hcju/isabelleps huggingface dataset:
expectation_gpv_\<I>_mono: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and fail :: "ennreal"   and f :: "'c \<Rightarrow> ennreal" assumes "\<I> \<le> \<I>'"   and "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv fail \<I> f gpv \<le> expectation_gpv fail \<I>' f gpv"
rotate: fixes t :: "'a"   and queue :: "'a list" assumes "t \<noteq> t0"   and "t0 \<in> set queue"   and "P (queue @ [t])" shows "P (t # queue)"
powsum_00: fixes x :: "'a" shows "x\<^bsub>0\<^esub>\<^bsup>0\<^esup> = (1::'a)"
pa2_disj_in1: shows "path_image c2 \<inter> inside (path_image c1 \<union> path_image c) = {}"
app_css_step: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and xs :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow> (cs', css', t)" shows "\<Gamma>\<turnstile> (cs, css @ xs, s) \<rightarrow> (cs', css' @ xs, t)"
mat_1_fun: fixes a :: "'b"   and b :: "'b" shows "mat (1::'a) $ a $ b = (if a = b then 1::'a else (0::'a))"
equiv_\<psi>: fixes x :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "pred_stream (\<lambda>s. s \<in> S) x" shows "pred_stream (\<lambda>s. \<psi> (reps (abss s)) = \<psi> s) (state cfg' ## x)"
length_psums: fixes xs :: "'a list" shows "length (psums xs) = length xs"
xt1: shows "(x, return_time {x \<in> S. s x = 0} x) \<in> Sigma Y existence_ivl0"
sound_weaken_appL: fixes SGS :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list" assumes "sound (SGS, C)" shows "sound (SG @ SGS, C)"
unset_break1: fixes P :: "'b control_state_scheme \<Rightarrow> bool" shows "\<lbrace>\<lambda>\<sigma>. P (\<sigma>\<lparr>break_status := False\<rparr>)\<rbrace> unset_break_status \<lbrace>\<lambda>r \<sigma>. P \<sigma> \<and> \<not> break_status \<sigma>\<rbrace>"
image_mset_lists: fixes A :: "'b set" shows "mset ` lists A = multisets A"
IH: fixes gpv :: "('a, 'e, 'f) gpv"   and s :: "'b" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv' gpv \<le> expectation_gpv2 (\<lambda>a. case a of (x, s) \<Rightarrow> f x) (inline callee gpv s)"
Diff_eq_empty_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A\<setminus>B = {}) = (A \<subseteq> B)"
abs_diff_eq1: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "\<bar>u - l\<bar> = u - l"
rewrite_syn_imp_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (rewrite_syn_imp \<phi>) \<subseteq> atoms_ltln \<phi>"
oinvariant_initD: fixes A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<times> 'f \<Rightarrow> bool"   and s :: "'e \<times> 'f" assumes "A \<Turnstile> (S, U \<rightarrow>) P"   and "s \<in> init A" shows "P s"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" assumes "l_known_ptrs known_ptr known_ptrs"   and "l_get_host_wf_axioms heap_is_wellformed known_ptrs type_wf get_host" shows "l_get_host_wf heap_is_wellformed known_ptr known_ptrs type_wf get_host"
comp_impl_tot_rec: fixes A :: "nat set" assumes "computable A" shows "total_recursive (chf A)"
match_list_helper_1: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m2 :: "'c match_expr"   and a :: "action"   and p :: "'d"   and m1' :: "'c match_expr list" assumes "\<not> matches \<gamma> m2 a p"   and "match_list \<gamma> (map (\<lambda>x. MatchAnd x m2) m1') a p" shows "False"
disconnected_nodes_eq2_h2: fixes doc_ptr :: "(_) document_ptr" assumes "doc_ptr \<noteq> owner_document" shows "|h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
validLG: shows "wf_list_graph (stateful_list_policy_to_list_graph \<lparr>hostsL = nodesL G, flows_fixL = edgesL G, flows_stateL = e # accu\<rparr>)"
lang_inorm: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "WS1S_Equivalence_Checking.wf \<Sigma> n r" shows "lang \<Sigma> n (inorm r) = lang \<Sigma> n r"
mat_kernel: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "mat_kernel A = {v \<in> carrier_vec nc. A *\<^sub>v v = 0\<^sub>v nr}"
