hcju/isabelleps huggingface dataset:
char_poly_four_block_zeros_col: fixes A1 :: "'b mat"   and A2 :: "'b mat"   and n :: "nat"   and A3 :: "'b mat" assumes "A1 \<in> carrier_mat 1 1"   and "A2 \<in> carrier_mat 1 n"   and "A3 \<in> carrier_mat n n" shows "char_poly (four_block_mat A1 A2 (0\<^sub>m n 1) A3) = char_poly A1 * char_poly A3"
nsub_RI_iconf: fixes e :: "'c exp"   and sh :: "sheap" assumes "\<not> sub_RI e" shows "iconf sh e"
f_def: shows "f \<equiv> \<lambda>S. \<Union>x\<in>set S. case x of receive\<langle>x\<rangle> \<Rightarrow> fv x | \<langle>assign: x \<doteq> t'\<rangle> \<Rightarrow> fv t' | \<langle>check: x \<doteq> t'\<rangle> \<Rightarrow> {} | insert\<langle>t,t'\<rangle> \<Rightarrow> fv t \<union> fv t' | _ \<Rightarrow> {}"
atomic_ipc_preserves_invariants: fixes s :: "state_t"   and tid :: "thread_id_t"   and dir :: "ipc_direction_t"   and stage :: "ipc_stage_t"   and partner :: "thread_id_t"   and page :: "page_t" assumes "atomic_step_invariant s" shows "atomic_step_invariant (atomic_step_ipc tid dir stage partner page s)"
represents_nth_root: fixes n :: "nat"   and y :: "'a"   and x :: "'a"   and p :: "int poly" assumes "n \<noteq> 0"   and "y ^ n = x"   and "p represents x" shows "poly_nth_root n p represents y"
OUT_g_A: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "d_OUT (\<lambda>(x, y). g (\<langle>y\<rangle>, \<langle>x\<rangle>)) x = d_OUT g \<langle>x\<rangle> - g (vertex.SINK, \<langle>x\<rangle>)"
not_contains_insertion_helper: fixes p :: "'a mpoly"   and var :: "nat"   and a :: "nat \<Rightarrow>\<^sub>0 nat"   and L :: "'d list"   and y :: "'d"   and x :: "'d" assumes "\<forall>x\<in>keys (mapping_of p). var \<notin> keys x"   and "\<And>k f. (k \<notin> keys f) = (lookup f k = (0::'c))" shows "lookup (mapping_of p) a = (0::'a) \<or> (\<Prod>aa. (if aa < length L then L[var := y] ! aa else (0::'d)) ^ lookup a aa) = (\<Prod>aa. (if aa < length L then L[var := x] ! aa else (0::'d)) ^ lookup a aa)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) FG.map C.map \<epsilon>.map"
decomposition_theorem_polyhedra: fixes P :: "'a vec set" shows "(\<exists>A b nr. A \<in> carrier_mat nr n \<and> b \<in> carrier_vec nr \<and> P = polyhedron A b) = (\<exists>Q X. Q \<union> X \<subseteq> carrier_vec n \<and> finite (Q \<union> X) \<and> P = convex_hull Q + cone X)"
dpd1: fixes x :: "'a" shows "x \<le> ad (ad x) \<cdot> x"
RedVar: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and V :: "char list"   and v :: "val"   and E :: "char list \<Rightarrow> ty option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "lcl s V = \<lfloor>v\<rfloor>" shows "P,E \<turnstile> \<langle>Var V,s\<rangle> \<rightarrow> \<langle>Val v,s\<rangle>"
1: shows "fmdom' (snd a) \<inter> (fmdom' s - vs) = {}"
icf: shows "wa_invar WAC = addi \<inter> {s. \<alpha> s \<in> wa_invar WAA}"
Gcd_dvd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "a \<in> A" shows "dvd.dvd times (Gcd A) a"
subspace: fixes A :: "'d set" assumes "closed_csubspace A" shows "csubspace A"
R_def: shows "R = {(V, V'). (V, V') \<in> R' \<and> set (PPV V) \<subseteq> set (PPc c) \<and> set (PPV V') \<subseteq> set (PPc c)} \<union> {([], [])}"
Cons\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and es :: "nat exp list"   and es' :: "nat exp list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>Val v # es',s\<^sub>2\<rangle>"
nth_mat_tensorlist_mult: fixes Ts :: "real tensor vec"   and ds :: "nat list"   and i :: "nat"   and A :: "real mat" assumes "\<And>A. A \<in> set\<^sub>v Ts \<Longrightarrow> dims A = ds"   and "i < dim_row A"   and "dim_vec Ts = dim_col A" shows "mat_tensorlist_mult A Ts ds $ i = listsum ds (map (\<lambda>j. A $$ (i, j) \<cdot> Ts $ j) [0..<dim_vec Ts])"
