hcju/isabelleps huggingface dataset:
T_intersec_meas_2: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat" assumes "A \<in> sets M"   and "B \<in> sets M" shows "A \<inter> (T ^^ n) -` B \<in> sets M"
inj_on_Int: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f A \<or> inj_on f B" shows "inj_on f (A \<inter> B)"
wary_subst_def: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm" shows "wary_subst \<rho> = (\<forall>x. wary (\<rho> x) \<and> \<infinity> \<le> arity (\<rho> x) \<and> local.ground_heads (head (\<rho> x)) \<subseteq> UNIV)"
empty_def: shows "TTree.empty \<equiv> Abs_ttree {[]}"
bind_btree_simps_2: fixes x :: "'b"   and k :: "'b \<rightarrow> 'a\<cdot>btree" shows "Leaf\<cdot>x \<bind> k = k\<cdot>x"
rank_take_cols_geq: fixes R :: "'a mat"   and nc :: "nat"   and ls :: "nat list"   and r :: "nat"   and B :: "'a mat" assumes "R \<in> carrier_mat n nc"   and "take_cols R ls \<in> carrier_mat n r"   and "B \<in> carrier_mat r nc"   and "R = take_cols R ls * B" shows "rank R \<le> rank (take_cols R ls)"
assms_4: shows "Cs \<equiv> ARG_MIN length Cs. gallery (C # Cs @ [D])"
mono_set: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" assumes "class.order less_eq less" shows "order.mono_set less_eq S = (\<forall>x y. less_eq x y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
biimp_simp: fixes \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula"   and G :: "'a formula" shows "\<A> \<Turnstile> F \<^bold>\<leftrightarrow> G = (\<A> \<Turnstile> F = \<A> \<Turnstile> G)"
zip_commute: fixes xs :: "'d list"   and ys :: "'e list" shows "xs || ys = map2 (\<lambda>x y. (y, x)) ys xs"
SeqThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
add_mset_eq_singleton_iff: fixes x :: "'b"   and M :: "'b multiset"   and y :: "'b" shows "(add_mset x M = \<LM> y  \<RM>) = (M = \<Empt> \<and> x = y)"
Fun_mkArr: fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't" assumes "S'.arr (S'.mkArr A B F)" shows "S'.Fun (S'.mkArr A B F) = restrict F A"
f4: shows "\<forall>s sa sb w c sc. \<not> low_equal s sa \<or> sb \<noteq> snd (fst (write_cpu w c s)) \<or> sc \<noteq> snd (fst (write_cpu w c sa)) \<or> low_equal sb sc"
word_size_bits_def: shows "word_size_bits = (2::'a)"
max_enn2ereal: fixes x :: "ennreal"   and y :: "ennreal" shows "max (enn2ereal x) (enn2ereal y) = enn2ereal (max x y)"
test_bit_bin: fixes w :: "'a word"   and n :: "nat" shows "w !! n = (n < LENGTH('a) \<and> bit (uint w) n)"
fmlookup_default_def: fixes m :: "('d, 'c) fmap"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "fmlookup_default m f x = (case fmlookup m x of None \<Rightarrow> f x | Some b \<Rightarrow> b)"
prems_1: shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t (a # S)"
sas_plus_formalism_and_induced_strips_formalism_are_equally_expressive_ii_a: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and k :: "nat" assumes "is_valid_problem_sas_plus \<Psi>" shows "finite (bounded_solution_set_sas_plus' \<Psi> k)"
Diff_insert2: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "A - insert a B = A - {a} - B"
imp_simps_4: fixes P :: "fm" shows "imp P F = not P"
op_subset_ivl_def: fixes a :: "'c set"   and b :: "'c set" shows "op_subset_ivl a b = (a \<subseteq> b)"
Abs_process_inverse: fixes y :: "('c list \<times> 'c set) set \<times> 'c list set" assumes "y \<in> process_set" shows "Rep_process (Abs_process y) = y"
strategyproofness_1: shows "pmf (sds R29) a + pmf (sds R29) b \<le> pmf (sds R36) d + pmf (sds R36) c"
