hcju/isabelleps huggingface dataset:
S_finite: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set" assumes "Finite_Markov_Decision_Process K S" shows "finite S"
FG_advice_b1: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and Y :: "'b ltln set" assumes "\<F>\<G> \<phi> w \<subseteq> Y"   and "w \<Turnstile>\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<mu>"
HNatInfinite_inverse_Infinitesimal: fixes n :: "nat star" assumes "n \<in> HNatInfinite" shows "inverse (hypreal_of_hypnat n) \<in> Infinitesimal"
typeI_two_cubes_trace_horizontal_boundaries: shows "two_chain_horizontal_boundary two_chain_typeI \<subseteq> one_chain_typeI"
in_quotientP_imp_non_empty: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "X \<noteq> {}"
2: shows "subterms\<^sub>s\<^sub>e\<^sub>t (pair ` setops\<^sub>s\<^sub>s\<^sub>t S) \<union> \<Union> ((set \<circ> fst \<circ> Ana) ` pair ` setops\<^sub>s\<^sub>s\<^sub>t S) - pair ` setops\<^sub>s\<^sub>s\<^sub>t S \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S)"
simps_1: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c" shows "rbt_map_entry k f Empty = Empty"
Inf_lower: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq (Inf A) x"
special_coset_eq_imp_eq_gensets: fixes S :: "'b set"   and T1 :: "'b set"   and T2 :: "'b set"   and w1 :: "'b"   and w2 :: "'b" assumes "PreCoxeterSystemWithDeletion S"   and "T1 \<in> Pow S"   and "T2 \<in> Pow S"   and "w1 +o \<langle>T1\<rangle> = w2 +o \<langle>T2\<rangle>" shows "T1 = T2"
smallo_1_iff: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real" assumes "eventually_nonneg F f" shows "((\<lambda>_. 1) \<in> o[F](f)) = filterlim f at_top F"
odd_mul_odd: fixes p :: "nat"   and q :: "nat" assumes "odd p"   and "odd q" shows "odd (p * q)"
firstocc: shows "\<forall>j<i. cs ! j \<noteq> cs ! i"
less_dag_Tip: fixes x :: "dag" shows "\<not> x < Tip"
assms_1: shows "inv_into M (ordermap M r) \<alpha> = inv_into M (ordermap M r) \<beta>"
scc_root_def: fixes E :: "('b \<times> 'b) set"   and s :: "('b, 'c) state_scheme"   and v :: "'b"   and scc :: "'b set" shows "scc_root' E s v scc = (is_scc E scc \<and> v \<in> scc \<and> v \<in> dom (discovered s) \<and> scc \<inter> dom (discovered s) \<subseteq> (tree_edges s)\<^sup>* `` {v})"
valid_behaviours_def: fixes state :: "'c \<Rightarrow> 'd set"   and msg :: "'d \<times> ('d, 'c) operation" shows "valid_behaviours state msg \<equiv> case msg of (i, Add j e) \<Rightarrow> i = j | (i, Rem is e) \<Rightarrow> is = state e"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data" shows "arr (R f) = arr f"
mono_SEQI2b: assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P `;` S) \<longrightarrow> Ra (P `;` S) s \<subseteq> Ra (Q `;` S) s"
HEndPhase0_Inv2a_dblock: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv1 s"   and "Inv2a_inner s p"   and "Inv2c_inner s p" shows "Inv2a_innermost s' p (dblock s' p)"
d1_def: fixes a :: "'a"   and b :: "'a" shows "d1 a b = (a l\<rightarrow> b) * (b l\<rightarrow> a)"
simps_5: fixes b :: "'a set"   and c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" shows "redexes (Cond b c\<^sub>1 c\<^sub>2) = {Cond b c\<^sub>1 c\<^sub>2}"
comp_right_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) * z = x * z \<squnion> y * z"
SAEvents_def: fixes SA :: "('b, 'a, 'c) seqauto" shows "SAEvents SA = (\<Union>l\<in>Label (Delta SA). fst (action l) \<union> ExprEvents (expr l))"
cf: shows "compl_fam (\<lambda>g. generate G {g}) (set gs)"
exhaustive_above_fusion_def: fixes g :: "('a, 'd) generator"   and a :: "'a"   and s :: "'d" shows "exhaustive_above_fusion g a s = exhaustive_above a (list.unfoldr g s)"
that: fixes s :: "nat \<Rightarrow> real" assumes "s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>"   and "(flow0 x \<circ> s) \<longlonglongrightarrow> p"   and "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x" shows "thesis"
range_subset_mem_starsetNat: fixes f :: "nat \<Rightarrow> 'a"   and A :: "'a set" assumes "range f \<subseteq> A" shows "(*f* f) whn \<in> *s* A"
