hcju/isabelleps huggingface dataset:
multiset_nonemptyE: fixes A :: "'a multiset" assumes "A \<noteq> \<Empt>" obtains x :: "'a" where "x \<in># A"
small_ranks_imp_small: fixes A :: "V set" assumes "small (rank ` A)" shows "small A"
set_maps_all_maps_list: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" assumes "pair_wf_digraph (list_digraph G_list)"   and "distinct (snd G_list)" shows "all_maps (list_digraph G_list) = set (maps_all_maps_list G_list)"
cases: fixes a1 :: "'b com option list"   and a2 :: "'b"   and a3 :: "'b com option list"   and a4 :: "'b" assumes "(a1, a2) -pc\<rightarrow> (a3, a4)" obtains i :: "nat" and Ps :: "'b com option list" and s :: "'b" and r :: "'b com option" and t :: "'b" where "a1 = Ps" and "a2 = s" and "a3 = Ps[i := r]" and "a4 = t" and "i < length Ps" and "(Ps ! i, s) -c\<rightarrow> (r, t)"
lincomb_in_finite_cone: fixes x :: "'a vec"   and l :: "'a vec \<Rightarrow> 'a"   and W :: "'a vec set" assumes "x = lincomb l W"   and "finite W"   and "\<forall>i\<in>W. (0::'a) \<le> l i"   and "W \<subseteq> carrier_vec n" shows "x \<in> finite_cone W"
trcl_unconcat: fixes c :: "'c"   and w1 :: "'d list"   and w2 :: "'d list"   and c' :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" assumes "(c, w1 @ w2, c') \<in> trcl t" shows "\<exists>ch. (c, w1, ch) \<in> trcl t \<and> (ch, w2, c') \<in> trcl t"
f_asymp_equiv: shows "\<theta> \<sim>[at_top] (\<lambda>x. x)"
dual_filters: shows "filters = ord.ideals (\<lambda>x y. y \<le> x)"
constant_in_extended_predist_pos_2: shows "0 \<le> epsilonG TYPE('a)"
proofTreeBounded: fixes A :: "(nat \<times> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list) set" assumes "proofTree A" shows "bounded A"
llist_of_tllist_eq_LNil: fixes xs :: "('a, 'b) tllist" shows "(llist_of_tllist xs = LNil) = is_TNil xs"
x'_y'_def_2: shows "y' \<equiv> max x y"
assms_2: shows "ivl \<subseteq> \<Union> (plane_of ` stops) \<times> UNIV"
normalize_bottom: shows "normalize \<^bold>\<bottom> = \<^bold>\<bottom>"
set_less_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" shows "ord.set_less less_eq A B = (ord.set_less_eq less_eq A B \<and> \<not> ord.set_less_eq less_eq B A)"
arev_in_arcs: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev M a \<in> arcs G"
inside_not_bound_2: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set"   and j :: "nat" assumes "[f[a .. b ..  c]X]"   and "j < card X"   and "j < card X - 1" shows "f j \<noteq> c"
onorm_def: fixes f :: "'b \<Rightarrow> 'c" shows "onorm f = (SUP x. \<parallel>f x\<parallel> / \<parallel>x\<parallel>)"
this_wall_in_crossingsI_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and D :: "'b set"   and C :: "'b set"   and Cs :: "'b set list" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "D \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "SimplicialComplex.maxsimpchain X (C # Cs @ [D])" shows "{f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X} \<in> set (ThinChamberComplex.wall_crossings X (C # Cs @ [D]))"
length_sort_sws_le: fixes k :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "length (sort_sws k xs) \<le> (length xs)\<^sup>2"
asymptotics5: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>0 \<le> x"   and "i < k" shows "ln (bs ! i * x - hb * x * ln x powr - (1 + e)) powr (- e / 2) < 1"
less_nat_zero_code: fixes n :: "nat" shows "(n < 0) = False"
Diff_triv_mset: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "M \<inter># N = {#}" shows "M - N = M"
