hcju/isabelleps huggingface dataset:
l_set_tag_name_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs \<and> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
c1: shows "(\<forall>p\<in>set qs. degree p = 0) \<longrightarrow> poly_f qs \<noteq> 0"
cinfinite_not_czero: fixes r :: "('c \<times> 'c) set" assumes "cinfinite r" shows "(r, czero) \<notin> ordIso"
succ_works_1: fixes va :: "'a" assumes "va \<in> VV p" shows "va \<rightarrow> succ va"
b: shows "[Inl A] \<cdot> [Inl B] -G\<rightarrow>\<^sup>* map Inr w"
the_inv_into_map_None: fixes V :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" shows "(the_inv_into_map V f x = None) = (x \<notin> f ` V)"
simps_9: fixes e\<^sub>1 :: "expr"   and F :: "char list"   and Cs :: "char list list"   and e\<^sub>2 :: "expr"   and A :: "char list set option" shows "\<D> (e\<^sub>1\<bullet>F{Cs} := e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
lab_def_3: shows "\<S>\<bullet>\<C> \<turnstile> es'' : ts'' _> t2s"
simps_5: fixes x :: "'c"   and \<phi> :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (Exists x \<phi>) \<sigma> = Exists x (Fsubst \<phi> \<sigma>)"
distinct_remdups_equal_3: fixes xs :: "real_alg_3 list" shows "distinct (map real_of_3 (Complex_Algebraic_Numbers.remdups_gen equal_3 xs))"
simps_2: fixes \<rho> :: "'b run"   and K :: "clock"   and n :: "nat" shows "#\<^sub>< \<rho> K Suc n = #\<^sub>\<le> \<rho> K n"
a_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> I"   and "y \<in> I" shows "x \<oplus> y \<in> I"
x_def: shows "x = ((1::'f) # map uminus (as @ bs)) \<bullet>\<cdot> (w # us @ vs)"
skl1_obs_iagreement_Resp: shows "oreach skl1 \<subseteq> l2_iagreement_Resp"
rsd_winners_manipulation_aux: fixes Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and agents' :: "'agent list" assumes "total_preorder_on alts Ri'"   and "i \<in> agents"   and "set agents' \<subseteq> agents"   and "finite agents"   and "finite alts"   and "alts \<noteq> {}" shows "rsd_winners (R(i := Ri')) alts agents' = rsd_winners R alts agents' \<or> RSD_set_rel (R i) (rsd_winners (R(i := Ri')) alts agents') (rsd_winners R alts agents')"
l_new_element_get_disconnected_nodes_def: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_disconnected_nodes get_disconnected_nodes_locs \<equiv> \<forall>h new_element_ptr h' r ptr'. h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr \<longrightarrow> h \<turnstile> new_element \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h'"
swapAbs_vsubstAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A" shows "A $[y1 // y]_ys $[z1 \<and> z2]_zs = A $[z1 \<and> z2]_zs $[y1 @ys[z1 \<and> z2]_zs // y @ys[z1 \<and> z2]_zs]_ys"
basis_in_ex1_supset_circuit_in: fixes \<E> :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> B"   and "x \<in> \<E> - B" shows "\<exists>!C. \<E>.circuit_in \<E> C \<and> C \<subseteq> insert x B"
mset_subset_eq_insertD: fixes x :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" assumes "add_mset x A \<subseteq># B" shows "x \<in># B \<and> A \<subset># B"
continuous_at_Sup_mono: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "mono f"   and "continuous (at_left (\<Squnion> S)) f"   and "S \<noteq> {}"   and "bdd_above S" shows "f (\<Squnion> S) = \<Squnion> (f ` S)"
ID_restricted_global_subset_unrestricted_2: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a" shows "RID\<^sub>G (s(r := None)) \<subseteq> RID\<^sub>G s"
induced_id_inj: shows "inj_on (GroupByPresentationInducedFun.induced_hom S P id) (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>)"
