hcju/isabelleps huggingface dataset:
bit_iff_odd_drop_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit a n = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (drop_bit n a))"
mset_merge: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "mset_tree (merge t1 t2) = mset_tree t1 + mset_tree t2"
subspace_scale: fixes S :: "'d set"   and x :: "'d"   and c :: "complex" assumes "csubspace S"   and "x \<in> S" shows "c *\<^sub>C x \<in> S"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a pmf" assumes "\<And>x. Ra x x" shows "rel_pmf Ra x x"
times_uexpr_def: fixes u :: "('a, 'b) uexpr"   and v :: "('a, 'b) uexpr" shows "u * v = bop (*) u v"
wim: shows "w \<in> ball z r - path_image (circlepath z r)"
pochhammer_same: fixes n :: "nat" shows "pochhammer (- of_nat n) n = (- (1::'a)) ^ n * fact n"
norm_le_zero_iff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x \<le> 0) = (x = zero)"
do_if_def: fixes v2 :: "v"   and e1 :: "exp0"   and e2 :: "exp0" shows "do_if v2 e1 e2 = (if v2 = Boolv True then Some e1 else if v2 = Boolv False then Some e2 else None)"
simps_1: shows "suffixes \<epsilon> = [\<epsilon>]"
comp: shows "l\<^sub>1\<^sup>c = l\<^sub>2"
monotone_case_prod_apply_iff: fixes orda :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'd"   and y :: "'e" shows "monotone orda ordb (\<lambda>x. (case x of (x, xa) \<Rightarrow> f x xa) y) = monotone orda ordb (\<lambda>(a, b). f a b y)"
subdivision_choose_rev: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and rev_H' :: "'d \<Rightarrow> 'd" assumes "subdivision (G, rev_G) (H, rev_H)"   and "bidirected_digraph H rev_H'" shows "\<exists>rev_G'. subdivision (G, rev_G') (H, rev_H')"
valid_lists_ccFromList: fixes xs :: "Vars.var list"   and S :: "Vars.var set"   and G :: "CoCalls" assumes "xs \<in> valid_lists S G" shows "ccFromList xs \<sqsubseteq> G G|` S"
typing_unitE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and \<tau> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> Unit : \<tau>" shows "\<tau> = TUnit"
A4a: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<P> x \<^bold>\<rightarrow> \<^bold>\<box>\<P> x) w)\<rfloor>"
subset_C_1: shows "\<forall>X\<in>P. X \<subseteq> C"
PolysI: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "keys p \<subseteq> .[X]" shows "p \<in> P[X]"
sqrt_even_pow2: fixes n :: "nat" assumes "even n" shows "sqrt (2 ^ n) = 2 ^ (n div 2)"
apsnd_apfst_commute: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'a"   and p :: "'d \<times> 'c" shows "apsnd f (apfst g p) = apfst g (apsnd f p)"
jkbpC_represents_uniquely: fixes T :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace set" assumes "SPR.represents T" shows "T = SPR.jkbpC"
wf_nFAll: fixes idx :: "'n"   and b :: "'k"   and \<phi> :: "('a, 'k) aformula" assumes "local.wf idx (FAll b \<phi>)" shows "local.wf idx (nFAll b \<phi>)"
emeasure_countably_additive: fixes M :: "'d measure" shows "countably_additive (sets M) (emeasure M)"
elims_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b tree" assumes "class.linorder less_eq less"   and "linorder.heap less_eq x" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and m :: "'b" and r :: "'b tree" where "x = \<langle>l, m, r\<rangle>" and "Ball (set_tree l \<union> set_tree r) (less_eq m) \<and> linorder.heap less_eq l \<and> linorder.heap less_eq r"
dynC_Prop: shows "G,statT \<turnstile> dynC valid_lookup_cls_for is_static emh"
otherassms_7: shows "(\<sigma>', q) \<in> oreachable A S U"
