hcju/isabelleps huggingface dataset:
items_le_Complete: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and I :: "item set"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.wellformed_items \<RR> Doc I" shows "items_le k (Complete k I) = Complete k (items_le k I)"
div_is_floor_divide_rat: fixes n :: "int"   and y :: "int" shows "n div y = \<lfloor>rat_of_int n / rat_of_int y\<rfloor>"
iFROM_inext_diff_const: fixes t :: "nat"   and n :: "nat" assumes "t \<in> [n\<dots>]" shows "inext t [n\<dots>] - t = Suc 0"
normalize_gauss_int_def: fixes z :: "gauss_int" shows "normalize z = (if z = 0 then 0 else if 0 \<le> ImZ z \<and> 0 < ReZ z then z else if ReZ z \<le> 0 \<and> 0 < ImZ z then - \<i>\<^sub>\<int> * z else if ImZ z \<le> 0 \<and> ReZ z < 0 then - z else \<i>\<^sub>\<int> * z)"
count_eq_zero_iff: fixes M :: "'k multiset"   and x :: "'k" shows "(count M x = 0) = (x \<notin># M)"
opartial_newpktTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and d :: "nat"   and di :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set"   and T :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set"   and P :: "bool" assumes "((\<sigma>, s), i:newpkt(d, di), \<sigma>', s') \<in> opnet_sos S T" shows "P"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>L 'd"   and xa :: "'c \<Rightarrow>\<^sub>L 'd" shows "blinfun_apply (x - xa) = (\<lambda>xb. blinfun_apply x xb - blinfun_apply xa xb)"
ubs_def: fixes R :: "'b set"   and S :: "'b set" shows "ubs R S = Collect (isUb R S)"
decseq_Suc_iff: fixes f :: "nat \<Rightarrow> 'a" shows "decseq f = (\<forall>n. f (Suc n) \<le> f n)"
ring_iso_imp_img_field: fixes h :: "'a \<Rightarrow> 'd"   and S :: "('d, 'e) ring_scheme" assumes "h \<in> ring_iso R S" shows "field (S\<lparr>zero := h \<zero>\<rparr>)"
Dube_aux: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F"   and "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> Dube_aux (card X) (poly_deg f) 1"
open_metric_completion_def: fixes U :: "'a metric_completion set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
rreq_rrep_sn_simps_3: fixes dests :: "nat \<Rightarrow> nat option"   and sip :: "nat" shows "rreq_rrep_sn (Rerr dests sip) = True"
antisym_def: fixes r :: "('c \<times> 'c) set" shows "antisym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
l_set_disconnected_nodes_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs \<equiv> l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
l3_inv7_derived: shows "l3_inv3 \<inter> l3_inv4 \<subseteq> l3_inv7"
RH_fp: assumes "x \<in> Rf CH fp" shows "thesis"
inverse_equality: fixes u :: "'b"   and v :: "'b" assumes "u \<cdot>' v = \<one>'"   and "v \<cdot>' u = \<one>'"   and "u \<in> M'"   and "v \<in> M'" shows "inverse' u = v"
zmset_subset_eq_insertD: fixes x :: "'b"   and A :: "'b zmultiset"   and B :: "'b zmultiset" assumes "add_zmset x A \<subseteq>#\<^sub>z B" shows "A \<subset>#\<^sub>z B"
initials_def: fixes G :: "diagram" shows "initials G = ffilter (\<lambda>v. \<forall>e\<in>set G^E. v |\<notin>| thd3 e) G^V"
healthy_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "healthy t = (feasible t \<and> mono_trans t \<and> scaling t)"
simps_1: fixes e :: "char list \<Rightarrow> 'b option" shows "env_dom (Env e) = dom e"
finite_pos_order: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G"   and "finite (carrier G)" shows "0 < order G"
