hcju/isabelleps huggingface dataset:
isomorphic_to_zero_right: fixes G :: "'b set"   and smult :: "('c, 'b) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and V :: "'d set"   and smult' :: "('c, 'b) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and T :: "'d \<Rightarrow> 'e"   and W :: "'e set" assumes "FGModuleIso G smult V smult' T W"   and "W = 0" shows "V = 0"
nsubst_hp_fml: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<alpha> :: "('sf + 'b, 'sc, 'sz) hp"   and \<phi> :: "('sf + 'b, 'sc, 'sz) formula" assumes "is_interp I" shows "(NPadmit \<sigma> \<alpha> \<longrightarrow> hpsafe \<alpha> \<longrightarrow> (\<forall>i. dsafe (\<sigma> i)) \<longrightarrow> (\<forall>\<nu> \<omega>. ((\<nu>, \<omega>) \<in> prog_sem I (PsubstFO \<alpha> \<sigma>)) = ((\<nu>, \<omega>) \<in> prog_sem (adjointFO I \<sigma> \<nu>) \<alpha>))) \<and> (NFadmit \<sigma> \<phi> \<longrightarrow> fsafe \<phi> \<longrightarrow> (\<forall>i. dsafe (\<sigma> i)) \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (FsubstFO \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (adjointFO I \<sigma> \<nu>) \<phi>)))"
interior_injective_linear_image: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "linear f"   and "inj f" shows "interior (f ` S) = f ` interior S"
event_occurs_on_unique: fixes p :: "'d"   and q :: "'d"   and ev :: "('d, 'e, 'f) event" assumes "p \<noteq> q"   and "occurs_on ev = p" shows "occurs_on ev \<noteq> q"
hom_relboundary_empty: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "hom_relboundary p X S {} = hom_boundary p X S"
init_tr_wf: fixes M :: "nat bdd list \<times> bool list" shows "wf_tr M (init_tr M)"
fls_lr_inverse_X_2: fixes y :: "'a" shows "fls_right_inverse fls_X y = fls_shift 1 (fls_const y)"
l_inv_ex: fixes x :: "int" assumes "x \<in> carrier (Z n)" shows "\<exists>y\<in>carrier (Z n). y \<otimes>\<^bsub>Z n\<^esub> x = \<one>\<^bsub>Z n\<^esub>"
add_mset_add_single: fixes a :: "'b"   and A :: "'b multiset" shows "add_mset a A = A + {#a#}"
that: fixes c :: "real" assumes "summable (\<lambda>x. (\<MM> (real x) - ln (real x) + c) / real x)" shows "thesis"
lookup_env_delete: fixes x' :: "'c"   and x :: "'c"   and m :: "'c \<Rightarrow> 'd" assumes "x' \<noteq> x" shows "env_delete x m x' = m x'"
open_imp_locally_compact: fixes S :: "'c set" assumes "open S" shows "locally compact S"
mult_eq_0_not_zero_divisor_left: fixes a :: "'b"   and b :: "'b" assumes "\<not> zero_divisor a" shows "(a * b = (0::'b)) = (b = (0::'b))"
isolate_variable_sparse_degree_eq_zero_iff: fixes p :: "'a mpoly"   and v :: "nat" shows "(isolate_variable_sparse p v (MPoly_Type.degree p v) = 0) = (p = 0)"
matrix_to_iarray_eq_of_fun: fixes A :: "(('b, 'c) vec, 'a) vec"   and f :: "nat \<Rightarrow> 'b iarray"   and n :: "nat" assumes "\<forall>i. vec_to_iarray (A $ i) = f (mod_type_class.to_nat i)"   and "n = IArray.length (matrix_to_iarray A)" shows "matrix_to_iarray A = IArray.of_fun f n"
converse_rtranclp_induct3: fixes r :: "'a \<times> 'b \<times> 'c \<Rightarrow> 'a \<times> 'b \<times> 'c \<Rightarrow> bool"   and ax :: "'a"   and ay :: "'b"   and az :: "'c"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay, az) (bx, by, bz)"   and "P bx by bz"   and "\<And>a aa b ab ac ba. \<lbrakk>r (a, aa, b) (ab, ac, ba); r\<^sup>*\<^sup>* (ab, ac, ba) (bx, by, bz); P ab ac ba\<rbrakk> \<Longrightarrow> P a aa b" shows "P ax ay az"
zero_power: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.semiring_1 one times plus zero"   and "0 < n" shows "power.power one times zero n = zero"
axioms_2: fixes G :: "('a, 'b) monoid_scheme" assumes "comm_group G" shows "Group.group G"
varTrue: fixes b :: "BExpr"   and t :: "Var \<Rightarrow> Val"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and r :: "Var \<Rightarrow> Val"   and s :: "Var \<Rightarrow> Val" assumes "evalB b t"   and "\<Phi> (r, t)"   and "\<forall>w.  r \<approx> w  \<longrightarrow> (b, \<Phi>, s, w) \<in> var" shows "(b, \<Phi>, s, t) \<in> var"
