hcju/isabelleps huggingface dataset:
elims_3: fixes x :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "\<not> qGoodAbs x" obtains xs :: "'varSort" and xa :: "'var" and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" where "x = qAbs xs xa X" and "\<not> qGood X"
tail_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.tail term_of_pair ord_term p \<equiv> ordered_term.lower ord_term p (ordered_term.lt term_of_pair ord_term p)"
intro: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" assumes "presheaf_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "presheaf_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str'"   and "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
RedNew: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and C :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and l :: "char list \<Rightarrow> val option" assumes "new_Addr h = \<lfloor>a\<rfloor>"   and "h' = h(a \<mapsto> (C, Collect (init_obj P C)))" shows "P,E \<turnstile> \<langle>new C,(h, l)\<rangle> \<rightarrow> \<langle>ref (a, [C]),(h', l)\<rangle>"
map_aux_in_Field: fixes x :: "'a Bonk_Schramm_extension_unfolded" shows "map_aux x \<in> Field wo_aux"
bitset_Suc: fixes n :: "nat" assumes "even n" shows "bitset (n + 1) = insert 0 (bitset n)"
mset_update: fixes i :: "nat"   and ls :: "'k list"   and v :: "'k" assumes "i < length ls" shows "mset (ls[i := v]) = add_mset v (mset ls - {#ls ! i#})"
f4: shows "module.lincomb (module_cpx_vec n) a A = \<zero>\<^bsub>module_cpx_vec n\<^esub>"
aborting_switch_independent: shows "\<forall>n s. aborting (cswitch n s) = aborting s"
empty_or_non_consec: shows "restrict v2 (res ts) c = bot \<or> maximum (restrict v1 (res ts) c) + 1 \<noteq> minimum (restrict v2 (res ts) c)"
inject_3: fixes x31 :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and x32 :: "name"   and x33 :: "(name, value) fmap"   and y31 :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and y32 :: "name"   and y33 :: "(name, value) fmap" shows "(Vrecabs x31 x32 x33 = Vrecabs y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
produces_imp_Max_atom: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "production N C = {A}" shows "A = Max (atms_of C)"
partList4_distinct_hlp: fixes a :: "'b set"   and ts :: "'b set list"   and s :: "'b set" assumes "a \<noteq> {}"   and "a \<notin> set ts"   and "disjoint (insert a (set ts))" shows "a \<notin> set (partList4 s ts)"
assms: shows "\<nexists>P scc. redundant_scc g P scc"
integrable_cts_step: fixes a :: "real"   and b :: "real" assumes "a < b" shows "integrable M (cts_step a b)"
crel_vs_iterate_state: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> ('mem, 'f) state"   and xs :: "'d list" assumes "((=) ===>\<^sub>T R) g f" shows "crel_vs (=) () (iterate_state f xs)"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) vec" shows "a *s (b *s x) = a * b *s x"
step_op: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and op :: "'op"   and ar :: "nat"   and \<Sigma> :: "'dyn list"   and x :: "'dyn"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IOp op"   and "\<AA>\<rr>\<ii>\<tt>\<yy> op = ar"   and "ar \<le> length \<Sigma>"   and "\<II>\<nn>\<ll> op (take ar \<Sigma>) = None"   and "\<OO>\<pp> op (take ar \<Sigma>) = x" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
induct: fixes P :: "int list \<Rightarrow> bool"   and a0 :: "int list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. (y \<le> x \<Longrightarrow> P (y # xs)) \<Longrightarrow> P (x # y # xs)" shows "P a0"
UN_simps_6: fixes A :: "'f \<Rightarrow> 'e set"   and B :: "'e set"   and C :: "'f set" shows "(\<Union>x\<in>C. A x - B) = \<Union> (A ` C) - B"
zip_commute: fixes xs :: "'c list"   and ys :: "'d list" shows "zip xs ys = map2 (\<lambda>x y. (y, x)) ys xs"