gbinomial_mult_1': fixes a :: "'a"   and k :: "nat" shows "(a gchoose k) * a = of_nat k * (a gchoose k) + of_nat (Suc k) * (a gchoose Suc k)"
a: shows "r1 = (( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) # ( \<LM> A  \<RM> \<Rightarrow>* \<Empt>) # ps, c) \<and> r1 \<in> upRules"
intros_2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and ys :: "'a list"   and z :: "'a" assumes "r x y"   and "rtrancl_path r y ys z" shows "rtrancl_path r x (y # ys) z"
disjE: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "dynamic_component.eval tCMP active cid t t' n (dynamic_component.disj \<gamma> \<gamma>')" shows "dynamic_component.eval tCMP active cid t t' n \<gamma> \<or> dynamic_component.eval tCMP active cid t t' n \<gamma>'"
cf_init_def: shows "cf_init \<equiv> return (op_mtx_new c)"
add_le_cancel_left: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" shows "(c + a \<subseteq># c + b) = (a \<subseteq># b)"
Equiv_Exec_stable_set_def: fixes A :: "'d set"   and equiv_states :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and localState1 :: "'g \<Rightarrow> 'e"   and localState2 :: "'h \<Rightarrow> 'f"   and input_fun1 :: "'d \<Rightarrow> 'i message_af"   and input_fun2 :: "'d \<Rightarrow> 'j message_af"   and output_fun1 :: "'g \<Rightarrow> 'k message_af"   and output_fun2 :: "'h \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and trans_fun2 :: "'j message_af \<Rightarrow> 'h \<Rightarrow> 'h"   and k1 :: "nat"   and k2 :: "nat"   and c1 :: "'g"   and c2 :: "'h" shows "Equiv_Exec_stable_set A equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 \<equiv> \<forall>input m. set input \<subseteq> A \<and> m \<in> A \<longrightarrow> Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 (f_Exec_Comp trans_fun1 (map input_fun1 input \<odot> k1) c1) (f_Exec_Comp trans_fun2 (map input_fun2 input \<odot> k2) c2)"
bisim1_ThrowD: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "pc < length (compE2 e) \<and> (xcp = \<lfloor>a\<rfloor> \<or> xcp = None) \<and> xs = loc"
mcont_LCons: fixes x :: "'c" shows "mcont lSup (\<sqsubseteq>) lSup (\<sqsubseteq>) (LCons x)"
lideal_axioms: shows "lideal \<ww> carrier_local_ring_at add_local_ring_at mult_local_ring_at zero_local_ring_at one_local_ring_at"
transP_def: fixes \<phi> :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "transP \<phi> \<equiv> \<forall>x y z. \<phi> x y \<and> \<phi> y z \<longrightarrow> \<phi> x z"
jozsa_algo_def: shows "jozsa_algo \<equiv> (H \<otimes>\<^bsup>n\<^esup> \<Otimes> Quantum.Id 1) * (U\<^sub>f * (H \<otimes>\<^bsup>n\<^esup> * |Deutsch.zero\<rangle> \<otimes>\<^bsup>n\<^esup> \<Otimes> H * |Deutsch.one\<rangle>))"
set_bit_def: fixes n :: "nat"   and m :: "nat" shows "set_bit n m \<equiv> m + (if \<not> test_bit m n then 2 ^ n else 0)"
SUP_inf: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and B :: "'b set"   and a :: "'a" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf (Sup (f ` B)) a = Sup ((\<lambda>b. inf (f b) a) ` B)"
rows_B: shows "{row B i |i. i < mr} \<subseteq> carrier_vec n"
gap_relabel_effect_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" shows "gap_relabel_effect f l u \<equiv> let l' = relabel_effect f l u in if gap_precond l' (l u) then gap_effect l' (l u) else l'"
diagvector_mult_left: fixes x :: "'c \<Rightarrow> 'a"   and A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "diagvector x ** A = (\<chi>i j. x i * A $h i $h j)"
F0: shows "res_lim ((\<bullet>) f \<circ> ns) = \<zero>"
