hcju/isabelleps huggingface dataset:
eqvtI_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and p :: "(name \<times> name) list" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "p \<bullet> P \<Longrightarrow>p \<bullet> \<alpha> \<prec> p \<bullet> P'"
inv_solve_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a = b + - c) = (b = a + c)"
isom_gch_units: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and h :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and f :: "nat \<Rightarrow> nat" assumes "Ugp E"   and "Gchain n g"   and "Gchain n h"   and "Gch_bridge n g h f" shows "card {i. i \<le> n \<and> g i \<cong> E} = card {i. i \<le> n \<and> h i \<cong> E}"
simps_1: fixes uu :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and uv :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and uw :: "'g" shows "c_tr uu uv uw [] = []"
Hash_Exp: fixes X :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Hash X \<noteq> Exp X' Y'"
AR_def: fixes S :: "'b set" shows "AR S \<equiv> \<forall>U S'. S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> S' retract_of U"
g: fixes y :: "'b" shows "mcont luba orda lub_spmf (ord_spmf (=)) (g y)"
prodrg_carrier: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)" shows "carrier (r\<Pi>\<^bsub>I\<^esub> A) = carrier (a\<Pi>\<^bsub>I\<^esub> A)"
cases: fixes a1 :: "'c"   and a2 :: "('c, 'd) bintree list" assumes "is_heap_list a1 a2" obtains h :: "'c" where "a1 = h__" and "a2 = []"   | h :: "'c" and ts :: "('c, 'd) bintree list" and t :: "('c, 'd) bintree" where "a1 = h" and "a2 = t # ts" and "is_heap_list h ts" and "is_heap_list (priority t) (children t)" and "h \<le> priority t"
distinct_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and x2 :: "'c" assumes "flow_attainability \<Delta>" shows "Inner x2 \<noteq> SOURCE"
simps_3: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) stateful_strand_step list" shows "\<lbrakk>M; D; receive\<langle>t\<rangle> # S\<rbrakk>\<^sub>s = (\<lambda>\<I>. \<lbrakk>insert (t \<cdot> \<I>) M; D; S\<rbrakk>\<^sub>s \<I>)"
skl1_refines_l1: shows "refines R11 med11 l1 skl1"
assms_1: shows "Negligible.negligible (\<lambda>\<eta>. local.ddh.advantage \<eta> (inter_S1_adversary \<eta> D msgs))"
right_neutral: fixes a :: "'e" shows "a + (0::'e) = a"
soundness: fixes k :: "nat"   and \<Psi> :: "'a TESL_atomic list"   and \<S> :: "'a constr list \<times> nat \<times> 'a TESL_atomic list \<times> 'a TESL_atomic list" assumes "[], 0 \<turnstile> \<Psi> \<triangleright> [] \<hookrightarrow>\<^bsup>k\<^esup> \<S>" shows "\<lbrakk> \<S> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<subseteq> \<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
sin_zero_abs_cos_one: fixes x :: "real" assumes "sin x = 0" shows "\<bar>cos x\<bar> = 1"
count: fixes y :: "'c"   and s' :: "'d"   and s :: "'d"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee s x)"   and "consider x"   and "x \<in> outs_\<I> \<I>" shows "count s' \<le> Suc (count s)"
Alm_all_GF_F: fixes w :: "nat \<Rightarrow> 'a set"   and \<psi> :: "'a ltln" shows "\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>) = suffix i w \<Turnstile>\<^sub>n F\<^sub>n \<psi>"
k_all: shows "\<forall>x y i. k i x = k i y"
semiring_norm_15: fixes m :: "num"   and n :: "num" shows "num.Bit1 m * num.Bit0 n = num.Bit0 (num.Bit1 m * n)"
nonzero_eq_divide_eq: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)" shows "(a = b / c) = (a * c = b)"
