hcju/isabelleps huggingface dataset:
list_before_trans: fixes l :: "'b list" assumes "distinct l" shows "trans (list_before_rel l)"
ty: shows "\<Gamma> \<turnstile> FIX [x].t : T"
closedin_connected_components_of: fixes C :: "'b set"   and X :: "'b topology" assumes "C \<in> connected_components_of X" shows "closedin X C"
ucast_id: fixes w :: "'b word" shows "UCAST('b \<rightarrow> 'b) w = w"
galois_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c \<Rightarrow> 'c"   and u :: "'c \<Rightarrow> 'c" shows "ord.galois less_eq l u \<equiv> \<forall>x y. less_eq (l x) y = less_eq x (u y)"
yinv_2: shows "y' \<otimes> y = \<one>"
arg_min_on_image_finite: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and B :: "'e set" assumes "inj_on f (g ` B)"   and "finite B"   and "B \<noteq> {}" shows "arg_min_on f (g ` B) = g (arg_min_on (f \<circ> g) B)"
elims_2: fixes x :: "exp"   and xa :: "obs" assumes "observe x xa" obtains n :: "nat" and n' :: "nat" where "x = ENat n" and "xa = ONat n'" and "n = n'"   | xb :: "nat" and e :: "exp" where "x = ELam xb e" and "xa = OFun"
preservation: shows "rel_preserves_binary_pred (indRelLTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
star_sum_power_slide: fixes m :: "nat"   and n :: "nat"   and x :: "'a" assumes "m \<le> n" shows "x\<^bsub>0\<^esub>\<^bsup>m\<^esup> \<cdot> (x\<^bsup>n\<^esup>)\<^sup>\<star> = (x\<^bsup>n\<^esup>)\<^sup>\<star> \<cdot> x\<^bsub>0\<^esub>\<^bsup>m\<^esup>"
infinite_remove: fixes S :: "'b set"   and a :: "'b" assumes "infinite S" shows "infinite (S - {a})"
mem_Collect_eq: fixes a :: "'x"   and P :: "'x \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
analz_cong: fixes G :: "msg set"   and G' :: "msg set"   and H :: "msg set"   and H' :: "msg set" assumes "analz G = analz G'"   and "analz H = analz H'" shows "analz (G \<union> H) = analz (G' \<union> H')"
whisker_right: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.ide V f"   and "partial_magma.arr V (V \<nu> \<mu>)" shows "H (V \<nu> \<mu>) f = V (H \<nu> f) (H \<mu> f)"
get_ancestors_si_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_ancestors_si ptr"
exhaust: fixes y :: "op" obtains "y = OpNeg"   | "y = OpAdd"   | "y = OpMul"
same_mset: shows "arr_mset mod_arr m r\<^sub>0 = arr_mset arr m r\<^sub>0"
llist_less_def: fixes s :: "'a llist"   and t :: "'a llist" shows "(s < t) = (s \<le> t \<and> s \<noteq> t)"
mult2_alt_ns_s_add: fixes X1 :: "'a multiset"   and Y1 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X2 :: "'a multiset"   and Y2 :: "'a multiset" assumes "(X1, Y1) \<in> mult2_alt_ns ns s"   and "(X2, Y2) \<in> mult2_alt_s ns s" shows "(X1 + X2, Y1 + Y2) \<in> mult2_alt_s ns s"
counter_b_aux: fixes u :: "nat \<Rightarrow> nat"   and xa :: "nat \<Rightarrow> bool"   and xb :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "u 0 = 0"   and "\<forall>n. (xa n \<longrightarrow> u (Suc n) = Suc (u n)) \<and> (\<not> xa n \<longrightarrow> u (Suc n) = u n) \<and> xb n = u (Suc n)" shows "count xa n = u (Suc n)"
nonzero: shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
a1: shows "s' = snd (fst (write_cpu (update_CWP cwp_val (cpu_reg_val PSR s)) PSR s)) \<and> get_S (cpu_reg_val PSR s) = 0"
injf_max_order_preserving2: fixes E :: "'a set" assumes "\<forall>x. \<exists>y\<in>E. x < y" shows "\<forall>n m. m < n \<longrightarrow> injf_max m E < injf_max n E"
