :: "nat" assumes "i \<le> p" shows "nth_default 0 (Bernstein_coeffs p c d P) i \<le> a * ((real i * d + real (p - i) * c) / real p) + b"
A_to_prodag_def: fixes A :: "('c, 'f) aGroup_scheme"   and I :: "'d set"   and S :: "'d \<Rightarrow> 'c \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> ('e, 'g) aGroup_scheme" shows "A_to_prodag A I S B = (\<lambda>a\<in>carrier A. \<lambda>k\<in>I. S k a)"
Sup_finite_4_def: fixes A :: "finite_4 set" shows "\<Squnion> A = (if a\<^sub>4 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>4 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>1)"
scaleC_collapse: fixes u :: "complex"   and a :: "'d" shows "(1 - u) *\<^sub>C a + u *\<^sub>C a = a"
neutral: fixes A :: "'k multiset" assumes "\<forall>x\<in>#A. x = (0::'k)" shows "\<Sum>\<^sub># A = (0::'k)"
potype_ext_def: fixes pset :: "'b set"   and order :: "('b \<times> 'b) set"   and more :: "'c" shows "\<lparr>pset = pset, order = order, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons potype_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple pset order) more"
fun_dual5: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f = \<partial> \<circ> g \<circ> \<partial>) = (\<partial> \<circ> f \<circ> \<partial> = g)"
arr_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.arr (B.VVV.cod f)"
Cond_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (if (e) e1 else e2, xs) (if (e') e1 else e2, xs')"
tranclp_less': fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "(\<lambda>x y. less y x)\<^sup>+\<^sup>+ = (\<lambda>x y. less y x)"
upper_asymptotic_density_0_diff: fixes A :: "nat set"   and B :: "nat set" assumes "A \<subseteq> B"   and "upper_asymptotic_density (B - A) = 0" shows "upper_asymptotic_density A = upper_asymptotic_density B"
non_empty_newQuery: fixes Q1 :: "(nat set \<times> 'a option list set) set"   and J :: "nat set"   and Q0 :: "(nat set \<times> 'a option list set) set"   and Q2 :: "(nat set \<times> 'a option list set) set"   and I :: "nat set"   and t :: "'a option list"   and n :: "nat" assumes "Q1 = filterQuery J Q0"   and "Q2 = newQuery J Q1 (I, t)"   and "\<forall>X\<in>Q0. wf_atable n X" shows "non_empty_query Q2"
emptyLeftSelf: fixes t :: "'c event list"   and A :: "'c event set" shows "(\<forall>t1. t1 \<in> set t \<longrightarrow> t1 \<notin> A) \<longrightarrow> t setinterleaves (([], t), A)"
eligible: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and DA :: "'a literal multiset"   and As :: "'a list"   and \<sigma> :: "'s" assumes "S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> maximal_wrt (As ! 0 \<cdot>a \<sigma>) (DA \<cdot> \<sigma>)" shows "eligible S \<sigma> As DA"
sort_key_eq_sort_key: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> 'c" assumes "class.linorder less_eq less"   and "mset xs = mset ys"   and "inj_on f (set xs)" shows "linorder.sort_key less_eq f xs = linorder.sort_key less_eq f ys"
word_mod_div_equality: fixes n :: "'b word"   and b :: "'b word" shows "n div b * b + n mod b = n"
order_code: fixes a :: "'a"   and p :: "'a poly" shows "Polynomial.order a p = (if p = 0 then Code.abort STR ''order of polynomial 0 undefined'' (\<lambda>_. Polynomial.order a p) else if poly p a \<noteq> (0::'a) then 0 else Suc (Polynomial.order a (p div [:- a, 1::'a:])))"
root_rat_def: fixes p :: "nat"   and x :: "rat" shows "root_rat p x \<equiv> case quotient_of x of (a, b) \<Rightarrow> concat (map (\<lambda>rb. map (\<lambda>ra. rat_of_int ra / rat_of_int rb) (root_int p a)) (take 1 (root_int p b)))"
ifte_def: fixes a :: "'b"   and x :: "'b"   and y :: "'b" shows "ifte a x y = a \<sqinter> x \<squnion> - a \<sqinter> y"
PreFunctor_def: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" shows "PreFunctor F \<equiv> ((\<forall>f g. f \<approx>>\<^bsub>CatDom F\<^esub> g \<longrightarrow> F ## f ;;\<^bsub>CatDom F\<^esub> g = F ## f ;;\<^bsub>CatCod F\<^esub> (F ## g)) \<and> (\<forall>X. X \<in> obj\<^bsub>CatDom F\<^esub> \<longrightarrow> (\<exists>Y\<in>obj\<^bsub>CatCod F\<^esub>. F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y))) \<and> Category (CatDom F) \<and> Category (CatCod F)"
