hcju/isabelleps huggingface dataset:
compl_mono: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq x y" shows "less_eq (uminus y) (uminus x)"
eventually_at_bot_linorder: fixes P :: "'a \<Rightarrow> bool" shows "eventually P at_bot = (\<exists>N. \<forall>n\<le>N. P n)"
is_full_pmdl_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_full_pmdl B = (\<forall>p. component_of_term ` keys p \<subseteq> component_of_term ` Keys B \<longrightarrow> p \<in> pmdl B)"
diagonal_to_Smith_aux_append2: fixes A :: "(('a, 'b) vec, 'c) vec"   and xs :: "nat list"   and ys :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_aux A (xs @ [ys]) bezout = Diagonal_to_Smith_row_i (diagonal_to_Smith_aux A xs bezout) ys bezout"
Sretr_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" shows "Sretr theta \<equiv> {(c, d). matchC_C theta c d}"
dim_unique: fixes B :: "('a \<Rightarrow> real) set"   and V :: "('a \<Rightarrow> real) set"   and n :: "nat" assumes "B \<subseteq> diff_fun_space"   and "V \<subseteq> diff_fun_space"   and "B \<subseteq> V"   and "V \<subseteq> sub.diff_fun_space.m1.span B"   and "\<not> sub.diff_fun_space.m1.dependent B"   and "card B = n" shows "diff_fun_space.m1.dim V = n"
Nmlize_in_Hom_1: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> \<in> HHom (Src t) (Trg t)"
dom_subid_aux2: fixes x :: "'a"   and y :: "'a" shows "ad (ad x) \<cdot> y \<le> y"
matricize_def: fixes rmodes :: "nat set"   and T :: "'a tensor" shows "matricize rmodes T = mat (prod_list (nths (dims T) rmodes)) (prod_list (nths (dims T) (- rmodes))) (\<lambda>(r, c). Tensor.lookup T (weave rmodes (digit_encode (nths (dims T) rmodes) r) (digit_encode (nths (dims T) (- rmodes)) c)))"
simps: fixes subsumes :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b \<Rightarrow> 'b list"   and todo :: "'b list"   and fin :: "'b list" shows "mk_rtrancl_list_main subsumes r todo fin = (case todo of [] \<Rightarrow> fin | a # tod \<Rightarrow> if \<exists>b\<in>set fin. subsumes b a then mk_rtrancl_list_main subsumes r tod fin else mk_rtrancl_list_main subsumes r (r a @ tod) (a # fin))"
Cidl: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) ;;\<^bsub>C\<^esub> f = f"
alg_multE_2: fixes x :: "'a"   and p :: "'a list"   and n :: "nat" assumes "x \<in> carrier R"   and "p \<in> carrier (poly_ring R)"   and "p \<noteq> []"   and "[\<one>, \<ominus> x] [^]\<^bsub>poly_ring R\<^esub> n pdivides p" shows "n \<le> alg_mult p x"
is_strongly_dom_component_safe_def: shows "Shadow_DOM_Components.is_strongly_dom_component_safe \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.to_tree_order"
set_tag_name_locs_impl: shows "set_tag_name_locs = local.a_set_tag_name_locs"
set_spmf_of_set: fixes A :: "'b set" shows "set_spmf (spmf_of_set A) = (if finite A then A else {})"
"1_3": assumes "n \<noteq> 0"   and "n \<noteq> 1"   and "a \<noteq> 1"   and "\<not> n < 0"   and "odd n"   and "a < 0" shows "Jacobi (- a) n = jacobi_code (- a) n"
inline_map_gpv: fixes callee :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b, 'c, 'd) gpv"   and f :: "'g \<Rightarrow> 'a"   and g :: "'h \<Rightarrow> 'e"   and gpv :: "('g, 'h, 'f) gpv"   and s :: "'b" shows "inline callee (map_gpv f g gpv) s = map_gpv (apfst f) id (inline (\<lambda>s x. callee s (g x)) gpv s)"
qFresh_qSwap_exchange1: fixes zs :: "'d"   and z2 :: "'e"   and z1 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "qFresh zs z2 (X #[[z1 \<and> z2]]_zs) = qFresh zs z1 X"
a2_map'_keys: shows "Mapping.keys a2_map' = {tp - len..tp + 1}"
isolated_verts_in_sccs: fixes u :: "'a" assumes "u \<in> H.isolated_verts" shows "{u} \<in> H.sccs_verts"
missing_cs_contradicts: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and n :: "nat"   and n' :: "nat" assumes "\<nexists>k'. cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k'"   and "k < n"   and "cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'" shows "\<exists>j'. (\<sigma>', j') \<cc> (\<sigma>, k)"
