hcju/isabelleps huggingface dataset:
equiv: fixes F :: "'edge set" assumes "F \<subseteq> E" shows "equiv V (connected F)"
adds_term_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_term pair_of_term u v = (term_powerprod.component_of_term pair_of_term u = term_powerprod.component_of_term pair_of_term v \<and> term_powerprod.pp_of_term pair_of_term u adds term_powerprod.pp_of_term pair_of_term v)"
of_int_sint: fixes a :: "'b Word.word" shows "of_int (sint a) = signed a"
term_subst_idle: fixes y :: "Syntax.var"   and t :: "Syntax.term"   and s :: "Syntax.term" assumes "atom y \<sharp> t" shows "t[s / y] = t"
keys_def: shows "keys \<equiv> map_fun mapping_rbt.impl_of id RBT_Impl.keys"
deep_model_correct_params_y_axioms: shows "deep_model_correct_params_y rs y"
preserves_arr: fixes f :: "'a" assumes "S.arr f" shows "S.arr (R f)"
perm_simps_3: fixes p :: "perm"   and tm1 :: "tm"   and tm2 :: "tm" shows "p \<bullet> Eats tm1 tm2 = Eats (p \<bullet> tm1) (p \<bullet> tm2)"
sfg_free: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "sfg A a"   and "n \<noteq> m" shows "n\<triangleright>a\<^bsub>A\<^esub> \<noteq> m\<triangleright>a\<^bsub>A\<^esub>"
res_lim_pushforward: fixes p :: "int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p"   and "padic_integers.is_Zp_continuous p f"   and "padic_integers.is_Zp_cauchy p s"   and "t = alt_seq s" shows "res_lim (f \<circ> t) = f (res_lim t)"
lift_xs: shows "lift 0 xs (replicate ar None) = Ok (ys, [None])"
fps_nth_conv_deriv: fixes f :: "complex fps"   and n :: "nat" assumes "0 < fps_conv_radius f" shows "fps_nth f n = (deriv ^^ n) (eval_fps f) 0 / fact n"
connectedD: fixes "open" :: "'a set \<Rightarrow> bool"   and A :: "'a set"   and U :: "'a set"   and V :: "'a set" assumes "class.topological_space open"   and "topological_space.connected open A"   and "open U"   and "open V"   and "U \<inter> V \<inter> A = {}"   and "A \<subseteq> U \<union> V" shows "U \<inter> A = {} \<or> V \<inter> A = {}"
simps: fixes a1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and a2 :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and a3 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" shows "(a1 \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>a2\<^esub> a3) = (\<exists>c T c'. a1 = c \<and> a2 = T \<and> a3 = c' \<and> c \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> c' \<and> (\<forall>c'\<in>set T. stack c \<lesssim> stack c') \<and> stack c' = stack c)"
bind_tm_def: fixes s :: "'b tm"   and f :: "'b \<Rightarrow> 'a tm" shows "s \<bind> f = (case s of TM u m \<Rightarrow> case f u of TM v n \<Rightarrow> TM v (m + n))"
simps_2: fixes ok :: "nat"   and f' :: "'b list"   and r' :: "'b list" shows "rem_steps (App ok f' r') = ok + 1"
return_time_cong: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and x :: "'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x" shows "Y.return_time A x = auto_ll_on_open.return_time g Y A x"
simps_5: fixes M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and \<rho> :: "complex mat" shows "denote (While M S) \<rho> = denote_while (M 0) (M 1) (denote S) \<rho>"
simps_5: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and va :: "32 word" shows "heap_base.typeof_h typeof_addr h (Intg va) = typeof (Intg va)"
