hcju/isabelleps huggingface dataset:
BInit_def: fixes ic :: "state \<Rightarrow> 'a"   and q :: "state \<Rightarrow> 'a list"   and oc :: "state \<Rightarrow> 'a" shows "BInit ic q oc \<equiv> q = #[]"
no_overlap: shows "\<forall>A\<in>(\<lambda>m. {i. \<exists>k. i = 2 ^ k * m \<and> i \<le> n}) ` {m. m \<le> n \<and> odd m}. \<forall>B\<in>(\<lambda>m. {i. \<exists>k. i = 2 ^ k * m \<and> i \<le> n}) ` {m. m \<le> n \<and> odd m}. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
nsum_minusTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. f j \<in> carrier A) \<longrightarrow> -\<^sub>a \<Sigma>\<^sub>e A f n = \<Sigma>\<^sub>e A (\<lambda>x\<in>{j. j \<le> n}. -\<^sub>a f x) n"
simps_1: shows "cast_and_box Ubx1 = map_option box_ubx1 \<circ> cast_Ubx1"
powhr_def: fixes x :: "real star"   and a :: "real star" shows "x powhr a = (*f2* (powr)) x a"
perm_simps_2: fixes p :: "perm"   and name :: "name" shows "p \<bullet> Var name = Var (p \<bullet> name)"
subst_list_def: fixes \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and ys :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "subst_list \<sigma> ys = map (\<lambda>p. (fst p \<cdot> \<sigma>, snd p \<cdot> \<sigma>)) ys"
K: shows "\<forall>\<^sub>F x in F. norm (f x) \<le> K"
zero_le: fixes x :: "'a multiset" shows "{#} \<subseteq># x"
Can_rep_can: fixes f :: "'c local.term set" assumes "\<F>C.can f" shows "Can (\<F>C.rep f)"
fst_Inf: fixes A :: "('c \<times> 'd) set" shows "fst (Inf A) = Inf (fst ` A)"
restrict0_in_fun_space: fixes f :: "'a \<Rightarrow> real" assumes "diff_fun k (charts_submanifold S) f" shows "restrict0 sub.carrier f \<in> sub.diff_fun_space"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>acc xs. (\<And>x. \<lbrakk>xs \<noteq> []; x \<in> set (remdups xs)\<rbrakk> \<Longrightarrow> P (x # acc) (remove1 x xs)) \<Longrightarrow> P acc xs" shows "P a0 a1"
addfeqsim: fixes t :: "nat \<Rightarrow> 'a" shows "first t ## t \<approx> t"
dehomo_subst_def: fixes x :: "'c" shows "dehomo_subst x = (\<lambda>y. if y = x then 1 else monomial (1::'d) (monomial 1 y))"
assms_2: shows "fv (\<Gamma>, e) \<subseteq> set L \<union> domA \<Gamma>"
IH: assumes "xs \<noteq> LNil" shows "\<exists>i. enat i < llength xs \<and> lnth xs i = llast xs \<and> (\<forall>j<llength xs. j \<le> enat i)"
winning_path_drop_add: fixes P :: "'a llist"   and p :: "Player"   and n :: "nat" assumes "valid_path P"   and "winning_path p (ldropn n P)"   and "enat n < llength P" shows "winning_path p P"
WHILET_rule: fixes R :: "('a \<times> 'a) set"   and I :: "'a \<Rightarrow> bool"   and s :: "'a"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "wf R"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> SPEC (\<lambda>s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s\<rbrakk> \<Longrightarrow> \<Phi> s" shows "WHILE\<^sub>T b f s \<le> SPEC \<Phi>"
map_bind_pmf: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'d pmf"   and g :: "'d \<Rightarrow> 'c pmf" shows "map_pmf f (M \<bind> g) = M \<bind> (\<lambda>x. map_pmf f (g x))"
simps: fixes ant :: "form multiset"   and suc :: "form multiset" shows "NominalSequents.set_of_seq ( ant \<Rightarrow>* suc) = set_mset (NominalSequents.mset ( ant \<Rightarrow>* suc))"
integrable_ln_Gamma_aux_real: fixes s :: "real"   and n :: "nat"   and N :: "nat" assumes "0 < s" shows "(\<lambda>x. pbernpoly n x / (x + s) ^ n) integrable_on {0..real N}"
monom_mult_atomize: fixes c :: "'c"   and t :: "'a"   and p :: "nat \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "monom_mult c t (atomize_poly p) = atomize_poly (MPoly_Type_Class.punit.monom_mult (monomial c t) 0 p)"
union_vars_poly_def: fixes xs :: "(char list list \<times> int) list"   and \<V> :: "char list set" shows "union_vars_poly xs \<V> = fold (\<lambda>(xs, uu_). union_vars_monom xs) xs \<V>"
