hcju/isabelleps huggingface dataset:
line_suitable_def: fixes Z :: "'b set"   and F :: "'b formula" shows "line_suitable Z F \<equiv> atoms F \<subseteq> Z"
times_divide_eq_right: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times a (divide b c) = divide (times a b) c"
strong_supplementation: fixes x :: "i"   and y :: "i" assumes "EM"   and "\<not> P x y" shows "\<exists>z. P z x \<and> \<not> O z y"
ap_state_def: fixes f :: "'e \<Rightarrow> ('h \<Rightarrow> 'f) \<times> 'i"   and x :: "'i \<Rightarrow> 'h \<times> 'g" shows "Applicative.ap f x = (\<lambda>s. case f s of (g, s') \<Rightarrow> case x s' of (y, s'') \<Rightarrow> (g y, s''))"
assms: shows "\<forall>x\<in>vars\<^sub>l l. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
ord_eq_less_trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a = b"   and "strict_border b c" shows "strict_border a c"
compact_segment: fixes a :: "'b"   and b :: "'b" shows "compact {a--b}"
inner_pasch: shows "\<forall>a b c p q. Bet a p c \<and> Bet b q c \<longrightarrow> (\<exists>x. Bet p x b \<and> Bet q x a)"
dickson_less_vD1: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'t"   and u :: "'t" assumes "dickson_less_v d m v u" shows "d (pp_of_term v) \<le> m"
ext: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
get_tag_name_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_tag_name element_ptr"
cinner_ge_zero: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "0 \<le> cinner x x"
is_TNil_tfinite: fixes xs :: "('a, 'b) tllist" assumes "is_TNil xs" shows "tfinite xs"
Inl_Inr_DataSpace: shows "DataSpace [Part UNIV Inl, Part UNIV Inr]"
last_coeffs_eq_coeff_degree: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "last (coeffs p) = lead_coeff p"
witness_1: fixes a :: "'c" assumes "a \<in> S" shows "i (j a) = a"
product_integral_prod: fixes M :: "'a \<Rightarrow> 'b measure"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "product_sigma_finite M"   and "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" shows "LINT x|Pi\<^sub>M I M. (\<Prod>i\<in>I. f i (x i)) = (\<Prod>i\<in>I. integral\<^sup>L (M i) (f i))"
prv_dsj_impR: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp \<chi> (dsj \<phi> \<chi>))"
inf_fun_def: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" shows "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
eventually_nonzero_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "(\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)) = (\<forall>\<^sub>F x in F. g x \<noteq> (0::'b))"
simps_4: fixes v :: "('c \<times> nat) tree"   and va :: "'c \<times> nat"   and vc :: "('c \<times> nat) tree"   and vd :: "'c \<times> nat" shows "split \<langle>v, va, \<langle>vc, vd, \<langle>\<rangle>\<rangle>\<rangle> = \<langle>v, va, \<langle>vc, vd, \<langle>\<rangle>\<rangle>\<rangle>"
RPCFail_Next_enabled: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "\<turnstile> Enabled RPCFail send rcv rst p \<longrightarrow> Enabled (<RPCNext send rcv rst p>_(rst!p, rtrner send!p, caller rcv!p))"
mult_eq_1: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) \<le> a"   and "a \<le> (1::'c)"   and "b \<le> (1::'c)" shows "(a * b = (1::'c)) = (a = (1::'c) \<and> b = (1::'c))"
coset_def: fixes xs :: "'a list" shows "List.coset xs = - set xs"
sum_f_non_negative: fixes X :: "(nat \<times> nat) set" shows "(0::'capacity) \<le> sum f X"
lipschitz_on_concat: fixes L :: "real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and c :: "real"   and g :: "real \<Rightarrow> 'b" assumes "L-lipschitz_on {a..b} f"   and "L-lipschitz_on {b..c} g"   and "f b = g b" shows "L-lipschitz_on {a..c} (\<lambda>x. if x \<le> b then f x else g x)"
