hcju/isabelleps huggingface dataset:
val_pos_n_val_pos: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(0 \<le> v x) = (0 \<le> n_val K v x)"
emb_zero: fixes f :: "'e \<Rightarrow> 'a"   and D :: "'e set"   and X :: "'e set" shows "emb f D \<in> X - D \<rightarrow> {\<zero>\<^bsub>K\<^esub>}"
SuccI: fixes kl :: "'d list"   and k :: "'d"   and Kl :: "'d list set" assumes "kl @ [k] \<in> Kl" shows "k \<in> BNF_Greatest_Fixpoint.Succ Kl kl"
CR_join_right_I: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "CR r"   and "(x, y) \<in> r\<^sup>\<down>"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>\<down>"
homeomorphism_symD: fixes S :: "'b set"   and t :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "homeomorphism S t f g" shows "homeomorphism t S g f"
bheight_joinR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invh l"   and "invh r"   and "bheight r \<le> bheight l" shows "bheight (joinR l x r) = bheight l"
distincts_Cons: fixes xs :: "'b list"   and xss :: "'b list list" shows "distincts (xs # xss) = (xs \<noteq> [] \<and> distinct xs \<and> distincts xss \<and> set xs \<inter> \<Union> (set ` set xss) = {})"
assms_1: shows "trace_callee_eq callee1 callee2 A p q"
HPair_eqvt: fixes p :: "perm"   and a :: "tm"   and b :: "tm" shows "p \<bullet> HPair a b = HPair (p \<bullet> a) (p \<bullet> b)"
complex_cone_def: fixes a :: "real"   and b :: "real" shows "complex_cone a b = {z. \<exists>y\<in>{a..b}. z = rcis (cmod z) y}"
subst_pp_alt: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat" shows "subst_pp f t = (\<Prod>x. f x ^ lookup t x)"
lnth_ldrop: fixes n :: "enat"   and m :: "nat"   and xs :: "'a llist" assumes "n + enat m < llength xs" shows "lnth (ldrop n xs) m = lnth xs (m + the_enat n)"
xfer_fair_run_aux: fixes cs :: "nat \<Rightarrow> 'c" assumes "C.is_fair_run cs" shows "\<exists>as. (\<forall>i. R (as i) (cs i)) \<and> A.is_fair_run as"
divideR_apply: fixes c :: "real"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(c *\<^sub>R f) x = c *\<^sub>R f x"
bi_total_fun: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool" assumes "bi_unique A"   and "bi_total B" shows "bi_total (A ===> B)"
rtrancl_path_appendE: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list"   and y :: "'k"   and ys :: "'k list"   and z :: "'k" assumes "rtrancl_path r x (xs @ y # ys) z" obtains "rtrancl_path r x (xs @ [y]) y" and "rtrancl_path r y ys z"
no_neg_cyc_reach_imp_path: fixes u :: "'a"   and v :: "'a"   and f :: "'b \<Rightarrow> real" assumes "u \<rightarrow>\<^sup>* v"   and "\<And>p. awalk u p v \<Longrightarrow> \<nexists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0" shows "\<exists>p. apath u p v \<and> \<mu> f u v = ereal (awalk_cost f p)"
kildall_def: fixes r :: "nat list \<Rightarrow> nat list \<Rightarrow> bool"   and f :: "nat list \<Rightarrow> nat list \<Rightarrow> nat list"   and step :: "nat \<Rightarrow> nat list \<Rightarrow> (nat \<times> nat list) list"   and \<tau>s :: "nat list list" shows "kildall r f step \<tau>s = fst (iter f step \<tau>s (unstables r step \<tau>s))"
Inequality: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c (update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
weak_universal_def: fixes uu :: "'d itself"   and R :: "('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph"   and G\<^sub>1 :: "('e, 'g) labeled_graph"   and G\<^sub>2 :: "('e, 'g) labeled_graph"   and f\<^sub>1 :: "('f \<times> 'g) set"   and f\<^sub>2 :: "('f \<times> 'g) set" shows "weak_universal uu R G\<^sub>1 G\<^sub>2 f\<^sub>1 f\<^sub>2 \<equiv> \<forall>h\<^sub>1 h\<^sub>2 G. graph_homomorphism (snd R) G h\<^sub>1 \<and> graph_homomorphism G\<^sub>1 G h\<^sub>2 \<and> f\<^sub>1 O h\<^sub>2 \<subseteq> h\<^sub>1 \<longrightarrow> (\<exists>h. graph_homomorphism G\<^sub>2 G h \<and> h\<^sub>2 \<subseteq> h)"
sys_finite_reachable: fixes prog :: "program" shows "finite (li.sa.reachable prog)"