const_on3: shows "const_on (FBinds \<Gamma>\<cdot>ae) (- (- domA \<Delta>)) TTree.empty"
isomorphic_group_triviality: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme" assumes "G \<cong> H"   and "Group.group G"   and "Group.group H" shows "trivial_group G = trivial_group H"
merge_index: fixes I :: "nat set"   and J :: "nat set"   and n :: "nat"   and tI :: "'a option list"   and tJ :: "'a option list"   and t :: "'a option list"   and i :: "nat" assumes "I \<inter> J = {}"   and "wf_tuple n I tI"   and "wf_tuple n J tJ"   and "t = merge tI tJ"   and "i < n" shows "i \<in> I \<and> t ! i = tI ! i \<or> i \<in> J \<and> t ! i = tJ ! i \<or> i \<notin> I \<and> i \<notin> J \<and> t ! i = None"
child_owner_document_c: shows "h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_owner_document) \<rightarrow>\<^sub>r child_owner_document_c"
transfer: shows "rel_fun cr_uint (rel_fun cr_uint cr_uint) (+) (+)"
WeakenContext_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta>' :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>'\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>(P, p, Q, A)\<in>\<Theta>'. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P Call p Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
count_mset_gt_0: fixes x :: "'k"   and xs :: "'k list" assumes "x \<in> set xs" shows "0 < count (mset xs) x"
assms_2: fixes r :: "complex"   and b :: "'a" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
diameter_def: fixes S :: "'b set" shows "diameter S = (if S = {} then 0 else SUP (x, y)\<in>S \<times> S. dist x y)"
map_getminE: fixes empty :: "'a"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and invar :: "'a \<Rightarrow> bool"   and lookup :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and is_empty :: "'a \<Rightarrow> bool"   and getmin :: "'a \<Rightarrow> 'b \<times> 'c"   and m :: "'a"   and k :: "'b"   and p :: "'c" assumes "PrioMap empty update delete invar lookup is_empty getmin"   and "getmin m = (k, p)"   and "invar m"   and "lookup m \<noteq> Map.empty" obtains "lookup m k = Some p" and "\<forall>k' p'. lookup m k' = Some p' \<longrightarrow> p \<le> p'"
bind_ref_tag_def: fixes x :: "'e"   and m :: "'e nres" shows "bind_ref_tag x m \<equiv> RETURN x \<le> m"
NATDistr: fixes N :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and F :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent \<Rightarrow> unit decision option"   and NAT :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent \<Rightarrow> 'b decision option" assumes "N \<noteq> []"   and "F = Cp (list2policyR N)" shows "(\<lambda>(x, y). x) o\<^sub>f (NAT \<Otimes>\<^sub>2 F \<circ> (\<lambda>x. (x, x))) = list2policy ((NAT \<Otimes>\<^sub>L map Cp N) (\<Otimes>\<^sub>2) (\<lambda>(x, y). x) (\<lambda>x. (x, x)))"
assms: shows "f \<in> borel_measurable (count_space UNIV)"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 \<mu> \<nu>"
assms_3: shows "\<forall>x. c < x \<and> x \<le> d2 \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
nodesI_1: fixes u :: "'a"   and v :: "'a"   and g :: "'a ugraph" assumes "(u, v) \<in> edges g" shows "u \<in> nodes g"
pos_divideC_less_eq: fixes c :: "complex"   and b :: "'a"   and a :: "'a" assumes "0 < c" shows "(b /\<^sub>C c < a) = (b < c *\<^sub>C a)"
sum_outgoing_pointwise: fixes g :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "sum g (outgoing u) = (\<Sum>v\<in>E `` {u}. g (u, v))"