TS_yx': fixes x :: "nat"   and y :: "nat"   and qs :: "nat list"   and h :: "nat list"   and h0 :: "nat list"   and r :: "nat list" assumes "x \<noteq> y"   and "qs \<in> lang (Star (Times (Atom y) (Atom x)))"   and "\<exists>hs. h = [x, y] @ hs" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) (qs @ r) = length qs + Partial_Cost_Model.T_on' (rTS h0) ([x, y], rev qs @ h) r \<and> (\<exists>hs. rev qs @ h = [x, y] @ hs) \<and> Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs = ([x, y], rev qs @ h)"
landau_product_preprocess_35: fixes k :: "num" shows "neg_numeral_class.dbl (numeral k) = numeral (num.Bit0 k)"
tmp: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "bool" assumes "\<forall>x\<in>A. P x \<or> Q" shows "(\<forall>x\<in>A. P x) \<or> Q"
T_mndet: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<noteq> {}" shows "T (mndet A P) = (\<Union>x\<in>A. T (x \<rightarrow> P x))"
S\<^sub>e: shows "S\<^sub>e \<subseteq> S"
carr_dsum_prod1: fixes I :: "'f set"   and M :: "'f \<Rightarrow> ('g, 'h) aGroup_scheme" shows "\<forall>x. x \<in> carr_dsumag I M \<longrightarrow> x \<in> carr_prodag I M"
mk_v_exists: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE"   and sol :: "(real, 'a) vec" shows "\<exists>\<omega>. Vagree \<omega> \<nu> (- semBV I ODE) \<and> Vagree \<omega> (mk_xode I ODE sol) (semBV I ODE)"
rA': shows "local.sub1 q k l (r, A) = (r', A')"
dom_char: fixes f :: "nat" shows "J.dom f = (if f \<in> {m} then f else J.null)"
last_nth: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "last xs = xs ! (length xs - Suc 0)"
inv_tries_tameEnumFilter: fixes p :: "nat"   and t :: "(nat, nat list list list) trie" assumes "tameEnumFilter p = Some t" shows "invar_trie t"
comp_ide_source: fixes a :: "'a"   and f :: "'a" assumes "a \<in> sources f"   and "ide f" shows "f \<star> a \<cong> f"
state_to_strips_state_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option" shows "\<phi>\<^sub>S \<Psi> s \<equiv> let defined = filter (\<lambda>v. s v \<noteq> None) (\<Psi>\<^sub>\<V>\<^sub>+) in map_of (map (\<lambda>(v, a). ((v, a), the (s v) = a)) (concat (map (possible_assignments_for \<Psi>) defined)))"
simps_7: shows "OUTfromCh data7 = {data5}"
insert_remove: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "finite A" shows "prod g (insert x A) = g x * prod g (A - {x})"
Cd_singleton: fixes x :: "'x"   and d :: "'d"   and X :: "'x set" shows "(x \<in> Cd d X) = (Cd d X = {x})"
inext_le_mono_fin_rev: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "inext a I \<le> inext b I"   and "a \<in> I"   and "b \<in> I"   and "finite I"   and "inext a I \<noteq> Max I" shows "a \<le> b"
pfinite_sim: shows "pfinite_converter \<I>_real \<I>_middle sim_outer"
Int_empty_right: fixes A :: "'e set" shows "A \<inter> {} = {}"
abs_conv_abscissa_mult_le: fixes f :: "'a fds"   and g :: "'a fds" shows "abs_conv_abscissa (f * g) \<le> max (abs_conv_abscissa f) (abs_conv_abscissa g)"
emeasure_empty: fixes M :: "'b measure" shows "emeasure M {} = 0"
span_subspace: fixes B :: "'b set"   and A :: "'b set" assumes "B \<subseteq> U\<^sub>V\<^sub>S"   and "A \<subseteq> B"   and "B \<subseteq> local.span A"   and "local.subspace B" shows "local.span A = B"
fmdom_restrict_fset: fixes A :: "'c fset"   and m :: "('c, 'd) fmap" shows "fmdom (fmrestrict_fset A m) |\<subseteq>| A"