satisfies_nFAll: fixes \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i"   and b :: "'k" assumes "lformula \<phi>" shows "(\<AA> \<Turnstile> nFAll b \<phi>) = (\<AA> \<Turnstile> FAll b \<phi>)"
minus_eq: fixes x :: "int"   and y :: "int" shows "x \<ominus>\<^bsub>\<Z>\<^esub> y = x + \<ominus>\<^bsub>\<Z>\<^esub> y"
constrains_Int_distrib: fixes C :: "'d set"   and A :: "'d set"   and B :: "'d set" shows "C co A \<inter> B = (C co A) \<inter> (C co B)"
gcp_lb2: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix (gcp xs ys) ys"
finite_fp: shows "finite {y \<in> K. f y = (0::'a)}"
power_mult_numeral: fixes a :: "'a"   and m :: "num"   and n :: "num" shows "(a ^ numeral m) ^ numeral n = a ^ numeral (m * n)"
inter_ov: fixes i :: "'b"   and j :: "'b" assumes "(i, j) \<in> (b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>) \<inter> (e \<union> b\<inverse> \<union> m\<inverse> \<union> ov\<inverse> \<union> ov \<union> s\<inverse> \<union> s \<union> f\<inverse> \<union> f \<union> d\<inverse> \<union> d) \<inter> (b \<union> m \<union> ov \<union> s \<union> d)" shows "(i, j) \<in> ov"
bounded_plan_set_finite: fixes ops :: "'c list"   and k :: "nat" shows "finite (bounded_plan_set ops k)"
Ri_spec_var: fixes x :: "'d"   and y :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<forall>z. Ri x y z \<le> f z" shows "x \<le> f y"
equal_rbt_def: fixes r :: "('a, 'b) RBT.rbt"   and r' :: "('a, 'b) RBT.rbt" shows "equal_class.equal r r' \<equiv> rbt.impl_of r = rbt.impl_of r'"
ruler_comp: fixes u :: "'b list"   and v :: "'b list"   and u' :: "'b list"   and v' :: "'b list" assumes "u \<le>p v"   and "u' \<le>p v'"   and "v \<bowtie> v'" shows "u \<bowtie> u'"
1: shows "finite {(e, f e) |e. f e \<noteq> bot}"
weakTransitiveCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "eqvt X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox><(X \<union> weakBisim)> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto><{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<union> weakBisim \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X \<union> weakBisim"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X \<union> weakBisim" shows "\<Psi> \<rhd> P \<approx> Q"
sum_Minfty: fixes f :: "'a \<Rightarrow> ereal"   and I :: "'a set" shows "(sum f I = - \<infinity>) = (finite I \<and> \<not> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<exists>i\<in>I. f i = - \<infinity>))"
RSubmodule_sum_list: fixes Ns :: "'m set list" assumes "\<forall>N\<in>set Ns. RSubmodule N" shows "RSubmodule (\<Sum>N\<leftarrow>Ns. N)"
cases: fixes a1 :: "value"   and a2 :: "value" assumes "\<turnstile>\<^sub>v a1 \<approx> a2" obtains ts :: "value list" and us :: "value list" and name :: "name" where "a1 = Vconstr namea__ ts" and "a2 = Vconstr namea__ us" and "list_all2 vrelated ts us"   | cs :: "(Term_Class.term \<times> sterm) list" and \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" where "a1 = Vabs cs \<Gamma>\<^sub>1''__" and "a2 = Vabs cs \<Gamma>\<^sub>2''__" and "fmrel_on_fset (frees (Sabs cs)) vrelated \<Gamma>\<^sub>1''__ \<Gamma>\<^sub>2''__" and "fmrel_on_fset (consts (Sabs cs)) vrelated (fmap_of_list rs) \<Gamma>\<^sub>2''__"   | \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" and css :: "(name, (Term_Class.term \<times> sterm) list) fmap" and name :: "name" where "a1 = Vrecabs css name \<Gamma>\<^sub>1" and "a2 = Vrecabs css name \<Gamma>\<^sub>2" and "pred_fmap (\<lambda>cs. fmrel_on_fset (frees (Sabs cs)) vrelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2 \<and> fmrel_on_fset (consts (Sabs cs)) vrelated (fmap_of_list rs) (\<Gamma>\<^sub>2 ++\<^sub>f mk_rec_env css \<Gamma>\<^sub>2)) css" and "name |\<in>| fmdom css"
list_combE_1: fixes f :: "'a"   and xs :: "'a list" assumes "pre_strong_term_class.wellformed (list_comb f xs)" shows "pre_strong_term_class.wellformed f"
convergenceTr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and b :: "'a"   and n :: "ant" assumes "valuation K v"   and "x \<in> carrier K"   and "b \<in> carrier K"   and "b \<in> vp K v\<^bsup> Vr K v n\<^esup>"   and "Abs (n_val K v x) \<le> n" shows "x \<cdot>\<^sub>r b \<in> vp K v\<^bsup> Vr K v n + n_val K v x\<^esup>"
