hcju/isabelleps huggingface dataset:
sup_ty_opt_trans: fixes P :: "'b prog"   and a :: "ty err"   and b :: "ty err"   and c :: "ty err" assumes "P \<turnstile> a \<le>\<^sub>\<top> b"   and "P \<turnstile> b \<le>\<^sub>\<top> c" shows "P \<turnstile> a \<le>\<^sub>\<top> c"
n'_def: shows "n' = smallest_prime_beyond (Suc n)"
assms: fixes x :: "nat" assumes "x \<in> free_vars e" shows "\<sigma> x = \<sigma>' x"
hol: shows "(\<lambda>z. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball \<xi> r"
R3: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and z :: "'b"   and zs :: "'b list"   and X :: "'b list \<times> 'b list \<times> 'b list" assumes "(x # xs, ys, z # zs) \<in> l_sort_set X"   and "\<not> x \<le> z" shows "(x # xs, ys @ [z], zs) \<in> l_sort_set X"
gm_int: shows "integrable M (\<lambda>x. f x * enn2real (nn_transfer_operator gm x))"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lookup (higherPoly\<^sub>0 x xa xb) = (\<lambda>mon. if \<exists>j\<in>{x..<x + xa}. 0 < lookup mon j then 0::'b else lookup xb (higherPowers x xa mon))"
in_latticeE: fixes f :: "'a vec"   and n :: "nat"   and fs :: "'a vec list" assumes "f \<in> vec_module.lattice_of n fs" obtains c :: "nat \<Rightarrow> int" where "f = abelian_monoid.sumlist (module_vec TYPE('a) n) (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs])"
seqI': fixes f :: "('obj, 'obj) concrete_category.arr"   and a :: "('obj, 'obj) concrete_category.arr"   and b :: "('obj, 'obj) concrete_category.arr"   and g :: "('obj, 'obj) concrete_category.arr"   and c :: "('obj, 'obj) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "seq g f"
less_mono_imp_le_mono: fixes f :: "nat \<Rightarrow> nat"   and i :: "nat"   and j :: "nat" assumes "\<And>i j. i < j \<Longrightarrow> f i < f j"   and "i \<le> j" shows "f i \<le> f j"
ennreal_mult_cancel_left: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a * b = a * c) = (a = top \<and> b \<noteq> 0 \<and> c \<noteq> 0 \<or> a = 0 \<or> b = c)"
I_qe_dlo: fixes \<phi> :: "atom fm"   and xs :: "'a list" shows "DLO.I (qe_dlo \<phi>) xs = DLO.I \<phi> xs"
"case": fixes f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and x :: "'e \<Rightarrow> 'f" shows "(case EnvT x of EnvT x \<Rightarrow> f x) = f x"
assms_2: shows "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"
switch_basechamber: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>"   and "C \<sim> g ` C" shows "OpposedThinChamberComplexFoldings X f g C"
invrules_1: shows "bp (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V})"
impl: assumes "length L \<le> k" shows "List.member (fin_string_upto k) (Abs_fin_string L)"
elims_3: fixes x :: "'b option"   and xa :: "'b option" assumes "\<not> x \<le> xa" obtains xb :: "'b" and y :: "'b" where "x = Some xb" and "xa = Some y" and "\<not> xb \<le> y"   | uu_ :: "'b" where "x = Some uu_" and "xa = None"
F_def: shows "F \<equiv> \<lambda>x. \<Pi>\<^sub>E i\<in>UNIV. if i \<le> p then {x i - e x<..<x i + e x} else UNIV"
Compl_subset_Compl_iff: fixes A :: "'a set"   and B :: "'a set" shows "(- A \<subseteq> - B) = (B \<subseteq> A)"
y0: shows "local.bad y0 \<and> p (local.worst_chain i) y0"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "finite_incidence_system point_set block_collection"   and "design_axioms block_collection" shows "design point_set block_collection"
as_needed_asses_submap_exec_xii: fixes v :: "'a"   and f :: "('a, 'b) fmap"   and g :: "('a, 'b) fmap" assumes "v \<in> fmdom' f" shows "fmlookup (f ++ g) v = fmlookup f v"
asym: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<prec> b"   and "b \<prec> a" shows "False"