relators: fixes S :: "'b set"   and P :: "('b \<times> bool) list set" assumes "GroupByPresentation S P" shows "order.greater_eq \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle> (Abs_freeword ` P)"
simps_12: fixes f :: "'f \<Rightarrow> 'e" shows "lmap f LNil = LNil"
select_convs_3: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "cTimeF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cTimeF"
smn_def: fixes n :: "nat"   and p :: "nat"   and cs :: "nat list" shows "smn n p cs \<equiv> quad_encode 3 n (encode (r_universal (n + length cs))) (Partial_Recursive.list_encode (code_constn n p # map (code_constn n) cs @ map (code_id n) [0..<n]))"
DF_Suc: fixes k :: "nat"   and i :: "nat" shows "DF k i \<lless> DF k (Suc i)"
mod_mod_trivial: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (modulo a b) b = modulo a b"
M_def: shows "M = Max {k. (T ^^ k) ((T ^^ n) x) \<in> A}"
assms_3: shows "\<forall>i\<in>W. (0::'a) \<le> l i"
SC_ZObis_Seq: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_ZObis tval aval indis (c1 ;; c2) = (PL_Indis.SC_siso tval aval indis c1 \<and> PL_Indis.SC_ZObis tval aval indis c2 \<or> PL_Indis.SC_ZObis tval aval indis c1 \<and> PL_Indis.SC_discr aval indis c2)"
par_int_ndet_distrib: fixes P :: "'a process"   and Q :: "'a process"   and A :: "'a set"   and M :: "'a process" shows "(P |-| Q \<lbrakk>A\<rbrakk> M) = ((P \<lbrakk>A\<rbrakk> M) |-| (Q \<lbrakk>A\<rbrakk> M))"
assms_5: fixes i :: "nat" assumes "i < n" shows "(0::'a) \<le> w $ i"
simps_4: fixes f :: "('d, 'c) trm"   and g :: "('d, 'c) trm" shows "FVT (Plus f g) = FVT f \<union> FVT g"
dist_le_zero_iff: 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 "(dist x y \<le> 0) = (x = y)"
induct: fixes x1 :: "('fun, 'var) Term.term set"   and x2 :: "('fun, 'var) Term.term \<Rightarrow> bool"   and x3 :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term set \<Rightarrow> (('fun, 'var) Term.term \<Rightarrow> bool) \<Rightarrow> ('fun, 'var) Term.term \<Rightarrow> bool" assumes "\<langle>x1;x2\<rangle> \<turnstile>\<^sub>r x3"   and "\<And>t M Q. t \<in> M \<Longrightarrow> P M Q t"   and "\<And>T f M Q. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> \<langle>M;Q\<rangle> \<turnstile>\<^sub>r t; \<And>t. t \<in> set T \<Longrightarrow> P M Q t; Q (Fun f T)\<rbrakk> \<Longrightarrow> P M Q (Fun f T)"   and "\<And>M Q t K T t\<^sub>i. \<lbrakk>\<langle>M;Q\<rangle> \<turnstile>\<^sub>r t; P M Q t; Ana t = (K, T); \<And>k. k \<in> set K \<Longrightarrow> \<langle>M;Q\<rangle> \<turnstile>\<^sub>r k; \<And>k. k \<in> set K \<Longrightarrow> P M Q k; t\<^sub>i \<in> set T\<rbrakk> \<Longrightarrow> P M Q t\<^sub>i" shows "P x1 x2 x3"
subst_eq: fixes k :: "nat"   and u :: "dB" shows "dB.Var k[u/k] = u"
mult_mm_assoc: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex"   and C :: "complex \<times> complex \<times> complex \<times> complex" shows "A *\<^sub>m\<^sub>m (B *\<^sub>m\<^sub>m C) = A *\<^sub>m\<^sub>m B *\<^sub>m\<^sub>m C"
in_set_conv_decomp_last: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in>. xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
word_not_le: fixes x :: "'a word"   and y :: "'a word" shows "(\<not> x \<le> y) = (y < x)"
out_next_subseqs: fixes base :: "'b"   and elements :: "'a list"   and n :: "nat"   and state :: "'a \<times> 'a iarray \<times> (nat \<times> 'b) list"   and out :: "'b list"   and state' :: "'a \<times> 'a iarray \<times> (nat \<times> 'b) list" assumes "invariant base elements n state"   and "local.next_subseqs state = (out, state')" shows "set out = S base elements (Suc n)"
