hcju/isabelleps huggingface dataset:
sorted_list_of_set_eq_Nil_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less"   and "finite A" shows "(linorder.sorted_list_of_set less_eq A = []) = (A = {})"
DirProds_is_group: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "Group.group (DirProds G I)"
Rep_run_inject: fixes x :: "'b run"   and y :: "'b run" shows "(Rep_run x = Rep_run y) = (x = y)"
assms: shows "LLL_Impl.reduce_basis \<alpha> fs_init = fs"
l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs Shadow_DOM.remove ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs"
natLeq_on_ordLeq_less_eq: fixes m :: "nat"   and n :: "nat" shows "(natLeq_on m \<le>o natLeq_on n) = (m \<le> n)"
mult_scalar_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "local.punit.monom_mult c t p \<odot> q = monom_mult c t (p \<odot> q)"
complex_eq_cvec_equivp: shows "equivp (\<approx>\<^sub>v)"
claim1: fixes S :: "nat set" assumes "finite S" shows "(\<Sum>i\<in>S. F (r i) - F (l i)) \<le> F b - F a"
floorlog_leD: fixes b :: "nat"   and x :: "nat"   and w :: "nat" assumes "floorlog b x \<le> w"   and "1 < b" shows "x < b ^ w"
Remove_def: shows "DFIG.Remove = (\<lambda>(A, A'). list_select DFIG_alphabet.x A < DFIG_alphabet.max A \<and> retidS A' = retidS A \<union> {list_select DFIG_alphabet.x A} \<and> DFIG_alphabet.max A' = DFIG_alphabet.max A)"
is_heap_list_simps_1: fixes h :: "'c" shows "is_heap_list h [] = True"
Lindelof_space_alt: fixes X :: "'a topology" shows "Lindelof_space X = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> topspace X \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \<U> \<and> topspace X \<subseteq> \<Union> \<V>))"
min_0_1_5: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq one (numeral.numeral one plus x) = one"
conforms_upd_gobj: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and r :: "oref"   and obj :: "obj"   and n :: "vn"   and T :: "ty"   and v :: "val" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "globs s r = Some obj"   and "var_tys G (tag obj) r n = Some T"   and "G,s\<turnstile>v\<Colon>\<preceq>T" shows "(x, upd_gobj r n v s)\<Colon>\<preceq>(G, L)"
tauLaw3CongSimLeft: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<alpha> :: "'a prefix" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi> P Q R S. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> Rel; \<Psi> \<rhd> R \<sim> S\<rbrakk> \<Longrightarrow> (\<Psi>, P, S) \<in> Rel"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> Rel \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<leadsto>\<guillemotleft>Rel\<guillemotright> \<alpha>\<cdot>Q"
qAFresh_qSwap_compose: fixes zs :: "'a"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and z :: "'b"   and x :: "'b" assumes "qAFresh zs y X"   and "qAFresh zs z X" shows "X #[[y \<and> x]]_zs #[[z \<and> y]]_zs = X #[[z \<and> x]]_zs"
simps: fixes a :: "(nat \<times> nat) convnet" shows "valid_net a = ((\<exists>M. a = Input M) \<or> (\<exists>m r1 r0. a = Conv (r0, r1) m \<and> output_size m = r1 \<and> valid_net m) \<or> (\<exists>m1 m2. a = Pool m1 m2 \<and> output_size m1 = output_size m2 \<and> valid_net m1 \<and> valid_net m2))"
domain_eq: shows "Domainp (pcr_ttree (=)) = (\<lambda>xss. [] \<in> xss \<and> downset xss)"
irfb_irbd_dual: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "fb\<^sup>-\<^sub>\<R> \<phi> = (\<smile> \<circ> bd\<^sup>-\<^sub>\<R> \<circ> \<partial>\<^sub>F) \<phi>"
is_dghm_axioms: shows "\<phi> i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i"
set_child_nodes_get_shadow_root_is_l_set_child_nodes_get_shadow_root: shows "l_set_child_nodes_get_shadow_root Shadow_DOM.set_child_nodes_locs get_shadow_root_locs"
af_letter_congruent: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" assumes "\<phi> \<sim> \<psi>" shows "af_letter \<phi> \<nu> \<sim> af_letter \<psi> \<nu>"