filter_set: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "Set.filter P (set xs) = set (filter P xs)"
p'_pos_iff: fixes x :: "real" shows "(0 < delayed_safe_distance.other.p' x) = (if 0 < a\<^sub>o then - other.s' \<delta> / a\<^sub>o < x else if a\<^sub>o < 0 then x < - other.s' \<delta> / a\<^sub>o else 0 < other.s' \<delta>)"
some_geodesic_connected_1: fixes x :: "'a"   and y :: "'a" shows "connected {x--y}"
Hermite_of_upt_row_i_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "nat"   and ass :: "'d \<Rightarrow> 'd"   and res :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "Hermite_of_upt_row_i A i ass res = foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<i])"
le_utrans_refl: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "le_utrans x x"
integrable: shows "interval_lebesgue_integrable lborel (min a (min b c)) (max a (max b c)) f"
image_Suc_atLeastLessThan: fixes i :: "nat"   and j :: "nat" shows "Suc ` {i..<j} = {Suc i..<Suc j}"
euler_approx_def: fixes n :: "nat" shows "euler_approx n = (\<Sum>k\<le>n. inverse (fact k))"
error_map_def: shows "error_map \<equiv> \<Lambda> e a. \<mu> uu. error_abs oo ssum_map\<cdot>(u_map\<cdot>e)\<cdot>(u_map\<cdot>a) oo error_rep"
assms_2: shows "proots_count (map_poly complex_of_real P) (circle_diam l r) = 0"
succ_in: fixes r :: "('a \<times> 'a) set"   and i :: "'a" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "(i, succ r i) \<in> r"
acute2_suma__nbet: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "Acute A B C"   and "Acute D E F"   and "A B C D E F SumA G H I" shows "\<not> Bet G H I"
WT_gpv_map_gpv': fixes g :: "'d \<Rightarrow> 'g"   and h :: "'h \<Rightarrow> 'e"   and \<I> :: "('g, 'h) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> 'i" assumes "map_\<I> g h \<I> \<turnstile>g gpv \<surd>" shows "\<I> \<turnstile>g map_gpv' f g h gpv \<surd>"
igOpIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igOpIPresIGWls MOD \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igWls MOD (stOf delta) (igOp MOD delta inp binp)"
trace_rest_closure_start: fixes rest1 :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme"   and rest2 :: "('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'h spmf" shows "trace_rest_closure rest1 rest2 IA IU p q p q"
simps_4: fixes I :: "interp"   and \<phi> :: "fml"   and \<psi> :: "fml" shows "fml_sem I (\<phi> && \<psi>) = fml_sem I \<phi> \<inter> fml_sem I \<psi>"
orientable_def: fixes x :: "'a" shows "orientable x \<equiv> \<exists>y. orientation x y"
map_of_remap_insertmap: fixes D :: "'f"   and fs :: "('e \<times> 'g \<times> 'h) list" shows "map_of (map ((\<lambda>((F, D), b, T). (F, D, b, T)) \<circ> (\<lambda>(F, y). ((F, D), y))) fs) = map_of (map (\<lambda>(F, y). (F, D, y)) fs)"
P_def: shows "P \<equiv> \<lambda>x n. x [^]\<^bsub>R\<^esub> CARD('a) ^ n = x"
deutsch_transform_dim_1: shows "dim_row U\<^sub>f = 4"
det_four_block_mat_upper_right_zero_col: fixes A1 :: "'b mat"   and n :: "nat"   and A2 :: "'b mat"   and A3 :: "'b mat"   and A4 :: "'b mat" assumes "A1 \<in> carrier_mat n n"   and "A2 = 0\<^sub>m n 1"   and "A3 \<in> carrier_mat 1 n"   and "A4 \<in> carrier_mat 1 1" shows "det (four_block_mat A1 A2 A3 A4) = det A1 * det A4"
distinct_15: fixes x5 :: "String.literal" shows "Boolean \<noteq> Class x5"
APPEND_EQ_APPEND_MID: fixes l1 :: "'b list"   and e :: "'b"   and l2 :: "'b list"   and m1 :: "'b list"   and m2 :: "'b list" shows "(l1 @ [e] @ l2 = m1 @ m2) = ((\<exists>l. m1 = l1 @ [e] @ l \<and> l2 = l @ m2) \<or> (\<exists>l. l1 = m1 @ l \<and> m2 = l @ [e] @ l2))"
inverts_mat_unique: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat"   and C :: "'c mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "inverts_mat A B"   and "inverts_mat A C" shows "B = C"
