hcju/isabelleps huggingface dataset:
nonzero_memE_1: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "domain R"   and "a \<in> nonzero R" shows "a \<in> carrier R"
times_iff: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V" shows "(\<langle>a, b\<rangle> \<in>\<^sub>\<circ> A \<times>\<^sub>\<circ> B) = (a \<in>\<^sub>\<circ> A \<and> b \<in>\<^sub>\<circ> B)"
kripke_def: fixes M :: "('d, 'e, 'f) KripkeStructure" shows "kripke M \<equiv> \<forall>a. relations M a \<subseteq> worlds M \<times> worlds M"
sp_append: fixes F :: "'q \<Rightarrow> 'r fundef option"   and xs :: "('s, 't, 'q, 'e, 'f, 'g, 'u, 'v) instr list"   and ys :: "('s, 't, 'q, 'e, 'f, 'g, 'u, 'v) instr list"   and \<Sigma> :: "Unboxed.type option list" shows "sp F (xs @ ys) \<Sigma> = sp F xs \<Sigma> \<bind> sp F ys"
of_nat_simps_5: fixes m :: "nat"   and n :: "nat" shows "of_nat (m * n) = of_nat m * of_nat n"
parallel_cases: fixes V :: "('exp, 'id) MWLfCom list" assumes "\<turnstile>\<^bsub>\<V>\<^esub> V" obtains "\<forall>i<length V. \<turnstile>\<^bsub>\<C>\<^esub> V ! i"
fsubsetD: fixes A :: "'x fset"   and B :: "'x fset"   and c :: "'x" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
step_assumption: fixes P :: "'a llist"   and v0 :: "'a" assumes "vmc_path_no_deadend G P v0 p \<sigma>"   and "v0 \<in> S"   and "Q P" shows "Q (ltl P) \<and> vmc_path_no_deadend.w0 P \<in> S"
DBM_reset_complete_empty: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and c :: "'c"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c"   and "DBM_reset (FW M n) n (v c) d M'"   and "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> FW M n" shows "\<not> u \<turnstile>\<^bsub>v,n\<^esub> M'"
powsum_prod: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "x\<^bsub>0\<^esub>\<^bsup>m\<^esup> \<cdot> x\<^bsub>0\<^esub>\<^bsup>n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>m + n\<^esup>"
relations_def: fixes A :: "'a set" shows "relations A = {f. (\<Otimes>a\<in>A. a [^] f a) = \<one>} \<inter> extensional A"
divmod_poly_one_main_i: fixes n :: "nat"   and Y :: "'a list"   and x :: "'i list"   and X :: "'a list"   and y :: "'i list"   and z :: "'i list"   and Z :: "'a list"   and N :: "nat" assumes "n \<le> length Y"   and "list_all2 R x X"   and "list_all2 R y Y"   and "list_all2 R z Z"   and "n = N" shows "rel_prod (list_all2 R) (list_all2 R) (divmod_poly_one_main_i ops x y z n) (divmod_poly_one_main_list X Y Z N)"
cases: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" assumes "upRulesp a1 a2" obtains c :: "('b, 'c) sequent" and R :: "'b" and Fs :: "('b, 'c) form list" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "ModalSequents.mset c = \<LM> Compound R Fs  \<RM>" and "ps \<noteq> []"
ht_size_def: fixes l :: "'c list list"   and n :: "nat" shows "ht_size l n \<equiv> n = sum_list (map length l)"
assms_2: shows "older_seniors x n \<subset> older_seniors y n"
lincomb_in_span: fixes X :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "X \<subseteq> carrier_vec n" shows "lincomb a X \<in> local.span X"
inj_on_inverseI: fixes A :: "'a set"   and g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
invertible_bij: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<forall>x. g (f x) = x"   and "\<forall>y. f (g y) = y" shows "bij f"
hindrance: fixes a :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "a \<in> A \<Gamma>"   and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "d_OUT f a < weight \<Gamma> a" shows "hindrance \<Gamma> f"
antimonoI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f y \<le> f x" shows "order.antimono less_eq f"