cases: fixes x :: "('b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('b \<times> 'c) list \<times> ('b \<times> 'c) list" obtains f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" and k :: "'b" and v :: "'c" and as :: "('b \<times> 'c) list" and k' :: "'b" and v' :: "'c" and bs :: "('b \<times> 'c) list" where "x = (f__, (k, v) # as__, (k', v') # bs__)"   | f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" and bs :: "('b \<times> 'c) list" where "x = (fa__, [], bs)"   | f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" and as :: "('b \<times> 'c) list" where "x = (f, as, [])"
leadsTo_Un_duplicate2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and C :: "'c set" assumes "F \<in> A \<longmapsto> A' \<union> C \<union> C" shows "F \<in> A \<longmapsto> A' \<union> C"
map_id: fixes t :: "'a multiset" shows "image_mset id t = t"
con3: fixes env :: "v sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "exp0 list"   and ck :: "bool"   and s :: "'b state"   and s' :: "'b state"   and err :: "v error_result" assumes "do_con_check (c env) cn (length es) \<and> evaluate_list ck env s (rev es) (s', Rerr err)" shows "evaluate ck env s (Con cn es) (s', Rerr err)"
sum_telescope_Ico: fixes a :: "nat"   and b :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "a \<le> b" shows "(\<Sum>i = a..<b. f i - f (Suc i)) = f a - f b"
a0i_agreement_def: shows "a0i_agreement \<equiv> {s. \<forall>A B M. signals s (Commit A B M) \<le> signals s (Running A B M)}"
asubg_mOp_closed: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set"   and x :: "'c" assumes "aGroup A"   and "A +> H"   and "x \<in> H" shows "-\<^sub>a\<^bsub>A\<^esub> x \<in> H"
valid_eq_valid': fixes l :: "color list" shows "valid' l = valid l"
top_greatest: fixes a :: "'a" shows "a \<le> (1::'a)"
set_shadow_root_get_child_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>Shadow_DOM.get_child_nodes_locs ptr'. r h h')"
ctr: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" shows "derivation_from St = LCons St (if is_final_dstate St then LNil else derivation_from (deterministic_RP_step St))"
YFtorNTCatCod: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" shows "NTCatCod (YFtorNT C f) = SET"
s_case_expand: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and s :: "'b \<times> 'c \<times> 'd" shows "(case s of (c, n, ts) \<Rightarrow> P c n ts) = P (cc s) (nn s) (tts s)"
SReal_hcmod_SComplex: fixes x :: "complex star" assumes "x \<in> SComplex" shows "hcmod x \<in> \<real>"
admits_diagonal_imp_admits_triangular_1xn: fixes n :: "nat" assumes "\<forall>A\<in>carrier_mat 1 2. admits_diagonal_reduction A" shows "\<forall>A\<in>carrier_mat 1 n. admits_triangular_reduction A"
assms_8: shows "\<pi> ` A \<subseteq> B"
sgn_2: shows "0 \<le> real_of_1 x"
1: shows "monom (1::'a) (n - Suc (degree u + i)) \<noteq> 0"
is_bal_l_bal: fixes l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0"   and n :: "'a" assumes "is_bal l"   and "is_bal r"   and "height l = height r + 2" shows "is_bal (l_bal\<^sub>0 n l r)"
fixf_po: shows "\<lparr>pset = P, order = induced P r\<rparr> \<in> PartialOrder"
hyps: shows "X = X @@ A ^^ Suc n \<union> (\<Union>m\<le>n. B @@ A ^^ m)"
nonneg_zcount_image_zmset: fixes M :: "'a zmultiset"   and f :: "'a \<Rightarrow> 'b"   and t :: "'b" assumes "\<And>t. 0 \<le> zcount M t" shows "0 \<le> zcount (Auxiliary.image_zmset f M) t"
Healthy_Sequence: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and P :: "'a hrel"   and Q :: "'a hrel" assumes "utp_theory_rel hcond"   and "P is hcond"   and "Q is hcond" shows "P ;; Q is hcond"
uminus_uint8_def: shows "uminus \<equiv> map_fun Rep_uint8 Abs_uint8 uminus"
pick_insert: fixes a :: "nat"   and I :: "nat set"   and i :: "nat"   and a' :: "nat" assumes "a \<notin> I"   and "i < card I"   and "pick (insert a I) a' = a"   and "i < a'"   and "a' < card I + 1" shows "pick (insert a I) i = pick I i"