Lindelof: fixes \<F> :: "'b set set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S" obtains \<F>' :: "'b set set" where "\<F>' \<subseteq> \<F>" and "countable \<F>'" and "\<Union> \<F>' = \<Union> \<F>"
\<omega>_limit_set_def: fixes x :: "'a" shows "rev.\<omega>_limit_set x = {p. rev.\<omega>_limit_point x p}"
prems_5: shows "\<forall>e\<in>pred c2. finite (Bexp.vars e)"
prj_in_hom_3: shows "\<guillemotleft>\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0 : HHH\<mu>\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<rho>.apex\<guillemotright>"
"2_1": shows "(xs, ys) \<in> \<langle>\<langle>A\<rangle>list_rel\<rangle>list_rel"
bij_lens_unrest_all: fixes X :: "'a \<Longrightarrow> 'b"   and P :: "('c, 'b) uexpr" assumes "bij_lens X"   and "X \<sharp> P" shows "\<Sigma> \<sharp> P"
lexext_singleton: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" shows "lexext gt [y] [x] = gt y x"
2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
lunit_char_1: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<l>[a] : \<I> \<otimes> a \<rightarrow> a\<guillemotright>"
bit_minus_1_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   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 "bit (uminus one) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
comm_ring_isom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_ring_isom hom \<equiv> ring_isom hom"
firststep: shows "TS_step_d ([x, y], h) y = ((0, []), y # h)"
M1_def: shows "M1 \<equiv> PM \<and> M1_axioms"
frag_extend_sum: fixes I :: "'c set"   and f :: "'e \<Rightarrow> 'd \<Rightarrow>\<^sub>0 int"   and g :: "'c \<Rightarrow> 'e \<Rightarrow>\<^sub>0 int" assumes "finite I" shows "frag_extend f (sum g I) = sum (frag_extend f \<circ> g) I"
cases: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list" assumes "class.linorder less_eq less" obtains "xa__ = []"   | x :: "'c" and ys :: "'c list" where "xa__ = x # ys"
set_category_given_img_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set" shows "set_category_given_img S img \<AA> \<equiv> set_category_data S \<and> set_category_given_img_axioms S img \<AA>"
reverse_append_below: fixes xs :: "['b]"   and ys :: "['b]" shows "reverse\<cdot>(xs ++ ys) \<sqsubseteq> reverse\<cdot>ys ++ reverse\<cdot>xs"
permutes_inv_eq: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and y :: "'e"   and x :: "'e" assumes "p permutes S" shows "(inv p y = x) = (p x = y)"
map_upt: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map (\<lambda>i. f (xs ! i)) [0..<length xs] = map f xs"
blichfeldt: fixes S :: "(real, 'a) vec set" assumes "S \<in> sets lebesgue"   and "1 < emeasure lebesgue S" obtains x :: "(real, 'a) vec" and y :: "(real, 'a) vec" where "x \<noteq> y" and "x \<in> S" and "y \<in> S" and "\<And>i. (x - y) $ i \<in> \<int>"
1: shows "x \<le> gfp (\<lambda>x. fst (F (x, gfp (\<lambda>y. snd (F (x, y))))))"
mult_exp_exp: fixes x :: "'a"   and y :: "'a" shows "exp x * exp y = exp (x + y)"
id1: fixes g :: "nat \<Rightarrow> 'a" shows "map (\<lambda>i. g i \<cdot>\<^sub>v map gso [0..<length fs] ! i) [0..<k] = map (\<lambda>i. g i \<cdot>\<^sub>v gso i) [0..<k]"
assms: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (imp (neg (neg \<phi>)) \<phi>)"
minimal_def: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.minimal M \<equiv> dfa.accessible M = dfa.states M \<and> inj_on (dfa.right_lang M) (dfa.states M)"
agree_supset: fixes B :: "('c + 'c) set"   and A :: "('c + 'c) set"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<nu>' :: "(real, 'c) vec \<times> (real, 'c) vec" assumes "B \<subseteq> A"   and "Vagree \<nu> \<nu>' A" shows "Vagree \<nu> \<nu>' B"
inj_image_eq_iff: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "inj f" shows "(f ` A = f ` B) = (A = B)"
