hcju/isabelleps huggingface dataset:
Un_Int_crazy: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<inter> B \<union> B \<inter> C \<union> C \<inter> A = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
asIMOD_igConsIPresIGWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "igConsIPresIGWls (asIMOD SEM)"
test_dvd_def: fixes p :: "int"   and n :: "nat"   and sl_impl :: "(int poly, int \<times> int poly list, 'a) subseqs_foldr_impl"   and sli :: "int \<times> int poly list \<Rightarrow> int poly list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "int poly"   and ws :: "int poly multiset" assumes "prime p"   and "p = p ^ n"   and "n \<noteq> 0"   and "correct_subseqs_foldr_impl (\<lambda>x. map_prod (p.mul_const x) ((#) x)) sl_impl sli" shows "p.test_dvd u ws = (\<forall>v l. v dvd u \<longrightarrow> 0 < degree v \<longrightarrow> degree v < degree u \<longrightarrow> p.Mp v \<noteq> p.Mp (Polynomial.smult l (\<Prod>\<^sub># ws)))"
Q_defined: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and v :: "'a" assumes "Prim_Invar2_ctd_loc w g r Q \<pi>"   and "(u, v) \<in> edges (Prim.rg g r)"   and "u \<notin> Prim.S r (A Q \<pi>)"   and "v \<in> Prim.S r (A Q \<pi>)" shows "Q u \<noteq> \<infinity>"
greaterThanLessThan_eq: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "ord.greaterThanLessThan less a b = ord.greaterThan less a \<inter> ord.lessThan less b"
one_inf_conv: fixes x :: "'a" shows "(1::'a) \<sqinter> x = (1::'a) \<sqinter> x\<^sup>T"
of_complex_inj: fixes x :: "complex"   and y :: "complex" assumes "of_complex x = of_complex y" shows "x = y"
chop_dense: fixes r :: "real_int" shows "0 < \<parallel>r\<parallel> \<longrightarrow> (\<exists>s t. R_Chop(r,s,t) \<and> 0 < \<parallel>s\<parallel> \<and> 0 < \<parallel>t\<parallel>)"
assms_1: shows "\<D>\<^sub>\<circ> r = set {a}"
permute_multiset_def: fixes p :: "perm"   and M :: "'a multiset" shows "p \<bullet> M = image_mset (permute p) M"
octo_of_real_sel_1: fixes x :: "real" shows "Ree (octo_of_real x) = x"
get_disconnected_nodes_locs_impl: fixes type_wf :: "(_) heap \<Rightarrow> bool"   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_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs" shows "get_disconnected_nodes_locs = l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes_locs"
coprime_imp_gcd_eq_1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a b" shows "gcd a b = one"
borel_measurable_inverse_ennreal: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. inverse (f x)) \<in> borel_measurable M"
\<tau>multithreaded_wf_axioms: shows "\<tau>multithreaded_wf final1 r1 \<tau>move1"
degreen_polyadd: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> max n0 n1" shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
MiniSC_eq: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "image_mset to_mini_formula \<Gamma> \<Rightarrow> image_mset to_mini_formula \<Delta> = \<Gamma> \<Rightarrow> \<Delta>"
prems_1: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S') \<theta>"
is_derivation_def: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" shows "is_derivation X p = (linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X \<and> (\<forall>f g. f \<in> diff_fun_space \<longrightarrow> g \<in> diff_fun_space \<longrightarrow> X (f * g) = f p * X g + g p * X f))"
res_eq_to_cong: fixes a :: "int"   and b :: "int" shows "(a mod m = b mod m) = [a = b] (mod m)"
weak_arrow_of_homs_axioms: shows "weak_arrow_of_homs (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) src\<^sub>D trg\<^sub>D H"
UP_ring_subring_is_ring: fixes S :: "'a set" assumes "subring S R" shows "cring (UP (R\<lparr>carrier := S\<rparr>))"
f_eq_then_id: assumes "\<forall>n. f n = n" shows "f = id"
z: shows "z \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
dvd_minus_iff: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "dvd.dvd times x (uminus y) = dvd.dvd times x y"
