hcju/isabelleps huggingface dataset:
cyclic_reduced_homology_group_nsphere: fixes p :: "int"   and n :: "nat" shows "cyclic_group (reduced_homology_group p (nsphere n))"
b: fixes k :: "nat" shows "\<integral>\<^sup>+ x. f ((T ^^ k) x) \<partial>M = integral\<^sup>N M f"
lfinite_ldropWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lfinite (ldropWhile P xs) = ((\<exists>x\<in>lset xs. \<not> P x) \<longrightarrow> lfinite xs)"
roots_of_coprime_r_location_property: fixes qs :: "rat poly list"   and sga :: "rat list"   and x1 :: "real"   and zer_list :: "real list" assumes "pairwise_coprime_list qs"   and "\<And>q. q \<in> set qs \<Longrightarrow> rsquarefree q"   and "sga = sign_vec qs x1"   and "has_no_zeros sga"   and "zer_list = sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}" shows "zer_list \<noteq> [] \<longrightarrow> x1 < zer_list ! 0 \<or> zer_list ! (length zer_list - 1) < x1 \<or> (\<exists>n<length zer_list - 1. zer_list ! n < x1 \<and> x1 < zer_list ! (n + 1))"
lrf_def: fixes ur :: "('s \<times> 's) set" shows "lrf ur \<equiv> \<forall>s\<in>S\<^bsub>SES\<^esub>. \<forall>s'\<in>S\<^bsub>SES\<^esub>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. reachable SES s \<and> s c\<longrightarrow>\<^bsub>SES\<^esub> s' \<longrightarrow> (s', s) \<in> ur"
semigroup_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "semigroup_add_hom hom \<equiv> plus_hom hom"
m1a_inv2r_serv_def: shows "m1a_inv2r_serv \<equiv> {s. \<forall>A B Rb Kab nlb. B \<notin> bad \<longrightarrow> runs s Rb = Some (Resp, [A, B], aKey Kab # nlb) \<longrightarrow> (\<exists>Rs Na. Kab = sesK (Rs $ sk) \<and> runs s Rs = Some (Serv, [A, B], aNon Na # take rs_len nlb))}"
0: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<guillemotleft>\<l> f : L f \<Rightarrow> local.map f\<guillemotright>"
rel_sum_eq: shows "rel_sum (=) (=) = (=)"
subst_cls_comp_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and C :: "'b literal multiset"   and \<tau> :: "'c"   and \<sigma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_cls subst_atm C (comp_subst \<tau> \<sigma>) = substitution_ops.subst_cls subst_atm (substitution_ops.subst_cls subst_atm C \<tau>) \<sigma>"
lst: shows "lst = matrix_to_lpolies (two_block_non_interfering A A\<^sup>T)"
set_cases_2: fixes e :: "'c"   and a :: "('d, 'c) Term.term" assumes "e \<in> fv a" obtains "a = Var e"   | z1 :: "'d" and z2 :: "('d, 'c) Term.term list" and x :: "('d, 'c) Term.term" where "a = Fun z1 z2" and "x \<in> set z2" and "e \<in> fv x"
trace_callee_eq_coinduct: fixes S :: "'h spmf \<Rightarrow> 'i spmf \<Rightarrow> bool"   and p :: "'h spmf"   and q :: "'i spmf"   and A :: "'j set"   and callee1 :: "'h \<Rightarrow> 'j \<Rightarrow> ('k \<times> 'h) spmf"   and callee2 :: "'i \<Rightarrow> 'j \<Rightarrow> ('k \<times> 'i) spmf" assumes "S p q"   and "\<And>p q a. \<lbrakk>S p q; a \<in> A\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"   and "\<And>p q a res res' b s'' s'. \<lbrakk>S p q; a \<in> A; res \<in> set_spmf p; (b, s'') \<in> set_spmf (callee1 res a); res' \<in> set_spmf q; (b, s') \<in> set_spmf (callee2 res' a)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q \<bind> (\<lambda>s. callee2 s a)) b)" shows "trace_callee_eq callee1 callee2 A p q"
dagger_unfoldl_eq: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<dagger> = x\<^sup>\<dagger>"
cl9_var: fixes R :: "('a \<times> 'a set) set" shows "Multirelations.d (R \<inter> 1\<^sub>\<sigma>) = R \<inter> 1\<^sub>\<sigma>"