of_int_closed: fixes A :: "'a set"   and n :: "int" assumes "ring_closed A" shows "of_int n \<in> A"
max_def: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.max a b = (if a \<le>s b then b else a)"
simps_7: fixes I :: "('e, 'f, 'd) interp"   and \<alpha> :: "('e, 'f, 'd) hp" shows "prog_sem I \<alpha>** = (prog_sem I \<alpha>)\<^sup>*"
from_nat_eq_imp_eq: fixes x :: "nat"   and xa :: "nat" assumes "mod_type_class.from_nat x = mod_type_class.from_nat xa"   and "x < CARD('b)"   and "xa < CARD('b)" shows "x = xa"
lift_RealIntVal_Int: fixes x :: "val"   and f :: "int \<Rightarrow> int"   and g :: "real \<Rightarrow> real" assumes "x \<in> space (stock_measure INTEG)" shows "lift_RealIntVal f g x = lift_IntVal f x"
assms: shows "C \<in> Liminf_llist (lmap P_of_state (lmap state_of_wstate Sts))"
product_two_squares_nat: fixes m :: "nat"   and n :: "nat" assumes "is_sum2sq_nat m"   and "is_sum2sq_nat n" shows "is_sum2sq_nat (m * n)"
dvd_add_triv_left_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus a b) = dvd.dvd times a b"
dbl_inc_simps_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one zero = one"
valid_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c Q, A \<equiv> \<forall>s t c'. \<Gamma> \<turnstile> (c, s) \<rightarrow>\<^sup>* (c', t) \<longrightarrow> final (c', t) \<longrightarrow> s \<in> Normal ` P \<longrightarrow> t \<notin> Fault ` F \<longrightarrow> c' = Skip \<and> t \<in> Normal ` Q \<or> c' = Throw \<and> t \<in> Normal ` A"
stableD: fixes F :: "'d program"   and A :: "'d set" assumes "F \<in> stable A" shows "F \<in> A co A"
variation_comm: fixes x :: "real"   and y :: "real" shows "variation x y = - variation y x"
finite_nlistsD: fixes A :: "'c set"   and n :: "nat" assumes "finite (nlists A n)" shows "finite A \<or> n = 0"
lpeak_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and xs :: "(accent \<times> 'a) list" shows "lpeak r a b xs = (\<exists>as b' cs a' bs. snd ` set as \<subseteq> under r a \<and> b' \<in> {[(Grave, b)], []} \<and> snd ` set cs \<subseteq> under r a \<union> under r b \<and> a' \<in> {[(Acute, a)], []} \<and> snd ` set bs \<subseteq> under r b \<and> xs = as @ b' @ cs @ a' @ bs)"
allow_deny_dom: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and po :: "int" shows "dom (C (AllowPortFromTo a b po)) \<subseteq> dom (C (DenyAllFromTo a b))"
setops\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
assms_2: shows "f ` ({..<q} - {0}) = {..<q} - {0}"
textbook_false: fixes \<tau> :: "'a state \<times> 'a state" shows "I\<lbrakk>false\<rbrakk> \<tau> = \<lfloor>\<lfloor>False\<rfloor>\<rfloor>"
max_Sup: fixes X :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "X \<noteq> {}"   and "bdd_above (f ` X)"   and "bdd_above (g ` X)" shows "Sup (pointwise_max f g ` X) = max (Sup (f ` X)) (Sup (g ` X))"
thin_def: fixes A :: "'a list set" shows "thin A \<equiv> \<nexists>x y. x \<in> A \<and> y \<in> A \<and> x \<noteq> y \<and> initial_segment x y"
object_ptr_kinds_preserved: fixes SW :: "((_) heap, 'n, 'o) prog set"   and setter :: "((_) heap, 'n, 'p) prog"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w object_ptr. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
arcosh_1: shows "arcosh (1::'a) = (0::'a)"
sourcesD_1: fixes a :: "'a"   and \<mu> :: "'a" assumes "a \<in> sources \<mu>" shows "ide a"
degree_m_le: fixes f :: "int poly" shows "pl.degree_m f \<le> degree f"
lspasl_a_der: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a"   and h4 :: "'a" assumes "h1,h2\<triangleright>h0"   and "h3,h4\<triangleright>h1" shows "\<exists>h5. (h3,h5\<triangleright>h0) \<and> (h2,h4\<triangleright>h5) \<and> (h1,h2\<triangleright>h0) \<and> (h3,h4\<triangleright>h1)"