hull_hull: fixes S :: "'c set \<Rightarrow> bool"   and s :: "'c set" shows "S hull (S hull s) = S hull s"
ec: shows "spr_simAbs ec = SPRsingle.sim_equiv_class a t"
kiljvm_def: fixes P :: "jvm_prog"   and mxs :: "nat"   and mxl :: "nat"   and T\<^sub>r :: "ty"   and "is" :: "bytecode"   and xt :: "ex_table" shows "kiljvm P mxs mxl T\<^sub>r is xt \<equiv> kildall (JVM_SemiType.le P mxs mxl) (JVM_SemiType.sup P mxs mxl) (exec P mxs T\<^sub>r xt is)"
factor: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "x divides\<^bsub>\<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>\<^esub> y = (\<exists>z. z \<noteq> zero \<and> times x z = y)"
claim': fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and i :: "nat"   and i' :: "nat"   and j :: "nat"   and j' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> i = \<pi>' i'"   and "\<pi> j = \<pi>' j'"   and "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"   and "j icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"   and "i' icd\<^bsup>\<pi>'\<^esup>\<rightarrow> 0"   and "j' icd\<^bsup>\<pi>'\<^esup>\<rightarrow> 0"   and "i < j"   and "\<pi> i \<noteq> local.return" shows "i' < j'"
fps_divide_self: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "f / f = 1"
unit_char_2: fixes a :: "'c" assumes "C.obj a" shows "D.iso (H.unit a)"
merges_incr: fixes xs :: "nat list list"   and ps :: "(nat \<times> nat list) list" assumes "xs \<in> list n A"   and "distinct (map fst ps)"   and "\<forall>(p, x)\<in>set ps. p < length xs \<and> x \<in> A" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] merges f ps xs"
cases: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and a1 :: "('a, 'c, 'd) gpv"   and a2 :: "('b, 'c, 'd) gpv" assumes "eq_\<I>_gpv A \<I> a1 a2" obtains gpv :: "('a, 'c, 'd) gpv" and gpv' :: "('b, 'c, 'd) gpv" where "a1 = gpv" and "a2 = gpv'" and "rel_spmf (eq_\<I>_generat A \<I> (eq_\<I>_gpv A \<I>)) (the_gpv gpv) (the_gpv gpv')"
reindex_pseudo_proj: fixes f :: "bool stream \<Rightarrow> 'a"   and c :: "bool"   and n :: "nat" shows "(\<Sum>w\<in>range (pseudo_proj_True n). f (c ## w)) = (\<Sum>y | \<exists>w\<in>range (pseudo_proj_True n). y = c ## w. f y)"
repeat_zmset_right: fixes a :: "nat"   and b :: "nat"   and A :: "'b zmultiset" shows "repeat_zmset a (repeat_zmset b A) = repeat_zmset (a * b) A"
prod_div2: fixes x1 :: "'c"   and y1 :: "'d"   and x2 :: "'c"   and y2 :: "'d" assumes "(x1, y1) \<preceq>\<^sub>R (x2, y2)" shows "y1 \<preceq>\<^sub>R y2"
path_rtrancl_edgesD: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path g u p v" shows "(u, v) \<in> (edges g)\<^sup>*"
ex_infinite_decides_subsets: fixes \<F> :: "nat set set"   and M :: "nat set" assumes "thin_set \<F>"   and "infinite M" obtains N :: "nat set" where "N \<subseteq> M" and "infinite N" and "decides_subsets \<F> N"
strict_mono_cancel_le: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "OrdinalInduct.strict_mono f" shows "(f x \<le> f y) = (x \<le> y)"
a_def: fixes x :: "'a" shows "a x = (1::'a) \<sqinter> - domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x"
simps_5: fixes I :: "('e, 'f, 'd) interp"   and \<alpha> :: "('e, 'f, 'd) hp"   and \<beta> :: "('e, 'f, 'd) hp" shows "prog_sem I (\<alpha> \<union>\<union> \<beta>) = prog_sem I \<alpha> \<union> prog_sem I \<beta>"
set_cases: fixes e :: "'a"   and a :: "'a option" assumes "e \<in> set_option a" obtains "a = Some e"