Inf_eq_bot_iff: 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 bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Inf A = bot) = (\<forall>x. less bot x \<longrightarrow> (\<exists>i\<in>A. less i x))"
a: fixes x :: "'a" assumes "x \<in> set xs" shows "x \<le> a"
canon_ang_minus_pi1: fixes \<alpha> :: "real" assumes "0 < \<alpha>"   and "\<alpha> \<le> 2 * pi" shows "\<downharpoonright>\<alpha> - pi\<downharpoonleft> = \<alpha> - pi"
cet_var': fixes t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat" assumes "t = \<Gamma> x" shows "\<Gamma> \<turnstile>\<^sub>c CVar x : t"
simps_3: fixes m :: "nat"   and M :: "nat" shows "FOV (FIn m M) = {m}"
mult_of_def: fixes R :: "('c, 'd) ring_scheme" shows "mult_of R \<equiv> \<lparr>carrier = carrier R - {\<zero>\<^bsub>R\<^esub>}, mult = (\<otimes>\<^bsub>R\<^esub>), one = \<one>\<^bsub>R\<^esub>\<rparr>"
worker_wrapper_body: fixes wrap :: "'c \<rightarrow> 'b"   and unwrap :: "'b \<rightarrow> 'c"   and body :: "'b \<rightarrow> 'b"   and computation :: "'b" assumes "wrap oo unwrap oo body = body"   and "computation = fix\<cdot>body" shows "computation = wrap\<cdot>(fix\<cdot>(unwrap oo body oo wrap))"
norm_scaleR: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "real"   and x :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (scaleR a x) = \<bar>a\<bar> * norm x"
frac_less: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and w :: "'b"   and z :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less x y"   and "less zero w"   and "less_eq w z" shows "less (divide x z) (divide y w)"
Inf_ne_def: fixes X :: "'a set" shows "Inf_ne X = FOLD_bin_ne inf X"
rep_eq: fixes x :: "unit2"   and xa :: "unit2" shows "Rep_unit2 (x + xa) = sp (Rep_unit2 x) (Rep_unit2 xa)"
simps_1: fixes A :: "'obj"   and uu :: "'obj"   and uv :: "'obj" shows "Dom (MkArr A uu uv) = A"
eqfset_imp_iff: fixes A :: "'b fset"   and B :: "'b fset"   and x :: "'b" assumes "A = B" shows "(x |\<in>| A) = (x |\<in>| B)"
in_atms_of_subst: fixes B :: "'a"   and C :: "'a literal multiset"   and \<sigma> :: "'s" assumes "B \<in> atms_of C" shows "B \<cdot>a \<sigma> \<in> atms_of (C \<cdot> \<sigma>)"
LLs_def: shows "LLs \<equiv> \<lambda>t1 t2. psubst Ls [(t1, xx), (t2, yy)]"
\<xi>'_def: fixes x' :: "Preliminaries.var" shows "\<xi>' x' = (if x = x' then a else \<xi> x')"
BlockSpec: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and bdy :: "('d, 'f, 'g) com" assumes "P \<subseteq> {s. \<exists>Z. init s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> return s t \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (R s t) c s t Q,A"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) bdy (Q' Z),(A' Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P block init bdy return c Q,A"
sigma_algebra_axioms: fixes i :: "nat" shows "sigma_algebra (space P) (sets.pullback_algebra M (\<lambda>f. f i) (space P))"
2: shows "prim_E w v e * - (1::'a) * prim_E w v e\<^sup>T \<le> bot"
p_or_not_p: fixes P :: "'a upred" shows "(P \<or> \<not> P) = true"
value_is_compatible_with_structure_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 f :: "'b trm \<Rightarrow> 'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.value_is_compatible_with_structure f = (\<forall>t s. f (t \<cdot> s) = f (f t \<cdot> f s))"
omega_closed_oexp: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<in> elts \<omega>"   and "\<beta> \<in> elts \<omega>" shows "\<alpha> \<up> \<beta> \<in> elts \<omega>"
simps_4: fixes x2 :: "String.literal"   and x1 :: "int" shows "Str x2 \<noteq> Num x1"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a"   and TRel :: "('b \<times> 'b) set" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRelT Enc TRel"
