hcju/isabelleps huggingface dataset:
intro: fixes fP :: "'a list \<Rightarrow> bool"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "fP [] = True"   and "\<And>x. fP [x] = True"   and "\<And>a b xs. fP (a # b # xs) = (P a b \<and> fP (b # xs))" shows "cnvSpec fP P"
dom_lc3: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d (x \<cdot> y) = d (x \<cdot> y)"
setmult_closed: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> rcosets H"   and "K2 \<in> rcosets H" shows "K1 <#> K2 \<in> rcosets H"
power2_eq_square: fixes a :: "'a \<times> 'b" shows "prod.power2 a = times_prod' a a"
cmp_simps_3: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "trg\<^sub>C (\<Phi>\<^sub>F (f, g)) = trg\<^sub>C (F f)"
SemTrue: fixes b :: "BExpr"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c1 :: "OBJ"   and n :: "nat"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c2 :: "OBJ" assumes "evalB b (fst s)"   and " s , c1 \<rightarrow>\<^sub>n  t " shows " s , Iff b c1 c2 \<rightarrow>\<^sub>n + 1  t "
cases: fixes x :: "tm \<times> tm \<times> tm \<times> tm" obtains t :: "name" and v :: "tm" and tm :: "tm" and y :: "tm" and y' :: "tm" and t' :: "name" and u :: "name" and u' :: "name" where "atom t \<sharp> (v, tm, y, y', t', u, u')" and "atom t' \<sharp> (v, tm, y, y', u, u')" and "atom u \<sharp> (v, tm, y, y', u')" and "atom u' \<sharp> (v, tm, y, y')" and "x = (v, tm, y, y')"
lcm_proj1_if_dvd_nat: fixes x :: "nat"   and y :: "nat" assumes "x dvd y" shows "lcm y x = y"
val_Zp_def: fixes p :: "int" assumes "padic_integers p" shows "padic_integers.val_Zp p = (\<lambda>x. if x = \<zero>\<^bsub>padic_int p\<^esub> then \<infinity> else eint (padic_val p x))"
emeasure_count_space_eq_0: fixes A :: "'a set"   and X :: "'a set" shows "(emeasure (count_space A) X = 0) = (X \<subseteq> A \<longrightarrow> X = {})"
IdD: fixes a :: "'b"   and b :: "'b" assumes "(a, b) \<in> Id" shows "a = b"
inout_arcs_arc_simps_1: fixes e :: "'c"   and G :: "('d, 'c) pre_digraph"   and u :: "'d"   and E :: "'c set" assumes "e \<in> arcs G"   and "tail G e = u" shows "out_arcs G u \<inter> insert e E = insert e (out_arcs G u \<inter> E)"
run_one_step_label_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x41 :: "nat"   and x42 :: "e list"   and x43 :: "e list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, Label x41 x42 x43) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>r rvs. res = RSBreak r rvs) \<or> (\<exists>rvs. res = RSReturn rvs) \<or> (\<exists>e. res = RSCrash e)"
exists_upto_of_nat: fixes p :: "'b \<Rightarrow> bool"   and n :: "nat"   and as :: "'b iarray" shows "exists_upto p (of_nat n) as = (\<exists>m<n. p (as !! m))"
append_Nil: fixes ys :: "'k list" shows "[] @ ys = ys"
i_Exec_Stream_expand_map_aggregate_append: fixes f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and input :: "nat \<Rightarrow> 'f message_af"   and k :: "nat"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" shows "i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun ((xs \<frown> input) \<odot> k) c) k ag = f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c)) k ag \<frown> i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) (f_Exec_Comp trans_fun (xs \<odot> k) c)) k ag"
div_dvd_sgn_abs: fixes l :: "int"   and k :: "int" assumes "l dvd k" shows "k div l = sgn k * sgn l * (\<bar>k\<bar> div \<bar>l\<bar>)"
in_interval_to_interval: fixes a :: "'a" shows "a \<in>\<^sub>i to_interval a"
regular_implies_mono: fixes f :: "'a set \<Rightarrow> 'a set" assumes "regular f" shows "mono f"
