hcju/isabelleps huggingface dataset:
least_def: fixes uu :: "'e itself"   and Rs :: "(('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph) set"   and S :: "('f, 'h) labeled_graph"   and G :: "('f, 'h) labeled_graph" shows "least uu Rs S G \<equiv> subgraph S G \<and> (\<forall>C. consequence_graph Rs C \<longrightarrow> maintained (S, G) C)"
assms_4: shows "\<exists>x\<in>closure S. f x < a"
that_2: assumes "x \<notin> A"   and "f x = undefined" shows "thesis"
obtain_qFresh: fixes V :: "'var set"   and XS :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and xs :: "'varSort" assumes "finite V \<or> |V| <o |UNIV|"   and "finite XS \<or> |XS| <o |UNIV|"   and "Ball XS qGood" shows "\<exists>x. x \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
space_uniform_measure: fixes M :: "'b measure"   and A :: "'b set" shows "space (uniform_measure M A) = space M"
inf_sup_ord_4: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
Dec: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "Enc X Y \<in> analz H"   and "Y \<in> synth (analz H)" shows "X \<in> analz H"
Listn_sl: fixes L :: "'b set \<times> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'b \<Rightarrow> 'b)"   and n :: "nat" assumes "semilat L" shows "semilat (Listn.sl n L)"
suf_cancel_eq: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" shows "ys \<cdot> xs \<le>s zs \<cdot> xs = ys \<le>s zs"
psubset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "(A \<subset> x \<triangleright> B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
real_Cauchy_convergent: fixes X :: "nat \<Rightarrow> real" assumes "Cauchy X" shows "convergent X"
fcomp_apply: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'b"   and x :: "'c" shows "(f ;; g) x = g (f x)"
pickFT_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "pickFT cl \<equiv> SOME n. pickFT_pred cl n"
FINITE_MPLS: fixes Pi :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" assumes "finite Pi" shows "finite (MPLS Pi)"
nrename_ImpR: fixes R :: "trm"   and u :: "name"   and v :: "name"   and x :: "name"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[u\<turnstile>n>v] = ImpR x.<c>.N d"   and "c \<sharp> (R, d)"   and "x \<sharp> (R, u, v)" shows "\<exists>N'. R = ImpR x.<c>.N' d \<and> N'[u\<turnstile>n>v] = N"
card_face_cycle_sets_da': shows "card H.face_cycle_sets = card G.face_cycle_sets - card {G.face_cycle_set a, G.face_cycle_set a'} + (if S = {} then 0 else 1)"
InvariantWatchesDifferNotifyWatchesLoop: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl newWl state in InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
intro: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G"   and "Group.group G"   and "normal_axioms H G" shows "H \<lhd> G"
abs_summable_finiteI: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "real" assumes "\<And>F. \<lbrakk>finite F; F \<subseteq> S\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>F. norm (f x)) \<le> B" shows "f abs_summable_on S"
vsv_vdomain_vsingleton_vrange_vsingleton: fixes a :: "V" assumes "\<D>\<^sub>\<circ> A = ZFC_in_HOL.set {a}" obtains b :: "V" where "\<R>\<^sub>\<circ> A = ZFC_in_HOL.set {b}"
bind_right_identity: fixes m :: "('a, 'b) state" shows "m \<bind> State_Monad.return = m"
le_trans_top: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "le_trans t (\<lambda>P s. bound_of P)"
runit_hcomp_3: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<r>[f \<star> g] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, src g] = f \<star> \<r>[g]"
weakly_Pareto_optimal_for_men_ex: shows "Ex weakly_Pareto_optimal_for_men"