add_neg_numeral_special_7: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and one :: "'e" assumes "class.neg_numeral minus plus zero uminus" shows "plus one (uminus one) = zero"
all_security_requirements_fulfilled_def: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "all_security_requirements_fulfilled M G \<equiv> \<forall>m\<in>set M. c_sinvar m G"
fbox_def: fixes F :: "'c \<Rightarrow> 'd set"   and P :: "'d \<Rightarrow> bool" shows "|F] P = (\<lambda>s. \<forall>s'. s' \<in> F s \<longrightarrow> P s')"
newQuery_equi_dev: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set"   and I :: "nat set"   and t :: "'b option list" shows "New_max.newQuery V Q (I, t) = New_max.projectTable V ` (\<lambda>tab. New_max.semiJoin tab (I, t)) ` Q"
paint_rbt_sorted: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) rbt"   and c :: "color" assumes "ord.rbt_sorted less t" shows "ord.rbt_sorted less (paint c t)"
wellformed_app: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and wellformed :: "'a \<Rightarrow> bool"   and all_frees :: "'a \<Rightarrow> name fset"   and u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" assumes "class.pre_strong_term size frees subst consts app unapp const unconst free unfree abs_pred wellformed all_frees" shows "wellformed (app u\<^sub>1 u\<^sub>2) = (wellformed u\<^sub>1 \<and> wellformed u\<^sub>2)"
component_of_refl: fixes F :: "'b program" shows "F component_of F"
Bonk_Schramm_extension_hyperbolic: shows "Gromov_hyperbolic_subset (deltaG TYPE('a)) UNIV"
one_is_normal: shows "{\<one>} \<lhd> G"
get_child_nodes_def: shows "Core_DOM_Functions.get_child_nodes \<equiv> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes"
Int_iff: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
igSwapIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igSwapIGOp MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
is_cnf_encode_negative_transition_frame_axiom: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and v :: "'b" shows "is_cnf (encode_negative_transition_frame_axiom \<Pi> t v)"
is_tdghmE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" obtains "\<Z> \<alpha>" and "vfsequence \<NN>" and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" and "vcard \<NN> = 5\<^sub>\<nat>" and "vsv (\<NN>\<lparr>NTMap\<rparr>)" and "\<D>\<^sub>\<circ> (\<NN>\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>" and "\<NN>\<lparr>NTDom\<rparr> = \<FF>" and "\<NN>\<lparr>NTCod\<rparr> = \<GG>" and "\<NN>\<lparr>NTDGDom\<rparr> = \<AA>" and "\<NN>\<lparr>NTDGCod\<rparr> = \<BB>" and "\<And>a. a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr> \<Longrightarrow> \<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<GG>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
AllFinite: shows "\<forall>p. finite {msg. msg \<in># msgs init \<and> (\<exists>v. <p, inM v> = msg)}"
rk_bc\<alpha>: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points"   and R :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {Q, R} = 2"   and "rk {R, A, a} = 2"   and "rk {R, B, b} = 2"   and "rk {R, C, c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {b, c, \<alpha>} = 2"
post_del_adjL: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree"   and l' :: "('b \<times> nat) tree"   and b :: "'b" assumes "invar \<langle>l, (a, lv), r\<rangle>"   and "pre_adjust \<langle>l', (b, lv), r\<rangle>" shows "post_del \<langle>l, (a, lv), r\<rangle> (adjust \<langle>l', (b, lv), r\<rangle>)"
