hcju/isabelleps huggingface dataset:
vareq_cong: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and t :: "real"   and x0 :: "'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x"   and "t \<in> Y.existence_ivl0 x0" shows "Y.vareq x0 t = c1_on_open_euclidean.vareq g f' Y x0 t"
Min_eqI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq x y"   and "x \<in> A" shows "linorder.Min less_eq A = x"
rev_cases: fixes xs :: "'e list" obtains "xs = []"   | ys :: "'e list" and y :: "'e" where "xs = ys @ [y]"
prj1_simps_3: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.cod \<p>\<^sub>1[f, g] = C.dom f"
preserves_reflects_arr: fixes f :: "'a" shows "arr \<d>[f] = arr f"
anti_mono: fixes l :: "nat"   and c :: "com"   and l' :: "nat" assumes "l \<turnstile> c"   and "l' \<le> l" shows "l' \<turnstile> c"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "span_vertical_category.src C \<mu> = span_vertical_category.trg C \<nu>" shows "two_composable_arrows_of_spans_axioms C \<mu> \<nu>"
assoc_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "assoc a b c \<equiv> tuple (pr1 a b \<cdot> pr1 (local.prod a b) c) (tuple (pr0 a b \<cdot> pr1 (local.prod a b) c) (pr0 (local.prod a b) c))"
FDOM_eff_subset_prob_dom: fixes p :: "('a, 'b) fmap"   and e :: "('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" assumes "(p, e) \<in> PROB" shows "fmdom' e \<subseteq> prob_dom PROB"
rep_eq: fixes x :: "'e \<Rightarrow> 'c"   and xa :: "'e set['d]" shows "set_bset (map_bset x xa) = x ` set_bset xa"
subst_simp_app: fixes A :: "'a trm"   and B :: "'a trm"   and x :: "'a"   and M :: "'a trm" shows "App A B[x ::= M] = App A[x ::= M] B[x ::= M]"
induct: fixes P :: "C \<Rightarrow> bool"   and a0 :: "C" assumes "P (C [])"   and "\<And>ns. P (C (C [] # ns))"   and "\<And>va vb ns. (\<And>x. P (C (va # vb))) \<Longrightarrow> P (C (C (va # vb) # ns))" shows "P a0"
ntrunc_Numb: fixes k :: "nat"   and i :: "nat" shows "ntrunc (Suc k) (Numb i) = Numb i"
valid_sb_reads: shows "valid_reads m\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
sorted_wrt_map_mono: fixes Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "sorted_wrt Q xs"   and "\<And>x y. Q x y \<Longrightarrow> P (f x) (f y)" shows "sorted_wrt P (map f xs)"
nn_integral_join: fixes f :: "'c \<Rightarrow> ennreal"   and N :: "'c measure"   and M :: "'c measure measure" assumes "f \<in> borel_measurable N"   and "sets M = sets (subprob_algebra N)" shows "integral\<^sup>N (join M) f = \<integral>\<^sup>+ M'. integral\<^sup>N M' f \<partial>M"
circ_extra_circ: fixes y :: "'a"   and x :: "'a" shows "(y * x\<^sup>\<circ>)\<^sup>\<circ> = (y * y\<^sup>\<circ> * x\<^sup>\<circ>)\<^sup>\<circ>"
disc_13: fixes x31 :: "'a local.term"   and x32 :: "'a local.term" shows "is_Tensor (x31 \<^bold>\<otimes> x32)"
seq_power_0: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice" assumes "seq_distrib_right (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "seq_distrib_right.seq_power (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (a::'b::refinement_lattice) 0 = nil"
Inf_top_conv_2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(top = Inf A) = (\<forall>x\<in>A. x = top)"
uminus_formula_def: fixes x :: "'a formula" shows "- x = Abs_formula (- Rep_formula x)"
b'Z: shows "emb b B \<in> A - B \<rightarrow> {\<zero>\<^bsub>K\<^esub>}"
mono_gamma_s: fixes f :: "'av st"   and g :: "'av st" assumes "f \<le> g" shows "\<gamma>\<^sub>s f \<subseteq> \<gamma>\<^sub>s g"
3: shows "Right_a.ide f \<and> Right_a.ide g"
bounded_clinear_minus: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "bounded_clinear (\<lambda>x. - f x)"
poly_eq_poly_eval': fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and x :: "'c"   and a :: "'d" assumes "p \<in> P[{x}]" shows "poly (poly_of_pm x p) a = poly_eval (\<lambda>_. a) p"