orbits_1: shows "pmf (sds R10) d = pmf (sds R10) a"
cts_iterates: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and i :: "nat" assumes "healthy (wp body)"   and "bd_cts (wp body)" shows "bd_cts (iterates body G i)"
assms: shows "freshEnv xs y (rho [x \<leftarrow> X]_xs)"
Scons_not_Numb: fixes M :: "('c, 'd) Old_Datatype.node set"   and N :: "('c, 'd) Old_Datatype.node set"   and k :: "nat" shows "Scons M N \<noteq> Numb k"
heap_is_wellformed_children_disc_nodes_different: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "set children \<inter> set disc_nodes = {}"
N1_leadsto_b: shows "\<turnstile> \<box>($PsiInv \<and> [(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2)) \<and> SF(N1)_(x, y, sem, pc1, pc2) \<and> \<box>SF(N2)_(x, y, sem, pc1, pc2) \<longrightarrow> (pc1 = #b \<or> pc1 = #g \<or> pc1 = #a \<leadsto> pc1 = #b)"
ext_surjective: fixes x :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>" shows "x = \<lparr>unit_sys = (Record.iso_tuple_fst Measurement_System_ext_Tuple_Iso \<circ> id) x, \<dots> = (Record.iso_tuple_snd Measurement_System_ext_Tuple_Iso \<circ> id) x\<rparr>"
a1: shows "low_equal s1 s2 \<and> (fst instr = load_store_type LDSB \<or> fst instr = load_store_type LDUB \<or> fst instr = load_store_type LDUH \<or> fst instr = load_store_type LD \<or> fst instr = load_store_type LDD) \<and> t1 = snd (fst (load_sub1 instr rd 0 s1)) \<and> t2 = snd (fst (load_sub1 instr rd 0 s2))"
part_reflexivity: fixes x :: "i" assumes "PM" shows "P x x"
\<pi>'_def: shows "\<pi>' = (\<lambda>x. if x \<notin> A then x else xs ! inv \<pi> (index xs x))"
span_li_not_depend_2: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N V" shows "LinearCombinations.module.lin_dep K (V.vs N) S = V.module.lin_dep S"
step_def: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and a :: "'d" shows "step A a = {(p, q). q \<in> next A a p}"
simps_2: fixes uu :: "'b" shows "changes [uu] = 0"
leg0_simps_3: shows "trg s\<^sub>0 = src s"
connected_by_path: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "Graph.connected G v w" obtains xs :: "'c list" where "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
unit_counit_adjunction_axioms: shows "unit_counit_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta> \<epsilon>"
mkPost_def: fixes T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool" shows "mkPost T = (\<lambda>s0 s t. s = mkState s0 \<longrightarrow> T s0 t)"
factorize_real_poly_def: fixes p :: "real poly" shows "factorize_real_poly p \<equiv> map_option (\<lambda>(c, ris). (Re c, complex_roots_to_real_factorization ris)) (factorize_complex_main (map_poly complex_of_real p))"
cases: fixes a1 :: "'a set"   and a2 :: "'a set"   and R :: "('a \<times> 'a) set" assumes "(a1, a2) \<in> max_ext R" obtains X :: "'a set" and Y :: "'a set" where "a1 = X" and "a2 = Y" and "finite X" and "finite Y" and "Y \<noteq> {}" and "\<And>x. x \<in> X \<Longrightarrow> \<exists>xa\<in>Y. (x, xa) \<in> R"
assms_7: fixes X :: "'a set" assumes "X \<in> sets M"   and "X \<subseteq> space M - space M'" shows "emeasure M X = 0"
product_lists_in_set_nth': fixes xs :: "'c list list"   and ys :: "'c list" assumes "\<forall>i<length xs. ys ! i \<in> set (xs ! i)"   and "length xs = length ys" shows "ys \<in> set (product_lists xs)"
axioms_1: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme" assumes "two_cats AA BB" shows "category AA"
imp_chain_zip_eq: fixes l :: "tm list"   and l' :: "tm list"   and e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "fol fm" assumes "equal_length l l'" shows "semantics e f g (foldr Imp (zip_eq l l') p) = (semantics_list e f l = semantics_list e f l' \<longrightarrow> semantics e f g p)"