div256: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "a < 256"   and "b < 256"   and "c < 256"   and "d < 256" shows "nat ((int d + int (256 * c) + int (65536 * b) + int (16777216 * a)) div 256 mod 256) = c"
gram_schmidt0_def: fixes n :: "nat"   and ws :: "'a vec list" shows "gram_schmidt0 n ws = gram_schmidt_sub0 n [] ws"
domains_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.domains (g \<cdot>\<^sub>C f) = C.domains f"
lang_ACI_norm: fixes r :: "'a rexp" shows "lang \<guillemotleft>r\<guillemotright> = lang r"
d2: shows "\<forall>p\<in>set Ps'. \<exists>p'\<in>set ps. p = extend ( \<Phi> + \<Gamma>' \<Rightarrow>* \<Psi>1 + \<Delta>') p'"
invh_baldL_invc: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "bheight l + 1 = bheight r"   and "invc r" shows "invh (baldL l a r) \<and> bheight (baldL l a r) = bheight r"
remainderexplemma: fixes y :: "nat"   and a :: "nat"   and z :: "nat"   and x :: "nat" assumes "y mod a = z mod a" shows "x * y mod a = x * z mod a"
assms_1: shows "set \<phi>s \<subseteq> fmla"
cet_val: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and v :: "val" shows "\<Gamma> \<turnstile>\<^sub>c CVal v : val_type v"
l_comb_add: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'b"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "t \<in> {j. j \<le> m} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> H" shows "l_comb R M (Suc (n + m)) (jointfun n s m t) (jointfun n f m g) = l_comb R M n s f \<plusminus> l_comb R M m t g"
sup_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
assms: shows "\<Psi> \<rhd> P \<approx>\<^sub>s Q"
runit_char_3: fixes f :: "'a" assumes "ide f" shows "PB.runit f \<star> src f = (f \<star> \<i>' (src f)) \<cdot> \<a>[f, src f, src f]"
card2_eI: fixes S :: "'a set" assumes "2 \<le> card S" shows "\<exists>x y. x \<noteq> y \<and> x \<in> S \<and> y \<in> S"
max_fun_diff: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> FINFUNC"   and "g \<in> FINFUNC" shows "\<exists>a b. a \<noteq> b \<and> a \<in> Field r \<and> b \<in> Field r \<and> f (s.max_fun_diff f g) = a \<and> g (s.max_fun_diff f g) = b"
class_changed_cons: fixes t :: "'b cdecl"   and P :: "'b prog"   and P' :: "'b prog" shows "fst t \<notin> classes_changed (t # P) (t # P')"
max_mapping_cobounded2: fixes P2 :: "'a \<Rightarrow> 'b option"   and P1 :: "'a \<Rightarrow> 'b option" assumes "dom P2 \<subseteq> dom P1" shows "rel_mapping (\<le>) P2 (max_mapping P1 P2)"
half_segments_of_aform_def: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "half_segments_of_aform X = (let x0 = lowest_vertex (fst X, nlex_pdevs (snd X)) in polychain_of x0 (map ((*\<^sub>R) 2) (ccw.selsort 0 (inl (snd X)))))"
Inf_hd_in_hd: fixes n :: "nat" shows "Inf (hd (F n)) \<in> hd (F n)"
butlast_strict_prefix: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "strict_prefix (butlast xs) xs"
checked_invariant_s: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) set"   and s :: "'s" assumes "checked_s t as s" shows "invariant_s t as s"
llength_eq_infty_conv_lfinite: fixes xs :: "'c llist" shows "(llength xs = \<infinity>) = (\<not> lfinite xs)"
in_box_eucl_less: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> box a b) = (a <e x \<and> x <e b)"
rep_eq: fixes x :: "'c monom" shows "monom_vars_list x = map fst (Rep_monom x)"
SN_rel_on_def: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" shows "SN_rel_on R S \<equiv> SN_on (S\<^sup>* O R O S\<^sup>*)"
iscale_def: shows "iscale \<equiv> \<lambda>z. (z + 1) /\<^sub>R 2"
equality: fixes r :: "('b, 'c) span_data_scheme"   and r' :: "('b, 'c) span_data_scheme" assumes "Leg0 r = Leg0 r'"   and "Leg1 r = Leg1 r'"   and "span_data.more r = span_data.more r'" shows "r = r'"
full_elims_1: fixes n :: "nat" assumes "full n \<langle>\<rangle>" obtains "n = 0"