create_store_commute: fixes i1 :: "'d"   and e1 :: "'c"   and e2 :: "'c"   and mo :: "'d"   and i2 :: "'d" shows "\<langle>Create i1 e1\<rangle> \<rhd> \<langle>Store e2 mo i2\<rangle> = \<langle>Store e2 mo i2\<rangle> \<rhd> \<langle>Create i1 e1\<rangle>"
lemma_conf_temporal: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and Tk :: "nat"   and K :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>\<rbrace>) \<in> set evs"   and "Key K \<in> analz (knows Spy evs)" shows "expiredK Tk evs"
dg_prod_is_arrI: fixes f :: "V"   and a :: "V"   and b :: "V" assumes "vsv f"   and "\<D>\<^sub>\<circ> f = I"   and "vsv a"   and "\<D>\<^sub>\<circ> a = I"   and "vsv b"   and "\<D>\<^sub>\<circ> b = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>" shows "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b"
antisym_conv2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "less_eq x y" shows "(\<not> less x y) = (x = y)"
resp_analz_image_freshK_lemma: fixes RB :: "msg"   and evs :: "event list"   and H :: "msg set" assumes "RB \<in> responses evs"   and "\<forall>K KK. KK \<subseteq> - range shrK \<longrightarrow> (Key K \<in> analz (Key ` KK \<union> H)) = (K \<in> KK \<or> Key K \<in> analz H)" shows "\<forall>K KK. KK \<subseteq> - range shrK \<longrightarrow> (Key K \<in> analz (insert RB (Key ` KK \<union> H))) = (K \<in> KK \<or> Key K \<in> analz (insert RB H))"
bigo_const_mult4: fixes c :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "c \<noteq> (0::'c)" shows "O(f) \<subseteq> O(\<lambda>x. c * f x)"
part_to_full: fixes C :: "com"   and s :: "char list \<Rightarrow> int"   and m :: "nat"   and s' :: "char list \<Rightarrow> int" assumes "(C, part s) \<Rightarrow>\<^sub>A m \<Down> part s'" shows "(C, s) \<Rightarrow> m \<Down> s'"
SubstAtomic_def: fixes v :: "hf"   and tm :: "hf"   and y :: "hf"   and y' :: "hf" shows "SubstAtomic v tm y y' \<equiv> \<exists>t u t' u'. SubstTerm v tm t t' \<and> SubstTerm v tm u u' \<and> (y = q_Eq t u \<and> y' = q_Eq t' u' \<or> y = q_Mem t u \<and> y' = q_Mem t' u')"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "'state" assumes "a1 \<rightarrow>*t a2" obtains cf :: "('test, 'atom) com \<times> 'state" and cf' :: "('test, 'atom) com \<times> 'state" and s'' :: "'state" where "a1 = cf" and "a2 = s''" and "cf \<rightarrow>*c cf'" and "cf' \<rightarrow>t s''"
lincomb_closed: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "S \<subseteq> carrier M"   and "a \<in> S \<rightarrow> carrier R" shows "lincomb a S \<in> carrier M"
rel_fun_def_butlast: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and T :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h \<Rightarrow> 'i" shows "(R ===> S ===> T) f g = (\<forall>x y. R x y \<longrightarrow> (S ===> T) (f x) (g y))"
beta_Upd: fixes l :: "Label"   and f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and t :: "sterm" assumes "l \<in> dom f"   and "lc (Obj f T)"   and "body t" shows "Upd (Obj f T) l t -> Obj (f(l \<mapsto> t)) T"
connect_def: fixes d :: "('c, 'd) distinguisher"   and res :: "('c, 'd) resource" shows "connect d res = run_gpv run_resource d res"
FCIA_def: fixes \<rho> :: "'a V_rec \<Rightarrow> 'a set"   and \<Gamma> :: "'a Gamma"   and \<V> :: "'a V_rec"   and Tr :: "'a list set" shows "FCIA \<rho> \<Gamma> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<beta> @ [v] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> Adm \<V> \<rho> Tr \<beta> c \<longrightarrow> (\<exists>\<alpha>' \<delta>'. set \<delta>' \<subseteq> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub> \<and> \<beta> @ [c] @ \<delta>' @ [v] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
proj_P2R: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" shows "proj \<phi> x = P2R \<phi> `` {x}"
assms_3: shows "B\<^sub>1 \<notin> P\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2"
