hcju/isabelleps huggingface dataset:
is_tm_tdghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> (\<Z> \<alpha> \<and> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>) \<and> \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB> \<and> \<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
u_max_eq: shows "u_max = ego.q \<delta> - v\<^sub>e\<^sup>2 / a\<^sub>e / 2"
sq_mtx_one_idty_2: fixes A :: "'a sq_mtx" shows "A * 1 = A"
finite_lprefix_def: shows "Coinductive_List.finite_lprefix = lprefix"
elim_interleave': fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "interleave' xs ys zs = (zs \<in> xs \<otimes> ys)"
comp_arr_ide: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.ide a"   and "B.VV.seq f a" shows "B.VV.comp f a = f"
simps: fixes rf :: "recf" shows "rec_accum rf = (let vl = arity rf in Pr (vl - 1) (Cn (vl - 1) rf (get_fstn_args (vl - 1) (vl - 1) @ [Cn (vl - 1) (constn 0) [recf.id (vl - 1) 0]])) (Cn (Suc vl) rec_mult [recf.id (Suc vl) vl, Cn (Suc vl) rf (get_fstn_args (Suc vl) (vl - 1) @ [Cn (Suc vl) s [recf.id (Suc vl) (vl - 1)]])]))"
fvi_less_nfv: fixes \<phi> :: "'c MFOTL.formula" shows "\<forall>i\<in>fv \<phi>. i < MFOTL.nfv \<phi>"
igSubstAbs_termMOD: shows "igSubstAbs termMOD = substAbs"
rep_eq: fixes x :: "'f cset" shows "cBall x = Ball (rcset x)"
poly_as_sum_of_monoms: fixes p :: "'b poly" shows "(\<Sum>i\<le>degree p. monom (coeff p i) i) = p"
one_is_same: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> = \<one>\<^bsub>G\<^esub>"
AbGroup_inner_dirsum_el_decomp_nth_in_nth: fixes Gs :: "'b set list"   and k :: "nat"   and n :: "nat"   and x :: "'b" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "k < length Gs"   and "n < length Gs"   and "x \<in> Gs ! n" shows "(if x \<in> (\<Oplus>G\<leftarrow>Gs. G) then \<Oplus>Gs\<leftarrow> x ! k else (0::'b)) = (if k = n then x else (0::'b))"
semilattice_set_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semilattice_set f \<equiv> semilattice f"
stable_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "PdXXX.stable_on ds X = (PdXXX.individually_rational_on ds X \<and> PdXXX.stable_no_blocking_on ds X)"
t_ne_1: shows "t_ne \<in> {t0--t1} \<inter> {t0--t2}"
gbm_delete_node_impl: shows "graph_delete_node gbm_\<alpha> gbm_invar gbm_delete_node"
depth_height_imp_sibling_ne: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t"   and "depth t a = height t"   and "0 < height t"   and "a \<in> alphabet t" shows "sibling t a \<noteq> a"
dagger_sum_unfold: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<dagger> + x\<^sup>\<dagger> \<cdot> y \<cdot> (x + y)\<^sup>\<dagger> = (x + y)\<^sup>\<dagger>"
c4: assumes "ip \<in> kD rt"   and "sqn rt ip = \<pi>\<^sub>2 r"   and "the (flag rt ip) = Aodv_Basic.inv" shows "P (rt(ip \<mapsto> r))"
rel_FGcontra_pos_distr_cond_def: fixes Co1 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co2' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Co3 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Co3' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Co4 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Co4' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Co5 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Co5' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra1 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra1' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra2 :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and Contra2' :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and Contra3 :: "'an \<Rightarrow> 'ao \<Rightarrow> bool"   and Contra3' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra4 :: "'aq \<Rightarrow> 'ar \<Rightarrow> bool"   and Contra4' :: "'ar \<Rightarrow> 'as \<Rightarrow> bool"   and Contra5 :: "'at \<Rightarrow> 'au \<Rightarrow> bool"   and Contra5' :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and uu :: "('aw \<times> 'ax \<times> 'ay \<times> 'az \<times> 'ba) itself" shows "rel_FGcontra_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' Contra5 Contra5' uu = (\<forall>L1 L1'. rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4 Contra5 OO rel_FGcontra L1' Co1' Co2' Co3' Co4' Co5' Contra1' Contra2' Contra3' Contra4' Contra5' \<le> rel_FGcontra (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Co4 OO Co4') (Co5 OO Co5') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') (Contra4 OO Contra4') (Contra5 OO Contra5'))"
