hcju/isabelleps huggingface dataset:
Morse_Gromov_theorem_aux1: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and lambda :: "real"   and C :: "real"   and G :: "'a set"   and z :: "real"   and delta :: "real" assumes "continuous_on {a..b} f"   and "lambda C -quasi_isometry_on {a..b} f"   and "a \<le> b"   and "geodesic_segment_between G (f a) (f b)"   and "z \<in> {a..b}"   and "deltaG TYPE('a) < delta" shows "infdist (f z) G \<le> lambda\<^sup>2 * (11 / 2 * C + 91 * delta)"
enabled'_correct: fixes s :: "nat \<Rightarrow> nat option"   and name :: "char list" assumes "s \<in> valid_states" shows "enabled' problem name s = enabled name s"
pq: fixes n :: "nat" assumes "n < length cl" shows "1 / real (length cl) = 1 / real (length dl)"
strict_mono_on_imp_inj_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "strict_mono_on f A" shows "inj_on f A"
not_bot_below_trans: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> \<bottom>"   and "a \<sqsubseteq> b" shows "b \<noteq> \<bottom>"
mk_all_list_fv_unchanged: fixes l :: "(variable \<times> typ) list"   and B :: "Core.term" shows "fv (mk_all_list l B) = fv B - set l"
dummy_Gcd_def: fixes x :: "'b set" shows "dummy_Gcd x = Gcd x"
continuous: fixes m :: "'b set" assumes "m open\<^bsub>T\<^esub>" shows "carrier \<inter> f -` m open"
simps_1: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "subfrmlsn (\<phi> and\<^sub>n \<psi>) = {\<phi> and\<^sub>n \<psi>} \<union> subfrmlsn \<phi> \<union> subfrmlsn \<psi>"
8: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * (e\<^sup>T * v * v\<^sup>T \<sqinter> g) = bot"
one_mono_ordered_semiring_1_axioms_def: fixes default :: "'b"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "one_mono_ordered_semiring_1_axioms default gt \<equiv> (\<forall>x y z. gt x y \<longrightarrow> gt (x + z) (y + z)) \<and> gt default (0::'b)"
arr_Par_arr_RelI: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T"   and "vsv (T\<lparr>ArrVal\<rparr>)" shows "arr_Par \<alpha> T"
simps_3: fixes x :: "name"   and a :: "name"   and P :: "pi" assumes "x \<sharp> a" shows "summands (a<x>.P) = {a<x>.P}"
corres: fixes v :: "'v"   and w :: "'w"   and u' :: "'v" shows "((v, w, u') \<in> E) = ((u', w, v) \<in> E)"
intros_5: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and ys :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "ms_closure P xs f basis"   and "ms_closure P ys g basis" shows "ms_closure P (times_ms_aux xs ys) (\<lambda>x. f x * g x) basis"
hom: shows "homotopic_with_canon (\<lambda>x. True) U S f f'"
asTerm_qSwap_swap: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGood qX" shows "asTerm (qX #[[x1 \<and> x2]]_xs) = asTerm qX #[x1 \<and> x2]_xs"
simps_2: fixes x :: "'b"   and v :: "'b"   and va :: "'b list" shows "max_list_non_empty (x # v # va) = max x (max_list_non_empty (v # va))"
induct: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "class.linorder less_eq less"   and "P []"   and "\<And>x ys. P ys \<Longrightarrow> P (x # ys)" shows "P a0"
TKR: shows "(rev (take n cl), l) \<in> \<langle>R\<rangle>list_rel"
orderI: fixes a :: "'d"   and b :: "'d" assumes "a = sup a b" shows "b \<le> a"
cases: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'c" assumes "r\<inverse>\<inverse> a1 a2" obtains a :: "'c" and b :: "'d" where "a1 = b" and "a2 = a" and "r a b"