subringE_2: fixes H :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subring H R" shows "\<zero>\<^bsub>R\<^esub> \<in> H"
hypnat_neq0_conv: fixes n :: "nat star" shows "(n \<noteq> 0) = (0 < n)"
ahS: shows "affine hull S - {a} \<subseteq> {x. closest_point (affine hull S) x \<noteq> a}"
lub_least: fixes cl :: "'b potype"   and S :: "'b set"   and L :: "'b" assumes "CL cl"   and "S \<subseteq> pset cl"   and "L \<in> pset cl"   and "\<forall>x\<in>S. (x, L) \<in> order cl" shows "(lub S cl, L) \<in> order cl"
event_stays_valid_if_no_occurrence: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> occurs_on ev'"   and "can_occur ev' c" shows "can_occur ev' c'"
axioms_2: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>" shows "digraph \<alpha> \<CC>"
is_le_rel_def: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "is_le_rel rel = (rel = (=) \<or> rel = (\<le>) \<or> rel = (<))"
le_add_same_cancel1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus a b) = less_eq zero b"
Gromov_extension_boundary_homeomorphism: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry f" shows "homeomorphism_on Gromov_boundary (Gromov_extension f)"
split_edges_sym: fixes g :: "'b ugraph" shows "\<exists>E. E \<inter> E\<inverse> = {} \<and> edges g = E \<union> E\<inverse>"
succ_BIT_simps_1: fixes x :: "int" shows "x BIT False + 1 = x BIT True"
ee_wfactorsD: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list"   and a :: "'c"   and b :: "'c" assumes "comm_monoid_cancel G"   and "essentially_equal G as bs"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a \<sim>\<^bsub>G\<^esub> b"
gcd_mult_left_left_cancel: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one b c" shows "gcd (times c a) b = gcd a b"
FAssRedsNull: fixes e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr"   and v :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>null,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1\<bullet>F{Cs} := e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<^sub>2\<rangle>"
iterative_override_on': shows "(\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>')(x := Abs_cfun (ESem_syn e)\<cdot>\<rho>')) = (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>''))(x := Abs_cfun (ESem_syn e)\<cdot>(\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>'')))"
im_sheaf_morphisms_def: fixes U :: "'a set"   and V :: "'a set" shows "imS.im_sheaf_morphisms U V \<equiv> \<rho> (identity S \<^sup>\<inverse> S U) (identity S \<^sup>\<inverse> S V)"
intros_2: fixes p :: "'node"   and S :: "'node set"   and p' :: "'node"   and Sl :: "'node set"   and p'' :: "'node" assumes "reachable_through p S p'"   and "p' \<in> W"   and "Sl \<in> slices p'"   and "Sl \<subseteq> S"   and "p'' \<in> Sl" shows "reachable_through p S p''"
abs_eq: fixes x :: "nat" shows "Word.of_nat x = Word.Word (int (take_bit LENGTH('k) x))"
simps_1: fixes bs :: "int list"   and c :: "int" shows "Inum bs (C c) = c"
mset_del_min: fixes h :: "'b tree"   and t :: "'c tree" assumes "is_root h"   and "t \<noteq> \<langle>\<rangle>" shows "mset_tree (del_min h) = mset_tree h - {#get_min h#}"
A: shows "\<forall>v\<in>sinks I D u ys. \<forall>z\<in>Z \<union> set zs. (v, D z) \<notin> I"
simps: fixes a :: "('a, 'b, 'c) expr" shows "redex a = ((\<exists>x e v. a = Apply (VE (Lambda x e)) (VE v)) \<or> (\<exists>e1 e2. a = Ite (VE (CV T)) e1 e2) \<or> (\<exists>e1 e2. a = Ite (VE (CV F)) e1 e2) \<or> (\<exists>v. a = Ref (VE v)) \<or> (\<exists>l. a = Read (VE (Loc l))) \<or> (\<exists>l v. a = Assign (VE (Loc l)) (VE v)) \<or> (\<exists>e. a = Rfork e) \<or> (\<exists>r. a = Rjoin (VE (Rid r))))"