simps: fixes u :: "'a Point"   and v :: "'a Point" shows "space2 u v = (xval u - xval v) * (xval u - xval v) + (yval u - yval v) * (yval u - yval v) + (zval u - zval v) * (zval u - zval v)"
X_poly_def: fixes R :: "('e, 'f) ring_scheme" shows "X_poly R = monom (UP R) \<one>\<^bsub>R\<^esub> 1"
prems_3: shows "a \<notin> acquired True (x # sb) X"
open_delete: fixes s :: "'c set"   and x :: "'c" assumes "open s" shows "open (s - {x})"
lexord_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "lexord_fusion g1 g2 s1 s2 = ord_class.lexordp (unstream g1 s1) (unstream g2 s2)"
simps_1: fixes uu :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and uv :: "'c option" shows "lift2_option uu None uv = None"
type_less_eq_x_Real: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<le> Real" obtains "\<tau> = OclVoid"   | "\<tau> = UnlimitedNatural"   | "\<tau> = Integer"   | "\<tau> = Real"
execute: fixes q :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a"   and a :: "nat \<times> 'a" assumes "q \<in> greachable A w V p"   and "case q of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> a \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V" shows "a \<in> greachable A w V p"
run_shift: fixes r :: "'action list"   and p :: "'state"   and s :: "'action stream" assumes "path r p"   and "run s (target r p)" shows "run (r @- s) p"
covering_def: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" shows "covering V Q = (V \<subseteq> (\<Union>(S, X)\<in>Q. S))"
square_bounded_less: fixes a :: "'b"   and b :: "'b" assumes "- a < b \<and> b < a" shows "b * b < a * a"
b: shows "[\<one>, \<ominus> x] \<in> carrier (poly_ring R)"
outer_measure_of_eq: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> sets M" shows "outer_measure_of M A = emeasure M A"
simps: fixes lx :: "'a"   and ux :: "'a"   and ly :: "'a"   and uy :: "'a" shows "Interval lx ux + Interval ly uy = Interval (lx + ly) (ux + uy)"
in_lists_conv_set: fixes xs :: "'a list"   and A :: "'a set" shows "(xs \<in> lists A) = (\<forall>x\<in>list.set xs. x \<in> A)"
simps_2: fixes \<theta> :: "(name \<times> trm) list"   and s :: "trm"   and t :: "trm" shows "\<theta><App s t> = App (\<theta>
) (\<theta><t>)"
config'_n_init: fixes init :: "('c list \<times> bool list \<times> 'c list) pmf"   and qs :: "'c list" shows "map_pmf (snd \<circ> snd) (config'_rand BIT init qs) = map_pmf (snd \<circ> snd) init"
mono_sup: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b"   and B :: "'b" assumes "mono f" shows "f A \<squnion> f B \<le> f (A \<squnion> B)"
max2_equals1: fixes a :: "nat + 'a Bonk_Schramm_extension_unfolded set"   and b :: "nat + 'a Bonk_Schramm_extension_unfolded set" assumes "a \<in> Field wo_aux"   and "b \<in> Field wo_aux" shows "(max2 wo_aux a b = a) = ((b, a) \<in> wo_aux)"
distinct_1: fixes x1 :: "'a" shows "ln x1 \<noteq> \<star>"
Enter_safe: fixes g :: "guest"   and r :: "room"   and c :: "key \<times> key"   and s :: "event list" assumes "hotel (Enter g r c # s)"   and "safe\<^sub>0 s r" shows "owns s r = \<lfloor>g\<rfloor>"
r_shift: fixes f :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "recfn (length xs) f" shows "eval (r_shift f) (x # xs) = eval f xs"
snd_echelon_form_of_column_k_det_eq: fixes bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and n :: "'c"   and A :: "(('c, 'd) vec, 'e) vec"   and i :: "nat"   and k :: "nat" shows "snd (echelon_form_of_column_k_det bezout (n, A, i) k) = echelon_form_of_column_k bezout (A, i) k"