WeakF_def: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "WF(F)_v \<equiv> \<diamond>\<box>Enabled \<langle>F\<rangle>_v \<longrightarrow> \<box>\<diamond>\<langle>F\<rangle>_v"
leD: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
intro: fixes invar :: "'e \<Rightarrow> bool"   and \<alpha> :: "'e \<Rightarrow> 'f"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g" assumes "\<exists>lc. (\<forall>xc\<in>set lc. invar xc) \<and> distinct (map \<alpha> lc) \<and> S0 = set (map \<alpha> lc) \<and> sorted_wrt R (map \<alpha> lc) \<and> iti = foldli lc" shows "set_iterator_abs_genord \<alpha> invar iti S0 R"
norm_Basis: fixes u :: "'a" assumes "u \<in> Basis" shows "norm u = 1"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "F (b1 + b2) = F b1 + F b2"
execute_return: fixes x :: "'b" shows "execute (return x) = Some \<circ> Pair x"
E4: shows "vpeq_subj_subj u (set_object_value x y s1) (set_object_value x y s2)"
h2: shows "\<forall>n. n < length l1 \<and> l1 ! n = m \<longrightarrow> (l1 @ l2) ! n = m"
hyps: shows "(x \<otimes> y) [^] n = x [^] n \<otimes> y [^] n"
DA0_\<eta>0_p: shows "DA0 \<in> M \<and> DA = DA0 \<cdot> \<eta>0 \<and> S DA0 \<cdot> \<eta>0 = S_M S M DA \<and> is_ground_subst \<eta>0"
is_word_append: fixes a :: "symbol list"   and b :: "symbol list" shows "is_word (a @ b) = (is_word a \<and> is_word b)"
ta_seq_consist_simps_2: fixes P :: "'a prog"   and vs :: "'b \<times> addr_loc \<Rightarrow> ('b val \<times> bool) option"   and ob :: "('b, 'c) obs_event action"   and obs :: "('b, 'c) obs_event action llist" shows "ta_seq_consist P vs (LCons ob obs) = ((case ob of NormalAction (ReadMem ad al v) \<Rightarrow> \<exists>b. vs (ad, al) = \<lfloor>(v, b)\<rfloor> | NormalAction _ \<Rightarrow> True | _ \<Rightarrow> True) \<and> ta_seq_consist P (mrw_value P vs ob) obs)"
evalS: shows "\<forall>p\<in>S. eval e f g p"
simps_25: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "char list \<Rightarrow> 'b" shows "(case NT of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f4"
semantic_causal_delivery: fixes i :: "'a"   and j :: "nat"   and s :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) message"   and i' :: "'a \<times> nat" assumes "event_at (i, j) (Receive s m)"   and "i' \<in> deps m" shows "\<exists>s' j' m'. event_at (i, j') (Receive s' (Insert m')) \<and> j' < j \<and> I m' = i'"
lambda_add_Galois: fixes m :: "'a"   and n :: "'a"   and x :: "(real, 'a) vec"   and y :: "(real, 'a) vec" assumes "m \<noteq> n" shows "(x = (\<chi>i. if i = m then y $ m + y $ n else y $ i)) = ((\<chi>i. if i = m then x $ m - x $ n else x $ i) = y)"
simps_3: fixes f1 :: "'b"   and f2 :: "'b" shows "(case sort_dir.Left of sort_dir.Left \<Rightarrow> f1 | sort_dir.Right \<Rightarrow> f2) = f1"
assms_1: shows "linear_split xs p = (ls @ [(sub, sep)], rs)"
nth_rotate1: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "rotate1 xs ! n = xs ! (Suc n mod length xs)"
quot_of_ratfps_def: shows "local.quot_of_ratfps \<equiv> map_fun Rep_ratfps id quot_of_fract"
val_Bool_simp1: fixes i :: "bool" shows "\<lbrakk>Def i\<rbrakk> = i"
phull_subset_module: fixes B :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "phull B \<subseteq> MPoly_Type_Class.punit.pmdl B"
D: shows "ExtAgrK (S (Card 0, 0, run')) = Some e"
res_diff_zero_fact_2: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "(a \<ominus> b) k = 0" shows "a k \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> b k = 0"