set_supp_mono: fixes L :: "'a list"   and L2 :: "'a list" assumes "set L \<subseteq> set L2" shows "supp L \<subseteq> supp L2"
sorted_nns: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and n :: "nat"   and kdt :: "'b kdt" assumes "sorted_wrt_dist p ps" shows "sorted_wrt_dist p (nearest_nbors n ps p kdt)"
to_it_def: fixes t :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" shows "to_it t = (SOME it. local_iwf it (fst (tree.root t)))"
Domain_set_zip: fixes A :: "'d list"   and B :: "'e list" assumes "length A = length B" shows "Domain (set (zip A B)) = set A"
trace_append_is_append_trace: fixes sig :: "'b signature"   and e' :: "'c \<times> ('b \<times> 'd) list"   and e :: "'e \<times> ('b \<times> 'd) list" shows "trace sig (append_exec e' e) = append (trace sig e) (trace sig e')"
free_gen_coeff_zero: fixes H :: "'a set"   and h :: "'a"   and a :: "'b" assumes "H \<subseteq> carrier M"   and "free_generator R M H"   and "h \<in> H"   and "a \<in> carrier R"   and "a \<cdot>\<^sub>s h = \<zero>" shows "a = \<zero>\<^bsub>R\<^esub>"
F_1: fixes i :: "'b" assumes "i \<in> I" shows "F i \<subseteq> events"
surjI: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. g (f x) = x" shows "surj g"
node_deliver_messages_Deliver: fixes m :: "'msgid \<times> 'op" shows "node_deliver_messages [Deliver m] = [m]"
cinsert_iff: fixes a :: "'d"   and b :: "'d"   and A :: "'d cset" shows "cin a (cinsert b A) = (a = b \<or> cin a A)"
cpayload_def: shows "cpayload \<equiv> {x. cpayloadp x}"
assms_2: shows "tfr\<^sub>s\<^sub>s\<^sub>t \<A>"
divide_gauss_int_def: fixes a :: "gauss_int"   and b :: "gauss_int" shows "a div b = round_complex (gauss2complex a / gauss2complex b)"
conversion_to_lconv: fixes u :: "'a"   and v :: "'a"   and M :: "'b set" assumes "(u, v) \<in> local.conversion' M" obtains xs :: "(accent \<times> 'b) list" where "snd ` set xs \<subseteq> M" and "(u, v) \<in> local.lconv xs"
mem_interior_cball: fixes x :: "'b"   and S :: "'b set" shows "(x \<in> interior S) = (\<exists>e>0. cball x e \<subseteq> S)"
simps: fixes ant :: "'b form multiset"   and suc :: "'b form multiset" shows "antec ( ant \<Rightarrow>* suc) = ant"
verts_slim_in_G: shows "pverts H.slim \<subseteq> pverts H"
alw_ev_stl: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "alw (ev P) (stl \<omega>) = alw (ev P) \<omega>"
sup_compl_top: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<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 x :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup x (uminus x) = top"
vw_v: shows "value_written P E (ws i) (ad, al) = v"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) = dist x y"
run_one_step_basic_current_memory_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Current_memory) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
assms_3: shows "[a] \<^sup>@ m \<cdot> [b] \<le>p z"
basis_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> X" shows "X \<subseteq> \<E>"
filtrationE1: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and t :: "'d" assumes "Filtration.filtration M F" shows "subalgebra M (F t)"
PReqUns: fixes evsPReqU :: "event list"   and C :: "agent"   and k :: "nat"   and KC1 :: "nat"   and Transaction :: "msg"   and M :: "agent"   and OrderDesc :: "nat"   and PurchAmt :: "nat"   and HOD :: "msg"   and OIData :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and Chall_C :: "nat"   and Chall_M :: "nat"   and PIHead :: "msg"   and P :: "agent"   and EKj :: "nat" assumes "evsPReqU \<in> set_pur"   and "C = Cardholder k"   and "CardSecret k = 0"   and "Key KC1 \<notin> used evsPReqU"   and "KC1 \<in> symKeys"   and "Transaction = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "HOD = Hash \<lbrace>Number OrderDesc, Number PurchAmt\<rbrace>"   and "OIData = \<lbrace>Number LID_M, Number XID, Nonce Chall_C, HOD, Nonce Chall_M\<rbrace>"   