assms: fixes x :: "'a" shows "monotone ord (ord_spmf (=)) (\<lambda>y. g y x)"
simps_3: fixes c1 :: "('a, 'b, 'c) cmd"   and c2 :: "('a, 'b, 'c) cmd" shows "finished (c1 ;; c2) = False"
weakenpre: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} c { Q}"   and "\<And>s. P' s \<Longrightarrow> P s" shows "\<turnstile>\<^sub>3\<^sub>' {P'} c { Q}"
to_pre_digraph_def: fixes G :: "('b, 'c) Graph_scheme" shows "to_pre_digraph G \<equiv> \<lparr>pre_digraph.verts = V\<^bsub>G\<^esub>, arcs = E\<^bsub>G\<^esub>, tail = fst, head = snd\<rparr>"
while_option_finite_subset_Some: fixes f :: "'a set \<Rightarrow> 'a set"   and C :: "'a set" assumes "mono f"   and "\<And>X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C"   and "finite C" shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f {} = Some P"
length_list_of_pdevs_pdevs_of_list: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> (0::'b)" shows "length (list_of_pdevs (pdevs_of_list xs)) = length xs"
finite_imp_finite_prob_proj: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" assumes "finite PROB" shows "finite (prob_proj PROB vs)"
assms_1: shows "Keys (set ps) \<subseteq> set ts"
assms_2: shows "\<lbrace>p''\<rbrace> T \<lbrace>q\<rbrace>\<^sub>u"
prems: shows "\<forall>a\<in>set (a # as). intra_kind (kind a)"
longer_falsifies\<^sub>l: fixes ds :: "bool list"   and l :: "fterm literal"   and d :: "bool list" assumes "falsifies\<^sub>l ds l" shows "falsifies\<^sub>l (ds @ d) l"
INF_superset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "B \<subseteq> A"   and "\<And>x. x \<in> B \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
mL_hom: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "\<forall>j\<le>n. mL K P I j \<in> Zset"
phl_export1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x \<cdot> y \<le> y \<cdot> x \<cdot> y"   and "\<lbrace>x \<cdot> y\<rbrace>z\<lbrace>w\<rbrace>" shows "\<lbrace>x\<rbrace>y \<cdot> z\<lbrace>w\<rbrace>"
assms: fixes x :: "'a"   and y :: "'b" assumes "fmlookup m x = Some y" shows "P x = Q x"
ncols_diagonal_to_Smith_i: fixes xs :: "nat list"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "ncols (diagonal_to_Smith_i xs A i bezout) = ncols A"
multiplicity_aprimedivisor_Suc_0_iff: fixes n :: "'a" assumes "primepow n" shows "(multiplicity (aprimedivisor n) n = Suc 0) = prime n"
imp: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (imp p q) = Ifm bbs bs (Imp p q)"
is_red_uminus: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "punit.is_red F (- p) = punit.is_red F p"
frequently_all: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'b filter" shows "(\<exists>\<^sub>F x in F. \<forall>y. P x y) = (\<forall>Y. \<exists>\<^sub>F x in F. P x (Y x))"
simps_1: fixes c :: "'a" shows "pderiv c Zero = {}"
intro: fixes p :: "'a process_name"   and p_ph_enabled :: "('a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state) \<Rightarrow> bool" assumes "eq_imp (\<lambda>_ s. s p) p_ph_enabled" shows "mark_object p p_ph_enabled"
continuous_on_closed_Un: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "closed s"   and "closed t"   and "continuous_on s f"   and "continuous_on t f" shows "continuous_on (s \<union> t) f"
regularCard_def: fixes r :: "('a \<times> 'a) set" shows "regularCard r \<equiv> \<forall>K. K \<subseteq> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