alpha_Tree_induct: fixes t1 :: "('a, 'b, 'c, 'd) Tree"   and t2 :: "('a, 'b, 'c, 'd) Tree"   and P :: "('a, 'b, 'c, 'd) Tree \<Rightarrow> ('a, 'b, 'c, 'd) Tree \<Rightarrow> bool" assumes "t1 =\<^sub>\<alpha> t2"   and "\<And>tset1 tset2. \<lbrakk>\<And>a b. \<lbrakk>a \<in> set_bset tset1; b \<in> set_bset tset2; a =\<^sub>\<alpha> b\<rbrakk> \<Longrightarrow> P a b; rel_bset (=\<^sub>\<alpha>) tset1 tset2\<rbrakk> \<Longrightarrow> P (tConj tset1) (tConj tset2)"   and "\<And>t1 t2. \<lbrakk>t1 =\<^sub>\<alpha> t2; P t1 t2\<rbrakk> \<Longrightarrow> P (tNot t1) (tNot t2)"   and "\<And>f \<phi>. P (tPred f \<phi>) (tPred f \<phi>)"   and "\<And>f1 \<alpha>1 t1 f2 \<alpha>2 t2. \<lbrakk>\<And>p. p \<bullet> t1 =\<^sub>\<alpha> t2 \<Longrightarrow> P (p \<bullet> t1) t2; f1 = f2; \<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)\<rbrakk> \<Longrightarrow> P (tAct f1 \<alpha>1 t1) (tAct f2 \<alpha>2 t2)" shows "P t1 t2"
inj_on_eq_image': fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "inj_on f A"   and "order.greater_eq A X"   and "order.greater_eq A Y"   and "order.greater_eq (f ` Y) (f ` X)" shows "order.greater_eq Y X"
distinct_15: shows "ToUpperCaseOp \<noteq> CharactersOp"
simE2: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs" assumes "P \<leadsto><Rel> Q"   and "Q \<Longrightarrow>\<alpha> \<prec> Q'"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto>\<^sup>^<Rel> S" obtains P' :: "ccs" where "P \<Longrightarrow>\<alpha> \<prec> P'" and "(P', Q') \<in> Rel"
P1_real_ideal_DDH_advantage_true: fixes \<sigma> :: "bool"   and M :: "'a \<times> 'a"   and z :: "'e"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'f \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'g spmf)"   and D :: "'g \<times> 'a \<Rightarrow> bool spmf" assumes "\<sigma> = True" shows "mal_def.adv_P1 M \<sigma> z (P1_S1, P1_S2) \<A> D = local.ddh.DDH_advantage (P1_DDH_mal_adv_\<sigma>_true M z \<A> D)"
partition: shows "\<exists>i<n. rec_exec (rcs ! i) xs = 1 \<and> (\<forall>j<n. j \<noteq> i \<longrightarrow> rec_exec (rcs ! j) xs = 0)"
Restr_rtrancl_mono: fixes v :: "'a"   and w :: "'a"   and E :: "('a \<times> 'a) set"   and U :: "'a set" assumes "(v, w) \<in> (Restr E U)\<^sup>*" shows "(v, w) \<in> E\<^sup>*"
prems: shows "degree (pCons a1 p1) = degree (pCons a2 p2)"
simps_5: fixes var :: "nat"   and p :: "real mpoly"   and as :: "atom list" shows "find_eq var (Neq p # as) = (let (A, B) = find_eq var as in (A, Neq p # B))"
integrable_dominated_convergence: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and w :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>i. s i \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"   and "\<And>i. AE x in M. norm (s i x) \<le> w x" shows "integrable M f"
mult_scalar_minus_mult_left: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "- p \<odot> q = - (p \<odot> q)"
tendstoI: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F x in F. dist (f x) l < e" shows "filterlim f (topological_space.nhds open l) F"
inverse_cancel: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'c)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)" shows "((\<lambda>x. inverse (f x)) \<in> O[F](\<lambda>x. inverse (g x))) = (g \<in> O[F](f))"
Derives1_bound': fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "i \<le> length b"
simps_2: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "('a, 'c) flow_entry_match"   and as :: "('a, 'c) flow_entry_match list" shows "no_overlaps \<gamma> (a # as) = (no_overlaps \<gamma> as \<and> (\<forall>b\<in>set as. ofe_prio a = ofe_prio b \<longrightarrow> \<not> (\<exists>p\<in>UNIV. \<gamma> (ofe_fields a) p \<and> \<gamma> (ofe_fields b) p)))"
update_convs_3: fixes getConflictFlag' :: "bool \<Rightarrow> bool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getConflictFlag_update getConflictFlag' \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag' getConflictFlag, getC = getC, \<dots> = more\<rparr>"