of_nat_less_of_nat_power_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (power.power one times (semiring_1.of_nat one plus zero b) w) (semiring_1.of_nat one plus zero x) = (b ^ w < x)"
assms: shows "{..0} \<subseteq> existence_ivl0 x"
lsd_def: fixes x :: "'a" shows "lsd x = (\<top> \<cdot> x \<le> x)"
Fun_Value_type_inv: fixes f :: "('fun, 'atom, 'sets) prot_fun"   and T :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term list" assumes "\<Gamma> (Fun f T) = Var Value" shows "(\<exists>n. f = Val n) \<or> (\<exists>bs. f = Abs bs)"
unfold_congs_2: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "(Literal \<times> bool) list"   and f :: "(Literal \<times> bool) list \<Rightarrow> (Literal \<times> bool) list"   and f' :: "(Literal \<times> bool) list \<Rightarrow> (Literal \<times> bool) list" assumes "r = r'"   and "getM r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "getM_update f r = getM_update f' r'"
inf_sup_aci_3: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (inf (y::'b::type) (z::'b::type)) = inf y (inf x z)"
primitive_part_eq_0_iff: fixes p :: "'b poly" shows "(primitive_part p = 0) = (p = 0)"
acrs1: fixes rs\<^sub>1 :: "'a rule list"   and rs\<^sub>2 :: "'a rule list" assumes "rs = rs\<^sub>1 @ rs\<^sub>2" shows "all_chains (no_call_to c) \<Gamma> rs\<^sub>1"
has_derivative_sin: fixes g :: "'b \<Rightarrow> real"   and g' :: "'b \<Rightarrow> real"   and x :: "'b"   and s :: "'b set" assumes "(g has_derivative g') (at x within s)" shows "((\<lambda>x. sin (g x)) has_derivative (\<lambda>x. g' x * cos (g x))) (at x within s)"
insert_not_empty: fixes a :: "'a::type"   and A :: "'a::type set" shows "insert (a::'a::type) (A::'a::type set) \<noteq> {}"
span_neg: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> More_Modules.ideal S" shows "- x \<in> More_Modules.ideal S"
enn2real_mono: fixes a :: "ennreal"   and b :: "ennreal" assumes "a \<le> b"   and "b < \<top>" shows "enn2real a \<le> enn2real b"
exhaust_sel: fixes hmultiset :: "hmultiset" obtains "hmultiset = HMSet (hmsetmset hmultiset)"
set_less_eq_aux'_into_set_less_eq_aux'': fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubseteq>'' B" shows "A \<sqsubseteq>''' B"
ucast_eq: fixes w :: "'d word" shows "UCAST('d \<rightarrow> 'c) w = word_of_int (uint w)"
injPermSubst: fixes b :: "name"   and P :: "pi"   and a :: "name" assumes "b \<sharp> P" shows "[(a, b)] \<bullet> P = P[a::=b]"
singleton: fixes x :: "'b" shows "\<Sum>\<^sub># {#x#} = x"
reachable_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'e"   and v :: "'e" assumes "H.digraph_isomorphism hom"   and "u \<in> iso_verts hom ` verts H"   and "v \<in> iso_verts hom ` verts H" shows "(u \<rightarrow>\<^sup>*\<^bsub>app_iso hom H\<^esub> v) = (iso_verts (H.inv_iso hom) u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> iso_verts (H.inv_iso hom) v)"
distinct_fresh_rename_ns: fixes G :: "variable set"   and n :: "nat"   and B :: "Core.term"   and insts :: "((variable \<times> typ) \<times> Core.term) list" assumes "finite G" shows "distinct (fresh_rename_idns n B insts G)"
Solve_def: fixes solve :: "('a \<times> constraint) list \<Rightarrow> 'a list + (nat \<Rightarrow> rat)" shows "Solve solve \<equiv> (\<forall>cs v. solve cs = Inr v \<longrightarrow> v \<Turnstile>\<^sub>c\<^sub>s (snd ` set cs)) \<and> (\<forall>cs I. solve cs = Inl I \<longrightarrow> minimal_unsat_core (set I) cs)"