filter_tab_range_subset: fixes c :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'d \<Rightarrow> 'c option" shows "range (filter_tab c t) \<subseteq> range t \<union> {None}"
Agent: fixes agt :: "agent"   and H :: "msg set" shows "msg.Agent agt \<in> synth H"
ee_refl: fixes as :: "'a list" assumes "set as \<subseteq> carrier (Ring_Divisibility.mult_of R)" shows "essentially_equal (Ring_Divisibility.mult_of R) as as"
list2FWpolicy_eq: fixes zs :: "'a list"   and x :: "((int \<times> int) set set, int) Combinators"   and y :: "((int \<times> int) set set, int) Combinators"   and z :: "((int \<times> int) set set, int) Combinators list"   and p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "zs \<noteq> []" shows "C (list2FWpolicy (x \<oplus> y # z)) p = C (x \<oplus> list2FWpolicy (y # z)) p"
cexpr_typing_marg_dens2_cexpr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat"   and y :: "nat"   and vs :: "nat list" assumes "\<Gamma> \<turnstile>\<^sub>c \<delta> : REAL" shows "case_nat (PRODUCT (\<Gamma> x) (\<Gamma> y)) \<Gamma> \<turnstile>\<^sub>c marg_dens2_cexpr \<Gamma> vs x y \<delta> : REAL"
d'_def: shows "d' = (\<lambda>x. dx x / 2)"
some_distinguishes: fixes p :: "perm" shows "(SOME x. supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) distinguishes (p \<bullet> P) from (p \<bullet> Q)"
intros: fixes C :: "'a sequent"   and F :: "'a"   and Fs :: "'a form list"   and Ps :: "'a sequent list" assumes "C = ( \<Empt> \<Rightarrow>* Compound F Fs)" shows "rightPrincipal (Ps, C) (Compound F Fs)"
sublist_imp_subseq: fixes xs :: "'b list"   and ys :: "'b list" assumes "sublist xs ys" shows "subseq xs ys"
subset_code_1: fixes xs :: "'a list"   and B :: "'a set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
interior_translation: fixes a :: "'a"   and S :: "'a set" shows "interior ((+) a ` S) = (+) a ` interior S"
arr_tensor: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "arr (f \<otimes> g)"
preserves_hom: fixes f :: "'d \<times> 'c"   and a :: "'d \<times> 'c"   and b :: "'d \<times> 'c" assumes "DopxC.in_hom f a b" shows "\<guillemotleft>Hom_FopxC.map f : Hom_FopxC.map a \<rightarrow>\<^sub>S Hom_FopxC.map b\<guillemotright>"
Hermite_Lindemann_list: fixes xs :: "(complex \<times> complex) list" assumes "\<forall>(x, y)\<in>set xs. algebraic x \<and> algebraic y"   and "distinct (map snd xs)"   and "(\<Sum>(c, \<alpha>)\<leftarrow>xs. c * exp \<alpha>) = 0" shows "\<forall>c\<in>fst ` set xs. c = 0"
sum_of_powers: fixes m :: "nat"   and n :: "nat" shows "(\<Sum>k\<le>n. real k ^ m) = (bernpoly (Suc m) (real n + 1) - bernpoly (Suc m) 0) / (real m + 1)"
encL: fixes S :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> SourceTerm S"
eq2: shows "(image_mset Mp fs = image_mset Mp gs) = (Fs = Gs)"
indspaceD: 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" assumes "f \<in> aezfun_scalar_mult.indspace smult G FH V" shows "f \<in> addfunset FH V \<inter> aezfun_scalar_mult.smultfunset smult G FH"
Abs_fps_inject: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> {f. True}"   and "y \<in> {f. True}" shows "(Abs_fps x = Abs_fps y) = (x = y)"
ge_game_eq: fixes G :: "game"   and H :: "game" shows "ge_game (G, H) = (\<forall>x. (zin x (right_options G) \<longrightarrow> \<not> ge_game (H, x)) \<and> (zin x (left_options H) \<longrightarrow> \<not> ge_game (x, G)))"
pr_var_def: fixes x :: "'b \<Longrightarrow> 'c" shows "&x = x"
