hcju/isabelleps huggingface dataset:
continuous_map_composeD: fixes T :: "'c topology"   and X :: "'d topology"   and g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" assumes "continuous_map T X (g \<circ> f)" shows "g \<in> f ` topspace T \<rightarrow> topspace X"
simps_7: fixes wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp" shows "alphabet.wf wf_atom n (Star r) = alphabet.wf wf_atom n r"
t\<^sub>p_sumofALG: fixes s :: "'c list \<times> 'd"   and a :: "nat \<times> nat list"   and qs :: "'c list"   and i :: "nat" assumes "distinct (fst s)"   and "snd a = []"   and "qs ! i \<in> set (fst s)" shows "t\<^sub>p (fst s) (qs ! i) a = (\<Sum>e\<in>set (fst s). ALG e qs i s)"
simps_9: fixes disconnected_nodes' :: "(_) node_ptr list \<Rightarrow> (_) node_ptr list"   and nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "disconnected_nodes_update disconnected_nodes' \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes' disconnected_nodes, \<dots> = more\<rparr>"
freshChainPerm: fixes xvec :: "name list"   and yvec :: "name list"   and C :: "'d"   and x :: "name" assumes "length xvec = length yvec"   and "yvec \<sharp>* C"   and "xvec \<sharp>* yvec"   and "x mem xvec"   and "distinct yvec" shows "x \<sharp> [xvec yvec] \<bullet>\<^sub>v C"
WT_converter_of_callee: fixes \<I> :: "('b, 'c) \<I>"   and \<I>' :: "('d, 'e) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a, 'd, 'e) gpv"   and s :: "'a" assumes "\<And>s q. q \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g callee s q \<surd>"   and "\<And>s q r s'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, s') \<in> results_gpv \<I>' (callee s q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q" shows "\<I>, \<I>' \<turnstile>\<^sub>C CNV callee s \<surd>"
assms: shows "a \<le> b \<or> b \<le> a"
devExt_bc: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat"   and n'' :: "nat" assumes "\<not> Option.is_none (devBC t n nid (n' + n''))" shows "devExt t n nid n' n'' = bc (\<sigma>\<^bsub>the (devBC t n nid (n' + n''))\<^esub>t (n' + n''))"
OclSelf_at_post_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e" shows "x@postH = OclSelf x H snd"
assms_2: shows "(a, b\<^sub>1) |\<in>| M"
mat1error: fixes env :: "v sem_env"   and e :: "exp0"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and err :: "v error_result" assumes "cupcake_evaluate_single env e (Rval v0)"   and "cupcake_match_result (c env) v0 pes Bindv = Rerr err" shows "cupcake_evaluate_single env (Mat e pes) (Rerr err)"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "i < length xs" shows "xs[i := x] ! i = x"
elims_1: fixes x :: "'a Matrix.mat"   and y :: "bool" assumes "square_mat x = y" obtains A :: "'a Matrix.mat" where "x = A" and "y = (dim_col A = dim_row A)"
list_all2_Cons: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and xs :: "'e list"   and y :: "'f"   and ys :: "'f list" shows "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
MutuallyDistinct_HA: fixes HA :: "('d, 'e, 'f) hierauto" shows "MutuallyDistinct (SAs HA)"
r_add_def: shows "r_add \<equiv> Pr 1 (recf.Id 1 0) (Cn 3 S [recf.Id 3 1])"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. T (\<I>, f)) local.map (\<lambda>f. if arr f then f \<cdot> \<l>[local.dom f] else null)"
\<phi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "\<phi>C (b, a) \<in> C.hom b a \<rightarrow> HomC.set (b, a)"
fun_upd_def: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'c" shows "f(a := b) = (\<lambda>x. if x = a then b else f x)"
finite_ne_induct: fixes F :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite F"   and "F \<noteq> {}"   and "\<And>x. P {x}"   and "\<And>x F. \<lbrakk>finite F; F \<noteq> {}; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
encode_problem_parallel_correct_iii_1: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat"   and op :: "'b strips_operator"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k < length (\<Phi>\<inverse> \<Pi> \<A> t)"   and "op \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)" shows "v \<in> set (add_effects_of op) \<longrightarrow> (\<Phi>\<^sub>S\<inverse> \<Pi> \<A> Suc k) v = Some True"