absolutely_integrable_Un: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'c set" assumes "f absolutely_integrable_on S"   and "f absolutely_integrable_on T" shows "f absolutely_integrable_on S \<union> T"
ldropWhile_simps_2: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist" shows "ldropWhile P (LCons x xs) = (if P x then ldropWhile P xs else LCons x xs)"
B\<^sub>X: shows "basis_in X B\<^sub>X"
hom_sum: fixes f :: "'c \<Rightarrow> 'b"   and X :: "'c set"   and d :: "nat" shows "monom (sum f X) d = (\<Sum>x\<in>X. monom (f x) d)"
assms: shows "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a) \<inter> subst_domain \<theta> = {}"
mr_cos_sprodTr: fixes H :: "'a set"   and a :: "'b"   and m :: "'a" assumes "submodule R M H"   and "a \<in> carrier R"   and "m \<in> carrier M" shows "mr_cos_sprod M H a (m \<uplus>\<^bsub>M\<^esub> H) = a \<cdot>\<^sub>s m \<uplus>\<^bsub>M\<^esub> H"
complex_interval_tendsto_neq: fixes f :: "nat \<Rightarrow> complex_interval"   and a :: "complex"   and b :: "complex" assumes "f \<longlonglongrightarrow>\<^sub>c a"   and "a \<noteq> b" shows "\<exists>n. \<not> b \<in>\<^sub>c f n"
ST_1: shows "ball a r \<inter> T \<subseteq> S"
bij_betwE: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "bij_betw f A B" shows "\<forall>a\<in>A. f a \<in> B"
good_qsbs_pick_X: shows "qGood pick X #[[pickE rho]]"
simps_4: fixes x :: "'a event"   and s :: "'a event list"   and X :: "'a event set"   and y :: "'a event"   and t :: "'a event list" shows "setinterleaving (x # s, X, y # t) = (if x \<in> X then if y \<in> X then if x = y then {x # u |u. u setinterleaves ((s, t), X)} else {} else {y # u |u. u setinterleaves ((x # s, t), X)} else if y \<notin> X then {x # u |u. u setinterleaves ((s, y # t), X)} \<union> {y # u |u. u setinterleaves ((x # s, t), X)} else {x # u |u. u setinterleaves ((s, y # t), X)})"
last_nth_length: fixes x :: "'a list" assumes "x \<noteq> []" shows "x ! (length x - Suc 0) = last x"
diffeomorphismD_2: fixes k :: "enat"   and S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "diffeomorphism k S T f g" shows "k-smooth_on T g"
sfs: shows "P,h,D \<turnstile>\<^sub>s sfs \<surd>"
class_add_check_types: fixes P :: "jvm_prog"   and mxs :: "nat"   and mxl :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option err list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "check_types P mxs mxl \<tau>s" shows "check_types (class_add P (C, cdec)) mxs mxl \<tau>s"
all_cong: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool" assumes "\<And>x. Q x \<Longrightarrow> P x = P' x" shows "(\<forall>x. Q x \<longrightarrow> P x) = (\<forall>x. Q x \<longrightarrow> P' x)"
simps_1: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "new_spairs_spp [] p = []"
102: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> x \<sqinter> - - z) = y \<squnion> x"
static_mhead: fixes m :: "methd" shows "static (mhead m) = static m"
c: shows "(c\<^sub>1 \<inter>\<^sub>g c\<^sub>2) = Some c"
derivative_def: fixes P :: "pi"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and Rel :: "(pi \<times> pi) set" shows "derivative P Q a x Rel \<equiv> case a of InputS b \<Rightarrow> \<forall>u. (P[x::=u], Q[x::=u]) \<in> Rel | BoundOutputS b \<Rightarrow> (P, Q) \<in> Rel"
set_intros: fixes x :: "'c" shows "x \<in> set_dual (dual x)"
image_is_empty: fixes f :: "'g \<Rightarrow> 'f"   and A :: "'g set" shows "(f ` A = {}) = (A = {})"
notin_segment_midpoint: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b" shows "a \<notin> closed_segment (midpoint a b) b"