PAC_checker_l_PAC_checker: fixes A :: "char list set \<times> (nat, (char list list \<times> int) list) fmap"   and B :: "nat set \<times> (nat, int mpoly) fmap"   and st :: "((char list list \<times> int) list, char list, nat) pac_step list"   and st' :: "(int mpoly, nat, nat) pac_step list"   and spec :: "(char list list \<times> int) list"   and spec' :: "int mpoly"   and b :: "char list code_status"   and b' :: "status" assumes "(A, B) \<in> \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(st, st') \<in> \<langle>p2rel (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel, var_rel\<rangle>pac_step_rel_raw)\<rangle>list_rel"   and "(spec, spec') \<in> sorted_poly_rel O mset_poly_rel"   and "(b, b') \<in> code_status_status_rel" shows "PAC_checker_l spec A b st \<le> \<Down> (code_status_status_rel \<times>\<^sub>r \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel) (PAC_checker spec' B b' st')"
prodM_mem_eq: fixes I :: "'f set"   and M :: "'f \<Rightarrow> ('g, 'a, 'h) Module_scheme"   and m1 :: "'f \<Rightarrow> 'g"   and m2 :: "'f \<Rightarrow> 'g" assumes "\<forall>i\<in>I. R module M i"   and "m1 \<in> carrier (m\<Pi>\<^bsub>R I\<^esub> M)"   and "m2 \<in> carrier (m\<Pi>\<^bsub>R I\<^esub> M)"   and "\<forall>i\<in>I. m1 i = m2 i" shows "m1 = m2"
chain_iterate: fixes F :: "'b \<rightarrow> 'b" shows "chain (\<lambda>i. iterate i\<cdot>F\<cdot>\<bottom>)"
max_mult_distrib_left: fixes p :: "'c"   and x :: "'c"   and y :: "'c" shows "p * max x y = (if (0::'c) \<le> p then max (p * x) (p * y) else min (p * x) (p * y))"
lift_Some: fixes h :: "'b \<Rightarrow> 'a"   and inp :: "'c \<Rightarrow> 'b option"   and i :: "'c" shows "(\<exists>v. lift h inp i = Some v) = (\<exists>v'. inp i = Some v')"
eq_Tuple3_def: shows "eq = (\<Lambda> \<langle>x1, y1, z1\<rangle> \<langle>x2, y2, z2\<rangle>. eq\<cdot>x1\<cdot>x2 andalso eq\<cdot>y1\<cdot>y2 andalso eq\<cdot>z1\<cdot>z2)"
opp_mult_def: fixes x :: "'a"   and y :: "'a" shows "times.opp_mult (\<odot>) x y \<equiv> y \<odot> x"
isReasonHoldsInPrefix: fixes literal :: "Literal"   and prefix :: "Literal list"   and valuation :: "Literal list"   and clause :: "Literal list" assumes "literal el prefix"   and "isPrefix prefix valuation"   and "isReason clause literal valuation" shows "isReason clause literal prefix"
lnth_lmap: fixes n :: "nat"   and xs :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "enat n < llength xs" shows "lnth (lmap f xs) n = f (lnth xs n)"
iTL_defs_4: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iSince I P Q \<equiv> \<diamond> t I. Q t \<and> (\<box> t' I \<down>> t. P t')"
select_result_E: fixes P :: "'a \<Rightarrow> bool"   and h :: "'c"   and f :: "('c, 'b, 'a) prog" assumes "P |h \<turnstile> f|\<^sub>r"   and "h \<turnstile> ok f" obtains x :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "P x"
min_gallery_simps_1: fixes X :: "'b set set" assumes "ChamberComplex X" shows "SimplicialComplex.min_maxsimpchain X [] = True"
range_subsetD: fixes f :: "'c \<Rightarrow> 'b"   and B :: "'b set"   and i :: "'c" assumes "range f \<subseteq> B" shows "f i \<in> B"
UN_E: fixes b :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'b" where "x \<in> A" and "b \<in> B x"
digraphD_1: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<Z> \<alpha>"