compute_SCC_correct: shows "compute_SCC \<le> compute_SCC_spec"
distinct_tl: fixes xs :: "'a list" assumes "distinct xs" shows "distinct (tl xs)"
le_principal: fixes F :: "'b filter"   and A :: "'b set" shows "(F \<le> principal A) = (\<forall>\<^sub>F x in F. x \<in> A)"
0: fixes t :: "('fun, 'var) Term.term" assumes "t \<in> SMP (trms\<^sub>s\<^sub>s\<^sub>t S \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t S) - range Var" shows "t \<in> SMP (trms\<^sub>s\<^sub>s\<^sub>t S) - range Var \<or> t \<in> SMP (pair ` setops\<^sub>s\<^sub>s\<^sub>t S) - range Var"
divide_right_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "(0::'b) \<le> c" shows "a / c \<le> b / c"
coalg_F1in: fixes A :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and x :: "'d" assumes "coalg A B2 s1 s2"   and "x \<in> A" shows "s1 x \<in> F1in UNIV A B2"
convexI: fixes s :: "'b set" assumes "\<And>x y u v. \<lbrakk>x \<in> s; y \<in> s; 0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" shows "convex s"
exec_assign': fixes \<sigma> :: "'a control_state_scheme"   and f :: "'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> assign f ;- M) = (\<sigma> \<Turnstile> M)"
hull_induct: fixes a :: "'b"   and Q :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> Q hull S"   and "\<And>x. x \<in> S \<Longrightarrow> P x"   and "Q {x. P x}" shows "P a"
iT_Plus_neg_cut_less2: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus>- k) \<down>< (t - k) = I \<down>< t \<oplus>- k"
Ostrowski_elem_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and vv :: "nat \<Rightarrow> 'c \<Rightarrow> ant"   and x :: "'c" shows "Ostrowski_elem K n vv x = (0 < vv 0 (1\<^sub>r\<^bsub>K\<^esub> \<plusminus>\<^bsub>K\<^esub> -\<^sub>a\<^bsub>K\<^esub> x) \<and> (\<forall>j\<in>nset (Suc 0) n. 0 < vv j x))"
conv_dist_comp: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "conv (times x y) = times (conv y) (conv x)"
set_mult_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set"   and K :: "'a set" shows "H <#>\<^bsub>G\<^esub> K = (\<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes>\<^bsub>G\<^esub> k})"
nodes_add_edge: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "nodes (add_edge v e v' g) = insert v (insert v' (nodes g))"
set_times_intro: fixes a :: "'b"   and C :: "'b set"   and b :: "'b"   and D :: "'b set" assumes "a \<in> C"   and "b \<in> D" shows "a * b \<in> C * D"
set_union: fixes xs :: "'b list"   and ys :: "'b list" shows "set (List.union xs ys) = set xs \<union> set ys"
snd_guaranteeStripPair: fixes f :: "'e"   and g :: "'d set" shows "snd (guaranteeStripPair f g) = g"
execlass_ex_def: fixes rel :: "('a \<times> 'a) list"   and c :: "'a" shows "execlass_ex rel c = List.member (exefield rel) c"
plus_commutative: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "Semilat A r f"   and "a \<in> A"   and "b \<in> A" shows "a \<squnion>\<^bsub>f\<^esub> b = b \<squnion>\<^bsub>f\<^esub> a"
contract_closure: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi> = Cn (K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi>)"
posssibleAllocationsRelCharacterization: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" shows "(a \<in> allAllocations N G) = (a \<in> injectionsUniverse \<and> Domain a \<subseteq> N \<and> Range a \<in> all_partitions G)"
Lex_inverse: fixes y :: "'b list" assumes "y \<in> {xs. True}" shows "unlex (Lex y) = y"
access_some_lookup: fixes root :: "('a att_scheme \<times> 'b, 'a att_scheme, 'c) env"   and path :: "'c list"   and uid :: "nat"   and perms :: "perm set"   and "file" :: "('a att_scheme \<times> 'b, 'a att_scheme, 'c) env" assumes "access root path uid perms = Some file" shows "lookup root path = Some file"
assert_def: fixes P :: "bool" shows "assert P \<equiv> if P then return () else fail"
wf_trm_subst_inv: fixes t :: "('fun, 'd) Term.term"   and \<delta> :: "'d \<Rightarrow> ('fun, 'c) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
m_widen: fixes C1 :: "'a acom"   and C2 :: "'a acom" assumes "P C1"   and "P C2"   and "\<not> C2 \<le> C1" shows "m (C1 \<nabla> C2) < m C1"