least_upper_bound: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "(- x \<le> - z \<and> - y \<le> - z) = (- x \<squnion> - y \<le> - z)"
tyenv_eq_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2 \<equiv> \<forall>x. type_max (to_total \<Gamma> x) mem\<^sub>1 = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
Sup_Assertion_def: fixes A :: "'c Assertion set" shows "\<Squnion>A = abs_wpt (\<Squnion>(assert ` A))"
vdomain_atE: fixes r :: "V"   and a :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" obtains b :: "V" where "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" and "r\<lparr>a\<rparr> = b"
extends_subst_fun_upd_matching: fixes \<sigma> :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and t :: "'a"   and \<tau> :: "'b \<Rightarrow> 'a option" assumes "\<sigma> x = Some t" shows "extends_subst (\<sigma>(x \<mapsto> t)) \<tau> = extends_subst \<sigma> \<tau>"
matricize_def: fixes rmodes :: "nat set"   and T :: "'b tensor" shows "matricize rmodes T = mat (prod_list (nths (dims T) rmodes)) (prod_list (nths (dims T) (- rmodes))) (\<lambda>(r, c). lookup T (weave rmodes (digit_encode (nths (dims T) rmodes) r) (digit_encode (nths (dims T) (- rmodes)) c)))"
cardFinestpart: fixes X :: "'c set" shows "card (finestpart X) = card X"
children_eq2: fixes ptr' :: "(_) object_ptr" assumes "ptr \<noteq> ptr'" shows "|h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h' \<turnstile> get_child_nodes ptr'|\<^sub>r"
arr_def: fixes f :: "Arr" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
wf_lderivs: fixes n :: "nat"   and r :: "'b rexp"   and ws :: "'a list" assumes "local.wf n r" shows "local.wf n (lderivs ws r)"
Sup_synlr_def: fixes A :: "('a, 'b) synlr set" shows "Sup A = Inf {y. \<forall>x\<in>A. x \<le> y}"
sterm_determines_frechet: fixes I :: "('b, 'c, 'd) interp"   and J :: "('e, 'f, 'd) interp"   and \<theta>1 :: "('b, 'd) trm"   and \<theta>2 :: "('e, 'd) trm"   and \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec" assumes "is_interp I"   and "is_interp J"   and "dfree \<theta>1"   and "dfree \<theta>2"   and "sterm_sem I \<theta>1 = sterm_sem J \<theta>2" shows "frechet I \<theta>1 (fst \<nu>) (snd \<nu>) = frechet J \<theta>2 (fst \<nu>) (snd \<nu>)"
odd_numeral: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (num.Bit1 n))"
nodes_it_def: fixes nodes_list_it :: "'d \<Rightarrow> ('c list \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'c list \<Rightarrow> 'c list) \<Rightarrow> 'c list \<Rightarrow> 'c list"   and g :: "'d" shows "graph_nodes_it_defs.nodes_it nodes_list_it g \<equiv> it_to_it (nodes_list_it g)"
dests_inv: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:9, PRreq-:17, PRrep-:8, PRerr-:1} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>))) \<and> (l \<in> {PAodv-:16..PAodv-:17} \<union> {PPkt-:8..PPkt-:9} \<union> {PRreq-:10..PRreq-:11} \<union> {PRreq-:18..PRreq-:19} \<union> {PRrep-:9..PRrep-:10} \<union> {PRerr-:2..PRerr-:4} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> iD (rt \<xi>) \<and> the (dests \<xi> ip) = sqn (rt \<xi>) ip)) \<and> (l = PPkt-:12 \<longrightarrow> dip \<xi> \<in> iD (rt \<xi>)))"
inject_4: fixes x51 :: "'a pfm"   and x52 :: "'a pfm"   and y51 :: "'a pfm"   and y52 :: "'a pfm" shows "((x51 \<^bold>\<longrightarrow>\<^sub>! x52) = (y51 \<^bold>\<longrightarrow>\<^sub>! y52)) = (x51 = y51 \<and> x52 = y52)"
execonst_type_of_def: fixes exesignature :: "exesignature" shows "execonst_type_of exesignature = (case exesignature of ExeSignature x1 x2 x3 \<Rightarrow> x1)"
Class_self: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c" assumes "equivalence S E"   and "a \<in> S" shows "a \<in> equivalence.Class S E a"
sort_snoc: fixes xs :: "'a list"   and x :: "'a" shows "sort (xs @ [x]) = insort x (sort xs)"
lexp_tail: fixes order :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "lexp order xs ys" shows "lexp order (x # xs) (x # ys)"