invert_side_of_triangle_2: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "C.arr h"   and "f \<cdot>\<^sub>C g = h"   and "C.iso g" shows "C.seq h (C.inv g) \<and> f = h \<cdot>\<^sub>C C.inv g"
abs_eq: fixes x :: "'a" shows "interval_of x = Interval (x, x)"
add_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "max c a \<le> max c b"
remove_from_disconnected_nodes_removes: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr"   and h' :: "(_) heap"   and disc_nodes' :: "(_) node_ptr list" assumes "l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs heap_is_wellformed parent_child_rel get_child_nodes"   and "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes"   and "h \<turnstile> set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes'" shows "node_ptr \<notin> set disc_nodes'"
antisym: fixes A :: "'e set"   and h :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "blinding_of_on A h bo"   and "bo x y"   and "bo y x"   and "x \<in> A" shows "x = y"
finmap_eq_iff: fixes P :: "'d \<Rightarrow>\<^sub>F 'e"   and Q :: "'d \<Rightarrow>\<^sub>F 'e" shows "(P = Q) = (domain P = domain Q \<and> (\<forall>i\<in>domain P. (P)\<^sub>F i = (Q)\<^sub>F i))"
abs_eq: fixes x :: "('c, 'd) sq_matrix" shows "cofactor x = of_fun (\<lambda>i j. Square_Matrix.det (minor x i j))"
V_imp_Nil: fixes \<phi> :: "('b, 'c, 'd) trans \<Rightarrow> bool"   and f :: "('b, 'c, 'd) trans \<Rightarrow> 'a"   and trn :: "('b, 'c, 'd) trans"   and tr :: "('b, 'c, 'd) trans list" assumes "BD_Security.V \<phi> f (trn # tr) = []" shows "BD_Security.V \<phi> f tr = []"
countableI_bij2: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and A :: "'c set" assumes "bij_betw f B A"   and "countable A" shows "countable B"
internal_branch: fixes ds1 :: "bool list"   and ds2 :: "bool list"   and d :: "bool"   and T :: "tree" assumes "branch (ds1 @ ds2 @ [d]) T" shows "internal ds1 T"
list_nth_split: fixes m :: "nat"   and cs :: "'b list"   and n :: "nat" assumes "m < length cs"   and "n < m"   and "1 < length cs" shows "\<exists>xs ys zs. cs = xs @ cs ! n # ys @ cs ! m # zs"
is_dependent_def: fixes t :: "Core.term" shows "is_dependent t \<equiv> loose_bvar1 t 0"
row_transpose: fixes i :: "'c"   and A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "row i (Finite_Cartesian_Product.transpose A) = column i A"
push_def: fixes v :: "'v"   and PDPE :: "'v set list \<times> 'v set \<times> ('v \<times> 'v) set" shows "push v PDPE \<equiv> let (p, D, pE) = PDPE; p = p @ [{v}]; pE = pE \<union> E \<inter> {v} \<times> UNIV in (p, D, pE)"
eval_def: fixes pred :: "'b Predicate.pred" shows "pred.eval pred = (case pred of pred.Pred x \<Rightarrow> x)"
comp_red_monic_basis_is_reduced_GB: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_Groebner_basis (set xs)" shows "is_reduced_GB (set (comp_red_monic_basis xs))"
bisims1Nil: fixes P :: "'a prog"   and h :: "'heap"   and xs :: "'addr val list" shows "P,[],h \<turnstile> ([], xs) [\<leftrightarrow>] ([], xs, 0, None)"
get_proc_intra: fixes a :: "'edge" assumes "valid_edge a"   and "intra_kind (kind a)" shows "get_proc (sourcenode a) = get_proc (targetnode a)"
graph_homomorphism_def2: fixes G\<^sub>1 :: "('b, 'c) labeled_graph"   and G\<^sub>2 :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> univalent f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union (map_graph f G\<^sub>1) G\<^sub>2 = G\<^sub>2)"
S_def: fixes h :: "'g \<times> 'a"   and e :: "'e" shows "\<Sigma>_AND.S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S_AND h e)"
