hcju/isabelleps huggingface dataset:
at_within_open: fixes "open" :: "'d set \<Rightarrow> bool"   and a :: "'d"   and S :: "'d set" assumes "class.topological_space open"   and "a \<in> S"   and "open S" shows "topological_space.at_within open a S = topological_space.at_within open a UNIV"
step_Normal_elim_cases_14: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Call p # cs, css, Normal s) \<rightarrow> u" obtains bdy :: "('e, 'd, 'f) com" where "u = ([bdy], (cs, Throw # cs) # css, Normal s)" and "\<Gamma> p = Some bdy"   | "u = (cs, css, Stuck)" and "\<Gamma> p = None"
\<Z>_srngD_6: fixes \<alpha> :: "V"   and \<SS> :: "V"   and a :: "V"   and b :: "V"   and c :: "V" assumes "\<Z>_srng \<alpha> \<SS>"   and "a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>" shows "a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c"
rule_charnDAFT: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and r :: "((int \<times> int) set set, int) Combinators" assumes "wellformed_policy1_strong p"   and "allNetsDistinct p"   and "singleCombinators p"   and "wellformed_policy3 p"   and "applied_rule_rev C x p = \<lfloor>DenyAllFromTo a b\<rfloor>"   and "r \<in> set (tl p)"   and "x \<in> dom (C r)" shows "r = DenyAllFromTo a b"
Nats_altdef1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.Nats one plus zero = {ring_1.of_int one plus zero minus n |n. 0 \<le> n}"
isomorphic_def: fixes a :: "'b"   and a' :: "'b" shows "(a \<cong>\<^sub>D a') = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f)"
domain_partitions_respects_range_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. (\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}) respects range_permutation A B"
AE_1_2: shows "AE x in M. (\<lambda>xa. ereal (- u xa x / real xa)) \<longlonglongrightarrow> - subcocycle_lim_ereal u x"
rel_spmf_eqI: fixes x :: "'b spmf" shows "rel_spmf (=) x x"
simps_3: fixes x :: "'b trm"   and y :: "'b trm" shows "is_a_variable (x \<cdot> y) = False"
vcpower_vdomain: fixes f :: "V"   and A :: "V"   and n :: "V" assumes "f \<in>\<^sub>\<circ> A ^\<^sub>\<times> n" shows "\<D>\<^sub>\<circ> f = n"
Collect_cong: fixes P :: "'ap \<Rightarrow> bool"   and Q :: "'ap \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
observable_transition_eqvt: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state"   and p :: "perm" assumes "P \<Rightarrow>{\<alpha>} P'" shows "p \<bullet> P \<Rightarrow>{p \<bullet> \<alpha>} p \<bullet> P'"
one_reducibility_is_many: fixes A :: "nat set"   and B :: "nat set" assumes "one_reducible_to A B" shows "many_reducible_to A B"
injection_eq: fixes f :: "'b \<rightarrow> 'a"   and g :: "'a \<rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x" shows "(g\<cdot>x = g\<cdot>y) = (x = y)"
IH_3: assumes "e \<notin> set (cycle_edges (enum_cycle j))" shows "h_minus_aux j e \<le> h' e"
that: fixes ns' :: "'node list" assumes "g \<turnstile> n-ns'\<rightarrow>m"   and "distinct ns'"   and "set ns' \<subseteq> set ns"   and "length ns' \<le> length ns"   and "n \<notin> set (tl ns')"   and "m \<notin> set (butlast ns')" shows "thesis"
linepath_def: fixes a :: "'c"   and b :: "'c" shows "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
finfun_apply_inverse: fixes x :: "'a \<Rightarrow>f 'b" shows "Abs_finfun (($) x) = x"
return_edges_only: 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 a :: "'g"   and Q' :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and p :: "'l"   and f' :: "('i \<Rightarrow> 'j option) \<Rightarrow> ('i \<Rightarrow> 'j option) \<Rightarrow> 'i \<Rightarrow> 'j option" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<forall>a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<longrightarrow> (\<exists>Qx fx. kind a' = Qx\<hookleftarrow>\<^bsub>p\<^esub>fx)"
terminates_in_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst" shows "substitution.terminates_in subst e s' \<equiv> \<exists>s. substitution.maximal_execution subst e s s'"
