hcju/isabelleps huggingface dataset:
elims_2: fixes atom_ordering :: "('a \<times> 'a) set"   and x :: "'a Literal"   and xa :: "'a Literal" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.literal_ordering atom_ordering x xa" obtains L1 :: "'a Literal" and L2 :: "'a Literal" where "x = L1" and "xa = L2" and "(atom L1, atom L2) \<in> atom_ordering"
Hcont: fixes Z :: "'a set" assumes "Z \<in> \<C>" shows "continuous_on (U - S) (H Z)"
maxR_def: fixes d :: "'b"   and x :: "'a"   and y :: "'a" shows "Bossy'.maxR d x y = (if (x, y) \<in> BPd' d then y else x)"
arc_set_balanced_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and A :: "'d set"   and v :: "'c" shows "pre_digraph.arc_set_balanced G u A v \<equiv> if u = v then \<forall>w\<in>verts G. pre_digraph.arc_set_balance G w A = 0 else (\<forall>w\<in>verts G. w \<noteq> u \<and> w \<noteq> v \<longrightarrow> pre_digraph.arc_set_balance G w A = 0) \<and> pre_digraph.arc_set_balance G u A = - 1 \<and> pre_digraph.arc_set_balance G v A = 1"
vsv_axioms: shows "vsv (\<GG>\<lparr>ArrMap\<rparr>)"
mult_left_identity: fixes P :: "rat mat" assumes "P \<in> carrier_mat m m"   and "P * of_int_hom.mat_hom (mat_of_rows n fs_init) = of_int_hom.mat_hom (mat_of_rows n fs_init)" shows "P = 1\<^sub>m m"
polymul_eq0_iff: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
fiter1: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> \<sqsubseteq> c"
sots1: fixes n :: "nat"   and k :: "nat" assumes "\<exists>a b. n = a\<^sup>2 + b\<^sup>2" shows "prime (4 * k + 3) \<longrightarrow> even (multiplicity (4 * k + 3) n)"
map_def: shows "D.map \<equiv> \<lambda>j. if j = J.Zero then local.dom f0 else if j = J.One then cod f0 else if j = J.j0 then f0 else if j = J.j1 then f1 else null"
Subset_SUCC: fixes H :: "fm set"   and a :: "tm" shows "H \<turnstile> a SUBS SUCC a"
lm051: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G" shows "Domain a \<subseteq> N \<and> Range a \<subseteq> Pow G - {{}}"
restrict_map_le: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "m |` S \<subseteq>\<^sub>m m"
activate: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and T :: "Syntax.type"   and c :: "cmd" assumes "\<Gamma> , \<Delta>\<langle>0:T\<rangle> \<turnstile>\<^sub>C c" shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>T \<mu>T:c : T"
d_3: shows "(x < 0 \<or> n < 0) = False"
pCons_0_add: fixes p :: "'a poly"   and q :: "'a poly" shows "pCons (0::'a) (p + q) = pCons (0::'a) p + pCons (0::'a) q"
vle_iff_le_Ord: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "(\<alpha> \<unlhd> \<beta>) = (\<alpha> \<le> \<beta>)"
dist_add_cancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "dist (a + b) (a + c) = dist b c"
g_def: shows "g = foldl Gauss_Jordan_column_k_det_P_iarrays (1::'a, 0, matrix_to_iarray A) [0..<Suc k]"
d: shows "is_div_op (\<lambda>a b. SOME k. k * b = a)"
hd_prefixes: fixes xs :: "'c list" shows "hd (prefixes xs) = []"
simps_1: fixes mds :: "Mode \<Rightarrow> 'Var set" shows "mds \<oplus> [] = mds"
poly_root_mult_length: fixes a :: "'a"   and b :: "'a"   and p :: "'a list" shows "length ([a, b] *** p) = Suc (length p)"
map_def: shows "local.map \<equiv> local.curry F.sym F.sym F.sym"
charts_submanifold_carrier: shows "charts_submanifold carrier = charts"
DISJNT: shows "\<Union> P\<^sub>2 \<inter> B\<^sub>2 = {}"
on_circline_cmat_cvec_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex" shows "on_circline_cmat_cvec H z = (quad_form z H = 0)"
fixed_point_char: fixes m :: "'c" assumes "m \<in> M" shows "(m \<in> fixed_points) = (\<forall>g\<in>carrier G. \<phi> g m = m)"
card_singletons_eq: fixes A :: "'b set" shows "card {{a} |a. a \<in> A} = card A"
simps_2: fixes x :: "'d"   and xs :: "'d list"   and ys :: "'d list" shows "(x # xs) @ ys = x # xs @ ys"
