hcju/isabelleps huggingface dataset:
A_trusts_secure_authenticator: fixes K :: "nat"   and A :: "agent"   and T :: "nat"   and evs :: "event list" assumes "Crypt K \<lbrace>Agent A, Number T\<rbrace> \<in> parts (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV" shows "\<exists>B X. Says A Tgs \<lbrace>X, Crypt K \<lbrace>Agent A, Number T\<rbrace>, Agent B\<rbrace> \<in> set evs \<or> Says A B \<lbrace>X, Crypt K \<lbrace>Agent A, Number T\<rbrace>\<rbrace> \<in> set evs"
at_set_avoiding: fixes Xs :: "'a set"   and c :: "'b" assumes "at TYPE('a)"   and "finite Xs"   and "finite (supp c)" obtains pi :: "('a \<times> 'a) list" where "(pi \<bullet> Xs) \<sharp>* c" and "set pi \<subseteq> Xs \<times> pi \<bullet> Xs" and "distinctPerm pi"
f13: shows "(s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r''), s \<otimes> s'' \<otimes> (s' \<otimes> s'')) \<in> carrier rel"
card2: shows "card {p. p partitions Suc m \<and> sum p {..Suc m} = Suc k \<and> p 1 = 0} = Partition (m - k) (Suc k)"
exhaust_sel: fixes complex_interval :: "complex_interval" obtains "complex_interval = Complex_Interval (Re_interval complex_interval) (Im_interval complex_interval)"
f2: shows "((\<exists>i ia. 0 \<le> i \<and> \<not> 0 \<le> i + - 1 * ia \<and> i mod ia \<noteq> i) \<or> \<not> 0 \<le> - 1 + 2 ^ LENGTH('b) \<or> 0 \<le> - 1 + 2 ^ LENGTH('b) + - 1 * 2 ^ LENGTH('b) \<or> (- 1 + 2 ^ LENGTH('b)) mod 2 ^ LENGTH('b) = - 1 + 2 ^ LENGTH('b)) = ((\<exists>i ia. 0 \<le> i \<and> \<not> 0 \<le> i + - 1 * ia \<and> i mod ia \<noteq> i) \<or> \<not> 1 \<le> 2 ^ LENGTH('b) \<or> 2 ^ LENGTH('b) + - 1 * ((- 1 + 2 ^ LENGTH('b)) mod 2 ^ LENGTH('b)) = 1)"
FF_def: shows "DN.FF \<equiv> \<lambda>\<mu>\<nu>. if VV.arr \<mu>\<nu> then (DN (fst \<mu>\<nu>), DN (snd \<mu>\<nu>)) else B.VV.null"
f13a: shows "\<not> (\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
cardOneImageCardOne: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "card A = 1" shows "card (f ` A) = 1"
sum_7: fixes f :: "7 \<Rightarrow> 'b" shows "sum f UNIV = f 1 + f 2 + f 3 + f 4 + f 5 + f 6 + f 7"
zeta_neg_of_nat: shows "zeta (- of_nat 1) = - complex_of_real (bernoulli' (Suc 1)) / of_nat (Suc 1)"
antipar_2: shows "src (tab\<^sub>0 g \<star> Tfg.p\<^sub>0)\<^sup>* = trg (tab\<^sub>0 g \<star> Tfg.p\<^sub>0)"
NTCompDefinedE: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "\<eta>1 \<approx>>\<bullet> \<eta>2" obtains "NatTrans \<eta>1" and "NatTrans \<eta>2" and "NTCatDom \<eta>2 = NTCatDom \<eta>1" and "NTCatCod \<eta>2 = NTCatCod \<eta>1" and "NTCod \<eta>1 = NTDom \<eta>2"
lambdas2: fixes l :: "nat"   and vs :: "(nat \<times> char list) list"   and c :: "call"   and x :: "lambda" assumes "Lambda l vs c \<in> lambdas x" shows "l \<in> labels x"
simps_20: shows "cospan_shape.TT \<noteq> cospan_shape.BB"
graph_def: fixes n :: "nat" shows "graph n \<equiv> let vs = [0..<n]; fs = [Face vs Final, Face (rev vs) Nonfinal] in Graph fs n (replicate n fs) (replicate n 0)"
guessed_frame_Init_xgnx: fixes R :: "fid_t" assumes "role (guessed_runs R) = Init" shows "guessed_frame R xgnx = Some (Exp Gen (NonceF (R $ nx)))"
Gramian_determinant_2: fixes k :: "nat" assumes "k \<le> m" shows "0 < gs.Gramian_determinant fs k"