Neg_Conj_E: fixes H :: "fm set"   and B :: "fm"   and A :: "fm"   and C :: "fm" assumes "H \<turnstile> B"   and "insert (Neg A) H \<turnstile> C" shows "insert (Neg (A AND B)) H \<turnstile> C"
has_fields_is_class_Object: fixes P :: "'a prog"   and D :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> D has_fields FDTs" shows "is_class P Object"
mono_Ndet_sym: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq> Q" shows "(S |-| P) \<sqsubseteq> (S |-| Q)"
direct_decompD_1: fixes A :: "'b set"   and ss :: "'b set list"   and qs :: "'b list" assumes "direct_decomp A ss"   and "qs \<in> listset ss" shows "sum_list qs \<in> A"
less_add_same_cancel2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus b a) = less zero b"
gen_lossless_gpv_intros: fixes p :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf"   and \<I> :: "('e, 'f) \<I>"   and co :: "bool" assumes "lossless_spmf p"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> gen_lossless_gpv co \<I> (c input)" shows "gen_lossless_gpv co \<I> (GPV p)"
eventually_in_atLeastAtMostI: fixes l :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and u :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in at_top. l x \<le> f x"   and "\<forall>\<^sub>F x in at_top. f x \<le> u x" shows "\<forall>\<^sub>F x in at_top. f x \<in> {l x..u x}"
axiom_true_def: shows "axiom_true \<equiv> Thm (Iff Truth (Imp Falsity Falsity))"
simps_5: fixes f1 :: "nat \<Rightarrow> tf \<Rightarrow> t list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f2 :: "tf \<Rightarrow> host \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "tf"   and x13 :: "t list"   and x14 :: "b_e list" shows "(case Func_native x11 x12 x13 x14 of Func_native x xa xb xc \<Rightarrow> f1 x xa xb xc | Func_host x xa \<Rightarrow> f2 x xa) = f1 x11 x12 x13 x14"
node_total_order_irrefl: fixes history :: "nat \<Rightarrow> 'c list"   and e :: "'c"   and i :: "nat" assumes "node_histories history" shows "\<not> node_histories.history_order history e i e"
converseI: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
emb_step_at_if_position: fixes t :: "('c, 'd) tm"   and p :: "dir list"   and d :: "dir" assumes "position_of t (p @ [d])" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b emb_step_at p d t"
simps: fixes S :: "'a st"   and X :: "char list set" shows "top_on\<^sub>s S X = (fun S = \<lambda>x. \<top> on X)"
SeqAuto_def: fixes S :: "'a set"   and I :: "'a"   and L :: "(('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) set"   and D :: "('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set" shows "SeqAuto S I L D = (I \<in> S \<and> S \<noteq> {} \<and> finite S \<and> finite D \<and> (\<forall>(s, l, t)\<in>D. s \<in> S \<and> t \<in> S \<and> l \<in> L))"
YFtorNT_YMapInv: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "YFtorNT C f = YMapInv C X (Hom\<^bsub>C\<^esub>[\<midarrow>,Y]) (m2z\<^bsub>C\<^esub>f)"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VV.seq g f = (VV.arr f \<and> VV.arr g \<and> CC.seq g f)"
cotrace_prepend: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and u :: "('a, 'b, 'c) event llist" assumes "trace c t c'"   and "cotrace c' u" shows "cotrace c (prepend t u)"
Gromov_product_at_diff1: fixes a :: "'a"   and x :: "'a"   and y :: "'a"   and b :: "'a" shows "\<bar>Gromov_product_at a x y - Gromov_product_at b x y\<bar> \<le> dist a b"
mat_size_alt: shows "M_mat_R signs subsets \<in> carrier_mat (length subsets) (length subsets)"
UN_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x" shows "\<Union> (f ` A) \<subseteq> \<Union> (g ` B)"
rep_eq: fixes x :: "'a set"   and xa :: "('a, 'b) fmap" shows "fmlookup (fmdrop_set x xa) = map_drop_set x (fmlookup xa)"
multi_self_add_other_not_self: fixes M :: "'a multiset"   and x :: "'a" shows "(M = x, M) = False"
lessalive_def: fixes s :: "Store"   and t :: "Store" shows "s \<lless> t = ((\<forall>x. alive x s \<longrightarrow> s \<equiv>[x] t) \<and> (\<forall>f. s@@staticLoc f = t@@staticLoc f))"