minimal_sequence_to_failure_extending_implies_Rep_Pre: fixes V :: "'d list set"   and M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V'' :: "('d \<times> 'e) list set"   and xs' :: "('d \<times> 'e) list" assumes "minimal_sequence_to_failure_extending V M1 M2 vs xs"   and "OFSM M1"   and "OFSM M2"   and "test_tools M2 M1 FAIL PM V \<Omega>"   and "V'' \<in> N (vs @ xs') M1 V"   and "prefix xs' xs" shows "\<not> Rep_Pre M2 M1 vs xs'"
indRelLSTPO_impl_SRel_and_TRel_reflect_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelLSTPO SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred SRel (\<lambda>b. hasBarb b SWB)"
hm_is_empty_impl: shows "imp_map_is_empty is_hashmap hm_isEmpty"
fold_congs_1: fixes r :: "'a control_state_scheme"   and r' :: "'a control_state_scheme"   and v' :: "bool"   and f :: "bool \<Rightarrow> bool"   and f' :: "bool \<Rightarrow> bool" assumes "r = r'"   and "break_status r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "break_status_update f r = break_status_update f' r'"
cases: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" assumes "(a1, a2) \<in> encoding.indRelLT Enc TRel" obtains S :: "'a" where "a1 = TargetTerm (Enc S)" and "a2 = SourceTerm S"   | T1 :: "'b" and T2 :: "'b" where "a1 = TargetTerm T1" and "a2 = TargetTerm T2" and "(T1, T2) \<in> TRel"
tranclp_into_rtranclp: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k" assumes "r\<^sup>+\<^sup>+ a b" shows "r\<^sup>*\<^sup>* a b"
coset_join2: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and H :: "'c set" assumes "Group.group G"   and "x \<in> carrier G"   and "subgroup H G"   and "x \<in> H" shows "H #>\<^bsub>G\<^esub> x = H"
in_set_real_minus_swapI: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and i :: "nat"   and a :: "float interval list"   and I :: "float interval list" assumes "x - e i \<in>\<^sub>r a ! i"   and "x \<in>\<^sub>r I ! i" shows "e i \<in>\<^sub>r I ! i - a ! i"
c: shows "t\<^sub>p (Partial_Cost_Model.steps init (take x qs) (take x Strat)) (qs ! x) (Strat ! x) = t\<^sub>p (Partial_Cost_Model.steps init (take x qs) (take x (take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat))) (qs ! x) ((take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat) ! x) + 1"
mset_bubble_min: fixes xs :: "'a list" shows "mset (bubble_min xs) = mset xs"
Quotient_rep_reflp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "Quotient R Abs Rep T" shows "R (Rep a) (Rep a)"
edge_to_inEdge: fixes n :: "'node"   and eD :: "'edgeD"   and m :: "'node"   and g :: "'g" assumes "(n, eD, m) \<in> \<alpha>e g"   and "invar g" obtains "(n, eD, m) \<in> set (inEdges g m)"
cfWriteNoExtend_def: fixes cfile :: "CFile"   and byteIndex :: "nat"   and "value" :: "byte" shows "cfWriteNoExtend cfile byteIndex value \<equiv> let i = byteIndex div blockSize; j = byteIndex mod blockSize; block = readCArray (data cfile) i in cfile\<lparr>data := writeCArray (data cfile) i (writeCArray block j value)\<rparr>"
assms_4: shows "a \<in> hyp2 \<union> S"
updown_eq_iff: fixes y :: "'c['b, 'd]"   and x :: "'c['a, 'd]" assumes "QD('a) = QD('b)"   and "y = toQ (fromQ x)" shows "x \<cong>\<^sub>Q y"
PF: shows "Ifm (x # bs) (plusinf (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))) = False"
pnn: shows "prv (neg (neg (wrepr.PP \<langle>\<phi>G\<rangle>)))"
sums_1: fixes z :: "complex" assumes "\<bar>Re z\<bar> < 1" shows "(\<lambda>n. f n * (a * z) ^ n) sums F z"
minus_divide_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "uminus (divide a b) = divide a (uminus b)"
IDirProd_comm: fixes A :: "'a set"   and B :: "'a set" shows "IDirProd G A B = IDirProd G B A"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'c" shows "initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> aodv_init i | Some s \<Rightarrow> s, snd \<sigma>)"