sep_implI: fixes h :: "'a"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>h'. \<lbrakk>h ## h'; P h'\<rbrakk> \<Longrightarrow> Q (h + h')" shows "(P \<longrightarrow>* Q) h"
FAssNull: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
valid_gc_join_mmsaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list"   and rel1 :: "'a option list set" assumes "valid_mmsaux args cur aux auxlist"   and "table (args_n args) (args_L args) rel1" shows "valid_mmsaux args cur (gc_join_mmsaux args rel1 aux) (map (\<lambda>(t, rel). (t, join rel (args_pos args) rel1)) auxlist)"
select_object\<^sub>S\<^sub>e\<^sub>q_def: shows "select_object\<^sub>S\<^sub>e\<^sub>q = select_object Sequence{} UML_Sequence.OclIncluding id"
axioms_1: fixes R :: "('c, 'd) ring_scheme" assumes "domain R" shows "cring R"
assms_3: shows "\<psi>1 \<in> fmla"
map_def: shows "\<Psi>.map = transformation_by_components.map (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F (\<lambda>a. S.inv (\<Phi> a))"
bij_betw_hom_hfun: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "bij_betw arr_to_hfun (hom a b) {F. hfun (ide_to_hf a) (ide_to_hf b) F}"
checker_r2_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "checker_r2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> \<equiv> let distance = s\<^sub>o - s\<^sub>e; precond = check_precond_r2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta>; safe_dist0 = safe_distance_1r a\<^sub>e v\<^sub>e \<delta>; safe_dist1 = safe_distance_2r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> in if \<not> precond then False else if safe_dist0 < distance then True else safe_dist1 < distance"
continuous_imp_mono: fixes f :: "'b set \<Rightarrow> 'c set" assumes "continuous f" shows "mono f"
assms_1: shows "P \<Longrightarrow>\<^sub>\<tau> P'"
unwind_to_def: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and \<Delta>s :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) set" shows "unwind_to \<Delta> \<Delta>s \<equiv> \<forall>s vl s1 vl1. reachNT s \<and> reach s1 \<and> \<Delta> s vl s1 vl1 \<longrightarrow> vl \<noteq> [] \<and> exit s (hd vl) \<or> iaction (disjAll \<Delta>s) s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> reaction (disjAll \<Delta>s) s vl s1 vl1"
llast_conv_lnth: fixes xs :: "'b llist"   and n :: "nat" assumes "llength xs = eSuc (enat n)" shows "llast xs = lnth xs n"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>C 'd"   and xa :: "'c \<Rightarrow>\<^sub>C 'd" shows "apply_bcontfun (x - xa) = (\<lambda>xb. apply_bcontfun x xb - apply_bcontfun xa xb)"
that_2: fixes y :: "'a" assumes "y \<in> set xs" shows "compare cmp x y \<noteq> Greater"
fold_congs_4: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "'b"   and f :: "'b \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b" assumes "r = r'"   and "hare r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "hare_update f r = hare_update f' r'"
right_total_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "right_total (BNF_Def.Grp A f) = (f ` A = UNIV)"
prime_factorization_Suc_0: shows "prime_factorization (Suc 0) = {#}"
countable_bipartite_web_axioms: shows "countable_bipartite_web \<Gamma>"
idhk: fixes y :: "'b" assumes "y \<in> t" shows "h (k y) = y"
word_clr_le: fixes w :: "'b word"   and n :: "nat" shows "set_bit_class.set_bit w n False \<le> w"
prj_chine_assoc_1: shows "f_src_src.Prj\<^sub>1 \<cdot> f_src_src.chine_assoc = f_src_src.Prj\<^sub>1\<^sub>1"
min_int_poly_unique: fixes p :: "int poly"   and x :: "'a" assumes "p represents x"   and "irreducible p"   and "0 < Polynomial.lead_coeff p" shows "min_int_poly x = p"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b" assumes "category C"   and "category D"   and "functor C D G"   and "equivalence_functor_axioms C D G" shows "equivalence_functor C D G"
F\<eta>_inverse_4: fixes b :: "'d" assumes "D.ide b" shows "F (\<eta> b) = C.inv (\<epsilon>' (F b))"