lattice_of_add: fixes fs :: "rat vec list"   and i :: "nat"   and j :: "nat"   and gs :: "rat vec list"   and l :: "int" assumes "set fs \<subseteq> Rn"   and "i < length fs"   and "j < length fs"   and "i \<noteq> j"   and "gs = fs[i := fs ! i + rat_of_int l \<cdot>\<^sub>v fs ! j]" shows "gs.lattice_of gs = gs.lattice_of fs"
surjective: fixes r :: "('b, 'c) partial_object_scheme" shows "r = \<lparr>carrier = carrier r, \<dots> = partial_object.more r\<rparr>"
igSwapAllIPresIGWlsAll_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSwapAllPresGWlsAll MOD" shows "igSwapAllIPresIGWlsAll (fromMOD MOD)"
cInf_lessD: 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 X :: "'a set"   and z :: "'a" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "less (Inf X) z" shows "\<exists>x\<in>X. less x z"
eq_integrals: shows "one_chain_line_integral F {j} \<V> = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
backward_terminating_iff3_aux: fixes x :: "'a" assumes "path x" shows "x\<^sup>T ; 1 \<cdot> 1 ; x\<^sup>T \<cdot> - (1 ; x) \<le> x\<^sup>T\<^sup>\<star>"
inf_le1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) x"
contg: shows "continuous_on (\<Union> \<F> - C) g"
of_rat_lec_zero: shows "of_rat_lec 0 = 0"
prod_mset_prod_list: fixes xs :: "'a list" shows "\<Prod>\<^sub># (mset xs) = prod_list xs"
simps_11: fixes b :: "'a"   and x :: "'a"   and c :: "'a"   and C :: "'a tree"   and AB :: "'a tree"   and D :: "'a tree" assumes "b < x"   and "x < c"   and "C \<noteq> \<langle>\<rangle>" shows "splay x \<langle>AB, b, \<langle>C, c, D\<rangle>\<rangle> = (case splay x C of \<langle>C1, c', C\<rangle> \<Rightarrow> \<langle>\<langle>AB, b, C1\<rangle>, c', \<langle>C, c, D\<rangle>\<rangle>)"
bit_double_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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (times (numeral.numeral one plus (num.Bit0 num.One)) a) n = (bit a (n - 1) \<and> n \<noteq> 0 \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
length_not_greater_eq_2_iff: shows "(\<not> 2 \<le> LENGTH('b)) = (LENGTH('b) = 1)"
Inr: shows "info_real_alg x = Inr (p, n)"
3: assumes "ll \<noteq> \<langle>\<rangle>" shows "real (T_dm ll) + \<Phi> (del_min ll) - \<Phi> ll \<le> 2 * \<phi> ll + 1"
djointish: shows "pairwise (\<lambda>x y. negligible (f x \<inter> f y)) S"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat, 'b) fmap" shows "fmlookup (swap\<^sub>f x xa xb) = (\<lambda>xc. fmlookup xb (swap x xa xc))"
sum_left_ideal: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "class.ring plus zero minus uminus times"   and "ring.left_ideal plus zero uminus times X"   and "U \<subseteq> X"   and "finite U" shows "comm_monoid_add.sum plus zero (\<lambda>i. times (f i) i) U \<in> X"
T_vrestr_0_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.vimage_restr M (T ^^ 0) A = A"
pars__os3412: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "C D OS A B"
tl_cCons: fixes x :: "'b"   and xs :: "'b list" shows "tl (x ## xs) = xs"
exec_step_ind_Invokestatic_Called: fixes ps :: "val list"   and n :: "nat"   and stk :: "val list"   and D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and m :: "jvm_method"   and f' :: "frame"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and frs :: "frame list"   and sh :: "sheap" assumes "ps = take n stk"   and "(D, b, Ts, T, mxs, mxl\<^sub>0, ins, xt) = method P C M"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D"   and "f' = ([], rev ps @ replicate mxl\<^sub>0 undefined, D, M, 0, No_ics)" shows "exec_step_ind (StepI (Invokestatic C M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc (Called Cs) frs sh (None, h, f' # (stk, loc, C\<^sub>0, M\<^sub>0, pc, No_ics) # frs, sh)"
