hcju/isabelleps huggingface dataset:
Conj\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]" shows "p \<bullet> Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = Conj\<^sub>\<alpha> (p \<bullet> tset\<^sub>\<alpha>)"
cases: fixes a1 :: "nat"   and a2 :: "Lholed"   and a3 :: "e list"   and a4 :: "e list" assumes "Lfilled_exact a1 a2 a3 a4" obtains lholed :: "Lholed" and es :: "e list" where "a1 = 0" and "a2 = lholed__" and "a3 = es__" and "a4 = es__" and "lholed__ = LBase [] []"   | vs :: "e list" and lholed :: "Lholed" and n :: "nat" and es' :: "e list" and l :: "Lholed" and es'' :: "e list" and k :: "nat" and es :: "e list" and lfilledk :: "e list" where "a1 = k + 1" and "a2 = lholed" and "a3 = es" and "a4 = vs @ [Label n es' lfilledk] @ es''" and "const_list vs" and "lholed = LRec vs n es' l es''" and "Lfilled_exact k l es lfilledk"
plusl_zero_infty: shows "[0]\<^sub>1 +\<^sub>1 \<infinity>\<^sub>1 = \<infinity>\<^sub>1"
P4: shows "Col A B D0 \<and> A B Perp P D0"
start_heap_obs_not_Read: fixes ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" shows "ReadMem ad al v \<notin> set h.start_heap_obs"
val_ifex: fixes b :: "'b"   and s :: "'a \<Rightarrow> bool" shows "val_ifex (ifex_of b) s = val b s"
AGM_FullContraction_axioms: shows "AGM_FullContraction Cn (\<div>) \<top> \<bottom> (.\<longrightarrow>.) .\<not> (.\<and>.) (.\<or>.) (.\<longleftrightarrow>.)"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV F)) e1 e2])"
finite_card_of_iff_card2: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "(|A| \<le>o |B|) = (card A \<le> card B)"
assoc_in_hom: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<guillemotleft>\<a>[a, b, c] : (a \<otimes> b) \<otimes> c \<rightarrow> a \<otimes> b \<otimes> c\<guillemotright>"
ideal_ofD: fixes f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and A :: "('c \<Rightarrow> 'd) set"   and a :: "'c \<Rightarrow> 'd" assumes "f \<in> \<I> A"   and "a \<in> A" shows "poly_eval a f = (0::'d)"
IMG_point_2: fixes p :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "\<guillemotleft>p : unity \<rightarrow> a\<guillemotright>" shows "IMG p = (UP \<circ> Map p \<circ> DOWN) unity"
pfixLe_refl: fixes x :: "nat list" shows "x pfixLe x"
hyps_3: shows "Some \<delta> = mgu (Fun f T) (Fun f U)"
foldl_transfer: fixes B :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "((B ===> A ===> B) ===> B ===> list_all2 A ===> B) foldl foldl"
euler_err_fas_def: fixes X0 :: "'c list"   and h :: "floatarith"   and CX :: "floatarith list" shows "euler_err_fas X0 h CX = map (euler_err_fas_nth X0 h CX) [0..<length X0]"
topological_space_def: fixes S :: "'a set"   and is_open :: "'a set \<Rightarrow> bool" shows "topological_space S is_open \<equiv> (is_open S \<and> is_open {}) \<and> (\<forall>U. is_open U \<longrightarrow> U \<subseteq> S) \<and> (\<forall>U V. is_open U \<longrightarrow> is_open V \<longrightarrow> is_open (U \<inter> V)) \<and> (\<forall>F. (\<forall>x. x \<in> F \<longrightarrow> is_open x) \<longrightarrow> is_open (\<Union>x\<in>F. x))"
FEx_upward: fixes n :: "nat"   and m :: "nat"   and g :: "form" assumes "infinite (deriv s)"   and "init s"   and "contains f n (m, FEx g)" shows "\<forall>m'. \<exists>n'. contains f n' (0, finst g m')"
Lcm_singleton: fixes a :: "'a" shows "Lcm {a} = normalize a"
relator_freeword: fixes s :: "'w"   and t :: "'w" assumes "s \<in> S"   and "t \<in> S" shows "pair_relator_freeword s t = relator_freeword (pair_relator_list s t)"
exhaust: fixes y :: "recording_state" obtains "y = NotStarted"   | "y = Recording"   | "y = Done"
cball_empty: fixes e :: "real"   and x :: "'b" assumes "e < 0" shows "cball x e = {}"
simps_2: fixes p :: "linear_poly"   and c :: "'a" shows "is_leq_ns (GEQ_ns p c) = False"
dests_bigger_than_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:15..PAodv-:17} \<union> {PPkt-:7..PPkt-:9} \<union> {PRreq-:9..PRreq-:11} \<union> {PRreq-:17..PRreq-:19} \<union> {PRrep-:8..PRrep-:10} \<union> {PRerr-:1..PRerr-:4} \<union> {PRerr-:6} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) ip \<le> the (dests \<xi> ip)))"
