hcju/isabelleps huggingface dataset:
I: shows "set_of_bv (nfa_steps (quantify_nfa i A) (nfa_startnode (quantify_nfa i A)) bss) = (\<Union>bs\<in>{bs. length bs = length bss}. set_of_bv (nfa_steps A (nfa_startnode (quantify_nfa i A)) (insertll i bs bss)))"
index_less_aux: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "index xs x < n"
join_bottom_1: fixes x :: "'a" shows "x \<squnion> \<bottom> = x"
match_some_induct: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "term"   and u :: "'c"   and env :: "(name, 'c) fmap"   and P :: "term \<Rightarrow> 'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> bool" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.match unapp unconst t u = Some env"   and "\<And>t\<^sub>1 t\<^sub>2 u\<^sub>1 u\<^sub>2 env\<^sub>1 env\<^sub>2. \<lbrakk>P t\<^sub>1 u\<^sub>1 env\<^sub>1; term.match unapp unconst t\<^sub>1 u\<^sub>1 = Some env\<^sub>1; P t\<^sub>2 u\<^sub>2 env\<^sub>2; term.match unapp unconst t\<^sub>2 u\<^sub>2 = Some env\<^sub>2\<rbrakk> \<Longrightarrow> P (t\<^sub>1 $ t\<^sub>2) (app u\<^sub>1 u\<^sub>2) (env\<^sub>1 ++\<^sub>f env\<^sub>2)"   and "\<And>name. P (Const name) (const name) fmempty"   and "\<And>name u. P (Free name) u (fmupd name u fmempty)" shows "P t u env"
signed_nonnegative_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and w :: "'b Word.word" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (ring_1.signed one plus zero minus w) = (0 \<le> sint w)"
hyps_1: fixes mr :: "mregex"   and ms :: "mregex" assumes "r = from_mregex mr \<phi>s"   and "safe_regex Past g (from_mregex mr \<phi>s)"   and "ms \<in> RPD mr" shows "safe_regex Past g (from_mregex ms \<phi>s)"
exhaust: fixes y :: "t" obtains "y = T_i32"   | "y = T_i64"   | "y = T_f32"   | "y = T_f64"
v_def: shows "v = (\<lambda>n x. u n x + birkhoff_sum (\<lambda>x. - u 1 x) n x)"
induct: fixes P :: "('e set \<times> 'e set) list \<Rightarrow> (('g, 'h, 'e) prot_fun, 'f) Term.term \<Rightarrow> (('g, 'h, 'e) prot_fun, 'f) Term.term \<Rightarrow> bool"   and a0 :: "('e set \<times> 'e set) list"   and a1 :: "(('g, 'h, 'e) prot_fun, 'f) Term.term"   and a2 :: "(('g, 'h, 'e) prot_fun, 'f) Term.term" assumes "\<And>TI x y. P TI (Var x) (Var y)"   and "\<And>TI f T g S. (\<And>z yb. \<lbrakk>z \<in> set T; yb \<in> set S\<rbrakk> \<Longrightarrow> P TI z yb) \<Longrightarrow> P TI (Fun f T) (Fun g S)"   and "\<And>uu_ v va vb. P uu_ (Fun v va) (Var vb)"   and "\<And>uu_ vb v va. P uu_ (Var vb) (Fun v va)" shows "P a0 a1 a2"
oalist_of_list_ntm_def: shows "oalist_of_list_ntm = OAlist_ntm"
sound_monitor: fixes i :: "nat"   and v :: "'b option list"   and \<pi> :: "'a prefix"   and \<sigma> :: "'a trace" assumes "(i, v) \<in> M \<pi>"   and "prefix_of \<pi> \<sigma>" shows "sat \<sigma> (map the v) i"
preservation1_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y \<le> x \<cdot> y \<cdot> x"   and "y \<cdot> x \<le> y" shows "(x \<cdot> y)\<^sup>\<star> \<cdot> x = x \<cdot> y\<^sup>\<star>"
distinct_30: fixes pi1' :: "pi"   and pi2' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "pi1' \<parallel> pi2' \<noteq> name1{name2}.pi"
valid_shallow_model': fixes Z :: "nat"   and M :: "nat"   and N :: "nat" shows "valid_net (shallow_model' Z M N)"
invariant_1_pos_6: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "invariant_1 (p, l, r)"   and "0 < l" shows "0 < r"
dim_solution_set_not_zero_imp_infinite_solutions_homogeneous: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "0 < vec.dim (solution_set A 0)" shows "infinite (solution_set A 0)"
rewrite_firstE: fixes cs :: "(Term_Class.term \<times> 'a) list"   and t :: "'a"   and t' :: "'a" assumes "rewrite_first cs t t'" obtains pat :: "Term_Class.term" and rhs :: "'a" and env :: "(name, 'a) fmap" where "(pat, rhs) \<in> set cs" and "match pat t = Some env" and "t' = subst rhs env"
assms: shows "Case_Labeling.CTXT IC CT OC (ovalidNF P (g x) Q)"
