hcju/isabelleps huggingface dataset:
Domainp_rel: fixes R1 :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and R3 :: "'f \<Rightarrow> 'i \<Rightarrow> bool" shows "Domainp (rel_generat R1 R2 R3) = pred_generat (Domainp R1) (Domainp R2) (Domainp R3)"
mem_simps_3: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
simps_15: fixes x2 :: "'b" shows "set_message_af (Msg x2) = {x2}"
Tree\<^sub>\<alpha>_abs_rep: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" shows "abs_Tree\<^sub>\<alpha> (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
Ainv: fixes x :: "'a" assumes "x \<in> space M" shows "(x \<in> A) = (T x \<in> A)"
prod_filter_commute: fixes F :: "'b filter"   and G :: "'c filter" shows "F \<times>\<^sub>F G = filtermap prod.swap (G \<times>\<^sub>F F)"
tr_def_1: shows "target ([xy] || tr) q1 = q2"
\<a>'_def: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] \<equiv> D.\<alpha>' (\<mu>, \<nu>, \<tau>)"
assms: shows "MATCH1 (P, \<sigma>) (P', \<sigma>')"
that: fixes \<T> :: "'a set set" assumes "simplicial_complex \<T>"   and "\<And>K. K \<in> \<T> \<Longrightarrow> aff_dim K \<le> d"   and "\<Union> \<T> = \<Union> \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F"   and "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C" shows "thesis"
span_base: fixes a :: "'c"   and S :: "'c set" assumes "a \<in> S" shows "a \<in> span S"
u_def: shows "u = Matrix.col (Tensor.mat_of_cols_list (length (l ! j)) l) j"
isSp_Append_cdep: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n''"   and "n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'" shows "n is-ns @ [n'']\<rightarrow>\<^sub>d* n'"
viewLn_empty: fixes t :: "('e, 'f) FingerTreeStruc" shows "(t \<noteq> Empty) = (viewLn t \<noteq> None)"
le_trans_antisym: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_trans x y"   and "le_trans y x" shows "equiv_trans x y"
keysFor_mono: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> H" shows "keysFor G \<subseteq> keysFor H"
eval_eval_body_eq: shows "eval = fix\<cdot>eval_body"
distributed_affineI: fixes X :: "'a \<Rightarrow> real"   and t :: "real"   and c :: "real"   and f :: "real \<Rightarrow> ennreal" assumes "distributed M lborel (\<lambda>x. (X x - t) / c) (\<lambda>x. ennreal \<bar>c\<bar> * f (x * c + t))"   and "c \<noteq> 0" shows "distributed M lborel X f"
pushout_step_def: fixes t :: "'d itself"   and R :: "('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph"   and G\<^sub>1 :: "('e, 'g) labeled_graph"   and G\<^sub>2 :: "('e, 'g) labeled_graph" shows "pushout_step t R G\<^sub>1 G\<^sub>2 \<equiv> subgraph G\<^sub>1 G\<^sub>2 \<and> (\<exists>f\<^sub>1 f\<^sub>2. graph_homomorphism (fst R) G\<^sub>1 f\<^sub>1 \<and> graph_homomorphism (snd R) G\<^sub>2 f\<^sub>2 \<and> f\<^sub>1 \<subseteq> f\<^sub>2 \<and> weak_universal t R G\<^sub>1 G\<^sub>2 f\<^sub>1 f\<^sub>2)"
typing_cond\<^sub>s\<^sub>s\<^sub>t_def: fixes \<A> :: "('fun, 'var) stateful_strand_step list" shows "typing_cond\<^sub>s\<^sub>s\<^sub>t \<A> \<equiv> wf\<^sub>s\<^sub>s\<^sub>t \<A> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t \<A>) \<and> tfr\<^sub>s\<^sub>s\<^sub>t \<A>"
swapBinp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and binp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option" shows "binp %%[x \<and> y]_xs \<equiv> lift (swapAbs xs x y) binp"
c_k_with_K: fixes i :: "int"   and j :: "int" assumes "i < j" shows "c i j = c_k i j (K i j)"
enabled_def: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "Enabled F \<equiv> %s. \<exists>t. F (first s ## t)"
subspace_add: fixes Sa :: "('a \<Rightarrow> real) set"   and x :: "'a \<Rightarrow> real"   and y :: "'a \<Rightarrow> real" assumes "Sa \<subseteq> diff_fun_space"   and "sub.diff_fun_space.m1.subspace Sa"   and "x \<in> Sa"   and "y \<in> Sa" shows "x + y \<in> Sa"