arev_eq: fixes a :: "'b" assumes "a \<notin> arcs G" shows "rev_G a = a"
truncate_up_mono: fixes x :: "real"   and y :: "real"   and p :: "nat" assumes "x \<le> y" shows "truncate_up p x \<le> truncate_up p y"
bidel_arc_head_isolated_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc_head_isolated G M a \<equiv> bidel_arc G M a \<and> bidel_arc_head_isolated_axioms G M a"
\<phi>: fixes F :: "nat \<Rightarrow> 'a set" assumes "\<And>n. closed (F n)"   and "\<And>n. \<phi> (F n)"   and "\<And>n. F (Suc n) \<subseteq> F n" shows "\<phi> (\<Inter> (range F))"
rep_eq: fixes x :: "'b regular"   and xa :: "'b regular" shows "Rep_regular (x \<sqinter> xa) = Rep_regular x \<sqinter> Rep_regular xa"
qPsubstInp_def: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" shows "qinp %[[qrho]] \<equiv> lift (qPsubst qrho) qinp"
hyp: shows "\<forall>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> \<Gamma>\<turnstile>cs,css \<Down> t \<and> \<not> AlternativeSmallStep.inf \<Gamma> cs css t"
zmultiset_eqI: fixes A :: "'b zmultiset"   and B :: "'b zmultiset" assumes "\<And>x. zcount A x = zcount B x" shows "A = B"
Least_Hermite_of_row_i2: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and k :: "'a"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<not> is_zero_row i A"   and "\<not> is_zero_row k A"   and "echelon_form A"   and "ass_function ass" shows "(LEAST n. Hermite_of_row_i ass res A k $ i $ n \<noteq> (0::'b)) = (LEAST n. A $ i $ n \<noteq> (0::'b))"
acomplete_if_wbalanced: fixes t :: "'a tree" assumes "wbalanced t" shows "acomplete t"
ordLeq_Func: fixes b1 :: "'a"   and b2 :: "'a"   and B :: "'a set"   and A :: "'b set" assumes "{b1, b2} \<subseteq> B"   and "b1 \<noteq> b2" shows "|A| \<le>o |Func A B|"
DF_SYSTEM: shows "DF UNIV \<sqsubseteq>\<^sub>F\<^sub>D SYSTEM"
subst_clss_single: fixes C :: "'a literal multiset"   and \<sigma> :: "'s" shows "{C} \<cdot>cs \<sigma> = {C \<cdot> \<sigma>}"
assms_5: shows "cis \<alpha> = cis \<alpha>'"
cmp_components_are_iso: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.iso (\<Phi> (f, g))"
C: shows "\<forall>x\<in>set xs. mi \<le> key x \<and> key x \<le> ma"
simps: fixes x :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" shows "neville_aitken_main x f i j = (if i < j then smult (inverse (x j - x i)) (X x i * neville_aitken_main x f (i + 1) j - X x j * neville_aitken_main x f i (j - 1)) else [:f i:])"
bind_cong: fixes x :: "'a option"   and y :: "'a option"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "x \<bind> f = y \<bind> g"
RedCastFail: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and v :: "'addr val"   and U :: "ty"   and P :: "'addr J_prog"   and T :: "ty"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id" assumes "typeof\<^bsub>hp s\<^esub> v = \<lfloor>U\<rfloor>"   and "\<not> P \<turnstile> U \<le> T" shows "extTA,P,t \<turnstile> \<langle>Cast T (Val v),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ClassCast),s\<rangle>"
eq_iff: fixes x :: "'a"   and y :: "'a"   and ko :: "'a key_order" shows "(x = y) = (le_of_key_order ko x y \<and> le_of_key_order ko y x)"
trunc_history: fixes xs :: "'a list"   and x :: "'a" assumes "op_history (xs @ [x])" shows "op_history xs"
enum_ex_prot_atom_def: fixes P :: "'b prot_atom \<Rightarrow> bool" shows "enum_class.enum_ex P \<equiv> list_ex P (map prot_atom.Atom enum_class.enum @ [Value, SetType, AttackType, Bottom, OccursSecType])"
lens_override_def: fixes X :: "'f \<Longrightarrow> 'e"   and S\<^sub>1 :: "'e"   and S\<^sub>2 :: "'e" shows "S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 = put\<^bsub>X\<^esub> S\<^sub>1 (get\<^bsub>X\<^esub> S\<^sub>2)"