set_Macaulay_red: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (Macaulay_red (Keys_to_list fs) fs) = set (Macaulay_list fs) - {p. lt p \<in> lt_set (set fs)}"
igFreshIGAbs2_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igFreshIGAbs2 MOD \<equiv> \<forall>ys y xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
typ_of_arg: fixes t :: "Core.term"   and u :: "Core.term"   and R :: "typ"   and T :: "typ" assumes "typ_of (t $ u) = Some R"   and "typ_of u = Some T" shows "typ_of t = Some (T \<rightarrow> R)"
inv_comp_right_2: fixes g :: "'d"   and f :: "'d" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso f" shows "D.iso g"
\<beta>: shows "\<beta> \<in> maps_over (labels p) UNIV"
HARootState_Range_ChiStar: fixes x :: "'d"   and S :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "x \<noteq> S"   and "S \<in> States (HARoot A)" shows "(x, S) \<notin> ChiStar A"
Inf_greatest: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and z :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq z x" shows "less_eq z (Inf A)"
induct: fixes P :: "nat set \<Rightarrow> (nat set \<times> 'a option list set) set \<Rightarrow> (nat set \<times> 'a option list set) set \<Rightarrow> bool"   and a0 :: "nat set"   and a1 :: "(nat set \<times> 'a option list set) set"   and a2 :: "(nat set \<times> 'a option list set) set" assumes "\<And>V Q_pos Q_neg. \<lbrakk>\<And>x xa y xb xc. \<lbrakk>\<not> card V \<le> 1; x = getIJ Q_pos Q_neg V; (xa, y) = x; xb = projectQuery xa (filterQuery xa Q_pos); xc = filterQueryNeg xa Q_neg\<rbrakk> \<Longrightarrow> P xa xb xc; \<And>x xa y xb xc xd xe xf xg xh. \<lbrakk>\<not> card V \<le> 1; x = getIJ Q_pos Q_neg V; (xa, y) = x; xb = projectQuery xa (filterQuery xa Q_pos); xc = filterQueryNeg xa Q_neg; xd = genericJoin xa xb xc; xe = Q_neg - xc; xf = filterQuery y Q_pos\<rbrakk> \<Longrightarrow> P y (newQuery y xf (xa, xh)) (newQuery y xe (xa, xh))\<rbrakk> \<Longrightarrow> P V Q_pos Q_neg" shows "P a0 a1 a2"
intros_2: fixes u :: "'d \<Rightarrow> 'c"   and c :: "'d"   and d :: "'c" assumes "- u c \<le> d" shows "dbm_entry_val u None (Some c) (Le d)"
coDhalf: shows "continuous_on ({0..1 / 2} - insert (1 / 2) S) (\<lambda>x. vector_derivative (g1 \<circ> (*) 2) (at x))"
dom_char: fixes f :: "'c local.term set" shows "local.dom f = (if arr f then mkarr (DOM f) else null)"
bijective_alt: fixes R :: "('d \<times> 'e) set" shows "bijective R = (single_valued R \<and> single_valued (R\<inverse>))"
finite_pred: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat" assumes "RedBlack prb"   and "finite_RedBlack prb"   and "rv \<in> red_vertices prb" shows "finite (pred (confs prb rv))"
tendsto_prod: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and L :: "'a \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F" shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> prod L S) F"
lhs_def: shows "lhs \<equiv> \<lambda>plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1. finite_dimensional_vector_space_pair_1_ow (Collect (Domainp B\<^sub>1)) plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 (Collect (Domainp B\<^sub>2))"
simps_185: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x91 :: "expr"   and x92 :: "char list"   and x93 :: "char list list"   and x94 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> x91\<bullet>x92{x93} := x94"
subset_self: fixes A :: "'e set" shows "A \<subseteq> A"
distinct_1: fixes x2 :: "'c set" shows "FAILi \<noteq> RES x2"
IH: assumes "a \<in> set xs" shows "a \<preceq>\<^sub>t list_max xs"
comp_assoc: fixes h :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" shows "S.comp (S.comp h g) f = S.comp h (S.comp g f)"
inj_on_def: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
sorted_list_of_set_filter: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite A" shows "sorted_list_of_set {x \<in> A. P x} = filter P (sorted_list_of_set A)"
C'_eq: shows "C' = (\<lambda>t. ball t (ee t / 3)) ` kk"