select_convs_1: fixes Calculus :: "'a processCalculus"   and HasBarb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and more :: "'c" shows "Calculus \<lparr>Calculus = Calculus, HasBarb = HasBarb, \<dots> = more\<rparr> = Calculus"
fresh_pVs: fixes a :: "atom" assumes "a \<sharp> p" shows "a \<sharp> p \<bullet> Vs"
det_fst_row_add_iterate_PA: fixes n :: "nat"   and A :: "(('b, 'c) vec, 'c) vec"   and P :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'c" assumes "n < nrows A" shows "det (fst (row_add_iterate_PA (P, A) n i j)) = det P"
simps_1: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "'b" shows "weight_fun.weight w w0 scf (Var x) = w0"
assms_1: shows "Q \<longmapsto> a<\<nu>x> \<prec> Q'"
usubst_fml: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<phi> :: "fml"   and I :: "interp" assumes "Uvariation \<nu> \<omega> U"   and "usubstappf \<sigma> U \<phi> \<noteq> undeff" shows "(\<nu> \<in> fml_sem I (the (usubstappf \<sigma> U \<phi>))) = (\<nu> \<in> fml_sem (USubst.adjoint \<sigma> I \<omega>) \<phi>)"
simps_9: shows "OUTfromCh data9 = {data8}"
assms: fixes x' :: "'a" assumes "P x"   and "P x'" shows "x' = x"
invariant: fixes s :: "'s \<times> ('c \<Rightarrow> real) set"   and D :: "('s \<times> ('c \<Rightarrow> real) set) pmf" assumes "s \<in> \<S>"   and "D \<in> \<K> s" shows "\<forall>s'\<in>set_pmf D. s' \<in> \<S>"
P3: shows "Bet E D D' \<and> Cong D D' B A"
fun_comp_eq_conv: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e"   and fg :: "'b \<Rightarrow> 'd" shows "(f \<circ> g = fg) = (\<forall>x. f (g x) = fg x)"
livenessI: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<And>\<alpha>. \<exists>\<sigma>. P (\<alpha> @- \<sigma>)" shows "liveness P"
convex_lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and Vs :: "'a vec list"   and b :: "'a vec" shows "convex_lincomb_list c Vs b = (nonneg_lincomb_list c Vs b \<and> sum c {0..<length Vs} = (1::'a))"
rep_eq: fixes x :: "'b interval" shows "set_of x = {fst (bounds_of_interval x)..snd (bounds_of_interval x)}"
subst_res_def2: fixes P :: "'b assn"   and w :: "vals"   and Y :: "vals" shows "(P\<leftarrow>w) Y = P w"
ord0_leq_refl: fixes x :: "ord0" shows "(x, x) \<in> ord0_leq"
know_composition: fixes m :: "KS"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "m \<notin> specKeysSecrets P"   and "m \<notin> specKeysSecrets Q"   and "know P m \<or> know Q m"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ" shows "know PQ m"
igSwapBinpIPresIGWlsBinp_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSwapBinpIPresIGWlsBinp MOD \<equiv> \<forall>zs z1 z2 delta binp. igWlsBinp MOD delta binp \<longrightarrow> igWlsBinp MOD delta (igSwapBinp MOD zs z1 z2 binp)"
power_strict_mono: fixes a :: "'d"   and b :: "'d"   and n :: "nat" assumes "a < b"   and "(0::'d) \<le> a"   and "0 < n" shows "a ^ n < b ^ n"
AE_iff_measurable: fixes N :: "'d set"   and M :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "N \<in> sets M"   and "{x \<in> space M. \<not> P x} = N" shows "almost_everywhere M P = (emeasure M N = 0)"
major: shows "rtrancl3p r (ax, (ay, az), aw) bs (cx, (cy, cz), cw)"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_uminus" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> vreal_uminus). -\<^sub>\<real> a = b"
assms_2: shows "A \<Longrightarrow>\<^sub>t P * F"
z_R_def: fixes index_list :: "nat list \<times> nat list"   and sign_asg :: "rat list" shows "z_R index_list sign_asg \<equiv> prod_list (map ((!) (map_sgas sign_asg)) (fst index_list)) * prod_list (map ((!) sign_asg) (snd index_list))"
simps_1: fixes M :: "(floatarith, nat) mapping"   and slp :: "floatarith list" shows "slp_of_fas' [] M slp = (M, slp)"
entails_frame: fixes P :: "'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and S :: "'a \<Rightarrow> real" assumes "P \<tturnstile> R"   and "Q \<tturnstile> S" shows "P && Q \<tturnstile> R && S"