\<Delta>\<Phi>_del_min: fixes hs :: "'b tree"   and x :: "'b" assumes "hs \<noteq> \<langle>\<rangle>" shows "\<Phi> (del_min \<langle>hs, x, \<langle>\<rangle>\<rangle>) - \<Phi> \<langle>hs, x, \<langle>\<rangle>\<rangle> \<le> 3 * log 2 (real (size hs)) - real (len hs) + 2"
enabledPrefix: fixes SES :: "('a, 'b) SES_rec"   and s :: "'a"   and t1 :: "'b list"   and t2 :: "'b list" assumes "enabled SES s (t1 @ t2)" shows "enabled SES s t1"
elims: fixes x :: "atom"   and y :: "atom fm" assumes "aplus_inf x = y" obtains uu_ :: "nat" where "x = Less 0 uu_" and "y = FalseF"   | v :: "nat" where "x = Less (Suc v__) 0" and "y = TrueF"   | i :: "nat" and j :: "nat" where "x = Less (Suc i__) (Suc j__)" and "y = Atom (Less i__ j__)"   | "x = Eq 0 0" and "y = TrueF"   | v :: "nat" where "x = Eq 0 (Suc va__)" and "y = FalseF"   | v :: "nat" where "x = Eq (Suc v) 0" and "y = FalseF"   | i :: "nat" and j :: "nat" where "x = Eq (Suc i) (Suc j)" and "y = Atom (Eq i j)"
synth_analz_idem_monoI: fixes c :: "msg"   and S :: "msg set"   and S' :: "msg set" assumes "c \<in> synth (analz S)"   and "S \<subseteq> synth (analz S')" shows "c \<in> synth (analz S')"
homeomorphic_map_openness: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and U :: "'a set" assumes "homeomorphic_map X Y f"   and "U \<subseteq> topspace X" shows "openin Y (f ` U) = openin X U"
nEq_neq_conv: fixes x :: "'a"   and y :: "'a" shows "(comp x y \<noteq> Eq) = (x \<noteq> y)"
simps: fixes ip_start :: "'a word"   and ip_end :: "'a word" shows "iprange_interval (ip_start, ip_end) = WordInterval ip_start ip_end"
distinct_45: fixes x6 :: "Arraytype" shows "ByteT \<noteq> ArrT x6"
polyn_n_m: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat"   and m :: "nat" assumes "pol_coeff S c"   and "n < m"   and "m \<le> fst c" shows "polyn_expr R X m (m, snd c) = polyn_expr R X n (n, snd c) \<plusminus> \<Sigma>\<^sub>f R (\<lambda>j. snd c j \<cdot>\<^sub>r X^\<^bsup>R j\<^esup>) Suc n m"
ide_is_iso: fixes a :: "'a" assumes "C.ide a" shows "C.iso a"
half: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (subst_bv (Fv x \<tau>') s) (Abs \<tau>' t $ Fv x \<tau>')"
IH_2: shows "insertion f (replace_coeff (insertion f) p2) = insertion f (reduce_nested_mpoly p2)"
assms_1: shows "path_conforms_with_strategy p P (\<sigma>(v := w))"
adjoint_mult: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and B :: "'a mat"   and l :: "nat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m l" shows "adjoint (A * B) = adjoint B * adjoint A"
B: shows "(\<Union>i. {..- inverse (real (Suc i))}) = {..<0}"
Longest_common_prefix_prefix: fixes xs :: "'b list"   and L :: "'b list set" assumes "xs \<in> L" shows "Longest_common_prefix L \<le>p xs"
mat_O_times_Q1: shows "adjoint (tensor_P mat_O (1\<^sub>m K)) * Q1 * tensor_P mat_O (1\<^sub>m K) = Q"
degree_abs_int_poly: fixes p :: "int poly" shows "degree (abs_int_poly p) = degree p"
vsv_vlrestriction_vrange: fixes s :: "V"   and r :: "V" assumes "vsv s"   and "vsv (r \<restriction>\<^sup>l\<^sub>\<circ> \<R>\<^sub>\<circ> s)" shows "vsv (r \<circ>\<^sub>\<circ> s)"
open_cov: fixes S :: "'a set" assumes "open S" shows "\<exists>K. S = \<Union> (B ` K)"
assms_4: shows "fst ` set txs \<subseteq> trm"
supset_circuit_in_imp_dep_in: fixes C :: "'a set"   and X :: "'a set" assumes "circuit_in \<E> C \<and> C \<subseteq> X" shows "\<not> indep_in \<E> X"
prems_4: shows "bvars\<^sub>s\<^sub>t (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S' @ [x]) \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