axioms_4: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "valid_ownership \<S> ts" shows "ownership_distinct ts"
sin_ms_aux'_def: fixes xs :: "('b \<times> real) msllist" shows "sin_ms_aux' xs = times_ms_aux xs (powser_ms_aux (msllist_of_msstream sin_series_stream) (times_ms_aux xs xs))"
prems_7: fixes x :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and y :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "x \<in> set (fst (gb_schema_dummy (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data')))"   and "y \<in> set (fst (gb_schema_dummy (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data')))"   and "(x, y) \<notin>\<^sub>p snd (gb_schema_dummy (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data'))"   and "fst x \<noteq> 0"   and "fst y \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` set (fst (gb_schema_dummy (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data')))) (fst x) (fst y)"
pos_oriented_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "pos_oriented_cmat H = (let (A, B, C, D) = H in 0 < Re A \<or> is_imag A \<and> (B \<noteq> 0 \<and> 0 < arg B \<or> B = 0 \<and> 0 < Re D))"
\<alpha>_\<beta>: shows "LeftDerivation \<alpha> (take n D) \<beta>"
face_cycle_pred_succ: fixes a :: "'b" shows "G.face_cycle_pred (G.face_cycle_succ a) = a"
simple_group_def: fixes G :: "('e, 'f) monoid_scheme" shows "simple_group G \<equiv> Group.group G \<and> simple_group_axioms G"
tappend_TNil2: fixes xs :: "('a, 'b) tllist" shows "tappend xs TNil = xs"
eventually_at_split: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "class.linorder_topology less_eq less open" shows "eventually P (topological_space.at_within open x UNIV) = (eventually P (topological_space.at_within open x (ord.lessThan less x)) \<and> eventually P (topological_space.at_within open x (ord.greaterThan less x)))"
fds_converges_def: fixes f :: "'b fds"   and s :: "'b" shows "fds_converges f s = summable (\<lambda>n. fds_nth f n / nat_power n s)"
inj_on_one_iff: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "inj_on h (carrier G) = (\<forall>x. x \<in> carrier G \<longrightarrow> h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>)"
closedin_topspace_empty: fixes T :: "'c topology"   and S :: "'c set" assumes "topspace T = {}" shows "closedin T S = (S = {})"
fa: shows "slp_of_fa fa M slp = (M1, slp1)"
gallery_chamber_system: fixes Cs :: "'a set list" assumes "gallery Cs" shows "order.greater_eq \<C> (set Cs)"
diag_block_upper_triangular: fixes As :: "'b mat list"   and j :: "nat"   and i :: "nat" assumes "\<And>A i j. \<lbrakk>A \<in> set As; j < i; i < dim_row A\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'b)"   and "Ball (set As) square_mat"   and "j < i"   and "i < dim_row (diag_block_mat As)" shows "diag_block_mat As $$ (i, j) = (0::'b)"
iMODb2_set_UN_set_eq: shows "iMODb2_set = iMODb2_UN_set"
nop_bij_vrange: shows "\<R>\<^sub>\<circ> f = A"
wrapA_unwrapA_id: shows "wrapA oo unwrapA = ID"
well_def_wp_healthy: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "well_def prog" shows "healthy (wp prog)"
lm084: fixes X :: "'b set"   and Y :: "'b set" assumes "finite X"   and "finite Y"   and "card (X \<inter> Y) = card X" shows "X \<subseteq> Y"
neq_dP: fixes p :: "'c ref"   and q :: "'c ref"   and h :: "'c \<Rightarrow> 'c ref"   and Ps :: "'c list" assumes "p \<noteq> q"   and "Path h p Ps q"   and "distinct Ps" shows "\<exists>a Qs. p = Ref a \<and> Ps = a # Qs \<and> a \<notin> set Qs"
RL: fixes ua :: "'a"   and pa :: "('a \<times> 'a) list"   and va :: "'a" assumes "gen_iapath V ua pa va" shows "gen_iapath V va (rev_path pa) ua"
countableI': fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "inj_on f S" shows "countable S"
assms_4: fixes formula :: "('a, 'b, 'c) formula"   and act :: "'c" assumes "P formula" shows "P (Act act formula)"
assms_5: shows "snd (rep_nat_term a) = snd (rep_nat_term b)"