swap\<^sub>w_def: shows "swap\<^sub>w = (swap_sum, swap_sum)"
simps_2: fixes l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" shows "wbt \<langle>l, uu, r\<rangle> = (wbt l \<and> wbt r \<and> (let ratio = real (size1 l) / real (size1 l + size1 r) in \<alpha> \<le> ratio \<and> ratio \<le> 1 - \<alpha>))"
allow_all_def: fixes p :: "'a" shows "A\<^sub>U p = \<lfloor>allow ()\<rfloor>"
prefix_def: fixes l1 :: "'a list"   and l2 :: "'a list" shows "l1 \<preceq> l2 \<equiv> \<exists>l3. l1 @ l3 = l2"
of_complex_add_of_complex: fixes x :: "complex"   and y :: "complex" shows "of_complex x +\<^sub>h of_complex y = of_complex (x + y)"
irrefl: fixes a :: "complex" shows "\<not> a <\<^sub>\<complex> a"
intro: fixes trsys1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and trsys2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and bisim :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and tlsim :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and \<tau>move1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and \<tau>move2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool" assumes "delay_bisimulation_obs trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "delay_bisimulation_diverge_axioms trsys1 trsys2 bisim \<tau>move1 \<tau>move2" shows "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"
induct: fixes P :: "iface \<Rightarrow> iface \<Rightarrow> bool"   and a0 :: "iface"   and a1 :: "iface" assumes "\<And>i1 i2. P (Iface i1) (Iface i2)" shows "P a0 a1"
permute_diff: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "perm"   and q :: "perm"   and x :: "'a" assumes "class.pt permute" shows "permute (p - q) x = permute p (permute (- q) x)"
sign_simps_30: fixes bm :: "'c"   and am :: "'c"   and ck :: "'c" shows "bm + (am + ck) = am + (bm + ck)"
OclOr_bot1: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" assumes "X \<tau> \<noteq> true \<tau>" shows "(UML_Types.bot_class.bot or X) \<tau> = UML_Types.bot_class.bot \<tau>"
that: fixes q :: "'alt pmf" assumes "q \<in> lotteries_on alts"   and "Pareto (SD \<circ> R) p q"   and "SD_efficient R q" shows "thesis"
subset_imp_leadsTo: fixes A :: "'c set"   and B :: "'c set"   and F :: "'c program" assumes "A \<subseteq> B" shows "F \<in> A \<longmapsto> B"
UNION_disjoint: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'e" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "sum g (\<Union> (A ` I)) = (\<Sum>x\<in>I. sum g (A x))"
fresh_subst: fixes z :: "name"   and s :: "Lambda_Z.term"   and y :: "name"   and t :: "Lambda_Z.term" assumes "atom z \<sharp> s"   and "z = y \<or> atom z \<sharp> t" shows "atom z \<sharp> subst y s t"
elem_in_nhd: fixes U :: "'a llist set"   and x :: "'a llist" assumes "U \<in> nhds x" shows "x \<in> U"
halloc_no_abrupt: fixes G :: "prog"   and s0 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>halloc oi\<succ>a\<rightarrow> s1"   and "normal s1" shows "normal s0"
subH1: shows "subgroup (H1 <#> H \<inter> K) (G\<lparr>carrier := H\<rparr>)"
add_strict_mono: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g"   and c :: "'g"   and d :: "'g" assumes "class.strict_ordered_ab_semigroup_add plus less_eq less"   and "less a b"   and "less c d" shows "less (plus a c) (plus b d)"
elims_3: fixes x :: "(String.literal \<times> String.literal) set \<times> ('a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "\<not> wf_osig x" obtains sub :: "(String.literal \<times> String.literal) set" and tcs :: "'a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" where "x = (sub, tcs)" and "\<not> (wf_subclass sub \<and> wf_tcsigs sub tcs)"
in_space_prob_algebra: fixes x :: "'c measure"   and M :: "'c measure" assumes "x \<in> space (prob_algebra M)" shows "emeasure x (space M) = 1"
