hcju/isabelleps huggingface dataset:
subcocycle_lim_Tinv: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "I.subcocycle u"   and "I.invertible_qmpt" shows "AE x in M. fmpt.subcocycle_lim M I.Tinv (\<lambda>n x. u n ((I.Tinv ^^ n) x)) x = I.subcocycle_lim u x"
multlist_dividesI: fixes f :: "'a"   and fs :: "'a list" assumes "f \<in> set fs"   and "set fs \<subseteq> carrier G" shows "f divides foldr (\<otimes>) fs \<one>"
stuck_bisim: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> R" shows "(a \<in> Domain a.E) = (b \<in> Domain b.E)"
refl: fixes a :: "'a" assumes "a \<in> U" shows "a \<le>\<^sub>o\<^sub>w a"
set_pmf_mix_eq: fixes a :: "real"   and p :: "'b pmf" assumes "a \<in> {0..1}" shows "mix_pmf a p p = p"
l_identity: fixes w :: "i"   and x :: "'b"   and y :: "'b"   and \<phi> :: "'b \<Rightarrow> \<o>" assumes "[x \<^bold>= y in w]"   and "[\<phi> x in w]" shows "[\<phi> y in w]"
size_balance_tree: fixes t :: "'b tree" shows "size (balance_tree t) = size t"
asian_effect_compute: fixes pr :: "real"   and matur :: "nat" shows "(\<Sum>w\<in>range (pseudo_proj_True matur). prod (prob_component pr w) {0..<matur} * discounted_value r (\<lambda>m. asian_option matur) matur w) = (\<Sum>y\<in>gener_bool_list matur. lprob_comp pr y * det_discount r matur * (\<Sum>i = 1..matur. lgeom_proc u d init (take i y)) / real matur)"
intros_6: fixes X :: "cmsg"   and Y :: "cmsg" assumes "X \<in> cpayload"   and "Y \<in> cpayload" shows "cPair X Y \<in> cpayload"
set_map: fixes f :: "'c \<Rightarrow> 'a"   and v :: "'b \<Rightarrow> 'c" shows "range (f \<circ> v) = \<P> f (range v)"
restrict_relp_restrict_relp: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and P' :: "'c \<Rightarrow> bool"   and Q' :: "'d \<Rightarrow> bool" shows "R \<upharpoonleft> P \<otimes> Q \<upharpoonleft> P' \<otimes> Q' = R \<upharpoonleft> P \<sqinter> P' \<otimes> Q \<sqinter> Q'"
adopt_node_children_subset: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and children' :: "(_) node_ptr list" assumes "h \<turnstile> adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M owner_document node \<rightarrow>\<^sub>h h'"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'"   and "known_ptrs h"   and "type_wf h" shows "set children' \<subseteq> set children"
trans_ile_iadd1: fixes i :: "enat"   and j :: "enat"   and m :: "enat" assumes "i \<le> j" shows "i \<le> j + m"
part_product_in_whole_product: fixes x :: "i"   and y :: "i"   and v :: "i"   and z :: "i" assumes "CM"   and "O x y"   and "P x v \<and> P y z" shows "P (x \<otimes> y) (v \<otimes> z)"
connected_sphere: fixes a :: "'a"   and r :: "real" assumes "2 \<le> DIM('a)" shows "connected (sphere a r)"
approx_minus_iff2: fixes x :: "'c star"   and y :: "'c star" shows "(x \<approx> y) = (- y + x \<approx> 0)"
simps_4: fixes f1 :: "vname \<Rightarrow> 'a"   and f2 :: "'a"   and x1 :: "vname" shows "(case VNam x1 of VNam x \<Rightarrow> f1 x | Res \<Rightarrow> f2) = f1 x1"
r'_def: shows "r' \<equiv> r \<union> {(x, y). x \<in> Z \<and> y \<in> X \<and> intv_const (I x) < k x \<and> isIntv (I' y)}"
primes_omega_primorial': fixes k :: "nat" shows "primes_omega (primorial' k) = k"
pre2: fixes g :: "nat list list" assumes "g \<in> set arch" shows "pre_iso_test g"
domAllow: shows "dom (C (AllowPortFromTo a b po)) \<noteq> {}"
Cons_5: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
simps_14: fixes Vs :: "char list list"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (e\<^sub>1;; e\<^sub>2) = compE\<^sub>1 Vs e\<^sub>1;; compE\<^sub>1 Vs e\<^sub>2"
preserves_arr: fixes f :: "'a" assumes "hom'.arr f" shows "hom.arr (d\<^sub>1 \<star> f \<star> e\<^sub>0)"