all_final_exceptD: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and Ts :: "'c set"   and t :: "'c" assumes "final_thread.all_final_except final s Ts"   and "final_thread.not_final_thread final s t" shows "t \<in> Ts"
ereal_bot: fixes x :: "ereal" assumes "\<And>B. x \<le> ereal B" shows "x = - \<infinity>"
stream_enc_enc_some: fixes I :: "(nat + nat set) list"   and w :: "'a list" assumes "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "stream_enc (w, I) = (SOME v. v \<in> enc (w, I)) @- sconst (any, replicate (length I) False)"
interv_sum_list_conv_sum_set_int: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "int \<Rightarrow> 'b"   and k :: "int"   and l :: "int" assumes "class.monoid_add plus zero" shows "sum_list (map f [k..l]) = sum f (set [k..l])"
induct: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x1 :: "'a language"   and x2 :: "'a language"   and P :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool" assumes "Plus_cong R x1 x2"   and "\<And>x y. x = y \<Longrightarrow> P x y"   and "\<And>x y. R x y \<Longrightarrow> P x y"   and "\<And>x y. \<lbrakk>Plus_cong R x y; P x y\<rbrakk> \<Longrightarrow> P y x"   and "\<And>x y z. \<lbrakk>Plus_cong R x y; P x y; Plus_cong R y z; P y z\<rbrakk> \<Longrightarrow> P x z"   and "\<And>x y x' y'. \<lbrakk>Plus_cong R x y; P x y; Plus_cong R x' y'; P x' y'\<rbrakk> \<Longrightarrow> P (Plus x x') (Plus y y')" shows "P x1 x2"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "l_known_ptr\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t.a_known_ptr ptr = (CharacterDataClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some document_ptr \<Rightarrow> is_document_ptr document_ptr))"
add_le_same_cancel1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus b a) b = less_eq a zero"
orderE: fixes a :: "'e multiset"   and b :: "'e multiset" assumes "a \<subseteq># b" obtains "a = a \<inter># b"
value_is_compatible_with_structure_def: fixes f :: "'a trm \<Rightarrow> 'a trm" shows "value_is_compatible_with_structure f = (\<forall>t s. f (t \<cdot> s) = f (f t \<cdot> f s))"
in_segment_inner_rot: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> {a--b}" shows "(x - a) \<bullet> rev.rot (b - a) = 0"
tSplice_es: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "tLength t = tLength t'" shows "es (tLast (t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t')) = es (tLast t)"
funpow_prop: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c"   and n :: "nat"   and x :: "'c" assumes "\<And>x. P (f x) = P x" shows "P ((f ^^ n) x) = P x"
Gromov_completion_dist_comparison_2: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "ereal (dist x y) \<le> eexp (ereal (- epsilonG TYPE('a)) * extended_Gromov_product_at basepoint x y)"
setVar_pState_inv: fixes prog :: "program"   and p :: "pState"   and v :: "varRef"   and val :: "integer"   and g :: "'a gState_scheme" assumes "pState_inv prog p" shows "pState_inv prog (snd (setVar v val g p))"
WeakI: fixes Gamma :: "formula list"   and R :: "(formula list \<times> formula list set) set"   and A :: "formula" assumes "Gamma \<in> deductions R"   and "Weaks \<subseteq> R" shows "A # Gamma \<in> deductions R"
simps_2: fixes e :: "'a edge_kind"   and es :: "'a edge_kind list"   and s :: "'a" shows "transfers (e # es) s = transfers es (transfer e s)"
sem\<^sub>e\<^sub>s\<^sub>t_c_decomp_rm\<^sub>e\<^sub>s\<^sub>t_deduct_aux: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> A"   and "t \<in> ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"   and "t \<notin> ik\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>" shows "ik\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t"
allAllocationsIntersectionSubset: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> allAllocations'' N G"