grid_child_level: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and dir :: "dir"   and ds :: "nat set" assumes "d < length b"   and "p \<in> grid (child b dir d) ds" shows "lv b d < lv p d"
line_left_inter_1: fixes P :: "Points"   and l :: "Lines"   and m :: "Lines" assumes "P \<in> line_ext l"   and "P \<notin> line_ext m" shows "line (Pascal_Property.inter l m) P = l"
abelian_groupE_2: fixes R :: "('c, 'd) ring_scheme" assumes "abelian_group R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
subprob_measure_le_1: fixes M :: "'a measure"   and X :: "'a set" assumes "subprob_space M" shows "Sigma_Algebra.measure M X \<le> 1"
dickson_gradingE: fixes d :: "'a \<Rightarrow> nat"   and seq :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "dickson_grading d"   and "\<And>i. d (seq i) \<le> n" obtains i :: "nat" and j :: "nat" where "i < j" and "seq i adds seq j"
att_andD1: fixes x1 :: "'a attree"   and x2 :: "'a attree list"   and s :: "'a set \<times> 'a set" assumes "\<turnstile>x1 # x2 \<oplus>\<^sub>\<and>\<^bsup>s\<^esup>" shows "\<turnstile>x1"
t_designI: fixes V :: "'b set"   and B :: "'b set multiset"   and k :: "int"   and t :: "int"   and \<Lambda>\<^sub>t :: "int" assumes "incomplete_design V B k"   and "t_wise_balance V B t \<Lambda>\<^sub>t"   and "t \<le> k" shows "t_design V B k t \<Lambda>\<^sub>t"
reachable_isRef: fixes s :: "Store"   and l :: "Location"   and x :: "Value" assumes "s\<turnstile> l reachable_from x" shows "isRefV x"
prefix_to_wordinterval_ipset_from_cidr: fixes pfx :: "'b prefix_match" assumes "valid_prefix pfx" shows "wordinterval_to_set (prefix_to_wordinterval pfx) = ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
augm_funcTr: fixes n :: "nat"   and B :: "nat \<Rightarrow> ('d, 'e) Ring_scheme"   and f :: "nat \<Rightarrow> 'd" assumes "\<forall>k\<le>Suc n. Ring (B k)"   and "f \<in> carr_prodag {i. i \<le> Suc n} B" shows "f = augm_func n (restrict f {i. i \<le> n}) (Un_carrier {i. i \<le> n} B) (Suc 0) (\<lambda>x\<in>{0}. f (x + Suc n)) (Un_carrier {0} (compose {0} B (slide (Suc n))))"
arr_def: fixes f :: "'a" shows "C.arr f \<equiv> C.domains f \<noteq> {} \<or> C.codomains f \<noteq> {}"
sort_key_inj_key_eq: fixes xs :: "'c list"   and ys :: "'c list"   and f :: "'c \<Rightarrow> 'a" assumes "mset xs = mset ys"   and "inj_on f (set xs)"   and "sorted (map f ys)" shows "sort_key f xs = ys"
succ_set_on_bound_1: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and dir :: "('i, 'a) Direction" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>i \<in> rvars (\<T> s')"   and "dir = Positive \<or> dir = Negative" shows "\<not> \<unrhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>i) (LB dir s x\<^sub>i) \<longrightarrow> \<langle>\<V> s'\<rangle> x\<^sub>i = the (LB dir s x\<^sub>i)"
Inter_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A"   and "bi_total A" shows "rel_fun (rel_set (rel_set A)) (rel_set A) \<Inter> \<Inter>"
inject_22: fixes x22a :: "('a, 'b, 'c) exp"   and y22a :: "('a, 'b, 'c) exp" shows "(throw x22a = throw y22a) = (x22a = y22a)"
hyps_3: fixes D :: "char list" assumes "z \<noteq> D" shows "\<not> (P \<turnstile> z \<preceq>\<^sup>* D \<and> P \<turnstile> D \<preceq>\<^sup>* z)"
skeleton_impl_def: shows "skeleton_impl \<equiv> stat_start_nres \<bind> (\<lambda>_. let I = Map.empty in FOREACH\<^bsup>\<lambda>it I. outer_invar it (oGS_\<alpha> I)\<^esup> V0 (\<lambda>v0 I0. if \<not> is_done_oimpl v0 I0 then let s = initial_impl v0 I0 in WHILE\<^sub>T\<^bsup>invar v0 (oGS_\<alpha> I0) \<circ> GS.\<alpha>\<^esup> (\<lambda>s. \<not> path_is_empty_impl s) (\<lambda>s. select_edge_impl s \<bind> (\<lambda>(vo, s). case vo of None \<Rightarrow> pop_impl s | Some v \<Rightarrow> if is_on_stack_impl v s then collapse_impl v s else if \<not> is_done_impl v s then RETURN (push_impl v s) else RETURN s)) s \<bind> (\<lambda>(S, B, I, P). RETURN I) else RETURN I0) I \<bind> (\<lambda>r. stat_stop_nres \<bind> (\<lambda>_. RETURN r)))"