C: shows "\<forall>x\<ge>x\<^sub>0. sgn (poly p x) = poly_inf p"
bindE: fixes R :: "'b Predicate.pred"   and Q :: "'b \<Rightarrow> 'a Predicate.pred"   and y :: "'a" assumes "pred.eval (R \<bind> Q) y" obtains x :: "'b" where "pred.eval R x" and "pred.eval (Q x) y"
zero_le_iff_zero_nprt: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq zero a = (lattice_ab_group_add.nprt zero inf a = zero)"
simps_8: fixes f :: "'d \<Rightarrow> 'c" shows "map_option f None = None"
le_ref1: fixes P :: "'c process"   and Q :: "'c process" assumes "P \<le> Q" shows "D Q \<subseteq> D P"
slim_paths_in_G_E: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "(u, p, v) \<in> slim_paths" obtains "gen_iapath (verts3 (with_proj G)) u p v" and "u \<noteq> v"
cont_discrete_cpo: fixes f :: "'c \<Rightarrow> 'd" shows "cont f"
resComm: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "<\<nu>a><\<nu>b>P \<sim>\<^sub>e <\<nu>b><\<nu>a>P"
divide_eq_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = one) = (b \<noteq> zero \<and> a = b)"
psubset_trans: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A \<subset> B"   and "B \<subset> C" shows "A \<subset> C"
measure_pmf_posI: fixes x :: "'c"   and p :: "'c pmf"   and A :: "'c set" assumes "x \<in> set_pmf p"   and "x \<in> A" shows "0 < measure_pmf.prob p A"
err: fixes s :: "'a"   and err :: "'c error_result" assumes "r = (s, Rerr err)" shows "thesis"
assms_2: shows "s = atomize_poly (idx_pm_of_pm bs s')"
get_attribute_defs: fixes ptr :: "(_) element_ptr"   and k :: "char list" shows "a_get_attribute ptr k = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. return (fmlookup m k))"
simps_5: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat" shows "typeof\<^bsub>h\<^esub> (Addr a) = (case h a of None \<Rightarrow> None | \<lfloor>(C, fs)\<rfloor> \<Rightarrow> \<lfloor>Class C\<rfloor>)"
alw3: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F"   and "STUTINV F" shows "\<turnstile> \<box>F"
mono: fixes c :: "'d"   and a :: "'d"   and d :: "'d"   and b :: "'d" assumes "c \<le> a"   and "d \<le> b" shows "c \<sqinter> d \<le> a \<sqinter> b"
bl2nat_cons_bk: fixes ks :: "cell list" shows "bl2nat (ks @ [Bk]) 0 = bl2nat ks 0"
rcosets_subset_PowG: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "rcosets\<^bsub>G\<^esub> H \<subseteq> Pow (carrier G)"
fset_mp: fixes A :: "'x fset"   and B :: "'x fset"   and c :: "'x" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "enat \<Rightarrow> 'a"   and enat :: "enat" shows "P (case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = ((enat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. enat = eSuc x2 \<longrightarrow> P (f2 x2)))"
odd_coeff_sign_changes_imp_pos_roots: fixes p :: "real poly" assumes "p \<noteq> 0"   and "odd (sign_changes (coeffs p))" obtains x :: "real" where "0 < x" and "poly p x = 0"
prob_eq_AE: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "AE x in M. P x = Q x"   and "{x \<in> space M. P x} \<in> events"   and "{x \<in> space M. Q x} \<in> events" shows "prob {x \<in> space M. P x} = prob {x \<in> space M. Q x}"
not_less0: fixes n :: "nat" shows "\<not> n < 0"
H: fixes x :: "'a option" assumes "\<And>y. y < x \<Longrightarrow> P y" shows "P x"
assms_4: shows "x \<in> C\<^sub>1 \<inter> C\<^sub>2"
pwd_spy_dec_def: fixes n :: "nat" shows "pwd_spy_dec n \<equiv> case pwd_owner_v n of (s, S, A, B, C, D) \<Rightarrow> (insert (Spy, Pwd n) s, S, A, B, C, D)"