forks_state_update_def: fixes N :: "nat"   and fs :: "nat list"   and e :: "dining_event" assumes "DiningPhilosophers N" shows "DiningPhilosophers.forks_state_update N fs e = (let i = fork e in fs[i := DiningPhilosophers.fork_state_update N i (fs ! i) e])"
B2_def: shows "B2 = {SOME x. x \<in> U |U. U \<in> A}"
sym_block_mult_one: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and bl :: "'a set" assumes "symmetric_bibd point_set block_collection u_block_size index"   and "bl \<in># block_collection" shows "count block_collection bl = 1"
inject_3: fixes x3 :: "nat"   and y3 :: "nat" shows "(Greater x3 = Greater y3) = (x3 = y3)"
A_FactGroup_inj_on: shows "inj_on (\<lambda>X. the_elem (h ` X)) (carrier (G A_Mod a_kernel G H h))"
pt_fun_eq: fixes pi :: "('b \<times> 'b) list"   and f :: "'a \<Rightarrow> 'c" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "(pi \<bullet> f = f) = (\<forall>x. pi \<bullet> f x = f (pi \<bullet> x))"
simps_1: fixes uu :: "Core.term"   and uv :: "Core.term" shows "is_app (uu $ uv) = True"
Cd_singleton: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" shows "(x \<in> Cd d X) = (Cd d X = {x})"
b_def: shows "b = res_lim (f \<circ> t)"
lta__nlea: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F" shows "\<not> Tarski_neutral_dimensionless.LeA Bet Cong D E F A B C"
unat_eq_nat_uint: fixes w :: "'b word" shows "unat w = nat (uint w)"
assms: shows "\<Psi> \<rhd> P \<lessapprox><Rel> Q"
wpo_ns_imp_NS: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" assumes "s \<succeq> t" shows "(s, t) \<in> NS"
lv_ref_rel_def: shows "lv_ref_rel = {(sa, sc). sc = \<lparr>opt_mru_state.next_round = v_state.next_round sa, mru_vote = process_mru (votes sa), decisions = v_state.decisions sa\<rparr>}"
lmap_lmap: fixes f :: "'c \<rightarrow> 'b"   and g :: "'d \<rightarrow> 'c"   and xs :: "'d llist" shows "lmap\<cdot>f\<cdot>(lmap\<cdot>g\<cdot>xs) = lmap\<cdot>(f oo g)\<cdot>xs"
pid_system_automaton'_simps_2: shows "g_E pid_system_automaton' = pid_gstep"
cmimi: shows "m\<inverse> O m\<inverse> \<subseteq> b\<inverse>"
hb_completionI: 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" assumes "\<And>ttas s' t x ta x' m' i. \<lbrakk>s -\<triangleright>ttas\<rightarrow>* s'; non_speculative P (w_values P (\<lambda>_. {}) (map snd E)) (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))); thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor>; t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>; actions_ok s' t ta; non_speculative P (w_values P (w_values P (\<lambda>_. {}) (map snd E)) (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>))\<rbrakk> \<Longrightarrow> \<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> take i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> = take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<and> ta_hb_consistent P (E @ concat (map (\<lambda>(t, ta). map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas) @ map (Pair t) (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) (llist_of (map (Pair t) (drop i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>))) \<and> (i < length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<longrightarrow> i < length \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>) \<and> (if \<exists>ad al v. \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i = NormalAction (ReadMem ad al v) then sim_action else (=)) (\<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i) (\<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> ! i)" shows "hb_completion s E"
pgen_lossless_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('b, 'c) \<I>"   and gpv :: "('d, 'b, 'c) gpv" shows "pgen_lossless_gpv fail \<I> gpv = (expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1)"
cinner_zero_left: 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 "cinner zero x = 0"
simps_2: fixes x :: "'b"   and t :: "'b bro" shows "ins x (N1 t) = n1 (ins x t)"