correct_DisjointSchedules1: fixes n :: "nat"   and nC :: "nat \<Rightarrow> Config"   and t :: "nat"   and i :: "nat"   and j :: "nat" assumes "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC i) mem schedule (nC i)"   and "i < n"   and "j < n"   and "i \<noteq> j" shows "\<not> t mod cycleLength (nC j) mem schedule (nC j)"
simps_4: fixes qe :: "'a fm \<Rightarrow> 'a fm"   and \<phi> :: "'a fm" shows "lift_nnf_qe qe (ExQ \<phi>) = qe (nnf (lift_nnf_qe qe \<phi>))"
infsetsum'_cong: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "infsetsum' f A = infsetsum' g A"
orbit_char: fixes m :: "'c" assumes "m \<in> M" shows "orbit m = {n. \<exists>g. g \<in> carrier G \<and> \<phi> g m = n}"
llist_le_refl: fixes s :: "'a llist" shows "s \<le> s"
5: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * (v * v\<^sup>T \<sqinter> g) \<le> r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star>"
iffs: fixes newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a"   and newOID' :: "CTypeId \<Rightarrow> nat"   and newAID' :: "Arraytype \<Rightarrow> nat"   and vals' :: "Location \<Rightarrow> Value"   and more' :: "'a" shows "(\<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = \<lparr>newOID = newOID', newAID = newAID', vals = vals', \<dots> = more'\<rparr>) = (newOID = newOID' \<and> newAID = newAID' \<and> vals = vals' \<and> more = more')"
eq_diff_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(a = minus c b) = (plus a b = c)"
non_Package_instance_method_inheritance_cases: fixes G :: "prog"   and old :: "qtname \<times> mdecl"   and C :: "qtname" assumes "G \<turnstile>Method old inheritable_in pid C "   and "accmodi old \<noteq> Package"   and "\<not> is_static old"   and "G\<turnstile>C\<prec>\<^sub>C declclass old"   and "G\<turnstile>Method old declared_in declclass old"   and "wf_prog G" obtains "G \<turnstile>Method old member_of C"   | new :: "qtname \<times> mdecl" where "G \<turnstile> new overrides\<^sub>S old" and "G \<turnstile>Method new member_of C"
sfree: fixes i :: "'sf"   and f' :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> i = Some f'" shows "dfree f'"
special_subgroup_genby_subset_ordering_iso: shows "OrderingSetIso dual_order.greater_eq dual_order.greater dual_order.greater_eq dual_order.greater (Pow S) genby"
sel_4: fixes x41 :: "'b mrexp2"   and x42 :: "'b mrexp2"   and x43 :: "bool"   and x44 :: "bool" shows "fin (Plus2 x41 x42 x43 x44) = x43"
act_emp_var: fixes x :: "'a" shows "\<alpha> x \<bottom> = \<bottom>"
bounded_linear_axioms_def: fixes f :: "'b \<Rightarrow> 'c" shows "bounded_linear_axioms f \<equiv> \<exists>K. \<forall>x. \<parallel>f x\<parallel> \<le> (\<parallel>x\<parallel>) * K"
asymmetric_bot_closed: shows "matrix_stone_relation_algebra.asymmetric mbot"
set_insert: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" obtains B :: "'a set" where "A = insert x B" and "x \<notin> B"
set_integrable_scaleR_right: fixes a :: "real"   and M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f" shows "set_integrable M A (\<lambda>t. a *\<^sub>R f t)"
integral_split: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and k :: "'a"   and c :: "real" assumes "f integrable_on cbox a b"   and "k \<in> Basis" shows "integral (cbox a b) f = integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f + integral (cbox a b \<inter> {x. c \<le> x \<bullet> k}) f"
perp_in_left_comm: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "X PerpAt B A C D "
Compl_eq: fixes A :: "'b set" shows "- A = {x. x \<notin> A}"
bind_UNION: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a set" shows "A \<bind> f = \<Union> (f ` A)"
subdiv_ends_1: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d" assumes "subdiv_step G rev_G H rev_H u v w uv uw vw" shows "tail G uv = u"
FrF: shows "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"