plus_hmultiset_def: fixes A :: "hmultiset"   and B :: "hmultiset" shows "A + B = HMSet (hmsetmset A + hmsetmset B)"
A: shows "dual_order.glbound_in_of \<P> \<langle>Tx\<rangle> ((- wx + wy) +o \<langle>Ty\<rangle>) A"
arrE: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.arr f" obtains "f \<noteq> A_B.null" and "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Dom f) (A_B.Cod f) (A_B.Map f)"
Iff_sym: fixes p :: "'a pfm"   and q :: "'a pfm" shows "(\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! q)) = (\<turnstile>\<^sub>! (q \<^bold>\<longleftrightarrow>\<^sub>! p))"
foldl_A1_eq: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "'a"   and i :: "'a"   and ww :: "'a list" assumes "\<And>e. f n e = e"   and "\<And>e. f e n = e"   and "\<And>a b c. f a (f b c) = f (f a b) c" shows "foldl f i ww = f i (foldl f n ww)"
widen_array_object: fixes P :: "'b prog"   and A :: "ty" shows "P \<turnstile> A\<lfloor>\<rceil> \<le> Class Object"
eq_\<I>_gpv_Pause: fixes out :: "'a"   and \<I> :: "('a, 'b) \<I>"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and rpv :: "('c, 'a, 'b) rpv"   and rpv' :: "('d, 'a, 'b) rpv" assumes "out \<in> outs_\<I> \<I>"   and "\<And>input. input \<in> responses_\<I> \<I> out \<Longrightarrow> eq_\<I>_gpv A \<I> (rpv input) (rpv' input)" shows "eq_\<I>_gpv A \<I> (Pause out rpv) (Pause out rpv')"
staticSecret_parts_Spy: fixes m :: "msg"   and evs :: "event list"   and A :: "agent" assumes "m \<in> parts (knows Spy evs)"   and "m \<in> staticSecret A" shows "A \<in> bad \<or> (\<exists>C B X. Says C B X \<in> set evs \<and> m \<in> parts {X}) \<or> (\<exists>C Y. Notes C Y \<in> set evs \<and> C \<in> bad \<and> m \<in> parts {Y})"
diff_less_eq: 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (minus a b) c = less a (plus c b)"
equality: fixes r :: "('a, 'b, 'c) graph_scheme"   and r' :: "('a, 'b, 'c) graph_scheme" assumes "nodes r = nodes r'"   and "edges r = edges r'"   and "more r = more r'" shows "r = r'"
linear_cf_def: fixes r :: "('a \<times> 'a) set"   and linear_cf :: "'a set \<Rightarrow> 'a set"   and X :: "'a set" assumes "linear_cf r linear_cf" shows "linear_cf X \<equiv> set_option (MaxR.MaxR_opt r X)"
additive_def: fixes f :: "'d \<Rightarrow> 'e" shows "additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
tseytin_tran_def: fixes F :: "'a formula" shows "tseytin_tran F \<equiv> \<^bold>\<And>let S = tseytin_toatom F in S F # tseytin_tran1 S F"
PrfP_fresh_iff: fixes a :: "atom"   and s :: "tm"   and k :: "tm"   and t :: "tm" shows "a \<sharp> PrfP s k t = (a \<sharp> s \<and> a \<sharp> k \<and> a \<sharp> t)"
a3: shows "as\<^sub>2 = map (\<lambda>v. (v, False)) (delete_effects_of op)"
n_in_res_v2: shows "n \<^bold>\<in> restrict w (clm ts) c"
proj_id: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "Ik_intT \<sigma> a" shows "proj \<sigma> a = a"
cospan_of_maps_in_bicategory_of_spans_axioms: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg h k"
AE_pair_measure: fixes P :: "'d \<times> 'e \<Rightarrow> bool" assumes "{x \<in> space (T \<Otimes>\<^sub>M P). P x} \<in> sets (T \<Otimes>\<^sub>M P)"   and "AE x in T. AE y in P. P (x, y)" shows "almost_everywhere (T \<Otimes>\<^sub>M P) P"
SFAssReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e,s,b\<rangle> \<rightarrow>* \<langle>C\<bullet>\<^sub>sF{D} := e',s',b'\<rangle>"
Skip: fixes s :: "char list \<Rightarrow> int" shows "(SKIP, s) \<Rightarrow> Suc 0 \<Down> s"
s_def: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" shows "s f x \<equiv> \<Sum>k\<in>Basis. (f k * (x \<bullet> k)) *\<^sub>R k"