CallStatic: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and ps :: "char list exp list"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> char list exp"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
simps_2: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list"   and x :: "'c" shows "sinks_aux_less I D U (xs @ [x]) = (if \<exists>v\<in>U \<union> sinks_aux_less I D U xs. (v, D x) \<in> I then insert (D x) (sinks_aux_less I D U xs) else sinks_aux_less I D U xs)"
collinear_midpoint: fixes a :: "'a"   and b :: "'a" shows "collinear {a, midpoint a b, b}"
insert_eq_iff: fixes a :: "'e"   and A :: "'e set"   and b :: "'e"   and B :: "'e set" assumes "a \<notin> A"   and "b \<notin> B" shows "(insert a A = insert b B) = (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
Z_Union: shows "local.Z_inj \<union> local.Z_not_inj = Z n m"
QIPI: fixes a :: "'b" assumes "QI a" shows "PI a"
nonneg: fixes n :: "'b"   and i :: "nat" shows "0 \<le> f n i"
left_commute: fixes b :: "'k multiset"   and a :: "'k multiset"   and c :: "'k multiset" shows "b \<inter># (a \<inter># c) = a \<inter># (b \<inter># c)"
simps_1: fixes at :: "('e, 'f) atm" shows "varsL (Pos at) = varsA at"
psubstT_num: fixes txs :: "('trm \<times> 'var) list"   and n :: "'trm" assumes "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "n \<in> num" shows "psubstT n txs = n"
lnull_lprefix: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lnull xs" shows "xs \<sqsubseteq> ys"
zip_option_zip_conv: fixes xs :: "'e list"   and ys :: "'f list"   and zs :: "('e \<times> 'f) list" shows "(zip_option xs ys = Some zs) = (length ys = length xs \<and> length zs = length xs \<and> zs = zip xs ys)"
not_lindepD: fixes S :: "rat vec set"   and A :: "rat vec set"   and f :: "rat vec \<Rightarrow> rat" assumes "gs.lin_indpt S"   and "finite A"   and "A \<subseteq> S"   and "f \<in> A \<rightarrow> UNIV"   and "gs.lincomb f A = 0\<^sub>v n" shows "f \<in> A \<rightarrow> {0}"
e2: shows "natUnion (\<lambda>u. items_le k (\<J> k u)) = natUnion (\<lambda>u. Gen (paths_le k (\<P> k u)))"
injective_scale: fixes c :: "complex" assumes "c \<noteq> 0" shows "inj ((*\<^sub>C) c)"
preserves_arr: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "C.hseq (fst f) (snd f)"
assms_1: shows "\<sigma> \<simeq>A\<simeq> \<tau>"
subset_Un_eq: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subseteq> B) = (A \<union> B = B)"
intros_8: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> \<sigma>" shows "Bag \<tau> \<sqsubset> Bag \<sigma>"
P_A: shows "P (lfp (\<lambda>S. W \<union> S \<inter> V \<union> directly_attracted p (S \<inter> V)) \<inter> V)"
is_cupcake_nsE: fixes ns :: "(char list, char list, v) namespace" assumes "is_cupcake_ns ns" obtains v :: "(char list \<times> v) list" where "ns = Bind v []" and "list_all (is_cupcake_value \<circ> snd) v"
dur_zero: fixes xs :: "('c \<Rightarrow> real) stream"   and \<omega> :: "('c \<Rightarrow> real) set stream"   and i :: "nat"   and x :: "'c" assumes "\<forall>i. xs !! i \<in> \<omega> !! i"   and "\<forall>j\<le>i. zero x (\<omega> !! j)"   and "x \<in> \<X>"   and "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)" shows "dur xs i = 0"
eq_N: shows "image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}})) = N"
list_slice2_append_mult: fixes xs :: "'b list"   and m :: "nat"   and k :: "nat"   and ys :: "'b list" assumes "length xs = m * k" shows "list_slice2 (xs @ ys) k = list_slice2 xs k @ list_slice2 ys k"