lex_two_refl: fixes ns1 :: "('b \<times> 'b) set"   and ns2 :: "('c \<times> 'c) set"   and s1 :: "('b \<times> 'b) set" assumes "refl ns1"   and "refl ns2" shows "refl (lex_two s1 ns1 ns2)"
simps_2: fixes n :: "nat"   and d :: "char"   and ds :: "char list" shows "string_to_nat_aux n (d # ds) = string_to_nat_aux (n * 10 + char_to_digit d) ds"
connected_distI: fixes v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "dist v d v'" shows "connected v v'"
resr_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<rightarrow> x \<le> z \<rightarrow> y"
omega_coinduct: fixes X :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and R :: "('b \<times> 'b) set" assumes "X x z"   and "\<And>x z. X x z \<Longrightarrow> \<exists>y. (x, y) \<in> R \<and> (X y z \<or> (y, z) \<in> Omega_Algebra_Models.omega R)" shows "(x, z) \<in> Omega_Algebra_Models.omega R"
zero_eq_1_divide_iff: fixes a :: "'c" shows "((0::'c) = (1::'c) / a) = (a = (0::'c))"
simps_1: fixes uu :: "'b set" shows "intersection_list uu [] = []"
some_right_adjoint_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "chosen_right_adjoints V H \<a> \<i> src trg" shows "chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg f \<equiv> SOME g. bicategory.adjoint_pair V H \<a> \<i> src trg f g"
negligible_iff_null_sets: fixes S :: "'a set" shows "negligible S = (S \<in> null_sets lebesgue)"
iterate_is_empty_def: fixes it :: "(bool \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool \<Rightarrow> bool) \<Rightarrow> bool \<Rightarrow> bool" shows "iterate_is_empty it = it (\<lambda>b. b) (\<lambda>_ _. False) True"
pdoSM1r: shows "partial_density_operator (DS (M1 * \<rho> * adjoint M1))"
bexCI: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "\<forall>x\<in>A. \<not> P x \<Longrightarrow> P a"   and "a \<in> A" shows "\<exists>x\<in>A. P x"
is_first_def: fixes x :: "'b"   and u :: "'b list" shows "is_first x u = (\<exists>v. u = [x] @ v)"
inst_local_respect: fixes s :: "rstate_t"   and u :: "thread_id_t"   and a :: "int_point_t" assumes "\<not> rifp (rcurrent s) u"   and "rprecondition s (rcurrent s) a" shows "rvpeq u s (rstep s a)"
assms_2: fixes c :: "'b"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "c \<noteq> (0::'b)"   and "t \<notin> keys p"   and "P p" shows "P (Poly_Mapping.single t c + p)"
comp_red_basis_GB: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_Groebner_basis (set xs)" shows "is_Groebner_basis (set (comp_red_basis xs))"
simps_2: fixes P :: "'state"   and t :: "('b, 'pred, 'act, 'effect) Tree" shows "FL_valid_Tree P (tNot t) = (\<not> FL_valid_Tree P t)"
prems_1: fixes x :: "'c" assumes "x \<in> insert x__ A" shows "finite (B x)"
div_smult_right: fixes a :: "'a"   and x :: "'a poly"   and y :: "'a poly" assumes "a \<noteq> (0::'a)" shows "x div smult a y = smult (inverse a) (x div y)"
sync: shows "\<forall>a t t'. jview a t = jview a t' \<longrightarrow> tLength t = tLength t'"
fcomp_comp: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'e" shows "fcomp f g = g \<circ> f"
Qg_unitTr: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G" shows "c_top G N N (N \<bullet> a) = N \<bullet> a"
degenerate_lotteries_def: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>" shows "degenerate_lotteries outcomes = {x \<in> \<P> outcomes. card (set_pmf x) = 1}"