projection_fixes_image: fixes M :: "'a set"   and x :: "'a" assumes "closed_csubspace M"   and "x \<in> M" shows "projection M x = x"
remove_nth_take_drop: fixes i :: "nat"   and xs :: "'b list" shows "remove_nth i xs = take i xs @ drop (Suc i) xs"
Cons: fixes a :: "('e, 'a) Node"   and t :: "('e, 'a) FingerTreeStruc" assumes "r = Some (a, t)" shows "thesis"
plus_monotone_strict: fixes s :: "'a \<Rightarrow>\<^sub>0 nat"   and t :: "'a \<Rightarrow>\<^sub>0 nat"   and u :: "'a \<Rightarrow>\<^sub>0 nat" assumes "s \<prec> t" shows "s + u \<prec> t + u"
isnormNum_int_2: shows "isnormNum (1)\<^sub>N"
on_triple_univ: fixes f :: "('e \<times> 'f) set" assumes "univalent f" shows "univalent (on_triple f)"
Ints_of_int: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus z \<in> ring_1.Ints one plus zero minus"
interior_of_eq_empty_complement: fixes X :: "'c topology"   and S :: "'c set" shows "(X interior_of S = {}) = (X closure_of (topspace X - S) = topspace X)"
extend_filter: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "frequently P F" shows "inf F (principal {x. P x}) \<noteq> bot"
modify_roundup_ge2: fixes emBits :: "nat" assumes "176 \<le> roundup emBits 8 * 8" shows "21 < roundup emBits 8"
put_effect_compute: fixes pr :: "real"   and matur :: "nat"   and K :: "real" shows "(\<Sum>w\<in>range (pseudo_proj_True matur). prod (prob_component pr w) {0..<matur} * discounted_value r (\<lambda>m. put_option matur K) matur w) = (\<Sum>y\<in>gener_bool_list matur. lprob_comp pr y * det_discount r matur * max (K - lgeom_proc u d init (take matur y)) 0)"
rel_interior_ball: fixes S :: "'a set" shows "rel_interior S = {x \<in> S. \<exists>e>0. ball x e \<inter> affine hull S \<subseteq> S}"
Eyz: shows "infdist (f w) {f y--f z} \<le> E"
nc_zero_closed: shows "(0::'a) \<sqinter> nc = (0::'a)"
cmp_simps_2: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "S.src (S.cmp\<^sub>U\<^sub>P (f, g)) = S.src (S.UP g)"
simps_3: fixes v :: "bool"   and va :: "bool list" shows "branch (v # va) Leaf = False"
Cantor_sum_0E: fixes \<alpha>s :: "V list"   and ms :: "nat list" assumes "Cantor_sum \<alpha>s ms = 0"   and "list.set \<alpha>s \<subseteq> ON"   and "list.set ms \<subseteq> {0<..}"   and "length \<alpha>s = length ms" shows "\<alpha>s = []"
R3_idem: shows "R3 \<circ> R3 = R3"
sim_trc_commute: fixes M :: "('b, 'c, 'd) KripkeStructure"   and M' :: "('b, 'c, 'e) KripkeStructure"   and f :: "'d \<Rightarrow> 'e"   and t :: "'d" assumes "kripke M"   and "sim M M' f"   and "t \<in> worlds M" shows "f ` (\<Union> (range (relations M)))\<^sup>* `` {t} = (\<Union> (range (relations M')))\<^sup>* `` {f t}"
path_from_toI: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c"   and w :: "'c" assumes "Digraph G"   and "Digraph.walk G xs \<and> distinct xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
bind_mono: fixes B :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'c option"   and C :: "'c \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'd option" assumes "monotone (flatf_ord None) option_ord B"   and "\<And>y. monotone (flatf_ord None) option_ord (C y)" shows "monotone (flatf_ord None) option_ord (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"
list_of_dnelist_inverse: fixes x :: "'a dnelist" shows "Abs_dnelist (list_of_dnelist x) = x"