Bool: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and \<phi> :: "('a, 'b) Language"   and i :: "bool" assumes "Interpretation I"   and "L\<lbrakk>\<phi>\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IBool i" shows "\<exists>A B e d. \<phi> = Vx : A \<turnstile> e\<equiv>d : B"
nat_pow_Suc2: fixes x :: "'a list"   and n :: "nat" assumes "x \<in> carrier (poly_ring R)" shows "x [^]\<^bsub>poly_ring R\<^esub> Suc n = x \<otimes>\<^bsub>poly_ring R\<^esub> x [^]\<^bsub>poly_ring R\<^esub> n"
UnE: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
rk2_param_impl_def: shows "rk2_param_impl = (let rk = rk2_param optns in if 0 < rk \<and> rk \<le> 1 then rk else 1)"
disjnt_insert2: fixes Y :: "'b set"   and a :: "'b"   and X :: "'b set" shows "disjnt Y (insert a X) = (a \<notin> Y \<and> disjnt Y X)"
Compl_lessThan: fixes k :: "'a" shows "- {..<k} = {k..}"
A: shows "xs \<in> permutations_of_multiset (image_mset f A)"
simple_fw_alt: fixes r :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme" shows "simple_fw r p = simple_fw_alt r p"
bvars_decomp_rm: fixes A :: "('fun, 'var) extstrand_step list" shows "bvars\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) = bvars\<^sub>e\<^sub>s\<^sub>t A"
lookup_single_not_eq: fixes k :: "'b"   and k' :: "'b"   and v :: "'c" assumes "k \<noteq> k'" shows "lookup (monomial v k) k' = (0::'c)"
ANDRIGHT_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set"   and Y :: "ctrm set" shows "pi \<bullet> ANDRIGHT (A AND B) X Y = ANDRIGHT (A AND B) (pi \<bullet> X) (pi \<bullet> Y)"
classes_changed_self: fixes P :: "'b prog" shows "classes_changed P P = {}"
continuous_map_Euclidean_space_iff: fixes m :: "nat"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> 'a" shows "continuous_map (Euclidean_space m) euclidean g = continuous_on (topspace (Euclidean_space m)) g"
contro: shows "\<nexists>v max_path. is_trail v max_path n \<and> (\<forall>v'. \<forall>e\<in>E. \<not> is_trail v' (e # max_path) n)"
simps: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and ax :: "Core.term list" shows "exe_wf_theory (ExeTheory (ExeSignature cto tao sa) ax) = (exe_sig_conds (ExeSignature cto tao sa) \<and> (\<forall>p\<in>set ax. term_ok (translate_theory (ExeTheory (ExeSignature cto tao sa) ax)) p \<and> typ_of p = Some propT) \<and> is_std_sig (translate_signature (ExeSignature cto tao sa)) \<and> exe_wf_sig (ExeSignature cto tao sa) \<and> eq_axs \<subseteq> set ax)"
space_empty_eq_bot: fixes a :: "'b measure" shows "(space a = {}) = (a = \<bottom>)"
compl_eq_compl_iff: fixes x :: "'a upred"   and y :: "'a upred" shows "((\<not> x) = (\<not> y)) = (x = y)"
vfiniteE1: fixes A :: "V" assumes "vfinite A" obtains "\<exists>n\<in>elts \<omega>. n \<approx>\<^sub>\<circ> A"
app\<^sub>iPop: fixes P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (Pop, P, pc, mxs, T\<^sub>r, s) = (\<exists>ts ST LT. s = (ts # ST, LT))"
simps_2: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and a :: "'c" shows "eval_tpoly \<alpha> (PNum a) = a"
cExt: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', S, T) \<in> Rel" shows "(\<Psi>' \<otimes> \<Psi>'', S, T) \<in> Rel"
weakBisimParAssoc: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> Q \<parallel> R \<approx> P \<parallel> (Q \<parallel> R)"
assms_3: shows "r \<cdot> p \<cdot> l \<le> l \<cdot> r"
space_T: fixes s :: "'b \<times> 'a" shows "space (T s) = space (stream_space (count_space UNIV))"
hyps: shows "(\<Sum> {0..n})\<^sup>2 = (\<Sum>i = 0..n. i ^ 3)"
is_monomial_uminus: fixes p :: "'d \<Rightarrow>\<^sub>0 'e" shows "is_monomial (- p) = is_monomial p"
cnf_of_encode_operator_precondition: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and op :: "'b strips_operator" shows "cnf (encode_operator_precondition \<Pi> t op) = (\<Union>v\<in>set (precondition_of op). {{(Operator t (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State t (index (\<Pi>\<^sub>\<V>) v))\<^sup>+}})"