face_of_Int_subface: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and D :: "'a set" assumes "A \<inter> B face_of A"   and "A \<inter> B face_of B"   and "C face_of A"   and "D face_of B" shows "C \<inter> D face_of C \<and> C \<inter> D face_of D"
pivot_tableau_eq_def: fixes t1 :: "(nat \<times> linear_poly) list"   and eq :: "nat \<times> linear_poly"   and t2 :: "(nat \<times> linear_poly) list"   and x :: "nat" shows "pivot_tableau_eq t1 eq t2 x \<equiv> let eq' = pivot_eq eq x; m = map (subst_var_eq_code x (rhs eq')) in (m t1, eq', m t2)"
augment_cf_impl_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and p :: "(nat \<times> nat) list"   and x :: "int" shows "augment_cf_impl cf p x \<equiv> (rec\<^sub>T D. (\<lambda>xa. case xa of ([], cf) \<Rightarrow> return cf | (e # p, cf) \<Rightarrow> augment_edge_impl cf e x \<bind> (\<lambda>cf. D (p, cf)))) (p, cf)"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xa :: "'c pdevs"   and xb :: "'d pdevs" shows "pdevs_apply (binop_pdevs x xa xb) = binop_pdevs_raw x (pdevs_apply xa) (pdevs_apply xb)"
dvd_nat_bounds: fixes p :: "nat"   and n :: "nat" assumes "0 < p"   and "n dvd p" shows "0 < n \<and> n \<le> p"
zeroth_root_nat: fixes n :: "nat" shows "nth_root_nat 0 n = 0"
conf_f_def2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ST :: "ty list"   and LT :: "ty err list"   and "is" :: "instr list"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat" shows "conf_f P h (ST, LT) is (stk, loc, C, M, pc) \<equiv> P,h \<turnstile> stk [:\<le>] ST \<and> P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT \<and> pc < length is"
simps: fixes A :: "('c, 'd) ioa"   and a :: "'c" shows "reachable A a = ((\<exists>s. a = s \<and> s \<in> start A) \<or> (\<exists>s aa t. a = t \<and> reachable A s \<and> s \<midarrow>aa\<midarrow>A\<longrightarrow> t))"
wf_listn: shows "wf {(y, x). x [\<sqsubset>\<^bsub>r\<^esub>] y}"
Target_def: fixes T :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" shows "Target T = target ` T"
rel_compp_Grp: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_pmf R = (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
rinit_throwE: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and e :: "char list exp"   and Cs :: "char list list"   and e\<^sub>0 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>RI (C,throw e) ; Cs \<leftarrow> e\<^sub>0,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "\<exists>a s\<^sub>t. e' = throw a \<and> P \<turnstile> \<langle>throw e,s\<rangle> \<Rightarrow> \<langle>throw a,s\<^sub>t\<rangle>"
vector_scalar_mult_def: fixes c :: "'c"   and x :: "('c, 'd) vec" shows "c *s x = (\<chi>i. c * x $ i)"
finite_Diff: fixes A :: "'b set"   and B :: "'b set" assumes "finite A" shows "finite (A\<setminus>B)"
restrict_disjoint_left: fixes ks :: "'c list"   and ms :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and ls :: "'c list"   and ns :: "'c list" assumes "disjoint ks ms" shows "ms \<langle>ks \<langle>f\<rangle> ls\<rangle> ns = mbot"
length_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "length (map f xs) = length xs"
sublist_append_both_I: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list"   and d :: "'a list" assumes "subseq a b"   and "subseq c d" shows "subseq (a @ c) (b @ d)"
ratfps_eq_0_iff: fixes x :: "'b ratfps" shows "(x = 0) = (fps_of_ratfps x = 0)"
select_convs_1: fixes NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "NonceS \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = NonceS"
singl_def: fixes a :: "'b" shows "singl a = [a]"