makes_compatible_invariant: 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 cms\<^sub>1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and cms\<^sub>2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list"   and k :: "nat"   and cms\<^sub>1' :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>1' :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.sound_mode_use dma \<C>_vars eval (cms\<^sub>1, mem\<^sub>1)"   and "sifum_security_init.sound_mode_use dma \<C>_vars eval (cms\<^sub>2, mem\<^sub>2)"   and "sifum_security_init.makes_compatible dma \<C>_vars eval (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "((cms\<^sub>1, mem\<^sub>1), k, cms\<^sub>1', mem\<^sub>1') \<in> sifum_security_init.meval eval" obtains cms\<^sub>2' :: "('c \<times> (Mode \<Rightarrow> 'a set)) list" and mem\<^sub>2' :: "'a \<Rightarrow> 'b" and mems' :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list" where "map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> ((cms\<^sub>2, mem\<^sub>2), k, cms\<^sub>2', mem\<^sub>2') \<in> sifum_security_init.meval eval \<and> sifum_security_init.makes_compatible dma \<C>_vars eval (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'"
lucas_lehmer_mult_cong_2: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "[snd (lucas_lehmer_mult m x y) = fst x * snd y + snd x * fst y] (mod m)"
well_behaved_composition: shows "wellBehavedComposition ES1 ES2 \<V> \<V>1 \<V>2"
lfp_mono: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "\<And>Z. f Z \<le> g Z" shows "lfp f \<le> lfp g"
wf_ipath_epath: fixes t :: "('sequent \<times> 'rule) tree"   and steps :: "('sequent \<times> 'rule) stream" assumes "local.wf t"   and "ipath t steps" shows "epath steps"
scaleR_blinfun_compose_left: fixes i :: "real"   and A :: "'g \<Rightarrow>\<^sub>L 'f"   and B :: "'e \<Rightarrow>\<^sub>L 'g" shows "i *\<^sub>R (A o\<^sub>L B) = i *\<^sub>R A o\<^sub>L B"
\<phi>_\<psi>: fixes b :: "'c"   and a :: "'c"   and x :: "'s" assumes "C.ide b"   and "C.ide a"   and "x \<in> local.set (b, a)" shows "\<phi> (b, a) (\<psi> (b, a) x) = x"
dmono_def: fixes D :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "dmono D = (\<forall>ij. mono (D ij))"
get_component_separates_tree_order: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'"   and "ptr' \<notin> set c" shows "set to \<inter> set c' = {}"
eq2: shows "local.span (\<Union> (some_basis ` set ss)) = sum_list ` listset ss"
Class_subset: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equivalence S E"   and "(a, b) \<in> E" shows "equivalence.Class S E a \<subseteq> equivalence.Class S E b"
conf_gext: fixes G :: "prog"   and s :: "st"   and v :: "val"   and T :: "ty"   and s' :: "st" assumes "G,s\<turnstile>v\<Colon>\<preceq>T"   and "s\<le>|s'" shows "G,s'\<turnstile>v\<Colon>\<preceq>T"
A: shows "x < y in fst (config\<^sub>p (rTS []) init qs) = x < y in fst (config\<^sub>p (rTS []) (Lxy init {x, y}) (Lxy qs {x, y}))"
L1_D_2: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> space\<^sub>N (\<LL> 1 M)" shows "integrable M f"
distinct_2: fixes x21 :: "'d tree"   and x22 :: "'d"   and x23 :: "'d tree" shows "\<langle>x21, x22, x23\<rangle> \<noteq> \<langle>\<rangle>"
measurable_integrable: fixes S :: "'b set" shows "(S \<in> lmeasurable) = (indicat_real S integrable_on UNIV)"
list_emb_Cons_iff1: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "P x y" shows "list_emb P (x # xs) (y # ys) = list_emb P xs ys"
convex_domain: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f" shows "convex (domain f)"
submoduleE_1: fixes H :: "'a vec set" assumes "Module.submodule H class_ring V" shows "H \<subseteq> carrier_vec n"
ln_bounds_1: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x"   and "x < 1" shows "(\<Sum>i = 0..<2 * n. (- 1) ^ i * (1 / real (i + 1)) * x ^ Suc i) \<le> ln (x + 1)"