simps_2: fixes \<mu> :: "'a" shows "Ide \<^bold>\<langle>\<mu>\<^bold>\<rangle> = ide \<mu>"
has_visible_field: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F:T in D" shows "P \<turnstile> C has F:T in D"
fv_trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_is_fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s: fixes F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "fv\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
id1: shows "smult c p = smult (content q) (primitive_part q) * smult (content r) (primitive_part r)"
preserves_cones: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and J :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and D :: "'d \<Rightarrow> 'a"   and a :: "'a"   and \<chi> :: "'d \<Rightarrow> 'a" assumes "functor A B F"   and "Limit.cone J A D a \<chi>" shows "Limit.cone J B (F \<circ> D) (F a) (F \<circ> \<chi>)"
list_of_mset_exi: fixes m :: "'a multiset" shows "\<exists>l. m = mset l"
face_cycle_set_def: fixes a :: "'b" shows "face_cycle_set a = orbit face_cycle_succ a"
matching_dsts: fixes rs :: "'a simple_rule list"   and c :: "parts_connection"   and s :: "'a word" assumes "\<forall>r\<in>set rs. simple_conn_matches (match_sel r) c" shows "wordinterval_to_set (matching_dsts s rs Empty_WordInterval) = {d. runFw s d c rs = Decision FinalAllow}"
full_checker_spec: fixes A :: "(nat, int mpoly) fmap"   and A' :: "int mpoly multiset"   and spec :: "int mpoly"   and pac :: "(int mpoly, nat, nat) pac_step list" assumes "(A, A') \<in> polys_rel" shows "full_checker spec A pac \<le> \<Down> {((st, G), st', G'). (st, st') \<in> status_rel \<and> (st \<noteq> FAILED \<longrightarrow> (G, G') \<in> polys_rel_full)} (PAC_checker_specification spec A')"
abc_steps_zero: fixes asm :: "nat \<times> nat list"   and ap :: "abc_inst list" shows "abc_steps_l asm ap 0 = asm"
zmset_of_empty: shows "zmset_of {#} = {#}\<^sub>z"
set_alist_of_remove_min: fixes q :: "('c, 'd) pq" assumes "\<not> PQ.is_empty q" shows "set (pq.alist_of (remove_min q)) = set (pq.alist_of q) - {(PQ.min q, the (PQ.priority q (PQ.min q)))}"
X: shows "X \<in> M \<rightarrow>\<^sub>M M'"
Q4_a_def: shows "Q4_a \<equiv> [: X, mrk \<leadsto> X', mrk' . \<exists>x\<in>X. (\<forall>y. (x, y) \<in> next \<longrightarrow> y \<in> mrk) \<and> X' = X - {x} \<and> mrk' = mrk :]"
root2_def: shows "root2 \<equiv> prod.fst (fix_syn (\<lambda>(root2, op2, rep2, left2, right2). (\<Lambda> pat. rep2\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root2\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op2\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>Null))) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_scons\<cdot>v\<cdot>(\<Lambda> u us. Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>(sfoldl\<cdot>(op2\<cdot>pat)\<cdot>(root2\<cdot>pat)\<cdot>us))))), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> us v. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>Null)) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> us va. match_scons\<cdot>va\<cdot>(\<Lambda> v vs. Fixrec.succeed\<cdot>(rep2\<cdot>pat\<cdot>(us :@ [:v:], vs))))))))"
emeasure_real: fixes p :: "'c spmf"   and A :: "'c set" shows "\<exists>r\<ge>0. emeasure (measure_spmf p) A = ennreal r"
existence_ivl_initial_time: fixes t0 :: "real"   and x0 :: "real" assumes "t0 \<in> T"   and "x0 \<in> X" shows "t0 \<in> existence_ivl t0 x0"
Sup_bot_conv_1: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup A = bot) = (\<forall>x\<in>A. x = bot)"
hyps: fixes l :: "bit list" shows "length (selecthelp l x) = x + 1"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "Right.seq g f" shows "Right.cod (Right.comp g f) = Right.cod g"
inv_inv: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> (\<ominus> x) = x"
timpl_closure_set_def: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" shows "timpl_closure_set M TI \<equiv> \<Union>t\<in>M. timpl_closure t TI"
subst_def: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm" shows "A[x ::= M] \<equiv> THE X. substitutes A x M X"