contractible_space_top_of_set: fixes S :: "'a set" shows "contractible_space (top_of_set S) = contractible S"
that: fixes x :: "'a" assumes "(THE x on U. P x) = Some x"   and "Q x" shows "thesis"
map_of_alist_product: fixes f :: "'d \<Rightarrow> 'f \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> 'c"   and xs :: "('d \<times> 'f) list"   and ys :: "('e \<times> 'g) list"   and a :: "'d"   and c :: "'e" shows "map_of (alist_product f xs ys) (a, c) = (case map_of xs a of None \<Rightarrow> None | Some b \<Rightarrow> map_option (f a b c) (map_of ys c))"
max_signed_def: shows "max_signed \<equiv> 2 ^ (word_len - 1) - 1"
fresh_finsert: fixes a :: "atom"   and x :: "'a"   and S :: "'a fset" shows "a \<sharp> finsert x S = (a \<sharp> x \<and> a \<sharp> S)"
gcd_unique_int: fixes d :: "int"   and a :: "int"   and b :: "int" shows "(0 \<le> d \<and> d dvd a \<and> d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d)) = (d = gcd a b)"
eqC: shows "f ` (S \<inter> f -` C) = C"
double_zero: fixes a :: "'a" shows "(a + a = (0::'a)) = (a = (0::'a))"
kleene_qfp_iff_least: fixes bot :: "'b"   and A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "bot \<in> A"   and "\<forall>q\<in>A. less_eq bot q"   and "omega_continuous A less_eq f"   and "omega_complete A less_eq"   and "\<forall>q\<in>A. \<forall>x\<in>A. sympartp less_eq (f q) q \<longrightarrow> less_eq x (f q) \<longrightarrow> less_eq x q"   and "\<forall>p\<in>A. \<forall>q\<in>A. \<forall>x\<in>A. sympartp less_eq p q \<longrightarrow> less_eq q x \<longrightarrow> less_eq p x" shows "extreme_bound A less_eq {f^n bot |. n} = extreme {s \<in> A. sympartp less_eq (f s) s} (\<lambda>x y. less_eq y x)"
cobounded2: fixes a :: "'b word"   and b :: "'b word" shows "signed.min a b \<le>s b"
dcharacter_fourier_expansion: fixes m :: "nat" shows "\<chi> m = (\<Sum>k = 1..n. complex_of_real (1 / real n) * gauss_sum_int (- int k) * unity_root n (int (m * k)))"
strict_prefix_butlast: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs (butlast ys)"   and "ys \<noteq> []" shows "strict_prefix xs ys"
strict_prefix_def: fixes xs :: "'c list"   and ys :: "'c list" shows "strict_prefix xs ys = (xs \<le> ys \<and> xs \<noteq> ys)"
mersenne_mod_nonneg: fixes k :: "int"   and n :: "nat" assumes "0 \<le> k" shows "0 \<le> mersenne_mod k n"
length_induct: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
wf_not_sym: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and x :: "'c" assumes "wf r"   and "(a, x) \<in> r" shows "(x, a) \<notin> r"
ZQ: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>Z {P} c {Q}" shows "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"
add_match_split_fst: fixes m :: "'c match_expr"   and m' :: "'c match_expr"   and a' :: "action"   and rs :: "'c rule list" shows "add_match m (Rule m' a' # rs) = Rule (MatchAnd m m') a' # add_match m rs"
is_acc_run_def: fixes r :: "nat \<Rightarrow> 'q \<times> 's" shows "prod.is_acc_run r \<equiv> prod.is_run r \<and> prod.is_acc r"
is_rat_rat_def: fixes x :: "rat" shows "is_rat x = True"
order_iff_strict: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" shows "(a \<sqsubseteq> b) = (a \<sqsubset> b \<or> a = b)"
notE: fixes P :: "bool"   and R :: "bool" assumes "\<not> P"   and "P" shows "R"
simps_1: fixes f :: "'d \<Rightarrow> 'c option" shows "omap_option f None = Some None"
pigeonhole: fixes T :: "'a set set"   and S :: "'a set" assumes "finite T"   and "S \<subseteq> \<Union> T"   and "card T < card S" shows "\<exists>x\<in>S. \<exists>y\<in>S. \<exists>X\<in>T. x \<noteq> y \<and> x \<in> X \<and> y \<in> X"
encode_problem_has_model_then_also_partial_encodings_4: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "\<A> \<Turnstile> encode_all_frame_axioms \<Pi> t"
