hcju/isabelleps huggingface dataset:
par_def: fixes ioa1 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and ioa2 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set" shows "ioa1 || ioa2 \<equiv> (asig_comp (asig_of ioa1) (asig_of ioa2), {pr. fst pr \<in> starts_of ioa1 \<and> snd pr \<in> starts_of ioa2}, {tr. let s = fst tr; a = fst (snd tr); t = snd (snd tr) in (a \<in> actions (asig_of ioa1) \<or> a \<in> actions (asig_of ioa2)) \<and> (if a \<in> actions (asig_of ioa1) then (fst s, a, fst t) \<in> trans_of ioa1 else fst t = fst s) \<and> (if a \<in> actions (asig_of ioa2) then (snd s, a, snd t) \<in> trans_of ioa2 else snd t = snd s)})"
vnat_eq_vreal: fixes x :: "nat" shows "x\<^sub>\<nat> = (CZH_Sets_ZQR.real x)\<^sub>\<real>"
finite_acyclic_wf: fixes r :: "('x \<times> 'x) set" assumes "finite r"   and "acyclic r" shows "wf r"
weights_adm: fixes weights :: "'a weights" assumes "class.weighted weights" shows "admissible_kbo (w weights) (w0 weights) (pr_strict weights) (pr_strict weights)\<^sup>=\<^sup>= (least weights) (scf weights)"
step: fixes y :: "V"   and z :: "V" assumes "y \<sqsubset> b"   and "y \<in> elts d"   and "z \<in> elts y" shows "z \<in> elts d"
odd_real_root_pow: fixes n :: "nat"   and x :: "real" assumes "odd n" shows "root n x ^ n = x"
neg_one_odd_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "odd n" shows "power.power one times (uminus one) n = uminus one"
finite_dimensional_vector_space_pair_1_on_axioms: shows "finite_dimensional_vector_space_pair_1_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>1) (*s\<^sub>2) basis1"
discounted_integrable: fixes M :: "'b measure"   and N :: "'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat"   and r :: "real" assumes "prob_space M"   and "integrable N (X n)"   and "- 1 < r"   and "space N = space M" shows "integrable N (discounted_value r X n)"
tfinite_map_tree: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b tree" shows "tfinite (map_tree f t) = tfinite t"
lins2_2: shows "set \<pi> = (fset G^V - fset S) <+> set G^E"
subsetCI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<notin> B\<rbrakk> \<Longrightarrow> False" shows "A \<subseteq> B"
actions_ok2_ex_actions_ok1: fixes s2 :: "('a, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'b wait_set_status option) \<times> 't set"   and t :: "'t"   and ta2 :: "('a, 't, 'x2, 'm2, 'b, 'c) thread_action"   and ta1 :: "('a, 't, 'x1, 'm1, 'b, 'c) thread_action" assumes "r2.actions_ok s2 t ta2"   and "ta1 \<sim>m ta2" obtains s1 :: "('a, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'b wait_set_status option) \<times> 't set" where "r1.actions_ok s1 t ta1"
eq3: shows "2 \<cdot>\<^sub>v (proj_psi *\<^sub>v psi'_l l) = 2 * (alpha_l l * complex_of_real (cos (\<theta> / 2)) - beta_l l * complex_of_real (sin (\<theta> / 2))) \<cdot>\<^sub>v \<psi>"
zero_less_Suc: fixes n :: "nat" shows "order.greater (Suc n) 0"
Y: fixes n :: "nat" assumes "0 < n"   and "n < funpow_dist1 G.face_cycle_succ a' a" shows "(H.face_cycle_succ ^^ n) a_neigh = (G.face_cycle_succ ^^ n) a'"
usubstappp_dual: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game" shows "usubstappp \<sigma> U \<alpha>^d = (fst (usubstappp \<sigma> U \<alpha>), Dualo (snd (usubstappp \<sigma> U \<alpha>)))"