path_conforms_with_strategy_prefix: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and P' :: "'a llist" assumes "path_conforms_with_strategy p P \<sigma>"   and "lprefix P' P" shows "path_conforms_with_strategy p P' \<sigma>"
path_cons_simp': fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and v :: "'a"   and v' :: "'a"   and e :: "('a \<times> 'b) \<times> 'a \<times> 'c"   and pth :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list" shows "Pre_Port_Graph.path edges v v' (e # pth) = (\<exists>b a ba. e = ((v, b), a, ba) \<and> ((v, b), a, ba) \<in> edges \<and> Pre_Port_Graph.path edges a v' pth)"
wf_has_field_mono2: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and E :: "String.literal"   and D :: "String.literal" assumes "wf_prog wf_md P"   and "P \<turnstile> C has F:T (fm) in E"   and "P \<turnstile> C \<preceq>\<^sup>* D"   and "P \<turnstile> D \<preceq>\<^sup>* E" shows "P \<turnstile> D has F:T (fm) in E"
predicate1D: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P \<le> Q"   and "P x" shows "Q x"
bot_least: fixes a :: "'a list" shows "Sublist.prefix [] a"
distinct_2: fixes name' :: "name"   and input' :: "('a, 'b, 'c) input"   and "term" :: "'a"   and psi :: "('a, 'b, 'c) psi" shows "\<nu>name'input' \<noteq> \<rparr>term.psi"
irreducible_dvd_pow: fixes p :: "'c poly"   and q :: "'c poly"   and n :: "nat" assumes "irreducible p"   and "p dvd q ^ n" shows "p dvd q"
lcoset_join2: fixes H :: "'a set"   and g :: "'a" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)"   and "g \<in> H" shows "g <#\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> H = H"
unclocked_single_eq': shows "Big_Step_Unclocked_Single.evaluate = BigStep.evaluate False"
times_subset_iff: fixes A :: "'a set"   and C :: "'b set"   and B :: "'a set"   and D :: "'b set" shows "(A \<times> C \<subseteq> B \<times> D) = (A = {} \<or> C = {} \<or> A \<subseteq> B \<and> C \<subseteq> D)"
NSLIMSEQ_def: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" shows "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L = (\<forall>N\<in>HNatInfinite. (*f* X) N \<approx> star_of L)"
nodes_def: shows "task.nodes = Helper ## map Assumption [] @- map Conclusion [ForallX (imp (Q x) (Q x))] @- smap Rule prop_rules"
welltyped_constant: fixes T :: "Unary_PCF.type" shows "constant T ::: \<B> \<rightarrow> T"
is_failed_alt_def: fixes st' :: "status" shows "is_failed st' = (\<not> is_success st' \<and> \<not> is_found st')"
constr_sem_c_def: fixes \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<I> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle> \<equiv> \<theta> supports \<I> \<and> \<lbrakk>{}; S\<rbrakk>\<^sub>c \<I>"
non_empty_cycle_root_loop_converse: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "r \<le> x\<^sup>+ ; r"
dg_Rel_Dom_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_Rel \<alpha>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> dg_Rel \<alpha>\<lparr>Obj\<rparr>"
map_def: fixes f :: "J.Arr" shows "E.map f = (if J.arr f then C.dom p0 else C.null)"
decided_determ: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and s :: "state"   and t :: "state"   and X :: "final_decision" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow>\<^sub>s t"   and "s = Decision X" shows "t = Decision X"
push_precond_eq_sat_or_nonsat: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" shows "push_precond f l e = (sat_push_precond f l e \<or> nonsat_push_precond f l e)"
atomic_sub_lemmas_8: fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> c\<^sub>1 kills c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> c\<^sub>1 kills c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
swap_5: fixes k :: "nat" assumes "k < m" shows "\<parallel>gs2.gso k\<parallel>\<^sup>2 = (if k = i - 1 then \<parallel>gs1.gso i\<parallel>\<^sup>2 + gs1.\<mu> i (i - 1) * gs1.\<mu> i (i - 1) * \<parallel>gs1.gso (i - 1)\<parallel>\<^sup>2 else if k = i then \<parallel>gs1.gso i\<parallel>\<^sup>2 * \<parallel>gs1.gso (i - 1)\<parallel>\<^sup>2 / \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 else \<parallel>gs1.gso k\<parallel>\<^sup>2)"