neg_one_power_add_eq_neg_one_power_diff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and k :: "nat"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "k \<le> n" shows "power.power one times (uminus one) (n + k) = power.power one times (uminus one) (n - k)"
Inf_to_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and dual :: "'a \<Rightarrow> 'a" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Inf = dual \<circ> Sup \<circ> (`) dual"
tendsto_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "F \<noteq> bot"   and "Limsup F f = Liminf F f" shows "(f \<longlongrightarrow> Limsup F f) F"
re_refine_aux: shows "gen_dfs \<le> \<Down> rel a.c.gen_dfs"
Upd_par_red: fixes t :: "sterm"   and l :: "Label"   and u :: "sterm"   and z :: "sterm" assumes "Upd t l u => z" obtains t' :: "sterm" and u' :: "sterm" and L :: "char list set" where "t => t'" and "finite L__" and "\<forall>s p. s \<notin> L__ \<and> p \<notin> L__ \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. u\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> u'__ = \<sigma>[s,p] t'')" and "z = Upd t' l u'__"   | f :: "Label \<Rightarrow> sterm option" and f' :: "Label \<Rightarrow> sterm option" and T :: "Sigma.type" and u' :: "sterm" and L :: "char list set" where "l \<in> dom f" and "Obj f T = t" and "Obj f T => Obj f' T" and "finite L" and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. u\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> u' = \<sigma>[s,p] t'')" and "z = Obj (f'(l \<mapsto> u')) T"
elims: fixes x :: "stack_elem list"   and y :: "Arity" assumes "Astack x = y" obtains "x = []" and "y = 0"   | xa :: "Vars.var" and S :: "stack_elem list" where "x = Arg xa__ # S__" and "y = inc\<cdot>(Astack S__)"   | e1 :: "exp" and e2 :: "exp" and S :: "stack_elem list" where "x = Alts e1 e2 # Sa__" and "y = 0"   | xa :: "Vars.var" and S :: "stack_elem list" where "x = Upd xaa__ # Sb__" and "y = 0"   | xa :: "Vars.var" and S :: "stack_elem list" where "x = Dummy xa # S" and "y = 0"
index_def: fixes s :: "'b" shows "index s = (SOME i. i \<in> I \<and> s \<in> S i)"
alg_dvd_add: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x alg_dvd y"   and "x alg_dvd z" shows "x alg_dvd y + z"
term_ok_imp_typ_ok: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and ty :: "typ" assumes "wf_theory thy"   and "term_ok thy t"   and "typ_of t = Some ty" shows "typ_ok thy ty"
del_Leaf: fixes x :: "'a" shows "del x \<langle>\<rangle> = None"
U_4: fixes i :: "nat" shows "closure (U i) \<subseteq> carrier"
ivl_disj_un_one_7: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "{l..u} \<union> {u<..} = {l..}"
index_unit_vec_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "unit\<^sub>v n i $ j = (if j = i then 1::'a else (0::'a))"
m_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (A [X])"   and "y \<in> carrier (A [X])" shows "x \<otimes>\<^bsub>A [X]\<^esub> y \<in> carrier (A [X])"
smap_bottom_iff: fixes f :: "'b \<rightarrow> 'a"   and xs :: "[:'b:]" shows "(smap\<cdot>f\<cdot>xs = \<bottom>) = (xs = \<bottom> \<or> (\<exists>x\<in>sset xs. f\<cdot>x = \<bottom>))"
on_triple_def: fixes R :: "('e \<times> 'f) set" shows "on_triple R \<equiv> {((l, s, t), l', s', t'). l = l' \<and> (s, s') \<in> R \<and> (t, t') \<in> R}"
locally_refl_def: fixes R :: "('a \<times> 'a) set"   and A :: "'a multiset" shows "locally_refl R A \<equiv> \<forall>a. a \<in># A \<longrightarrow> (a, a) \<in> R"
CCfix_nonrec_def: fixes x :: "Vars.var"   and e :: "exp" shows "CCfix_nonrec x e = (\<Lambda> i. ccBind x e\<cdot>(Aheap_nonrec x e\<cdot>i, snd i) \<squnion> fv e G\<times> (ccNeighbors x (snd i) - (if isVal e then {} else {x})) \<squnion> snd i)"
c_hd_aux0: fixes u :: "nat" assumes "c_len u = 1" shows "nat_to_list u = [c_snd (u - 1)]"
conv1: shows "\<I>1,\<I>1' \<turnstile>\<^sub>C conv1 \<sim> map_converter id id f1 g1 1\<^sub>C"