indspaceD_transform: fixes f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set"   and g :: "'b"   and x :: "('a, 'b) aezfun"   and a :: "'a" assumes "f \<in> aezfun_scalar_mult.indspace smult G FH V"   and "g \<in> G"   and "x \<in> FH" shows "f (a \<delta>\<delta> g * x) = smult (a \<delta>\<delta> g) (f x)"
a_antitone': fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "ar y \<le> ar x"
path_hd: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list" assumes "local.path g (n # ns)" shows "local.path g [n]"
T_on_rand'_nn: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and A :: "('b \<Rightarrow> 'e pmf) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) pmf)"   and s :: "('b \<times> 'e) pmf"   and qs :: "'c list" shows "0 \<le> On_Off.T_on_rand' step t A s qs"
st_neg_numeral: fixes w :: "num" shows "st (- numeral w) = - numeral w"
inv_closed: fixes x :: "'c" assumes "x \<in> carrier H" shows "inv\<^bsub>H\<^esub> x \<in> carrier H"
inv_ordsetmap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "a \<in> f ` P"   and "b \<in> f ` P"   and "less_eq' a b" shows "less_eq (the_inv_into P f a) (the_inv_into P f b)"
ftor: shows "Ftor (YFtor C @@ Y) : Op C \<longrightarrow> SET"
VDM_Sound_n: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> c : A" shows "\<forall>n. G \<Turnstile>\<^sub>n c : A"
nat_le_omega: fixes n :: "nat" shows "ordinal_of_nat n \<le> \<omega>"
lexext_simps_3: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" shows "lexext gt (y # ys) (x # xs) = (gt y x \<or> x = y \<and> lexext gt ys xs)"
put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ptr_in_heap: fixes object_ptr :: "(_) object_ptr"   and object :: "(_) Object"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put object_ptr object h = h'" shows "object_ptr |\<in>| object_ptr_kinds h'"
ts_c_def_2: shows "ts' = ts_c @ [t]"
extract_viewres_def: fixes r :: "('a, 'b) viewres" shows "extract_viewres r \<equiv> if extract_viewres_isNone r then None else Some (extract_viewres_a r, extract_viewres_t r)"
steps_steps': fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and s :: "'c list" assumes "length qs = length as" shows "Partial_Cost_Model.steps s qs as = steps' s qs as (length as)"
squarefree': fixes x :: "real" shows "\<not> (poly (p div d) x = 0 \<and> poly (sturm_squarefree' p ! 1) x = 0)"
assms: fixes x :: "'a"   and y :: "'b"   and z :: "'c" shows "mf x y z \<equiv> ASSERT (P x y z) \<bind> (\<lambda>_. RETURN (f x y z))"
SC_discr_imp_SC_ZObis: fixes c :: "('test, 'atom) com" assumes "SC_discr c" shows "SC_ZObis c"
cases: fixes P :: "'a prog"   and a1 :: "ty"   and a2 :: "ty" assumes "P \<turnstile> a1 \<le> a2" obtains T :: "ty" where "a1 = T" and "a2 = T"   | C :: "String.literal" and D :: "String.literal" where "a1 = Class C__" and "a2 = Class D" and "P \<turnstile> C__ \<preceq>\<^sup>* D"   | C :: "String.literal" where "a1 = NT" and "a2 = Class C"   | A :: "ty" where "a1 = NT" and "a2 = A__\<lfloor>\<rceil>"   | A :: "ty" where "a1 = Aa__\<lfloor>\<rceil>" and "a2 = Class Object"   | A :: "ty" and B :: "ty" where "a1 = A\<lfloor>\<rceil>" and "a2 = B\<lfloor>\<rceil>" and "P \<turnstile> A \<le> B"
cond_iff: fixes P :: "'a \<Rightarrow> bool"   and X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "IF P THEN X ELSE Y FI = cond_sugar P X Y"
fps_def: fixes L :: "('a, 'b) eq_object_scheme"   and f :: "'a \<Rightarrow> 'a" shows "fps L f = {x. x \<in> carrier L \<and> f x .=\<^bsub>L\<^esub> x}"
dividesI': fixes b :: "'c"   and G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and c :: "'c" assumes "b = a \<otimes>\<^bsub>G\<^esub> c"   and "c \<in> carrier G" shows "a divides\<^bsub>G\<^esub> b"
absolute_safe_distance_def: shows "absolute_safe_distance = - v\<^sub>e\<^sup>2 / (2 * a\<^sub>e)"
lt_plus_le_max: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "lt (p + q) \<preceq>\<^sub>t ord_term_lin.max (lt p) (lt q)"
E22: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> \<diamond>F = \<diamond>G"