simps_3: fixes xs :: "'c"   and x :: "'d"   and delta' :: "'e"   and inp' :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and binp' :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option" shows "(qVar xs x #= qOp delta' inp' binp') = False"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys x = {k. lookup x k \<noteq> (0::'b)}"
transp_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "transp r = trans {(x, y). r x y}"
proj2_pt_scalar: fixes v :: "(real, 2) vec" shows "\<exists>c. c \<noteq> 0 \<and> proj2_rep (proj2_pt v) = c *\<^sub>R vector2_append1 v"
vec_of_basis_enum_to_inverse: fixes \<psi> :: "'a" shows "vec_of_basis_enum (inverse \<psi>) = vec_of_list [inverse (vec_of_basis_enum \<psi> $ 0)]"
foldr_And_map_rlfm2: fixes f :: "'a \<Rightarrow> fm"   and xs :: "'a list" assumes "\<forall>n. isrlfm (f n)" shows "isrlfm (foldr MIR.conj (map f xs) T)"
ord_eq_le_trans: fixes a :: "('c, 'd) Term.term"   and b :: "('c, 'd) Term.term"   and c :: "('c, 'd) Term.term" assumes "a = b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
uminus_eq: fixes x :: "'a" shows "- x = x \<leadsto> bot"
in_set_conv_nth: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = (\<exists>i<List.length xs. xs ! i = x)"
pell_valuation_eq_iff: fixes z1 :: "int \<times> int"   and z2 :: "int \<times> int" shows "(pell_valuation z1 = pell_valuation z2) = (z1 = z2)"
RedFAccNull: fixes F :: "char list"   and D :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>null\<bullet>F{D},s\<rangle> \<rightarrow> \<langle>THROW NullPointer,s\<rangle>"
Auth_PriK_def: shows "Auth_PriK \<equiv> SOME f. infinite (- range f)"
sep_disj_prod_conv: fixes a :: "'a"   and x :: "'b"   and b :: "'a"   and y :: "'b" shows "(a, x) ## (b, y) = (a ## b \<and> x ## y)"
reverse_multi_step_pres_denot: fixes e :: "exp"   and e' :: "exp"   and v :: "val"   and \<rho> :: "(nat \<times> val) list" assumes "e \<longrightarrow>* e'"   and "v \<in> E e' \<rho>" shows "v \<in> E e \<rho>"
match_iface_set: fixes i :: "char list"   and p_iface :: "char list" shows "match_iface (Iface i) p_iface = (p_iface \<in> internal_iface_name_to_set i)"
valid_empty_edges_iff_exists_offending_flows: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" assumes "sinvar_mono"   and "wf_graph G"   and "\<not> sinvar G nP" shows "sinvar \<lparr>nodes = nodes G, edges = {}\<rparr> nP = (set_offending_flows G nP \<noteq> {})"
opt_def: fixes A :: "'b set" shows "opt A = insert None {Some y |y. y \<in> A}"
in_homE: fixes f :: "(bool, bool list) concrete_category.arr"   and a :: "(bool, bool list) concrete_category.arr"   and b :: "(bool, bool list) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" obtains "J.arr f" and "J.dom f = a" and "J.cod f = b"
llast_ldropn: fixes n :: "nat"   and xs :: "'b llist" assumes "enat n < llength xs" shows "llast (ldropn n xs) = llast xs"
comm_monoid_mult_axioms: shows "class.comm_monoid_mult (*) (1::'a)"
insort_key_left_comm: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'d"   and xs :: "'d list" assumes "class.linorder less_eq less"   and "f x \<noteq> f y" shows "linorder.insort_key less_eq f y (linorder.insort_key less_eq f x xs) = linorder.insort_key less_eq f x (linorder.insort_key less_eq f y xs)"
del_ops_def: fixes rects :: "'a rectangle list" shows "del_ops rects = list (del_op rects) (length rects)"
Sigma_fm_All2_Var: fixes A :: "fm"   and j :: "name"   and i :: "name" assumes "Sigma_fm A"   and "atom j \<sharp> (i, A)" shows "Sigma_fm (All2 i (Var j) A)"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'd"   and xa :: "'c \<Rightarrow>\<^sub>0 'd" shows "poly_mapping.lookup (x - xa) = (\<lambda>k. poly_mapping.lookup x k - poly_mapping.lookup xa k)"
asimp_def: fixes a :: "atom" shows "asimp a = (case a of Less r cs \<Rightarrow> if \<forall>c\<in>set cs. c = 0 then if r < 0 then TrueF else FalseF else Atom a | Eq r cs \<Rightarrow> if \<forall>c\<in>set cs. c = 0 then if r = 0 then TrueF else FalseF else Atom a)"