split_div: fixes P :: "nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" shows "P (m div n) = ((n = 0 \<longrightarrow> P 0) \<and> (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<longrightarrow> m = n * i + j \<longrightarrow> P i)))"
select_convs_1: fixes nrm :: "lname set"   and brk :: "label \<Rightarrow> lname set"   and more :: "'a" shows "nrm \<lparr>nrm = nrm, brk = brk, \<dots> = more\<rparr> = nrm"
sum_defs_1: fixes A :: "V"   and B :: "V" shows "A \<Uplus> B \<equiv> vtimes (set {0}) A \<squnion> vtimes (set {1}) B"
support_impl: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (\<lambda>l s. P s \<longrightarrow> Q l s) \<subseteq> support Q"
last_message_Msg_eq_last: fixes xs :: "'a message_af list" assumes "xs \<noteq> []"   and "last xs \<noteq> NoMsg" shows "last_message xs = last xs"
Abs_cfun_inverse2: fixes f :: "'b \<Rightarrow> 'c" assumes "cont f" shows "Rep_cfun (Abs_cfun f) = f"
ex_moebius_circline_x_axis: fixes H :: "circline" assumes "circline_type H < 0" shows "\<exists>M. moebius_circline M H = x_axis"
bigomega_def: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "\<Omega>[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)}"
assms: shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
U_1: shows "\<forall>u\<in>set U. \<exists>y. u = Var y"
Domain_iff: fixes a :: "'b"   and r :: "('b \<times> 'c) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
n_2: shows "\<forall>m\<le>n. b <\<^bsub>F (\<lambda>i. if h i < m then Lab else Lba)\<^esub> a"
card_1_imp_singleton: fixes A :: "'a set" assumes "card A = Suc 0" shows "\<exists>a. A = {a}"
assms_2: shows "\<forall>x\<in>\<UU>A. f x = c"
disj_matrices_def: fixes A :: "'a matrix"   and B :: "'a matrix" shows "disj_matrices A B = ((\<forall>j i. Rep_matrix A j i \<noteq> (0::'a) \<longrightarrow> Rep_matrix B j i = (0::'a)) \<and> (\<forall>j i. Rep_matrix B j i \<noteq> (0::'a) \<longrightarrow> Rep_matrix A j i = (0::'a)))"
subst\<^sub>0: fixes aneg :: "'c \<Rightarrow> 'c fm"   and anormal :: "'c \<Rightarrow> bool"   and I\<^sub>a :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and depends\<^sub>0 :: "'c \<Rightarrow> bool"   and decr :: "'c \<Rightarrow> 'c"   and solvable\<^sub>0 :: "'c \<Rightarrow> bool"   and trivial :: "'c \<Rightarrow> bool"   and subst\<^sub>0 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and eq :: "'c"   and x :: "'d"   and xs :: "'d list"   and a :: "'c" assumes "ATOM_EQ aneg anormal I\<^sub>a depends\<^sub>0 decr solvable\<^sub>0 trivial subst\<^sub>0"   and "solvable\<^sub>0 eq"   and "\<not> trivial eq"   and "I\<^sub>a eq (x # xs)"   and "depends\<^sub>0 a" shows "I\<^sub>a (subst\<^sub>0 eq a) xs = I\<^sub>a a (x # xs)"
ih1: shows "eval (change_int f i) p1 = change_tv f (eval i p1)"
has_lock_has_locks_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "has_lock l t = (\<exists>n. has_locks l t = Suc n)"
interior_cball: fixes x :: "'b"   and e :: "real" shows "interior (cball x e) = ball x e"
cr_fls_def: shows "cr_fls \<equiv> \<lambda>x y. x = fls_nth y"
Quotient3_rep_reflp: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g"   and a :: "'h" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep a)"
eq_omega_sum_less_iff: fixes ms :: "nat list"   and ns :: "nat list" assumes "length ms = length ns" shows "((omega_sum ms, omega_sum ns) \<in> VWF) = ((ms, ns) \<in> lenlex less_than)"