rel_def: shows "rel = {(A, B). (case (A, B) of (Inl (m, x), Inl (m', xa)) \<Rightarrow> 0 < m' \<and> m' < m \<and> m \<le> snd fund_sol | (Inl (m, x), Inr ba) \<Rightarrow> False | (Inr x, Inl (m', xa)) \<Rightarrow> m' \<le> snd fund_sol | (Inr x, Inr ba) \<Rightarrow> False) \<and> A = f B}"
finite_measure_lebesgue_on: fixes S :: "'c set" assumes "S \<in> lmeasurable" shows "finite_measure (lebesgue_on S)"
rwof_leof_step: fixes m :: "'c nres"   and c :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and s :: "'c" assumes "rwof m c f s"   and "c s" shows "f s \<le>\<^sub>n SPEC (rwof m c f)"
vsubsetD: fixes A :: "V"   and B :: "V"   and x :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "x \<in>\<^sub>\<circ> A" shows "x \<in>\<^sub>\<circ> B"
m_weakly_prefers_Pm: fixes \<mu> :: "('m \<times> 'w) set"   and m :: "'m"   and w :: "'w"   and w' :: "'w" assumes "match \<mu>"   and "(m, w) \<in> \<mu>" shows "(w' \<in> m_weakly_prefers m \<mu>) = ((w, w') \<in> Pm m)"
a_get_root_node_locs_def: shows "get_root_node_locs = get_ancestors_locs"
intros_33: fixes vs :: "e list"   and n :: "nat"   and i :: "nat"   and lholed :: "Lholed"   and LI :: "e list"   and es :: "e list" assumes "const_list vs"   and "length vs = n"   and "Lfilled i lholed (vs @ [$Br i]) LI" shows "\<lparr>[Label n es LI]\<rparr> \<leadsto> \<lparr>vs @ es\<rparr>"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "encoding.indRelLPOp Enc (TargetTerm (Enc S)) (SourceTerm S)"
subseteq_zmset_def: fixes A :: "'b zmultiset"   and B :: "'b zmultiset" shows "(A \<subseteq>#\<^sub>z B) = (\<forall>a. zcount A a \<le> zcount B a)"
assms: fixes xs :: "'a list"   and ys :: "'a list" assumes "js = xs @ ys" shows "set xs \<inter> set ys = {}"
simps_1: fixes A :: "'j \<Rightarrow> 'c"   and uu :: "'j \<Rightarrow> 'c"   and uv :: "'j \<Rightarrow> 'c" shows "J_C.Dom (J_C.MkArr A uu uv) = A"
length_remdups_concat: fixes xss :: "'a list list" shows "|remdups (concat xss)| = card (\<Union> (set ` set xss))"
star2: fixes x :: "'a" shows "(1' + x)\<^sup>\<star> = x\<^sup>\<star>"
P2_ideal_model_rewrite: fixes M :: "'a \<times> 'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "(bool \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a \<times> 'a) \<times> nat) \<times> 'g) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> 'e spmf)" shows "P2_ideal_model M \<sigma> z \<A> = P2_ideal_model' M \<sigma> z \<A>"
has_vector_derivative_at_split: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real" shows "(f has_vector_derivative f') (at x) = ((f has_vector_derivative f') (at_left x) \<and> (f has_vector_derivative f') (at_right x))"
rel_resource_eq: shows "rel_resource (=) (=) = (=)"
lt_in_keys: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "i.lpp p \<in> keys p"
ex_countable_subset_image: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and P :: "'a set \<Rightarrow> bool" shows "(\<exists>T. countable T \<and> T \<subseteq> f ` S \<and> P T) = (\<exists>T. countable T \<and> T \<subseteq> S \<and> P (f ` T))"
that: fixes ti :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> ti"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, ti\<rangle> \<Rightarrow> t" shows "thesis"
disjoint_sum: fixes f :: "'c \<Rightarrow> 'd" assumes "finite E" shows "sum (sum f) (orbits G E \<phi>) = sum f E"
arr_char: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.arr f = (A1.arr (fst f) \<and> A2.arr (snd f))"
sep_add_disj_eq: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y ## z" shows "x ## y + z = (x ## y \<and> x ## z)"
mon_w_uncons: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and e :: "('h, 'k) label"   and w :: "('h, 'k) label list" shows "mon_w fg (e # w) = mon_e fg e \<union> mon_w fg w"
distinct_11: fixes x3 :: "v list"   and x4 :: "e list" shows "RSReturn x3 \<noteq> RSNormal x4"
