hcju/isabelleps huggingface dataset:
extension_from_component: fixes S :: "'a set"   and U :: "'b set"   and C :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "locally connected S \<or> compact S"   and "ANR U"   and "C \<in> components S"   and "continuous_on C f"   and "f ` C \<subseteq> U" obtains g :: "'a \<Rightarrow> 'b" where "continuous_on S g" and "g ` S \<subseteq> U" and "\<And>x. x \<in> C \<Longrightarrow> g x = f x"
f_1: shows "(f \<longlongrightarrow> a) F"
prems: shows "\<forall>i<Suc k. \<not> (CS (f i) = cs \<and> CSS (f i) = css)"
simps_4: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
convert_perm_compose: fixes \<pi> :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> nat" assumes "\<pi> permutes {..<n}"   and "\<sigma> permutes {..<n}" shows "convert_perm (\<pi> \<circ> \<sigma>) = convert_perm \<pi> \<circ> convert_perm \<sigma>"
label_stab_map_surj_on_vertices: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0" shows "(\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v}) ` \<Union> X = \<Union> (CoxeterComplex.TheComplex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}))"
lower: fixes x :: "'a measure"   and A :: "'a measure set" assumes "x \<in> A" shows "Inf A \<le> x"
\<phi>_simps_3: shows "trg t\<^sub>0u\<^sub>1.\<phi> = trg u\<^sub>1"
lookup_operator_works_3: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and name' :: "char list"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat" assumes "is_standard_operator' (name, pres, effs, layer)"   and "(name, pres, effs, layer) \<in> set ast\<delta>"   and "lookup_action (abs_ast_operator (name, pres, effs, layer)) = Some (name', pres', effs', layer')" shows "effs = effs'"
min_height_size1_if_incomplete: fixes t :: "'c tree" assumes "\<not> complete t" shows "2 ^ min_height t < size1 t"
transition_eqvt': fixes P :: "('state, 'effect) L_state"   and \<alpha> :: "('act, 'effect) L_action"   and Q :: "('state, 'effect) L_state"   and p :: "perm" assumes "P \<rightarrow>\<^sub>L \<langle>\<alpha>,Q\<rangle>" shows "p \<bullet> P \<rightarrow>\<^sub>L \<langle>p \<bullet> \<alpha>,p \<bullet> Q\<rangle>"
poly_vars: fixes p :: "('c monom \<times> 'd) list"   and f :: "'c \<Rightarrow> ('e monom \<times> 'd) list"   and g :: "'c \<Rightarrow> ('e monom \<times> 'd) list" assumes "\<And>w. w \<in> poly_vars p \<Longrightarrow> f w = g w" shows "poly_subst f p = poly_subst g p"
perpendicular_sym: fixes H1 :: "circline"   and H2 :: "circline" shows "perpendicular H1 H2 = perpendicular H2 H1"
cartesian_monoidal_category_axioms: shows "cartesian_monoidal_category (\<cdot>) T \<alpha> \<iota> \<delta> \<tau>"
open_cofinite_def: shows "open = (\<lambda>X. finite (- X) \<or> X = {})"
ordertype_infinite_\<omega>: fixes A :: "V set" assumes "A \<subseteq> elts \<omega>"   and "infinite A" shows "ordertype A VWF = \<omega>"
rel_sel: fixes R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "('f, 'h, 'j) Resumption.resumption"   and b :: "('g, 'i, 'j) Resumption.resumption" shows "Resumption.resumption.rel_resumption R1 R2 a b = (Resumption.resumption.is_Done a = Resumption.resumption.is_Done b \<and> (Resumption.resumption.is_Done a \<longrightarrow> Resumption.resumption.is_Done b \<longrightarrow> rel_option R1 (Resumption.resumption.result a) (Resumption.resumption.result b)) \<and> (\<not> Resumption.resumption.is_Done a \<longrightarrow> \<not> Resumption.resumption.is_Done b \<longrightarrow> R2 (Resumption.resumption.output a) (Resumption.resumption.output b) \<and> rel_fun (=) (Resumption.resumption.rel_resumption R1 R2) (Resumption.resumption.resume a) (Resumption.resumption.resume b)))"