simps_3: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat"   and n :: "nat" shows "DBM_upd M f i (Suc j) n = DBM_update (DBM_upd M f i j n) i (Suc j) (f i (Suc j))"
QQ6_a_def: shows "QQ6_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . p \<noteq> nil \<and> \<not> gg mrk atom left p \<and> \<not> gg mrk atom right p \<and> t \<noteq> nil \<and> atom t \<and> p' = t \<and> t' = right t \<and> mrk' = mrk \<and> left' = left \<and> right' = right(t := p) \<and> atom' = atom(t := False) :]"
map_ide: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and S :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and \<iota> :: "('ag \<Rightarrow> 'ah) \<Rightarrow> 'ai"   and a :: "'ah" assumes "cones_functor J C D S \<iota>"   and "partial_magma.ide C a" shows "cones_functor.map J C D S \<iota> a = set_category_given_img.mkIde S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (\<iota> ` {\<chi>. Limit.cone J C D a \<chi>})"
snd_to_last_max_normal: assumes "finite (carrier G)"   and "1 < length \<GG>" shows "max_normal_subgroup (\<GG> ! (length \<GG> - 2)) G"
ltake_lmap: fixes n :: "enat"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "ltake n (lmap f xs) = lmap f (ltake n xs)"
mult_isor: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y" shows "x \<cdot> z \<le> y \<cdot> z"
comparable: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "connex A less_eq"   and "x \<in> A"   and "y \<in> A" shows "less_eq x y \<or> less_eq y x"
vintersectionI: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A"   and "x \<in>\<^sub>\<circ> B" shows "x \<in>\<^sub>\<circ> A \<inter>\<^sub>\<circ> B"
elims_1: fixes x :: "'b list \<times> 'b list"   and y :: "bool" assumes "invar x = y" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)" and "y = (fs = [] \<longrightarrow> rs = [])"
disc_1: fixes x1 :: "nat" shows "\<not> is_Const (Var x1)"
abs_summable_on_nat_iff: fixes f :: "nat \<Rightarrow> 'a"   and A :: "nat set" shows "(f abs_summable_on A) = summable (\<lambda>n. if n \<in> A then norm (f n) else 0)"
lowner_le_Q: shows "tensor_P proj_psi (proj_k 0) \<le>\<^sub>L adjoint exM0 * P' * exM0 + adjoint exM1 * Q * exM1"
eval\<Pi>\<^sub>1: fixes x :: "\<Pi>\<^sub>1" shows "eval\<Pi>\<^sub>1 x \<in> UNIV"
sublinear_subadd: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "sublinear t"   and "feasible t" shows "sub_add t"
runit_in_vhom: fixes f :: "'c"   and a :: "'c" assumes "C.ide f"   and "src\<^sub>C f = a" shows "\<guillemotleft>\<r>\<^sub>C[f] : f \<star>\<^sub>C a \<Rightarrow>\<^sub>C f\<guillemotright>"
Ints_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<int>" shows "sum f A \<in> \<int>"
Exit_no_return_target: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "knd a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "trg a = NewExit" shows "False"
quote_all_MonPon2_PfP_ssubst: fixes \<alpha>1 :: "fm"   and \<alpha>2 :: "fm"   and \<beta> :: "fm"   and V :: "name set" assumes "{} \<turnstile> \<alpha>1 IMP \<alpha>2 IMP \<beta>"   and "V \<subseteq> Vs"   and "supp \<alpha>1 \<subseteq> atom ` Vs"   and "supp \<alpha>2 \<subseteq> atom ` Vs"   and "supp \<beta> \<subseteq> atom ` Vs" shows "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<alpha>1\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<alpha>2\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