ps'_def: shows "ps' = filter (\<lambda>p. dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps"
alt_group_carrier_as_three_cycles: fixes n :: "nat" shows "carrier (alt_group n) = generate (alt_group n) (three_cycles n)"
linear_scale: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and r :: "'b"   and b :: "('b, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (r *s b) = r *s f b"
coherence_key_fact: fixes a :: "'c local.term"   and b :: "'c local.term"   and c :: "'c local.term" assumes "Ide a \<and> Nml a"   and "Ide b \<and> Nml b"   and "Ide c \<and> Nml c"   and "Src a = Trg b"   and "Src b = Trg c" shows "\<lbrace>(a \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> b) \<^bold>\<Down> c\<rbrace> \<cdot> (\<lbrace>a \<^bold>\<Down> b\<rbrace> \<star> \<lbrace>c\<rbrace>) = (\<lbrace>a \<^bold>\<Down> b \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> c\<rbrace> \<cdot> (\<lbrace>a\<rbrace> \<star> \<lbrace>b \<^bold>\<Down> c\<rbrace>)) \<cdot> \<a>[\<lbrace>a\<rbrace>, \<lbrace>b\<rbrace>, \<lbrace>c\<rbrace>]"
simple_paths_def: fixes x :: "'a" shows "simple_paths x \<equiv> \<exists>y. y \<squnion> y\<^sup>T = x \<and> pd_kleene_allegory_class.acyclic y \<and> injective y \<and> univalent y"
poly_eq_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(p = q) = (\<forall>n. coeff p n = coeff q n)"
real_power_add: fixes d :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < d" shows "real_power d (a + b) = real_power d a * real_power d b"
UP_m_assoc: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> nat \<Rightarrow> int"   and r :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "p \<in> carrier (UP Zp)"   and "q \<in> carrier (UP Zp)"   and "r \<in> carrier (UP Zp)" shows "p \<otimes>\<^bsub>UP Zp\<^esub> q \<otimes>\<^bsub>UP Zp\<^esub> r = p \<otimes>\<^bsub>UP Zp\<^esub> (q \<otimes>\<^bsub>UP Zp\<^esub> r)"
pw_ref_iff: fixes S :: "'a nres"   and R :: "('a \<times> 'b) set"   and S' :: "'b nres" shows "(S \<le> \<Down> R S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> (\<exists>s'. (x, s') \<in> R \<and> inres S' s')))"
last_is_acc_impl_def: fixes s :: "nat set list \<times> 'c" shows "last_is_acc_impl s \<equiv> let (a, uu_) = s in ASSERT (a \<noteq> []) \<bind> (\<lambda>_. RETURN (\<forall>i<num_acc. i \<in> last a))"
HInv3_R_sym: fixes s :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "HInv3_R s p q d" shows "HInv3_R s q p d"
free_in_quad_fm: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and A :: "atom fm" assumes "var \<notin> vars a"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "var \<notin> vars d" shows "freeIn var (quadratic_sub_fm var a b c d A)"
insert_preserves_order: fixes ops :: "('b \<times> 'b option) list"   and rest :: "('b \<times> 'b option) list"   and before :: "('b \<times> 'b option) list"   and after :: "('b \<times> 'b option) list"   and oid :: "'b"   and ref :: "'b option" assumes "insert_ops ops"   and "insert_ops rest"   and "rest = before @ after"   and "ops = before @ (oid, ref) # after" shows "\<exists>xs ys zs. interp_ins rest = xs @ zs \<and> interp_ins ops = xs @ ys @ zs"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "monoid_add_hom hom"   and "inj_semigroup_add_hom hom" shows "inj_monoid_add_hom hom"
exiting: fixes s :: "'s" assumes "s \<notin> X" shows "\<exists>t\<in>X. (s, t) \<in> acc"
t_def: shows "integrate_insert m s (P m) (S m) = Inr t \<and> set t = to_woot_character (M \<union> {Insert m}) ` insert_messages (M \<union> {Insert m})"
LESS_SUC_1: fixes k :: "'k"   and l :: "nat"   and idx :: "'n" shows "LEQ k (Suc l) idx = LESS k l idx"
Abs_G_perm_def: fixes g :: "'a" shows "\<pp> g \<equiv> Abs_permutation (G_perm g)"