and "PIHead = \<lbrace>Number LID_M, Number XID, HOD, Number PurchAmt, Agent M\<rbrace>"   and "Gets C (sign (priSK M) \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Nonce Chall_M, cert P EKj onlyEnc (priSK RCA)\<rbrace>) \<in> set evsPReqU"   and "Says C M \<lbrace>Number LID_M, Nonce Chall_C\<rbrace> \<in> set evsPReqU"   and "Notes C \<lbrace>Number LID_M, Transaction\<rbrace> \<in> set evsPReqU" shows "Says C M \<lbrace>EXHcrypt KC1 EKj \<lbrace>PIHead, Hash OIData\<rbrace> (Pan (pan C)), OIData, Hash \<lbrace>PIHead, Pan (pan C)\<rbrace>\<rbrace> # Notes C \<lbrace>Key KC1, Agent M\<rbrace> # evsPReqU \<in> set_pur"
igFreshIGVar_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igFreshIGVar MOD \<equiv> \<forall>ys y xs x. ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igFresh MOD ys y (igVar MOD xs x)"
Zp_residue_add_zero_2: fixes b :: "nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and k :: "nat" assumes "b \<in> carrier Zp"   and "a k = 0" shows "(b \<oplus> a) k = b k"
Subst_imp_subst: fixes p :: "hf"   and q :: "hf"   and e :: "(name, hf) finfun" assumes "Subst p q"   and "Form p" shows "\<exists>A i t. p = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> q = \<lbrakk>\<guillemotleft>A(i::=t)\<guillemotright>\<rbrakk>e"
ShuffleLR_PlusL: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "ShuffleLR (Plus r s) t = Plus (ShuffleLR r t) (ShuffleLR s t)"
data_refinement: fixes R :: "'e \<Rightarrow> 'f"   and p :: "'e"   and S :: "'g \<Rightarrow> 'e"   and q :: "'g"   and R' :: "'g \<Rightarrow> 'h"   and S' :: "'h \<Rightarrow> 'f" assumes "mono R"   and "\<Turnstile> p{| S |}q"   and "DataRefinement S R R' S'" shows "\<Turnstile> R p{| S' |}R' q"
transitive: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<approx> Q"   and "Q \<approx> R" shows "P \<approx> R"
pdevs_apply_pdevs_of_ivl: fixes l :: "'a"   and u :: "'a"   and i :: "nat" shows "pdevs_apply (pdevs_of_ivl l u) i = (if i < length Basis_list then ((u - l) \<bullet> Basis_list ! i / 2) *\<^sub>R Basis_list ! i else (0::'a))"
app_vlrestrictionE: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> A" and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
distinct_3: fixes x1 :: "nat"   and x31 :: "type_scheme"   and x32 :: "type_scheme" shows "FVar x1 \<noteq> x31 =-> x32"
assms_3: shows "col \<in> {a..b} \<rightarrow> {..<k}"
is_red_monomial_lt: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "0 \<notin> B" shows "is_red (monomial (1::'c) ` lt ` B) = is_red B"
cases: fixes a1 :: "'b"   and a2 :: "'c"   and r :: "('c \<times> 'b) set" assumes "(a1, a2) \<in> r\<inverse>" obtains a :: "'c" and b :: "'b" where "a1 = b" and "a2 = a" and "(a, b) \<in> r"
code: shows "\<i>\<^sub>\<int> = Gauss_Int 0 1"
semiring_norm_45: shows "\<bar>1::'e\<bar> = (1::'e)"
sorted_sort_id: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = xs"
circ_separate_mult_1: fixes y :: "'b"   and x :: "'b" assumes "y * x \<le> x * y" shows "(x * y)\<^sup>\<star> \<le> x\<^sup>\<star> * y\<^sup>\<star>"
uminus_carrier_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "- A \<in> carrier_mat nr nc"
filtration_measurable: fixes f :: "'a \<Rightarrow> 'c"   and t :: "nat"   and N :: "'c measure" assumes "f \<in> F t \<rightarrow>\<^sub>M N" shows "random_variable N f"
div_diff1_eq1: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a mod m \<le> b mod m" shows "(b - a) div m = b div m - a div m"
nres_relI: fixes c :: "'b nres"   and R :: "('b \<times> 'c) set"   and a :: "'c nres" assumes "c \<le> \<Down> R a" shows "(c, a) \<in> \<langle>R\<rangle>nres_rel"
red: shows "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e'',s''\<rangle>"