union_lgrid_eq: shows "{p} \<union> (\<Union>d\<in>{d \<in> ds. d < length p}. lgrid (child p left d) {d' \<in> ds. d' \<le> d} (level p + Suc l) \<union> lgrid (child p right d) {d' \<in> ds. d' \<le> d} (level p + Suc l)) = lgrid p ds (level p + Suc l)"
in_epsclosure_steps: fixes p :: "'b"   and q :: "'b"   and A :: "'b \<times> ('c option \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and w :: "'c list"   and r :: "'b" assumes "(p, q) \<in> NAe.steps A w"   and "(q, r) \<in> (eps A)\<^sup>*" shows "(p, r) \<in> NAe.steps A w"
maintained_holds_iff: fixes G :: "('c, 'd) labeled_graph"   and e\<^sub>L :: "'c allegorical_term"   and e\<^sub>R :: "'c allegorical_term" assumes "graph G" shows "maintained (translation e\<^sub>L, translation (A_Int e\<^sub>L e\<^sub>R)) G = G \<Turnstile> e\<^sub>L \<sqsubseteq> e\<^sub>R"
isomorphic_implies_hpar_1: fixes f :: "'a"   and f' :: "'a" assumes "S.isomorphic f f'" shows "S.ide f"
h3b: assumes "\<forall>x y. (\<Sum>i\<le>n. k i x * y ^ i) = 0" shows "\<exists>y. \<forall>x<y. (\<Sum>i\<le>n. k i x * x ^ i) = 0"
vrrestriction_vsubset_rel: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>r\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> r"
restricted_agree_imp_agree: fixes s2 :: "('d, 'e) fmap"   and vs :: "'d set"   and s1 :: "('d, 'e) fmap" assumes "fmdom' s2 \<subseteq> vs"   and "agree (fmrestrict_set vs s1) s2" shows "agree s1 s2"
vrestr_UN: fixes f :: "'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "f --` \<Union> (B ` A) = (\<Union>x\<in>A. f --` B x)"
equivalence_map_def: fixes f :: "'a" shows "equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. internal_equivalence f g \<eta> \<epsilon>"
\<phi>: shows "\<guillemotleft>\<phi> : src f \<Rightarrow> w' \<star> w\<guillemotright> \<and> local.iso \<phi>"
sumdig_def: fixes n :: "nat" shows "sumdig n = (\<Sum>x<nlen n. n div 10 ^ x mod 10)"
denote_measure_expand: fixes m :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and \<rho> :: "complex mat" assumes "m \<le> n"   and "well_com (Measure n M S)" shows "denote (Measure m M S) \<rho> = matrix_sum d (\<lambda>k. denote (S ! k) (M k * \<rho> * adjoint (M k))) m"
adj_not_same: fixes a :: "'a" assumes "(a, a) \<in> parcs slim" shows "False"
lemScale0: fixes u :: "'a Vector" shows "(0::'a) ** u = 0"
healthy_meet_cont: fixes P :: "'a hrel"   and Q :: "'a hrel" assumes "P is \<H>"   and "Q is \<H>" shows "P \<^bold>\<sqinter> Q = P \<sqinter> Q"
N_triangle_ineq: fixes n :: "int"   and m :: "int" shows "N (n + m) \<le> N n + N m"
simps: fixes x :: "'a"   and t :: "'a tree"   and n :: "nat"   and dl :: "nat" shows "insert2_d x (t, n, dl) = (case ins2 (n + dl) 0 x t of Same2 \<Rightarrow> (t, n, dl) | Bal2 t' \<Rightarrow> (t', n + 1, dl))"
assms: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and m :: "nat"   and data :: "nat \<times> 'b"   and p :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and q :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "dickson_grading d"   and "fst ` (set gs \<union> set bs \<union> set hs) \<subseteq> dgrad_p_set d m"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "crit data gs bs hs p q" shows "crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs \<union> set hs)) (fst p) (fst q)"