pl_1: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<phi>)]]"
delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_pointer_ptr_in_heap: fixes ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr h = Some h'" shows "ptr |\<in>| shadow_root_ptr_kinds h"
INF_identity_eq: fixes Inf :: "'b set \<Rightarrow> 'b"   and A :: "'b set" shows "Inf ((\<lambda>x. x) ` A) = Inf A"
assms: shows "low_equal s1 s2 \<and> t1 = user_reg_mod w curr_win rd s1 \<and> t2 = user_reg_mod w curr_win rd s2"
stutter_reduced_equivalent: fixes \<sigma> :: "nat \<Rightarrow> 'f" shows "\<natural>\<sigma> \<approx> \<sigma>"
bfs_def: fixes src :: "nat"   and dst :: "nat" shows "bfs src dst \<equiv> WHILE\<^bsup>bfs_invar src dst\<^esup> (\<lambda>(f, V, C, N, d). f = False \<and> C \<noteq> {}) (\<lambda>(f, V, C, N, d). SPEC (\<lambda>v. v \<in> C) \<bind> (\<lambda>v. let C = C - {v} in if v = dst then RETURN (True, {}, {}, {}, d) else FOREACH\<^bsup>FE_invar V N v\<^esup> ((ls.\<alpha> \<circ> succ) v) (\<lambda>w (V, N). if w \<notin> V then ASSERT (w \<notin> N) \<bind> (\<lambda>_. RETURN (insert w V, insert w N)) else RETURN (V, N)) (V, N) \<bind> (\<lambda>(V, N). ASSERT (bfs_invar' src dst (f, V, C, N, d)) \<bind> (\<lambda>_. if C = {} then let C = N; N = {}; d = d + 1 in RETURN (f, V, C, N, d) else RETURN (f, V, C, N, d))))) (False, {src}, {src}, {}, 0) \<bind> (\<lambda>(f, uu_, uu_, uu_, d). if f then RETURN (Some d) else RETURN None)"
is_lasso_def: fixes L :: "'Q lasso" shows "is_lasso L \<equiv> is_lasso_pre L \<and> (\<forall>A\<in>F. set (lasso_cycle L) \<inter> A \<noteq> {})"
degree_eq_zeroE: fixes p :: "'a poly" assumes "degree p = 0" obtains a :: "'a" where "p = [:a:]"
insert_after_ref: fixes xs :: "'a list"   and ref :: "'a"   and ys :: "'a list"   and oid :: "'a" assumes "distinct (xs @ ref # ys)" shows "insert_spec (xs @ ref # ys) (oid, Some ref) = xs @ ref # oid # ys"
simps_2: fixes x :: "'b"   and ts :: "'b bintree list" shows "t_ins_aux \<lparr>t_type.folding = False, item = x, subtrees = Leaf # ts\<rparr> = t_ins_aux \<lparr>t_type.folding = True, item = x, subtrees = Branch x Leaf Leaf # ts\<rparr>"
graph_homomorphism_semantics: fixes A :: "('b, 'c) labeled_graph"   and B :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set"   and a :: "'c"   and b :: "'c"   and e :: "'b allegorical_term"   and a' :: "'d"   and b' :: "'d" assumes "graph_homomorphism A B f"   and "(a, b) \<in> :A:\<lbrakk>e\<rbrakk>"   and "(a, a') \<in> f"   and "(b, b') \<in> f" shows "(a', b') \<in> :B:\<lbrakk>e\<rbrakk>"
poly_identity_mod_p: fixes v :: "'a mod_ring poly" shows "v ^ CARD('a) - v = (\<Prod>x\<in>UNIV. v - [:x:])"
subset: shows "unforwarded_non_volatile_reads (r # sb) W \<subseteq> A"
flatten_uminus: fixes p :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c" shows "flatten (- p) = - flatten p"
bh'_1: shows "f =m Polynomial.smult (lead_coeff f) (\<Prod>\<^sub># (mset hs)) \<and> (\<forall>f\<in>#mset hs. irreducible\<^sub>d_m f \<and> monic (Mp f))"
RedCondT: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" shows "redp P E (if (true) e\<^sub>1 else e\<^sub>2) s e\<^sub>1 s"
cod_src_eq_dom_src: shows "\<nu>'.cod.src = \<nu>'.dsrc"
inD_def: shows "inD \<equiv> S.UP \<circ> Inr"
wary_App: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wary s"   and "wary t"   and "enat (num_args s) < arity_hd (head s)" shows "wary (App s t)"
more_arith_simps_6: fixes a :: "'a" shows "a * (1::'a) = a"