residual_eq_iff_perm: fixes \<alpha>1 :: "'b"   and P1 :: "'c"   and \<alpha>2 :: "'b"   and P2 :: "'c" shows "(\<langle>\<alpha>1,P1\<rangle> = \<langle>\<alpha>2,P2\<rangle>) = (\<exists>p. supp (\<alpha>1, P1) - bn \<alpha>1 = supp (\<alpha>2, P2) - bn \<alpha>2 \<and> (supp (\<alpha>1, P1) - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> (\<alpha>1, P1) = (\<alpha>2, P2) \<and> p \<bullet> bn \<alpha>1 = bn \<alpha>2)"
other_def: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "other a x y = (if a = x then y else x)"
conj: fixes F :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "is_filter F"   and "F P"   and "F Q" shows "F (\<lambda>x. P x \<and> Q x)"
\<theta>\<^sub>g: shows "\<guillemotleft>\<theta>\<^sub>g : tab\<^sub>0 g \<star> w\<^sub>g \<rightarrow> u\<^sub>g\<guillemotright>"
rep_eq: fixes x :: "nat \<Rightarrow> bool" shows "Rep_uint8 (set_bits x) = set_bits x"
assms_4: shows "s (f \<triangleright> n) = s (g \<triangleright> n)"
C4: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "(\<Psi>', S, T) \<in> Rel"   and "xvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel"
slow_select_correct: fixes k :: "nat"   and xs :: "'a list" shows "slow_select k xs = select k xs"
assms_5: shows "A2 \<in> carrier_mat n1 nc1"
mod_poly_code: fixes f :: "'b poly"   and g :: "'b poly" shows "f mod g = (let cg = coeffs g in if cg = [] then f else let cf = coeffs f; ilc = inverse (last cg); ch = map ((*) ilc) cg; r = mod_poly_one_main_list (rev cf) (rev ch) (1 + length cf - length cg) in poly_of_list (rev r))"
reachable_successors: fixes w :: "'b stream"   and A :: "('b, 'a) nba"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a" shows "gsuccessors A w V p \<subseteq> greachable A w V p"
carrier_eq: shows "carrier = {x. of_weak_ranking (weak_ranking_aux carrier) x x}"
IH: fixes vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "\<forall>a. ListMem a as \<longrightarrow> fmdom' (fmrestrict_set vs (snd a)) = {}" shows "fmrestrict_set vs (exec_plan s as) = fmrestrict_set vs s"
return_returns_heap: fixes h :: "'a"   and x :: "'c"   and h' :: "'a" shows "h \<turnstile> return x \<rightarrow>\<^sub>h h' = (h = h')"
le_add_diff_inverse: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
nth_solution_sound: fixes D :: "nat"   and n :: "nat" assumes "pell D" shows "pell.solution D (pell.nth_solution D n)"
map_bind_spmf: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'d spmf"   and g :: "'d \<Rightarrow> 'c spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
metric_LIM_compose2: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'b"   and a :: "'a"   and g :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "f \<midarrow>a\<rightarrow> b"   and "g \<midarrow>b\<rightarrow> c"   and "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b" shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
power2_norm_eq_cinner': fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<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 dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(norm x)\<^sup>2 = Re (cinner x x)"
HFail_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HFail s s' p"   and "HInv4d s p" shows "HInv4d s' p"
leg1_simps_3: shows "src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1) = src (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0)"
simps_2: fixes uv :: "'c" shows "weak_gt_arctic_delta MinInfty_delta (Num_arc_delta uv) = False"
wf_trm_arity: fixes f :: "'fun"   and T :: "('fun, 'b) Term.term list" assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)" shows "length T = arity f"
rep_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "count (factorial_semiring.prime_factorization zero times one normalize x) = (\<lambda>p. if normalization_semidom.prime zero times one normalize p then factorial_semiring.multiplicity times one p x else 0)"
nonempty_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "\<gamma>\<^sub>A\<^sub>G\<^sub>M K \<phi> \<noteq> {}"