img_is_add_subgroup: fixes H :: "'a set" assumes "subgroup H (add_monoid R)" shows "subgroup (h ` H) (add_monoid S)"
simps_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C" shows "monoidal_language.Cod C (t \<^bold>\<otimes> u) = monoidal_language.Cod C t \<^bold>\<otimes> monoidal_language.Cod C u"
intf: fixes a :: "'a"   and b :: "'a" shows "(\<lambda>x. if x \<in> S then f x else (0::'b)) integrable_on cbox a b"
trm_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b"   and f :: "'b" assumes "category_with_terminal_object C"   and "partial_magma.in_hom C t (partial_magma.dom C f) (category_with_terminal_object.some_terminal C)" shows "t = category_with_terminal_object.trm C f"
bexE: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" obtains x :: "'c" where "x \<in> A" and "P x"
SN_on_imp_accp: fixes g :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "SN_on {(y, z). g z y} A" shows "\<forall>x\<in>A. Wellfounded.accp g x"
orderI: fixes a :: "'d"   and b :: "'d" assumes "a = a \<sqinter> b" shows "a \<le> b"
real_of_nat_eq_numeral_power_cancel_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and y :: "nat"   and x :: "num"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero y = power.power one times (numeral.numeral one plus x) n) = (y = numeral x ^ n)"
sum_is_submodule: fixes N1 :: "'c set"   and N2 :: "'c set" assumes "LinearCombinations.submodule K N1 V"   and "LinearCombinations.submodule K N2 V" shows "LinearCombinations.submodule K (subspace_sum N1 N2) V"
assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a \<squnion> b \<squnion> c = a \<squnion> (b \<squnion> c)"
hyps_3: shows "f (sum g F) = (\<Sum>a\<in>F. f (g a))"
F2: shows "\<forall>w u. {R \<in> u. R \<in> w} = u \<inter> w"
nsDistr: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows " n \<times>\<^bsub>R\<^esub> x \<plusminus>  m \<times>\<^bsub>R\<^esub> x =  (n + m) \<times>\<^bsub>R\<^esub> x"
mbisim_delay_bisimulation: shows "delay_bisimulation_diverge r1.redT r2.redT mbisim mta_bisim m\<tau>move1 m\<tau>move2"
assms_5: fixes y :: "'a"   and z :: "'a" assumes "r\<^sup>*\<^sup>* a y"   and "r y z"   and "I y"   and "I z"   and "P y" shows "P z"
commute: fixes a :: "'d upred"   and b :: "'d upred" shows "(a \<or> b) = (b \<or> a)"
otherwithE: fixes Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'e set"   and P :: "('e \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd"   and a :: "'f"   and R :: "('e \<Rightarrow> 'd) \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool" assumes "otherwith Q I P \<sigma> \<sigma>' a"   and "\<lbrakk>P \<sigma> a; \<forall>j. j \<notin> I \<longrightarrow> Q (\<sigma> j) (\<sigma>' j)\<rbrakk> \<Longrightarrow> R \<sigma> \<sigma>' a" shows "R \<sigma> \<sigma>' a"
basic_monos_1: fixes A :: "'b set" shows "A \<subseteq> A"
fmlookup_default_if_1: fixes ys :: "('b, 'a) fmap"   and a :: "'b"   and r :: "'a"   and d :: "'a" assumes "fmlookup ys a = Some r" shows "fmlookup_default d ys a = r"
z_def: shows "z = embed_pmf (\<lambda>(x, y). enn2real (h x y))"
aasc_seqn: fixes a :: "ant"   and A :: "ant set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "aasc_seq A a n < aasc_seq A a (Suc n)"
monad_resumption_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and pause :: "'d \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_resumption return bind pause" shows "monad_resumption (return_writer return) (bind_writer return bind) (pause_writer pause)"
mono: fixes x :: "'a" shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) (\<lambda>xs. f x xs (LCons x xs))"
iNodeOf_outPorts: 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 "is" :: "nat list" assumes "iwf fc t ent"   and "is \<in> it_paths t"   and "outPorts (iNodeOf (tree_at t is)) = {||}" shows "False"