ide_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.ide (B.VVV.dom f)"
lprefix_down_linear: fixes xs :: "'c llist"   and zs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> zs"   and "ys \<sqsubseteq> zs" shows "xs \<sqsubseteq> ys \<or> ys \<sqsubseteq> xs"
treap_def: fixes t :: "('a \<times> 'b) tree" shows "treap t = (bst (map_tree fst t) \<and> heap (map_tree snd t))"
assms_1: shows "N \<subseteq> Pow \<Omega>"
rawpsubst_imp: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubst (imp \<phi>1 \<phi>2) txs = imp (rawpsubst \<phi>1 txs) (rawpsubst \<phi>2 txs)"
subset_code_3: shows "(List.coset [] \<subseteq> set []) = False"
intros: fixes C :: "'a sequent"   and F :: "'a"   and Fs :: "'a form list"   and Ps :: "'a sequent list" assumes "C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* \<Empt>)" shows "leftPrincipal (Ps, C) (Compound F Fs)"
unbordered_pref_Lyndon: fixes u :: "'a list"   and v :: "'a list" assumes "Lyndon (u \<cdot> v)"   and "u \<noteq> \<epsilon>"   and "\<not> bordered u" shows "Lyndon u"
blinding_of_sum_hash: fixes boa :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and rha :: "'c \<Rightarrow> 'd"   and bob :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rhb :: "'e \<Rightarrow> 'f" assumes "boa \<le> vimage2p rha rha (=)"   and "bob \<le> vimage2p rhb rhb (=)" shows "rel_sum boa bob \<le> vimage2p (map_sum rha rhb) (map_sum rha rhb) (=)"
padic_simps_3: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int"   and n :: "nat" shows "padic_mult p f g n = f n \<otimes>\<^bsub>residue_ring (p ^ n)\<^esub> g n"
bind_returns_error_I3: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and x :: "'c"   and h' :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and e :: "'b" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>e e" shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>e e"
pre_addSnd: shows "pre_subdivFace' g f v v 0 (tl is)"
rp: shows "\<forall>b\<in>Basis. real_polynomial_function (\<lambda>x. f x \<bullet> b)"
DirProds_obtain_elem_carr: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and i :: "'e"   and x :: "'f" assumes "Group.group (DirProds G I)"   and "i \<in> I"   and "x \<in> carrier (G i)" obtains k :: "'e \<Rightarrow> 'f" where "k \<in> carrier (DirProds G I)" and "k i = x"
nonEmptyListFiltered: fixes P :: "'b \<Rightarrow> bool"   and l :: "'b list" assumes "P -` {True} \<inter> set l \<noteq> {}" shows "concat (map (\<lambda>n. if P (l ! n) then [n] else []) [0..<length l]) \<noteq> []"
ord_isom_Ssegment_Ssegment: fixes D :: "'c Order"   and E :: "'d Order"   and f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "Order D"   and "Order E"   and "ord_isom D E f"   and "a \<in> carrier D" shows "ord_isom (SIod D (Ssegment D a)) (SIod E (Ssegment E (f a))) (restrict f (carrier (SIod D (Ssegment D a))))"
add_left_imp_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.cancel_semigroup_add plus"   and "plus a b = plus a c" shows "b = c"
xz: shows "rtrancl_path r x (xs @ y # ys) z"
check'_def: fixes a :: "nat list"   and b :: "nat list" shows "check' a b = filter (check_cond a b)"
sees_methods_sub_Obj: fixes P :: "'b prog"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "P \<turnstile> C sees_methods Mm" shows "P \<turnstile> C \<preceq>\<^sup>* Object"
scaleR_sum_left: fixes f :: "'e \<Rightarrow> real"   and A :: "'e set"   and x :: "'d" shows "sum f A *\<^sub>R x = (\<Sum>a\<in>A. f a *\<^sub>R x)"