refl: fixes P :: "conc_interface" shows "P \<simeq> P"
divide_inverse_commute: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a b = times (inverse b) a"
p': shows "fps_deriv p = fps_deriv F * (1 - fps_X) ^ (k + 1) - F * (1 - fps_X) ^ k * of_nat (k + 1)"
qPsubstBinp_def: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and qbinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option" shows "qbinp %%[[qrho]] \<equiv> lift (qPsubstAbs qrho) qbinp"
accept_Language: fixes s :: "'s1" assumes "wellformed1 s" shows "accept1 s = \<oo> (Language1 s)"
tagged_division_Un_interval_real: fixes p1 :: "(real \<times> real set) set"   and a :: "real"   and b :: "real"   and k :: "real"   and c :: "real"   and p2 :: "(real \<times> real set) set" assumes "p1 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<le> c}"   and "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x \<bullet> k}"   and "k \<in> Basis" shows "p1 \<union> p2 tagged_division_of {a..b}"
mod_ex2: fixes a :: "nat" shows "is_even a = (\<exists>q. a = 2 * q)"
Max_ge: fixes A :: "'d DBMEntry set"   and x :: "'d DBMEntry" assumes "finite A"   and "x \<in> A" shows "x \<preceq> linordered_monoid.Max A"
ennreal_times_divide: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "a * (b / c) = a * b / c"
aodv_wf: shows "wellformed \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V"
ind: shows "trace init (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) final \<and> (\<forall>k\<ge>j - 1 + 1. S (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) k = S (t[j - 1 := t ! j, j := t ! (j - 1)]) k) \<and> (\<forall>k\<le>i. S (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) k = S (t[j - 1 := t ! j, j := t ! (j - 1)]) k) \<and> prerecording_event (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) i \<and> postrecording_event (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) (i + 1) \<and> (\<forall>k. i + 1 < k \<and> k < j - 1 + 1 \<longrightarrow> \<not> regular_event (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)]) ! k))"
widen_PrimT2: fixes G :: "prog"   and S :: "ty"   and x :: "prim_ty" assumes "G\<turnstile>S\<preceq>PrimT x" shows "\<exists>y. S = PrimT y"
cases: fixes a :: "diagram" assumes "wf_dia a" obtains E :: "(node fset \<times> command_gadget \<times> node fset) list" and V :: "node fset" and \<Lambda> :: "node \<Rightarrow> assertion_gadget" where "a = Graph V \<Lambda> E" and "\<forall>e\<in>set E. wf_com (snd3 e)" and "\<forall>v\<in>fset V. wf_ass (\<Lambda> v)" and "acyclicity E" and "linearity E" and "\<forall>e\<in>set E. fst3 e |\<union>| thd3 e |\<subseteq>| V"
cInf_insert_If: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and a :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_below less_eq X" shows "Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
"2_2": shows "(!!) (smap int (q ## trace v q)) \<approx> (!!) (smap int (q ## trace v' q))"
tdec_b_def: shows "tdec_b \<equiv> TMDec"
br'_inner_invar_step: fixes q :: "'d"   and W :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and it :: "('d, 'e) ta_rule set"   and \<Sigma>' :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and r :: "('d, 'e) ta_rule" assumes "q \<in> W"   and "br'_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) it \<Sigma>'"   and "r \<in> it"   and "it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}" shows "br'_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) (it - {r}) (br'_inner_step r \<Sigma>')"