det_discounted: fixes r :: "real"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat"   and w :: "'b" shows "discounted_value r X n w = det_discount r n * X n w"
pcr_cr_eq: shows "pcr_pos (=) = cr_pos"
indoc_appendI2: fixes a :: "'item"   and u :: "'item list"   and v :: "'item list" assumes "a \<notin> set u"   and "Ind {a} (set u)"   and "indoc a v" shows "indoc a (u @ v)"
assms_2: shows "G1 \<in> {G \<times> I |G. G \<in> \<G>}"
simps_2: fixes i :: "nat"   and divs :: "'b mod_ring poly list"   and n :: "nat" shows "berlekamp_factorization_main i divs [] n = divs"
ClassD: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and b :: "'c"   and a :: "'c" assumes "equivalence S E"   and "b \<in> equivalence.Class S E a"   and "a \<in> S" shows "(b, a) \<in> E"
take_bit_or: fixes n :: "nat"   and a :: "'a"   and b :: "'a" shows "take_bit n (a OR b) = take_bit n a OR take_bit n b"
c_remove_is_pr: shows "c_remove \<in> PrimRec2"
domsMT_notND_DD2: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set" assumes "dom (Cp (DenyAllFromTo a b)) \<inter> dom (Cp (DenyAllFromTo c d)) \<noteq> {}" shows "\<not> netsDistinct b d"
nat_pow_closed: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> carrier P" shows "x [^]\<^bsub>P\<^esub> n \<in> carrier P"
hd_tl_lens_indep: shows "hd\<^sub>L \<bowtie> tl\<^sub>L"
iT_Div_card_ge: fixes I :: "nat set"   and d :: "nat" shows "card I div d + (if card I mod d = 0 then 0 else Suc 0) \<le> card (I \<oslash> d)"
additive_subgroup_def: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme" shows "additive_subgroup H G \<equiv> subgroup H (add_monoid G)"
monom_mult_assoc: fixes c :: "'d"   and s :: "'c"   and d :: "'d"   and t :: "'c"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "punit.monom_mult c s (punit.monom_mult d t p) = punit.monom_mult (c * d) (s + t) p"
tendsto_imp_eventually_ne: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and c :: "'b"   and F :: "'c filter"   and c' :: "'b" assumes "class.t1_space open"   and "filterlim f (topological_space.nhds open c) F"   and "c \<noteq> c'" shows "\<forall>\<^sub>F z in F. f z \<noteq> c'"
BAD_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "mbs.BAD A P = {f \<in> SEQ A. bad P f}"
uint_mod_def: fixes x :: "uint"   and y :: "uint" shows "uint_mod x y = snd (uint_divmod x y)"
select_convs_5: fixes Obj :: "'d set"   and Mor :: "'e set"   and Dom :: "'e \<Rightarrow> 'd"   and Cod :: "'e \<Rightarrow> 'd"   and Id :: "'d \<Rightarrow> 'e"   and Comp :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and more :: "'f" shows "Category.Id \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Id"
non_npD: fixes v :: "val"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and C :: "char list" assumes "v \<noteq> Null"   and "P,h \<turnstile> v :\<le> Class C" shows "\<exists>a C' fs. v = Addr a \<and> h a = \<lfloor>(C', fs)\<rfloor> \<and> P \<turnstile> C' \<preceq>\<^sup>* C"
rref_condition4_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c" assumes "reduced_row_echelon_form A"   and "\<not> is_zero_row i A"   and "i \<noteq> j" shows "A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) = (0::'a)"
interfree_aux_rule2: fixes a :: "'a ann_com"   and q :: "'a set" assumes "\<forall>(R, r)\<in>atomics a. \<parallel>- (q \<inter> R) r q" shows "interfree_aux (None, q, Some a)"
simWeakSim: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" assumes "P \<leadsto>[Rel] Q" shows "P \<leadsto><Rel> Q"
D_mono: fixes A :: "'f set option"   and A' :: "'f set option"   and e :: "('f, 'g, 'h) exp" assumes "A \<sqsubseteq> A'"   and "\<D> e A" shows "\<D> e A'"