two_set_categories_bij_betw_Univ_axioms: shows "two_set_categories_bij_betw_Univ (\<cdot>) (\<cdot>\<acute>) \<AA> \<phi>"
diff_Union_pairwise_disjoint: fixes \<A> :: "'b set set"   and \<B> :: "'b set set" assumes "disjoint \<A>"   and "\<B> \<subseteq> \<A>" shows "\<Union> \<A> - \<Union> \<B> = \<Union> (\<A> - \<B>)"
bit_not_int_iff: fixes k :: "int"   and n :: "nat" shows "bit (NOT k) n = (\<not> bit k n)"
Perp_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D \<or> Tarski_neutral_dimensionless.Perp Bet Cong B A C D \<or> Tarski_neutral_dimensionless.Perp Bet Cong A B D C \<or> Tarski_neutral_dimensionless.Perp Bet Cong B A D C \<or> Tarski_neutral_dimensionless.Perp Bet Cong C D A B \<or> Tarski_neutral_dimensionless.Perp Bet Cong C D B A \<or> Tarski_neutral_dimensionless.Perp Bet Cong D C A B \<or> Tarski_neutral_dimensionless.Perp Bet Cong D C B A" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B C D"
sum_decode_inverse: fixes n :: "nat" shows "sum_encode (sum_decode n) = n"
parts_of_LtKeys: fixes K :: "msg set" assumes "K \<subseteq> range LtK" shows "parts K = K"
cod_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.cod (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
isArrLenLoc_simps_2: fixes f :: "AttId" shows "isArrLenLoc (staticLoc f) = False"
right_inverse: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "Groups.group f z inverse" shows "f a (inverse a) = z"
\<delta>_prod_def: fixes \<delta>1 :: "('a, 'c) ta_rule set"   and \<delta>2 :: "('b, 'c) ta_rule set" shows "\<delta>_prod \<delta>1 \<delta>2 \<equiv> {r_prod q1 \<rightarrow> l qs1 q2 \<rightarrow> l qs2 |q1 q2 l qs1 qs2. length qs1 = length qs2 \<and> q1 \<rightarrow> l qs1 \<in> \<delta>1 \<and> q2 \<rightarrow> l qs2 \<in> \<delta>2}"
Frame: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3 {P} C { Q}" shows "\<turnstile>\<^sub>3 {P \<and>* F} C { Q \<and>* F}"
gfp_mono: fixes f :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice"   and g :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice" assumes "\<And>Z::'b::complete_lattice. (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) Z \<le> (g::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) Z" shows "gfp (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) \<le> gfp (g::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)"
delete_only_m_def: fixes x :: "('a, 'c option) insert_message" shows "delete_only_m x = (if DeleteMessage (I x) = m then delete x else x)"
degree_uminus_pdevs: fixes x :: "'b pdevs" shows "degree (uminus_pdevs x) = degree x"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "llist_of (x # xs) = LCons x (llist_of xs)"
AbGroup_sum_list: fixes Gs :: "'a set list" assumes "\<forall>G\<in>set Gs. AbGroup G" shows "AbGroup (\<Sum>G\<leftarrow>Gs. G)"
simps: fixes xs :: "'b list" shows "C_nmsort xs = C_runs xs + C_merge_all (runs xs)"
lesspoll_def: fixes A :: "'b set"   and B :: "'c set" shows "A \<prec> B \<equiv> A \<lesssim> B \<and> \<not> A \<approx> B"
a2_map'_def: shows "a2_map' = Mapping.update (tp + 1) Mapping.empty (upd_nested a2_map new_tstp (max_tstp new_tstp) tmp)"
cc_restr_mono2: fixes G :: "CoCalls"   and G' :: "CoCalls"   and S :: "Vars.var set" assumes "G \<sqsubseteq> G'" shows "G G|` S \<sqsubseteq> G' G|` S"
Cons_1: assumes "\<exists>a\<in>set xs. P a" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>y\<in>set ys. \<not> P y)"
l_set_mode_get_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_disconnected_nodes type_wf set_mode set_mode_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_mode type_wf set_mode set_mode_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_mode_get_disconnected_nodes_axioms set_mode_locs get_disconnected_nodes_locs"