borel_measurable_lebesgue_preimage_borel: fixes f :: "'c \<Rightarrow> 'd" shows "(f \<in> borel_measurable lebesgue) = (\<forall>T. T \<in> sets borel \<longrightarrow> {x. f x \<in> T} \<in> sets lebesgue)"
simps: fixes a1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "a1 \<approx> a2 = (\<exists>\<R> lc\<^sub>1 lc\<^sub>2. a1 = lc\<^sub>1 \<and> a2 = lc\<^sub>2 \<and> strong_low_bisim_mm \<R> \<and> (lc\<^sub>1, lc\<^sub>2) \<in> \<R>)"
r_not_DA_in_tl: fixes p :: "('a, 'b) Combinators list"   and a :: "('a, 'b) Combinators" assumes "wellformed_policy1_strong p"   and "a \<in> set p"   and "a \<noteq> DenyAll" shows "a \<in> set (tl p)"
elts_vsum: fixes a :: "V"   and b :: "V" shows "elts (a \<Uplus> b) = ZFC_Cardinals.Inl ` elts a \<union> ZFC_Cardinals.Inr ` elts b"
if_SE_D2: fixes P :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d"   and B\<^sub>1 :: "'d \<Rightarrow> (bool \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> (bool \<times> 'd) option" assumes "\<not> P \<sigma>" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2) = (\<sigma> \<Turnstile> B\<^sub>2)"
execn_to_execn_mark_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "\<not> isFault t" shows "\<Gamma>\<turnstile> \<langle>mark_guards f c,s\<rangle> =n\<Rightarrow> t"
le_adm: fixes u :: "'d \<Rightarrow> 'e process"   and v :: "'d \<Rightarrow> 'e process" assumes "cont u"   and "monofun v" shows "adm (\<lambda>x. u x \<le> v x)"
internal_iface_name_wildcard_longest_correct_Some: fixes X :: "char list" assumes "internal_iface_name_wildcard_longest i1 i2 = Some X" shows "(internal_iface_name_match i1 p_i \<and> internal_iface_name_match i2 p_i) = internal_iface_name_match X p_i"
fconj2: fixes n :: "nat"   and m :: "nat"   and f :: "form"   and g :: "form"   and xs :: "(nat \<times> form) list"   and nfs :: "(nat \<times> form) list" assumes "(n, (m, FConj f g) # xs) \<in> deriv nfs"   and "\<not> is_axiom (s_of_ns ((m, FConj f g) # xs))" shows "(Suc n, xs @ [(0, g)]) \<in> deriv nfs"
val_bounded_2: shows "h.val \<le> sum c (outgoing s)"
uptoR0: shows "d_Bisimulation_Up_To_USdB d R0"
sum_unfold_sum_mset: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum plus zero f A = comm_monoid_add.sum_mset plus zero (image_mset f (mset_set A))"
upd_incoming_eq_old_next_pair: fixes ns :: "('b, 'c) node_scheme set"   and n :: "('b, 'd) node_scheme" shows "old_next_pair ` ns = old_next_pair ` upd_incoming n ns"
goes_wrong_denot_empty: fixes e :: "exp" assumes "goes_wrong e"   and "FV e = {}" shows "E e [] = {}"
mat1I_def: fixes ze :: "'b"   and on :: "'b"   and n :: "nat" shows "mat1I ze on n \<equiv> map (vec1I ze on n) [0..<n]"
domain_restrict_chart: fixes S :: "'c set"   and c :: "('c, 'd) chart" assumes "open S" shows "domain (restrict_chart S c) = domain c \<inter> S"
L6_Controller_Zero: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l (Suc t) = Zero"   and "z (Suc t) = []" shows "l t = Zero"
r2f_surj: shows "top_pres (\<P> \<F>)"
assms_2: shows "SIDE_PRECOND (X \<noteq> {})"
eventuallyI: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "\<And>x. P x" shows "eventually P F"
Complex_divide: fixes x :: "complex"   and y :: "complex" shows "x / y = Complex ((Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)) ((Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))"
sub_consts_And: fixes cs :: "'a list"   and p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "sub_consts cs (And p q) = And (sub_consts cs p) (sub_consts cs q)"
chainTransitionAppend_3: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"
sots2: fixes n :: "nat" assumes "\<And>k. prime (4 * k + 3) \<longrightarrow> even (multiplicity (4 * k + 3) n)" shows "is_sum2sq_nat n"
butlast_append: fixes xs :: "'a list"   and ys :: "'a list" shows "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