groupF_set: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "\<forall>X\<in>set (groupF f xs). \<forall>x\<in>set X. x \<in> set xs"
region_unique: fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and u :: "'d \<Rightarrow> real"   and R :: "('d \<Rightarrow> real) set" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "u \<in> R"   and "R \<in> \<R>" shows "[u]\<^sub>\<R> = R"
run_step_return_imp_not_trap_const_list_2: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and es :: "e list"   and s' :: "s"   and vs' :: "v list"   and res :: "v list" assumes "run_step d i (s, vs, es) = (s', vs', RSReturn res)" shows "\<not> const_list es"
nextss_closed_dfs': fixes g :: "(node \<times> node) list"   and ys :: "node list"   and xs :: "node list" assumes "nextss g ys \<subseteq> set xs \<union> set ys" shows "nextss g (dfs g xs ys) \<subseteq> set (dfs g xs ys)"
ass: shows "n \<notin> (g_E G)\<^sup>* `` {0}"
const: fixes c :: "real" shows "(\<lambda>_. c) \<in> R"
get_tag_name_def: shows "get_tag_name = (\<lambda>element_ptr. get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name)"
L1_D_3: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> space\<^sub>N (\<LL> 1 M)" shows "Norm (\<LL> 1 M) f = LINT x|M. \<bar>f x\<bar>"
map_iterateoi_I: fixes invar :: "'c \<Rightarrow> bool"   and itoi :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e \<Rightarrow> 'f option" assumes "\<And>m. invar m \<Longrightarrow> map_iterator_linord (itoi m) (\<alpha> m)" shows "map_iterateoi \<alpha> invar itoi"
differentiable_rGamma_aux2: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and rGamma :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.Gamma dist uniformity open divide inverse times one plus zero minus uminus scaleR norm sgn rGamma" shows "let z = uminus (semiring_1.of_nat one plus zero n) in LIM y topological_space.at_within open z UNIV. scaleR (inverse (norm (minus y z))) (minus (minus (rGamma y) (rGamma z)) (times (times (power.power one times (uminus one) n) (comm_monoid_mult.prod times one (semiring_1.of_nat one plus zero) {1..n})) (minus y z))) :> topological_space.nhds open zero"
hf_cases: fixes y :: "hf" obtains "y = 0"   | a :: "hf" and b :: "hf" where "y = b \<triangleleft> a" and "a \<^bold>\<notin> b"
add_neg_nonpos: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subset># {#}"   and "b \<subseteq># {#}" shows "a + b \<subset># {#}"
subset_hull: fixes S :: "'b set \<Rightarrow> bool"   and t :: "'b set"   and s :: "'b set" assumes "S t" shows "(S hull s \<subseteq> t) = (s \<subseteq> t)"
rec_ci_not_null: fixes f :: "recf"   and a :: "nat"   and b :: "nat" shows "rec_ci f \<noteq> ([], a, b)"
not_equal_is_parallel: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "xs \<parallel> ys"
ContraAssume: fixes B :: "fm"   and H :: "fm set"   and A :: "fm" assumes "Neg B \<in> H" shows "insert B H \<turnstile> A"
image_ident: fixes Y :: "'n set" shows "(\<lambda>x. x) ` Y = Y"
valid_bind'_cong: fixes f :: "'b \<Rightarrow> ('a \<times> 'b) option"   and \<sigma> :: "'b"   and g :: "'b \<Rightarrow> ('a \<times> 'b) option"   and M :: "'b \<Rightarrow> (bool \<times> 'b) option" assumes "f \<sigma> = g \<sigma>" shows "(\<sigma> \<Turnstile> f ;- M) = (\<sigma> \<Turnstile> g ;- M)"
mat_delete_index: fixes A :: "'a mat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "A \<in> carrier_mat (Suc n) (Suc n)"   and "i < Suc n"   and "j < Suc n"   and "i' < n"   and "j' < n" shows "A $$ (insert_index i i', insert_index j j') = mat_delete A i j $$ (i', j')"
interior_interior: fixes S :: "'b set" shows "interior (interior S) = interior S"
Hermite_reduce_above_Least_eq: fixes ass :: "'a \<Rightarrow> 'a"   and i :: "'b"   and A :: "(('a, 'c) vec, 'b) vec"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "ass_function ass"   and "\<not> is_zero_row i A" shows "(LEAST n. A $ i $ n \<noteq> (0::'a)) = (LEAST n. Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res $ i $ n \<noteq> (0::'a))"