finite_imp_closed: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.t1_space open"   and "finite S" shows "topological_space.closed open S"
l_get_host_axioms: shows "l_get_host get_host get_host_locs"
bound: fixes x :: "'a"   and i :: "nat" assumes "x \<in> space M" shows "norm (s i x) \<le> 2 * norm (f x)"
iso_iff_section_and_retraction: fixes f :: "'a" shows "C.iso f = (C.section f \<and> C.retraction f)"
tobd_F12_inj: fixes x :: "'d"   and xa :: "('e, 'd, 'f) F1"   and y :: "'d"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" assumes "x \<in> F1set2 xa"   and "y \<in> F1set2 xa" shows "(toCard (F1set2 xa) (dir_image (bd_F1 +c r2) Abs_bd_type_F) x = toCard (F1set2 xa) (dir_image (bd_F1 +c r2) Abs_bd_type_F) y) = (x = y)"
power_both_mono: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and m :: "nat"   and n :: "nat"   and b :: "'d" assumes "class.semiring_1_real_line one times plus zero minus less_eq less"   and "less_eq one a"   and "m \<le> n"   and "less_eq a b" shows "less_eq (power.power one times a m) (power.power one times b n)"
l_get_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
measurable_restrict_space2: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and \<Omega> :: "'d set"   and N :: "'d measure" assumes "f \<in> space M \<rightarrow> \<Omega>"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M restrict_space N \<Omega>"
adjacent_atomics: fixes i :: "nat"   and j :: "nat"   and as :: "'b list" shows "adjacent (i, j) (atomics as i j)"
in_image_mset: fixes y :: "'c"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset" shows "(y \<in># image_mset f M) = (y \<in> f ` set_mset M)"
plus_game_zero_left: fixes G :: "game" shows "plus_game zero_game G = G"
dvdnumcoeff_aux: fixes t :: "Ferrack.num"   and m :: "int" assumes "ismaxcoeff t m"   and "0 \<le> m"   and "1 < numgcdh t m" shows "dvdnumcoeff t (numgcdh t m)"
treap_ins: fixes t :: "('c \<times> 'd) tree"   and k :: "'c"   and p :: "'d" assumes "treap t" shows "treap (ins k p t)"
disjoint_image: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set set" assumes "inj_on f (\<Union> A)"   and "disjoint A" shows "disjoint ((`) f ` A)"
LI_compose_measure_lt: fixes S :: "('a, 'b) strand_step list"   and T :: "('a, 'b) Term.term list"   and S' :: "('a, 'b) strand_step list"   and \<theta>\<^sub>1 :: "'b \<Rightarrow> ('a, 'b) Term.term"   and f :: "'a"   and \<theta>\<^sub>2 :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "((S @ map Send T @ S', \<theta>\<^sub>1), S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S', \<theta>\<^sub>2) \<in> measure\<^sub>s\<^sub>t"
weak_conv_imp_integral_bdd_continuous_conv: fixes \<mu> :: "nat \<Rightarrow> real measure"   and M :: "real measure"   and f :: "real \<Rightarrow> 'b"   and B :: "real" assumes "\<And>n. real_distribution (\<mu> n)"   and "real_distribution M"   and "weak_conv_m \<mu> M"   and "\<And>x. isCont f x"   and "\<And>x. norm (f x) \<le> B" shows "(\<lambda>n. integral\<^sup>L (\<mu> n) f) \<longlonglongrightarrow> integral\<^sup>L M f"
det_eq: fixes A :: "('c, 'd) sq_matrix" shows "Square_Matrix.det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. to_fun A i (p i)))"
inject_2: fixes x21 :: "'a"   and x22 :: "'a factor_tree"   and x23 :: "'a factor_tree"   and y21 :: "'a"   and y22 :: "'a factor_tree"   and y23 :: "'a factor_tree" shows "(Factor_Node x21 x22 x23 = Factor_Node y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
range_prefix_match_snm: fixes pfx :: "'b prefix_match"   and rg :: "'b wordinterval" shows "wordinterval_to_set (snd (range_prefix_match pfx rg)) = snd (ipset_prefix_match pfx (wordinterval_to_set rg))"
simps_2: fixes x :: "'a"   and ys :: "'a list" shows "strict_sorted (x # ys) = (Ball (set ys) ((<) x) \<and> strict_sorted ys)"
proots_sphere_card_def: fixes p :: "complex poly"   and z0 :: "complex"   and r :: "real" shows "proots_sphere_card p z0 r = card (proots_within p (sphere z0 r))"
fls: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls" shows "fls \<in> fmla"