update_convs_1: fixes xpos' :: "nat \<Rightarrow> nat"   and xpos :: "nat"   and ypos :: "nat"   and more :: "'a" shows "xpos_update xpos' \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr> = \<lparr>xpos = xpos' xpos, ypos = ypos, \<dots> = more\<rparr>"
second_P_k_pre_not_in_Q: shows "hd (tl (P_k_pre @ [y])) \<notin> Q.second_vertices"
deriv: fixes x :: "real" assumes "x \<in> {a..b} - s" shows "(g has_real_derivative g' x) (at x within {a..b})"
A_objects: fixes \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>" shows "[[\<^bold>\<exists>x. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)]]"
charpoly_def: fixes A :: "('c, 'd) sq_matrix" shows "Cayley_Hamilton.charpoly A = Square_Matrix.det (\<^bold>X - \<^bold>C A)"
snd_pseudo_divmod_main_cong: fixes a1 :: "'b"   and b1 :: "'b"   and a3 :: "'b poly"   and b3 :: "'b poly"   and a4 :: "'b poly"   and b4 :: "'b poly"   and a5 :: "nat"   and b5 :: "nat"   and a6 :: "nat"   and b6 :: "nat"   and a2 :: "'b poly"   and b2 :: "'b poly" assumes "a1 = b1"   and "a3 = b3"   and "a4 = b4"   and "a5 = b5"   and "a6 = b6" shows "snd (pseudo_divmod_main a1 a2 a3 a4 a5 a6) = snd (pseudo_divmod_main b1 b2 b3 b4 b5 b6)"
bit_push_bit_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (push_bit m a) n = (m \<le> n \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit a (n - m))"
hyps_2: shows "(P, S - {Q}) \<in> sumComposeSet"
const_name_simps_1: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.const_name unconst (const name) = name"
nonzero: fixes h :: "'a \<Rightarrow> real" assumes "h \<in> powr_closure f" shows "\<forall>\<^sub>F x in F. h x \<noteq> 0"
Limit_vcomp_in_VsetI: fixes \<alpha> :: "V"   and r :: "V"   and s :: "V" assumes "Limit \<alpha>"   and "r \<in>\<^sub>\<circ> Vset \<alpha>"   and "s \<in>\<^sub>\<circ> Vset \<alpha>" shows "r \<circ>\<^sub>\<circ> s \<in>\<^sub>\<circ> Vset \<alpha>"
pick_index: fixes a :: "nat"   and I :: "nat set"   and a' :: "nat" assumes "a \<in> I"   and "a' < card I" shows "(pick I a' = a) = (index (sorted_list_of_set I) a = a')"
val_nonzero_z_unique: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "\<exists>!z. v x = ant z"
simps_2: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and v :: "'f"   and va :: "bool" shows "redT_updT' ts (ThreadExists v va) = ts"
prime_ideal_def: fixes R :: "('c, 'd) Ring_scheme"   and p :: "'c set" shows "prime_ideal R p = (ideal R p \<and> 1\<^sub>r\<^bsub>R\<^esub> \<notin> p \<and> (\<forall>x\<in>carrier R. \<forall>y\<in>carrier R. x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> p \<longrightarrow> x \<in> p \<or> y \<in> p))"
assms_2: fixes x :: "'b" assumes "x \<in> set (coeffs p)"   and "x \<noteq> (0::'b)" shows "f x \<noteq> (0::'a)"
strict_mono_interval: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "strict_mono f"   and "f 0 \<le> n" obtains k :: "nat" where "f k \<le> n" and "n < f (Suc k)"