simps_7: fixes binop :: "binop"   and e1 :: "expr"   and e2 :: "expr" shows "assignsE (BinOp binop e1 e2) = (if binop = CondAnd \<or> binop = CondOr then assignsE e1 else assignsE e1 \<union> assignsE e2)"
alg_multE_1: fixes x :: "'a list"   and p :: "'a list list" assumes "x \<in> carrier (poly_ring R)"   and "p \<in> carrier (poly_ring (poly_ring R))"   and "p \<noteq> []" shows "[\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> x] [^]\<^bsub>poly_ring (poly_ring R)\<^esub> UP.alg_mult p x pdivides\<^bsub>poly_ring R\<^esub> p"
assms_4: shows "ws = map ((!) fs) [i..<m]"
monoid_homomorphism_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" shows "monoid_homomorphism \<eta> M composition unit M' composition' unit' \<equiv> (Set_Theory.map \<eta> M M' \<and> Group_Theory.monoid M composition unit) \<and> Group_Theory.monoid M' composition' unit' \<and> monoid_homomorphism_axioms \<eta> M composition unit composition' unit'"
maximal_wrt_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and A :: "'b"   and DA :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.maximal_wrt A DA = (DA = {#} \<or> A = Max (atms_of DA))"
W_def: fixes k :: "nat" shows "W k = wlp_while_n M0 M1 (wlp S) k P"
presheaf_of_rings_axioms: shows "presheaf_of_rings S is_open idim.im_sheaf idim.im_sheaf_morphisms b idim.add_im_sheaf idim.mult_im_sheaf idim.zero_im_sheaf idim.one_im_sheaf"
polyhedron_eq_finite_exposed_faces: fixes S :: "'b set" shows "polyhedron S = (closed S \<and> convex S \<and> finite {F. F exposed_face_of S})"
cge: fixes x :: "'a" assumes "x \<in># p1" shows "2 \<le> (\<Sum>p\<in>#P. count p x)"
vec_index_conjugate: fixes i :: "nat"   and v :: "'a vec" assumes "i < dim_vec v" shows "conjugate v $ i = conjugate (v $ i)"
less_le_not_le: fixes x :: "'a"   and y :: "'a" shows "(x < y) = strict (\<le>) x y"
assms_1: shows "\<langle>M;A;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m m"
errorable_def: fixes x :: "'b" shows "x\<^sub>\<bottom> = Abs_errorable (Some x)"
imult_infinity_right: fixes n :: "eint" assumes "0 < n" shows "n * \<infinity> = \<infinity>"
sym_Un: fixes theta1 :: "('b \<times> 'b) set"   and theta2 :: "('b \<times> 'b) set" assumes "sym theta1"   and "sym theta2" shows "sym (theta1 \<union> theta2)"
valofn_ge_2p: fixes x :: "('a, 'b) IEEE.float" shows "2 ^ IEEE.exponent x / 2 ^ bias TYPE(('a, 'b) IEEE.float) \<le> valofn x"
less_numeral_extra_4: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one one"
simps_19: fixes x51 :: "'d set"   and x52 :: "('d, 'e, 'f) com"   and x53 :: "('d, 'e, 'f) com" shows "Cond x51 x52 x53 \<noteq> Skip"
pow_Suc_subset_trancl: fixes n :: "nat"   and R :: "('b \<times> 'b) set" shows "R ^^ Suc n \<subseteq> R\<^sup>+"
nn_integral_snd: fixes f :: "'a \<times> 'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>M1 \<partial>M2 = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
intro: fixes U\<^sub>2 :: "'a set"   and le\<^sub>2 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>2 :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2" shows "ord_order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2"
eq_\<I>_attach: fixes \<I>' :: "('h, 'i) \<I>"   and res :: "('h, 'i) resource"   and conv :: "('j, 'k, 'h, 'i) converter"   and conv' :: "('j, 'k, 'h, 'i) converter" assumes "\<I>' \<turnstile>res res \<surd>"   and "\<I>_full,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'" shows "conv \<rhd> res = conv' \<rhd> res"
assms: shows "[\<lparr>F,x\<rparr> in v]"
continuous_at_imp_continuous_within: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "isCont f x" shows "continuous (at x within s) f"