op_dg_components_2: fixes \<CC> :: "V" shows "op_dg \<CC>\<lparr>Arr\<rparr> = \<CC>\<lparr>Arr\<rparr>"
Bot_def: fixes po :: "'b potype" shows "Bot po = least (\<lambda>x. True) po"
fst_snd: shows "fst (snd f) = fst (snd g)"
minimum_spanning_forest_def: fixes F :: "('a, 'b) graph"   and G :: "('a, 'b) graph" shows "minimum_spanning_forest F G \<equiv> spanning_forest F G \<and> optimal_forest F G"
norm_V_preservation': fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> \<subseteq> V"   and "canonical M n"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "[DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> \<subseteq> V"
H_tensor_iter_tensor_of_H: fixes n :: "nat" shows "H \<Otimes> Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>Suc n\<^esub>  j / sqrt 2 ^ Suc n))"
semIntAbs_prSwapAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prSwapAbs (semIntAbs SEM) SEM"
ord_eq_le_trans: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "a = b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
agree': shows "Vagree \<nu> \<nu>' (\<Union> (primify ` FVT t1) \<union> \<Union> (primify ` FVT t2))"
kat_prop: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<cdot> y \<le> y \<cdot> d z) = (d x \<cdot> y \<cdot> ad z = (0::'a))"
Qigen_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'b option"   and u :: "'v"   and adjs :: "('v \<times> 'c) list"   and v :: "'v" shows "Qigen Q \<pi> u adjs v = (if v \<notin> fst ` set adjs then Q v else Qinter Q \<pi> u v)"
sub1: shows "P \<turnstile> C \<prec>\<^sup>1 D"
add_leaves_bdd_all_eq': fixes P :: "'a \<Rightarrow> bool"   and tr :: "'a bdd" shows "bdd_all P tr = list_all P (add_leaves tr [])"
ide_cod: fixes f :: "'a2" assumes "C2.arr f" shows "C2.ide (C2.cod f)"
interleaves_prefix_fst: fixes ws :: "'b list"   and P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "\<And>n. n < length ws \<Longrightarrow> P (ws ! n) (drop (Suc n) ws @ xs)" shows "xs \<simeq> {ys, zs, P} = ws @ xs \<simeq> {ws @ ys, zs, P}"
bind_lub_spmf: fixes Y :: "'c spmf set"   and f :: "'c \<Rightarrow> 'd spmf" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y" shows "lub_spmf Y \<bind> f = lub_spmf ((\<lambda>p. p \<bind> f) ` Y)"
fset_of_list_remdups: fixes l :: "'b list" shows "fset_of_list (remdups l) = fset_of_list l"
Arcsin_in_Reals: fixes z :: "complex" assumes "z \<in> \<real>"   and "\<bar>Re z\<bar> \<le> 1" shows "Arcsin z \<in> \<real>"
echelon_form_of_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "echelon_form_of A bezout = echelon_form_of_upt_k A (ncols A - 1) bezout"
t_gr: shows "\<forall>t'>t. local.prefixes t' j \<up>"
cod_char: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.cod f = (if \<F>\<^sub>SC.arr f then \<F>C.cod f else \<F>C.null)"
A3: shows "\<forall>TP SQ. TP \<in>T P \<and> SQ \<in>S Q \<longrightarrow> False"
Quotient_crel_typecopy: fixes Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient (=) Abs Rep T" shows "T \<equiv> \<lambda>x y. x = Rep y"
x_u: shows "x = (1 - u) *\<^sub>R lb + u *\<^sub>R ub"
lm155: fixes f :: "('a \<times> 'b) set"   and X :: "'a set" assumes "runiq f"   and "X \<subseteq> Domain f" shows "graph X (toFunction f) = f || X"
List_def: fixes h :: "'b \<Rightarrow> 'b ref"   and x :: "'b ref"   and as :: "'b list" shows "List h x as = Path h x as Null"
first_summand_in: fixes x :: "i"   and y :: "i" shows "P x (x \<oplus> y)"
Joints_ne_empty_2: fixes xs :: "('a \<times> 'a pdevs) list" shows "{} \<noteq> Joints xs"