otherassms_3: shows "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"
eq_zero_iff: fixes x :: "nat" shows "(\<Phi> x = 0) = (\<not> coprime x d)"
bex_empty: fixes P :: "'g \<Rightarrow> bool" shows "Bex {} P = False"
emb: shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s"
eq_obs_intra_in_rv: fixes n :: "'node SDG_node"   and S :: "'node SDG_node set"   and n' :: "'node SDG_node"   and x :: "'var" assumes "obs_intra (parent_node n) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = obs_intra (parent_node n') \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "x \<in> rv S n" shows "x \<in> rv S n'"
bounded_clinearI: fixes f :: "'b \<Rightarrow> 'c"   and K :: "real" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b"   and "\<forall>x. norm (f x) \<le> norm x * K" shows "bounded_clinear f"
inf_right_idem: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<sqinter> y = x \<sqinter> y"
C1_def: fixes \<psi> :: "'a ell2" shows "C1 \<psi> = Abs_ell2 (C0 (Rep_ell2 \<psi>))"
nodes_list_it_correct: fixes g :: "('a, 'b, 'c) gen_cfg_wf" assumes "True" shows "set_iterator (foldri (gen_wf_\<alpha>n g)) (nodes (gen_wf.\<alpha> g))"
unique_Nb: fixes B :: "agent"   and M :: "nat"   and A :: "agent"   and X :: "msg"   and Nb :: "nat"   and evs :: "event list"   and M' :: "nat"   and A' :: "agent"   and X' :: "msg" assumes "Says B Server \<lbrace>Nonce M, Agent A, Agent B, X, Crypt (shrK B) \<lbrace>Nonce Nb, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"   and "Says B Server \<lbrace>Nonce M', Agent A', Agent B, X', Crypt (shrK B) \<lbrace>Nonce Nb, Nonce M', Nonce M', Agent A', Agent B\<rbrace>\<rbrace> \<in> set evs"   and "B \<notin> bad"   and "evs \<in> orb" shows "M = M' \<and> A = A' \<and> X = X'"
fps_X_dvd_fps_ln: fixes c :: "'a" shows "fps_X dvd fps_ln c"
f: shows "f \<in> K \<rightarrow>\<^sub>M R"
red_ord: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F p q" shows "punit.ord_strict_p q p"
induct: fixes P :: "(int \<times> int) list \<Rightarrow> bool"   and a0 :: "(int \<times> int) list" assumes "P []"   and "\<And>p\<^sub>0. P [p\<^sub>0]"   and "\<And>p\<^sub>0 p\<^sub>1. P [p\<^sub>0, p\<^sub>1]"   and "\<And>p\<^sub>0 v vb vc. P (v # vb # vc) \<Longrightarrow> P (p\<^sub>0 # v # vb # vc)" shows "P a0"
empty_table: fixes n :: "nat"   and V :: "nat set" shows "table n V empty_table"
not_is_Pure_conv: fixes r :: "('d, 'e, 'f) generat" shows "(\<not> is_Pure r) = (\<exists>out c. r = IO out c)"
linum: fixes t :: "Reflection_Examples.num"   and bs :: "nat list" shows "Inum (linum t) bs = Inum t bs"
promela_props_ltl_map_aprops: fixes \<phi> :: "expr ltlc"   and APs :: "expr iarray"   and \<phi>\<^sub>i :: "nat ltlc" assumes "ltl_convert \<phi> = (APs, \<phi>\<^sub>i)" shows "promela_props_ltl APs = map_props (idx (IArray.list_of APs)) \<circ> promela_props"
summable_LIMSEQ: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "(\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> suminf f"
inv_solve_right': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier \<G>"   and "b \<in> carrier \<G>"   and "c \<in> carrier \<G>" shows "(b \<otimes> inv c = a) = (b = a \<otimes> c)"
env_restr_below_self: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "f f|` S \<sqsubseteq> f"
numadd_nb: fixes t :: "Cooper.num"   and s :: "Cooper.num" assumes "numbound0 t"   and "numbound0 s" shows "numbound0 (numadd t s)"
f10: shows "\<not> (\<exists>(a', b', c')\<in>set d. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