poincare_ray_meets_line_def: fixes a :: "complex_homo"   and b :: "complex_homo"   and c :: "complex_homo"   and d :: "complex_homo" shows "poincare_ray_meets_line a b c d = (\<exists>x\<in>unit_disc. poincare_on_ray x a b \<and> poincare_on_line x c d)"
disc_filtr_prob_space_axioms: shows "disc_filtr_prob_space M F"
LeastI_ex: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x" shows "P (ord.Least less_eq P)"
trans: shows "\<forall>x. f (f x) \<le> f x"
star_cases: fixes x :: "'c star" obtains X :: "nat \<Rightarrow> 'c" where "x = star_n X"
one_uint64_def: shows "1 \<equiv> Abs_uint64 1"
simps_2: fixes f :: "'s" shows "arity_hd (Sym f) = \<infinity>"
spec_refine: fixes P :: "'b upred"   and R :: "'b upred"   and Q :: "'b upred" assumes "Q \<sqsubseteq> (P \<and> R)" shows "(P \<Rightarrow> Q) \<sqsubseteq> R"
Rep_\<I>_inject: fixes x :: "\<I>"   and y :: "\<I>" shows "(Rep_\<I> x = Rep_\<I> y) = (x = y)"
labeling_consistent: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c, 'd) sa_rec_scheme"   and B :: "('b, 'c, 'e) sa_rec_scheme"   and a :: "'a"   and b :: "'b" assumes "lsimulation R A B"   and "(a, b) \<in> R" shows "sa_L A a = sa_L B b"
eq_numeral_extra_2: shows "(1::'a) \<noteq> (0::'a)"
simps_18: fixes x51 :: "bexp"   and x52 :: "com"   and x21 :: "char list"   and x22 :: "aexp" shows "WHILE x51 DO x52 \<noteq> x21 ::= x22"
A: shows "\<not> (\<exists>v\<in>D ` (X \<union> set zs). (D y, v) \<in> I)"
sublist_imp_len_filter_le: fixes l' :: "'a list"   and l :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "subseq l' l" shows "length (filter P l') \<le> length (filter P l)"
single_singleton: fixes A :: "'a pred"   and default :: "unit \<Rightarrow> 'a" assumes "\<exists>!x. eval A x" shows "single (singleton default A) = A"
in_pdevs_domain: fixes i :: "nat"   and x :: "'a pdevs" shows "(i \<in> pdevs_domain x) = (pdevs_apply x i \<noteq> (0::'a))"
Lim_emeasure_incseq: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Union> (range A))"
connected_iff_eq_connected_component_set: fixes S :: "'a set" shows "connected S = (\<forall>x\<in>S. connected_component_set S x = S)"
norm_column_le_onorm: fixes i :: "'b"   and A :: "((real, 'b) vec, 'a) vec" shows "norm (column i A) \<le> onorm ((*v) A)"
eqvt: shows "eqvt (\<lambda>S. \<Union> (supp ` S))"
wf_Tree_wf\<^sub>p: shows "wf Tree_wf\<^sub>p"
h: shows "h = fpxs_compose_power (fpxs_of_fls h') (1 / rat_of_nat d)"
IH: fixes l :: "'a list" assumes "l \<in> set (subseqs l')" shows "set l \<subseteq> set l'"
exact_aux_1: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs" shows "valid_decomp X (exact_aux k m qs)"
simps_6: fixes \<Sigma> :: "nat \<Rightarrow> 'b set"   and project :: "'b \<Rightarrow> 'b"   and lookup :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp"   and s :: "'c rexp" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n (Times r s) = project.lang \<Sigma> project lookup n r @@ project.lang \<Sigma> project lookup n s"
LeastI: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool"   and k :: "'f" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
union_of_set: fixes x :: "hf" shows "\<exists>z. \<forall>u. (u \<^bold>\<in> z) = (\<exists>y. y \<^bold>\<in> x \<and> u \<^bold>\<in> y)"
is_cnf_disj: fixes f1 :: "'a formula"   and f2 :: "'a formula" assumes "is_cnf (f1 \<^bold>\<or> f2)" obtains f :: "'a formula" where "f1 \<^bold>\<or> f2 = f" and "is_disj f"