FactRing_zeroideal_2: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "R \<simeq> R Quot {\<zero>\<^bsub>R\<^esub>}"
id_remove_1_mset_iff_notin: fixes M :: "'b multiset"   and a :: "'b" shows "(M = remove1_mset a M) = (a \<notin># M)"
ltrm_sub: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "0 < deg R g" shows "monom P (Cring_Poly.compose R f g (deg R (Cring_Poly.compose R f g))) (deg R (Cring_Poly.compose R f g)) = monom P (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g))) (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g))"
interchange_rows_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f"   and b :: "'f" shows "interchange_rows A a b = (\<chi>i j. if i = a then A $ b $ j else if i = b then A $ a $ j else A $ i $ j)"
rfn_tool8: fixes r :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "compseries G r f"   and "0 < r" shows "d_gchain G r f"
inner_zero_right: fixes x :: "real" shows "x * 0 = 0"
WT_gpv_mono: fixes \<I>1 :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>2 :: "('a, 'b) \<I>" assumes "\<I>1 \<turnstile>g gpv \<surd>"   and "outs_\<I> \<I>1 \<subseteq> outs_\<I> \<I>2"   and "\<And>x. x \<in> outs_\<I> \<I>1 \<Longrightarrow> responses_\<I> \<I>2 x \<subseteq> responses_\<I> \<I>1 x" shows "\<I>2 \<turnstile>g gpv \<surd>"
SN_rel_imp_SN: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "SN_rel R S" shows "SN R"
euclid_ext_poly_mod_main_def: fixes p :: "int"   and ff_ops :: "'a arith_ops_record"   and f :: "int poly"   and g :: "int poly" shows "euclid_ext_poly_mod_main p ff_ops f g = (case bezout_coefficients_i ff_ops (of_int_poly_i ff_ops f) (of_int_poly_i ff_ops g) of (a, b) \<Rightarrow> (to_int_poly_i ff_ops a, to_int_poly_i ff_ops b))"
hn_refine_def: fixes \<Gamma> :: "assn"   and c :: "'c Heap"   and \<Gamma>' :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and m :: "'d nres" shows "hn_refine \<Gamma> c \<Gamma>' R m \<equiv> nofail m \<longrightarrow> <\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t"
magnetism_def: shows "magnetism = R\<^sub>e 25001 / R\<^sub>e (10 ^ 5) + Power (Var 0) 2 + R\<^sub>e 2 * Power (Var 1) 2 + R\<^sub>e 2 * Power (Var 2) 2 + R\<^sub>e 2 * Power (Var 3) 2 + R\<^sub>e 2 * Power (Var 4) 2 + R\<^sub>e 2 * Power (Var 5) 2 + R\<^sub>e 2 * Power (Var 6) 2 - Var 0"
empty_dom: fixes g :: "'g \<Rightarrow> 'h option" assumes "{} = dom g" shows "g = Map.empty"
hyps_2: fixes b :: "'b" shows "\<Psi> \<otimes> b \<rhd> P \<parallel> !P \<longmapsto> Rs"
GB_imp_reducibility: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "punit.is_Groebner_basis G"   and "f \<noteq> 0"   and "f \<in> local.punit.pmdl G" shows "punit.is_red G f"
acyclic_on_def: fixes U :: "'c set" shows "acyclic_on U = (\<lambda>r. \<forall>x\<in>U. (x, x) \<notin> on U r\<^sup>+)"
WriteResult: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs" shows "\<turnstile> Write ch mm rs p l \<longrightarrow> (rs!p)$ \<noteq> #NotAResult"
that: fixes p :: "rat poly" assumes "\<And>a. rat_of_int (f a) = poly p (rat_of_int a)" shows "thesis"
exp_times_scaleR_commute: fixes t :: "real"   and A :: "'b" shows "exp (t *\<^sub>R A) * A = A * exp (t *\<^sub>R A)"
schroeder_1_var: fixes x :: "'a" shows "is_conjugation ((;) x) ((;) (x\<^sup>\<smile>))"
S_intersection2_def: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection2 p q \<equiv> proj2_abs (S_intersection2_rep p q)"
path_through_components: fixes H :: "'a"   and e :: "'a" assumes "(top \<le> H \<and> H \<sqinter> H \<le> H) \<and> local.comp_inf.symmetric H"   and "(e \<sqinter> top \<sqinter> id (e \<sqinter> top) \<le> top \<and> top \<sqinter> (e \<sqinter> top) = top) \<and> id e \<sqinter> top \<sqinter> id (id e \<sqinter> top) \<le> top \<and> top \<sqinter> (id e \<sqinter> top) = top" shows "top = top \<squnion> top \<sqinter> H \<sqinter> e \<sqinter> top"