xy_in_set_cases: fixes x :: "'a"   and l :: "'a list"   and y :: "'a" assumes "x \<in> set l"   and "y \<in> set l" obtains l1 :: "'a list" and l2 :: "'a list" where "x = y" and "l = l1__ @ y # l2__"   | l1 :: "'a list" and l2 :: "'a list" and l3 :: "'a list" where "x \<noteq> y" and "l = l1a__ @ x # l2a__ @ y # l3__"   | l1 :: "'a list" and l2 :: "'a list" and l3 :: "'a list" where "x \<noteq> y" and "l = l1 @ y # l2 @ x # l3"
lincomb_as_lincomb_list_distinct: fixes ws :: "'a vec list"   and f :: "'a vec \<Rightarrow> 'a" assumes "set ws \<subseteq> carrier_vec n"   and "distinct ws" shows "lincomb f (set ws) = lincomb_list (\<lambda>i. f (ws ! i)) ws"
assms: shows "\<mu> \<in> \<K> (abss s)"
WtFT_WtNFT: fixes cl :: "('d, 'e, 'f) cmd list" assumes "0 < length cl" shows "WtFT cl + WtNFT cl = 1"
assms: shows "bl1 \<in># \<B>\<^sup>C"
RD_bounds_sublistD_i_a: fixes Pi :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" assumes "finite Pi" shows "finite {length p - 1 |p. valid_path Pi p \<and> distinct p}"
bigtheta_trans1': fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)"   and "h \<in> \<Theta>[F](g)" shows "f \<in> \<Theta>[F](h)"
presAtmSyntactic: fixes v :: "Concrete.var"   and expr :: "exp" assumes "sec v = Hi" shows "Example_PL.presAtm (Assign v expr)"
ord_dvd_group_order: fixes a :: "'a" assumes "a \<in> carrier G" shows "ord a dvd order G"
uses_writes: fixes n :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "\<forall>v\<in>local.reads n. \<sigma> v = \<sigma>' v" shows "\<forall>v\<in>local.writes n. sem n \<sigma> v = sem n \<sigma>' v"
Mapping_keys_fold_upd_set: fixes k :: "'a"   and Z :: "'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and xs :: "('b \<times> 'c) list"   and m :: "('a, 'b) mapping" assumes "k \<in> Mapping.keys (fold (\<lambda>(t, X) m. upd_set m (\<lambda>_. t) (Z t X)) xs m)" shows "k \<in> Mapping.keys m \<or> (\<exists>(t, X)\<in>set xs. k \<in> Z t X)"
less_imp_neq: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "less x y" shows "x \<noteq> y"
elims_2: fixes x :: "'at Literal" assumes "negative x" obtains A :: "'at" where "x = Neg A"
bres_anti: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" assumes "less_eq_prod x y" shows "less_eq_prod (prod.bres y z) (prod.bres x z)"
assms_1: shows "m \<le>\<^sub>n SPEC \<Psi>"
borel_measurable_AE: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable lebesgue"   and "AE x in lebesgue. f x = g x" shows "g \<in> borel_measurable lebesgue"
trans_reach: fixes T :: "('e, 'f) TS_scheme" shows "{reach T} TS.trans T {> reach T}"
Gramian_determinant_1: fixes n :: "nat"   and fs :: "int vec list"   and k :: "nat" assumes "fs_int_indpt n fs"   and "k \<le> length fs" shows "rat_of_int (gram_schmidt.Gramian_determinant n fs k) = (\<Prod>j<k. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) j\<parallel>\<^sup>2)"
\<iota>_\<o>: fixes t :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "t \<in> range \<iota>" shows "\<iota> (\<o> t) = t"
matrix_transpose_mul: fixes A :: "(('d, 'g) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and B :: "(('d, 'f) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (A ** B) = Finite_Cartesian_Product.transpose B ** Finite_Cartesian_Product.transpose A"