transC_MtransC: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state" assumes "cf \<rightarrow>c cf'" shows "cf \<rightarrow>*c cf'"
distinct_15: fixes x9 :: "('a, String.literal) phantom" shows "OclAny \<noteq> Enum x9"
assms_1: fixes u :: "'a" assumes "u \<in> Basis" shows "((\<lambda>x. f x \<bullet> u) has_vector_derivative f' \<bullet> u) (at x within S)"
principalidealI: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "\<exists>i\<in>carrier R. I = Idl\<^bsub>R\<^esub> {i}" shows "principalideal I R"
one_in_subset: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<forall>a\<in>H. inv a \<in> H"   and "\<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H" shows "\<one> \<in> H"
all_imp_alw: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" assumes "\<And>xs. \<phi> xs" shows "alw \<phi> xs"
ircI: fixes f :: "'d set \<Rightarrow> 'd set" assumes "\<And>B a. a \<notin> f (insert a B) \<Longrightarrow> f (insert a B) = f B" shows "irc f"
cbf: shows "b O f \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
if_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'e"   and u :: "'e"   and y :: "'e"   and v :: "'e" assumes "b = c"   and "c \<Longrightarrow> x = u"   and "\<not> c \<Longrightarrow> y = v" shows "(if b then x else y) = (if c then u else v)"
random_variable_small_tails: fixes alpha :: "real"   and f :: "real \<Rightarrow> real"   and K :: "real" assumes "0 < alpha"   and "PZ.random_variable borel f" shows "\<exists>C. PZ.prob {x \<in> space Z. C \<le> \<bar>f x\<bar>} < alpha \<and> K \<le> C"
order_trans_rules_21: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y < z" shows "x < z"
lookup_phis'_code: fixes g :: "'g"   and "next" :: "'node \<times> 'val"   and v :: "'node \<times> 'val" shows "Mapping.lookup (phis'_code g next) v = (if snd v = snd next then None else map_option (map (uninst_code.substNext_code phis g next)) (Mapping.lookup (phis g) v))"
length_lower: fixes a :: "bit list"   and b :: "bit list" assumes "length a < length b"   and "hd b \<noteq> \<zero>" shows "bv_to_nat a < bv_to_nat b"
map_sel_2: fixes a :: "'g list"   and f :: "'g \<Rightarrow> 'h" assumes "a \<noteq> []" shows "tl (map f a) = map f (tl a)"
subst_apply_fv_unfold: fixes t :: "('e, 'f) Term.term"   and \<delta> :: "'f \<Rightarrow> ('e, 'd) Term.term" shows "fv (t \<cdot> \<delta>) = fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` fv t)"
pfun_minus_common_subset: fixes h :: "('a, 'b) pfun"   and f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" assumes "h \<subseteq>\<^sub>p f"   and "h \<subseteq>\<^sub>p g" shows "(f - h = g - h) = (f = g)"
smap_siterate: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "smap f (siterate f x) = siterate f (f x)"
evenE: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" obtains b :: "'b" where "a = times (numeral.numeral one plus (num.Bit0 num.One)) b"
dpWeak: fixes \<Gamma> :: "'b form multiset"   and \<Delta> :: "'b form multiset"   and n :: "nat"   and R :: "('b sequent list \<times> 'b sequent) set"   and R' :: "('b sequent list \<times> 'b sequent) set"   and \<Gamma>' :: "'b form multiset"   and \<Delta>' :: "'b form multiset" assumes "( \<Gamma> \<Rightarrow>* \<Delta>, n) \<in> derivable R*"   and "R' \<subseteq> upRules"   and "R = Ax \<union> R'" shows "( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', n) \<in> derivable R*"
lemma6: fixes r :: "lam" assumes "WN r" shows "\<exists>r'. r \<rightarrow>\<^sub>l r'"
st_Infinitesimal_add_SReal: fixes x :: "real star"   and e :: "real star" assumes "x \<in> \<real>"   and "e \<in> Infinitesimal" shows "st (x + e) = x"
1: shows "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F y x f"
exec_to_exec_strip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "\<not> isFault t" shows "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