is_valid_operator_sas_plus_then_6: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (effect_of op). \<forall>(v', a')\<in>set (effect_of op). v \<noteq> v' \<or> a = a'"
open_openin_trans: fixes S :: "'a set"   and T :: "'a set" assumes "open S"   and "open T"   and "T \<subseteq> S" shows "openin (top_of_set S) T"
sup_inf_distrib1: fixes 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 x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.distrib_lattice inf less_eq less sup" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
count_roots_rat_def: fixes p :: "int poly" shows "count_roots_rat p = count_roots (real_of_int_poly p)"
restrict_top_id: fixes e :: "'b ifex"   and v :: "'b"   and v' :: "'b"   and val :: "bool" assumes "ifex_ordered e"   and "ifex_top_var e = Some v"   and "v' < v" shows "restrict_top e v' val = e"
intF: fixes f :: "('fsym, 'tp) efsym"   and al :: "univ list" assumes "GE_wtFsym f"   and "list_all2 eintT (GE_arOf f) al" shows "eintT (GE_resOf f) (eintF f al)"
cosh_dist_swap: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2" shows "cosh_dist a b = cosh_dist b a"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.dom f1, f2)) (\<lambda>f2. F (A1.dom f1, f2)) (\<lambda>f2. F (A1.dom f1, f2))"
wary_sub: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "sub s t"   and "wary t" shows "wary s"
bijective_eq_bij: fixes f :: "'c \<Rightarrow> 'd" shows "bijective f = bij f"
unfold: shows "eval_body_final = (\<Lambda> r v s f. Fixrec.run\<cdot>(match_Val\<cdot>v\<cdot>(\<Lambda> n. Fixrec.succeed\<cdot>(s\<cdot>n)) +++ match_Add\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(r\<cdot>x\<cdot>(\<Lambda> n. r\<cdot>y\<cdot>(\<Lambda> m. s\<cdot>(n + m))\<cdot>f)\<cdot>f)) +++ match_Throw\<cdot>v\<cdot>(Fixrec.succeed\<cdot>f) +++ match_Catch\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(r\<cdot>x\<cdot>s\<cdot>(r\<cdot>y\<cdot>s\<cdot>f)))))"
init: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.init M \<in> dfa.states M"
distinct_Keys_to_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "distinct (Keys_to_list ps)"
I: fixes b :: "'a"   and a :: "'a" shows "b * a r\<rightarrow> b = (1::'a)"
hb_completion_shift: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list"   and ttas :: "('thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action) list"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" assumes "hb_completion s E"   and "s -\<triangleright>ttas\<rightarrow>* s'"   and "non_speculative P (w_values P (\<lambda>_. {}) (map snd E)) (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))" shows "hb_completion s' (E @ concat (map (\<lambda>(t, ta). map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))"
cinner_sup_norm: fixes \<psi> :: "'c" shows "norm \<psi> = (SUP \<phi>. cmod \<langle>\<phi>, \<psi>\<rangle> / norm \<phi>)"
lhs_def: shows "lhs \<equiv> \<lambda>plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1. linear_ow (Collect (Domainp B\<^sub>1)) plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 (Collect (Domainp B\<^sub>2))"
doesnt_read_mutually_exclusive': fixes c :: "'Com"   and x :: "'Var"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "doesnt_read c x"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "\<not> (\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>)"
is_zero_row_upt_ncols: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec" shows "is_zero_row_upt_k i (ncols A) A = (\<forall>j. A $ i $ j = (0::'b))"
intro: fixes block_collection :: "'a set multiset"   and point_set :: "'a set" assumes "\<And>b. b \<in># block_collection \<Longrightarrow> b \<subseteq> point_set" shows "incidence_system point_set block_collection"
act_rews_1: fixes v :: "state \<Rightarrow> 'a"   and s :: "state"   and t :: "state" shows "($v) (s, t) \<equiv> v s"