distr_lift_IntVal: fixes M :: "val measure"   and \<delta> :: "val \<Rightarrow> ennreal"   and f :: "int \<Rightarrow> int"   and g :: "(int \<Rightarrow> ennreal) \<Rightarrow> int \<Rightarrow> ennreal" assumes "has_density M (stock_measure INTEG) \<delta>"   and "\<And>M \<delta>. has_density M (count_space UNIV) \<delta> \<Longrightarrow> has_density (distr M (count_space UNIV) f) (count_space UNIV) (g \<delta>)" shows "has_density (distr M (stock_measure INTEG) (lift_IntVal f)) (stock_measure INTEG) (g (\<lambda>x. \<delta> (IntVal x)) \<circ> extract_int)"
conjl1_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<lhd> y \<le> z) = (x \<le> - (- z \<cdot> y))"
step': fixes b :: "int"   and a :: "int" assumes "b < 0" shows "P (Fract a b)"
fmfilter_alt_defs_3: fixes B :: "'d fset" shows "fmdrop_fset B = fmfilter (\<lambda>a. a |\<notin>| B)"
scf_nonempty: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "social_choice_function agents alts scf"   and "pref_profile_wf agents alts R" shows "scf R \<noteq> {}"
nextVertex_in_edges: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "(v, f \<bullet> v) \<in> \<E> f"
assms: shows "\<forall>m \<sigma> p. stateful_matcher' \<sigma> m p = stateful_matcher_tagged' m (packet_tagger' \<sigma> p)"
direct_lim_is_ring: fixes U :: "'a set" assumes "U \<in> {U. is_open\<^sub>X U \<and> x \<in> U}" shows "ring stx.carrier_direct_lim stx.add_rel stx.mult_rel (stx.class_of U (zero_str\<^sub>X U)) (stx.class_of U (one_str\<^sub>X U))"
A3: shows "Esigma (2 ^ k) * Esigma p * Esigma q = Esigma (2 ^ k) * Esigma r"
distinct_43: fixes x4 :: "nat"   and x81 :: "nat"   and x82 :: "msg" shows "Pan x4 \<noteq> Crypt x81 x82"
deriv_2: fixes x :: "real"   and A :: "real set" assumes "x \<noteq> a" shows "(F3 has_vector_derivative c / (x - a) ^ n) (at x within A)"
less_eq_fin_string_def: shows "(\<le>) \<equiv> map_fun Rep_fin_string (map_fun Rep_fin_string id) lleq_charlist"
eval_liftFm: fixes I :: "real list"   and amount :: "nat"   and d :: "nat"   and L :: "real list"   and F :: "atom fm" assumes "length I = amount"   and "d \<le> length L" shows "eval F L = eval (liftFm d amount F) (insert_into L d I)"
write_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"
Map_cod: fixes f :: "('o, 'a) arr" assumes "arr f" shows "Map (cod f) = Id (Cod f)"
mat_delete_carrier: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat m n" shows "mat_delete A i j \<in> carrier_mat (m - 1) (n - 1)"
relf_congruent_rel: fixes b :: "'a" shows "refl (congruent_rel b)"
prefix_same_cases: fixes xs\<^sub>1 :: "'e list"   and ys :: "'e list"   and xs\<^sub>2 :: "'e list" assumes "prefix xs\<^sub>1 ys"   and "prefix xs\<^sub>2 ys" shows "prefix xs\<^sub>1 xs\<^sub>2 \<or> prefix xs\<^sub>2 xs\<^sub>1"
inext_min_step: fixes n :: "nat"   and k :: "nat"   and I :: "nat set" assumes "n < k"   and "k < inext n I" shows "k \<notin> I"
simps_9: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>n \<psi> \<or> (\<exists>j<i. suffix j \<xi> \<Turnstile>\<^sub>n \<phi>))"
compl_sup: fixes x :: "'a"   and y :: "'a" shows "- sup_class.sup x y = inf_class.inf (- x) (- y)"
P: fixes Y :: "'a set" shows "finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> B \<longrightarrow> P (Y - A)"
drop_Suc: fixes n :: "nat"   and xs :: "'c list" shows "drop (Suc n) xs = drop n (tl xs)"
LocalSecretsDef: fixes A :: "specID" shows "LocalSecrets A = {m. m \<notin> specKeysSecrets A \<and> ((\<exists>x y. x \<in> loc A \<and> m = kKS y \<and> exprChannel x (kE y)) \<or> (\<exists>x z. x \<in> loc A \<and> m = sKS z \<and> exprChannel x (sE z)))} \<union> \<Union> (LocalSecrets ` subcomponents A)"
pow_mbc: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "length (bc (\<sigma>\<^bsub>MBC t n\<^esub>t n)) = PoW t n"