ivl_disj_un_singleton_5: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{l} \<union> {l<..u} = {l..u}"
Cons_1: fixes u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}" shows "lookupDef g n ` set us = lookup_multimap u' n"
d_IN_ge_point: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and y :: "'a"   and x :: "'a" shows "f (y, x) \<le> d_IN f x"
distinct_143: fixes x5 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Cos x5 \<noteq> Power x151 x152"
ord_1_right: fixes n :: "nat" shows "ord n 1 = 1"
intros_3: fixes M :: "'b measure"   and N :: "'b measure" assumes "space M = space N"   and "sets M = sets N"   and "emeasure M \<le> emeasure N" shows "M \<le> N"
simps_2: fixes p :: "char list"   and ts :: "'a list" shows "get_atom (Neg p ts) = (p, ts)"
rulesInPCs_18: shows "CutFreePC \<subseteq> PC"
transfer: fixes p :: "'c pmf"   and s :: "'c set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "pmf_as_measure.cr_pmf (uniform_measure (measure_pmf p) s) (cond_pmf p s)"
in_mono: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
b: shows "cs\<^bsup>\<pi>\<^esup> k = xs @ b # bs"
path_image_sym: fixes p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "pathfinish p = pathstart q"   and "pathfinish q = pathstart p" shows "path_image (p +++ q) = path_image (q +++ p)"
round_robin_step_invar_None: fixes final :: "'c \<Rightarrow> bool"   and r :: "('b, 'a, 'c, 'd, 'e, 'f) semantics"   and convert_RA :: "'b \<Rightarrow>f nat \<Rightarrow> 'f list"   and n0 :: "nat"   and \<sigma> :: "'a list \<times> nat"   and t' :: "'a"   and s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('c \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set"   and t :: "'a"   and \<sigma>' :: "'a list \<times> nat" assumes "multithreaded_base.step_thread final r convert_RA (Round_Robin.round_robin_update_state n0 \<sigma> t') s t' = \<lfloor>(t, None, \<sigma>')\<rfloor>"   and "Round_Robin.round_robin_invar \<sigma> (dom (thr s))" shows "Round_Robin.round_robin_invar \<sigma>' (dom (thr s))"
leg1_simps_2: shows "hseq r\<^sub>1 r\<^sub>0s\<^sub>1.p\<^sub>1"
Cons_1: fixes B :: "Core.term"   and lev :: "nat" assumes "list_all (\<lambda>a. case a of (a, b) \<Rightarrow> (case a of (idx, ty) \<Rightarrow> \<lambda>t. typ_of t = Some ty) b) xs"   and "\<not> loose_bvar B lev" shows "fold (\<lambda>a. case a of (a, b) \<Rightarrow> (case a of (idx, ty) \<Rightarrow> \<lambda>t B. subst_bv1 (bind_fv2 (idx, ty) lev B) lev t) b) xs B = fold (\<lambda>single. subst_term [single]) xs B"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y + z = x + (y + z)"
scale_sum_right: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "a * sum f A = (\<Sum>x\<in>A. a * f x)"
listrel1_converse: fixes r :: "('g \<times> 'g) set" shows "listrel1 (r\<inverse>) = (listrel1 r)\<inverse>"
fdisjnt_swap: fixes M :: "'b fset"   and N :: "'b fset" assumes "fdisjnt M N" shows "fdisjnt N M"
asBinp_equal_iff_alphaBinp: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option"   and qbinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" assumes "qGoodBinp qbinp" shows "(asBinp qbinp = asBinp qbinp') = ((qbinp, qbinp') \<in> alphaBinp)"
unit_factor_1_imp_normalized: fixes x :: "'a" assumes "unit_factor x = (1::'a)" shows "normalize x = x"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "normal_bicategory V H \<a> \<i> src trg"   and "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_normal_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
Abs_bit0_def: fixes x :: "int" shows "Abs x = Abs_bit0' x"
not_zero_j_H: shows "\<not> is_zero_row j H"
swap_msgs_Send_RecvMarker: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) 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"