tc_lower: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "tt p \<prec>\<^sub>t v" shows "tc (lower p v) = tc p"
ring_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "ring_ow U plus zero minus uminus times \<equiv> semiring_ow U plus times \<and> ab_group_add_ow U plus zero minus uminus"
assms_1: fixes y :: "'a" shows "cont (\<lambda>x. c x y)"
T_vrestr_0_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "(T ^^ 0) --` A = A"
trnr\<^sub>\<epsilon>_def: fixes u :: "'a"   and \<nu> :: "'a" shows "trnr\<^sub>\<epsilon> u \<nu> \<equiv> \<r>[u] \<cdot> (u \<star> \<epsilon>) \<cdot> \<a>[u, f, g] \<cdot> (\<nu> \<star> g)"
disjI2_PL: fixes A :: "'a set"   and q :: "'a"   and p :: "'a" assumes "A \<turnstile> q" shows "A \<turnstile> p .\<or>. q"
HPhase1or2ReadThen_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4a s q"   and "Inv2b s" shows "HInv4a s' q"
seq_compact_Int_closed: fixes s :: "'b set"   and t :: "'b set" assumes "seq_compact s"   and "closed t" shows "seq_compact (s \<inter> t)"
distinct_5: fixes x1 :: "'a" shows "prot_atom.Atom x1 \<noteq> AttackType"
fsmult_Gmult_comm: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and g :: "'a"   and v :: "'c"   and a :: "'b" assumes "FGModule G smult V"   and "g \<in> G"   and "v \<in> V" shows "aezfun_scalar_mult.fsmult smult a (aezfun_scalar_mult.Gmult smult g v) = aezfun_scalar_mult.Gmult smult g (aezfun_scalar_mult.fsmult smult a v)"
method_exit_inner_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFGExit.method_exit sourcenode kind valid_edge Exit n"   and "CFG.intra_path sourcenode targetnode kind valid_edge n as n'" shows "as = []"
dagger_unfoldr_eq: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x = x\<^sup>\<star>"
lim_i_2: shows "Liminf_upto_llist (lmap snd TNs) (enat i) = lnth (lmap snd TNs) i"
inverse_divide: 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 a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "inverse (divide a b) = divide b a"
map: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x :: "'d" shows "map_phantom f1 f2 (Phantom('c) x) = Phantom('a) (f2 x)"
to_nat_inj: fixes a :: "'a"   and b :: "'a" assumes "Bij_Nat.to_nat a = Bij_Nat.to_nat b" shows "a = b"
Domain_fst: fixes r :: "('h \<times> 'i) set" shows "Domain r = fst ` r"
hseq_char': fixes \<nu> :: "'b"   and \<mu> :: "'b" shows "B.hseq \<nu> \<mu> = (\<nu> \<star>\<^sub>B \<mu> \<noteq> B.null)"
l_adopt_node_wf_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_adopt_node_wf type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes known_ptrs adopt_node \<equiv> l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes \<and> l_known_ptrs known_ptr known_ptrs \<and> l_adopt_node_wf_axioms type_wf heap_is_wellformed get_child_nodes get_disconnected_nodes known_ptrs adopt_node"
uncountable_infinite: fixes A :: "'a set" assumes "uncountable A" shows "infinite A"
uppert_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "??.Jordan_Normal_Form_Existence.uppert A i j \<equiv> j < i \<longrightarrow> A $$ (i, j) = (0::'b)"
floor_less_numeral: fixes x :: "'b"   and v :: "num" shows "(\<lfloor>x\<rfloor> < numeral v) = (x < numeral v)"
weakCongSimClosedAux: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "eqvt Rel"   and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<leadsto>\<guillemotleft>Rel\<guillemotright> p \<bullet> Q"
preSetpreList_eq: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and C :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList xs C l s = preSet (set xs) C l s"
\<iota>_\<o>: fixes t :: "'s" assumes "t \<in> range \<iota>" shows "\<iota> (\<o> t) = t"
connected_root_move_root: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "connected_root r x"   and "q \<le> x\<^sup>\<star> ; r" shows "connected_root q x"
comm_groupE_3: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"