homeomorphism_cont2: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphism S T f g" shows "continuous_on T g"
assms: shows "x \<in>\<^sub>\<circ> fconst_on A c"
UnionI: fixes X :: "'b set"   and C :: "'b set set"   and A :: "'b" assumes "X \<in> C"   and "A \<in> X" shows "A \<in> \<Union> C"
fmlookup_cases: fixes y :: "'c \<Rightarrow> 'd option" assumes "y \<in> {m. finite (dom m)}" obtains x :: "('c, 'd) fmap" where "y = fmlookup x"
word_rel_append: fixes v :: "'b \<Rightarrow> ('a \<times> 'a) set"   and w :: "'b list"   and w' :: "'b list" shows "word_rel v w O word_rel v w' = word_rel v (w @ w')"
assms_3: shows "i < card {i. i < m \<and> i \<in> I}"
apply_power_eq_self_iff: fixes f :: "'a perm"   and n :: "nat"   and a :: "'a" shows "((f ^ n) \<langle>$\<rangle> a = a) = (order f a dvd n)"
rerename_rename_noop: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and "from" :: "nat"   and V :: "'a set"   and form :: "'b"   and to :: "nat"   and f :: "'a \<Rightarrow> 'a" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "freshenLC from ` V \<inter> lconsts form = {}" shows "renameLCs (Abstract_Formulas.rerename freshenLC V from to f) form = renameLCs f form"
assms_3: shows "\<beta> \<in> {0<..<1}"
permutations_of_set_aux_altdef: fixes acc :: "'a list"   and A :: "'a set" shows "permutations_of_set_aux acc A = (\<lambda>xs. rev xs @ acc) ` permutations_of_set A"
assms_2: shows "\<rho> \<in> space (state_measure (set vs') \<Gamma>)"
rep_eq: shows "coeff 0 = (\<lambda>_. 0::'d)"
lucas_lehmer_add'_def: shows "lucas_lehmer_add' = (\<lambda>(a, b) (c, d). (a + c, b + d))"
real_of_word_simps_5: fixes w :: "'a Word.word" shows "(0 \<le> signed_real_of_word w) = (0 \<le> sint w)"
Ex_mono: fixes A :: "fm"   and H :: "fm set"   and B :: "fm"   and i :: "name" assumes "insert A H \<turnstile> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (SyntaxN.Ex i A) H \<turnstile> SyntaxN.Ex i B"
comp_inv_arr: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "S.inverse_arrows f g" shows "g \<cdot>\<^sub>S f = S.dom f"
ccw'_sortedP_snd_half_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "ccw'.sortedP (lowest_vertex (fst X, nlex_pdevs (snd X))) (map snd (half_segments_of_aform X))"
basis_inI_aux: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.basis \<E> (indep_system.indep_in indep \<E>) X" shows "indep_system.basis_in indep \<E> X"
hypreal_sqrt_approx_zero2: fixes x :: "real star" assumes "0 \<le> x" shows "((*f* sqrt) x \<approx> 0) = (x \<approx> 0)"
case_prod_beta: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and p :: "'d \<times> 'e" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
IH1: fixes s :: "'a net_state"   and a :: "'b node_action"   and s' :: "'a net_state" assumes "(s, a, s') \<in> automaton.trans (pnet np n1)" shows "\<forall>i d di. a \<noteq> i:newpkt(d, di)"
nu_add: fixes x :: "'a"   and y :: "'a" shows "\<nu> (x + y) = \<nu> x + \<nu> y"
simps_3: fixes uv :: "nat" shows "- extN.N uv = extN.Bot"
measurable_Pair1_compose: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and M :: "'c measure"   and M1 :: "'d measure"   and M2 :: "'e measure"   and h :: "'f \<Rightarrow> 'c"   and N :: "'f measure" assumes "(\<lambda>x. (f x, g x)) \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"   and "h \<in> N \<rightarrow>\<^sub>M M" shows "(\<lambda>x. f (h x)) \<in> N \<rightarrow>\<^sub>M M1"
succ_subset: fixes v :: "'V" shows "succ G v \<subseteq> UNIV \<times> V"
disj_simps_4: fixes P :: "fm" shows "Ferrack.disj P F = P"
top_greatest: fixes a :: "'a" shows "a \<le> \<top>"