vsubst_compose1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y2 :: "'var"   and y :: "'var"   and y1 :: "'var" assumes "good X" shows "X #[y1 // y]_ys #[y2 // y]_ys = X #[y1 @ys[y2 / y]_ys // y]_ys"
fmfilter_cong: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x = Q x" shows "fmfilter P m = fmfilter Q m"
lemma_NBseq_def: fixes X :: "'a \<Rightarrow> 'b" shows "(\<exists>K>0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N))"
lift_Ri'_def: fixes x :: "'alt"   and y :: "'alt" shows "lift_Ri' x y = (x \<in> alts \<and> y \<in> alts \<and> (x = y \<or> x \<notin> alts' \<or> y \<in> alts' \<and> Ri' x y))"
5: shows "\<turnstile> concl (Thm (Imp p p))"
auto_ll_on_open_def: fixes f :: "'c \<Rightarrow> 'c"   and X :: "'c set" shows "auto_ll_on_open f X \<equiv> local_lipschitz UNIV X (\<lambda>_. f) \<and> open X"
pow_int_1: fixes x :: "'b" shows "pow_int x 1 = x"
sup_ge1: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x (sup x y)"
root_bound_def: fixes p :: "int poly" shows "root_bound p \<equiv> let n = degree p; m = 1 + div_ceiling (max_list_non_empty (map (\<lambda>i. \<bar>coeff p i\<bar>) [0..<n])) \<bar>lead_coeff p\<bar> in rat_of_int (2 ^ log_ceiling 2 m)"
simps_2: fixes y :: "'a option" shows "None \<squnion> y = y"
no_marker_and_snapshotted_implies_no_more_markers: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a"   and cid :: "nat"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "ps c p \<noteq> None"   and "Marker \<notin> set (msgs c cid)"   and "channel cid = Some (p, q)" shows "Marker \<notin> set (msgs c' cid)"
finite_axes_cross_def: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "finite_axes_cross g z = finite {t. (Re (g t - z) = 0 \<or> Im (g t - z) = 0) \<and> 0 \<le> t \<and> t \<le> 1}"
f_Exec_eq_f_Exec_Stream_Init_last: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp trans_fun xs c = last (f_Exec_Comp_Stream_Init trans_fun xs c)"
AndI: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F"   and "\<Gamma> \<turnstile> G" shows "\<Gamma> \<turnstile> F \<^bold>\<and> G"
tensor_mat_carrier: fixes dims :: "nat list"   and vars :: "nat set"   and m1 :: "'a mat"   and m2 :: "'a mat" shows "tensor_mat dims vars m1 m2 \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
bv_finite: fixes n :: "nat" shows "finite (set_pmf (bv n))"
precedesAppend: fixes a :: "'b"   and b :: "'b"   and l :: "'b list"   and l' :: "'b list" assumes "precedes a b l" shows "precedes a b (l @ l')"
t_def_2: shows "\<S>\<bullet>\<C>' \<turnstile> [x] : [] _> [t]"
FAssRedsNull: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>null,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s\<^sub>2,b\<^sub>2\<rangle>"
WT_gpv_mono: fixes \<I>1 :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and \<I>2 :: "('d, 'e) \<I>" assumes "\<I>1 \<turnstile>g gpv \<surd>"   and "outs_\<I> \<I>1 \<subseteq> outs_\<I> \<I>2"   and "\<And>x. x \<in> outs_\<I> \<I>1 \<Longrightarrow> responses_\<I> \<I>2 x \<subseteq> responses_\<I> \<I>1 x" shows "\<I>2 \<turnstile>g gpv \<surd>"
split_inf_Catch: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, s) \<rightarrow> \<dots>(\<infinity>)" shows "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> \<dots>(\<infinity>) \<or> (\<exists>s'. \<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow>\<^sup>* (Throw, Normal s') \<and> \<Gamma>\<turnstile> (c\<^sub>2, Normal s') \<rightarrow> \<dots>(\<infinity>))"
disjoint_Nv1_Vv2: shows "N\<^bsub>\<V>1\<^esub> \<inter> V\<^bsub>\<V>2\<^esub> = {}"
simps_2: fixes t :: "nat"   and volatile :: "bool"   and a :: "nat" shows "eval_expr t (Mem volatile a) = ({t}, \<lambda>\<theta>. the (\<theta> t))"
order_full_node: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "order k (full_node k a h)"
of_rat_2_1: fixes x :: "rat" shows "real_of_2 (of_rat_2 x) = real_of_rat x"