add_neg_nonpos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'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_comm_monoid_add plus zero less_eq less"   and "less a zero"   and "less_eq b zero" shows "less (plus a b) zero"
inject_5: fixes x61 :: "com"   and x62 :: "com"   and y61 :: "com"   and y62 :: "com" shows "(x61;; x62 = y61;; y62) = (x61 = y61 \<and> x62 = y62)"
insert_if: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "sum g (insert x A) = (if x \<in> A then sum g A else g x + sum g A)"
simps_3: fixes a :: "ereal" shows "a + \<infinity> = \<infinity>"
succss_def: fixes xs :: "'a set" shows "succss xs \<equiv> \<Union>x\<in>xs. set (succs x)"
continuous_within_open: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "a \<in> A"   and "open A" shows "continuous (at a within A) f = isCont f a"
scene_override_commute: fixes S\<^sub>2 :: "'a"   and S\<^sub>1 :: "'a"   and X :: "'a scene" shows "S\<^sub>2 \<oplus>\<^sub>S S\<^sub>1 on (- X) = S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on X"
strict_from_inj_image: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> local.F_inj" shows "strict_from_inj n f ` {0..<n} = f ` {0..<n}"
simps_2: fixes \<sigma> :: "'a" shows "GE_arOf (Wit \<sigma>) = []"
that: fixes D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N> \<langle>l', D'\<rangle>"   and "[D']\<^bsub>v,n\<^esub> = Z"   and "valid_dbm D'" shows "thesis"
assms_5: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> pmdl B"   and "f \<in> dgrad_p_set d m"   and "f \<noteq> 0" shows "\<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]"   and f :: "'a \<rightarrow> 'b" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "smap\<cdot>f\<cdot>(x :# xs) = f\<cdot>x :# smap\<cdot>f\<cdot>xs"
DisjI2: fixes A :: "'a set"   and B :: "'a set"   and X :: "'a set" assumes "A \<inter> B = {}" shows "A \<inter> (B - X) = {}"
lenext_trans: fixes gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list" assumes "\<lbrakk>gts zs ys; gts ys xs\<rbrakk> \<Longrightarrow> gts zs xs"   and "lenext gts zs ys"   and "lenext gts ys xs" shows "lenext gts zs xs"
delayed_pool_add_def: fixes dw :: "int \<times> word32 \<times> CPU_register"   and s :: "'b sparc_state" shows "delayed_pool_add dw s \<equiv> let (i, v, cr) = dw in if i = 0 then cpu_reg_mod v cr s else let curr_pool = get_delayed_pool s in s\<lparr>dwrite := curr_pool @ [dw]\<rparr>"
hom_mult: fixes x :: "real poly"   and y :: "real poly" shows "map_poly cor (x * y) = map_poly cor x * map_poly cor y"
fst_uminus: fixes x :: "'a \<times> 'b" shows "fst (- x) = - fst x"
RedNewFail: fixes h :: "'heap"   and C :: "String.literal"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "allocate h (Class_type C) = {}" shows "extTA,P,t \<turnstile> \<langle>new C,(h, l)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt OutOfMemory),(h, l)\<rangle>"
partial_vector_derivative_def: fixes F :: "'b \<Rightarrow> 'a"   and base_vec :: "'b"   and a :: "'b" shows "partial_vector_derivative F base_vec a \<equiv> vector_derivative (\<lambda>x. F (a - (a \<bullet> base_vec) *\<^sub>R base_vec + x *\<^sub>R base_vec)) (at (a \<bullet> base_vec))"
IH: shows "Liminf_llist (ltl xs) = (if lnull (ltl xs) then {} else llast (ltl xs))"
real_root_commute: fixes m :: "nat"   and n :: "nat"   and x :: "real" shows "root m (root n x) = root n (root m x)"
RSubmodule_inner_dirsum: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and Ns :: "'b set list" assumes "RModule R smult M"   and "\<forall>N\<in>set Ns. RModule R smult N \<and> N \<subseteq> M" shows "RModule R smult (\<Oplus>N\<leftarrow>Ns. N) \<and> (\<Oplus>N\<leftarrow>Ns. N) \<subseteq> M"
inv_mult_group: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> (x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y) = inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x"
ordertype_eq_ordertype_iso_Restr: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and s :: "('b \<times> 'b) set"   and B :: "'b set" assumes "wf r"   and "total_on A r"   and "trans r"   and "small A"   and "Field (Restr r A) = A"   and "wf s"   and "total_on B s"   and "trans s"   and "small B"   and "Field (Restr s B) = B" shows "(ordertype A r = ordertype B s) = (\<exists>f. iso (Restr r A) (Restr s B) f)"