gcollapse_impl_def: fixes v :: "'q"   and s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gcollapse_impl v s \<equiv> let (A, S, B, I, P) = s in GS.idx_of_impl (S, B, I, P) v \<bind> (\<lambda>i. ASSERT (i + 1 \<le> length B) \<bind> (\<lambda>_. let B = take (i + 1) B in ASSERT (i < length A) \<bind> (\<lambda>_. Un_set_drop_impl i A \<bind> (\<lambda>us. let A = take i A @ [us] in RETURN (A, S, B, I, P)))))"
one_step_implies_mult: fixes J :: "'c multiset"   and K :: "'c multiset"   and r :: "('c \<times> 'c) set"   and I :: "'c multiset" assumes "J \<noteq> {#}"   and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "(I + K, I + J) \<in> mult r"
FL_valid_Tree\<^sub>\<alpha>_eqvt: fixes P :: "'state"   and t :: "('b, 'pred, 'act, 'effect) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "FL_valid_Tree\<^sub>\<alpha> P t" shows "FL_valid_Tree\<^sub>\<alpha> (p \<bullet> P) (p \<bullet> t)"
linear: fixes x :: "'t"   and y :: "'t" shows "x \<preceq>\<^sub>t y \<or> y \<preceq>\<^sub>t x"
assms_3: shows "rk {A, B, M} \<le> 2"
assms_3: fixes b :: "'a" assumes "b dvd a" shows "a dvd b \<or> is_unit b"
f_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "map of_int_hom.vec_hom fs ! i \<in> Rn"
PO_refines_implies_Range_trans: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme" assumes "PO_refines R Ta Tc" shows "{Range R} TS.trans Tc {> Range R}"
Polygamma_def: fixes n :: "nat"   and z :: "'a" shows "Polygamma n z = (if n = 0 then (\<Sum>k. inverse (of_nat (Suc k)) - inverse (z + of_nat k)) - euler_mascheroni else (- (1::'a)) ^ Suc n * fact n * (\<Sum>k. inverse ((z + of_nat k) ^ Suc n)))"
redT_updLs_Some_thread_idD: fixes ls :: "('e, 'd) locks"   and t :: "'d"   and las :: "'e lock_actions"   and l :: "'e"   and t' :: "'d" assumes "has_lock (redT_updLs ls t las $ l) t'"   and "t \<noteq> t'" shows "has_lock (ls $ l) t'"
face_of_singleton: fixes x :: "'b"   and S :: "'b set" shows "({x} face_of S) = (x extreme_point_of S)"
tree_iterate_fusion: fixes h :: "'b \<Rightarrow> 'c"   and l :: "'b \<Rightarrow> 'b"   and l' :: "'c \<Rightarrow> 'c"   and r :: "'b \<Rightarrow> 'b"   and r' :: "'c \<Rightarrow> 'c"   and x :: "'b" assumes "h \<circ> l = l' \<circ> h"   and "h \<circ> r = r' \<circ> h" shows "map_tree h (tree_iterate l r x) = tree_iterate l' r' (h x)"
res_ring_car: fixes k :: "nat" shows "carrier (residue_ring (p ^ k)) = {0..p ^ k - 1}"
Trans: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "(P, Q) \<in> Rel"   and "(Q, R) \<in> Rel" shows "(P, R) \<in> Rel"
lastp_un_D_closed: assumes "p \<noteq> []"   and "pE \<inter> last p \<times> UNIV = {}" shows "E `` (last p \<union> D) \<subseteq> last p \<union> D"
r_eq_orig': fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and c :: "char list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" assumes "\<forall>rs\<in>ran \<Gamma>. no_call_to c rs" shows "\<Gamma>,\<gamma>,p\<turnstile> [Rule MatchAny (Call c)] \<Rightarrow>\<^sub>r t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule MatchAny (Call c)], Undecided\<rangle> \<Rightarrow> t"
irreflp_onI: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>a. a \<in> A \<Longrightarrow> \<not> P a a" shows "irreflp_on P A"
iNextWeak_True: fixes t0 :: "nat"   and I :: "nat set" shows "\<circle>\<^sub>W t t0 I. True"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set xs" shows "x \<le> y \<or> y \<le> x"
intro: fixes R :: "('a, 'b) ring_scheme" assumes "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<And>a b. \<lbrakk>a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>; a \<in> carrier R; b \<in> carrier R\<rbrakk> \<Longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>" shows "domain_axioms R"