4: fixes y :: "'b" assumes "y \<in> T" shows "g y \<in> S \<and> (g y, y) \<in> A"
rep_eq: fixes x :: "('a, 'b) fmap"   and xa :: "('a, 'b) fmap" shows "(x \<subseteq>\<^sub>f xa) = (fmlookup x \<subseteq>\<^sub>m fmlookup xa)"
tan_eq_arctan_Ex: fixes x :: "real"   and y :: "real" shows "(tan x = y) = (\<exists>xa. x = arctan y + real_of_int xa * pi \<or> x = pi / 2 + real_of_int xa * pi \<and> y = 0)"
finite_dimensional_vector_space_euclidean: shows "finite_dimensional_vector_space (*\<^sub>C) CBasis"
eq2: shows "dirichlet_prod' mangoldt F x = dirichlet_prod' (dirichlet_prod moebius_mu (\<lambda>n. of_real (ln (real n)))) F x"
l2_inv7_def: shows "l2_inv7 \<equiv> {s. extr (bad s) (ik s) (chan s) \<subseteq> synth (analz generators)}"
sub_branch_nominals: fixes f :: "'d \<Rightarrow> 'b"   and branch :: "(('c, 'd) fm list \<times> 'd) list" shows "branch_nominals (sub_branch f branch) = f ` branch_nominals branch"
case_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case A of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f1"
fv_subst_int2: fixes x :: "Vars.var"   and S :: "Vars.var set"   and y :: "Vars.var"   and e :: "exp" assumes "x \<notin> S"   and "y \<notin> S" shows "S \<inter> fv e[y::=x] = S \<inter> fv e"
rr: shows "FV e \<turnstile> \<rho> \<sqsubseteq> \<rho>'"
continuous_at_extension_sequentially: fixes a :: "'c"   and T :: "'c set"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "a \<in> T"   and "T \<subseteq> closure S"   and "\<And>u b. \<lbrakk>\<forall>n. u n \<in> S; b \<in> T; u \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b" shows "continuous (at a within S \<union> T) f"
ltl_WeakUntil_StrongRelease_con_2: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> M\<^sub>c \<psi> = (\<not> \<xi> \<Turnstile>\<^sub>c not\<^sub>c \<phi> W\<^sub>c not\<^sub>c \<psi>)"
bind_rzero: fixes x :: "'b option" shows "x \<bind> Map.empty = None"
add_mset_remove_trivial_eq: fixes N :: "'a multiset"   and a :: "'a" shows "(N = add_mset a (N - \<LM> a  \<RM>)) = (a \<in># N)"
nth_root_nat_def: fixes k :: "nat"   and n :: "nat" shows "nth_root_nat k n = (if k = 0 then 0 else Max {m. m ^ k \<le> n})"
simps_1: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form" shows "extend S C f 0 = S"
pfp_def: fixes f :: "'c \<Rightarrow> 'c" shows "pfp f = while_option (\<lambda>x. \<not> f x \<sqsubseteq> x) f"
index_col: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "col A j $ i = A $$ (i, j)"
ind: fixes ly :: "nat list"   and tp :: "(action \<times> nat) list" assumes "ly = layout_of xs"   and "tp = tm_of xs" shows "length tp = 2 * sum_list (take (length xs) (layout_of xs))"
subst_mono_fv: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and s :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "x \<in> fv t" shows "s x \<sqsubseteq> t \<cdot> s"
rel_map_2: fixes Sa :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c llist"   and g :: "'e \<Rightarrow> 'd"   and y :: "'e llist" shows "llist_all2 Sa x (lmap g y) = llist_all2 (\<lambda>x y. Sa x (g y)) x y"
cccFix_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "cccFix \<Gamma>\<cdot>i = (Afix \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), CCfix \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), snd i))"
lookup_unicity: fixes x :: "name"   and a :: "coname"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and y :: "name"   and b :: "coname" assumes "lookup x a \<theta>_n \<theta>_c = Ax y b"   and "b \<sharp> (\<theta>_c, \<theta>_n)"   and "y \<sharp> (\<theta>_c, \<theta>_n)" shows "x = y \<and> a = b"