hseq_char: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "B.hseq \<nu> \<mu> = (B.arr \<mu> \<and> B.arr \<nu> \<and> src\<^sub>B \<nu> = trg\<^sub>B \<mu>)"
Int_absorb2: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> B" shows "A \<inter> B = A"
wf\<^sub>s\<^sub>t_simple_induct: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and P :: "('d, 'c) strand_step list \<Rightarrow> bool" assumes "wf\<^sub>s\<^sub>t V S"   and "simple S"   and "P []"   and "\<And>v S. \<lbrakk>wf\<^sub>s\<^sub>t V S; simple S; P S\<rbrakk> \<Longrightarrow> P (S @ [send\<langle>Var v\<rangle>\<^sub>s\<^sub>t])"   and "\<And>t S. \<lbrakk>wf\<^sub>s\<^sub>t V S; simple S; P S; fv t \<subseteq> V \<union> \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S))\<rbrakk> \<Longrightarrow> P (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"   and "\<And>X F S. \<lbrakk>wf\<^sub>s\<^sub>t V S; simple S; P S\<rbrakk> \<Longrightarrow> P (S @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])" shows "P S"
transitive_stepwise_le: fixes m :: "nat"   and n :: "nat"   and R :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "m \<le> n"   and "\<And>x. R x x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"   and "\<And>n. R n (Suc n)" shows "R m n"
choiceFn_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "choiceFn A r \<equiv> \<forall>A'\<subseteq>A. A' \<noteq> {} \<longrightarrow> choiceSet A' r \<noteq> {}"
f: shows "f \<in> state_measure V' \<Gamma> \<rightarrow>\<^sub>M subprob_algebra N"
concat_eq_Nil_conv: fixes xss :: "'g list list" shows "(concat xss = []) = (\<forall>xs\<in>set xss. xs = [])"
horner_schema: fixes f :: "nat \<Rightarrow> nat"   and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and F :: "nat \<Rightarrow> nat"   and s :: "nat"   and n :: "nat"   and j' :: "nat"   and x :: "real" assumes "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)" shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum>j = 0..<n. (- 1) ^ j * (1 / real (f (j' + j))) * x ^ j)"
simps_1: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and I :: "('b, 'c, 'a) interp"   and uu :: "'b"   and i :: "'a"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and x :: "(real, 'a) vec" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.extendf_deriv I uu (Var i) \<nu> x = (\<lambda>_. 0)"
le_shiftr: fixes u :: "'b word"   and v :: "'b word"   and n :: "nat" assumes "u \<le> v" shows "u >> n \<le> v >> n"
simps_2: fixes edge_rev :: "'e \<Rightarrow> 'e"   and edge_succ :: "'e \<Rightarrow> 'e"   and more :: "'f" shows "edge_succ \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_succ"
det_addrow_mat: fixes k :: "nat"   and l :: "nat"   and n :: "nat"   and a :: "'b" assumes "k \<noteq> l" shows "det (addrow_mat n a k l) = (1::'b)"
antisym_conv: fixes y :: "('c, 'd) Term.term"   and x :: "('c, 'd) Term.term" assumes "y \<sqsubseteq> x" shows "(x \<sqsubseteq> y) = (x = y)"
ltl_simps_2: fixes x21 :: "'a"   and x22 :: "'a llist" shows "ltl (LCons x21 x22) = x22"
edges_of_comp2: fixes ys :: "'a list"   and xs :: "'a list" shows "edges_of_walk ys \<subseteq> edges_of_walk (xs @ ys)"
assms_1: shows "equivalent_on \<sigma> \<eta> (vars_of_cl C) I"
adm: shows "ccpo.admissible lSup (\<sqsubseteq>) P"
span_is_subset: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N V" shows "local.module.span S \<subseteq> N"