zero: fixes w :: "complex" assumes "w \<notin> S" shows "winding_number \<gamma> w = 0"
select_method: shows "P \<turnstile> (C,Cs @\<^sub>p Ds) selects M = (Ts, T, pns, body) via Cs'"
remove1_commute: fixes x :: "'a"   and y :: "'a"   and zs :: "'a list" shows "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)"
axioms_3: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>H :: "'d \<times> 'd \<Rightarrow> 'e"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudonatural_transformation_whisker_left V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
hk: shows "homeomorphism (sphere b s) (sphere 0 1) h k"
reduce_rotate: fixes Ds :: "'a literal list list"   and C :: "'a literal list"   and L :: "'a literal"   and E :: "'a literal list" shows "reduce Ds (C @ [L]) E = reduce Ds (L # C) E"
emeasure_PiM_Collect: fixes J :: "'i set"   and X :: "'i \<Rightarrow> 'a set" assumes "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> M.events i" shows "emeasure (Pi\<^sub>M I M) {x \<in> space (Pi\<^sub>M I M). \<forall>i\<in>J. x i \<in> X i} = (\<Prod>i\<in>J. emeasure (M i) (X i))"
af_G_letter_sat_core: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl"   and \<nu> :: "'b set" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P \<phi>" shows "\<G> \<Turnstile>\<^sub>P af_G_letter \<phi> \<nu>"
hyps: assumes "distinct (map fst xs)" shows "distinct (map fst (map_pair f xs))"
array_foldr_foldr: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'b list"   and b :: "'a" shows "array_foldr (\<lambda>n. f) (Array a) b = foldr f a b"
in_complex_interval_def: fixes y :: "complex"   and x :: "complex_interval" shows "y \<in>\<^sub>c x \<equiv> case x of Complex_Interval r i \<Rightarrow> Re y \<in>\<^sub>i r \<and> Im y \<in>\<^sub>i i"
map_invar_impl_les: fixes m :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b option"   and s :: "'a"   and s' :: "'a" assumes "map_invar_impl m s"   and "les s s'" shows "map_invar_impl m s'"
SS_def: shows "SS = {y. x - y \<in> supp f \<and> y \<in> supp g}"
assms_3: fixes vs :: "nat \<Rightarrow> nat" assumes "vs v = 0" shows "Q vs vs"
args_to_set_subset: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "ap_spec ap"   and "ab_spec ab" shows "args_to_set (gs, ab gs bs hs data, ap gs bs ps hs data) \<subseteq> fst ` (set gs \<union> set bs \<union> fst ` set ps \<union> snd ` set ps \<union> set hs)"
expand: fixes dual :: "'b dual"   and dual' :: "'b dual" assumes "\<partial>\<^sup>- dual = \<partial>\<^sup>- dual'" shows "dual = dual'"
qe_cooper_def: shows "qe_cooper = Z.lift_nnf_qe (qe_cooper\<^sub>1 \<circ> hd_coeffs1)"
ih_7: shows "InvariantCnCharacterization (getCn state') (getC state') (getM state')"
intro: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.orthocomplemented_lattice minus uminus inf less_eq less sup bot top"   and "class.orthomodular_lattice_axioms uminus inf less_eq sup" shows "class.orthomodular_lattice minus uminus inf less_eq less sup bot top"
opp_fold_A_chamber_system_image: shows "opp_fold_A \<turnstile> ChamberComplex.chamber_system A = {F \<in> ChamberComplex.chamber_system A. face_distance z F = chamber_distance D F}"
Suc_0_and_eq: fixes n :: "nat" shows "Suc 0 AND n = n mod 2"
dWWP: shows "wlp S (wlp (While M S) P) \<in> carrier_mat d d"
has_integral_iff_emeasure_lborel: fixes A :: "'c set"   and r :: "real" assumes "A \<in> sets borel"   and "0 \<le> r" shows "((\<lambda>x. 1) has_integral r) A = (emeasure lborel A = ennreal r)"