add_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "D x y \<and> D (x \<oplus> y) z" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
tolerate_error_def: fixes Y :: "'c set"   and E :: "'c set" shows "tolerate_error Y E = op_ivl_rep_of_set E \<bind> (\<lambda>(ei, es). op_ivl_rep_of_set Y \<bind> (\<lambda>(yi, ys). let ea = sup \<bar>ei\<bar> \<bar>es\<bar>; ya = sup \<bar>yi\<bar> \<bar>ys\<bar> in adaptive_rtol_spec \<bind> (\<lambda>rtol. adaptive_atol_spec \<bind> (\<lambda>atol. let errtol = sup (rtol *\<^sub>R ya) (atol *\<^sub>R sum_list Basis_list) in RETURN (ea \<le> errtol, infnorm ea)))))"
distinct_34: fixes pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "!pi' \<noteq> name1{name2}.pi"
disc_iff: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" shows "\<not> lnull (derivation_from St)"
eventually_at_filter: fixes "open" :: "'e set \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and a :: "'e"   and s :: "'e set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<forall>\<^sub>F x in topological_space.nhds open a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
NOT_VS_IN_DOM_PROJ_PRE_EFF: fixes v :: "'d"   and vs :: "'d set"   and a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "v \<notin> vs"   and "a \<in> PROB" shows "(v \<in> fmdom' (fst a) \<longrightarrow> v \<in> fmdom' (fst (action_proj a (prob_dom PROB - vs)))) \<and> (v \<in> fmdom' (snd a) \<longrightarrow> v \<in> fmdom' (snd (action_proj a (prob_dom PROB - vs))))"
step_Normal_elim_cases_7: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and b :: "'b set"   and c :: "('b, 'a, 'c) com"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (While b c, Normal s) \<rightarrow> u" obtains "u = (Seq c (While b c), Normal s)" and "s \<in> b"   | "u = (Skip, Normal s)" and "s \<notin> b"
assms_4: shows "h \<turnstile> adopt_node document_ptr node_ptr \<rightarrow>\<^sub>h h'"
share_all_until_volatile_write_commute: fixes 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"   and \<S> :: "nat \<Rightarrow> bool option"   and L :: "nat set"   and A :: "nat set"   and R :: "nat set"   and W :: "nat set" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> L = {}"   and "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> A = {}"   and "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> R = {}"   and "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> R = {}"   and "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> R = {}" shows "share_all_until_volatile_write ts \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L = share_all_until_volatile_write ts (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
times_divide_eq_right: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times a (divide b c) = divide (times a b) c"
VWF_def: shows "VWF \<equiv> VWO - Id"
ideD_1: fixes a :: "'b" assumes "C'.ide a" shows "C'.arr a"
\<mu>_bound_rowI: fixes i :: "nat"   and fs :: "int vec list"   and bnd :: "rat" assumes "\<And>j. j \<le> i \<Longrightarrow> (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)\<^sup>2 \<le> bnd" shows "\<mu>_bound_row fs bnd i"
mset_case_Suc: fixes a :: "nat"   and b :: "nat" shows "{#case x of 0 \<Rightarrow> c | Suc x \<Rightarrow> N ! x. x \<in># mset_set {Suc a..<Suc b}#} = {#N ! (x - 1). x \<in># mset_set {Suc a..<Suc b}#}"