A: shows "\<forall>yps'. length yps' < length yps \<longrightarrow> (\<forall>Y'. (yp # yps', Y') \<in> futures (c_process step out s\<^sub>0) xps \<longrightarrow> (ipurge_tr I (c_dom D) (c_dom D yp) yps', ipurge_ref I (c_dom D) (c_dom D yp) yps' Y') \<in> futures (c_process step out s\<^sub>0) xps)"
lex_comp_aux_def: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.nat_term rep_nat_term splus" shows "nat_term.lex_comp_aux rep_nat_term = (\<lambda>x y. case comp_of_ord lex_pp (fst (rep_nat_term x)) (fst (rep_nat_term y)) of Eq \<Rightarrow> comparator_of (snd (rep_nat_term x)) (snd (rep_nat_term y)) | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
cod_eqI: fixes b :: "'a"   and f :: "'a" assumes "A.ide b"   and "A.seq b f" shows "A.cod f = b"
ex231e: fixes x :: "'a" shows "x \<sqinter> top \<sqinter> x \<sqinter> top = x \<sqinter> top"
transfer_preconditions: fixes Q :: "'c \<Rightarrow> nat option"   and V :: "'c \<Rightarrow> nat option"   and D :: "'c \<Rightarrow> enat"   and S :: "'c set"   and u :: "'c"   and du :: "nat" assumes "coupling Q V D S" shows "(Q u = Some du) = (D u = enat du \<and> u \<notin> S)"
analz_insert_MPair: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "analz (insert \<lbrace>X, Y\<rbrace> H) = insert \<lbrace>X, Y\<rbrace> (analz (insert X (insert Y H)))"
per_perp_in: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "Per A B C" shows "B PerpAt A B B C "
cases: fixes a :: "'a list" assumes "steps a" obtains x :: "'a" where "a = [xb__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "a = x # y # xs" and "E x y" and "steps (y # xs)"
Tau: fixes P :: "pi" shows "\<tau>.P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P"
distinct_9: shows "OclAny \<noteq> UnlimitedNatural"
node\<^sub>i_order_helper: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "k \<le> length ts"   and "length ts \<le> 4 * k + 1"   and "\<forall>x\<in>set (subtrees ts). order k x"   and "order k t" shows "case node\<^sub>i k ts t of T\<^sub>i t \<Rightarrow> order k t | Up\<^sub>i btree1 b btree2 \<Rightarrow> True"
trace_callee_closure_sim: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf"   and p' :: "'d spmf"   and q' :: "'g spmf"   and a :: "'e"   and b :: "'f" assumes "trace_callee_closure callee1 callee2 A p q p' q'"   and "a \<in> A" shows "trace_callee_closure callee1 callee2 A p q (cond_spmf_fst (p' \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q' \<bind> (\<lambda>s. callee2 s a)) b)"
converse_rtranclpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b" assumes "r\<^sup>*\<^sup>* x z" obtains "x = z"   | y :: "'b" where "r x y" and "r\<^sup>*\<^sup>* y z"
lcm_eq_1_iff: fixes a :: "'a"   and b :: "'a" shows "(lcm a b = (1::'a)) = (is_unit a \<and> is_unit b)"
arrE: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f" obtains "f \<noteq> J_C.null" and "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) (J_C.Dom f) (J_C.Cod f) (J_C.Map f)"
vector_matrix_mul_assoc: fixes v :: "('a, 'd) vec"   and M :: "(('a, 'c) vec, 'd) vec"   and N :: "(('a, 'b) vec, 'c) vec" shows "v v* M v* N = v v* (M ** N)"
Ex_def: fixes P :: "'b \<Rightarrow> bool" shows "Ex P \<equiv> \<forall>Q. (\<forall>x. P x \<longrightarrow> Q) \<longrightarrow> Q"
bet__lt1213: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and C :: "'a"   and A :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "B \<noteq> C"   and "Bet A B C" shows "Tarski_neutral_dimensionless.Lt Bet Cong A B A C"
axioms_3: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "limit_cone J C D a \<chi>" shows "diagram J C D"
that: assumes "nop_onto A (2\<^sub>\<nat>) f" shows "thesis"
minus_diff_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "minus (uminus b) a = minus (uminus a) b"
one_less_inverse: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less a one" shows "less one (inverse a)"