cases: fixes a1 :: "'d"   and a2 :: "'d \<Rightarrow> real"   and a3 :: "intv" assumes "intv_elem a1 a2 a3" obtains u :: "'d \<Rightarrow> real" and x :: "'d" and d :: "nat" where "a1 = x__" and "a2 = ub__" and "a3 = Const d__" and "ub__ x__ = real d__"   | d :: "nat" and u :: "'d \<Rightarrow> real" and x :: "'d" where "a1 = xa__" and "a2 = uc__" and "a3 = Intv d" and "real d < uc__ xa__" and "uc__ xa__ < real (d + 1)"   | c :: "nat" and u :: "'d \<Rightarrow> real" and x :: "'d" where "a1 = x" and "a2 = u" and "a3 = Greater c" and "real c < u x"
executable: fixes p :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and initial :: "'a \<Rightarrow> bool"   and a :: "'b" assumes "p \<in> transition_system_initial.nodes execute enabled initial"   and "enabled a p" shows "a \<in> transition_system_initial.executable execute enabled initial"
mult_in_1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](\<lambda>_. 1::'b)"   and "g \<in> O[F](\<lambda>_. 1::'b)" shows "(\<lambda>x. f x * g x) \<in> O[F](\<lambda>_. 1::'b)"
elims_3: fixes x :: "bool"   and xa :: "'b \<times> 'c set" assumes "\<not> pure_assn_raw x xa" obtains b :: "bool" and h :: "'b" and as :: "'c set" where "x = b" and "xa = (h, as)" and "\<not> (as = {} \<and> b)"
comp_cod_arr: fixes f :: "'d"   and b :: "'d" assumes "Dop.arr f"   and "Dop.cod f = b" shows "Dop.comp b f = f"
snga_assn_def: fixes r :: "'b array"   and a :: "'b list" shows "r \<mapsto>\<^sub>a a \<equiv> Abs_assn (snga_assn_raw r a)"
yoneda_lemma_axioms: shows "yoneda_lemma (\<cdot>) (\<cdot>\<^sub>S) \<phi> (Y a') a"
t: shows "t \<noteq> null \<and> natural_isomorphism (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom t) (Cod t) (Map t)"
mat_entry_explicit: fixes M :: "'a mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "M \<in> carrier_mat m n"   and "i < m"   and "j < n" shows "(M *\<^sub>v unit_vec n j) $ i = M $$ (i, j)"
commute: fixes a :: "'b"   and b :: "'b" shows "sup (a::'b) (b::'b) = sup b a"
check_extension_l_new_var_multiple_err_def: fixes v :: "char list"   and p :: "(char list list \<times> int) list" shows "check_extension_l_new_var_multiple_err v p = SPEC (\<lambda>_. True)"
t_non_zero: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "t_covering_design point_set block_collection u_block_size grouping min_index" shows "1 \<le> grouping"
HMA_invertible_matrix_mod_type: shows "(Mod_Type_Connect.HMA_M ===> (=)) invertible_mat invertible"
is_process9: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>s. s @ [tick] \<notin> DIVERGENCES P \<or> s \<in> DIVERGENCES P"
flow_summation_aux: shows "(\<Sum>v\<in>E `` {u}. f' (u, v)) - (\<Sum>v\<in>E `` {u}. f' (v, u)) = (\<Sum>v\<in>E\<inverse> `` {u}. f' (v, u)) - (\<Sum>v\<in>E\<inverse> `` {u}. f' (u, v))"
inject_8: fixes x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly"   and y81 :: "'b poly"   and y82 :: "nat"   and y83 :: "'b poly" shows "(CN x81 x82 x83 = CN y81 y82 y83) = (x81 = y81 \<and> x82 = y82 \<and> x83 = y83)"
splay_Leaf_iff: fixes a :: "'a"   and t :: "'a tree" shows "(splay a t = \<langle>\<rangle>) = (t = \<langle>\<rangle>)"
distinct_disc_7: fixes event :: "('d, 'e, 'f) event" assumes "isTrans event" shows "\<not> isRecvMarker event"
cancels_to_1_unfold: fixes x :: "(bool \<times> 'e) list"   and y :: "(bool \<times> 'e) list" assumes "cancels_to_1 x y" obtains xs1 :: "(bool \<times> 'e) list" and x1 :: "bool \<times> 'e" and x2 :: "bool \<times> 'e" and xs2 :: "(bool \<times> 'e) list" where "x = xs1 @ x1 # x2 # xs2" and "y = xs1 @ xs2" and "canceling x1 x2"
finsum_smult_ldistr: fixes A :: "'f set"   and a :: "'a"   and f :: "'f \<Rightarrow> 'c" assumes "finite A"   and "a \<in> carrier R"   and "f \<in> A \<rightarrow> carrier M" shows "a \<odot>\<^bsub>M\<^esub> finsum M f A = (\<Oplus>\<^bsub>M\<^esub>i\<in>A. a \<odot>\<^bsub>M\<^esub> f i)"
continuous_on_mult: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x * g x)"