r_one: fixes x :: "'a mod_ring poly" assumes "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> = x"
convergent_subseq_res': fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and l :: "nat"   and n :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "l < n" shows "convergent_subseq s n l = convergent_subseq s l l"
hs_R_def: fixes post :: "'a \<Rightarrow> 'a dlist" shows "hs_R post \<equiv> {(q, q'). q' \<in> ls.\<alpha> (post q)}"
extractableE: fixes X :: "msg"   and K :: "msg set"   and I :: "msg set" assumes "X \<in> extractable K I" obtains A :: "agent" and B :: "agent" and M :: "msg" where "X = implInsec A__ B__ M__" and "X \<in> I"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implAuth Aa__ Ba__ Ma__" and "X \<in> I"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implConfid Ab__ Bb__ Mb__" and "X \<in> I" and "(Ab__, Bb__) \<in> broken K"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implSecure A B M" and "X \<in> I" and "(A, B) \<in> broken K"
PPsD: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set" assumes "t \<in> .[X]" shows "keys t \<subseteq> X"
hyps_3: fixes b :: "binding list" assumes "\<turnstile> b ok"   and "b extends \<Gamma>" shows "b\<turnstile>T\<^sub>1<:S\<^sub>1"
take_def: shows "take \<equiv> \<lambda>uu uua. rec_list (\<lambda>n. []) (\<lambda>x xs xsa n. case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # xsa m) uua uu"
simps_3: fixes ao :: "'a option"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "set_less_eq_aux_Compl ao (x # xs) (y # ys) = (if x < y then proper_interval ao (Some x) \<or> set_less_eq_aux_Compl (Some x) xs (y # ys) else if y < x then proper_interval ao (Some y) \<or> set_less_eq_aux_Compl (Some y) (x # xs) ys else proper_interval ao (Some y))"
nz: fixes z :: "complex"   and n :: "nat" assumes "z \<in> S" shows "\<F> n z \<noteq> 0"
perm_assoc_switch_r: fixes as :: "'a list"   and bs :: "'a list"   and cs :: "'a list" assumes "as <~~> bs"   and "bs [\<sim>] cs" shows "\<exists>bs'. as [\<sim>] bs' \<and> bs' <~~> cs"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "uniformly_continuous_on (closure X) g"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x"   and "\<And>Y h x. \<lbrakk>X \<subseteq> Y; Y \<subseteq> closure X; continuous_on Y h; \<And>x. x \<in> X \<Longrightarrow> f x = h x; x \<in> Y\<rbrakk> \<Longrightarrow> h x = g x" shows "thesis"
splitset_def: fixes xs :: "'c list" shows "splitset xs \<equiv> {(p, q). xs = p @ q}"
abs_diff_triangle_ineq2: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "\<bar>a - b\<bar> \<le> \<bar>a - c\<bar> + \<bar>c - b\<bar>"
eq_fin_concat_eq_fin_start: fixes u :: "'item list"   and v\<^sub>1 :: "'item list"   and v\<^sub>2 :: "'item list" assumes "u @ v\<^sub>1 =\<^sub>F u @ v\<^sub>2" shows "v\<^sub>1 =\<^sub>F v\<^sub>2"
put_get: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'e" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
1: shows "distributed M lborel (\<lambda>x. - \<mu> + X x) (\<lambda>x. ennreal (normal_density 0 \<sigma> x))"
uJ_proj: fixes n :: "nat"   and x :: "'a" shows "u n x = uJ n (projJ x)"
diff_freshAbs_vsubstAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "goodAbs A"   and "y \<noteq> y1" shows "freshAbs ys y (A $[y1 // y]_ys)"
assm: shows "\<parallel>len v ts c\<parallel> = 0 \<and> v' \<le> v"
wls_swapAbs_sym: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wlsAbs (us, s) A" shows "A $[x \<and> y]_zs = A $[y \<and> x]_zs"