strip_guards_simps_7: fixes F :: "'f set"   and p :: "'e" shows "strip_guards F (Call p) = Call p"
simplex_likeI: fixes P :: "'b set"   and f :: "'b \<Rightarrow> nat set"   and A :: "nat set" assumes "finite P"   and "OrderingSetIso dual_order.greater_eq dual_order.greater dual_order.greater_eq dual_order.greater P f"   and "f ` P = Pow A" shows "order.simplex_like P"
weak_ranking_eqD: fixes alts :: "'b set"   and R1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite_total_preorder_on alts R1"   and "finite_total_preorder_on alts R2"   and "weak_ranking R1 = weak_ranking R2" shows "R1 = R2"
image_mset_is_empty_iff: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" shows "(image_mset f M = \<Empt>) = (M = \<Empt>)"
Radj_repv2: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and r :: "real" shows "Radj x y (repv \<nu> x r) = repv (Radj x y \<nu>) y r"
nxtAct_active: fixes c :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>i\<^esub> = i"
id: shows "d__ * gcd G1 G2 = Polynomial.smult (unit_factor b__) (normalize (primitive_part Gk))"
d: fixes k :: "nat" assumes "k \<le> m"   and "k \<noteq> i" shows "d fs k = d fs' k"
game2_a_bad_def: shows "game2_a_bad = spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. exec_gpv rf_encrypt \<A>1 Map.empty \<bind> (\<lambda>(((p0, p1), \<sigma>), s1). return_spmf (r \<in> dom s1)))"
openin_Union: fixes K :: "'b set set"   and U :: "'b topology" assumes "\<And>S. S \<in> K \<Longrightarrow> openin U S" shows "openin U (\<Union> K)"
iso_assoc: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "local.iso \<a>[f, g, h]"
simps_2: fixes G :: "('c, 'b) labeled_graph"   and a :: "'c allegorical_term"   and b :: "'c allegorical_term" shows ":G:\<lbrakk>A_Cmp a b\<rbrakk> = (:G:\<lbrakk>a\<rbrakk>) O (:G:\<lbrakk>b\<rbrakk>)"
tagged_partial_division_of_def: fixes s :: "('b \<times> 'b set) set"   and i :: "'b set" shows "(s tagged_partial_division_of i) = (finite s \<and> (\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow> interior K1 \<inter> interior K2 = {}))"
Union_image_insert: fixes f :: "'e \<Rightarrow> 'd set"   and a :: "'e"   and B :: "'e set" shows "\<Union> (f ` insert a B) = f a \<union> \<Union> (f ` B)"
left_unit: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> G" shows "compose S (identity S) a = a"
HF_axP_sf: fixes t :: "tm" shows "Sigma_fm (HF_axP t)"
constructive_security_aux_obsf: shows "constructive_security_aux_obsf real1 ideal1 sim1 (\<I>_real1_core \<oplus>\<^sub>\<I> \<I>_real1_rest) (\<I>_ideal1_core \<oplus>\<^sub>\<I> \<I>_ideal1_rest) (\<I>_common1_core \<oplus>\<^sub>\<I> \<I>_common1_rest) adv1"
cfni_def: fixes p :: "nat"   and f :: "nat \<Rightarrow> real" shows "cfni p f = sum (fiX f p) {..<np} / real np"
mult_stalk_def: shows "stx.mult_stalk \<equiv> stx.mult_rel"
le_utransD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "le_utrans t u"   and "unitary P" shows "t P \<tturnstile> u P"
integral_dominated_convergence_at_top: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and s :: "real \<Rightarrow> 'a \<Rightarrow> 'b"   and w :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>t. s t \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. ((\<lambda>i. s i x) \<longlongrightarrow> f x) at_top"   and "\<forall>\<^sub>F i in at_top. AE x in M. norm (s i x) \<le> w x" shows "((\<lambda>t. integral\<^sup>L M (s t)) \<longlongrightarrow> integral\<^sup>L M f) at_top"