assms_6: shows "pf stk (Suc n) w \<noteq> 0 \<or> pf risk_free_asset (Suc n) w \<noteq> 0"
h1: shows "all_list_constr_R (get_subsets_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (length (qs1 @ qs2))"
card_bit0: shows "CARD('b bit0) = 2 * CARD('b)"
Lambert_W_exp_times_self': fixes x :: "real" assumes "- 1 \<le> x" shows "Lambert_W (exp x * x) = x"
coeff_add: fixes p :: "'a poly"   and q :: "'a poly"   and n :: "nat" shows "coeff (p + q) n = coeff p n + coeff q n"
simps_8: fixes a :: "MIR.num" shows "numbound0 (Floor a) = numbound0 a"
orderI: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "Semilat A r f" shows "order r A"
nb\<^sub>1: shows "cnf ` set (map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index (\<Pi>\<^sub>\<O>) op))) \<^bold>\<or> Atom (State (Suc t) (index (\<Pi>\<^sub>\<V>) v))) (add_effects_of op)) = (\<lambda>v. {{(Operator t (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State (Suc t) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+}}) ` set (add_effects_of op)"
left_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b * c) = a * b * c"
prems_1: shows "\<exists>s. (a # p') @ s = M"
ide_MkIde: fixes A :: "'a \<Rightarrow> 'b" assumes "A \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))" shows "ide (MkIde A)"
fmdom'_drop_fset: fixes A :: "'a fset"   and m :: "('a, 'b) fmap" shows "fmdom' (fmdrop_fset A m) = fmdom' m - fset A"
inf_idem: fixes x :: "'c multiset" shows "x \<inter># x = x"
ts_def_3: shows "\<S>\<bullet>\<C> \<turnstile> [$Store t tp a off] : ts''' _> ts'"
opp_res_fold_A: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "ChamberComplexFolding A (restrict1 (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A C \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B' D) (\<Union> A))"
kruskal_acyclic_inv_1: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and f :: "'b"   and e :: "'b" assumes "class.bounded_distrib_kleene_allegory star one times sup less_eq less bot inf top conv"   and "less_eq (times f (conv f)) one"   and "times (times e (times (star (conv f)) (star f))) e = bot" shows "times (times (conv (inf f (times (times top e) (star (conv f))))) (star f)) e = bot"
setim_cmpfn: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and C :: "'f set"   and A1 :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C"   and "A1 \<subseteq> A" shows "compose A g f ` A1 = g ` f ` A1"
tensor_vec_def: fixes dims :: "nat list"   and vars :: "nat set"   and v1 :: "'a vec"   and v2 :: "'a vec" shows "partial_state.tensor_vec dims vars v1 v2 = Matrix.vec (state_sig.d dims) (\<lambda>i. v1 $ partial_state.encode1 dims vars i * v2 $ partial_state.encode2 dims vars i)"
f_aggregate_one: fixes k :: "nat"   and xs :: "'d list"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k"   and "length xs = k" shows "f_aggregate xs k ag = [ag xs]"
rec1: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set" assumes "ps \<noteq> []"   and "sps = sel gs bs ps (snd data)"   and "fst data \<le> count_const_lt_components (fst (compl gs bs (ps -- sps) sps (snd data)))" shows "P data D bs ps (full_gb (gs @ bs), D)"
append_eq_conv_conj: fixes xs :: "'k list"   and ys :: "'k list"   and zs :: "'k list" shows "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
equals0I: fixes A :: "'a set" assumes "\<And>y. y \<in> A \<Longrightarrow> False" shows "A = {}"
not_le_less: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "(\<not> a \<preceq>\<^bsub>D\<^esub> b) = b \<prec>\<^bsub>D\<^esub> a"
nprv_flsE: fixes F :: "'fmla set"   and \<phi> :: "'fmla" assumes "nprv F fls"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla" shows "nprv F \<phi>"