m\<mu>_def: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "m\<mu> s s' = (shr s = shr s' \<and> m\<mu>t (shr s) (thr s) (thr s'))"
fund_sol_is_arg_min: shows "is_arg_min (\<lambda>x. pell_valuation (case x of (x, y) \<Rightarrow> (int x, int y))) nontriv_solution fund_sol"
assoc_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "S.ide f"   and "S.ide g"   and "S.ide h"   and "S.src f = S.trg g"   and "S.src g = S.trg h" shows "S.arr (S.\<a> f g h)"
BinOpThrow1: fixes e :: "char list exp"   and bop :: "bop"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>throw e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
open_left: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "open S"   and "x \<in> S"   and "y < x" shows "\<exists>b<x. {b<..x} \<subseteq> S"
inj_on_imp_Ker0: assumes "inj_on T V" shows "Ker = 0"
convex_on_bounded_continuous: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real"   and b :: "real" assumes "open S"   and "convex_on S f"   and "\<forall>x\<in>S. \<bar>f x\<bar> \<le> b" shows "continuous_on S f"
\<Phi>_Eps: fixes NL :: "nat set list" assumes "infinite (hd NL)"   and "Inf ` set NL \<union> hd NL \<subseteq> M" shows "\<Phi> NL (Eps (\<Phi> NL))"
arity_simps_1: fixes \<zeta> :: "('s, 'v) hd" shows "arity (Hd \<zeta>) = arity_hd \<zeta>"
first_countable_topology_axioms_def: fixes "open" :: "'c set \<Rightarrow> bool" shows "class.first_countable_topology_axioms open \<equiv> \<forall>x. \<exists>A. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
min_inv_strict: fixes r :: "'d \<rightarrow> 'd" shows "\<delta>\<cdot>r\<cdot>\<bottom> = \<bottom>"
times_fun_var: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(f \<oplus> g) x = \<Squnion>{f y \<oplus> g z |y z. D y z \<and> x = y \<oplus> z}"
list_combE_2: fixes f :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "pre_strong_term_class.wellformed (list_comb f xs)"   and "x \<in> set xs" shows "pre_strong_term_class.wellformed x"
bij_betwI': fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and Y :: "'b set" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y)"   and "\<And>x. x \<in> X \<Longrightarrow> f x \<in> Y"   and "\<And>y. y \<in> Y \<Longrightarrow> \<exists>x\<in>X. y = f x" shows "bij_betw f X Y"
associative: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<in> B"   and "b \<in> B"   and "c \<in> B" shows "addB (addB a b) c = addB a (addB b c)"
reachable_snapshot_inv_def: shows "reachable_snapshot_inv = (\<lambda>s. \<forall>x. reachable x s \<longrightarrow> in_snapshot x s)"
VUnion_least: fixes A :: "V"   and B :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> x \<subseteq>\<^sub>\<circ> B" shows "\<Union>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> B"
Pi_split_insert_domain: fixes x :: "'h \<Rightarrow> 'i"   and i :: "'h"   and I :: "'h set"   and X :: "'h \<Rightarrow> 'i set" shows "(x \<in> Pi (insert i I) X) = (x \<in> Pi I X \<and> x i \<in> X i)"
intros_9: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>2 :: "('d, 'e, 'f) com"   and c\<^sub>1 :: "('d, 'e, 'f) com" assumes "s \<notin> b"   and "\<Gamma>\<turnstile>c\<^sub>2 \<down> Normal s" shows "\<Gamma>\<turnstile>Cond b c\<^sub>1 c\<^sub>2 \<down> Normal s"
rbt_interwk_is_rbt: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t1 :: "('c, 'd) rbt"   and t2 :: "('c, 'd) rbt"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t1"   and "ord.rbt_sorted less t2" shows "ord.is_rbt less (ord.rbt_inter_with_key less f t1 t2)"
perfect_sec_P2_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool" shows "xor_sim_det.perfect_sec_P2 m1 m2 \<equiv> R2_xor m1 m2 = xor_funct m1 m2 \<bind> (\<lambda>(s1, s2). S2_xor m2 s2)"