pos_add_strict: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.strict_ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less b c" shows "less b (plus a c)"
25: fixes x :: "'a"   and y :: "'a" shows "- - x \<squnion> - (- x \<sqinter> y) = - (- x \<sqinter> y)"
assms: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
rep_eq: fixes bound :: "nat" shows "generator (upt_generator bound) = (\<lambda>n. n < bound, \<lambda>n. (n, Suc n))"
InfModelI: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "MonotModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> (MonotProblem.intTI wtFsym wtPsym arOf resOf parOf \<Phi>) (MonotProblem.intFI wtFsym wtPsym arOf resOf parOf \<Phi>) (MonotProblem.intPI wtFsym wtPsym arOf resOf parOf \<Phi>)"
kas: shows "x0 = x \<and> y0 = y \<or> y0 = x \<and> x0 = y"
decr_grading_p_sum: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and f :: "'d \<Rightarrow> 't \<Rightarrow>\<^sub>0 'c"   and A :: "'d set" shows "decr_grading_p d n (sum f A) = (\<Sum>a\<in>A. decr_grading_p d n (f a))"
empty_map_add: fixes m :: "'a \<Rightarrow> 'b option" shows "m \<Oplus> \<emptyset> = m"
intros_3: shows "OclVoid \<sqsubset>\<^sub>B String"
1: shows "- - v\<^sup>T * w \<le> - - v\<^sup>T * (w \<sqinter> - v\<^sup>T) * w\<^sup>\<star> \<squnion> - - v\<^sup>T"
vertex_bipartite_web_2: fixes x :: "'v" shows "vertex (bipartite_web_of \<Gamma>) (Inr x) = (vertex \<Gamma> x \<and> x \<notin> A \<Gamma>)"
sup_ge2: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and y :: "'b::type"   and x :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (y::'b::type) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) y)"
path_connected_complement_homeomorphic_convex_compact: fixes S :: "'c set"   and T :: "'d set" assumes "S homeomorphic T"   and "convex T"   and "compact T"   and "2 \<le> DIM('c)" shows "path_connected (- S)"
fm_image_measurable_finite: fixes M :: "'d measure"   and X :: "('a \<Rightarrow> 'd) set" assumes "space M = UNIV"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M))" shows "fm ` X \<in> sets (Pi\<^sub>F (Collect finite) (\<lambda>_. M))"
E_wfD_2: fixes G :: "'c graph"   and v :: "'c"   and v' :: "'c" assumes "wf_graph G"   and "(v, v') \<in> edges G" shows "v' \<in> nodes G"
root_hash_G_def: fixes rha :: "'e \<Rightarrow> 'f"   and rhb :: "'g \<Rightarrow> 'h" shows "root_hash_G rha rhb = root_hash_G' \<circ> map_G\<^sub>m rha id rhb id"
F_def: shows "F \<equiv> rec_nat [M0] (\<lambda>n NL. Eps (\<Phi> NL) # NL)"
nvT_vars: fixes x :: "Preliminaries.var"   and T :: "'c trm" assumes "x \<in> nvT T" shows "x \<in> vars T"
disjointD: fixes A :: "'c set set"   and a :: "'c set"   and b :: "'c set" assumes "disjoint A"   and "a \<in> A"   and "b \<in> A"   and "a \<noteq> b" shows "a \<inter> b = {}"
finite_ET: fixes ST :: "('d, 'e, 'f) status" shows "finite (ET ST)"
simps_3: fixes p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int" shows "closest_pair_bf_code [p\<^sub>0, p\<^sub>1] = (dist_code p\<^sub>0 p\<^sub>1, p\<^sub>0, p\<^sub>1)"
3: shows "stake (Suc i) w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1"
g: fixes x :: "'a" assumes "x \<in> t - s" shows "(0::'b) \<le> g x"
fstate_def: fixes tr :: "'c \<times> 'b" shows "fstate tr \<equiv> snd tr"
pell_valuation_pos_neg_aux: fixes x :: "int"   and y :: "int" assumes "0 < x"   and "0 < y"   and "solution (x, y)" shows "pell_valuation (x, - y) \<in> {0<..<1}"
foldseq_distr_unary: fixes g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<And>a b. g (f a b) = f (g a) (g b)" shows "g (foldseq f s n) = foldseq f (\<lambda>x. g (s x)) n"
cf_h_polyn: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "pol_coeff S (n, f)" shows "polyn_expr A Y n (cf_h h (n, f)) = polyn_expr A Y n (n, cmp h f)"
power2_diff: fixes x :: "'a"   and y :: "'a" shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - (2::'a) * x * y"