valid_refs_invD_9: fixes m :: "'e"   and x :: "'f"   and s :: "'e process_name \<Rightarrow> ('g, 'e, 'h, 'f) Model.local_state" assumes "mut_m.reachable m x s"   and "valid_refs_inv s" shows "\<exists>obj. sys_heap s x = Some obj"
Aexp_eq: fixes e :: "exp"   and a :: "Arity" shows "Aexp_syn' a e = fst (cCCexp e\<cdot>a)"
get_dom_component_separates_tree_order: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'"   and "ptr' \<notin> set c" shows "set to \<inter> set c' = {}"
entail_union: fixes N :: "'f set"   and N1 :: "'f set"   and N2 :: "'f set" shows "(N \<Turnstile> N1 \<and> N \<Turnstile> N2) = (N \<Turnstile> N1 \<union> N2)"
rep_eq: fixes x :: "'b"   and xa :: "'c" shows "lookup (Poly_Mapping.single x xa) = (\<lambda>k'. xa when x = k')"
assms_2: shows "sat_precond_as s1 (a # as)"
assms: shows "(x, xs) \<in># ms_of_greek ys"
dia_diff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x\<rangle> y \<cdot> ad ( |x\<rangle> z) \<le>  |x\<rangle> (d y \<cdot> ad z)"
assms_2: shows "p \<in> sterms \<Gamma> (p1 \<oplus> p2)"
skew_case: fixes t :: "('b \<times> nat) tree" shows "skew t = (case t of \<langle>\<rangle> \<Rightarrow> \<langle>\<rangle> | \<langle>\<langle>\<rangle>, a, t3\<rangle> \<Rightarrow> \<langle>\<langle>\<rangle>, a, t3\<rangle> | \<langle>\<langle>t1, (y, lvy), t2\<rangle>, (x, lvx), t3\<rangle> \<Rightarrow> if lvx = lvy then \<langle>t1, (y, lvx), \<langle>t2, (x, lvx), t3\<rangle>\<rangle> else t)"
trim_def: fixes rs :: "'rule stream"   and s :: "'state" shows "trim rs s = sdrop_while (\<lambda>r. \<not> enabled r s) rs"
eq2: shows "\<phi> ((24, 0) [^]\<^bsub>R\<^esub> ((M - 1) div 2)) = \<ominus>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub>"
axioms_2: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "proper_sub_design \<U> \<A> \<V> \<B>" shows "design \<V> \<B>"
image_uminus_atLeastLessThan: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.atLeastLessThan less_eq less x y = ord.greaterThanAtMost less_eq less (uminus y) (uminus x)"
chine_simps_1: shows "C.arr assoc\<nu>\<pi>\<rho>.chine"
dvd_minus_self: fixes m :: "nat"   and n :: "nat" shows "(m dvd n - m) = (n < m \<or> m dvd n)"
ESem_fresh_cong: fixes ESem :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<rightarrow> 'c"   and e :: "'a"   and \<rho> :: "'b \<Rightarrow> 'c"   and \<rho>' :: "'b \<Rightarrow> 'c" assumes "has_ignore_fresh_ESem ESem"   and "\<rho> f|` fv e = \<rho>' f|` fv e" shows "ESem e\<cdot>\<rho> = ESem e\<cdot>\<rho>'"
rbt_sorted_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'c) list" assumes "n \<le> Suc (length kvs)"   and "sorted (map fst kvs)"   and "distinct (map fst kvs)" shows "rbt_sorted (fst (rbtreeify_g n kvs))"
F1_def: shows "F1 = filter (\<lambda>f. \<exists>v\<in>set V1. f \<in> set (facesAt g v)) (faces g)"
simps_2: fixes x :: "('b, 'a) Node"   and xs :: "('b, 'a) Node list" shows "FingerTreeStruc.gmnl (x # xs) = FingerTreeStruc.gmn x + FingerTreeStruc.gmnl xs"
int_of_uint64_inv: fixes x :: "int" assumes "0 \<le> x"   and "x < 18446744073709551616" shows "int_of_uint64 (uint64_of_int x) = x"
Ln_times: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0" shows "Ln (w * z) = (if Im (Ln w + Ln z) \<le> - pi then Ln w + Ln z + \<i> * complex_of_real (2 * pi) else if pi < Im (Ln w + Ln z) then Ln w + Ln z - \<i> * complex_of_real (2 * pi) else Ln w + Ln z)"
assms_6: shows "sKS s \<notin> LocalSecrets Q"
asIMOD_igWlsAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and us :: "'varSort"   and s :: "'sort"   and A :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs" shows "igWlsAbs (asIMOD SEM) (us, s) A = (\<forall>val. (sWlsVal SEM val \<or> A val = undefined) \<and> (sWlsVal SEM val \<longrightarrow> sWlsAbs SEM (us, s) (A val)))"