pell_valuation_mult: fixes z1 :: "int \<times> int"   and z2 :: "int \<times> int" shows "pell_valuation (pell_mul z1 z2) = pell_valuation z1 * pell_valuation z2"
YFtorNTMor: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "YFtorNT C f \<in> mor\<^bsub>CatExp (Op C) SET\<^esub>"
anonymous_profile_update: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and Ri' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pref_profile_wf agents alts R"   and "i \<in> agents"   and "finite agents"   and "total_preorder_on alts Ri'" shows "anonymous_profile (R(i := Ri')) = anonymous_profile R - {#weak_ranking (R i)#} + {#weak_ranking Ri'#}"
boolean_algebra_axioms: shows "boolean_algebra (AND) (OR) NOT (0::'a) (- (1::'a))"
lsc_hull_ex_unique: fixes f :: "'b \<Rightarrow> ereal" shows "\<exists>!g. Epigraph UNIV g = closure (Epigraph UNIV f)"
m_assoc: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f"   and y :: "'f"   and z :: "'f" assumes "Group.monoid 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)"
WT_callee: fixes s :: "'s" assumes "I s" shows "\<I>r \<turnstile>c right s \<surd>"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>x. \<lbrakk>P x; P (Suc x)\<rbrakk> \<Longrightarrow> P (Suc (Suc x))" shows "P a0"
lcm_mod_ring_def: shows "lcm = normalization_euclidean_semiring_class.lcm"
full_chain_tranclp_imp_exists_full_chain: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "full_chain R\<^sup>+\<^sup>+ xs" shows "\<exists>ys. full_chain R ys \<and> emb xs ys \<and> lhd ys = lhd xs \<and> llast ys = llast xs"
ldrop_inf: shows "llength (ldrop (enat k) Sts) = \<infinity> \<and> \<not> lfinite (ldrop (enat k) Sts)"
distinct_6: fixes x3 :: "error"   and x2 :: "jump" shows "Error x3 \<noteq> Jump x2"
galois_aux: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y = 0) = (x \<le> - y)"
idem_FD: fixes P :: "'a process" shows "P \<sqsubseteq>\<^sub>F\<^sub>D P"
add_less_same_cancel2: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus a b) b = less a zero"
analytic_imp_holomorphic': fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and t :: "complex set" assumes "f analytic_on s"   and "t \<subseteq> s" shows "f holomorphic_on t"
sgn_divide: fixes a :: "'a"   and b :: "'a" shows "sgn (a / b) = sgn a / sgn b"
lq: shows "iszlfm (And (Dvd (\<zeta> (zlfm p)) (CN 0 1 (C 0))) (a_\<beta> (zlfm p) (\<zeta> (zlfm p))))"
splus_mono_left: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s \<preceq> t" shows "s \<oplus> v \<preceq>\<^sub>t t \<oplus> v"
new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr" assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')" shows "get ptr h = get ptr h'"
assms: shows "run_one_step d i (s, vs, ves, $Br_table js j) = (s', vs', res)"
X_minus_plus: fixes R :: "('e, 'f) ring_scheme"   and a :: "'e" assumes "UP_cring R"   and "a \<in> carrier R" shows "X_poly_minus R a = X_poly_plus R (\<ominus>\<^bsub>R\<^esub> a)"
a_le_suc_n: shows "mod_type_class.to_nat a \<le> Suc n"
echelon_form_of_column_k_efficient: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and k :: "nat" shows "echelon_form_of_column_k bezout (A, i) k = echelon_form_of_column_k_efficient bezout (A, i) k"
g_to_list_def: fixes m :: "'s" shows "g_to_list m \<equiv> iterate m (#) []"
pair_sigma_finite_def: fixes M1 :: "'b measure"   and M2 :: "'c measure" shows "pair_sigma_finite M1 M2 \<equiv> sigma_finite_measure M1 \<and> sigma_finite_measure M2"
choose_int_suc: fixes n :: "nat"   and i :: "int" shows "choose_int (Suc n) i = choose_int n (i - 1) + choose_int n i"
plurality_rule_def: fixes A :: "'c set"   and Is :: "'d set"   and F :: "'d \<Rightarrow> 'c" shows "plurality_rule A Is F \<equiv> {x \<in> A. \<forall>y\<in>A. card {i \<in> Is. F i = y} \<le> card {i \<in> Is. F i = x}}"
simps_2: fixes x :: "'b memref"   and xs :: "'b memref list"   and S :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" shows "release (x # xs) S \<R> = (case x of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then release xs (S \<union> R - L) Map.empty else release xs S \<R> | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> release xs (S \<union> R - L) (augment_rels S R \<R>) | _ \<Rightarrow> release xs S \<R>)"