assms_2: shows "(cmod \<langle>v|w\<rangle>)\<^sup>2 = Re (\<langle>v|v\<rangle> * \<langle>w|w\<rangle>)"
assert_cont: fixes x :: "'b \<Rightarrow> 'b" assumes "x \<le> id"   and "x \<in> Apply.disjunctive" shows "x = assert_fun (x \<top>)"
validE_NF_def: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool"   and Q :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and E :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>, \<lbrace>E\<rbrace>! \<equiv> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>, \<lbrace>E\<rbrace> \<and> no_fail P f"
quasi_isometry_on_subset: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'b set" assumes "lambda C -quasi_isometry_on X f"   and "Y \<subseteq> X" shows "lambda C -quasi_isometry_on Y f"
strong_induct: fixes P :: "'a \<Rightarrow> act \<Rightarrow> bool"   and z :: "'a"   and act :: "act" assumes "\<And>name z. P z (\<lparr>name\<rparr>)"   and "\<And>name z. P z (\<langle>name\<rangle>)"   and "\<And>z. P z (\<tau>)" shows "P z act"
getRel_dom_2: fixes G :: "('c, 'd) labeled_graph"   and a :: "'d"   and b :: "'d"   and l :: "'c" assumes "graph G"   and "(a, b) \<in> getRel l G" shows "b \<in> vertices G"
e2xs_def: fixes e :: "nat" shows "e2xs e \<equiv> pdec12 (e2frame e)"
that: fixes T :: "'a set" assumes "finite T"   and "T \<subseteq> S"   and "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. t adds s" shows "thesis"
tab_in_hom_2: shows "\<guillemotleft>\<rho> : r\<^sub>1 \<Rightarrow> r \<star> r\<^sub>0\<guillemotright>"
Map_comp': fixes t' :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and t :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.seq t' t" shows "A_B.Map (t' \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] t) = vertical_composite.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Map t) (A_B.Map t')"
dec_seqn1: fixes a :: "int"   and A :: "int set"   and n :: "nat" assumes "a \<in> A"   and "A \<subseteq> Zset"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "dec_seq A a (Suc n) \<le> dec_seq A a n - 1"
Cantor_sum_Nil: fixes ms :: "nat list" shows "Cantor_sum [] ms = 0"
safety_prop_Acts_iff: fixes X :: "'c program set"   and G :: "'c program" assumes "safety_prop X" shows "(Acts G \<subseteq> insert Id (\<Union> (Acts ` X))) = (G \<in> X)"
disc_1: fixes x1 :: "'d" shows "is_Var (Var x1)"
ex_W_cmpserTr0m: fixes m :: "nat" shows "\<forall>f. w_cmpser G m f \<longrightarrow> (\<exists>g. W_cmpser G (card (f ` {i. i \<le> m}) - 1) g \<and> g ` {i. i \<le> card (f ` {i. i \<le> m}) - 1} = f ` {i. i \<le> m})"
evalmat_unfold: fixes P :: "'c poly"   and A :: "(('c, 'd) vec, 'd) vec" shows "evalmat P A = (\<Sum>i = 0..degree P. coeff P i *k matpow A i)"
product_crit_def: fixes data :: "nat \<times> 'e"   and gs :: "(('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and p :: "('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('a \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" shows "product_crit data gs bs hs p q = (gcs (punit.lt (fst p)) (punit.lt (fst q)) = (0::'a))"
Gauss_Jordan_in_ij_PA_def: fixes A' :: "(('b, 'c) vec, 'c) vec \<times> (('b, 'd) vec, 'c) vec"   and i :: "'c"   and j :: "'d" shows "Gauss_Jordan_in_ij_PA A' i j = (let P = fst A'; A = snd A'; n = LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n; interchange_A = interchange_rows A i n; interchange_P = interchange_rows P i n; P' = mult_row interchange_P i ((1::'b) / interchange_A $ i $ j) in (\<chi>s. if s = i then P' $ s else row_add P' s i (- interchange_A $ s $ j) $ s, Gauss_Jordan_in_ij A i j))"
rep_eq: fixes x :: "complex"   and xa :: "'d conjugate_space" shows "from_conjugate_space (x *\<^sub>C xa) = cnj x *\<^sub>C from_conjugate_space xa"
l_1: shows "(\<lambda>n. - u n / real n) \<longlonglongrightarrow> l"