mult_cancel_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times a c = times b c) = (c = zero \<or> a = b)"
simps: fixes rs :: "'b wordinterval" shows "wordinterval_CIDR_split_prefixmatch rs = (if \<not> wordinterval_empty rs then case wordinterval_CIDR_split1 rs of (None, u) \<Rightarrow> [] | (Some s, u) \<Rightarrow> s # wordinterval_CIDR_split_prefixmatch u else [])"
basis_drop_emp: fixes G :: "'b list set" shows "\<BB> G\<^sub>+ = \<BB> G"
rel_list_R1_list_to_list_R1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" shows "rel_list_R1 R (list_to_list_R1 xs) (list_to_list_R1 ys) = list_all2 R xs ys"
smult_assoc_simp: fixes x :: "'a vec"   and a :: "'a"   and b :: "'a" assumes "True"   and "True"   and "x \<in> carrier_vec n" shows "a \<cdot>\<^sub>v (b \<cdot>\<^sub>v x) = a * b \<cdot>\<^sub>v x"
order_less_imp_le: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "x \<le> y"
that: fixes Q' :: "'state" assumes "Q \<Rightarrow>\<langle>\<alpha>\<rangle> Q'"   and "P' \<approx>\<cdot> Q'" shows "thesis"
rep_eq: fixes x :: "complex"   and xa :: "'d ccsubspace" shows "space_as_set (x *\<^sub>C xa) = (*\<^sub>C) x ` space_as_set xa"
G_Inf_overapprox_F_Inf: fixes \<iota>\<^sub>0 :: "'a literal multiset inference"   and M :: "'a literal multiset set" assumes "\<iota>\<^sub>0 \<in> G.Inf_from M (\<Union> (\<G>_F ` M))" shows "\<exists>\<iota>\<in>F.Inf_from M. \<iota>\<^sub>0 \<in> \<G>_I M \<iota>"
tail_add_vert: fixes u :: "'a" shows "tail (S.add_vert u) = fst"
cases: fixes a1 :: "'Var \<Rightarrow> Sec option"   and a2 :: "'BExp"   and a3 :: "Sec" assumes "a1 \<turnstile>\<^sub>b a2 \<in> a3 " obtains \<Gamma> :: "'Var \<Rightarrow> Sec option" and e :: "'BExp" where "a1 = \<Gamma>" and "a2 = e" and "a3 = max_dom (to_total \<Gamma> ` bexp_vars e)"
eucl_rel_poly_remainderI: fixes y :: "'b poly"   and r :: "'b poly"   and x :: "'b poly"   and q :: "'b poly" assumes "y \<noteq> 0"   and "degree r < degree y"   and "x = q * y + r" shows "eucl_rel_poly x y (q, r)"
tau_mpj_bilin3: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and P :: "('e \<times> 'g, 'a, 'f) Module_scheme"   and m :: "'e"   and n :: "'g"   and a :: "'a" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "m \<in> carrier M"   and "n \<in> carrier N"   and "a \<in> carrier R" shows "mpj P (TR\<^bsub>R\<^esub> M N P) (a \<cdot>\<^sub>s\<^bsub>M\<^esub> m, n) = a \<cdot>\<^sub>s\<^bsub>M \<^bsub>P\<^esub>\<Otimes>\<^bsub>R\<^esub> N\<^esub> mpj P (TR\<^bsub>R\<^esub> M N P) (m, n)"
assms_3: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and e :: "'event" assumes "S p q"   and "e \<in> E" shows "trace_eq_simcl S (mk_lossless (p \<bind> (\<lambda>s. cpoke core1 s e))) (mk_lossless (q \<bind> (\<lambda>s. cpoke core2 s e)))"
F2set2_bd_incr: fixes x :: "('e, 'd, 'f) F2" shows "|F2set2 x| \<le>o F2bd'"
abs_neg_numeral: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (uminus (numeral.numeral one plus n)) = numeral.numeral one plus n"
consec_un_element1: fixes i :: "nat_int"   and j :: "nat_int"   and n :: "nat" shows "consec i j \<and> n \<^bold>\<in> i \<longrightarrow> n \<^bold>\<in> i \<squnion> j"
intro: fixes T :: "'b set set" assumes "\<forall>x\<in>carrier\<^bsub>T\<^esub>. \<forall>y\<in>carrier\<^bsub>T\<^esub>. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds\<^bsub>T\<^esub> x. y \<notin> u) \<or> (\<exists>v\<in>nhds\<^bsub>T\<^esub> y. x \<notin> v)" shows "T0_axioms T"