ind_hyp: fixes xl :: "'a list" shows "(xl \<in> insert_in_list_at_arbitrary_pos x l) = (\<exists>l1 l2. l = l1 @ l2 \<and> xl = l1 @ [x] @ l2)"
set_intros_1: fixes x1 :: "'c" shows "x1 \<in> set1_blindable\<^sub>m (Unblinded x1)"
compact_space_def: fixes X :: "'a topology" shows "compact_space X \<equiv> compactin X (topspace X)"
that: fixes x :: "'a"   and y :: "'a" assumes "xy = x + y"   and "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "thesis"
front_tickFree_def: fixes s :: "'b event list" shows "front_tickFree s = (s = [] \<or> tickFree (tl (rev s)))"
UnCI: fixes c :: "'e"   and B :: "'e set"   and A :: "'e set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
Join_commute: fixes F :: "'d program"   and G :: "'d program" shows "F \<squnion> G = G \<squnion> F"
round_unique': fixes x :: "'a"   and n :: "int" assumes "\<bar>x - of_int n\<bar> < (1::'a) / (2::'a)" shows "round x = n"
tyenv_wellformed_mode_update: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and mds :: "Mode \<Rightarrow> 'a set"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set"   and upd :: "'a ModeUpd" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.tyenv_wellformed bexp_vars \<C>_vars mds \<Gamma> \<S> P"   and "sifum_types.anno_type_stable bexp_vars \<C>_vars \<Gamma> \<S> upd" shows "sifum_types.tyenv_wellformed bexp_vars \<C>_vars (sifum_lang_no_dma.update_modes upd mds) (sifum_types.add_anno \<C>_vars dma_type \<Gamma> \<S> upd) (sifum_types.add_anno_stable \<S> upd) (sifum_lang_no_dma.restrict_preds_to_vars bexp_vars P {v. sifum_lang_no_dma.stable (sifum_types.add_anno_stable \<S> upd) v})"
open_subopen: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open" shows "open S = (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
dependent_mono: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'d set"   and A :: "'d set" assumes "module scale"   and "module.dependent scale B"   and "B \<subseteq> A" shows "module.dependent scale A"
rightreg_scalar_mult_def: fixes r :: "'r"   and f :: "'r \<Rightarrow> 'b" shows "r \<currency> f = (\<lambda>x. if x \<in> R then f (x * r) else (0::'b))"
dirvec_nth_polychain_of: fixes n :: "nat"   and xs :: "'a list"   and p :: "'a" assumes "n < length xs" shows "dirvec (polychain_of p xs ! n) = xs ! n"
v: shows "take i u \<cdot> [v ! i] \<cdot> drop (Suc i) v = v"
RModHom_composite_left: fixes N :: "'w set"   and smult'' :: "'f \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'w \<Rightarrow> 'b" assumes "ImG \<subseteq> N"   and "RModuleHom UNIV (\<star>) N smult'' S" shows "RModuleHom UNIV (\<cdot>) V smult'' (S \<circ> T)"
is_Write\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Write\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = True"
weak_ranking_index_unique: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'c set list"   and i :: "nat"   and j :: "nat"   and x :: "'c" assumes "finite_total_preorder_on carrier le"   and "is_weak_ranking xs"   and "i < length xs"   and "j < length xs"   and "x \<in> xs ! i"   and "x \<in> xs ! j" shows "i = j"
mult_row_code_nth: fixes A :: "(('b, 'c) vec, 'a) vec"   and a :: "'a"   and q :: "'b" shows "($) (mult_row A a q) = mult_row_row A a q"
assms_2: shows "g \<in> elts (VPi A B)"
relation_of_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "relation_of P A \<equiv> {(a, b). (a, b) \<in> A \<times> A \<and> P a b}"
b: shows "a = (ln l, b) \<or> a = (\<star>, b)"
valof_valof1: fixes e' :: "exp"   and s :: "loc \<Rightarrow> nat"   and v :: "nat"   and s' :: "loc \<Rightarrow> nat"   and c1 :: "com"   and c2 :: "com"   and e :: "exp" assumes "(e', s) -|-> (v, s')"   and "e' = VALOF c1 RESULTIS VALOF c2 RESULTIS e" shows "(VALOF c1;;c2 RESULTIS e, s) -|-> (v, s')"
poly_weak_mono_def: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" shows "poly_weak_mono p v \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> \<beta> v \<le> \<alpha> v \<longrightarrow> eval_poly \<beta> p \<le> eval_poly \<alpha> p"