dbproj_subseq_in_subseqs: fixes Di :: "('a \<times> 'b) list"   and i :: "'a"   and D :: "('a \<times> 'b) list" assumes "Di \<in> set (subseqs (dbproj i D))" shows "Di \<in> set (subseqs D)"
r_dec: fixes a :: "nat" shows "eval r_dec [a] \<down>= a - 1"
Rep_mult_matrix: fixes fmul :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and fadd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d matrix"   and B :: "'b matrix"   and j :: "nat"   and i :: "nat" assumes "\<forall>a. fmul (0::'d) a = (0::'c)"   and "\<forall>a. fmul a (0::'b) = (0::'c)"   and "fadd (0::'c) (0::'c) = (0::'c)" shows "Rep_matrix (mult_matrix fmul fadd A B) j i = foldseq fadd (\<lambda>k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) (max (ncols A) (nrows B))"
prim_impl_correct_2: fixes \<pi>i :: "'m" assumes "prim_impl = \<pi>i" shows "is_MST (G_\<alpha>w g) (component_of (G_\<alpha>g g) r) (\<alpha>_MST \<pi>i)"
transitive: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and s :: "'a"   and t :: "'a" assumes "distinct5 p q r s t"   and "in5 S p q r s t"   and "ccw t s p"   and "ccw t s q"   and "ccw t s r"   and "ccw t p q"   and "ccw t q r" shows "ccw t p r"
fc1: shows "c \<sharp> (x, t, \<Gamma>1, \<Gamma>2)"
HFinite_add: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "x + y \<in> HFinite"
simps_6: fixes x1 :: "'b" shows "\<infinity> \<noteq> Le x1"
hyps: assumes "\<forall>g\<in>set (take n gs). terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))" shows "{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci (take n gs)) ft {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
Quotient_alt_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)))"
le_fininf_suffixI: fixes w :: "'action stream"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream" assumes "w =\<^sub>I w\<^sub>1 @- w\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w"
term_variants_pred_inv_2: fixes P :: "'c \<Rightarrow> 'c list"   and f :: "'c"   and T :: "('c, 'd) Term.term list"   and h :: "'c"   and S :: "('c, 'd) Term.term list"   and i :: "nat" assumes "term_variants_pred P (Fun f T) (Fun h S)"   and "i < length T" shows "term_variants_pred P (T ! i) (S ! i)"
aff_dim_eq_dim: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> affine hull S" shows "aff_dim S = int (dim ((+) (- a) ` S))"
smallstepV_ps_to_suffix: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and X :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> V\<rightarrow> \<lless>\<pi>' @ X, e'\<ggreater>" obtains \<pi>'' :: "Syntax.term list" where "\<pi> = \<pi>'' @ X"
simps_109: fixes x101 :: "stmt"   and x102 :: "stmt"   and x61 :: "label"   and x62 :: "expr"   and x63 :: "stmt" shows "x101 Finally x102 \<noteq> x61\<bullet> While(x62) x63"
length_factorization_lattice: fixes u :: "'a poly"   and d :: "nat"   and m :: "'a" shows "length (factorization_lattice u d m) = degree u + d"
induct: fixes P :: "('c, 'd) FingerTreeStruc \<Rightarrow> ('c, 'd) Node \<Rightarrow> bool"   and a0 :: "('c, 'd) FingerTreeStruc"   and a1 :: "('c, 'd) Node" assumes "\<And>a. P Empty a"   and "\<And>b a. P (Single b) a"   and "\<And>uu_ pr m b a. P (Deep uu_ pr m (One b)) a"   and "\<And>uv_ pr m b c a. P (Deep uv_ pr m (Two b c)) a"   and "\<And>uw_ pr m b c d a. P (Deep uw_ pr m (Three b c d)) a"   and "\<And>ux_ pr m b c d e a. P m (node3 b c d) \<Longrightarrow> P (Deep ux_ pr m (Four b c d e)) a" shows "P a0 a1"