min_trms_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and E :: "('b trm \<times> 'b trm) set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.min_trms trm_ord E = {x. \<exists>pair. (pair \<in> E \<and> (\<forall>pair'\<in>E. (snd pair', snd pair) \<notin> trm_ord)) \<and> x = fst pair}"
snd_to_last_max_normal: assumes "finite (carrier G)"   and "1 < length \<HH>" shows "max_normal_subgroup (\<HH> ! (length \<HH> - 2)) G"
Inr_not_Inl: fixes b :: "'d"   and a :: "'c" shows "Inr b \<noteq> Inl a"
comp_converter_parallel2: fixes conv1l :: "('a, 'c, 'i, 'k) converter"   and conv1r :: "('b, 'd, 'j, 'l) converter"   and conv2l :: "('i, 'k, 'e, 'g) converter"   and conv2r :: "('j, 'l, 'f, 'h) converter" shows "comp_converter (parallel_converter2 conv1l conv1r) (parallel_converter2 conv2l conv2r) = parallel_converter2 (comp_converter conv1l conv2l) (comp_converter conv1r conv2r)"
total_on_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
lexl_not_refl: fixes r :: "('a \<times> 'a) set"   and x :: "'a list" assumes "irrefl r" shows "(x, x) \<notin> lex r"
ordered_resolvent_def: fixes atom_ordering :: "('a \<times> 'a) set"   and P1 :: "'a Literal set"   and P2 :: "'a Literal set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.ordered_resolvent atom_ordering P1 P2 C \<equiv> \<exists>A. C = P1 - {Pos A} \<union> (P2 - {Neg A}) \<and> propositional_atoms.strictly_maximal_literal atom_ordering P1 (Pos A) \<and> propositional_atoms.strictly_maximal_literal atom_ordering P2 (Neg A)"
add_right_cancel: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.cancel_semigroup_add plus" shows "(plus b a = plus c a) = (b = c)"
d_mono: fixes n :: "nat" shows "sum_integral_diff_series (Suc n) \<le> sum_integral_diff_series n"
pseudo_proj_True_inverse_induct: fixes w :: "bool stream"   and n :: "nat" assumes "w \<in> range (pseudo_proj_True n)" shows "pseudo_proj_True n -` {w} = pseudo_proj_True (Suc n) -` {w} \<union> pseudo_proj_True (Suc n) -` {pseudo_proj_False n w}"
Lcm_mult: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and c :: "'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "A \<noteq> {}" shows "Lcm (times c ` A) = normalize (times c (Lcm A))"
rimp_trace_def: fixes X :: "('a \<times> ('b \<times> 'a) list) set"   and Z :: "('a \<times> ('b \<times> 'a) list) set" shows "rimp_trace X Z = \<Union> {Y. t_prod X Y \<subseteq> Z}"
aform_sup_inner_def: fixes prec :: "nat"   and X :: "(real \<times> real pdevs) list"   and n :: "real list" shows "aform_sup_inner prec X n = (case inner_aforms' prec X (map pdevs_of_real n) of Some Xn \<Rightarrow> Sup_aform' prec (hd Xn))"
subnet_reachable_1: fixes s1 :: "'c net_state"   and s2 :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree" assumes "SubnetS s1 s2 \<in> reachable (pnet np (p1 \<parallel> p2)) TT" shows "s1 \<in> reachable (pnet np p1) TT"
Min_add_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and k :: "'a" assumes "class.linordered_ab_semigroup_add plus less_eq less"   and "finite S"   and "S \<noteq> {}" shows "linorder.Min less_eq ((\<lambda>x. plus (f x) k) ` S) = plus (linorder.Min less_eq (f ` S)) k"
sub_left_inj_on: fixes k :: "nat" shows "inj_on (\<lambda>x. x - k) {k..}"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> vint_plus \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> vint_plus \<in>\<^sub>\<circ> Vset \<alpha>" shows "vint_plus \<in>\<^sub>\<circ> Vset \<alpha>"