frameImpResChainPres: fixes F :: "'b frame"   and G :: "'b frame"   and xvec :: "name list" assumes "F \<hookrightarrow>\<^sub>F G" shows "\<lparr>\<nu>*xvec\<rparr>F \<hookrightarrow>\<^sub>F \<lparr>\<nu>*xvec\<rparr>G"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   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" assumes "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs"   and "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs" shows "l_set_tag_name_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs"
of_list_repr_def: fixes ba :: "'b list \<times> 'b list" shows "of_list_repr ba = rev (fst ba) @ snd ba"
set_sort: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "set (sort_key f xs) = set xs"
generate_valid_topology_SOME_edges: fixes M :: "'a SecurityInvariant_configured list"   and V :: "'a set"   and E :: "('a \<times> 'a) set" shows "edges (generate_valid_topology_SOME M \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> E"
equal_eq: fixes x :: "'a"   and y :: "'a" shows "equal_class.equal x y = (x = y)"
step_Cn_correct: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "valid ((Cn n f gs, xs, []) # rest)"   and "\<And>xs rest. valid ((f, xs, []) # rest) \<Longrightarrow> correct ((f, xs, []) # rest, None)"   and "\<And>g xs rest. \<lbrakk>g \<in> set gs; valid ((g, xs, []) # rest)\<rbrakk> \<Longrightarrow> correct ((g, xs, []) # rest, None)" shows "correct ((Cn n f gs, xs, []) # rest, None)"
absorb_iff2: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"
out_arcs_w: shows "out_arcs H w = {rev_H uw, rev_H vw}"
minWait_def: fixes rs :: "'rule stream"   and s :: "'state" shows "minWait rs s \<equiv> LEAST n. enabled (shd (sdrop n rs)) s"
dest_def: shows "dest = fst \<circ> snd \<circ> snd"
sum_two_rw: fixes f :: "nat \<Rightarrow> 'a" shows "sum f {0..<2} = sum f {0, 1}"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Oldf x1 = Oldf y1) = (x1 = y1)"
ex_gt_or_lt: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 a :: "'b" assumes "class.linear_continuum Inf Sup inf less_eq less sup" shows "\<exists>b. less a b \<or> less b a"
cart2_pt_def: fixes p :: "proj2" shows "cart2_pt p \<equiv> vector [proj2_rep p $ 1 / proj2_rep p $ 3, proj2_rep p $ 2 / proj2_rep p $ 3]"
roofedI: fixes G :: "('k, 'l) graph_scheme"   and x :: "'k"   and B :: "'k set"   and S :: "'k set" assumes "\<And>p y. \<lbrakk>path G x p y; y \<in> B\<rbrakk> \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S" shows "x \<in> roofed_gen G B S"
CK_nf_neg_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "CK_nf_neg f = (\<exists>g. CK_nf_pos g \<and> f = C \<circ> g)"
imp_true: fixes p :: "'b upred" shows "(p \<Rightarrow> true) = true"
image_subset_iff: fixes f :: "'h \<Rightarrow> 'g"   and A :: "'h set"   and B :: "'g set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
subgroup_self: fixes G :: "('d, 'e) monoid_scheme" assumes "Group.group G" shows "subgroup (carrier G) G"
enc_trm: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "\<langle>\<phi>\<rangle> \<in> trm"
WT_gpv_ContD: fixes \<Gamma> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv"   and out :: "'b"   and c :: "('e, 'b, 'd) rpv"   and input :: "'d" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out" shows "\<Gamma> \<turnstile>g c input \<surd>"
sys_block_sizes_def: shows "sys_block_sizes \<equiv> {int (card bl) |bl. bl \<in># \<B>}"
abs_idempotent: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (abs a) = abs a"
of_nat_mask_eq: 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 bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "semiring_1.of_nat one plus zero (mask n) = mask n"
finite_facetrel_card: fixes x :: "'b set"   and y :: "'b set" assumes "finite x"   and "y \<lhd> x" shows "card x = Suc (card y)"
strongly_fair_imp_weakly_fair: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "strongly_fair enabled taken \<sigma>" shows "weakly_fair enabled taken \<sigma>"
recurrent_acc_1: fixes x :: "'s"   and y :: "'s" assumes "recurrent x"   and "(x, y) \<in> acc" shows "U y x = 1"