point_mult_add: fixes a :: "'a"   and b :: "'a"   and p :: "'a point"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p"   and "nonsingular a b" shows "point_mult a (m + n) p = local.add a (point_mult a m p) (point_mult a n p)"
F2set3_image_Lev2: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xb :: "'e"   and xd :: "bd_type_F"   and x :: "'e" assumes "xe \<in> Lev2 s1 s2 n xb"   and "xe @ [Inr xd] \<in> Lev2 s1 s2 (Suc n) xb"   and "rv2 s1 s2 xe xb = Inr x" shows "xd \<in> tobd_F23 s2 x ` F2set3 (s2 x)"
cases: fixes a1 :: "'b list"   and a2 :: "'c sequent" assumes "upRulesp a1 a2" obtains c :: "'c sequent" and R :: "'c" and Fs :: "'c form list" and ps :: "'b list" where "a1 = ps" and "a2 = c" and "SRCTransforms.mset c \<equiv> \<LM> Compound R Fs  \<RM>" and "ps \<noteq> []"
line_integral_exists_joinD2: fixes f :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "line_integral_exists f basis (g1 +++ g2)"   and "valid_path g2" shows "line_integral_exists f basis g2"
eq_iff: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "(a = b) = (le a b \<and> le b a)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>\<t>[f] : local.map a \<rightarrow> \<Omega>.map b\<guillemotright>"
drop_stake: fixes n :: "nat"   and m :: "nat"   and s :: "'b stream" shows "drop n (stake m s) = stake (m - n) (sdrop n s)"
sqmatrix_add_assoc: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix"   and C :: "('c, 'd) sqmatrix" shows "A + B + C = A + (B + C)"
igSubstAbsIPresIGWlsAbs_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstAbsIPresIGWlsAbs MOD \<equiv> \<forall>ys Y y us s A. isInBar (us, s) \<and> igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSubstAbs MOD ys Y y A)"
distribution_def: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "f \<in> rv M" shows "law M f \<equiv> Measure.measure M \<circ> (-`) f"
eq_onp_def: fixes R :: "'h \<Rightarrow> bool" shows "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
godel_code'_nonzero: fixes nl :: "nat list"   and n :: "nat" shows "0 < godel_code' nl n"
isGB_I_is_red: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "\<And>f. \<lbrakk>f \<in> pmdl G; f \<in> dgrad_p_set d m; f \<noteq> 0\<rbrakk> \<Longrightarrow> is_red G f" shows "is_Groebner_basis G"
ideal_pow_vp: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I"   and "I \<noteq> carrier (Vr K v)"   and "I \<noteq> {\<zero>\<^bsub>Vr K v\<^esub>}" shows "I = vp K v \<^bsup>\<diamondsuit>Vr K v na (n_val K v (Ig K v I))\<^esup>"
extend_act_Id: fixes h :: "'d \<times> 'e \<Rightarrow> 'f" assumes "Extend h" shows "extend_act h Id = Id"
gt_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wt t \<ge>\<^sub>p wt s"   and "head t = head s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "t >\<^sub>t s"
c_tr_failures: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and xs :: "'e list" shows "(c_tr step out s\<^sub>0 xs, {(x, p). p \<noteq> out (foldl step s\<^sub>0 xs) x}) \<in> c_failures step out s\<^sub>0"
l_def: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "o[F](g) = {f. \<forall>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
steps: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c\<^sub>f, s\<^sub>f)"
rel_nondetT'_phantom: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_nondetT' A = rel_nondetT' \<top>"
min_dnf_atoms_dnf: fixes \<Phi> :: "'b ltln fset"   and \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<Phi> \<in> min_dnf \<psi>"   and "\<phi> \<in> fset \<Phi>" shows "dnf \<phi> = {{|\<phi>|}}"