cases: fixes a1 :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and a2 :: "('s \<times> ('c \<Rightarrow> real)) cfg stream" assumes "enabled_repcs a1 a2" obtains xs :: "('s \<times> ('c \<Rightarrow> real)) cfg stream" and st' :: "'s \<times> ('c \<Rightarrow> real)" and cfg' :: "('s \<times> ('c \<Rightarrow> real) set) cfg" and st :: "'s \<times> ('c \<Rightarrow> real)" and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" where "a1 = repcs st cfg" and "a2 = xs" and "enabled_repcs (shd xs) (stl xs)" and "shd xs = repcs st' cfg'" and "st' \<in> set_pmf (rept st (action cfg))" and "abss st' = state cfg'" and "cfg' \<in> R_G.valid_cfg"
less_irrefl: fixes x :: "complex" shows "\<not> x <\<^sub>\<complex> x"
sets_return_val: fixes x :: "val" shows "sets (return_val x) = sets (stock_measure (val_type x))"
lcoeff_nonzero_deg: fixes p :: "nat \<Rightarrow> 'a" assumes "deg R p \<noteq> 0"   and "p \<in> carrier P" shows "coeff P p (deg R p) \<noteq> \<zero>"
lemma_1_reachability_s_i: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set" assumes "s \<in> valid_states PROB" shows "s \<in> reachable_s PROB s"
hyps_8: fixes b :: "(name \<times> ty) list"   and T :: "ty" assumes "b \<turnstile> e\<^sub>2 : T" shows "b \<turnstile> e\<^sub>2' : T"
countable_insert: fixes A :: "'a set"   and a :: "'a" assumes "countable A" shows "countable (insert a A)"
lfinite_ldropn: fixes n :: "nat"   and xs :: "'a llist" shows "lfinite (ldropn n xs) = lfinite xs"
AE_exponential: fixes l :: "real" assumes "0 < l" shows "almost_everywhere (exponential l) ((<) 0)"
greaterThan_eq_iff: fixes x :: "'b"   and y :: "'b" shows "({x<..} = {y<..}) = (x = y)"
diff_right_commute: fixes M :: "'k multiset"   and N :: "'k multiset"   and Q :: "'k multiset" shows "M - N - Q = M - Q - N"
fold_insert2: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "comp_fun_commute f"   and "finite A"   and "x \<notin> A" shows "Finite_Set.fold f z (insert x A) = Finite_Set.fold f (f x z) A"
eq: shows "{x. poly p x = 0} = {x. poly p x = 0 \<and> poly q x < 0} \<union> {x. poly p x = 0 \<and> 0 < poly q x} \<union> {x. poly p x = 0 \<and> poly q x = 0}"
elims_1: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list"   and y :: "bool" assumes "sorted_wrt x xa = y" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = P__" and "xa = []" and "y"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xb :: "'a" and ys :: "'a list" where "x = P" and "xa = xb # ys" and "y = (Ball (set ys) (P xb) \<and> sorted_wrt P ys)"
F_to_idx_refine: assumes "\<forall>A\<in>F. finite A" shows "F_to_idx_impl F \<le> \<Down> Id (F_to_idx F)"
fls_shifted_times_transfer: fixes n :: "int"   and f :: "'a fls"   and g :: "'a fls" shows "fls_shift n f * g = f * fls_shift n g"
tl_nempty_length: fixes s :: "'a list" assumes "s \<noteq> []"   and "tl s \<noteq> []" shows "0 < length s - 1"
r2: assumes "is_sublist [x, y] vs" shows "nextElem vs c x = y"
assms_1: shows "Counterclockwise_2D_Arbitrary.lex p q"
c: shows "lookup p ((lp p - lp f) \<oplus> lt f) = lc p"
domain_atlas_subset_carrier: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and c :: "('c, 'd) chart" assumes "c_manifold charts k"   and "c \<in> c_manifold.atlas charts k" shows "domain c \<subseteq> manifold.carrier charts"