succ_not_in: fixes i :: "'a" assumes "local.aboveS i \<noteq> {}" shows "(succ i, i) \<notin> r"
dom_eq_ran: fixes index :: "nat \<times> nat" shows "local.domain index = local.range index"
dot_rowvector_columnvector: fixes A :: "(('a, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "('a, 'd) Finite_Cartesian_Product.vec" shows "columnvector (A *v v) = A ** columnvector v"
HRP_HPair_quot: fixes x :: "tm"   and y :: "tm" shows "{HRP x \<guillemotleft>x\<guillemotright>, HRP y \<guillemotleft>y\<guillemotright>} \<turnstile> HRP (HPair x y) \<guillemotleft>HPair x y\<guillemotright>"
ipurge_ref_aux_def: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list"   and X :: "'a set" shows "ipurge_ref_aux I D U xs X \<equiv> {x \<in> X. \<forall>v\<in>sinks_aux I D U xs. (v, D x) \<notin> I}"
sample_return_pmf: fixes x :: "'p"   and f :: "'p \<Rightarrow> 'm" shows "sample (return_pmf x) f = f x"
nn_integral_has_integral_lborel: fixes f :: "'b \<Rightarrow> real"   and I :: "real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "(f has_integral I) UNIV" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal I"
deg_le1: fixes n :: "nat" shows "degree (hensel_fpxs1 $ n) \<le> degree g"
simps: fixes n :: "nat"   and u :: "nat"   and ns :: "nat list"   and xs :: "'c list" shows "add_inv n (u, ns, xs) = (foldl (+) 0 ns = n \<and> length xs = n)"
fbox_mult: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x \<cdot> y] z =  |x]  |y] z"
pair_return_pmf1: fixes x :: "'c"   and y :: "'d pmf" shows "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
cl_ivl_Suc: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "prod g {m..Suc n} = (if Suc n < m then 1::'a else prod g {m..n} * g (Suc n))"
maps_over_finite: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "finite (maps_over A B)"
brn_index: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat"   and cl :: "('c, 'd, 'a) cmd list"   and i :: "nat" assumes "PL cval"   and "n < length cl"   and "i < PL.brn (cl ! n)" shows "PL.brnL cl n + i < PL.brnL cl (length cl)"
add_roots: fixes w :: "'b list"   and z :: "'b list"   and w' :: "'b list" assumes "w \<in> z*"   and "w' \<in> z*" shows "w \<cdot> w' \<in> z*"
that: fixes r :: "'a"   and s :: "'a" assumes "r \<in> R"   and "s \<in> S"   and "X = rel.Class (r, s)" shows "thesis"
consistentE: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list \<times> Vars.var list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "consistent (ae, ce, a, as) (\<Gamma>, e, S)" obtains as :: "Arity list" and r :: "Vars.var list" where "asb__ = (as, r)" and "a_consistent (ae, a, as) (restrictA (- set r) \<Gamma>, e, restr_stack (- set r) S)" and "edom ae = edom ce" and "prognosis ae as a (\<Gamma>, e, S) \<sqsubseteq> ce" and "\<And>x. \<lbrakk>x \<in> thunks \<Gamma>; many \<sqsubseteq> ce x\<rbrakk> \<Longrightarrow> ae x = up\<cdot>0" and "set r \<subseteq> domA \<Gamma> \<union> upds S - edom ce"
simps_1: fixes l :: "'b llist" shows "l \<up> 0 = l"
assms_3: shows "dist p c \<le> dist (p $ k) v"
weakening_version2: fixes \<Gamma>1 :: "(name \<times> ty) list"   and t :: "lam"   and T :: "ty"   and \<Gamma>2 :: "(name \<times> ty) list" assumes "\<Gamma>1 \<turnstile> t : T"   and "valid \<Gamma>2"   and "\<Gamma>1 \<subseteq> \<Gamma>2" shows "\<Gamma>2 \<turnstile> t : T"