vdomain_small: fixes r :: "V" shows "small {a. \<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
adj_sub_ode_fact: fixes i :: "'d"   and x1 :: "('d, 'e) ODE"   and j :: "'e + 'e"   and \<sigma> :: "('d, 'f, 'e, 'g) subst_scheme"   and x2 :: "('d, 'f, 'e) formula" assumes "Inl i \<in> SIGO x1"   and "j \<in> (case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Inl i \<in> ({Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}) \<inter> (SIGF x2 \<union> {Inl x |x. Inl x \<in> SIGO x1} \<union> {Inr (Inr x) |x. Inr x \<in> SIGO x1})"
siso0_Atm: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and atm :: "'c" assumes "PL_Indis indis" shows "PL_Indis.siso0 tval aval indis (Atm atm) = PL_Indis.compatAtm aval indis atm"
Iod_less: fixes T :: "'a set"   and a :: "'a"   and b :: "'a" assumes "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T" shows "a \<prec>\<^bsub>Iod D T\<^esub> b = a \<prec> b"
extend_bound: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm"   and m :: "nat" shows "\<Union> (extend S f ` {..m}) = extend S f m"
X: shows "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
is_interval_path_connected: fixes S :: "'a set" assumes "is_interval S" shows "path_connected S"
coinduct: fixes f :: "'c \<Rightarrow> 'c"   and X :: "'c" assumes "mono f"   and "X \<le> f (X \<squnion> gfp f)" shows "X \<le> gfp f"
reduce_rules_subset: fixes \<delta> :: "('a, 'b) ta_rule set"   and P :: "'a set" shows "reduce_rules \<delta> P \<subseteq> \<delta>"
normal_imp_subgroup: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G" shows "subgroup H G"
ff_def: shows "ff \<equiv> \<lambda>x. deriv f x * h x / f x"
holds_eq1: fixes x :: "'c" shows "holds ((=) x) = HLD {x}"
ideal_generated_UNIV_obtain_pair: fixes a :: "'a"   and b :: "'a"   and d :: "'a" assumes "ideal_generated {a, b} = ideal_generated {1::'a}" shows "\<exists>p q. p * a + q * b = d"
bexE: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" obtains x :: "'a" where "x \<in> A" and "P x"
times_fps_def: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps" shows "f \<cdot> g = Abs_fps (\<lambda>n. \<Sum> {f $ y \<cdot> g $ z |y z. n = y @ z})"
minimum_def: fixes i :: "nat_int" assumes "i \<noteq> bot" shows "minimum i = Min (Rep_nat_int i)"
emeasure_compact_finite: fixes A :: "'b set" assumes "compact A" shows "emeasure lborel A < \<infinity>"
reachable_awalk: fixes u :: "'c"   and v :: "'c" shows "(u \<rightarrow>\<^sup>*\<^bsub>app_iso hom G\<^esub> v) = (\<exists>p. H.awalk u p v)"
bdd_below_image_dist: fixes x :: "'c"   and A :: "'c set" shows "bdd_below (dist x ` A)"
assms_2: shows "Cond Rel \<and> Rel \<subseteq> RelA"
nth_upt: fixes i :: "nat"   and k :: "nat"   and j :: "nat" assumes "i + k \<sqsubset> j" shows "[i..<j] ! k = i + k"
semilat_opt: fixes L :: "'b set \<times> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'b \<Rightarrow> 'b err)" assumes "err_semilat L" shows "err_semilat (Opt.esl L)"
double_zero_sym: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(zero = plus a a) = (a = zero)"
inv_naturality: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "(fst f \<star>\<^sub>C snd f) \<cdot>\<^sub>C C.inv (fst (C.VV.dom f) \<star>\<^sub>C snd (C.VV.dom f)) = C.inv (fst (C.VV.cod f) \<star>\<^sub>C snd (C.VV.cod f)) \<cdot>\<^sub>C (fst f \<star>\<^sub>C snd f)"
\<pi>_idempotent': fixes k :: "nat"   and T :: "(symbol \<times> character list) set"   and I :: "item set" shows "\<pi> k {} (\<pi> k T I) = \<pi> k T I"
elems: fixes X :: "'a set" assumes "X \<in># Cs" shows "\<exists>x. P x \<and> X = assocs G x"