elims_2: fixes x :: "bool list"   and xa :: "bool list" assumes "x \<preceq>\<^sub>b xa" obtains "x = []" and "xa = []"   | xb :: "bool" and xs :: "bool list" and y :: "bool" and ys :: "bool list" where "x = xb # xs" and "xa = y # ys" and "(xb \<longrightarrow> y) \<and> xs \<preceq>\<^sub>b ys"
prefs_wf: fixes i :: "'b" assumes "i \<in> agents" shows "finite_total_preorder_on alts ((R2 \<circ> \<pi>) i)"
finprod_empty: fixes f :: "'e \<Rightarrow> 'a" shows "finprod G f {} = \<one>"
summable_LIMSEQ: fixes f :: "nat \<Rightarrow> 'c" assumes "summable f" shows "(\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> suminf f"
eq_on_def: fixes A :: "'d set" shows "eq_on A = (\<lambda>x y. x \<in> A \<and> x = y)"
retract_cancel1: fixes ys :: "'d list"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" assumes "cancel1 ys (map f xs)" shows "\<exists>zs. equivclp cancel1 xs zs \<and> ys = map f zs"
op_in1c: shows "open (inside (path_image c1 \<union> path_image c))"
insert_iff: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" shows "(a \<in> b \<triangleright> A) = (a = b \<or> a \<in> A)"
bernoulli_num_def: fixes n :: "nat" shows "bernoulli_num n = \<lfloor>bernoulli n * real (bernoulli_denom n)\<rfloor>"
differentiable_at_imp_differentiable_on: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> s \<Longrightarrow> f differentiable at x" shows "f differentiable_on s"
path_connectedin_Union: fixes \<A> :: "'a set set"   and X :: "'a topology" assumes "\<And>S. S \<in> \<A> \<Longrightarrow> path_connectedin X S"   and "\<Inter> \<A> \<noteq> {}" shows "path_connectedin X (\<Union> \<A>)"
hpair_def': fixes a :: "hf"   and b :: "hf" shows "\<langle>a, b\<rangle> = \<lbrace>\<lbrace>a, a\<rbrace>, \<lbrace>a, b\<rbrace>\<rbrace>"
permutation_mat_dim_2: fixes n :: "nat"   and p :: "nat \<Rightarrow> nat" shows "dim_row (permutation_mat n p) = n"
splitAt_in_fst: fixes v :: "'c"   and ram :: "'c"   and vs :: "'c list" assumes "v \<in> set (fst (splitAt ram vs))" shows "v \<in> set vs"
integral_conv: fixes x :: "real"   and y :: "real" assumes "x < y" shows "(\<lambda>n. integral\<^sup>L (M_seq n) (cts_step x y)) \<longlonglongrightarrow> expectation (cts_step x y)"
drop_emp_nel_1: fixes u :: "'b list"   and S :: "'b list set" assumes "u \<in> S\<^sub>+" shows "u \<noteq> \<epsilon>"
connectedin_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "connectedin (subtopology X S) T = (connectedin X T \<and> T \<subseteq> S)"
rec_fourtimes_def: shows "rec_fourtimes = Cn 1 rec_mult [recf.id 1 0, constn 4]"
inv_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y ** x = one"   and "x ** y' = one" shows "y = y'"
T_cruel: fixes s :: "'b list"   and n :: "nat"   and A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and "is" :: "'c" assumes "s \<noteq> []"   and "distinct s" shows "n * length s \<le> T s (cruel A (s, is) n) (off2 A (s, is) (cruel A (s, is) n))"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "MIR.num" shows "dvdnumcoeff (CN n c t) = (\<lambda>g. g dvd c \<and> dvdnumcoeff t g)"
simps_3: fixes uv :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt"   and q :: "('Var, 'AExp, 'BExp) Stmt" shows "has_annotated_stop (Stmt.If uv p q) = (has_annotated_stop p \<or> has_annotated_stop q)"
a: fixes x :: "'a" assumes "x \<in> A" shows "x \<in> C"