comm_neutral: fixes a :: "'a" shows "a \<squnion> bot = a"
root_is_take: fixes w :: "'b list"   and r :: "'b list" assumes "w \<le>p r\<^sup>\<omega>"   and "\<^bold>|r\<^bold>| \<le> \<^bold>|w\<^bold>|" shows "r \<le>p w"
clauseContainsItsLiteralsVariable: fixes literal :: "Literal"   and clause :: "Literal list" assumes "literal el clause" shows "var literal \<in> vars clause"
OF_match_fields_safe_def: fixes m :: "of_match_field set" shows "OF_match_fields_safe m \<equiv> (the \<circ>\<circ> OF_match_fields) m"
mds_consistent_def: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" shows "mds_consistent mds \<Gamma> \<S> P \<equiv> \<S> = (mds AsmNoWrite, mds AsmNoReadOrWrite) \<and> dom \<Gamma> = {x. x \<notin> \<C> \<and> stable \<S> x} \<and> type_stable \<S> P"
suc_above_AboveS: fixes B :: "'b set" assumes "B \<subseteq> Field s"   and "s.AboveS B \<noteq> {}" shows "s.AboveS B = s.above (s.suc B)"
FD_F: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q" shows "P \<sqsubseteq>\<^sub>F Q"
C1: fixes y :: "name"   and R :: "ccs" assumes "y \<sharp> R" shows "(\<lparr>\<nu>y\<rparr>R, R) \<in> Rel"
lfilter_eq_LNil: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(lfilter P xs = LNil) = (\<forall>x\<in>lset xs. \<not> P x)"
prv_dsj_mono: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (imp \<phi>1 \<chi>1)"   and "prv (imp \<phi>2 \<chi>2)" shows "prv (imp (dsj \<phi>1 \<phi>2) (dsj \<chi>1 \<chi>2))"
continuous_on_imp_piecewise_continuous_on: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on {a..b} f" shows "piecewise_continuous_on a b {} f"
simps: fixes a1 :: "'g LDCFG_node"   and a2 :: "'h"   and Def :: "'g \<Rightarrow> 'h set"   and E :: "'g"   and X :: "'g"   and H :: "'h set"   and L :: "'h set" shows "((a1, a2) \<in> lift_Def_set Def E X H L) = ((\<exists>V n. a1 = Node n \<and> a2 = V \<and> V \<in> Def n) \<or> (\<exists>V. a1 = Node E \<and> a2 = V \<and> V \<in> H))"
has_integral_const_real: fixes c :: "'b"   and a :: "real"   and b :: "real" shows "((\<lambda>x. c) has_integral Henstock_Kurzweil_Integration.content {a..b} *\<^sub>R c) {a..b}"
assms_2: shows "(minarc\<^sub>M f) d \<noteq> bot"
empty_state_def: fixes e :: "'c" shows "empty_state e \<equiv> \<lparr>counter = 0, discovered = Map.empty, finished = Map.empty, pending = {}, stack = [], tree_edges = {}, back_edges = {}, cross_edges = {}, \<dots> = e\<rparr>"
bou: shows "bounded (range (\<lambda>k. integral (cbox a b) (f k)))"
cases: fixes x :: "mode" obtains "x = Nop"   | "x = Fast"   | "x = Slow"
le_fun_def: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" shows "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
pseudofunctor_def: fixes V\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>C :: "'b \<Rightarrow> 'b"   and src\<^sub>C :: "'b \<Rightarrow> 'b"   and trg\<^sub>C :: "'b \<Rightarrow> 'b"   and V\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>D :: "'c \<Rightarrow> 'c"   and src\<^sub>D :: "'c \<Rightarrow> 'c"   and trg\<^sub>D :: "'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and \<Phi> :: "'b \<times> 'b \<Rightarrow> 'c" shows "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi> \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<and> weak_arrow_of_homs V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) \<and> (composite_functor (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) V\<^sub>C V\<^sub>D (\<lambda>\<mu>\<nu>. H\<^sub>C (fst \<mu>\<nu>) (snd \<mu>\<nu>)) F \<and> composite_functor (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) (subcategory.comp (product_category.comp V\<^sub>D V\<^sub>D) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>D (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>D (snd \<mu>\<nu>) \<and> src\<^sub>D (fst \<mu>\<nu>) = trg\<^sub>D (snd \<mu>\<nu>))) V\<^sub>D (weak_arrow_of_homs.FF V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) (\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<and> natural_isomorphism (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) V\<^sub>D ((\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<circ> weak_arrow_of_homs.FF V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) (F \<circ> (\<lambda>\<mu>\<nu>. H\<^sub>C (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<Phi> \<and> pseudofunctor_axioms V\<^sub>C H\<^sub>C \<a>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D F \<Phi>"
case_prod_conv: fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and a :: "'b"   and b :: "'d" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
A: shows "(if \<exists>t. (\<exists>p. (c, s) \<Rightarrow> p \<Down> t) \<and> (\<exists>i. Q t = enat i) then enat (\<down>\<^sub>t (c, s)) + Q (\<down>\<^sub>s (c, s)) else \<infinity>) \<le> P s"
cod_trg_eq_dom_trg: shows "\<mu>.cod.trg = \<mu>.dtrg"
1: shows "smap f (gtrace (sdrop n r) (gtarget (stake n r) v)) = sconst k"
ge: fixes y :: "'a" assumes "f y \<le> y" shows "x \<le> y"
slice_def: fixes "from" :: "nat"   and to :: "nat"   and list :: "'c list" shows "slice from to list = take (to - from) (drop from list)"
Fun_le_FunD: fixes sch1 :: "type_scheme"   and sch2 :: "type_scheme"   and sch1' :: "type_scheme"   and sch2' :: "type_scheme" assumes "sch1 =-> sch2 \<le> sch1' =-> sch2'" shows "sch1 \<le> sch1' \<and> sch2 \<le> sch2'"
Cons: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "x \<in> Field r"   and "\<forall>y\<in>set xs. (x, y) \<in> r"   and "sorted_on r xs" shows "sorted_on r (x # xs)"
independent_injective_image: fixes s :: "('b, 'c) vec set"   and A :: "(('b, 'c) vec, 'd) vec" assumes "vec.independent s"   and "inj_on ((*v) A) (vec.span s)" shows "vec.independent ((*v) A ` s)"
holo: shows "f holomorphic_on eball 0 (ereal r)"
carr_def: fixes T :: "'b set set" shows "carrier\<^bsub>T\<^esub> = \<Union> T"
space_as_set_inject: fixes x :: "'a ccsubspace"   and y :: "'a ccsubspace" shows "(space_as_set x = space_as_set y) = (x = y)"
zero_in_unit_disc: shows "0\<^sub>h \<in> unit_disc"
that: fixes x :: "V" assumes "x \<in> elts A"   and "p = \<langle>x, app f x\<rangle>" shows "thesis"
commute_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "commute R S = square R S S R"
oppOrdering_thenOrdering: fixes x :: "Ordering"   and y :: "Ordering" shows "oppOrdering\<cdot>(thenOrdering\<cdot>x\<cdot>y) = thenOrdering\<cdot>(oppOrdering\<cdot>x)\<cdot>(oppOrdering\<cdot>y)"
family_oracle_apply: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'c) spmf"   and s :: "'b \<Rightarrow> 'c"   and i :: "'b"   and x :: "'d" shows "family_oracle f s (i, x) = map_spmf (apsnd (fun_upd s i)) (f i (s i) x)"
R_a0im1_ri_def: shows "R_a0im1_ri \<equiv> {(s, t). signals s = ri_runs2sigs (runs t) \<and> corrupted s = Domain (m1_state.leak t) \<times> UNIV}"
reachable_par_subnet_induct: fixes s :: "'c net_state"   and t :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree"   and I :: "'d node_action \<Rightarrow> bool"   and P :: "'c net_state \<Rightarrow> 'c net_state \<Rightarrow> bool" assumes "SubnetS s t \<in> reachable (pnet np (p1 \<parallel> p2)) I"   and "\<And>s t. SubnetS s t \<in> init (pnet np (p1 \<parallel> p2)) \<Longrightarrow> P s t"   and "\<And>s t s' t' a. \<lbrakk>SubnetS s t \<in> reachable (pnet np (p1 \<parallel> p2)) I; P s t; (SubnetS s t, a, SubnetS s' t') \<in> automaton.trans (pnet np (p1 \<parallel> p2)); I a\<rbrakk> \<Longrightarrow> P s' t'" shows "P s t"
suminf_cmult_ereal: fixes f :: "nat \<Rightarrow> ereal"   and a :: "ereal" assumes "\<And>i. 0 \<le> f i"   and "0 \<le> a" shows "(\<Sum>i. a * f i) = a * suminf f"
yatmostonceinbs: shows "count_list bs (cs ! i) \<le> 1"
conj_chaos: fixes a :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<iinter> chaos = a"