sift_up_step_aux1: fixes heap :: "int \<Rightarrow> int"   and l :: "int"   and r :: "int"   and x :: "int"   and k :: "int" assumes "is_heap_except_up (\<le>) heap l r x"   and "l \<le> parent l x"   and "heap x \<le> heap (parent l x)"   and "bounded l r k"   and "k \<noteq> parent l x"   and "bounded l r (parent l k)" shows "swap heap (parent l x) x (parent l k) \<le> swap heap (parent l x) x k"
ind_is_open_def: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and U :: "'c set" assumes "ind_topology X is_open S" shows "ind_topology.ind_is_open X is_open S U \<equiv> U \<subseteq> S \<and> (\<exists>V\<subseteq>X. is_open V \<and> U = S \<inter> V)"
equiv_def: shows "(\<^bold>\<equiv>) \<equiv> \<lambda>x y. (x \<^bold>\<rightarrow> y) \<^bold>& (y \<^bold>\<rightarrow> x)"
mu_def: shows "mu = monomial (lc (q g1 \<odot> g1)) (lp (q g1 \<odot> g1) - lcs (lp g1) (lp g2))"
cons2: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and err :: "v error_result"   and es :: "exp0 list" assumes "evaluate ck env s e (s', Rerr err)" shows "evaluate_list ck env s (e # es) (s', Rerr err)"
closed_nat_set_iff: fixes A :: "nat set" assumes "\<forall>m n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A" shows "A = UNIV \<or> (\<exists>n. A = {0..<n})"
it_to_sorted_list_def: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g set" shows "it_to_sorted_list R s \<equiv> SPEC (\<lambda>l. distinct l \<and> s = set l \<and> sorted_wrt R l)"
power_eq_iff_eq_base: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "nat"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "0 < n"   and "less_eq zero a"   and "less_eq zero b" shows "(power.power one times a n = power.power one times b n) = (a = b)"
at_res_inst: fixes d :: "cars"   and c :: "cars" shows "\<forall>ts v. (\<forall>v'. v = d > v' \<longrightarrow> 0 < \<parallel>ext v'\<parallel> \<and> len v' ts c = ext v' \<and> restrict v' (res ts) c = lan v' \<and> |lan v'| = 1) \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1"
Block\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and i :: "nat"   and T :: "ty" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>{i:T; e},s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>1\<rangle>"
sset_sconst: fixes x :: "'c" shows "sset (sconst x) = {x}"
simps: fixes events :: "'a \<Rightarrow> ('a, 'b) event list"   and i :: "'a \<times> nat"   and j :: "'a \<times> nat" assumes "dist_execution_preliminary TYPE('a)" shows "dist_execution_preliminary.happened_immediately_before events i j = (dist_execution_preliminary.is_valid_event_id events i \<and> dist_execution_preliminary.is_valid_event_id events j \<and> (fst i = fst j \<and> Suc (snd i) = snd j \<or> dist_execution_preliminary.is_reception events i j))"
ln_upper_13_pos: fixes x :: "real" assumes "1 \<le> x" shows "ln x \<le> ln_upper_13 x"
simps_1: fixes V :: "Vars.var set" shows "restr_stack V [] = []"
Afix_unroll: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "Afix \<Gamma>\<cdot>ae = ABinds \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>ae) \<squnion> ae"
simps_3: fixes c1 :: "'a acom"   and c2 :: "'a acom" shows "strip (c1;; c2) = strip c1;; strip c2"
minGraphProps5: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "f \<in> \<F> g"
elts_ord_of_nat: fixes k :: "nat" shows "elts (ord_of_nat k) = ord_of_nat ` {..<k}"
assms_1: shows "valid_unMultigraph.is_Eulerian_trail (del_unEdge v2 w v3 G) v1 ps v2"