frequently_cofinite: fixes P :: "'c \<Rightarrow> bool" shows "Inf_many P = infinite {x. P x}"
calculus_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" shows "calculus Bot Inf entails Red_I Red_F \<equiv> consequence_relation Bot entails \<and> calculus_axioms Bot Inf entails Red_I Red_F"
reals_power_lt_ex: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "1 < y" shows "\<exists>k>0. (1 / y) ^ k < x"
submonoid_inverse_closed: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> G" shows "inverse u \<in> G"
KL_same_eq_0: fixes b :: "real" shows "KL_divergence b M M = 0"
induct: fixes P :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a1 :: "'a list"   and a2 :: "'b list" assumes "\<And>Pa. P Pa [] []"   and "\<And>Pa a as c cs. P Pa as cs \<Longrightarrow> P Pa (a # as) (c # cs)"   and "\<And>uu_ v va. P uu_ (v # va) []"   and "\<And>uu_ v va. P uu_ [] (v # va)" shows "P a0 a1 a2"
deriv_bound_mono: fixes m :: "nat"   and n :: "nat"   and r :: "('a \<times> 'a) set"   and a :: "'a" assumes "m \<le> n"   and "deriv_bound r a m" shows "deriv_bound r a n"
uminus_uminus_vec: fixes v :: "'b vec" shows "- (- v) = v"
stoch_proc_filt_filt_equiv: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and P :: "'d measure"   and n :: "nat" assumes "filt_equiv F M N" shows "stoch_proc_filt M f P n = stoch_proc_filt N f P n"
viewL_def: fixes t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.viewL t = (case FingerTreeStruc.viewLn t of None \<Rightarrow> None | Some (a, t2) \<Rightarrow> Some (FingerTreeStruc.n_unwrap a, t2))"
wf_execute: fixes \<pi> :: "plan_action"   and s :: "object atom formula set" assumes "plan_action_enabled \<pi> s"   and "wf_world_model s" shows "wf_world_model (execute_plan_action \<pi> s)"
Conj\<^sub>\<alpha>_def': fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha> set['a]" shows "Formula.Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = Formula.abs_Tree\<^sub>\<alpha> (Formula.Tree.tConj (map_bset Formula.rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>))"
assms_1: shows "set ps \<subseteq> set bs \<times> set bs"
umformung_OPT: fixes qs :: "'c list"   and init :: "'c list"   and Strat :: "(nat \<times> nat list) list" assumes "set qs \<subseteq> set init"   and "distinct init"   and "T\<^sub>p_opt init qs = T\<^sub>p init qs Strat"   and "length qs = length Strat"   and "init \<noteq> []" shows "T\<^sub>p_opt init qs = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALGxy_det Strat qs init x y + ALG_Pxy Strat qs init x y)"
h3a: assumes "\<exists>y. \<forall>x<y. (\<Sum>i\<le>n. k i x * x ^ i) = 0" shows "\<forall>x y. (\<Sum>i\<le>n. k i x * y ^ i) = 0"
lambda_ex: fixes v :: "i"   and \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and x :: "\<nu>" shows "[\<lparr>\<^bold>\<lambda>x. \<phi> (x\<^sup>P),x\<^sup>P\<rparr> in v] = (\<exists>y. \<nu>\<upsilon> y = \<nu>\<upsilon> x \<and> [\<phi> (y\<^sup>P) in v])"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<longrightarrow>\<^sub>1 x2"   and "\<And>M y. P y M M"   and "\<And>t1 t2 s1 s2 y. \<lbrakk> t1 \<longrightarrow>\<^sub>1 t2; \<And>z. P z t1 t2;  s1 \<longrightarrow>\<^sub>1 s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App t1 s1) (App t2 s2)"   and "\<And>s1 s2 a y. \<lbrakk>a \<sharp> y;  s1 \<longrightarrow>\<^sub>1 s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (Lam [a].s1) (Lam [a].s2)"   and "\<And>a s1 s2 t1 t2 y. \<lbrakk>a \<sharp> y; a \<sharp> (s1, s2);  s1 \<longrightarrow>\<^sub>1 s2; \<And>z. P z s1 s2;  t1 \<longrightarrow>\<^sub>1 t2; \<And>z. P z t1 t2\<rbrakk> \<Longrightarrow> P y (App (Lam [a].t1) s1) (t2[a::=s2])" shows "P z x1 x2"