index_eq_none: fixes l :: "'a list"   and n :: "nat" shows "(index l n = None) = (length l \<le> n)"
intros: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "hindrance \<Gamma> f"   and "current \<Gamma> f"   and "wave \<Gamma> f" shows "hindered \<Gamma>"
gauss2complex_of_nat: fixes n :: "nat" shows "gauss2complex (of_nat n) = of_nat n"
dist_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and e :: "'a \<Rightarrow> real" assumes "\<And>i. i \<in> I \<Longrightarrow> dist (f i) (g i) \<le> e i" shows "dist (sum f I) (sum g I) \<le> sum e I"
vec_first_add: fixes n :: "nat"   and x :: "'b vec"   and y :: "'b vec" assumes "n \<le> dim_vec x"   and "n \<le> dim_vec y" shows "vec_first (x + y) n = vec_first x n + vec_first y n"
upper_iso: shows "isotone \<Y> \<X> \<pi>\<^sub>*"
rep_eq: fixes x :: "ocircline"   and xa :: "complex_homo" shows "on_ocircline x xa = on_circline_clmat_hcoords (rep_ocircline x) (rep_complex_homo xa)"
greatest_ge_nonzero_row': fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and i :: "'f" assumes "echelon_form_upt_k A k"   and "i \<le> (GREATEST m. \<not> is_zero_row_upt_k m k A)"   and "\<not> (\<forall>a. is_zero_row_upt_k a k A)" shows "\<not> is_zero_row_upt_k i k A"
inpcp: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<bottom>" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F"
assms_2: shows "\<not> (AE x in M. f x = 0)"
ts_length_hint1: fixes x :: "nat \<Rightarrow> 'a list"   and i :: "nat" assumes "ts x" shows "x i \<noteq> []"
filterlim_at_infinity_times: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "filterlim f at_infinity F"   and "filterlim g at_infinity F" shows "LIM x F. f x * g x :> at_infinity"
power2_commute: fixes 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 x :: "'c"   and y :: "'c" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "power.power one times (minus x y) 2 = power.power one times (minus y x) 2"
hom_nat_pow: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "h (x [^] n) = h x [^]\<^bsub>S\<^esub> n"
g: fixes z :: "complex" assumes "z \<in> S" shows "(g has_field_derivative deriv f z / f z) (at z)"
add_spairs_spp_def: fixes ps :: "(('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat) list"   and bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "add_spairs_spp ps bs p = merge_wrt pair_ord_spp (new_spairs_spp bs p) ps"
set2_tllist_induct: fixes x :: "'a"   and xs :: "('b, 'a) tllist"   and P :: "'a \<Rightarrow> ('b, 'a) tllist \<Rightarrow> bool" assumes "x \<in> set2_tllist xs"   and "\<And>xs. is_TNil xs \<Longrightarrow> P (terminal xs) xs"   and "\<And>xs y. \<lbrakk>\<not> is_TNil xs; y \<in> set2_tllist (ttl xs); P y (ttl xs)\<rbrakk> \<Longrightarrow> P y xs" shows "P x xs"
setVar'_def: fixes gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and val :: "integer"   and g :: "'b gState_scheme"   and p :: "pState" shows "setVar' gl v idx val g p = (if gl then if v = STR ''_'' then (g, p) else case lm.lookup v (gState.vars g) of None \<Rightarrow> abortv STR ''Unknown global variable: '' v (\<lambda>_. (g, p)) | Some x \<Rightarrow> (g\<lparr>gState.vars := lm.update v (editVar x idx val) (gState.vars g)\<rparr>, p) else case lm.lookup v (pState.vars p) of None \<Rightarrow> abortv STR ''Unknown proc variable: '' v (\<lambda>_. (g, p)) | Some x \<Rightarrow> (g, p\<lparr>pState.vars := lm.update v (editVar x idx val) (pState.vars p)\<rparr>))"