convex_hull_affinity: fixes a :: "'b"   and c :: "real"   and S :: "'b set" shows "convex hull (\<lambda>x. a + c *\<^sub>R x) ` S = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
r_neg: fixes x :: "complex vec" assumes "x \<in> carrier_vec d" shows "x + \<ominus>\<^bsub>V\<^esub> x = 0\<^sub>v d"
minus_eq: fixes x :: "'a set"   and y :: "'a set" shows "x \<ominus>\<^bsub>R Quot J\<^esub> y = x \<oplus>\<^bsub>R Quot J\<^esub> \<ominus>\<^bsub>R Quot J\<^esub> y"
F: shows "set F = \<Union> (funas_term ` {t, t \<cdot> \<sigma>})"
collapse_2: fixes literal :: "'a literal" assumes "\<not> sign literal" shows "Neg (get_pred literal) (get_terms literal) = literal"
main_nat: fixes a :: "nat"   and b :: "nat" shows "(a + b) * valid_countings a b = (a - b) * all_countings a b"
A7: shows "\<forall>a b c p q. B a p c \<and> B b q c \<longrightarrow> (\<exists>x. B p x b \<and> B q x a)"
sgn_0: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn zero = zero"
digraph_isomorphism_invI: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "wf_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "pre_digraph.digraph_isomorphism (app_iso hom G) (pre_digraph.inv_iso G hom)"
Sup_distr: fixes X :: "('a \<times> 'b) set"   and y :: "'a \<times> 'b" shows "times_prod' (Sup_prod X) y = Sup_prod ((\<lambda>x. times_prod' x y) ` X)"
faithful_functors_compose: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> 'f" assumes "faithful_functor A B F"   and "faithful_functor B C G" shows "faithful_functor A C (G \<circ> F)"
ncols_def: fixes A :: "'b matrix" shows "ncols A \<equiv> if nonzero_positions (Rep_matrix A) = {} then 0 else Suc (Max (snd ` nonzero_positions (Rep_matrix A)))"
remainder_bound: fixes s :: "'a" assumes "c \<le> s \<bullet> (1::'a)" shows "norm (remainder s) \<le> A / real (Suc N) powr (s \<bullet> (1::'a))"
deg_of_mono: fixes x :: "real"   and y :: "real" shows "(deg_of x \<le> deg_of y) = (x \<le> y)"
A_1: shows "\<forall>k\<le>n. M k k \<le> \<one>"
scalingI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P c x. \<lbrakk>sound P; 0 \<le> c\<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x" shows "scaling t"
rep_eq: fixes x :: "'f" shows "Rep_formula (Pred x) = Pred\<^sub>\<alpha> x"
simps_3: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'b"   and T :: "ty" shows "heap_base.conf_extRet typeof_addr P h RetStaySame T = True"
homeomorphic_imp_injective_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "inj_on f (topspace X)"
induced_subgraph_altdef: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" shows "induced_subgraph H G = (subgraph H G \<and> (\<forall>H'. subgraph H' G \<longrightarrow> verts H' \<noteq> verts H \<or> arcs H' \<subseteq> arcs H))"
star_unfoldr: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x \<le> x\<^sup>\<star>"
ys_2: shows "remove_cycles bs k [k] = k # ys"
final_notin_to_finaln: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and T :: "('e, 'f) xstate set"   and n :: "nat" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>T" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>T"
valid_Exit_path_intra_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and n :: "'b"   and as :: "'a list" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as Exit" obtains as' :: "'a list" and pex :: "'b" where "CFG.intra_path sourcenode targetnode kind valid_edge n as' pex" and "CFGExit.method_exit sourcenode kind valid_edge Exit pex" and "set (CFG.sourcenodes sourcenode as') \<subseteq> set (CFG.sourcenodes sourcenode as)"
simps_3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = plus (plus (numeral.numeral one plus n) (numeral.numeral one plus n)) one"
eq_\<I>_gpv_map_gpv1: shows "eq_\<I>_gpv A\<inverse>\<inverse> \<I> (map_gpv f id gpv') gpv = eq_\<I>_gpv (\<lambda>x. A\<inverse>\<inverse> (f x)) \<I> gpv' gpv"