simps_2: fixes a :: "'a"   and w :: "'a list" shows "LynSuf (a # w) = (if Lyndon (a # w) then a # w else LynSuf w)"
assms_6: assumes "m < n \<and> x < y"   and "x \<noteq> m"   and "x \<noteq> n"   and "y = n" shows "P (arr m) (arr x)"
distinct_19: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi' :: "pi" shows "name1{name2}.pi \<noteq> \<tau>.pi'"
assms: fixes p :: "real poly" assumes "p \<in> set (poly_list fml)" shows "ls ! index_of ps p = poly p x"
det_upper_triangular: fixes A :: "'b mat"   and n :: "nat" assumes "Matrix.upper_triangular A"   and "A \<in> carrier_mat n n" shows "Determinant.det A = prod_list (diag_mat A)"
euler_genus_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map" shows "pre_digraph_map.euler_genus G M \<equiv> (int (2 * card (pre_digraph.sccs G)) - int (card (pre_digraph.isolated_verts G)) - pre_digraph_map.euler_char G M) div 2"
indicator_def: fixes S :: "'d set"   and x :: "'d" shows "indicator S x = (if x \<in> S then 1::'c else (0::'c))"
subst_size_form: fixes p :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term"   and i :: "nat" shows "FOL_Fitting.size_form (p[t/i]) = FOL_Fitting.size_form p"
rtrancl_listrel1_if_listrel: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set" assumes "(xs, ys) \<in> listrel r" shows "(xs, ys) \<in> (listrel1 r)\<^sup>*"
closed_Int: fixes S :: "'b set"   and T :: "'b set" assumes "closed S"   and "closed T" shows "closed (S \<inter> T)"
real_sqrt_less_mono: fixes x :: "real"   and y :: "real" assumes "x < y" shows "sqrt x < sqrt y"
q_unfold: shows "q = pq * xx ^ min op oq * (qq div pq * xx ^ (oq - min op oq))"
hyps_3: assumes "swapidseq_ext A n p" shows "swapidseq_ext (A \<union> B) n p"
iptables_goto_bigstep_to_undecided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> Undecided" shows "s = Undecided"
row_carrier_vec: fixes i :: "nat"   and nr :: "nat"   and A :: "'c mat"   and nc :: "nat" assumes "i < nr"   and "A \<in> carrier_mat nr nc" shows "row A i \<in> carrier_vec nc"
inl_isinl: fixes op :: "op"   and xs :: "dynamic list"   and opinl :: "opinl" assumes "inl op xs = Some opinl" shows "isinl opinl xs"
encrypt_def: fixes key :: "bool list"   and plain :: "bool list" shows "encrypt key plain = spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. return_spmf (r, plain [\<oplus>] f key r))"
read_tmps_distinct_nth_update: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and sb :: "'b memref list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "read_tmps_distinct ts"   and "i < length ts"   and "distinct_read_tmps sb" shows "read_tmps_distinct (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
coarse_run_def: fixes rho :: "nat \<Rightarrow> ('f, 'g, 'h, 'i) config_scheme"   and r :: "nat"   and p :: "'g" shows "coarse_run rho r p \<equiv> state (rho (fg_start_round rho p r)) p"
assms_1: shows "(return_time P \<longlongrightarrow> return_time P x) (at x within S)"
order_1_eq_0: fixes x :: "'b" shows "order x 1 = 0"
rev_exhaust: fixes xs :: "'b list" obtains "xs = []"   | ys :: "'b list" and y :: "'b" where "xs = ys @ [y]"
diff_closed: fixes G :: "'a set"   and g :: "'a"   and h :: "'a" assumes "Group G"   and "g \<in> G"   and "h \<in> G" shows "g - h \<in> G"
inject_5: fixes x5 :: "load_store_type"   and y5 :: "load_store_type" shows "(load_store_type x5 = load_store_type y5) = (x5 = y5)"
finite_cartesian_product_iff: fixes A :: "'b set"   and B :: "'c set" shows "finite (A \<times> B) = (A = {} \<or> B = {} \<or> finite A \<and> finite B)"
IH: fixes w :: "'a"   and c :: "'a list" assumes "path E w ps x"   and "v \<in> set ps"   and "dropWhile ((\<noteq>) v) ps = c" shows "path E v c x"