sum_component: fixes f :: "'c \<Rightarrow> ('a, 'b) Finite_Cartesian_Product.vec"   and S :: "'c set"   and i :: "'b" shows "sum f S $h i = (\<Sum>x\<in>S. f x $h i)"
root_hash_G_unfold: fixes rha :: "'e \<Rightarrow> 'f"   and rhb :: "'g \<Rightarrow> 'h" shows "root_hash_G rha rhb = G\<^sub>h \<circ> root_hash_F (root_hash_T rha) rhb \<circ> the_G\<^sub>m"
Mh: shows "(\<lambda>(x, y). h x y) \<in> M \<Otimes>\<^sub>M M' \<rightarrow>\<^sub>M N"
p: shows "\<turnstile>\<^sub>p {wlp S (wlp (While M S) P)} S {adjoint (M 0) * P * M 0 + adjoint (M 1) * wlp S (wlp (While M S) P) * M 1}"
real_of_ereal_pos: fixes x :: "ereal" assumes "0 \<le> x" shows "0 \<le> real_of_ereal x"
real_a_cnj_ge_0: fixes a :: "'c" shows "0 \<le> real_of (a * cnj_class.cnj a)"
rel_map_2: fixes Sa :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a set['c]"   and g :: "'d \<Rightarrow> 'b"   and y :: "'d set['c]" shows "rel_bset Sa x (map_bset g y) = rel_bset (\<lambda>x y. Sa x (g y)) x y"
discriminant_cubic_depressed_def: fixes e :: "'a"   and f :: "'a" shows "discriminant_cubic_depressed e f = - ((4::'a) * e ^ 3 + (27::'a) * f\<^sup>2)"
AA: shows "swaprows I (Suc I) A \<in> carrier_mat n m"
mapping_conv_bijective: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.mapping x = matrix_stone_relation_algebra.bijective (x\<^sup>t)"
bij_betw_function_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A" shows "bij_betw (function_of A enum) {xs. set xs \<subseteq> B \<and> length xs = card A} (A \<rightarrow>\<^sub>E B)"
atMost_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {..n} = g 0 + (\<Sum>i<n. g (Suc i))"
nn_integral_count_space_nn_integral: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and M :: "'d measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ i. f i x \<partial>count_space I \<partial>M = \<integral>\<^sup>+ i. integral\<^sup>N M (f i) \<partial>count_space I"
image_comp: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'h"   and r :: "'i set" shows "f ` g ` r = (f \<circ> g) ` r"
ass_function_int_def: fixes n :: "int" shows "ass_function_int n = \<bar>n\<bar>"
assms_4: fixes s :: "'s" shows "S s \<noteq> {}"
coreProperties: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and Q_pos :: "(nat set \<times> 'b option list set) set"   and Q_neg :: "(nat set \<times> 'b option list set) set"   and I :: "nat set"   and J :: "nat set" assumes "getIJ getIJ"   and "2 \<le> card V"   and "getIJ Q_pos Q_neg V = (I, J)" shows "1 \<le> card I \<and> 1 \<le> card J \<and> V = I \<union> J \<and> I \<inter> J = {}"
eval_var_monom: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "eval_monom \<alpha> (var_monom x) = \<alpha> x"
inf_cases: fixes c :: "com"   and cs :: "com list"   and s :: "state" assumes "PsHoareTotal.inf (c # cs) s" shows "PsHoareTotal.inf [c] s \<or> (\<exists>t. s -c\<rightarrow> t \<and> PsHoareTotal.inf cs t)"
Right_Coset_def: fixes H :: "'a set"   and x :: "'a" shows "H |\<cdot> x = {h \<cdot> x |h. h \<in> H}"
decode_instr_result_4: fixes v1 :: "word32" assumes "\<nexists>e. decode_instruction v1 = Inl e" shows "\<exists>v2. decode_instruction v1 = Inr v2"
option_ord_Some1_iff: fixes x :: "'f"   and y :: "'f option" shows "option_ord (Some x) y = (y = Some x)"
sconst_cycle: fixes x :: "'a" shows "sconst x = cycle [x]"