cases: fixes a1 :: "'d"   and a2 :: "'e"   and r :: "('d \<times> 'f) set"   and s :: "('f \<times> 'e) set" assumes "(a1, a2) \<in> r O s" obtains a :: "'d" and b :: "'f" and c :: "'e" where "a1 = a" and "a2 = c" and "(a, b) \<in> r" and "(b, c) \<in> s"
update_convs_1: fixes obj_mark' :: "bool \<Rightarrow> bool"   and obj_mark :: "bool"   and obj_fields :: "'a \<Rightarrow> 'c option"   and obj_payload :: "'a \<Rightarrow> 'b option"   and more :: "'d" shows "obj_mark_update obj_mark' \<lparr>obj_mark = obj_mark, obj_fields = obj_fields, obj_payload = obj_payload, \<dots> = more\<rparr> = \<lparr>obj_mark = obj_mark' obj_mark, obj_fields = obj_fields, obj_payload = obj_payload, \<dots> = more\<rparr>"
real_of_1_0_2: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "invariant_1 (p, l, r)"   and "l = 0" shows "r = 0"
success_heapI: fixes f :: "heap \<Rightarrow> 'b \<times> heap"   and h :: "heap" shows "success (Heap_Monad.heap f) h"
noContr_implies_neg_PP_fls: assumes "noContr" shows "prv (neg (PP \<langle>fls\<rangle>))"
distinct_51: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt"   and x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt" shows "While x61 x62 \<noteq> Await x71 x72"
sel_rews_5: fixes slist :: "[:'r:]"   and a :: "'r" assumes "slist \<noteq> \<bottom>" shows "shead\<cdot>(a :# slist) = a"
simps_10: shows "OUTfromV data10 = {stA1}"
assms_1: shows "OpposedThinChamberComplexFoldings X f' g' C0"
subgroup_of_normal_set_mult: fixes N :: "'a set"   and H :: "'a set" assumes "N \<lhd> G"   and "subgroup H G" shows "subgroup H (G\<lparr>carrier := N <#> H\<rparr>)"
S_def: shows "S = (sylvester_mat p q)\<^sup>T"
homeomorphism_locally_imp: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and Q :: "'c set \<Rightarrow> bool" assumes "locally P S"   and "homeomorphism S T f g"   and "\<And>S S'. \<lbrakk>P S; homeomorphism S S' f g\<rbrakk> \<Longrightarrow> Q S'" shows "locally Q T"
le_fin_prefix_fininf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2"   and "w\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>3" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>3"
Cons_5: shows "\<forall>l s. PProbs s \<and> PProbl l \<and> list_all P1 l \<longrightarrow> (\<exists>l'. f s l' = f s l \<and> length (filter P2 l') \<le> k1 \<and> length (filter P3 l') \<le> length (filter P3 l) \<and> list_all P1 l' \<and> subseq l' l)"
open_image_ereal: shows "open (UNIV - {\<infinity>, - \<infinity>})"
Perms_def: shows "Perms = {(Gamma, {Gamma'}) |Gamma Gamma'. Gamma <~~> Gamma'}"
mat_mult_assoc: fixes nr :: "nat"   and n1 :: "nat"   and m1 :: "'a list list"   and n2 :: "nat"   and m2 :: "'a list list"   and nc :: "nat"   and m3 :: "'a list list" assumes "mat nr n1 m1"   and "mat n1 n2 m2"   and "mat n2 nc m3" shows "mat_mult nr (mat_mult nr m1 m2) m3 = mat_mult nr m1 (mat_mult n1 m2 m3)"
set_times_rearrange3: fixes a :: "'a"   and B :: "'a set"   and C :: "'a set" shows "a *o B * C = a *o (B * C)"
assms_5: shows "h \<turnstile> get_parent ptr' \<rightarrow>\<^sub>r Some parent"
at_within_open: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and S :: "'b set" assumes "class.topological_space open"   and "a \<in> S"   and "open S" shows "topological_space.at_within open a S = topological_space.at_within open a UNIV"