tendsto_within_subset: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b filter"   and x :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "filterlim f l (at x within S)"   and "T \<subseteq> S" shows "filterlim f l (at x within T)"
order_consistent_def: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "order_consistent r s = (\<forall>a a'. (a, a') \<in> r \<longrightarrow> (a', a) \<in> s \<longrightarrow> a = a')"
continuous_map_axioms_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and is_open' :: "'d set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map_axioms S is_open is_open' f \<equiv> \<forall>U. is_open' U \<longrightarrow> is_open (f \<^sup>\<inverse> S U)"
AE_iff_emeasure_eq_1: fixes M :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "prob_space M"   and "Measurable.pred M P" shows "almost_everywhere M P = (emeasure M {x \<in> space M. P x} = 1)"
ereal_infty_less_eq2_2: fixes a :: "ereal"   and b :: "ereal" assumes "a \<le> b"   and "b = - \<infinity>" shows "a = - \<infinity>"
oInv_def: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" shows "oInv F a = (if F 0 \<le> a then THE x. F x \<le> a \<and> a < F (oSuc x) else 0)"
set_transfer: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (fmrel R) (rel_set R) fmran' fmran'"
assms_2: shows "KD_Tree.size_kdt kdt = 2 ^ h"
last_ConsR: fixes xs :: "'b list"   and x :: "'b" assumes "xs \<noteq> []" shows "last (x # xs) = last xs"
rep_eq: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.Tree\<^sub>\<alpha>" shows "L_transform_Tree\<^sub>\<alpha> x = L_transform_Tree (FL_Formula.rep_Tree\<^sub>\<alpha> x)"
T_mtf_eq: shows "T_mtf (length rs) = int (T_on MTF s0 rs)"
frontier_Union_subset_closure: fixes F :: "'a set set" shows "frontier (\<Union> F) \<subseteq> closure (\<Union> (frontier ` F))"
span_extend_basis: fixes B :: "'c set" assumes "l.vs2.independent B" shows "l.vs2.span (l.vs2.extend_basis B) = UNIV"
af_ltl_continuation_suffix: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl"   and i :: "nat" shows "w \<Turnstile> \<phi> = suffix i w \<Turnstile> af \<phi> (prefix i w)"
gAbsInj_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gAbsInj MOD \<equiv> \<forall>xs s x X' X X1' X1. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<and> wls s X1' \<and> gWls MOD s X1 \<and> gAbs MOD xs x X' X = gAbs MOD xs x X1' X1 \<longrightarrow> X = X1"
tl_Q_hit_pre: shows "tl (Q_hit_pre @ [y]) \<noteq> []"
valOf: fixes c :: "com"   and s :: "loc \<Rightarrow> nat"   and s0 :: "loc \<Rightarrow> nat"   and e :: "exp"   and n :: "nat"   and s1 :: "loc \<Rightarrow> nat" assumes "(c, s) -[eval]-> s0"   and "(e, s0) -|-> (n, s1)" shows "(VALOF c RESULTIS e, s) -|-> (n, s1)"
continuous_openin_preimage_eq: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on S f = (\<forall>T. open T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
fps_mult_subdegree_ge: fixes f :: "'a fps"   and g :: "'a fps" assumes "f * g \<noteq> 0" shows "subdegree f + subdegree g \<le> subdegree (f * g)"
rbt_min_opt_in_set: fixes t :: "('b, unit) RBT_Impl.rbt" assumes "t \<noteq> rbt.Empty" shows "rbt_min_opt t \<in> set (RBT_Impl.keys t)"
enum_inj: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(s.enum i = s.enum j) = (i = j)"
llist_all2_lmap2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and g :: "'c \<Rightarrow> 'b"   and ys :: "'c llist" shows "llist_all2 P xs (lmap g ys) = llist_all2 (\<lambda>x y. P x (g y)) xs ys"
c_in_upper_bound: fixes x :: "nat"   and u :: "nat" assumes "c_in x u = 1" shows "2 ^ x \<le> u"