map_transform_restrA: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>"   and S :: "Vars.var set"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "map_transform t ae (restrictA S \<Gamma>) = restrictA S (map_transform t ae \<Gamma>)"
ccw_system123_axioms: fixes S :: "(real \<times> real) set" shows "ccw_system123 ccw S"
splay_elemsD: fixes x :: "'a"   and t :: "'a tree"   and l :: "'a tree"   and a :: "'a"   and r :: "'a tree" assumes "splay x t = \<langle>l, a, r\<rangle>"   and "Sorted_Less.sorted (inorder t)" shows "(x \<in> set (inorder t)) = (x = a)"
\<alpha>_def: shows "\<alpha> = (\<lambda>(u, w, v). Upair u v)"
alt_definition_of_degenerate_lotteries: shows "{return_pmf x |x. x \<in> outcomes} = local.degenerate_lotteries"
g_ind: shows "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (xb - Suc xa) @ xc}))"
produced_by_scan_step: fixes x :: "item"   and k :: "nat"   and u :: "nat" assumes "x \<in> \<J> k u"   and "k < item_end x" shows "produced_by_scan_step x k u"
IH_3: fixes x :: "nat"   and xa :: "nat"   and xaa :: "'a"   and xb :: "nat \<Rightarrow> 'a" assumes "x = dim_row A"   and "xa = dim_col A"   and "i < x \<and> j < xa"   and "xaa = A $$ (i, j)"   and "xaa \<noteq> (0::'a)"   and "xaa = (1::'a)"   and "xb = (\<lambda>i. A $$ (i, j))"   and "eliminate_entries xb A i j \<in> carrier_mat nr nc"   and "eliminate_entries xb B i j \<in> carrier_mat nr nc2" shows "gauss_jordan_main (mat\<^sub>h (eliminate_entries xb A i j)) (mat\<^sub>h (eliminate_entries xb B i j)) (Suc i) (Suc j) = map_prod mat\<^sub>h mat\<^sub>h (gauss_jordan_main (eliminate_entries xb A i j) (eliminate_entries xb B i j) (Suc i) (Suc j))"
either_above_arg1: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "t \<sqsubseteq> t \<oplus>\<oplus> t'"
lift_CFG_wf: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list"   and H :: "'i set"   and L :: "'i set" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "CFG_wf src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) (lift_ParamDefs ParamDefs) (lift_ParamUses ParamUses)"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "a *s (b *s x) = a * b *s x"
s_2: shows "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
fls_base_factor_mult_fls_X_intpow_2: fixes f :: "'a fls"   and i :: "int" shows "fls_base_factor (f * fls_X_intpow i) = fls_base_factor f"
nth_default_def: fixes dflt :: "'a"   and xs :: "'a list"   and n :: "nat" shows "nth_default dflt xs n = (if n < length xs then xs ! n else dflt)"
step: fixes Mkt :: "('c, 'd) discrete_market"   and v :: "real"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and asset :: "'d"   and n :: "nat" shows "remaining_qty Mkt v pf asset (Suc (Suc n)) = (\<lambda>w. remaining_qty Mkt v pf asset (Suc n) w + (cls_val_process Mkt pf (Suc n) w - val_process Mkt pf (Suc n) w) / prices Mkt asset (Suc n) w)"
array_rel_def_internal: fixes R :: "('a \<times> 'b) set" shows "array_rel R \<equiv> {(Array xs, Array ys) |xs ys. (xs, ys) \<in> \<langle>R\<rangle>list_rel}"
subcls_superD: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and c :: "class" assumes "G\<turnstile>C\<prec>\<^sub>C D"   and "class G C = Some c" shows "G\<turnstile>super c\<preceq>\<^sub>C D"
acute_per__lta: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Acute Bet Cong A B C"   and "D \<noteq> E"   and "E \<noteq> F"   and "Tarski_neutral_dimensionless.Per Bet Cong D E F" shows "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F"
is_parallel_solution_for_problem_plan_operator_set: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi>" shows "\<forall>ops\<in>set \<psi>. \<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