insert_map_delete_map_eq: fixes i :: "nat"   and x :: "'c"   and g :: "nat \<Rightarrow> 'c" shows "insert_map i x (delete_map i g) = g(i := x)"
bv_or_is_node: fixes A :: "bool list bdd list \<times> bool list"   and p :: "bool list"   and q :: "bool list" assumes "nfa_is_node A p"   and "nfa_is_node A q" shows "nfa_is_node A (bv_or p q)"
sub_single: fixes A :: "'c set"   and a :: "'c" assumes "A \<noteq> {}"   and "A \<subseteq> {a}" shows "A = {a}"
PDERIVS_incr: fixes R :: "'b rexp set" shows "R \<subseteq> PDERIVS R"
gen_dfs_pre_def: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a" shows "gen_dfs_pre E U S V u0 \<equiv> E `` U \<subseteq> U \<and> finite U \<and> V \<subseteq> U \<and> u0 \<in> U \<and> E `` (V - S) \<subseteq> V \<and> u0 \<notin> V \<and> S \<subseteq> V \<and> (\<forall>v\<in>S. (v, u0) \<in> (E \<inter> S \<times> UNIV)\<^sup>*)"
tpick_in: fixes V :: "Preliminaries.var set"   and \<sigma> :: "'tp" assumes "tinfinite V" shows "tpick \<sigma> V \<in> V"
sup_compl_top_left1: 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"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup (uminus x) (sup x y) = top"
deg_pm_single: fixes x :: "'e"   and k :: "'d" shows "deg_pm (monomial k x) = k"
b_e_type_current_memory: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Current_memory" shows "\<exists>sec n. ts' = ts @ [T_i32] \<and> memory \<C> = Some n"
sel_1: fixes x21 :: "'a"   and x22 :: "'a llist" shows "lhd (LCons x21 x22) = x21"
major: shows "silent_moves t (x, shr s) (x', m')"
eSwapAbs_nchotomy: fixes eA :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" shows "(\<exists>us s A. eA = OK A \<and> igWlsAbs MOD (us, s) A) \<or> eSwapAbs MOD zs z1 z2 eA = ERR"
iH: fixes X :: "bool list" assumes "y < length X" shows "flip y X ! y = (\<not> X ! y)"
dist_in_closed_segment: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> closed_segment a b" shows "dist x a \<le> dist a b \<and> dist x b \<le> dist a b"
hd_take: fixes j :: "nat"   and xs :: "'a list" assumes "0 < j" shows "hd (take j xs) = hd xs"
diffeomorphism_def: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and f' :: "'e \<Rightarrow> 'c" shows "Differentiable_Manifold.diffeomorphism k charts1 charts2 f f' \<equiv> diff k charts1 charts2 f \<and> diff k charts2 charts1 f' \<and> diffeomorphism_axioms charts1 charts2 f f'"
that: assumes "arr_Par \<alpha> T"   and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>" shows "thesis"
node_tau_deliver_unchanged: fixes i :: "nat"   and \<zeta>\<^sub>i :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" shows "\<langle>i : \<zeta>\<^sub>i : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (S, U \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a = \<tau> \<or> (\<exists>i d. a = i:deliver(d)) \<longrightarrow> (\<forall>j. j \<noteq> i \<longrightarrow> \<sigma>' j = \<sigma> j))"
empty_def: shows "local.empty \<equiv> Abs_BinomialHeap []"
filter_option_eq_Some_iff: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option"   and y :: "'c" shows "(filter_option P x = Some y) = (x = Some y \<and> P y)"
sorted_list_of_set_merge_distinct: fixes xs :: "'c list"   and ys :: "'c list" assumes "sorted xs"   and "distinct xs"   and "sorted ys"   and "distinct ys" shows "merge_distinct xs ys = sorted_list_of_set (set xs \<union> set ys)"
closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N"