unitary11_inv: fixes k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and a :: "complex"   and b :: "complex" assumes "k \<noteq> 0"   and "M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a)"   and "mat_det (a, b, cnj b, cnj a) \<noteq> 0" shows "\<exists>k' a' b'. k' \<noteq> 0 \<and> mat_inv M = k' *\<^sub>s\<^sub>m (a', b', cnj b', cnj a') \<and> mat_det (a', b', cnj b', cnj a') \<noteq> 0"
Ccod: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "cod\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
coprimeI_primes: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> (0::'b) \<or> b \<noteq> (0::'b)"   and "\<And>p. \<lbrakk>prime p; p dvd a; p dvd b\<rbrakk> \<Longrightarrow> False" shows "coprime a b"
Derives1_drop_prefixword: fixes w :: "symbol list"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "is_word w"   and "Derives1 (w @ a) i r b" shows "Derives1 a (i - length w) r (drop (length w) b)"
While_tE3: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int option"   and x :: "nat"   and t :: "char list \<Rightarrow> int option" assumes "(WHILE b DO c, s) \<Rightarrow>\<^sub>A x \<Down> t" obtains "x = Suc 0" and "t = s" and "vars b \<subseteq> dom s" and "\<not> pbval b s"   | xa :: "nat" and s2 :: "char list \<Rightarrow> int option" and y :: "nat" where "pbval b s" and "vars b \<subseteq> dom s" and "(c, s) \<Rightarrow>\<^sub>A xa \<Down> s2" and "(WHILE b DO c, s2) \<Rightarrow>\<^sub>A y \<Down> t" and "Suc (xa + y) = x"
join_related2: fixes y :: "int"   and x :: "int" assumes "y \<le> x" shows "int.join x y = x"
floor_power: fixes x :: "'c"   and n :: "nat" assumes "x = of_int \<lfloor>x\<rfloor>" shows "\<lfloor>x ^ n\<rfloor> = \<lfloor>x\<rfloor> ^ n"
subseq_sum_weights_le: fixes xs :: "'lbl list"   and ys :: "'lbl list" assumes "xs \<preceq> ys" shows "sum_weights xs \<le> sum_weights ys"
snapshot_subset: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" shows "snapshot PROB s \<subseteq> PROB"
NB_Crypt_imp_Server_msg: fixes B :: "agent"   and A :: "agent"   and evs :: "event list"   and NB :: "msg"   and K :: "nat" assumes "B \<notin> bad"   and "A \<noteq> B"   and "evs \<in> otway"   and "Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace> \<in> parts (knows Spy evs)" shows "\<exists>NA. Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs"
S: shows "maximal S (\<lambda>T. T \<subseteq> set (cols (conjugate A)) \<and> lin_indpt T)"
vimage2p_id: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "vimage2p id id R = R"
Node: fixes l :: "'link" assumes "cwf (pointsTo l)" shows "cwf (Link l)"
q_def: fixes t :: "real" shows "ego.q t = s\<^sub>e + v\<^sub>e * t"
space_closed: shows "G ` I \<subseteq> Pow \<Omega>"
simps_2: fixes var :: "nat"   and a :: "'a" shows "atom_var (Geq var a) = var"
adds_minus_splus: fixes u :: "'t"   and t :: "'a" assumes "pp_of_term u adds t" shows "(t - pp_of_term u) \<oplus> u = term_of_pair (t, component_of_term u)"
T: shows "\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'a}"
pmf_of_set_UN: fixes f :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and n :: "nat" assumes "finite (\<Union> (f ` A))"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> card (f x) = n"   and "disjoint_family_on f A" shows "pmf_of_set (\<Union> (f ` A)) = pmf_of_set A \<bind> (\<lambda>x. pmf_of_set (f x))"
distinct_117: fixes x4 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Inverse x4 \<noteq> x151 ^\<^sub>e x152"
monic_degree_one_root_condition: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "ring R"   and "a \<in> carrier R" shows "ring.is_root R [\<one>\<^bsub>R\<^esub>, \<ominus>\<^bsub>R\<^esub> a] b = (a = b)"
bounded_clinear_scaleC_left: fixes x :: "'b" shows "bounded_clinear (\<lambda>c. c *\<^sub>C x)"
SINK_mono': fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "f \<le> g" shows "MFMC_Network.SINK g \<subseteq> MFMC_Network.SINK f"