def_hashmap_size_bool_def: shows "def_hashmap_size = (\<lambda>_. 2)"
explode_mem_zet: fixes z :: "ZF" shows "explode z \<in> zet"
hyps_1: fixes gpv :: "('a, 'b, 'c) gpv" shows "interaction_bound' gpv \<le> interaction_bound consider gpv"
distinct_5: fixes x2 :: "dB"   and x31 :: "'a itrm"   and x32 :: "'a itrm" shows "Pure x2 \<noteq> x31 \<diamondop> x32"
domintros_6: fixes e :: "exp0"   and va :: "locn \<times> locn" assumes "is_value_dom e" shows "is_value_dom (Lannot e va)"
rewriteL_comp_comp2: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and l1 :: "'f \<Rightarrow> 'd"   and l2 :: "'c \<Rightarrow> 'f"   and h :: "'g \<Rightarrow> 'c"   and r :: "'g \<Rightarrow> 'f" assumes "f \<circ> g = l1 \<circ> l2"   and "l2 \<circ> h = r" shows "f \<circ> (g \<circ> h) = l1 \<circ> r"
real_add_le_0_iff: fixes x :: "real"   and y :: "real" shows "(x + y \<le> 0) = (y \<le> - x)"
filterlim_at_top_at_top: fixes Q :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'a" assumes "\<And>x y. \<lbrakk>Q x; Q y; x \<le> y\<rbrakk> \<Longrightarrow> f x \<le> f y"   and "\<And>x. P x \<Longrightarrow> f (g x) = x"   and "\<And>x. P x \<Longrightarrow> Q (g x)"   and "eventually Q at_top"   and "eventually P at_top" shows "filterlim f at_top at_top"
map_of_eq_Some_iff: fixes xys :: "('d \<times> 'e) list"   and x :: "'d"   and y :: "'e" assumes "distinct (map fst xys)" shows "(map_of xys x = Some y) = ((x, y) \<in> set xys)"
assms_4: shows "v \<in> ball a r \<inter> T"
simps_5: fixes n :: "nat"   and m :: "nat"   and F :: "'b formula"   and G :: "'b formula" shows "cut_bound n m (F \<^bold>\<or> G) = cut_bound (cut_bound n m F) m G"
intro_nofail_1: fixes S :: "'c nres" shows "(S \<noteq> FAIL) = nofail S"
swap: fixes C :: "('c \<times> 'd) set"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "finite C"   and "{a. \<exists>b. g a b \<noteq> (0::'e)} \<times> {b. \<exists>a. g a b \<noteq> (0::'e)} \<subseteq> C" shows "(\<Sum>a. Sum_any (g a)) = (\<Sum>b. \<Sum>a. g a b)"
inv_comm: fixes x :: "complex vec"   and y :: "complex vec" assumes "x + y = 0\<^sub>v d"   and "x \<in> carrier_vec d"   and "y \<in> carrier_vec d" shows "y + x = 0\<^sub>v d"
simply_connected_eq_contractible_loop_some: fixes S :: "'c set" shows "simply_connected S = (path_connected S \<and> (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a))))"
assms_5: shows "R (xs ! lo) (xs ! (lo + 1))"
entails_def: fixes uu :: "'c itself"   and C :: "'d set"   and T :: "('d Standard_Constant allegorical_term \<times> 'd Standard_Constant allegorical_term) set"   and S :: "'d Standard_Constant allegorical_term \<times> 'd Standard_Constant allegorical_term" shows "entails uu C T S \<equiv> \<forall>G. model C G T \<longrightarrow> G \<Turnstile> S"
prems_1: shows "ab = [a, b]\<^sub>\<circ>"
distinct_length_filter: fixes xs :: "'k list"   and P :: "'k \<Rightarrow> bool" assumes "distinct xs" shows "length (filter P xs) = card ({x. P x} \<inter> set xs)"
intro: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "RModuleHom R smult M smult T"   and "RModuleEnd_axioms M T" shows "RModuleEnd R smult M T"
simps: fixes t :: "'t"   and a :: "'x \<times> 'm" shows "\<tau>diverge t a = (\<exists>s s'. a = s \<and> silent_move t s s' \<and> \<tau>diverge t s')"
independent_mono: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "module scale"   and "\<not> module.dependent scale A"   and "B \<subseteq> A" shows "\<not> module.dependent scale B"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "R.seq g f" shows "R.dom (R.comp g f) = R.dom f"