convert_semantics_lem: fixes fml :: "rat poly fml"   and ls :: "real list"   and ps :: "rat poly list"   and x :: "real" assumes "\<And>p. p \<in> set (poly_list fml) \<Longrightarrow> ls ! index_of ps p = rpoly p x" shows "fml_sem fml x = lookup_sem (map_fml (index_of ps) fml) ls"
C_def: shows "C \<equiv> \<Lambda> C. C_abs\<cdot>(up\<cdot>C)"
invc_join: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree" assumes "invc l"   and "invc r" shows "(color l = Black \<and> color r = Black \<longrightarrow> invc (join l r)) \<and> invc2 (join l r)"
H_eq_one: fixes x :: "'b" assumes "x \<in> cball (0::'b) 1" shows "H x = 1"
external_actions_not_new: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> external_actions E"   and "is_new_action (action_obs E a)" shows "False"
cp_OclNot: fixes X :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'d state \<times> 'd state" shows "not X \<tau> = not (\<lambda>_. X \<tau>) \<tau>"
put_get: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d"   and v :: "'c" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
iT_cut_le1_2: fixes t :: "nat"   and n :: "nat" assumes "t \<in> [n\<dots>]" shows "[n\<dots>] \<down>\<le> t = [n\<dots>,t - n]"
mset_link: fixes t :: "'b tree" shows "mset_tree (link t) = mset_tree t"
node_ptr_kinds_commutes: fixes node_ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr |\<in>| object_ptr_kinds h) = (node_ptr |\<in>| node_ptr_kinds h)"
elims: fixes x :: "('h + 'i) + 'j"   and y :: "'h + 'i + 'j" assumes "rsuml x = y" obtains a :: "'h" where "x = Inl (Inl a)" and "y = Inl a"   | b :: "'i" where "x = Inl (Inr b)" and "y = Inr (Inl b)"   | c :: "'j" where "x = Inr c" and "y = Inr (Inr c)"
invertible_lf_axioms_def: fixes f :: "'a \<Rightarrow> 'b" shows "invertible_lf_axioms f \<equiv> \<exists>g. g \<circ> f = id \<and> f \<circ> g = id"
seq_split: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"   and "rs = rs\<^sub>1 @ rs\<^sub>2" obtains t' :: "state" where "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t'" and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t" and "no_matching_Goto \<gamma> p rs\<^sub>1"   | "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t" and "\<not> no_matching_Goto \<gamma> p rs\<^sub>1"
suminf_minus_initial_segment: fixes f :: "nat \<Rightarrow> 'c"   and k :: "nat" assumes "summable f" shows "(\<Sum>n. f (n + k)) = (\<Sum>n. f n) - sum f {..<k}"
Si_at_top: shows "(Si \<longlongrightarrow> pi / 2) at_top"
simps_1: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "sorted_wrt P [] = True"
comp_functor_identity: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c" assumes "functor A B F" shows "F \<circ> identity_functor.map A = F"
join\<^sub>\<epsilon>: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('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"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and r' :: "'a" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = None" shows "substitution.revision_step subst r s \<epsilon>"
HInfinite_HFinite_iff: fixes x :: "'a star" shows "(x \<in> HInfinite) = (x \<notin> HFinite)"
b_e_typing_imp_list_types_agree: fixes \<C> :: "t_context"   and vs :: "v list"   and ts' :: "t list"   and ts :: "t list" assumes "\<C> \<turnstile> map EConst vs : ts' _> ts' @ ts" shows "list_all2 types_agree ts vs"
zip_truncate_right: fixes xs :: "'a list"   and ys :: "'b list" shows "zip xs ys = zip xs (take (length xs) ys)"
inducts: fixes x1 :: "(C, BD) sequent list"   and x2 :: "(C, BD) sequent"   and P :: "(C, BD) sequent list \<Rightarrow> (C, BD) sequent \<Rightarrow> bool" assumes "(x1, x2) \<in> g3mod2"   and "\<And>A. P [ \<Empt> \<Rightarrow>* \<LM> A  \<RM>] ( \<Empt> \<Rightarrow>* \<LM> \<diamond> A  \<RM>)"   and "\<And>A. P [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>] ( \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>)" shows "P x1 x2"