execute_def: fixes h :: "'f"   and p :: "('f, 'd, 'e) prog" shows "h \<turnstile> p = the_prog p h"
simps_1: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "removeShadowRules1 (x # xs) = (if DenyAll \<in> set xs then removeShadowRules1 xs else x # xs)"
rel_witness_pmf1: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf" assumes "rel_pmf A p q" shows "rel_pmf (\<lambda>a (a', b). a = a' \<and> A a' b) p (rel_witness_pmf A (p, q))"
hyps_3: shows "n \<in> set (\<alpha>n g)"
assms_1: shows "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>"
fps_ext: fixes p :: "('c, 'd) fps"   and q :: "('c, 'd) fps" assumes "\<And>n. p $ n = q $ n" shows "p = q"
rel_pred_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a set"   and B :: "'b set" shows "rel_pred R A B = rel_fun R (=) (\<lambda>x. x \<in> A) (\<lambda>y. y \<in> B)"
assms_1: shows "cash_flow \<in> borel_measurable (G T)"
Seq_sound1: fixes x :: "('b com option \<times> 'b) list"   and P :: "'b com"   and Q :: "'b com"   and s :: "'b" assumes "x \<in> cptn_mod"   and "x ! 0 = (Some (Seq P Q), s)"   and "\<And>i. i < length x \<Longrightarrow> fst (x ! i) \<noteq> Some Q" shows "\<exists>xs\<in>cp (Some P) s. x = map (lift Q) xs"
ys_a: shows "set_mset (mset ys) \<subseteq> A"
L2_set_strict_mono: fixes K :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "finite K"   and "K \<noteq> {}"   and "\<And>i. i \<in> K \<Longrightarrow> f i < g i"   and "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i" shows "L2_set f K < L2_set g K"
eq_1: shows "(\<Prod>i\<in>UNIV - {k}. (if i = k then a i + b i else c i) $ p i) = (\<Prod>i\<in>UNIV - {k}. (if i = k then a i else c i) $ p i)"
ball_insert: fixes a :: "'c"   and B :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(\<forall>x\<in>insert a B. P x) = (P a \<and> (\<forall>x\<in>B. P x))"
balanced_Node_if_wbal1: fixes l :: "'a kdt"   and r :: "'a kdt"   and k :: "'a"   and v :: "real" assumes "balanced l"   and "balanced r"   and "KD_Tree.size_kdt l = KD_Tree.size_kdt r + 1" shows "balanced (Node k v l r)"
eval_nembed_false: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "\<not> P s" shows "\<guillemotleft> \<N> P \<guillemotright> s = 1"
Catch_decomp_Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p\<^sub>2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and p' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Catch Skip p\<^sub>2, s) \<rightarrow> (p', s')" shows "s' = s \<and> p' = Skip"
g_2: shows "set_mset fs \<subseteq> Irr_Mon"
le_diff_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (minus c b) = less_eq (plus a b) c"
argmax_non_empty_iff: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite X"   and "X \<noteq> {}" shows "argmax f X \<noteq> {}"
f: fixes x :: "'a" assumes "x \<in> A" shows "R x (f x)"
strict_monoI_Suc: fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n < f (Suc n)" shows "strict_mono f"
lex_ord_raw_valI: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and aux :: "order option"   and k :: "'a"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> order option"   and ko :: "'o" assumes "oalist_inv xs"   and "oalist_inv ys"   and "aux \<noteq> Some Eq"   and "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)"   and "aux = f k (lookup_raw xs k) (lookup_raw ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set (fst xs) \<union> fst ` set (fst ys); lt ko k' k\<rbrakk> \<Longrightarrow> f k' (lookup_raw xs k') (lookup_raw ys k') = Some Eq" shows "lex_ord_raw ko f xs ys = aux"
index_add_mat_2: fixes A :: "'c mat"   and B :: "'c mat" shows "dim_row (A + B) = dim_row B"
IH: assumes "\<forall>l<k. v \<notin> local.writes (path \<sigma> l)" shows "(\<sigma>\<^bsup>k\<^esup>) v = \<sigma> v"
TESL_interp_unfold_stepwise_tagrelgen: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool" shows "\<lbrakk> time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> {Y. \<exists>n. Y = \<lbrakk> time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"