eager_def: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "int pmf \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and p :: "'c \<Rightarrow> int pmf"   and e :: "'c exp" shows "prob_exp_base.eager return bind fail get put sample p e = prob_exp_base.sample_vars return bind get put sample p (vars e) (exp_base.eval return bind fail (memo_exp_base.lookup return fail get) e)"
assms_2: fixes x :: "'b" assumes "x \<in> space M" shows "g x \<le> f x"
par_INF_distrib: fixes X :: "'c::type set"   and c :: "'a::refinement_lattice"   and d :: "'c::type \<Rightarrow> 'a::refinement_lattice" assumes "(X::'c::type set) \<noteq> {}" shows "(c::'a::refinement_lattice) \<parallel> \<Sqinter> ((d::'c::type \<Rightarrow> 'a::refinement_lattice) ` (X::'c::type set)) = (\<Sqinter>x::'c::type\<in>X. c \<parallel> d x)"
scaleR_simps_7: fixes a :: "real"   and x :: "'a" shows "(a *\<^sub>R x = (0::'a)) = (a = 0 \<or> x = (0::'a))"
lens_equiv_def: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" shows "X \<approx>\<^sub>L Y = (X \<subseteq>\<^sub>L Y \<and> Y \<subseteq>\<^sub>L X)"
one_mod_2_pow_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "modulo one (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = zero_neq_one.of_bool one zero (0 < n)"
b: fixes v :: "'variable" assumes "v \<in> set (precondition_of op')" shows "v \<notin> set (delete_effects_of op)"
lve_Exit_edge: fixes e :: "'f LDCFG_node \<times> 'g edge_kind \<times> 'f LDCFG_node"   and X :: "'f"   and valid_edge :: "'h \<Rightarrow> bool"   and src :: "'h \<Rightarrow> 'f"   and trg :: "'h \<Rightarrow> 'f"   and knd :: "'h \<Rightarrow> 'g edge_kind"   and E :: "'f" assumes "e = (Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit)" shows "lift_valid_edge valid_edge src trg knd E X e"
can_runit: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a" shows "free_monoidal_category.can C (free_monoidal_category.runit\<^sub>F\<^sub>M\<^sub>C C a)"
traces_def: fixes P :: "'e process" shows "traces P \<equiv> Domain (failures P)"
iprev_less_mono: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a < b"   and "a \<in> I"   and "b \<in> I"   and "\<exists>x\<in>I. x < a" shows "iprev a I < iprev b I"
analytic_zeta: fixes A :: "complex set" assumes "1 \<notin> A" shows "zeta analytic_on A"
Spy_dont_know_private_keys: fixes b :: "keymode"   and A :: "agent"   and evs :: "event list" assumes "Key (privateKey b A) \<in> parts (knows Spy evs)"   and "evs \<in> certified_mail" shows "A \<in> bad"
not_eq_extremum: fixes a :: "'a word" shows "(a \<noteq> - 1) = (a < - 1)"
intros_3: fixes KS :: "key list"   and X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "foldr Crypt KS \<lbrace>X, Y\<rbrace> \<in> crypts H" shows "foldr Crypt KS X \<in> crypts H"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "type_wf = ShadowRootClass.type_wf"   and "get_mode = get_mode"   and "get_mode_locs = get_mode_locs" shows "l_get_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_mode get_mode_locs type_wf"
measurable_walk: fixes s :: "'s" shows "random_variable (stream_space (count_space UNIV)) (walk s)"
t: shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t declassified\<^sub>l\<^sub>s\<^sub>t A I"