vdiff_vsingleton_nin: fixes a :: "V"   and A :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "set {a} -\<^sub>\<circ> A = set {a}"
ofailE: fixes s :: "'d"   and v :: "'c"   and P :: "bool" assumes "ofail s = Some v" shows "P"
extremum_strict: fixes a :: "'b list" shows "\<not> a <p \<epsilon>"
wa_intro: fixes T1 :: "('d, 'f) tree_automaton_rec"   and T2 :: "('e, 'f) tree_automaton_rec"   and addi :: "(('d \<times> 'e) set \<times> ('d \<times> 'e) list \<times> ('d \<times> 'e, 'f) ta_rule set) set" assumes "while_algo (dfs_algo (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2))))"   and "wa_invar (frp_algo T1 T2) = addi \<inter> {s. frp_\<alpha> s \<in> wa_invar (dfs_algo (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2))))}"   and "\<And>s s'. \<lbrakk>s \<in> addi; s \<in> wa_cond (frp_algo T1 T2); frp_\<alpha> s \<in> wa_invar (dfs_algo (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2)))); (s, s') \<in> wa_step (frp_algo T1 T2)\<rbrakk> \<Longrightarrow> s' \<in> addi"   and "wa_initial (frp_algo T1 T2) \<subseteq> addi" shows "while_algo (frp_algo T1 T2)"
fcompare_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fcompare a b = (if is_nan a \<or> is_nan b then Und else if is_infinity a \<and> sign a = 1 then if is_infinity b \<and> sign b = 1 then Eq else Lt else if is_infinity a \<and> sign a = 0 then if is_infinity b \<and> sign b = 0 then Eq else Gt else if is_infinity b \<and> sign b = 1 then Gt else if is_infinity b \<and> sign b = 0 then Lt else if valof a < valof b then Lt else if valof a = valof b then Eq else Gt)"
sWlsVal_def: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and val :: "'varSort \<Rightarrow> 'h \<Rightarrow> 'f" shows "sWlsVal SEM val \<equiv> \<forall>ys y. sWls SEM (asSort ys) (val ys y)"
AbGroup_inner_dirsum_el_decomp_nth_onto_nth: fixes Gs :: "'a set list"   and n :: "nat" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "n < length Gs" shows "(\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>Gs. G) then \<Oplus>Gs\<leftarrow> a ! n else (0::'a)) ` (\<Oplus>G\<leftarrow>Gs. G) = Gs ! n"
wf_stateful_policy_def: fixes \<T> :: "'a stateful_policy" shows "wf_stateful_policy \<T> \<equiv> (fst ` flows_fix \<T> \<subseteq> hosts \<T> \<and> snd ` flows_fix \<T> \<subseteq> hosts \<T>) \<and> flows_state \<T> \<subseteq> flows_fix \<T> \<and> finite (hosts \<T>)"
simps_1: fixes X :: "char list set" shows "bury SKIP X = SKIP"
xw: shows "xw = \<lbrakk>\<guillemotleft>Aw\<guillemotright>\<rbrakk>e"
max_ext_wf: fixes r :: "('e \<times> 'e) set" assumes "wf r" shows "wf (max_ext r)"
intro: fixes S :: "'b set set"   and T :: "'b set set" assumes "\<And>s. (s open\<^bsub>S\<^esub>) = (\<exists>t. t open\<^bsub>T\<^esub> \<and> s = t \<inter> carrier\<^bsub>S\<^esub>)" shows "subtopology S T"
that: fixes f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "f \<in> F"   and "t \<in> keys (rep_list p)"   and "rep_list f \<noteq> 0"   and "punit.lt (rep_list f) adds t"   and "sing_reg (t \<oplus> lt f) (punit.lt (rep_list f) \<oplus> lt p)"   and "top_tail t (punit.lt (rep_list p))" shows "thesis"
floor_less_cancel: fixes x :: "'c"   and y :: "'c" assumes "\<lfloor>x\<rfloor> < \<lfloor>y\<rfloor>" shows "x < y"
pow_divides_pow_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "0 < n" shows "dvd.dvd times (power.power one times a n) (power.power one times b n) = dvd.dvd times a b"
dims2_def: shows "d.dims2 = nths dims (vars2 \<union> vars3)"
set_pmf_of_set: fixes S :: "'c set" assumes "S \<noteq> {}"   and "finite S" shows "set_pmf (pmf_of_set S) = S"
ide_cod: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.cod C f)"
getFrN_FvarsT: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and t :: "'trm"   and n :: "nat" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla"   and "t \<in> set ts" shows "set (getFrN xs ts \<phi>s n) \<inter> FvarsT t = {}"
comp_tot_rec: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "total_recursive f"   and "total_recursive g" shows "total_recursive (f \<circ> g)"
L2_set_le_sum: fixes A :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" shows "L2_set f A \<le> sum f A"