quat_inner_i_right: fixes x :: "quat" shows "x \<bullet> \<i> = Im1 x"
insert_compr: fixes a :: "'b"   and B :: "'b set" shows "insert a B = {x. x = a \<or> x \<in> B}"
bij_betwI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> A"   and "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> B \<Longrightarrow> f (g y) = y" shows "bij_betw f A B"
insort_insert_insort_key: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" assumes "f x \<notin> f ` set xs" shows "insort_insert_key f x xs = insort_key f x xs"
S6Clerk: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MClkNext memCh crCh cst p \<and> $S6 rmhist p \<longrightarrow> MClkRetry memCh crCh cst p \<or> MClkReply memCh crCh cst p"
implies_itself_top: fixes y :: "'a"   and x :: "'a" shows "y \<le> x \<leadsto> x"
more_temp_simps3_8: fixes F :: "'a \<Rightarrow> bool" shows "\<box>\<diamond>\<box>F \<equiv> \<diamond>\<box>F"
f_accessible_subset: fixes q :: "'d"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set" assumes "q \<in> f_accessible \<delta> Q" shows "q \<in> Q \<union> \<delta>_states \<delta>"
fvdiff_plus1: fixes t1 :: "('d, 'c) trm"   and t2 :: "('d, 'c) trm" shows "FVDiff (Plus t1 t2) = FVDiff t1 \<union> FVDiff t2"
Overrides: fixes dynM :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some dynM"   and "dynM \<noteq> statM"   and "G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM" shows "thesis"
zeroX_nextX: fixes a :: "vbl" shows "zeroX \<noteq> nextX a"
union_le_mono2: fixes B :: "'k multiset"   and D :: "'k multiset"   and C :: "'k multiset" assumes "B < D" shows "C + B < C + D"
get_spec_zip_simp: shows "get_spec (zip (TopoS_Composition_Theory_impl.get_IFS (get_impl M)) (TopoS_Composition_Theory.get_IFS (get_spec M))) = TopoS_Composition_Theory.get_IFS (get_spec M)"
kills_e2: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>"
simps_1: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'e" shows "run \<delta> q\<^sub>0 w 0 = q\<^sub>0"
upd_rev_cs_Cons_Call_Cons: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and cs :: "'edge list"   and as :: "'edge list"   and c' :: "'edge"   and cs' :: "'edge list" assumes "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "upd_rev_cs cs as = c' # cs'" shows "upd_rev_cs cs (a # as) = cs'"
sup_left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
perm_append_swap: fixes xs :: "'b list"   and ys :: "'b list" shows "xs @ ys <~~> ys @ xs"
finite_sets: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "finite_incidence_system point_set block_collection" shows "finite point_set"
irreducible_PFM_PM: fixes p :: "'b poly" assumes "irreducible (map_poly to_fract p)"   and "content_ff (map_poly to_fract p) =dff 1" shows "irreducible p"
elims_2: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list" assumes "sorted_wrt x xa" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = P__" and "xa = []"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xb :: "'a" and ys :: "'a list" where "x = P" and "xa = xb # ys" and "Ball (set ys) (P xb) \<and> sorted_wrt P ys"
diff_delta_arctan_upper_41_def: shows "diff_delta_arctan_upper_41 \<equiv> \<lambda>x. 256 / (x\<^sup>2 * (35 + 315 * x\<^sup>2 + 693 * x ^ 4 + 429 * x ^ 6)\<^sup>2 * (1 + x\<^sup>2))"
distinct_65: fixes x4 :: "nat"   and x101 :: "nat"   and x102 :: "msg" shows "Key x4 \<noteq> Crypt x101 x102"