poly_mult_replicate_zero_1: fixes p :: "'a list"   and n :: "nat" assumes "set p \<subseteq> carrier R" shows "poly_mult (replicate n \<zero>) p = []"
index_distinct_group_num_alt_def: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "size (filter_mset ((\<subseteq>) ps) (mset_set \<G>)) = 0" shows "int (\<B> index ps) = \<Lambda>"
put_eq: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d"   and k :: "'c"   and u :: "'c"   and \<rho> :: "'d"   and v :: "'c" assumes "vwb_lens x"   and "get\<^bsub>x\<^esub> \<sigma> = k"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "put\<^bsub>x\<^esub> \<rho> k = \<sigma>"
elims_2: fixes x :: "'b literal set" assumes "ground_clause x" obtains C :: "'b literal set" where "x = C" and "vars_of_cl C = {}"
False: shows "\<not> (0 < CARD('a) \<and> 0 < CARD('b) \<or> CARD('b) = 1)"
nonneg_dirichlet_series_def: fixes f :: "'a fds" shows "nonneg_dirichlet_series f \<equiv> \<forall>n>0. fds_nth f n \<in> \<real>\<^sub>\<ge>\<^sub>0"
transformation_group_closed: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "\<alpha> \<in> G"   and "x \<in> S" shows "\<alpha> x \<in> S"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>B g\<guillemotright>" shows "trg\<^sub>B \<mu> = trg\<^sub>B f"
Nil_is_append_conv: fixes xs :: "'c list"   and ys :: "'c list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
compute_bound_tm_def: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and tm :: "taylor_model" shows "compute_bound_tm prec I a tm = compute_bound_poly prec (map_poly interval_of (tm_poly tm)) I a + tm_bound tm"
rule_sub_remark_3_autosubst: fixes v :: "i" assumes "\<And>v x. [\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"   and "[\<^bold>\<exists>x. \<lparr>A!,x\<^sup>P\<rparr> in v]" shows "[\<^bold>\<exists>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
H_on_ket_zero: shows "H * |Deutsch.zero\<rangle> = Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, 1 / sqrt 2]])"
qfree_asubst_peps: fixes k :: "nat"   and a :: "atom" shows "qfree (asubst\<^sub>+ k a)"
lisetD_2: fixes i :: "nat"   and A :: "'b set"   and w :: "'b llist" assumes "i \<in> liset A w" shows "w ?! i \<in> A"
weakly_preferred_by_men_def: fixes \<mu> :: "('m \<times> 'w) set"   and \<mu>' :: "('m \<times> 'w) set" shows "weakly_preferred_by_men \<mu> \<mu>' = (\<forall>m. \<forall>w\<in>{w. (m, w) \<in> \<mu>}. \<exists>w'\<in>{w. (m, w) \<in> \<mu>'}. w' \<in> m_weakly_prefers m \<mu>)"
vsv_vimageE: fixes b :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V" assumes "b \<in>\<^sub>\<circ> VLambda Aa f `\<^sub>\<circ> A" obtains x :: "V" where "VLambda Aa f\<lparr>x\<rparr> = b" and "x \<in>\<^sub>\<circ> A"
Xd1_Xd2'_stable: shows "BSI.stable_on UNIV {X5.Xd1, X5.Xd2'}"
FvarsT': fixes t :: "'trm"   and x :: "'var" assumes "t \<in> trm"   and "x \<in> FvarsT t" shows "x \<in> var"
bex_imageD: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and P :: "'e \<Rightarrow> bool" assumes "\<exists>x\<in>f ` A. P x" shows "\<exists>x\<in>A. P (f x)"
closed_Int_compact: fixes s :: "'b set"   and t :: "'b set" assumes "closed s"   and "compact t" shows "compact (s \<inter> t)"
nmod2: fixes n :: "int" shows "n mod 2 = 0 \<or> n mod 2 = 1"
loop_backward: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'sc, 'a) interp"   and id1 :: "'a" shows "\<nu> \<in> fml_sem I (Pc pid1 && [[$\<alpha> id1]][[$\<alpha> id1**]]Pc pid1) \<longrightarrow> \<nu> \<in> fml_sem I ([[$\<alpha> id1**]]Pc pid1)"