Max_wrt_among_map_relation_bij_subset: fixes f :: "'a \<Rightarrow> 'c"   and A :: "'a set" assumes "bij f" shows "f ` Max_wrt_among le A \<subseteq> Max_wrt_among (map_relation (inv f) le) (f ` A)"
FGModule_FG: shows "FGModule G (*) ActingGroup.group_ring"
a_l_coset_carrier: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <+ H"   and "x \<in> carrier R"   and "subgroup H (add_monoid R)" shows "y \<in> carrier R"
constrains_reachable_Int: fixes F :: "'b program"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> B co B'" shows "F \<in> reachable F \<inter> B co reachable F \<inter> B'"
c: fixes mr :: "'a simple_match"   and ar :: "char list"   and mf :: "'a simple_match"   and af :: "simple_action"   and a :: "'a simple_match" assumes "(mr, ar) \<in> set (lr_of_tran_s1 rt)"   and "(mf, af) \<in> simple_rule_dtor ` set fw"   and "simple_match_and mr mf = Some a" shows "oiface a = ifaceAny"
span_eq_dim: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set"   and T :: "'e set" assumes "vector_space scale"   and "module.span scale S = module.span scale T" shows "vector_space.dim scale S = vector_space.dim scale T"
eval: shows "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>"
assoc: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term"   and b :: "'a \<Rightarrow> ('b, 'a) Term.term"   and c :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "a \<circ>\<^sub>s b \<circ>\<^sub>s c = a \<circ>\<^sub>s (b \<circ>\<^sub>s c)"
meet_join_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<squnion> y) = x"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set1_T\<^sub>m x" shows "m a b = m' a b"
simps_2: fixes l :: "('c \<times> 'd) tree"   and a :: "'c"   and uu :: "'d"   and r :: "('c \<times> 'd) tree" shows "Tree2.inorder \<langle>l, (a, uu), r\<rangle> = Tree2.inorder l @ a # Tree2.inorder r"
bind_returns_heap_E2: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and h' :: "'a" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "pure f h" obtains x :: "'d" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
backtrack_cs: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and c :: "(nat, 'a \<times> QDelta) mapping \<times> (nat, 'a \<times> QDelta) mapping"   and K :: "'a set"   and s' :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and s'' :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" assumes "checked_simplex' cs J s"   and "checkpoint_simplex' s = c"   and "invariant_simplex' cs K s'"   and "backtrack_simplex' c s' = s''"   and "J \<subseteq> K" shows "invariant_simplex' cs J s''"
PR1: fixes p :: "'b pfm"   and q :: "'b pfm" assumes "\<turnstile>\<^sub>! p"   and "\<turnstile>\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q)" shows "\<turnstile>\<^sub>! q"
is_volatile_def: fixes r :: "'b memref" shows "is_volatile r = (case r of Write\<^sub>s\<^sub>b volatile a x v xa xb xc xd \<Rightarrow> volatile | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> volatile | _ \<Rightarrow> False)"
finite_subset_Union_chain: fixes A :: "'c set"   and \<B> :: "'c set set"   and \<A> :: "'c set set" assumes "finite A"   and "A \<subseteq> \<Union> \<B>"   and "\<B> \<noteq> {}"   and "subset.chain \<A> \<B>" obtains B :: "'c set" where "B \<in> \<B>" and "A \<subseteq> B"
to_fun_to_poly: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_fun (to_polynomial R a) b = a"
h_OUT: fixes n :: "nat"   and x :: "'v" shows "d_OUT (\<lambda>x. ennreal (h n x)) x = (if x \<in> A_n n then weight \<Gamma> x else 0)"
cyclic_group_assoc: fixes G :: "('d, 'e) cyclic_group_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "Cyclic_Group.cyclic_group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<otimes>\<^bsub>G\<^esub> b \<otimes>\<^bsub>G\<^esub> c = a \<otimes>\<^bsub>G\<^esub> (b \<otimes>\<^bsub>G\<^esub> c)"
A: shows "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> wr"