gfinprod_cong: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and g :: "'b \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "A = B"   and "f \<in> B \<rightarrow> carrier (homology_group p X)"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "gfinprod (homology_group p X) f A = gfinprod (homology_group p X) g B"
int_shiftl_BIT_2: fixes x :: "int"   and n :: "nat" shows "x << Suc n = 2 * (x << n)"
elims: fixes x :: "'h \<Rightarrow> 'g list"   and xa :: "'h LDCFG_node"   and y :: "'g list" assumes "lift_ParamDefs x xa = y" obtains ParamDefs :: "'h \<Rightarrow> 'g list" and n :: "'h" where "x = ParamDefsa__" and "xa = Node n" and "y = ParamDefsa__ n"   | ParamDefs :: "'h \<Rightarrow> 'g list" where "x = ParamDefsb__" and "xa = NewEntry" and "y = []"   | ParamDefs :: "'h \<Rightarrow> 'g list" where "x = ParamDefs" and "xa = NewExit" and "y = []"
finite_set_to_list: fixes Y :: "'a set" shows "\<exists>xs. distinct xs \<and> set xs = Y"
reparam_weak_eq_refl: fixes \<gamma>1 :: "real \<Rightarrow> 'a" shows "reparam_weak \<gamma>1 \<gamma>1"
of_int_le_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus z) zero = (z \<le> 0)"
\<C>_vars_subset_\<C>: fixes x :: "'Var" shows "\<C>_vars x \<subseteq> \<C>"
trans_less_nmultiset: fixes X :: "'b nmultiset"   and Y :: "'b nmultiset"   and Z :: "'b nmultiset" assumes "X < Y"   and "Y < Z" shows "X < Z"
matrix_to_iarray_Gram_Schmidt_matrix_efficient: fixes A :: "((real, 'a) vec, 'b) vec" shows "matrix_to_iarray (Gram_Schmidt_matrix A) = Gram_Schmidt_matrix_iarrays_efficient (matrix_to_iarray A)"
odd_p: fixes p :: "nat" assumes "noninert_gauss_int_prime p" shows "odd p"
semigroup_axioms: shows "Type_Semigroups.semigroup f\<^sub>a"
prv_exi_cong: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var"   and "prv (imp \<phi> \<chi>)" shows "prv (imp (exi x \<phi>) (exi x \<chi>))"
simps_4: fixes v :: "'b" shows "toplevel_summands (Atom v) = {Atom v}"
assms: shows "learn_cp \<psi> U (\<phi> i)"
perm_sym: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <~~> ys" shows "ys <~~> xs"
inc_i_def: fixes ba :: "'b list \<times> 'b list" shows "inc_i ba = (case ba of (b, a) \<Rightarrow> (hd a # b, tl a))"
DOM_cod: fixes f :: "'c local.term set" assumes "arr f" shows "DOM (cod f) = COD f"
vec_inv_closed: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "vec_inv v \<in> carrier_vec n"
is_lasso_def: fixes G :: "('e, 'f) igb_graph_rec_scheme"   and L :: "'e lasso" assumes "igb_graph G" shows "igb_graph.is_lasso G L \<equiv> graph.is_lasso_pre G L \<and> (\<forall>i<igbg_num_acc G. \<exists>q\<in>set (lasso_cycle L). i \<in> igbg_acc G q)"
cases: fixes a :: "'d"   and ps :: "'d \<times> 'b" assumes "a \<in> scope ps" obtains v :: "'d" where "a = v" and "v |\<in>| vertices" and "\<And>pth v'. \<lbrakk>path v v' pth; terminal_vertex v'\<rbrakk> \<Longrightarrow> ps \<in> snd ` set pth"
assms_4: shows "(\<lambda>x. ln (b x)) \<in> o(f)"
image_cong_simp: fixes M :: "'c set"   and N :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "M = N"   and "\<And>x. x \<in> N =simp=> f x = g x" shows "f ` M = g ` N"
in_set_independent_pdevsE: fixes y :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "y \<in> set (independent_pdevs xs)" obtains x :: "real \<times> real" where "x \<in> set xs" and "coll 0 x y"