path_image_compose: fixes f :: "'d \<Rightarrow> 'c"   and p :: "real \<Rightarrow> 'd" shows "path_image (f \<circ> p) = f ` path_image p"
assms_2: shows "h \<turnstile> w \<rightarrow>\<^sub>h h'"
subalg: fixes M :: "'c measure"   and F :: "'c measure" assumes "sigma_finite_subalgebra M F" shows "subalgebra M F"
inv_left_eq_iff: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "bijection f" shows "(inv f a = b) = (f b = a)"
transpose_def: fixes A :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose A = (\<chi>i j. A $h j $h i)"
domain_nonnegative: fixes x :: "brick" shows "0 \<le> domain x"
lm09: fixes pair :: "'c \<times> 'd set"   and N :: "'c set"   and \<Omega> :: "'d set"   and a :: "('c \<times> 'd set) set" assumes "fst pair \<in> N"   and "snd pair \<in> Pow \<Omega> - {{}}" shows "summedBidSecond (pseudoAllocation a <| (N \<times> finestpart \<Omega>)) pair = summedBidSecond (bidMaximizedBy a N \<Omega> Elsee 0) pair"
Inf_set_def: fixes A :: "'d set set" shows "\<Inter> A = {x. Inf ((\<in>) x ` A)}"
gen_llength_cons_fusion: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and s :: "'c" shows "gen_llength_cons g 0 s = llength (lunstream g s)"
mat_incr_mult_adjoint_mat_incr: fixes n :: "nat" shows "mat_incr n * adjoint (mat_incr n) = 1\<^sub>m n"
right_connected: fixes \<mu> :: "'a"   and \<mu>' :: "'a"   and \<nu> :: "'a" assumes "C.seq \<mu> \<mu>'" shows "(\<nu> \<star>\<^sub>C \<mu> \<noteq> C.null) = (\<nu> \<star>\<^sub>C \<mu>' \<noteq> C.null)"
set_merge_distinct: fixes xs :: "'c list"   and ys :: "'c list" shows "set (merge_distinct xs ys) = set xs \<union> set ys"
convol_o: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and h :: "'d \<Rightarrow> 'g" shows "<f , g> \<circ> h = <f \<circ> h , g \<circ> h>"
part_equivp_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "part_equivp R = ((\<exists>x. R x x) \<and> (\<forall>x y. R x y = (R x x \<and> R y y \<and> R x = R y)))"
supt_def: shows "{\<rhd>} \<equiv> {(xa, x). ??.Subterm_and_Context.suptp xa x}"
in_scc_of_self: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c" assumes "wf_digraph G"   and "u \<in> verts G" shows "u \<in> pre_digraph.scc_of G u"
irreducible_m_def: fixes f :: "int poly" shows "irreducible_m f = (Mp f \<noteq> Mp 0 \<and> \<not> f dvdm 1 \<and> (\<forall>a b. f =m a * b \<longrightarrow> a dvdm 1 \<or> b dvdm 1))"
lindep_same: fixes S :: "'a vec set" assumes "S \<subseteq> mat_kernel D" shows "KD.lin_dep S = KD.NC.lin_dep S"
not_arr_Null: shows "\<not> arrow_of_spans (\<cdot>) Null"
last_prog_hd_prog_append': fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and sb :: "'a memref list"   and sb' :: "'a memref list" assumes "history_consistent \<theta> (hd_prog p (sb @ sb')) (sb @ sb')"   and "last_prog p (sb @ sb') = p" shows "last_prog (hd_prog p sb') sb = hd_prog p sb'"
nn_integral_try_spmf: fixes p :: "'a spmf"   and q :: "'a spmf"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_spmf (TRY p ELSE q)) f = integral\<^sup>N (measure_spmf p) f + integral\<^sup>N (measure_spmf q) f * ennreal (pmf p None)"
arr_iff_ARR: fixes f :: "'c local.term set" shows "\<F>C.arr f = \<F>C.ARR f"
Key_analz_image_Key: fixes evs :: "event list"   and SK :: "nat"   and KK :: "nat set" assumes "evs \<in> kerbIV_gets"   and "SK \<in> symKeys \<and> KK \<subseteq> - range shrK"   and "\<forall>K\<in>KK. \<not> AKcryptSK K SK evs" shows "(Key SK \<in> analz (Key ` KK \<union> knows Spy evs)) = (SK \<in> KK \<or> Key SK \<in> analz (knows Spy evs))"
in_gamma_option_iff: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and x :: "'c"   and r :: "'d \<Rightarrow> 'c set"   and u :: "'d option" assumes "Gamma num' plus' \<gamma>" shows "(x \<in> \<gamma>_option r u) = (\<exists>u'. u = Some u' \<and> x \<in> r u')"