zero_notin_list_of_oalist: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalist :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_list :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and xs :: "'c" assumes "oalist_abstract rep_key_order list_of_oalist oalist_of_list" shows "(0::'d) \<notin> snd ` set (fst (list_of_oalist xs))"
\<A>_def: shows "\<A> = {(A, j). A \<subseteq> X \<and> card A \<le> k \<and> j \<in> X \<and> (card A = k \<longrightarrow> j \<in> A)}"
Lt_lt_conv: fixes x :: "'a"   and y :: "'a" shows "(comp x y = Lt) = lt x y"
is_poincare_line_conjugate_circline_iff: fixes H :: "circline" shows "is_poincare_line (conjugate_circline H) = is_poincare_line H"
cases: fixes a1 :: "nat"   and a2 :: "((nat \<times> nat) \<times> transition) fset"   and a3 :: "nat"   and a4 :: "nat \<Rightarrow>f value option"   and a5 :: "(String.literal \<times> value list) list" assumes "visits a1 a2 a3 a4 a5" obtains s :: "nat" and e :: "((nat \<times> nat) \<times> transition) fset" and r :: "nat \<Rightarrow>f value option" where "a1 = s__" and "a2 = e__" and "a3 = s__" and "a4 = r__" and "a5 = []"   | e :: "((nat \<times> nat) \<times> transition) fset" and s :: "nat" and r :: "nat \<Rightarrow>f value option" and l :: "String.literal" and i :: "value list" and target :: "nat" and t :: "(String.literal \<times> value list) list" where "a1 = target" and "a2 = e" and "a3 = s" and "a4 = r" and "a5 = (l, i) # t" and "\<exists>(s', T)|\<in>|possible_steps e s r l i. visits target e s' (evaluate_updates T i r) t"
src_ipPart: fixes rs :: "'a simple_rule list"   and A :: "'a word set set"   and B :: "'a word set"   and s1 :: "'a word"   and s2 :: "'a word"   and p :: "('a, 'b) simple_packet_scheme" assumes "ipPartition (set (map wordinterval_to_set (extract_IPSets_generic0 src rs))) A"   and "B \<in> A"   and "s1 \<in> B"   and "s2 \<in> B" shows "simple_fw rs (p\<lparr>p_src := s1\<rparr>) = simple_fw rs (p\<lparr>p_src := s2\<rparr>)"
set_tree_union: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t2 :: "('b \<times> 'c) tree"   and t1 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "Tree2.bst t2" shows "Tree2.set_tree (Set2_Join.union join t1 t2) = Tree2.set_tree t1 \<union> Tree2.set_tree t2"
inverse_functors_sym: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "inverse_functors B A G F"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vrat_inverse" shows "a\<inverse>\<^sub>\<rat> = b"
sat_inf_imp_ground_red: fixes N :: "'f set"   and \<iota>' :: "'f inference"   and \<iota> :: "'g inference" assumes "saturated N"   and "\<iota>' \<in> Inf_from N"   and "\<G>_I \<iota>' \<noteq> None \<and> \<iota> \<in> the (\<G>_I \<iota>')" shows "\<iota> \<in> Red_I_G (\<G>_Fset N)"
lcf_scalar_mult: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "p \<in> carrier P"   and "a \<in> carrier R" shows "(a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p)) = a \<otimes> p (deg R p)"
removeProcs_def: fixes ps :: "'b pState_scheme list" shows "removeProcs ps = foldr (\<lambda>p (dead, sd, ps, dcs). if dead \<and> pc p = 0 then (True, True, ps, pState.channels p @ dcs) else (False, sd, p # ps, dcs)) ps (True, False, [], [])"
wpt_is_pre: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile>\<^sub>t {wp\<^sub>t c Q} c {Q}"
group_left_neutral: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group_ow U f z inverse"   and "a \<in> U" shows "f z a = a"
matchT_T_def2: fixes d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchT_T d c = (\<forall>s t t'. s \<approx> t \<and> (d, t) \<rightarrow>t t' \<longrightarrow> (\<exists>s'. (c, s) \<rightarrow>t s' \<and> s' \<approx> t'))"