converse_join: fixes r :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "(sup r s)\<inverse>\<inverse> = sup r\<inverse>\<inverse> s\<inverse>\<inverse>"
before_id: fixes xs :: "'c list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "xs ! i < xs ! j in xs = (i < j)"
ir_valid_decomp1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P c c' Q" shows "ir_valid P c SKIP (decomp P c c' Q) \<and> ir_valid (decomp P c c' Q) SKIP c' Q"
replicate_mset_subset_iff_lt: fixes m :: "nat"   and x :: "'b"   and n :: "nat" shows "(replicate_mset m x \<subset># replicate_mset n x) = (m < n)"
assms: fixes s :: "'a" assumes "fds_converges g s" shows "fds_converges f s"
unique_ctran_Await: fixes x :: "('b com option \<times> 'b) list"   and b :: "'b set"   and c :: "'b com"   and s :: "'b"   and i :: "nat"   and j :: "nat" assumes "x \<in> cptn"   and "x ! 0 = (Some (Await b c), s)"   and "Suc i < length x"   and "x ! i -c\<rightarrow> x ! Suc i"   and "Suc j < length x"   and "i \<noteq> j" shows "x ! j -e\<rightarrow> x ! Suc j"
induct: fixes P :: "('b \<Rightarrow> 'b list \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list"   and a3 :: "'b list" assumes "\<And>P x xs y ys z zs. \<lbrakk>P x xs \<Longrightarrow> P P xs ys (z # zs); \<not> P x xs \<Longrightarrow> P P xs (y # ys) zs\<rbrakk> \<Longrightarrow> P P (x # xs) (y # ys) (z # zs)"   and "\<And>P x xs y ys. P P xs ys [] \<Longrightarrow> P P (x # xs) (y # ys) []"   and "\<And>P x xs z zs. P P xs [] zs \<Longrightarrow> P P (x # xs) [] (z # zs)"   and "\<And>uu_ uv_ uw_. P uu_ (uv_ # uw_) [] []"   and "\<And>ux_ uy_ uz_ va_. P ux_ [] (uy_ # uz_) va_"   and "\<And>vb_ vd_ ve_. P vb_ [] [] (vd_ # ve_)"   and "\<And>vf_. P vf_ [] [] []" shows "P a0 a1 a2 a3"
greek_less_app_mono2: fixes r :: "('a \<times> 'a) set"   and s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list"   and p :: "(accent \<times> 'a) list" assumes "trans r"   and "(s, t) \<in> greek_less r" shows "(p @ s, p @ t) \<in> greek_less r"
l_coset_swap: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <# H"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> y <# H"
cyc_free_diag_dest': fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len m i i xs"   and "i \<le> n" shows "(0::'b) \<le> m i i"
NI_bind_def: shows "NI_bind \<equiv> nondetM_base.bind_nondet bind_id merge_id"
extreme_point_not_in_interior: fixes x :: "'b"   and S :: "'b set" assumes "x extreme_point_of S" shows "x \<notin> interior S"
rep_eq: fixes x :: "'d pmf" shows "rcset (cset_pmf x) = set_pmf x"
in_pairs_sorted_list_of_set_in_set: fixes AAA :: "'c set set"   and AB_pairs :: "('c \<times> 'c) list"   and A :: "'c"   and B :: "'c" assumes "finite AAA"   and "\<forall>AA\<in>AAA. finite AA"   and "AB_pairs \<in> (pairs \<circ> compare_linorder.sorted_list_of_set) ` AAA"   and "(A, B) \<in> set AB_pairs" shows "\<exists>AA. AA \<in> AAA \<and> A \<in> AA \<and> B \<in> AA"