\<Phi>_def: fixes a :: "presb" shows "\<Phi> a = (case a of presb.Eq is n z \<Rightarrow> {FBase (presb.Eq is n i) |i. i \<in> {min (- h is) (min n z)..max (- l is) (max n z)}} \<union> {FBool False})"
simple_match_valid_fbs: fixes rt :: "('b, 'c) routing_rule_scheme list"   and fw :: "'b simple_rule list"   and ifs :: "'d" assumes "valid_prefixes rt"   and "simple_fw_valid fw" shows "list_all simple_match_valid (map fst (lr_of_tran_fbs rt fw ifs))"
rel_spmf_characterisation_axioms: shows "rel_spmf_characterisation TYPE('c) TYPE('d)"
rel_induct: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c list_R1"   and y :: "'d list_R1"   and Q :: "'c list_R1 \<Rightarrow> 'd list_R1 \<Rightarrow> bool" assumes "rel_list_R1 R x y"   and "\<And>a b. rel_sum (=) (rel_prod R Q) a b \<Longrightarrow> Q (list_R1 a) (list_R1 b)" shows "Q x y"
mult_col_div_row_carrier_1: fixes a :: "'b"   and k :: "nat"   and A :: "'b mat" shows "dim_row (mult_col_div_row a k A) = dim_row A"
kbo_std_basis_axioms: shows "kbo_std_basis ground_heads_var (>\<^sub>s) (\<lambda>_. \<infinity>) (\<lambda>_. \<infinity>) wt_sym \<epsilon> 0 extf"
span_scale: fixes x :: "'a poly"   and S :: "'a poly set"   and c :: "'a" assumes "x \<in> poly_vs.span S" shows "smult c x \<in> poly_vs.span S"
lam_in_hom: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and g :: "'a" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>g : a \<otimes> b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>\<Lambda> a b c g : a \<rightarrow> exp b c\<guillemotright>"
dickson_gradingD2: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat" assumes "dickson_grading d" shows "almost_full_on (adds) {x. d x \<le> n}"
\<sigma>_3: shows "subst_range \<sigma> \<subseteq> (\<lambda>c. Fun c []) ` ({f. public f} \<inter> \<C>) - T"
2: shows "Broadcast (i2, Insert e2 n2) \<in> set (history k)"
of_nat_less_two_power: fixes n :: "nat" shows "of_nat n < (2::'a) ^ n"
ae_g: fixes x :: "'a" shows "AE y in g x. Sigma_Algebra.measure (g x) {y} \<noteq> 0"
simps: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" shows "rbtreeify_g n kvs = (if n = 0 \<or> n = 1 then (Empty, kvs) else if n mod 2 = 0 then case rbtreeify_g (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow> apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs') else case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow> apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs'))"
facet_of_def: fixes F :: "'b set"   and S :: "'b set" shows "(F facet_of S) = (F face_of S \<and> F \<noteq> {} \<and> aff_dim F = aff_dim S - 1)"
one_always_in: fixes n_vars :: "nat"   and M :: "nat \<Rightarrow> bool" assumes "1 < n_vars" shows "1 \<in> set (model_to_dimacs_model M [1..<n_vars]) \<or> - 1 \<in> set (model_to_dimacs_model M [1..<n_vars])"
cols_sort1: fixes xss :: "'a list list"   and n :: "nat"   and f :: "'a list \<Rightarrow> 'a" assumes "cols xss n" shows "cols (sort1 f xss) n"
preserves_seq: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "seq g f" shows "seq (trg g) (trg f)"
distinct_18: fixes x41 :: "trm"   and x42 :: "trm"   and x21 :: "POPLmarkRecord.type"   and x22 :: "trm" shows "x41 \<bullet> x42 \<noteq> (\<lambda>:x21. x22)"
Tag_Pair: fixes X :: "tag"   and X' :: "msg"   and Y' :: "msg" shows "Tag X \<noteq> \<langle>X', Y'\<rangle>"
card_1_element: fixes E :: "'b set" assumes "card E = 1" shows "\<exists>a. E = {a}"
assms: shows "i < 2 ^ n \<or> j < 2 ^ n"
simps_1: fixes p :: "nat" shows "Pratt_Certificate.size_pratt (Prime p) = log 2 (real p)"
append_simpl2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and i :: "nat"   and ys :: "'c list" assumes "Matrix_Tensor.mult id f" shows "length xs \<le> i \<longrightarrow> (xs @ ys) ! i = ys ! (i - length xs)"