those_eq_SomeD: fixes f :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list" assumes "those (map f xs) = Some ys" shows "ys = map (the \<circ> f) xs \<and> (\<forall>i. \<exists>y. i < length xs \<longrightarrow> f (xs ! i) = Some y)"
Func_emp2: fixes A :: "'d set" assumes "A \<noteq> {}" shows "Func A {} = {}"
step: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and c' :: "'c"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and lc :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and c'' :: "'c"   and mds'' :: "Mode \<Rightarrow> 'a set"   and mem'' :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "((c', mds'), mem') \<in> sifum_security_init.loc_reach dma eval lc"   and "(((c', mds'), mem'), (c'', mds''), mem'') \<in> eval" shows "((c'', mds''), mem'') \<in> sifum_security_init.loc_reach dma eval lc"
measurable_Pair2: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and M1 :: "'c measure"   and M2 :: "'d measure"   and M :: "'e measure"   and x :: "'c" assumes "f \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M"   and "x \<in> space M1" shows "(\<lambda>y. f (x, y)) \<in> M2 \<rightarrow>\<^sub>M M"
int_nat_two_exp: fixes k :: "nat" shows "2 ^ k = int (2 ^ k)"
l1_learn_def: fixes m :: "msg" shows "l1_learn m \<equiv> {(s, s'). synth (analz (insert m (ik s))) \<inter> secret s = {} \<and> s' = s\<lparr>ik := ik s \<union> {m}\<rparr>}"
bd_def: fixes x :: "'a"   and y :: "'a" shows "\<langle>x| y = range_op (y \<cdot> x)"
inv: shows "Partial_Cost_Model.config'_rand BIT s qs = type0 [x0, y0] x y"
tree_singleton: fixes u :: "'b" shows "tree (ins_node u graph_empty)"
swap_msgs_Send_RecvMarker: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
Gromov_extension_id_1: shows "Gromov_extension id = id"
square2_def: fixes n :: "nat"   and i :: "nat"   and j :: "nat" shows "square2 n i j = {i..<2 ^ n + i} \<times> {j..<2 ^ n + j}"
prems_2: shows "\<forall>y\<in>#SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q). eprio (findMin q) \<le> snd y"
ts\<^sub>2\<^sub>3: shows "instance_of\<^sub>t\<^sub>s ts\<^sub>2 ts\<^sub>3"
msb_big: fixes a :: "'a Word.word" shows "msb a = (2 ^ (LENGTH('a) - Suc 0) \<le> a)"
map\<^sub>1_simps_5: fixes f :: "'c" assumes "C.ide f" shows "D.cod (\<sigma>\<^sub>1 f) = \<sigma>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f"
dist_not_less_zero: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open" shows "\<not> dist x y < 0"
ccompatible_def: fixes \<R> :: "('d \<Rightarrow> 'e) set set"   and cc :: "('d, 'e) cconstraint" shows "ccompatible \<R> cc \<equiv> \<forall>R\<in>\<R>. R \<subseteq> \<lbrace>cc\<rbrace> \<or> \<lbrace>cc\<rbrace> \<inter> R = {}"
disjoint_list_partitioningIps: fixes ts :: "'a wordinterval list"   and ss :: "'a wordinterval list" assumes "{} \<notin> set (map wordinterval_to_set ts)"   and "disjoint_list (map wordinterval_to_set ts)"   and "wordinterval_list_to_set ss \<subseteq> wordinterval_list_to_set ts" shows "disjoint_list (map wordinterval_to_set (partitioningIps ss ts))"
cinner_eq_zero_iff: fixes x :: "'a" shows "(cinner x x = 0) = (x = (0::'a))"
nonempty_set_sub: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "{x \<in> A. P x} \<noteq> {}" shows "{x \<in> A. P x} \<subseteq> A"
P'_2: fixes P'' :: "'a list set \<times> 'a list" assumes "paths_good P''" shows "paths_weight P' \<le> paths_weight P''"
matpow_Suc: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and n :: "nat" shows "matpow A (Suc n) = matpow A n ** A"
simps_17: fixes v :: "\<I>"   and va :: "formula Regex.regex" shows "get_and_list (MatchP v va) = [MatchP v va]"
strip_map_acom: fixes f :: "'b \<Rightarrow> 'a"   and C :: "'b acom" shows "strip (ACom.map_acom f C) = strip C"
rank_eq_state_rank: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "\<MM>.rank x n = \<MM>.state_rank (\<MM>.token_run x n) n"