cases: fixes G :: "('a, 'b) cfg"   and a1 :: "('a + 'b) list"   and a2 :: "'a list" assumes "in_cfl G a1 a2" obtains "a1 = []" and "a2 = []"   | \<alpha> :: "('a + 'b) list" and w :: "'a list" and a :: "'a" where "a1 = Inl a # \<alpha>__" and "a2 = a # w__" and "in_cfl G \<alpha>__ w__"   | N :: "'b" and \<alpha> :: "('a + 'b) list" and w :: "'a list" where "a1 = Inr N # \<alpha>" and "a2 = w" and "fBex (cfg.prod G N) (\<lambda>\<beta>. in_cfl G (\<beta> @ \<alpha>) w)"
not_in_vars_mset_subst_mset: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and M :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "x \<notin> vars_term t" shows "x \<notin> vars_mset (subst_mset (subst x t) M)"
R\<^sub>C_of_local_preservation: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>1\<^sub>C :: "'Com\<^sub>C"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>2\<^sub>C :: "'Com\<^sub>C"   and mem\<^sub>2\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>1\<^sub>C' :: "'Com\<^sub>C"   and mds\<^sub>C' :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "secure_refinement \<R>\<^sub>A \<R> P"   and "abs.strong_low_bisim_mm \<R>\<^sub>A"   and "(\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> R\<^sub>C_of \<R>\<^sub>A \<R> P"   and "\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C" shows "\<exists>c\<^sub>2\<^sub>C' mem\<^sub>2\<^sub>C'. \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>2\<^sub>C', mds\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C \<and> (\<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C', mds\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C) \<in> R\<^sub>C_of \<R>\<^sub>A \<R> P"
seq1_def: shows "seq1 = (\<lambda>j. seq (Suc (i + j)))"
exr: shows "x \<in> set (map (consistent_sign_vec (qs1 @ qs2)) (characterize_root_list_p p)) \<longrightarrow> (\<exists>r\<in>set (characterize_root_list_p p). x = consistent_sign_vec (qs1 @ qs2) r)"
lead_coeff_of_nat: fixes n :: "nat" shows "Polynomial.lead_coeff (of_nat n) = of_nat n"
Seq_not_s\<^sub>2: fixes s\<^sub>2 :: "stmt"   and s\<^sub>1 :: "stmt" shows "s\<^sub>2 \<noteq> Seq s\<^sub>1 s\<^sub>2"
Rep_formula_inject: fixes x :: "'a formula"   and y :: "'a formula" shows "(Rep_formula x = Rep_formula y) = (x = y)"
Rf: shows "f \<in> {..n} \<rightarrow> carrier R"
right: fixes \<sigma>' :: "'state"   and coll' :: "'coll" assumes "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> (Suc (Suc n))" shows "\<exists>\<sigma>1 coll1 coll2. (\<sigma>1, coll1) \<in> csmall P \<sigma> \<and> (\<sigma>', coll2) \<in> csmall_nstep P \<sigma>1 (Suc n) \<and> combine coll1 coll2 = coll'"
stypes_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "stypes s i j = types (inst s ! i) ! j"
boundedByIncLevel: fixes n :: "nat"   and subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b" shows "boundedBy n (incLevel ` tree subs gamma) = (\<exists>m. n = Suc m \<and> boundedBy m (tree subs gamma))"
IH_2: fixes M :: "('fun, 'var) Term.term set" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>d \<theta>" shows "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>); S'\<rbrakk>\<^sub>d \<theta>"
evalS_equiv: fixes Gamma :: "formula list"   and f :: "vbl \<Rightarrow> object"   and g :: "vbl \<Rightarrow> object"   and M :: "model" assumes "equalOn (freeVarsFL Gamma) f g" shows "evalS M f Gamma = evalS M g Gamma"
simps_1: fixes w :: "nat"   and a :: "'a" shows "cost (Leaf w a) = 0"