cont_prodD1: fixes luba :: "'b set \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubc :: "'d set \<Rightarrow> 'd"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'd"   and x :: "'b" assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc f"   and "class.preorder orda (mk_less orda)"   and "lub_singleton luba" shows "cont lubb ordb lubc ordc (\<lambda>y. f (x, y))"
sba2_idempotent: fixes x :: "'a" shows "x \<squnion> x = x"
inf_top_left: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf top x = x"
\<delta>ss_lhs_def: fixes f :: "'L"   and ss :: "'Q set list" shows "\<delta>ss_lhs f ss \<equiv> {uu_. \<exists>q qs. uu_ = q \<and> q \<rightarrow> f qs \<in> \<delta> \<and> list_all_zip (\<in>) qs ss}"
quotients_def: shows "normal\<GG>butlast.quotients = map (\<lambda>i. G\<lparr>carrier := \<GG> ! (n - 1), carrier := take n \<GG> ! (i + 1)\<rparr> Mod take n \<GG> ! i) [0..<length (take n \<GG>) - 1]"
function_convergent_subsequence: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "real" assumes "countable S"   and "\<And>n x. x \<in> S \<Longrightarrow> norm (f n x) \<le> M" obtains k :: "nat \<Rightarrow> nat" where "strict_mono k" and "\<And>x. x \<in> S \<Longrightarrow> \<exists>l. (\<lambda>n. f (k n) x) \<longlonglongrightarrow> l"
\<psi>_\<phi>: fixes f :: "'c"   and b :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : b \<rightarrow> a\<guillemotright>" shows "\<psi> (b, a) ((S.UP \<circ> inl) f) = f"
refpoint_eq_s: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and c' :: "'f list multiset" assumes "valid fg c" shows "((c, e, c') \<in> refpoint fg) = ((c, e, c') \<in> tr fg)"
obj_simps_3: fixes a :: "'a" assumes "B.obj a" shows "trg a = a"
unprot_invt: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "unprot \<sigma> \<or> protFw \<sigma>" shows "invt \<sigma> a = a"
oaodv_trans: fixes i :: "nat" shows "automaton.trans (opaodv i) = oseqp_sos \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i"
frp1_4: fixes p :: "int poly"   and qs :: "int poly list" assumes "factors_of_int_poly p = qs" shows "distinct qs"
nnfis_times: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and z :: "real" assumes "measure_space M"   and "a \<in> nnfis f M"   and "0 \<le> z" shows "z * a \<in> nnfis (\<lambda>w. z * f w) M"
set_memb_axioms: shows "set_memb rs.\<alpha> rs.invar rs.memb"
cases: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat" assumes "interaction_bounded_by consider gpv n" obtains "interaction_bound consider gpv \<le> n"
at_bot_le_at_infinity: shows "at_bot \<le> at_infinity"
findim: shows "\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) ms = V"
inj_on_to_nat_on: fixes A :: "'a set" assumes "countable A" shows "inj_on (to_nat_on A) A"
pCons_2: shows "ipoly (p \<circ>\<^sub>p q) x = ipoly p (ipoly q x)"
rbt_max_eq_rbt_max_opt: fixes t :: "('b, unit) RBT_Impl.rbt" assumes "t \<noteq> rbt.Empty"   and "is_rbt t" shows "rbt_max t = rbt_max_opt t"
inv_is_inverse: fixes f :: "'a1" assumes "C1.iso f" shows "C1.inverse_arrows f (C1.inv f)"
simps_1: fixes xs :: "'b list list" shows "vecsumlist' 0 xs = []"
assms_2: shows "net_ips s1 = net_tree_ips n1"
path_from_toE: fixes v :: "'c"   and xs :: "'c list"   and w :: "'c" assumes "v \<leadsto>xs\<leadsto>\<^bsub>T\<^esub> w" shows "T.path xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"