healthy_nnegD2: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "real"   and P :: "'a \<Rightarrow> real" assumes "healthy t"   and "bounded_by b P"   and "nneg P" shows "nneg (t P)"
gcd_add2: fixes m :: "'a"   and n :: "'a" shows "gcd m (m + n) = gcd m n"
NFOR_disjuncts: fixes \<psi> :: "('a, 'b) aformula" assumes "nf_ACI \<psi>" shows "NFOR (disjuncts \<psi>) = \<psi>"
s_mul_ext_bottom: fixes A :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "A \<noteq> {#}" shows "(A, {#}) \<in> s_mul_ext ns s"
sint_word_ariths_8: shows "sint 1 = signed_take_bit (LENGTH('b) - 1) 1"
le_imp_power_dvd: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "m \<le> n" shows "dvd.dvd times (power.power one times a m) (power.power one times a n)"
sem_4a: shows "\<forall>w. av w \<sqsubseteq> pv w"
l_one: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "\<one>\<^bsub>DirProds G I\<^esub> \<otimes>\<^bsub>DirProds G I\<^esub> x = x"
o_case_sum: fixes h :: "'g \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'g" shows "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
assms_3: shows "k \<le> 13 * n - 2"
last_appendR: fixes ys :: "'d list"   and xs :: "'d list" assumes "ys \<noteq> []" shows "last (xs @ ys) = last ys"
stepsize_def: fixes j :: "nat" shows "stepsize j = t (Suc j) - t j"
nth_equal_first_eq: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" assumes "x \<notin> set xs"   and "n \<le> length xs" shows "((x # xs) ! n = x) = (n = 0)"
Qc: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'a option"   and u :: "'v"   and v :: "'v"   and adjs :: "('v \<times> nat) list"   and x :: "'v" shows "Qigen Q \<pi> u ((v, w {u, v}) # adjs) x = (if x = v then Qinter Q \<pi> u v else Qigen Q \<pi> u adjs x)"
valid_proc_def: fixes P :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool"   and p :: "'d process_name"   and Q :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> p \<lbrace>Q\<rbrace> = (\<forall>(c, afts)\<in>vcg_fragments (gc_coms p). valid_syn gc_coms p afts P c Q)"
res_neg_eq: fixes x :: "int" shows "\<ominus> x = - x mod int p"
ih_7: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
MultiportPorts_single_not_rewrite_MatchAnd: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and ports :: "ipt_l4_ports"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (MatchNot (Match (MultiportPorts ports))) a p = matches (\<beta>, \<alpha>) (MatchAnd (MatchNot (Match (Src_Ports ports))) (MatchNot (Match (Dst_Ports ports)))) a p"
fimage_vintersection: fixes r :: "V"   and A :: "V"   and B :: "V" shows "r `\<^sub>\<bullet> (A \<inter>\<^sub>\<circ> B) \<subseteq>\<^sub>\<circ> r `\<^sub>\<bullet> A \<inter>\<^sub>\<circ> r `\<^sub>\<bullet> B"
topological_basis_def: fixes "open" :: "'d set \<Rightarrow> bool"   and B :: "'d set set" assumes "class.topological_space open" shows "topological_space.topological_basis open B = ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union> B' = x)))"
tuple_in_Some_Some: fixes as :: "'a option list"   and t :: "nat" assumes "Mapping.lookup tuple_in as = Some t"   and "\<And>X. (t, X) \<in> set discard \<Longrightarrow> as \<notin> X" shows "Mapping.lookup tuple_in' as = Some t"
rep_eq: fixes x :: "'b" shows "\<lbrakk>\<guillemotleft>x\<guillemotright>\<rbrakk>\<^sub>e = (\<lambda>b. x)"
P3'_def: shows "P3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P3"