do_con_check_build_conv: fixes env :: "'b sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "'c list"   and vs :: "v list" assumes "do_con_check (c env) cn (length es)" shows "build_conv (c env) cn vs \<noteq> None"
ts_prop_2: fixes active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and k :: "cnf" assumes "singleton active" shows "active (singleton.the_singleton active) k"
cases: fixes x :: "('b \<times> 'b) set \<times> ('c \<Rightarrow> 'b) \<times> 'b set \<times> 'c list" obtains uu_ :: "('b \<times> 'b) set" and uv_ :: "'c \<Rightarrow> 'b" and uw_ :: "'b set" where "x = (uu_, uv_, uw_, [])"   | I :: "('b \<times> 'b) set" and D :: "'c \<Rightarrow> 'b" and U :: "'b set" and xs :: "'c list" and xa :: "'c" where "x = (I, D, U, xs @ [xa])"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "B.ide (B.L a)"
delete_def: fixes x :: "'b"   and t :: "('b \<times> color) tree" shows "delete x t = paint Black (del x t)"
A_nonzero_fixed_i: fixes i :: "'n" shows "\<exists>j. A $h i $h j \<noteq> 0"
horiz_vector_helper_neg_R: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list"   and I :: "nat list \<times> nat list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x"   and "list_constr (fst I) (length qs)"   and "list_constr (snd I) (length qs)" shows "((\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> poly (prod_list (retrieve_polys qs (snd I))) x < 0) = (z_R I sign = - 1)"
LEQ_SUC: fixes k :: "'k"   and k' :: "'k"   and i :: "nat"   and idx :: "'n" assumes "k \<noteq> k'" shows "LEQ k i (SUC k' idx) = LEQ k i idx"
linear_gcd: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "0 < a"   and "0 < b"   and "0 < c"   and "0 < d"   and "coprime a c"   and "coprime b d" shows "gcd (a * b) (c * d) = gcd a d * gcd b c"
list_all3_mapI: fixes P :: "'j \<Rightarrow> 'k \<Rightarrow> 'l \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'j"   and g :: "'h \<Rightarrow> 'k"   and h :: "'i \<Rightarrow> 'l"   and xs :: "'g list"   and ys :: "'h list"   and zs :: "'i list" assumes "list_all3 (\<lambda>x y z. P (f x) (g y) (h z)) xs ys zs" shows "list_all3 P (map f xs) (map g ys) (map h zs)"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree" shows "weight (Node w t\<^sub>1 t\<^sub>2) = weight t\<^sub>1 + weight t\<^sub>2"
linear_cf_def: fixes d :: "'d"   and X :: "'x set" shows "Cd d X \<equiv> set_option (MaxR.MaxR_opt (Pd d) X)"
simps_4: fixes f1 :: "'c"   and f2 :: "'c" shows "(case check of assign \<Rightarrow> f1 | check \<Rightarrow> f2) = f2"
atLeastAtMost_singleton: fixes a :: "'b" shows "{a..a} = {a}"
cinner_simps_2: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "\<langle>x, y + z\<rangle> = \<langle>x, y\<rangle> + \<langle>x, z\<rangle>"
sum_eq_0_iff: fixes F :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "finite F" shows "(comm_monoid_add.sum (+) {#} f F = {#}) = (\<forall>a\<in>F. f a = {#})"
is_VarE: fixes t :: "('a, 'b) Term.term" assumes "is_Var t" obtains x :: "'b" where "t = Var x"
conv_galois_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and 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 bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "less_eq (composition y x) z = less_eq y (uminus (composition (uminus z) (converse x)))"
matchrel_rtrancl_matchers: fixes x :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and y :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)" assumes "(x, y) \<in> matchrel\<^sup>*" shows "matchers_map (snd x) \<inter> matchers (set_mset (fst x)) = matchers_map (snd y) \<inter> matchers (set_mset (fst y))"
take_rev: fixes n :: "nat"   and xs :: "'b list" shows "take n (rev xs) = rev (drop (length xs - n) xs)"
elemrcos_carrier: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and a' :: "'c" assumes "subgroup H G"   and "Group.group G"   and "a \<in> carrier G"   and "a' \<in> H #>\<^bsub>G\<^esub> a" shows "a' \<in> carrier G"
termFSwImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Iteration.model_scheme" shows "termFSwImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSwapAll h hA MOD"