sel: fixes x :: "char list" shows "iface_sel (Iface x) = x"
limit_range_suffix_incr: fixes r :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "limit r = range (suffix i r)"   and "i \<le> j" shows "limit r = range (suffix j r)"
v_props_2: shows "\<gamma> = \<V> \<union> two_chain_horizontal_boundary two_chain"
rec_newrght_def: shows "rec_newrght = (let g0 = Cn 3 rec_newrgt0 [recf.id 3 0, recf.id 3 1]; g1 = Cn 3 rec_newrgt1 [recf.id 3 0, recf.id 3 1]; g2 = Cn 3 rec_newrgt2 [recf.id 3 0, recf.id 3 1]; g3 = Cn 3 rec_newrgt3 [recf.id 3 0, recf.id 3 1]; g4 = recf.id 3 1; r0 = Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 0) [recf.id 3 0]]; r1 = Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 1) [recf.id 3 0]]; r2 = Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 2) [recf.id 3 0]]; r3 = Cn 3 rec_eq [recf.id 3 2, Cn 3 (constn 3) [recf.id 3 0]]; r4 = Cn 3 rec_less [Cn 3 (constn 3) [recf.id 3 0], recf.id 3 2]; gs = [g0, g1, g2, g3, g4]; rs = [r0, r1, r2, r3, r4] in rec_embranch (zip gs rs))"
obtain_point_with_rep: fixes r :: "int" assumes "r \<in> d.replication_numbers" shows "\<exists>x. x \<in> \<V> \<and> \<B>\<^sup>C rep x = r"
liftAll_lift_comp: fixes phi :: "'a \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and inp :: "'b \<Rightarrow> 'c option" shows "liftAll phi (lift f inp) = liftAll (phi \<circ> f) inp"
norm_euclidean_space_def: fixes x :: "'a euclidean_space" shows "norm x = sqrt (x \<bullet> x)"
mult_mat_vec_carrier: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and v :: "'a Matrix.vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "A *\<^sub>v v \<in> carrier_vec nr"
assms_1: shows "v \<notin> subst_domain \<I>"
hvzk_z1_r1_tuple_rewrite: fixes r1 :: "nat"   and r2 :: "nat"   and c :: "nat"   and x1 :: "nat"   and x2 :: "nat" assumes "r1 < order \<G>" shows "(\<^bold>g [^] r1 \<otimes> g' [^] r2, c, (r1 + c * x1) mod order \<G>, (r2 + c * x2) mod order \<G>) = (\<^bold>g [^] (((r1 + c * x1) mod order \<G> + order \<G> * c * x1 - c * x1) mod order \<G>) \<otimes> g' [^] r2, c, (r1 + c * x1) mod order \<G>, (r2 + c * x2) mod order \<G>)"
IH: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and vs :: "'a set" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "exec_plan s (as_proj as vs) \<in> valid_states PROB"
a: shows "Vagree \<omega> \<omega>' (FVT \<theta>)"
ALG_Pxy_def: fixes Strat :: "(nat \<times> nat list) list"   and qs :: "'b list"   and init :: "'b list"   and x :: "'b"   and y :: "'b" shows "ALG_Pxy Strat qs init x y = (\<Sum>i<length qs. ALG_P' Strat qs init i x y)"
col_AAB: fixes A :: "Points"   and B :: "Points" shows "col A A B"
Says_Server_message_form: fixes A :: "agent"   and K' :: "nat"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and Ticket :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs"   and "evs \<in> bankerberos" shows "K' = shrK A \<and> K \<notin> range shrK \<and> Ticket = Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<and> Key K \<notin> used before Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) on evs \<and> Tk = CT before Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) on evs"
k: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>G\<in>(\<lambda>x. ball x (d x (e / 2))) ` S. ball x k \<subseteq> G"
Inter_nonempty_distrib: fixes A :: "'b set set"   and B :: "'b set" assumes "A \<noteq> {}" shows "\<Inter> A \<inter> B = (\<Inter>C\<in>A. C \<inter> B)"
SD_preorderD_2: fixes q :: "'a pmf"   and p :: "'a pmf" assumes "SD le q p" shows "q \<in> lotteries_on carrier"