gen_span_closed: fixes gens :: "(bool \<times> 'a) list set" assumes "gens \<subseteq> carrier \<F>\<^bsub>gens\<^esub>" shows "\<langle>gens\<rangle>\<^bsub>\<F>\<^bsub>gens\<^esub>\<^esub> \<subseteq> carrier \<F>\<^bsub>gens\<^esub>"
dependent_def: fixes P :: "('a, 'b) vec set" shows "vec.dependent P = (\<exists>a\<in>P. a \<in> vec.span (P - {a}))"
fcompare_double_Eq2: fixes y :: "double"   and z :: "double"   and x :: "double"   and c :: "ccode" assumes "fcompare_double y z = ccode.Eq"   and "fcompare_double x y = c" shows "fcompare_double x z = c"
surj_plus: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "surj (plus a)"
preserves_iso: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.iso (fst f \<star>\<^sub>C snd f)"
mult_left_le: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq c one"   and "less_eq zero a" shows "less_eq (times a c) a"
int_vertices_simps_2: fixes vv :: "'a \<times> 'a"   and p :: "('a \<times> 'a) list" shows "int_vertices (vv # p) = insert (fst vv) (int_vertices p)"
elims_2: fixes x :: "'b list option"   and xa :: "'b list option" assumes "proper_interval x xa" obtains "x = None" and "xa = None"   | xs :: "'b list" where "x = None" and "xa = Some xsa__" and "xsa__ \<noteq> []"   | xs :: "'b list" where "x = Some xsb__" and "xa = None"   | xs :: "'b list" and ys :: "'b list" where "x = Some xs" and "xa = Some ys" and "List_Proper_Interval.proper_interval_list_aux xs ys"
relcomppE: fixes r :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and s :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and a :: "'n"   and c :: "'p" assumes "(r OO s) a c" obtains b :: "'o" where "r a b" and "s b c"
LIM_D: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and a :: "'a"   and r :: "real" assumes "f \<midarrow>a\<rightarrow> L"   and "0 < r" shows "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
neqE: fixes x :: "'c word"   and y :: "'c word" assumes "x \<noteq> y" obtains "x <s y"   | "y <s x"
val: fixes s :: "'a"   and v :: "'b" assumes "r = (s, Rval v)" shows "thesis"
VDMConseq: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "G \<rhd> c : A"   and "\<forall>s t. A s t \<longrightarrow> B s t" shows "G \<rhd> c : B"
Myhill_Nerode2: fixes r :: "'a rexp" shows "finite (UNIV // \<approx>lang r)"
list_order_insert_ref: fixes ops :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a" assumes "insert_ops (ops @ [(oid, Some ref)])"   and "ref \<in> set (interp_ins ops)" shows "list_order (ops @ [(oid, Some ref)]) ref oid"
fi: fixes i :: "nat" shows "f i \<le> f 0 - \<delta> * of_nat i"
bot_meet: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<bottom> \<sqinter> x = \<bottom>"
ccSquare_eqvt: fixes \<pi> :: "perm"   and S :: "Vars.var set" shows "\<pi> \<bullet> S\<^sup>2 = (\<pi> \<bullet> S)\<^sup>2"
flat_ord_compat_2: fixes x :: "'b"   and y :: "'b" assumes "flat_ge x y" shows "y \<le> x"
that: fixes ops :: "'a strips_operator list"   and \<pi>' :: "'a strips_operator list list" assumes "\<pi> = ops # \<pi>'" shows "thesis"
finite_conv_nat_seg_image: fixes A :: "'k set" shows "finite A = (\<exists>n f. A = f ` {i. i < n})"
map_default_inject_2: fixes b :: "'b"   and b' :: "'b"   and g :: "'a \<Rightarrow> 'b option"   and g' :: "'a \<Rightarrow> 'b option" assumes "infinite UNIV \<or> b = b'"   and "finite (dom g)"   and "b \<notin> ran g"   and "finite (dom g')"   and "b' \<notin> ran g'"   and "map_default b g = map_default b' g'" shows "g = g'"
that: assumes "xs \<in>\<^sub>\<circ> vproduct (vcard xs) A"   and "x \<in>\<^sub>\<circ> A (vcard xs)"   and "n = vcard (xs #\<^sub>\<circ> x)" shows "thesis"
that: fixes f :: "V"   and f' :: "V"   and f'' :: "V"   and a :: "V"   and a' :: "V"   and a'' :: "V"   and b :: "V"   and b' :: "V"   and b'' :: "V" assumes "F = [f, f', f'']\<^sub>\<circ>"   and "A = [a, a', a'']\<^sub>\<circ>"   and "B = [b, b', b'']\<^sub>\<circ>"   and "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"   and "f' : a' \<mapsto>\<^bsub>\<BB>\<^esub> b'"   and "f'' : a'' \<mapsto>\<^bsub>\<CC>\<^esub> b''" shows "thesis"