minus_closed: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (R Quot I)"   and "y \<in> carrier (R Quot I)" shows "x \<ominus>\<^bsub>R Quot I\<^esub> y \<in> carrier (R Quot I)"
\<gamma>_le: shows "\<gamma> - 0 \<le> d (\<Union> (range F'))"
assms_4: shows "arr = arr' on - {l\<^sub>0..<m}"
interior_closure: fixes S :: "'b set" shows "interior S = - closure (- S)"
leg0_commutes: shows "trg_\<mu>.cod.leg0 \<cdot> trg_\<mu>.chine = trg_\<mu>.dom.leg0"
inf_aci_1: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<inter># y = y \<inter># x"
simps_1: fixes Vs :: "char list list" shows "compEs\<^sub>1 Vs [] = []"
rep_eq: fixes x :: "'b coreflexive"   and xa :: "'b coreflexive" shows "(x \<le> xa) = (Rep_coreflexive x \<le> Rep_coreflexive xa)"
assms_2: shows "(s, i:deliver(d), s') \<in> automaton.trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
dense: fixes x :: "'a"   and z :: "'a" assumes "x < z" shows "\<exists>y>x. y < z"
Regions'_axioms: shows "Regions' X v n not_in_X"
path_valid_node_2: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'" shows "valid_node n'"
\<Phi>_hps_ge0: fixes hs :: "'b heap list" shows "0 \<le> \<Phi>_hps hs"
BVG_choice: fixes \<alpha> :: "game"   and \<beta> :: "game" shows "BVG (\<alpha> \<union>\<union> \<beta>) \<subseteq> BVG \<alpha> \<union> BVG \<beta>"
unit_imp_no_prime_divisors: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 x :: "'a"   and p :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times x one"   and "comm_semiring_1.prime_elem times one zero p" shows "\<not> dvd.dvd times p x"
finvimage_fid_on: fixes A :: "V"   and B :: "V" shows "fid_on A -`\<^sub>\<bullet> B = A \<inter>\<^sub>\<circ> B"
assms_3: shows "the (rt1 dip) \<sqsubseteq> the (rt2 dip)"
one_in_subset: fixes H :: "'a vec set" assumes "H \<subseteq> carrier_vec n"   and "H \<noteq> {}"   and "\<forall>a\<in>H. \<ominus>\<^bsub>V\<^esub> a \<in> H"   and "\<forall>a\<in>H. \<forall>b\<in>H. a + b \<in> H" shows "0\<^sub>v n \<in> H"
fun_mult_scalar_commute: fixes f :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 't \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "f 0 = 0"   and "\<And>x y. f (x + y) = f x + f y"   and "\<And>c t. f (monom_mult c t p) = monom_mult c t (f p)" shows "f (q \<odot> p) = q \<odot> f p"
ZFfunCod_def: fixes f :: "ZF" shows "|cod|f \<equiv> Snd (Fst f)"
IH: shows "List_Supplement.embed (xs @ ys) = List_Supplement.embed xs @ List_Supplement.embed ys"
insert: fixes A :: "'e set"   and x :: "'e"   and g :: "'e \<Rightarrow> 'f" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
dvdm_imp_degree_le: fixes p :: "int poly"   and q :: "int poly" assumes "p.dvdm p q"   and "monic p"   and "p.Mp q \<noteq> 0"   and "1 < p" shows "degree p \<le> degree q"
invar1: shows "prim_invar1 (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u))"
comp_null_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C" shows "C f (partial_magma.null C) = partial_magma.null C"
map_disc_iff: fixes f :: "'l \<Rightarrow> 'k"   and a :: "'l list" shows "(map f a = []) = (a = [])"
in_vertices_graph: fixes v :: "nat"   and g :: "graph" shows "(v \<in> \<V> g) = (v < countVertices g)"