rref_condition3_equiv: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'d" assumes "reduced_row_echelon_form A"   and "i < j"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row j A" shows "(LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ j $ n \<noteq> (0::'b))"
carrier_append_rows: fixes A :: "'a mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'a mat"   and nr2 :: "nat" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc" shows "A @\<^sub>r B \<in> carrier_mat (nr1 + nr2) nc"
FP_weakest: fixes F :: "'b program"   and A :: "'b set" assumes "\<And>B. F \<in> stable (A \<inter> B)" shows "A \<subseteq> FP F"
cases: fixes a1 :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and a2 :: "('a, 'b) form"   and a3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "leftPrincipal a1 a2 a3" obtains C :: "('a, 'b) sequent" and A :: "('a, 'b) form" and Ps :: "('a, 'b) sequent list" and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" where "a1 = (Ps, C)" and "a2 = A" and "a3 = R" and "C = ( \<LM> A  \<RM> \<Rightarrow>* \<Empt>)" and "A \<noteq> ff" and "(Ps, C) \<in> R"
fg: shows "homeomorphism (sphere (0::'n) 1 - {i}) {x. i \<bullet> x = 0} f g"
fun_type_length_eq: fixes f :: "'fun"   and X :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and g :: "'fun"   and Y :: "('fun, 'atom) Term.term list" assumes "\<Gamma> (Fun f X) = Fun g Y" shows "length X = length Y"
SetToIndex: fixes a :: "'a"   and liste :: "'a list" assumes "a \<in> set liste" shows "\<exists>index<length liste. a = liste ! index"
minus_1_power_even: fixes k :: "nat" shows "(- (1::'a)) ^ k = (if even k then 1::'a else - (1::'a))"
wellformed_finished_item: shows "wellformed_item (finished_item \<alpha>)"
ee: fixes t :: "real" assumes "t \<in> {0..1}" shows "0 < ee t \<and> (\<forall>t1 t2. t1 \<in> {0..1} \<longrightarrow> t2 \<in> {0..1} \<longrightarrow> \<bar>t1 - t\<bar> < ee t \<longrightarrow> \<bar>t2 - t\<bar> < ee t \<longrightarrow> (\<exists>d>0. \<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and> (\<forall>u\<in>{0..1}. cmod (g1 u - k (t1, u)) < d \<and> cmod (g2 u - k (t2, u)) < d) \<and> linked_paths atends g1 g2 \<longrightarrow> contour_integral g2 f = contour_integral g1 f))"
sum_mset_0_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "\<And>x. less_eq zero (f x)"   and "comm_monoid_add.sum_mset plus zero (image_mset f A) = zero" shows "\<forall>x\<in>#A. f x = zero"
a': shows "( \<Gamma> \<oplus> Compound F Fs \<Rightarrow>* \<delta>, n) \<in> derivable R*"
mom_select_correct_aux: fixes k :: "nat"   and xs :: "'a list" assumes "mom_select_dom (k, xs)"   and "k < length xs" shows "mom_select k xs = select k xs"
frag_induction: fixes c :: "'b \<Rightarrow>\<^sub>0 int"   and S :: "'b set"   and P :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> bool" assumes "Poly_Mapping.keys c \<subseteq> S"   and "P 0"   and "\<And>x. x \<in> S \<Longrightarrow> P (frag_of x)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (a - b)" shows "P c"
simps_1: fixes d :: "'dyn" shows "norm_unboxed (OpDyn d) = d"
WT_sim: shows "\<I>_real, \<I>_ideal \<turnstile>\<^sub>C sim \<surd>"
compI_1: fixes f :: "'b"   and g :: "'b" assumes "f \<in> Ar"   and "g \<in> Ar"   and "Cod f = Dom g" shows "g \<bullet> f \<in> Ar"
nosplit_simps2_1: shows "nosplit [] = [] @ [1::'a]"
currentD_OUT_IN: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f"   and "x \<notin> A \<Gamma>" shows "d_OUT f x \<le> d_IN f x"