hcomp_can: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term"   and v :: "'b bicategorical_language.term"   and w :: "'b bicategorical_language.term" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Ide V src trg v"   and "bicategorical_language.Ide V src trg w"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u"   and "bicategorical_language.Nmlize V v = bicategorical_language.Nmlize V w"   and "bicategorical_language.Src src t = bicategorical_language.Trg trg v"   and "bicategorical_language.Src src u = bicategorical_language.Trg trg w" shows "H (bicategory.can V H \<a> \<i> src trg u t) (bicategory.can V H \<a> \<i> src trg w v) = bicategory.can V H \<a> \<i> src trg (bicategorical_language.Hcomp u w) (bicategorical_language.Hcomp t v)"
nrename_ImpL: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and M :: "trm"   and u :: "name"   and N :: "trm"   and z :: "name" assumes "R[x\<turnstile>n>y] = ImpL <c>.M u.N z"   and "c \<sharp> (N, R)"   and "u \<sharp> (y, x, z, M, R)"   and "z \<sharp> (x, y)" shows "\<exists>M' N'. R = ImpL <c>.M' u.N' z \<and> M'[x\<turnstile>n>y] = M \<and> N'[x\<turnstile>n>y] = N \<and> c \<sharp> N' \<and> u \<sharp> M'"
set_sel_1: fixes a :: "'a list" assumes "a \<noteq> []" shows "hd a \<in> set a"
secure_def: fixes P :: "'e process"   and I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f" shows "secure P I D \<equiv> \<forall>xs y ys Y zs Z. (y # ys, Y) \<in> futures P xs \<and> (zs, Z) \<in> futures P xs \<longrightarrow> (ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y) \<in> futures P xs \<and> (y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> futures P xs"
case_prodE: fixes c :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c \<times> 'd" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'c" and y :: "'d" where "p = (x, y)" and "c x y"
shift_append: fixes xs :: "'b list"   and ys :: "'b list"   and s :: "'b stream" shows "(xs @ ys) @- s = xs @- ys @- s"
2: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and oldDefs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and oldUses :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and phis :: "'a \<Rightarrow> 'b \<times> 'e \<Rightarrow> 'e list option"   and var :: "'a \<Rightarrow> 'e \<Rightarrow> 'd"   and g :: "'a"   and p :: "'e"   and q :: "'e"   and r :: "'e" assumes "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var"   and "CFG_SSA_wf_base.phiArg \<alpha>n defs phis g p q"   and "CFG_SSA_wf_base.phiArg \<alpha>n defs phis g p r"   and "distinct [p, q, r]"   and "p \<in> CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g" shows "\<not> (graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g (CFG_SSA_wf_base.defNode \<alpha>n defs phis g q) (CFG_SSA_wf_base.defNode \<alpha>n defs phis g p) \<and> graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g (CFG_SSA_wf_base.defNode \<alpha>n defs phis g r) (CFG_SSA_wf_base.defNode \<alpha>n defs phis g p))"
inf_sup_distrib1: fixes 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"   and y :: "'a"   and z :: "'a" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
sds_anonymous_neutral: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R1"   and "is_pref_profile R2"   and "anonymous_profile R1 = image_mset (map ((`) \<sigma>)) (anonymous_profile R2)" shows "sds R1 = map_pmf \<sigma> (sds R2)"
foo2: shows "(\<chi>i. x) \<in> UNIV"
assms: shows "X \<in> obj\<^bsub>NTCatDom (YFtorNT C f)\<^esub>"
one_reducible_to_via_trans: fixes A :: "nat set"   and B :: "nat set"   and f :: "nat \<Rightarrow> nat"   and C :: "nat set"   and g :: "nat \<Rightarrow> nat" assumes "one_reducible_to_via A B f"   and "one_reducible_to_via B C g" shows "one_reducible_to_via A C (g \<circ> f)"
direct_lim_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set" shows "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I \<equiv> sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> direct_lim_axioms is_open I"
dgrad_p_set_le_fst_crit_pair: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d" shows "dgrad_p_set_le d {fst (crit_pair p q)} {p, q}"