eval\<^sub>G_respectfulness_1: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and S :: "'b ltl set" assumes "\<phi> \<longrightarrow>\<^sub>P \<psi>" shows "eval\<^sub>G S \<phi> \<longrightarrow>\<^sub>P eval\<^sub>G S \<psi>"
ordered_powerprod_axioms: shows "ordered_powerprod ord ord_strict"
hyperb_ineq: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and deltaG :: "'a itself \<Rightarrow> real"   and e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "min (Gromov_product_at e x y) (Gromov_product_at e y z) - deltaG TYPE('a) \<le> Gromov_product_at e x z"
topology_finer_continuous_id: fixes X :: "'c topology"   and Y :: "'c topology" assumes "topspace X = topspace Y" shows "(\<forall>S. openin X S \<longrightarrow> openin Y S) = continuous_map Y X id"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<le> a + b"
abelian_monoidE_4: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_monoid R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
func_times: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f * g = (\<lambda>x. f x * g x)"
1: shows "(\<lambda>k. ln (real (primorial' k))) \<sim>[sequentially] (\<lambda>x. real x * ln (real x))"
k_def: shows "k = Max (multiplicity p ` {1..n})"
mult_nonneg_nonneg: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "(0::'b) \<le> b" shows "(0::'b) \<le> a * b"
counit_in_hom_2: shows "\<guillemotleft>\<psi> : F f \<star>\<^sub>D g' \<Rightarrow>\<^sub>D trg\<^sub>D (F f)\<guillemotright>"
min_step_inext: fixes x :: "nat"   and y :: "nat"   and I :: "nat set" assumes "x < y"   and "x \<in> I"   and "y \<in> I"   and "\<And>k. \<lbrakk>x < k; k < y\<rbrakk> \<Longrightarrow> k \<notin> I" shows "inext x I = y"
l_coset_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and H :: "'c set" shows "a <#\<^bsub>G\<^esub> H = (\<Union>h\<in>H. {a \<otimes>\<^bsub>G\<^esub> h})"
simps_8: fixes x2 :: "'b" shows "PVar x2 \<noteq> PUnit"
fw_upd_def: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "linordered_monoid.fw_upd m k i j \<equiv> upd m i j (linordered_monoid.min (m i j) (dbm_add (m i k) (m k j)))"
\<Psi>_def: shows "\<Psi> \<equiv> \<lambda>l m n M N j. m < n \<and> N \<subseteq> M \<and> n \<in> M \<and> (\<forall>U. Form l U \<and> U \<subseteq> WW \<and> [n] < inter_scheme l U \<and> list.set (inter_scheme l U) \<subseteq> N \<longrightarrow> g U = j)"
directly_subsumes_reflexive: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and s2 :: "nat"   and t :: "transition" shows "directly_subsumes e1 e2 s1 s2 t t"
is_done_refine: fixes s :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list"   and v :: "'a" assumes "GS_invar s" shows "is_done_impl v s = (v \<in> GS.D_\<alpha> s)"
cOpen: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a[x] \<prec> P'"   and "a \<noteq> x" shows "F C (<\<nu>x>P) a x P'"
sym: fixes x :: "'a"   and y :: "'a" shows "invert_order (cmp x y) = cmp y x"
subst_Mu: fixes T :: "Syntax.type"   and c :: "cmd"   and s :: "trm"   and k :: "nat" shows "(\<mu>T:c)[s/k]\<^sup>T = \<mu>T:c[liftM_trm s 0/k]\<^sup>C"
unit_triangle_1: fixes a :: "'a" assumes "obj a" shows "\<i>[a] \<star> a = (a \<star> \<i>[a]) \<cdot> \<a>[a, a, a]"
card: shows "card ((*\<^sub>v) B ` bas) = card bas"
assms_2: shows "b2 \<in># remove1_mset b1 \<B>"
IH1_1: assumes "M \<Turnstile> F" shows "M(p := True) \<Turnstile> F"
eval_ubx_correct: fixes opubx :: "opubx"   and xs :: "(dynamic, nat, bool) unboxed list"   and z :: "(dynamic, nat, bool) unboxed" assumes "eval_ubx opubx xs = Some z" shows "eval_inl (deubx opubx) (map unboxed_dynamic.norm_unboxed xs) = unboxed_dynamic.norm_unboxed z"
root_ptr: shows "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root_ptr"
insert_not_empty: fixes a :: "'d"   and A :: "'d set" shows "insert a A \<noteq> {}"