equivalentFormulae_def: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list" shows "equivalentFormulae formula1 formula2 \<equiv> \<forall>valuation. model valuation formula1 = model valuation formula2"
parallel_fixp_induct: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'c" assumes "class.ccpo luba orda (mk_less orda)"   and "class.ccpo lubb ordb (mk_less ordb)"   and "ccpo.admissible (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. P (fst x) (snd x))"   and "monotone orda orda f"   and "monotone ordb ordb g"   and "P (luba {}) (lubb {})"   and "\<And>x y. P x y \<Longrightarrow> P (f x) (g y)" shows "P (ccpo.fixp luba orda f) (ccpo.fixp lubb ordb g)"
sint_up_scast: fixes w :: "'a Word.word" assumes "is_up SCAST('a \<rightarrow> 'b)" shows "sint (SCAST('a \<rightarrow> 'b) w) = sint w"
A_2: shows "a \<otimes> f (deg R f) = \<zero>"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d multiset" shows "image_mset g (image_mset f v) = image_mset (g \<circ> f) v"
inverse_fpxs_def: fixes f :: "'a fpxs" shows "inverse f = fpxs_compose_power (fpxs_of_fls (inverse (fls_of_fpxs f))) (1 / rat_of_nat (fpxs_root_order f))"
insertion_fun_def: fixes f :: "nat \<Rightarrow> 'b"   and p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'b" shows "insertion_fun f p = (\<Sum>m. p m * (\<Prod>v. f v ^ poly_mapping.lookup m v))"
new_eq: fixes s1 :: "Store"   and t :: "New"   and s2 :: "Store" shows "(new s1 t = new s2 t) = (\<forall>x. \<tau> x = typeofNew t \<longrightarrow> alive x s1 = alive x s2)"
simps_3: fixes a :: "'a"   and uv :: "'a list" shows "sorted_list_subset (a # uv) [] = Some a"
normalize_alt_simps_14: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and iter :: "iterator"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and call :: "'a call_expr" shows "((\<Gamma>, \<tau>) \<turnstile>\<^sub>C Iterator iter its its_ty body \<Rrightarrow> call) = ((\<exists>\<sigma>. its_ty = Some \<sigma> \<and> (\<exists>body\<^sub>2. call = Iterator iter its (Some \<sigma>) body\<^sub>2 \<and> \<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, \<sigma>)) its) \<turnstile> body \<Rrightarrow> body\<^sub>2)) \<or> its_ty = None \<and> (\<exists>\<sigma> body\<^sub>2. call = Iterator iter its (Some \<sigma>) body\<^sub>2 \<and> element_type \<tau> \<sigma> \<and> \<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, \<sigma>)) its) \<turnstile> body \<Rrightarrow> body\<^sub>2))"
amd2_update_refine: shows "(uncurry2 (amd_update dflt), uncurry2 (\<lambda>x xa. RETURN \<circ> (\<lambda>f. f(x := xa)))) \<in> nat_assn\<^sup>k *\<^sub>a id_assn\<^sup>k *\<^sub>a (is_nff dflt)\<^sup>d \<rightarrow>\<^sub>a is_nff dflt"
intro: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int" assumes "k_\<Lambda>_PBD point_set block_collection 1 u_block_size" shows "k_PBD point_set block_collection u_block_size"
bounded_degree_dimension: fixes K :: "'a list set"   and n :: "nat" assumes "subfield K (K [X])" shows "ring.dimension (K [X]\<^bsub>K [X]\<^esub>) n (UP.poly_of_const ` K) {q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>). length q \<le> n}"
consistent_def2: shows "consistent = (\<exists>\<phi>\<in>fmla. \<not> prv \<phi>)"
assms_2: fixes i :: "'a" assumes "i \<in> I" shows "0 \<le> f i"
doesnt_read_or_modify_doesnt_modify: fixes c :: "'Com"   and x :: "'Var" assumes "doesnt_read_or_modify c x" shows "doesnt_modify c x"