exp_aform_err: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and p :: "nat"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "x \<in> aform_err e X"   and "exp_aform_err p X = Some Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "exp x \<in> aform_err e Y"
cases: fixes x :: "('b, 'a) Term.term" obtains xa :: "'a" where "x = Var xa"   | f :: "'b" and ts :: "('b, 'a) Term.term list" where "x = Fun f ts"
assms_1: shows "Pascal_Property.inter l m \<in> line_ext n"
conjugate_involution: fixes z :: "complex_homo" shows "conjugate (conjugate z) = z"
prod_rel_id: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('b \<times> 'b) set" assumes "R1 = Id"   and "R2 = Id" shows "R1 \<times>\<^sub>r R2 = Id"
g1: fixes z :: "real" assumes "0 \<le> z"   and "z * 2 < 1"   and "z * 2 \<notin> s1" shows "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z) = 2 *\<^sub>R vector_derivative g1 (at (z * 2))"
id_2: shows "M.sumlist (map (\<lambda>i. h i \<cdot>\<^sub>v gs ! i) [0..<Suc k]) = M.sumlist (map (\<lambda>i. h i \<cdot>\<^sub>v gs ! i) [0..<k]) + h k \<cdot>\<^sub>v gs ! k"
inv_equality: fixes G :: "('h, 'i) monoid_scheme"   and y :: "'h"   and x :: "'h" assumes "Group.group G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> x = y"
c1l: shows "set (labels_conv (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) \<subseteq> r \<down>s {fst \<beta>_step, fst \<alpha>_step}"
g': shows "g' = snd (snd (splitFace g v u f ws))"
smult_distrib_right: fixes r :: "'f"   and s :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "s \<in> UNIV"   and "m \<in> V" shows "(r + s) \<cdot> m = r \<cdot> m + s \<cdot> m"
subset: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "semilattice_set f"   and "finite A"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "f (semilattice_set.F f B) (semilattice_set.F f A) = semilattice_set.F f A"
connected_components_of_subset_sing: fixes X :: "'c topology"   and S :: "'c set" shows "(connected_components_of X \<subseteq> {S}) = (connected_space X \<and> (topspace X = {} \<or> topspace X = S))"
mset_subset_eq_add_left: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A \<subseteq># A + B"
pos_cost_pos_awalk_cost: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and c :: "'b \<Rightarrow> real" assumes "awalk u p v"   and "\<And>e. e \<in> arcs G \<Longrightarrow> 0 \<le> c e" shows "0 \<le> awalk_cost c p"
rconjby_lconjby: fixes x :: "'a"   and y :: "'a" shows "rconjby x (lconjby x y) = y"
sameCC_reachable: fixes E' :: "'edge set"   and u :: "'a"   and v :: "'a" assumes "E' \<subseteq> E"   and "u \<in> V"   and "v \<in> V" shows "(CC E' u = CC E' v) = ((u, v) \<in> connected E')"
apply_cltn2_def: fixes x :: "proj2"   and A :: "cltn2" shows "apply_cltn2 x A \<equiv> proj2_abs (proj2_rep x v* cltn2_rep A)"
assms_4: shows "c < (1 - b) * real x\<^sub>1"
nodes_join: fixes g\<^sub>1 :: "'b ugraph"   and g\<^sub>2 :: "'b ugraph" shows "nodes (graph_join g\<^sub>1 g\<^sub>2) = nodes g\<^sub>1 \<union> nodes g\<^sub>2"
cInf_singleton: fixes x :: "'a multiset" shows "Inf {x} = x"
N1_def: shows "N1 = {u. (x, u) \<in> E'\<^sup>*}"
lookup_witness': shows "lookup (tensors_from_net (witness' (rs ! 1) (tl (tl rs))) $ 0) is = (if nths is (Collect even) = nths is {n. odd n} \<and> (\<forall>i\<in>set is. i < last (butlast rs)) then 1 else 0)"
h_def: fixes a :: "'b" shows "h a = (if a = y then 1::'a else g a)"
periodic_finite_ex: fixes d :: "int"   and P :: "int \<Rightarrow> bool" assumes "0 < d"   and "\<forall>x k. P x = P (x - k * d)" shows "(\<exists>x. P x) = (\<exists>j\<in>{0..d - 1}. P j)"