meet_ord: fixes x :: "'a"   and y :: "'a" shows "(ad x \<le> ad y) = (ad x \<cdot> ad y = ad x)"
collapse: fixes tm :: "'a tm" shows "TM (tm.val tm) (un_TM2 tm) = tm"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and exp :: "exp" assumes "P2 \<Gamma>"   and "P1 exp" shows "P1 (Let \<Gamma> exp)"
distinct_4: fixes x1 :: "'d" shows "sys \<noteq> mutator x1"
MtransC_mayDiverge: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>*c (c', s')"   and "mayDiverge c' s'" shows "mayDiverge c s"
cinner_scaleC_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and r :: "complex"   and y :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (scaleC r y) = r * cinner x y"
values_def: fixes q :: "('c, 'd) pq" shows "|q| = map fst (alist_of q)"
assms_5: shows "path_image \<gamma> \<subseteq> U - {z}"
simps: fixes t :: "'b trm" shows "pos_of t = {p. position_in p t}"
clinear_def: fixes f :: "'d \<Rightarrow> 'e" shows "clinear f \<equiv> Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) f"
lm055: fixes N :: "'c set"   and G :: "'d set"   and a :: "('c \<times> 'd set) set" assumes "finite N"   and "finite G"   and "a \<in> allAllocations N G" shows "finite a"
is_ladder_def: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list" shows "is_ladder D L = (L \<noteq> [] \<and> (\<forall>u<length L. ladder_n L u \<le> length D) \<and> (\<forall>u v. u < v \<and> v < length L \<longrightarrow> ladder_n L u < ladder_n L v) \<and> ladder_last_n L = length D)"
polysubst0_I: fixes b :: "'b"   and bs :: "'b list"   and a :: "'b poly"   and t :: "'b poly" shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly (Ipoly (b # bs) a # bs) t"
simps_1: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and r :: "String.literal"   and ts :: "trm list" shows "sat \<sigma> V v i (Pred r ts) = (case V r of None \<Rightarrow> (r, map (eval_trm v) ts) \<in> \<Gamma> \<sigma> i | Some X \<Rightarrow> map (eval_trm v) ts \<in> X i)"
pochhammer_eq_0_mono: fixes a :: "'a"   and n :: "nat"   and m :: "nat" assumes "pochhammer a n = (0::'a)"   and "n \<le> m" shows "pochhammer a m = (0::'a)"
simps_1: fixes x :: "('b set set, 'c) Combinators"   and xs :: "('b set set, 'c) Combinators list" shows "separated (x # xs) = ((\<forall>s. s \<in> set xs \<longrightarrow> disjSD_2 x s) \<and> separated xs)"
same_hyp_different_init_not_cons: fixes f :: "nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and g :: "nat \<Rightarrow> nat option"   and n :: "nat"   and s :: "nat \<Rightarrow> nat option" assumes "f \<in> U"   and "g \<in> U"   and "f \<triangleright> n \<noteq> g \<triangleright> n"   and "s (f \<triangleright> n) = s (g \<triangleright> n)" shows "\<not> learn_cons \<phi> U s"
leq_trans: fixes leq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.quasi_order leq"   and "leq x y"   and "leq y z" shows "leq x z"
assms_7: shows "b = (\<Otimes>x\<in>A. x [^] f x)"
CONS_I: fixes a :: "('b, unit) Old_Datatype.node set"   and A :: "('b, unit) Old_Datatype.node set set"   and M :: "('b, unit) Old_Datatype.node set" assumes "a \<in> A"   and "M \<in> list A" shows "CONS a M \<in> list A"
ord_isom_onto: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Order D"   and "Order E"   and "ord_isom D E f" shows "f ` carrier D = carrier E"