skl1_inv2I: fixes x :: "skl1_state" assumes "\<And>gnx A B Rb Ni. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; in_progressS (progress x Rb) {xny, xni, xnr, xgnx, xgny, xsk}; sklvl1.guessed_frame Rb xgnx = Some gnx; sklvl1.guessed_frame Rb xni = Some Ni; can_signal x A B\<rbrakk> \<Longrightarrow> 0 < signalsInit x (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)" shows "x \<in> skl1_inv2"
above_Field: fixes r :: "('c \<times> 'c) set"   and a :: "'c" shows "above r a \<subseteq> Field r"
Hadjicostas_integral_integrable: fixes z :: "complex" assumes "- 2 < Re z" shows "complex_interval_lebesgue_integrable lborel 0 1 (f z)"
finite_state_bound: fixes bp :: "instr array"   and s0 :: "integer" shows "finite (state_bound bp s0)"
prod_ide_eq: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "hfsetcat.prod a b = HF'.prod a b"
\<R>\<^sub>2_closed_glob_consistent: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma>' :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_security.closed_glob_consistent dma (sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>')"
devExt_bc: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n' :: "nat"   and n'' :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> Option.is_none (Blockchain.devBC active cmp bc mining t n nid (n' + n''))" shows "Blockchain.devExt active cmp bc mining t n nid n' n'' = bc (cmp (the (Blockchain.devBC active cmp bc mining t n nid (n' + n''))) (t (n' + n'')))"
intersection_numbers_def: shows "source.intersection_numbers \<equiv> {b1 |\<inter>| b2 |b1 b2. b1 \<in># \<B> \<and> b2 \<in># remove1_mset b1 \<B>}"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexIsomorphism X X f"   and "ChamberComplexAutomorphism_axioms X f" shows "ChamberComplexAutomorphism X f"
assms_6: fixes S :: "'a set"   and T :: "'a set" assumes "P S"   and "Q T" shows "S \<inter> T = {}"
sqrt_int_floor: fixes x :: "int" shows "sqrt_int_floor x = \<lfloor>sqrt (real_of_int x)\<rfloor>"
assms_2: fixes y :: "'b" shows "monotone ord (ord_spmf (=)) (g y)"
assms_2: shows "\<FF> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
gt_trans: fixes u :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "u >\<^sub>t t"   and "t >\<^sub>t s" shows "u >\<^sub>t s"
impl_test: fixes ifex_of :: "'a \<Rightarrow> 'b ifex"   and val :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool"   and b1 :: "'a"   and b2 :: "'a" assumes "reduced_bdt_checkers ifex_of val" shows "impl_test ifex_of b1 b2 = (\<forall>s. val b1 s \<longrightarrow> val b2 s)"
Pd_singletons_for_ds_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and X :: "'d set"   and ds :: "'e set"   and d :: "'e" assumes "Contracts Xd Xh Pd Ch" shows "Field (Contracts.Pd_singletons_for_ds Xd Pd X ds d) \<subseteq> {x. Xd x = d}"
set_case_simps_4: fixes s :: "'a set"   and c_empty :: "'b"   and c_sing :: "'a \<Rightarrow> 'b"   and c_else :: "'b" assumes "infinite s" shows "set_case s c_empty c_sing c_else = c_else"
infinite_descent_measure: fixes P :: "'a \<Rightarrow> bool"   and V :: "'a \<Rightarrow> nat"   and x :: "'a" assumes "\<And>x. \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y" shows "P x"
filter_empty_ivls_eucl: shows "(filter_empty_ivls_impl (\<le>), filter_empty_ivls) \<in> clw_rel (\<langle>rnv_rel\<rangle>ivl_rel) \<rightarrow> clw_rel (\<langle>rnv_rel\<rangle>ivl_rel)"
IH: shows "wfvarsoccs\<^sub>s\<^sub>t (unlabel A) = (\<Union>l. wfvarsoccs\<^sub>s\<^sub>t (proj_unl l A))"
a': shows "( \<Gamma> \<Rightarrow>* E, n) \<in> derivable R*"
inf_greatest: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y"   and "less_eq x z" shows "less_eq x (inf y z)"
filter_sup_right_isotone: fixes G :: "'a set"   and H :: "'a set"   and F :: "'a set" assumes "G \<subseteq> H" shows "filter_sup F G \<subseteq> filter_sup F H"