rtranclp_False: shows "(\<lambda>i j. False)\<^sup>*\<^sup>* = (=)"
less_bot_Bot_Value_code: fixes x :: "'b" shows "(bot.Bot < bot.Value x) = True"
simps_2: fixes x :: "int" shows "Challenge1A.take [x] = [x]"
diff_f: shows "f differentiable_on sphere (0::'a) 1 \<inter> S"
rem_UnPath_parity_others: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v"   and n :: "'v" assumes "finite E"   and "is_trail v ps v'"   and "n \<notin> {v, v'}" shows "even (degree n (rem_unPath ps G)) = even (degree n G)"
stable_wtc: fixes pc :: "nat" assumes "stable r step \<tau>s pc"   and "pc < length \<tau>s" shows "wtc c pc (\<tau>s ! pc) \<noteq> \<top>"
infinity_eq_eSuc_iff: fixes n :: "enat" shows "(\<infinity> = eSuc n) = (n = \<infinity>)"
ACI_nPlus_ACI_norm: fixes r :: "'c rexp"   and s :: "'c rexp" shows "ACI_nPlus \<guillemotleft>r\<guillemotright> \<guillemotleft>s\<guillemotright> = \<guillemotleft>Plus r s\<guillemotright>"
eps_def: shows "eps = (MIN i. (A *v pow_A_1 u - sr *s pow_A_1 u) $h i / pow_A_1 u $h i)"
intro: fixes G :: "('c, 'd) graph" assumes "fst ` edges G \<subseteq> nodes G"   and "snd ` snd ` edges G \<subseteq> nodes G" shows "valid_graph G"
jordan_nf_def: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list" shows "jordan_nf A n_as \<equiv> 0 \<notin> fst ` set n_as \<and> similar_mat A (jordan_matrix n_as)"
that: fixes c :: "real" assumes "c \<in> {a..b}"   and "((\<lambda>x. g x * f x) has_integral g a * integral {a..c} f + g b * integral {c..b} f) {a..b}" shows "thesis"
isSimplePath_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isSimplePath c u p v \<equiv> Graph.isPath c u p v \<and> distinct (pathVertices u p)"
is_constant_constructors_simps_1: fixes n :: "nat"   and x :: "'a ltln" shows "(X\<^sub>n' n x = false\<^sub>n) = (x = false\<^sub>n)"
toplevel_summands_lderivs_Pr: fixes xs :: "'a list"   and r :: "'b rexp"   and s :: "'b rexp" assumes "xs \<noteq> []"   and "lderivs xs (Pr r) = Pr s" shows "toplevel_summands s \<subseteq> {Zero} \<or> toplevel_summands s \<subseteq> (\<Union>xs. toplevel_summands (lderivs xs r))"
n_galois1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<sqsubseteq> !y \<oplus> !z) = (!x \<cdot> t y \<sqsubseteq> !z)"
eval_embed_false: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "\<not> P s" shows "\<guillemotleft> P \<guillemotright> s = 0"
lit_ord_irrefl: fixes L :: "'a literal" shows "(L, L) \<notin> lit_ord"
mfinal2_simulation: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "r2.mfinal s2" shows "\<exists>s1'. \<tau>mRed1 s1 s1' \<and> s1' \<approx>m s2 \<and> r1.mfinal s1' \<and> shr s1' = shr s1"
induct: fixes P :: "(nat \<times> 'a) list \<Rightarrow> (nat \<times> 'a) list \<Rightarrow> bool"   and a0 :: "(nat \<times> 'a) list"   and a1 :: "(nat \<times> 'a) list" assumes "P [] []"   and "\<And>uu_ a as. P as [] \<Longrightarrow> P ((uu_, a) # as) []"   and "\<And>uv_ b bs. P [] bs \<Longrightarrow> P [] ((uv_, b) # bs)"   and "\<And>i a as j b bs. \<lbrakk>i < j \<Longrightarrow> P as ((j, b) # bs); \<lbrakk>\<not> i < j; j < i\<rbrakk> \<Longrightarrow> P ((i, a) # as) bs; \<lbrakk>\<not> i < j; \<not> j < i\<rbrakk> \<Longrightarrow> P as bs\<rbrakk> \<Longrightarrow> P ((i, a) # as) ((j, b) # bs)" shows "P a0 a1"
assms_4: shows "op \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
simps_1: fixes h :: "'heap"   and a :: "'addr" shows "typeof\<^bsub>h\<^esub> (Addr a) = map_option ty_of_htype (typeof_addr h a)"
C_def: fixes k :: "nat" shows "C k = {c \<in> ucycles (edge_ugraph S_edges). uwalk_length c = k}"