row_space_eq_col_space_transpose: fixes A :: "(('b, 'c) vec, 'd) vec" shows "row_space A = col_space (Finite_Cartesian_Product.transpose A)"
a: shows "!z \<cdot> x = x \<cdot> !z"
wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t_singleton: fixes s :: "('d, 'c) stateful_strand_step" shows "wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t [s] = wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p s"
one_strongly_connected_implies_6_eq: fixes x :: "'a" assumes "one_strongly_connected x" shows "x ; 1 ; x = x ; x\<^sup>+"
inv1: shows "opnet onp p\<^sub>1 \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p\<^sub>1) \<rightarrow>) globala (\<lambda>a. case a of (\<sigma>, aa, \<sigma>') \<Rightarrow> castmsg (I \<sigma>) aa \<and> (aa \<noteq> \<tau> \<and> (\<forall>i d. aa \<noteq> i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>1. S (\<sigma> i) (\<sigma>' i))) \<and> (aa = \<tau> \<or> (\<exists>i d. aa = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>1. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p\<^sub>1 \<longrightarrow> \<sigma>' i = \<sigma> i)))"
KBasic2_2: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<not>\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<diamond>\<^bold>\<not>\<phi> in v]"
distinct_1: fixes x1 :: "('a, 'b) atm"   and x2 :: "('a, 'b) atm" shows "Pos x1 \<noteq> Neg x2"
terminates_normalize_to_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>Language.normalize c \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
inv_dghm_ArrMap_vdomain: shows "\<D>\<^sub>\<circ> (inv_dghm \<FF>\<lparr>ArrMap\<rparr>) = \<BB>\<lparr>Arr\<rparr>"
bernoulli_stream_pref_prob_pos: fixes n :: "nat"   and x :: "bool stream" assumes "0 < p"   and "p < 1" shows "0 < emeasure M {w \<in> space M. stake n w = stake n x}"
max2: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> getRel idt G" shows "x \<in> vertices G"
select_convs_2: fixes init :: "'c set"   and trans :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e" shows "automaton.trans \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = trans"
hd_neg: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Minus $$ e \<Rightarrow> (\<lambda>\<sigma> x. f \<sigma> (op_sem Minus x))"
comp_eq_dest: fixes a :: "'e \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'e"   and c :: "'f \<Rightarrow> 'd"   and d :: "'c \<Rightarrow> 'f"   and v :: "'c" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
map_scale_one_left: fixes t :: "'a \<Rightarrow>\<^sub>0 'b" shows "(1::'b) \<cdot> t = t"
vsubset_vsingleton_rightD: fixes A :: "V"   and a :: "V" assumes "A \<subseteq>\<^sub>\<circ> ZFC_in_HOL.set {a}" shows "A = ZFC_in_HOL.set {a} \<or> A = []\<^sub>\<circ>"
bgauge_existence_lemma: fixes s :: "'b set"   and q :: "real \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<forall>x\<in>s. \<exists>d>0. q d x) = (\<forall>x. \<exists>d>0. x \<in> s \<longrightarrow> q d x)"
inangle__lea: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "P InAngle A B C" shows "A B P LeA A B C"
simps: fixes a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" shows "ord_resolve a1 a2 a3 a4 a5 a6 = (\<exists>CAs n Cs AAs As \<sigma> D. a1 = CAs \<and> a2 = D + negs (mset As) \<and> a3 = AAs \<and> a4 = As \<and> a5 = \<sigma> \<and> a6 = (\<Sum>\<^sub># (mset Cs) + D) \<cdot> \<sigma> \<and> length CAs = n \<and> length Cs = n \<and> length AAs = n \<and> length As = n \<and> n \<noteq> 0 \<and> (\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)) \<and> (\<forall>i<n. AAs ! i \<noteq> {#}) \<and> Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs)) \<and> eligible \<sigma> As (D + negs (mset As)) \<and> (\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)) \<and> (\<forall>i<n. S (CAs ! i) = {#}))"
Keys_def: fixes F :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "Keys F = \<Union> (keys ` F)"
add_local_ring_at_def: shows "pi.add_local_ring_at \<equiv> local.add_rel"
mult_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times c a) (times c b)"