carrier_subgroup_generated_subset: fixes A :: "'a set" shows "carrier (subgroup_generated G A) \<subseteq> carrier G"
oppositeSymmetry: fixes literal1 :: "Literal"   and literal2 :: "Literal" shows "(opposite literal1 = literal2) = (opposite literal2 = literal1)"
eventually_at_top_linorderI: fixes c :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "\<And>x. c \<le> x \<Longrightarrow> P x" shows "eventually P at_top"
fold_insert_idem: fixes A :: "'v set"   and X :: "'v \<Rightarrow> int pmf"   and x :: "'v"   and z :: "'m" assumes "finite A" shows "sample_vars X (insert x A) z = bind (memo (sample_var X) x) (\<lambda>_. sample_vars X A z)"
ordinal_0_minus: fixes x :: "ordinal" shows "0 - x = 0"
rel_fun_conversep_grp_grp: fixes f :: "'c \<Rightarrow> 'a"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'd" shows "rel_fun (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp B g) = BNF_Def.Grp {x. range (x \<circ> f) \<subseteq> B} (map_fun f g)"
IH_2: fixes z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and i :: "nat" assumes "\<not> ([] \<in> set buf \<or> nts = [])"   and "mbufnt_take f (f (map hd buf) (hd nts) z) (map tl buf) (tl nts) = (z', buf', nts')"   and "wf_mbufn i js Ps (map tl buf)"   and "list_all2 R [i..<j] (tl nts)"   and "\<And>k. k \<in> set js \<Longrightarrow> k \<le> j" shows "list_all2 R [Mini (i + length (tl nts)) js..<j] nts'"
c_iop_welldefTr2: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G" shows "N \<bullet> \<rho> a \<subseteq> c_iop G N (N \<bullet> a)"
S5Basic_14: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>) \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
simps_6: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Is :: "char list set"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and x :: "char list set"   and Es :: "char list \<Rightarrow> char list set"   and b :: "bexp"   and C :: "acom"   and ux :: "char list set" shows "qdeps ({((I, Is), S, E, x, Es)} WHILE b DO C) ux = Is"
to_nat_mono: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "mod_type_class.to_nat a < mod_type_class.to_nat b"
simps_2: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "asym.implConfid A B M = Aenc \<langle>Messages.Agent A, M\<rangle> (pubK B)"
isomorphic_implies_hpar_2: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong>\<^sub>C f'" shows "C.ide f'"
semiring_0_ow: shows "class.semiring_0 = semiring_0_ow UNIV"
1: shows "snd (action_proj a vs) = fmrestrict_set vs (snd a)"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "antilinear f"   and "bounded_antilinear_axioms f" shows "bounded_antilinear f"
ir_sym: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare (flip P) c c' (flip Q)" shows "ir_hoare P c' c Q"
statement65_special_case: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "p \<in> S"   and "q \<in> S"   and "r \<in> S"   and "p \<noteq> q"   and "r \<notin> {p, q}" shows "\<exists>J. is_K2_isometry J \<and> apply_cltn2 east J = p \<and> apply_cltn2 west J = q \<and> apply_cltn2 north J = r \<and> apply_cltn2 far_north J = proj2_intersection (polar p) (polar q)"
elims: fixes x :: "'a acom"   and y :: "'a" assumes "post x = y" obtains P :: "'a" where "x = SKIP {P__}" and "y = P__"   | xa :: "char list" and e :: "aexp" and P :: "'a" where "x = xa ::= e {Pa__}" and "y = Pa__"   | c1 :: "'a acom" and c2 :: "'a acom" where "x = c1__;; c2__" and "y = post c2__"   | b :: "bexp" and c1 :: "'a acom" and c2 :: "'a acom" and P :: "'a" where "x = IF b__ THEN c1 ELSE c2 {Pb__}" and "y = Pb__"   | Inv :: "'a" and b :: "bexp" and c :: "'a acom" and P :: "'a" where "x = {Inv} WHILE b DO c {P}" and "y = P"