add_mset_eq_add_mset: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a"   and M' :: "'a multiset" shows "(add_mset a M = add_mset b M') = (a = b \<and> M = M' \<or> a \<noteq> b \<and> b \<in># M \<and> add_mset a (remove1_mset b M) = M')"
finite_measure_axioms: fixes p :: "'c spmf" shows "finite_measure (measure_spmf p)"
sep_conj_conj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "((\<lambda>s. P s \<and> Q s) \<and>* R) h" shows "(P \<and>* R) h \<and> (Q \<and>* R) h"
Span_is_add_subgroup: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "subgroup (Span K Us) (add_monoid R)"
simps: fixes i :: "nat"   and n :: "rat" shows "index_geq_n i n = GEQ (lp_monom 1 i) n"
ends_in_segment_1: fixes a :: "'c"   and b :: "'c" shows "a \<in> closed_segment a b"
trans: fixes y :: "'b"   and x :: "'b"   and z :: "'b" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
hyps_along_Cons: fixes fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and ent :: "'form fset \<times> 'form"   and i :: "nat"   and "is" :: "nat list" assumes "iwf fc t ent"   and "i # is \<in> it_paths t" shows "hyps_along t (i # is) = (\<lambda>h. subst (iSubst t) (freshen (iAnnot t) (labelsOut (iNodeOf t) h))) ` fset (hyps_for (iNodeOf t) (inPorts' (iNodeOf t) ! i)) \<union> hyps_along (iAnts t ! i) is"
assms_4: shows "continuous (at x within {a..b} - s) f"
eq_upto_seq_inconsist_appendI: fixes P :: "'d prog"   and obs :: "('e, 'f) obs_event action list"   and OBS :: "('e, 'f) obs_event action list"   and vs :: "'e \<times> addr_loc \<Rightarrow> ('e val \<times> bool) option"   and obs' :: "('e, 'f) obs_event action list"   and OBS' :: "('e, 'f) obs_event action list" assumes "eq_upto_seq_inconsist P obs OBS vs"   and "ta_seq_consist P vs (llist_of obs) \<Longrightarrow> eq_upto_seq_inconsist P obs' OBS' (mrw_values P vs OBS)" shows "eq_upto_seq_inconsist P (obs @ obs') (OBS @ OBS') vs"
free_tv_app_subst_te: fixes S :: "nat \<Rightarrow> typ"   and t :: "typ" shows "free_tv ($ S t) \<subseteq> cod S \<union> free_tv t"
igSwapBinp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'b, 'j) model_scheme"   and zs :: "'e"   and z1 :: "'h"   and z2 :: "'h"   and binp :: "'a \<Rightarrow> 'b option" shows "igSwapBinp MOD zs z1 z2 binp \<equiv> lift (igSwapAbs MOD zs z1 z2) binp"
As'_p: shows "map (\<lambda>L. L \<cdot>l \<eta>) (map Neg As') = map Neg As \<and> mset (map Neg As') = SDA'"
elims_2: fixes x :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool"   and xa :: "('c, 'd) chain" assumes "chain_all x xa" obtains \<phi> :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool" and \<sigma> :: "'c" where "xa__ = \<phi>__" and "xa = \<lbrace> \<sigma>__ \<rbrace>"   | \<phi> :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool" and \<sigma> :: "'c" and x :: "'d" and \<Pi> :: "('c, 'd) chain" where "xa__ = \<phi>" and "xa = \<lbrace> \<sigma> \<rbrace> \<cdot> x \<cdot> \<Pi>" and "\<phi> (\<sigma>, x, pre \<Pi>) \<and> chain_all \<phi> \<Pi>"
linear_injective_0: fixes f :: "'b \<Rightarrow> 'c" assumes "linear f" shows "inj f = (\<forall>x. f x = (0::'c) \<longrightarrow> x = (0::'b))"
id: fixes x :: "'a" assumes "x \<in> S" shows "poly p x = poly q x"
funpower_endomorphism: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "ChamberComplexEndomorphism X f" shows "ChamberComplexEndomorphism X (f ^^ n)"
Args_min_simps_2: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a" shows "is_arg_min_on f S x = (x \<in> S \<and> (\<forall>y\<in>S. f x \<le> f y))"
mono_neutral_right: fixes T :: "'d set"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'e)" shows "prod g T = prod g S"
cod_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>" shows "cod (mkIde A) = mkIde A"
exp_ln: fixes x :: "real" assumes "0 < x" shows "exp (ln x) = x"
ax_SkipI: fixes P :: "'b assn"   and Q :: "'b assn"   and G :: "prog"   and A :: "'b triple set" assumes "P \<Rightarrow> Q\<leftarrow>\<diamondsuit>" shows "G,A\<turnstile>{P} .Skip. {Q}"
linrqe_def: fixes p :: "fm" shows "linrqe p = qelim (prep p) ferrack"