S_def: shows "S = {PIdl r |r. r \<in> carrier R - {\<zero>} \<and> r \<notin> Units R \<and> (\<nexists>fs. set fs \<subseteq> carrier (Ring_Divisibility.mult_of R) \<and> wfactors (Ring_Divisibility.mult_of R) fs r)}"
theorem_5_inv_def: fixes ds :: "'b set"   and fp :: "'a set" shows "theorem_5_inv ds fp = (Rf CH fp \<inter> \<Union> {X. stable_on ds X} = {})"
Node_3: shows "wbt \<langle>l, a, r\<rangle>"
leftmost_labels: fixes c :: "cmd"   and n :: "nat"   and c' :: "cmd" assumes "labels c n c'" shows "subcmd c (leftmostCmd c')"
assms_3: fixes x :: "constraint" assumes "x \<in> set cs" shows "\<nexists>a b. LTPP a b = x"
card_of_empty: fixes A :: "'d set" shows "|{}| \<le>o |A|"
collapse_1: fixes sterm :: "sterm" assumes "is_Sconst sterm" shows "Sconst (un_Sconst sterm) = sterm"
preserves_reflects_arr: fixes f :: "'d" shows "C.arr (F f) = D.arr f"
carr_prodag_mem_eq: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and X :: "'c \<Rightarrow> 'd"   and Y :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A"   and "\<forall>l\<in>I. X l = Y l" shows "X = Y"
nu_commute_fixpoint_1: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "isotone f"   and "has_greatest_fixpoint (f \<circ> g)"   and "f \<circ> g = g \<circ> f" shows "is_fixpoint f (\<nu> (f \<circ> g))"
Gromov_product_isometry: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "isometry_on UNIV f" shows "Gromov_product_at (f x) (f y) (f z) = Gromov_product_at x y z"
hypreal_sqrt_hrabs: fixes x :: "real star" shows "(*f* sqrt) (x\<^sup>2) = \<bar>x\<bar>"
approx_int_pos: fixes n :: "nat" shows "0 < \<bar>x - real_of_int (p n) / real_of_int (q n)\<bar>"
env_restr_below_self: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "f f|` S \<sqsubseteq> f"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<sigma>"
nneg_const: fixes c :: "'d" shows "nneg (\<lambda>x. c) = ((0::'d) \<le> c)"
poly_mod_2_axioms: shows "poly_mod_2 (p ^ l)"
least_aux: fixes f :: "nat \<Rightarrow> com list \<times> state"   and c :: "com"   and cs :: "com list"   and s :: "state"   and k :: "nat" assumes "f 0 = (c # cs, s)"   and "\<forall>i. f i \<rightarrow> f (Suc i)"   and "fst (f k) = cs"   and "\<forall>i<k. fst (f i) \<noteq> cs" shows "\<forall>i\<le>k. \<exists>p. (p \<noteq> []) = (i < k) \<and> fst (f i) = p @ cs"
B: shows "B = Matrix.mat n n (\<lambda>(i, j). A $$ (p i, p j))"
finite_UnionD: fixes A :: "'a set set" assumes "finite (\<Union> A)" shows "finite A"
PcalM_def: shows "PcalM = {s. s \<subseteq> carrier (G\<lparr>carrier := P\<rparr>) \<and> card s = p ^ a}"
funcset_mem: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and x :: "'b" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
path_components_eq_connected_components_of: fixes X :: "'a topology" assumes "locally_path_connected_space X" shows "path_components_of X = connected_components_of X"
fresh_fun_upd: fixes a :: "atom"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" assumes "a \<sharp> f"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> f(x := y)"
image_Collect_subsetI: fixes P :: "'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f"   and B :: "'f set" assumes "\<And>x. P x \<Longrightarrow> f x \<in> B" shows "f ` {x. P x} \<subseteq> B"
m\<tau>move_silentD: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "m\<tau>move s (t, ta) s'" shows "ta = \<lbrace>\<rbrace>"
Red_preserves_sconf: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "P,E,hp s,shp s \<turnstile> e : T"   and "iconf (shp s) e"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"