uint_xor: fixes x :: "'b word"   and y :: "'b word" shows "uint (x XOR y) = uint x XOR uint y"
last: shows "last \<HH> = carrier (G\<lparr>carrier := H\<rparr>)"
shunting_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> - y \<le> z) = (x \<le> z \<squnion> y)"
succsr_is_node: fixes x :: "int"   and y :: "int"   and n :: "nat"   and ks :: "int list"   and l :: "int" assumes "(x, y) \<in> (succsr (dioph_ineq_succs n ks))\<^sup>*"   and "dioph_is_node ks l x" shows "dioph_is_node ks l y"
mod_mod_power_cancel: fixes m :: "nat"   and n :: "nat"   and p :: "nat" assumes "m \<le> n" shows "p mod 2 ^ n mod 2 ^ m = p mod 2 ^ m"
ind_hyp_1: assumes "\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>" shows "line_integral_exists F basis \<gamma>"
that: fixes z :: "complex" assumes "(\<Sum>i\<le>n. a i * z ^ i) = 0" shows "thesis"
local: fixes p :: "('b, 'c, 'a, 'd) seqp"   and pn :: "'a"   and l :: "'d"   and \<sigma> :: "nat \<Rightarrow> 'b"   and a :: "'c seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'b"   and q :: "('b, 'c, 'a, 'd) seqp"   and l' :: "'d"   and pp :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "p \<in> ctermsl (\<Gamma> pn)"   and "not_call p"   and "l \<in> labels \<Gamma> p"   and "P (\<sigma>, l)"   and "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   and "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> q"   and "(\<sigma>, pp) \<in> oreachable A S U"   and "p \<in> sterms \<Gamma> pp"   and "(\<sigma>', q) \<in> oreachable A S U"   and "S \<sigma> \<sigma>' a" shows "P (\<sigma>', l')"
path_from_root_Vr: fixes v :: "'a" assumes "v \<in> verts G"   and "dist v \<noteq> \<infinity>" shows "s \<rightarrow>\<^sup>* v"
dagger_trans_eq: fixes x :: "'a ppath set" shows "pp_prod (pp_star x) (pp_star x) = pp_star x"
assms_2: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "(g has_derivative g' x) (at x within S)"
has_visible_field: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "P \<turnstile> C sees F:T (fm) in D" shows "P \<turnstile> C has F:T (fm) in D"
vrestriction_vsubset_rel: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> r"
simply_connected_imp_path_connected: fixes S :: "'a set" assumes "simply_connected S" shows "path_connected S"
elims_1: fixes x :: "Core.term"   and y :: "bool" assumes "eta_reducible x = y" obtains uu_ :: "typ" and t :: "Core.term" where "x = Abs uu_ (t__ $ Bv 0)" and "y = (\<not> is_dependent t__ \<or> eta_reducible t__)"   | uv_ :: "typ" and v :: "String.literal" and va :: "typ" where "x = Abs uv___ (Ct v__ va__)" and "y = eta_reducible (Ct v__ va__)"   | uv_ :: "typ" and v :: "variable" and va :: "typ" where "x = Abs uva___ (Fv vb__ vaa__)" and "y = eta_reducible (Fv vb__ vaa__)"   | uv_ :: "typ" and v :: "nat" where "x = Abs uvb___ (Bv vc__)" and "y = eta_reducible (Bv vc__)"   | uv_ :: "typ" and v :: "typ" and va :: "Core.term" where "x = Abs uvc___ (Abs vd__ vab__)" and "y = eta_reducible (Abs vd__ vab__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "String.literal" and vc :: "typ" where "x = Abs uvd___ (ve__ $ Ct vba__ vca__)" and "y = eta_reducible (ve__ $ Ct vba__ vca__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "variable" and vc :: "typ" where "x = Abs uve___ (vf__ $ Fv vbb__ vcb__)" and "y = eta_reducible (vf__ $ Fv vbb__ vcb__)"   | uv_ :: "typ" and v :: "Core.term" and