cnf2bhv_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n \<equiv> the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 + (n - \<langle>c \<and> t\<rangle>)"
witness_l1: fixes Y :: "nat"   and r0 :: "nat"   and M :: "nat" shows "witness Y r0 [M] = Conv (all1_matrix Y r0) (witness' r0 [M])"
intro: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and B :: "nat bdd list \<times> bool list" assumes "wf_dfa A n"   and "wf_dfa B n" shows "prod_DFS A B n"
t: fixes i :: "nat"   and x :: "nat" shows "eval (r_constn 1 d) [i, x] \<down>= d"
assms_2: shows "\<forall>b'\<in>ran \<beta>. Q b'"
subcls_asym: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   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 C :: "char list"   and D :: "char list" assumes "wf_prog wf_md P"   and "(C, D) \<in> (subcls1 P)\<^sup>+" shows "(D, C) \<notin> (subcls1 P)\<^sup>+"
dim_gen_is_basis: fixes A :: "'c set" assumes "finite A"   and "A \<subseteq> carrier V"   and "gen_set A"   and "card A \<le> dim" shows "basis A"
CONSTRAINT_D: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "CONSTRAINT P x" shows "P x"
compile_env_def: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'fun \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and e :: "'a" shows "compile_env \<A> \<O> e \<equiv> map_result id Subx.Fenv.from_list (Result.those (map (compile_env_entry \<A> \<O>) (Finca_to_list e)))"
crepresentation_raw_def: shows "crepresentation \<equiv> module.representation (*\<^sub>C)"
acute_not_per: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Acute Bet Cong A B C" shows "\<not> Tarski_neutral_dimensionless.Per Bet Cong A B C"
minus_mult_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring plus zero minus uminus times" shows "uminus (times a b) = times a (uminus b)"
image_mset_const_eq: fixes c :: "'c"   and M :: "'d multiset" shows "{#c. a \<in># M#} = replicate_mset (size M) c"
new_Addr_SomeD: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat" assumes "new_Addr h = \<lfloor>a\<rfloor>" shows "h a = None"
LeastI_ex: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (Least P)"
eval_Method_i_i_i_o_o_o_o_conv: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal" shows "pred.eval (Method_i_i_i_o_o_o_o P C M) = (\<lambda>(Ts, T, m, D). P \<turnstile> C sees M: Ts\<rightarrow>T = m in D)"
compact_blinfun_lemma: fixes f :: "nat \<Rightarrow> 'd \<Rightarrow>\<^sub>L 'e" assumes "bounded (range f)" shows "\<forall>d\<subseteq>Basis. \<exists>x r. strict_mono r \<and> (\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (blinfun_apply (f (r n)) i) (blinfun_apply x i) < e)"
invertible_lf: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "invertible_lf s1 s2 f" shows "\<exists>g. linear s2 s1 g \<and> g \<circ> f = id \<and> f \<circ> g = id"
flowD_capacity_IN: fixes \<Delta> :: "('k, 'l) network_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "flow \<Delta> f" shows "d_IN f x \<le> d_IN (capacity \<Delta>) x"
sgn_div_norm: fixes sgn :: "'c \<Rightarrow> 'c"   and norm :: "'c \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.sgn_div_norm sgn norm scaleR" shows "sgn x = scaleR (inverse (norm x)) x"
eqd_comp: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v" shows "x \<bowtie> u"
leg0_uniquely_isomorphic_2: shows "\<exists>!\<theta>. \<guillemotleft>\<theta> : t\<^sub>0 \<star> \<omega>.chine \<Rightarrow> r\<^sub>0\<guillemotright>"
"3_5": shows "length (a # b # xs) \<le> n"
generated_ring_Int: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> generated_ring"   and "b \<in> generated_ring" shows "a \<inter> b \<in> generated_ring"