snd_Gauss_Jordan_in_ij_det_P_is_snd_Gauss_Jordan_in_ij_PA: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'c"   and P :: "(('b, 'd) vec, 'd) vec" shows "snd (Gauss_Jordan_in_ij_det_P A i j) = snd (Gauss_Jordan_in_ij_PA (P, A) i j)"
simps_1: fixes v :: "'v"   and v' :: "'v" shows "is_path v [] v' = (v = v' \<and> v' \<in> V)"
pair_no_loops: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "fst e \<noteq> snd e"
cobounded2: fixes a :: "'k multiset"   and b :: "'k multiset" shows "a \<inter># b \<subseteq># b"
fun_ord_ge: shows "fun_ord (\<lambda>x y. y \<le> x) = (\<lambda>x y. y \<le> x)"
equality: fixes r :: "('a, 'b, 'c) pre_digraph_scheme"   and r' :: "('a, 'b, 'c) pre_digraph_scheme" assumes "verts r = verts r'"   and "arcs r = arcs r'"   and "tail r = tail r'"   and "head r = head r'"   and "pre_digraph.more r = pre_digraph.more r'" shows "r = r'"
basechambers_half_chamber_systems_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "C0 \<in> f \<turnstile> ChamberComplex.chamber_system X"
llcp_same_conv_length: fixes xs :: "'c llist" shows "llcp xs xs = llength xs"
real_of_ereal_add: fixes a :: "ereal"   and b :: "ereal" shows "real_of_ereal (a + b) = (if \<bar>a\<bar> = \<infinity> \<and> \<bar>b\<bar> = \<infinity> \<or> \<bar>a\<bar> \<noteq> \<infinity> \<and> \<bar>b\<bar> \<noteq> \<infinity> then real_of_ereal a + real_of_ereal b else 0)"
bij_iff: fixes f :: "'b \<Rightarrow> 'c" shows "bij f = (\<forall>x. \<exists>!y. f y = x)"
moebius_mu_eq_zero_iff: fixes n :: "nat" shows "(moebius_mu n = (0::'a)) = (\<not> squarefree n)"
inf_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "inf f g x = inf (f x) (g x)"
span_span: fixes A :: "'b set" shows "ideal (ideal A) = ideal A"
SUP_least: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and u :: "'a" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> u" shows "Sup (f ` A) \<le> u"
replace_Goto_with_Call_in_terminal_chain_Undecided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and t :: "state" assumes "\<Gamma> chain = Some rs"   and "terminal_chain rs" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Goto chain)], Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"
simps: fixes a1 :: "'a OCL_Types.type"   and a2 :: "'a OCL_Types.type" shows "to_unique_collection a1 a2 = ((\<exists>\<tau>. a1 = Collection \<tau> \<and> a2 = Set \<tau>) \<or> (\<exists>\<tau>. a1 = Set \<tau> \<and> a2 = Set \<tau>) \<or> (\<exists>\<tau>. a1 = OrderedSet \<tau> \<and> a2 = OrderedSet \<tau>) \<or> (\<exists>\<tau>. a1 = Bag \<tau> \<and> a2 = Set \<tau>) \<or> (\<exists>\<tau>. a1 = Sequence \<tau> \<and> a2 = OrderedSet \<tau>))"
exec_bind_SE_success2: fixes A :: "'b \<Rightarrow> ('a \<times> 'b) option"   and \<sigma> :: "'b"   and b :: "'a"   and \<sigma>' :: "'b"   and M :: "'b \<Rightarrow> (bool \<times> 'b) option" assumes "A \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> A ;- M) = (\<sigma>' \<Turnstile> M)"
eq_zero: fixes a :: "nat" assumes "\<not> coprime a n" shows "\<chi>\<^sub>0 a = 0"
A: shows "set (divisors_nat 12285) = {1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095, 12285}"
tau_meet: fixes x :: "'a"   and y :: "'a" shows "\<tau> (x \<sqinter> y) = \<tau> x \<sqinter> \<tau> y"
at_right_ereal: fixes r :: "real" shows "at_right (ereal r) = filtermap ereal (at_right r)"
assms: shows "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 < n"
assms_1: shows "stream_all2 (\<in>) xs (cycle as)"
op_mtx_new_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a" shows "op_mtx_new c \<equiv> c"
cring_pow_ltrm: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "p (deg R p) [^] n \<noteq> \<zero>" shows "monom P ((p [^]\<^bsub>P\<^esub> n) (deg R (p [^]\<^bsub>P\<^esub> n))) (deg R (p [^]\<^bsub>P\<^esub> n)) = monom P (p (deg R p)) (deg R p) [^]\<^bsub>P\<^esub> n"
ide_char: fixes a :: "'t" shows "S'.ide a = (S'.arr a \<and> S'.dom a = a \<and> S'.cod a = a)"