vc :: "nat" where "x = Abs uvf___ (vg__ $ Bv (Suc vcc__))" and "y = eta_reducible (vg__ $ Bv (Suc vcc__))"   | uv_ :: "typ" and v :: "Core.term" and vb :: "typ" and vc :: "Core.term" where "x = Abs uvg___ (vh__ $ Abs vbc__ vcd__)" and "y = eta_reducible (vh__ $ Abs vbc__ vcd__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "Core.term" and vc :: "Core.term" where "x = Abs uv_ (vi__ $ (vb $ vc))" and "y = eta_reducible (vi__ $ (vb $ vc))"   | t :: "Core.term" and u :: "Core.term" where "x = t $ u" and "y = (eta_reducible t \<or> eta_reducible u)"   | v :: "String.literal" and va :: "typ" where "x = Ct vj__ vac__" and "\<not> y"   | v :: "variable" and va :: "typ" where "x = Fv vk__ va" and "\<not> y"   | v :: "nat" where "x = Bv v" and "\<not> y"
preserves_ide: fixes a :: "'a" assumes "A.ide a" shows "B.ide (F a)"
simple_integral_indicator: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "A \<in> sets M"   and "simple_function M f" shows "\<integral>\<^sup>S x. f x * indicator A x \<partial>M = (\<Sum>x\<in>f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
weight_eq_Sum_freq: fixes t :: "'a tree" assumes "consistent t" shows "weight t = sum (freq t) (alphabet t)"
wt_t_ge_s: shows "wt s \<le> wt t"
assoc: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" shows "a \<union># b \<union># c = a \<union># (b \<union># c)"
reduced_ifex: fixes b :: "'b ltln" shows "reduced (ifex_of_ltl b) {}"
gcd_isgcd: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "somegcd G a b gcdof a b"
transp_llist_all2: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "transp R" shows "transp (llist_all2 R)"
set_cases: fixes e :: "'b"   and a :: "'b tree" assumes "e \<in> set_of a" obtains z2 :: "'b tree" and z3 :: "'b tree" and z4 :: "nat" where "a = MKT e z2__ z3__ z4__"   | z1 :: "'b" and z2 :: "'b tree" and z3 :: "'b tree" and z4 :: "nat" where "a = MKT z1__ z2a__ z3a__ z4a__" and "e \<in> set_of z2a__"   | z1 :: "'b" and z2 :: "'b tree" and z3 :: "'b tree" and z4 :: "nat" where "a = MKT z1 z2 z3 z4" and "e \<in> set_of z3"
inverse_eq_imp_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> '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.division_ring divide inverse one times plus zero minus uminus"   and "inverse a = inverse b" shows "a = b"
index_image: fixes xs :: "'c list"   and X :: "'c set" assumes "distinct xs"   and "set xs = X" shows "index xs ` X = {0..<length xs}"
filter_inter: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> F"   and "B \<in> F" shows "A \<inter> B \<in> F"
wf_BaseC: shows "wf_cdecl tprg (Base, BaseCl)"
known_ptrs_subset: fixes h' :: "(_) heap"   and h :: "(_) heap" assumes "object_ptr_kinds h' |\<subseteq>| object_ptr_kinds h"   and "DocumentClass.known_ptrs h" shows "DocumentClass.known_ptrs h'"
marker_must_stay_if_no_snapshot: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and cid :: "nat" assumes "trace init t final"   and "ps (S t i) p \<noteq> None"   and "\<not> ps (S t i) q \<noteq> None"   and "channel cid = Some (p, q)" shows "Marker \<in> set (msgs (S t i) cid)"
abs: fixes x :: "name"   and y :: "name"   and env :: "(name, name) fmap"   and n1 :: "nterm"   and n2 :: "nterm" assumes "alpha_equiv (fmupd x y env) n1 n2" shows "alpha_equiv env (\<Lambda>\<^sub>n x. n1) (\<Lambda>\<^sub>n y. n2)"