d'_2: fixes x :: "'a"   and s :: "real"   and y :: "'a" assumes "x \<in> {a--b}"   and "s = 0 \<longrightarrow> y \<noteq> x"   and "dist (s, y) (0, x) < d' x" shows "flow0 y s \<in> S"
valid_arbitrary_strategy: fixes p :: "Player" shows "strategy p \<sigma>_arbitrary"
is_coext_def: fixes D :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b"   and e :: "'b"   and x' :: "'a"   and f :: "'b"   and g :: "'a" assumes "arrow_from_functor D C F x y e" shows "arrow_from_functor.is_coext D C F x e x' f g \<equiv> partial_magma.in_hom D g x' x \<and> f = C e (F g)"
numeral_le_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "num"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (numeral.numeral one plus m) (numeral.numeral one plus n) = (m \<le> n)"
dependent_biggerset_general: fixes S :: "('b, 'c) vec set" assumes "finite S \<Longrightarrow> vec.dim S < card S" shows "vec.dependent S"
x: shows "\<not> (x = (1::'a) \<or> n = 0)"
relative_to_Int: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and C :: "'c set"   and D :: "'c set" assumes "(P relative_to S) C"   and "(P relative_to S) D"   and "\<And>X Y. \<lbrakk>P X; P Y\<rbrakk> \<Longrightarrow> P (X \<inter> Y)" shows "(P relative_to S) (C \<inter> D)"
WTrt_binop_NotEq: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty" shows "P \<turnstile> T1\<guillemotleft>NotEq\<guillemotright>T2 : Boolean"
Gcd_minors_A_dvd_Gcd_minors_PAQ: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and P :: "'b mat"   and Q :: "'b mat"   and k :: "nat" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n" shows "Gcd (minors A k) dvd Gcd (minors (P * A * Q) k)"
cases: fixes x :: "tm" obtains ax :: "name" and p :: "tm" and v :: "name" and w :: "name" and xa :: "name" and x0 :: "name" and xw :: "name" and xevw :: "name" and allw :: "name" and allvw :: "name" where "atom ax \<sharp> (p, v, w, xa, x0, xw, xevw, allw, allvw)" and "atom allvw \<sharp> (p, v, w, xa, x0, xw, xevw, allw)" and "atom allw \<sharp> (p, v, w, xa, x0, xw, xevw)" and "atom xevw \<sharp> (p, v, w, xa, x0, xw)" and "atom xw \<sharp> (p, v, w, xa, x0)" and "atom x0 \<sharp> (p, v, w, xa)" and "atom xa \<sharp> (p, v, w)" and "atom w \<sharp> (p, v)" and "atom v \<sharp> p" and "x = p"
Lindelof_space_Union: fixes \<U> :: "'a set set"   and X :: "'a topology" assumes "countable \<U>"   and "\<And>U. U \<in> \<U> \<Longrightarrow> Lindelof_space (subtopology X U)" shows "Lindelof_space (subtopology X (\<Union> \<U>))"
lcs_unique: fixes s :: "'a"   and l :: "'a"   and t :: "'a" assumes "s adds l"   and "t adds l"   and "\<And>u. \<lbrakk>s adds u; t adds u\<rbrakk> \<Longrightarrow> l adds u" shows "l = lcs s t"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Dom (fst Fg))"
uint_set_bit_def: fixes x :: "uint"   and n :: "integer"   and b :: "bool" shows "uint_set_bit x n b = (if n < 0 \<or> dflt_size_integer \<le> n then undefined set_bit_class.set_bit x n b else set_bit_class.set_bit x (nat_of_integer n) b)"
not_outside: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq> y" shows "x \<in> carrier"
lipschitz_on_cmult_nonneg: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "real" assumes "C-lipschitz_on U f"   and "0 \<le> a" shows "(a * C)-lipschitz_on U (\<lambda>x. a *\<^sub>R f x)"
trms\<^sub>e\<^sub>s\<^sub>t_ikI: fixes t :: "('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list" assumes "t \<in> ik\<^sub>e\<^sub>s\<^sub>t A" shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A)"