hcju/isabelleps huggingface dataset:
backward_finite_path_root_terminating_or_cycle: fixes r :: "'a"   and x :: "'a" shows "backward_finite_path_root r x = (backward_terminating_path_root r x \<or> non_empty_cycle_root r x)"
is_rbt_impl_of: fixes t :: "'a set_rbt" shows "ord.is_rbt cless (mapping_rbt.impl_of t)"
CL: fixes nd :: "('a, 'b) node_scheme" assumes "nd \<in> nds" shows "incoming nd \<subseteq> insert expand_init (name ` nds)"
assms_4: shows "h [(k, v)] = map_val_pair (\<lambda>k. f k (0::'b)) [(k, v)]"
OAlist_def: fixes xs :: "('a \<times> 'b) list \<times> 'a key_order" shows "OAlist xs = Abs_oalist (sort_oalist_ko xs)"
intro: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier \<X>\<^bsub>G\<^esub>; y \<in> carrier \<Y>\<^bsub>G\<^esub>\<rbrakk> \<Longrightarrow> (\<pi>\<^sup>*\<^bsub>G\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> y)" shows "galois_connection_axioms G"
Id_on_int: fixes A :: "'b set"   and f :: "('b \<times> 'c) set" shows "Id_on A O f = A \<times> UNIV \<inter> f"
singleCombinatorsConc: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list" assumes "singleCombinators (x # xs)" shows "singleCombinators xs"
Step_def: fixes A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and s :: "'b list \<times> 'c"   and r :: "'b" shows "Step A s r = (let (a, is') = snd A s r in (step (fst s) r a, is'))"
cont_bot: shows "cont (flat_lub b) (flat_ord b) lub ord bot"
axioms_2: fixes hom :: "'c \<Rightarrow> 'd" assumes "inj_comm_ring_hom hom" shows "zero_hom_0 hom"
flushed_values_mem_independent: fixes a :: "nat"   and sb :: "'b memref list"   and m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat" assumes "a \<in> outstanding_refs is_Write\<^sub>s\<^sub>b sb" shows "flush sb m' a = flush sb m a"
assms_5: shows "NextRecvUpd' c0 c1 p q"
addr_of_sys_xcpt_def: fixes s :: "char list" shows "addr_of_sys_xcpt s \<equiv> if s = NullPointer then 0 else if s = ClassCast then 1 else if s = OutOfMemory then 2 else undefined"
exhaust: fixes y :: "'a seq" obtains "y = Empty"   | x21 :: "'a" and x22 :: "'a pred" where "y = Insert x21 x22"   | x31 :: "'a pred" and x32 :: "'a seq" where "y = Join x31 x32"
f1_def: fixes x1 :: "'a"   and x2 :: "'b" shows "f1 x1 x2 = fst (f x1 x2)"
nomatch: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d" assumes "\<not> matches \<gamma> m a p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided"
is_OK_returns_result_E: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog" assumes "h \<turnstile> ok f" obtains x :: "'ar" where "h \<turnstile> f \<rightarrow>\<^sub>r x"
neutral_const: fixes A :: "'b set" shows "(\<Sum>_\<in>A. (0::'a)) = (0::'a)"
sum_def: fixes a :: "integer \<Rightarrow> 'b narrowing_cons"   and b :: "integer \<Rightarrow> 'b narrowing_cons"   and d :: "integer" shows "Quickcheck_Narrowing.sum a b d = (case a d of Narrowing_cons (Narrowing_sum_of_products ssa) ca \<Rightarrow> case b d of Narrowing_cons (Narrowing_sum_of_products ssb) cb \<Rightarrow> Narrowing_cons (Narrowing_sum_of_products (ssa @ ssb)) (ca @ cb))"
Collect_mem_eq: fixes A :: "'a set" shows "{x. x \<in> A} = A"
disj_2: shows "h\<^sub>0 \<bottom> h\<^sub>1'"
sup_bot_right: fixes x :: "'d multiset" shows "x \<union># {#} = x"
bij_k_words: fixes k :: "nat" shows "bij_betw (\<lambda>wi. fst wi # snd wi) (L \<times> {w. length w = k \<and> set w \<subseteq> L}) {w. length w = Suc k \<and> set w \<subseteq> L}"
value_noninf_nonzero: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x \<noteq> \<infinity>" shows "x \<noteq> \<zero>"
n2m_dtor_corec_coG_coF_def: fixes s1 :: "'g \<Rightarrow> ('f, 'f coG coF + 'e) coG0"   and s2 :: "'e \<Rightarrow> ('f coG + 'g, 'f coG coF + 'e) coF0" shows "n2m_dtor_corec_coG_coF s1 s2 = dtor_corec_coF (BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf \<circ> map_pre_coGcoF\<^sub>M id (case_sum id (n2m_dtor_corec_coG s1 s2)) id \<circ> s2)"
\<phi>_naturality: fixes f :: "'b"   and x :: "'b"   and x' :: "'b"   and g :: "'c"   and y' :: "'c"   and y :: "'c"   and h :: "'b" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>B x'\<guillemotright>"   and "\<guillemotleft>g : y' \<rightarrow>\<^sub>C y\<guillemotright>"   and "\<guillemotleft>h : F' y \<rightarrow>\<^sub>B x\<guillemotright>" shows "\<phi>' y' (f \<cdot>\<^sub>B h \<cdot>\<^sub>B F' g) = G' f \<cdot>\<^sub>C \<phi>' y h \<cdot>\<^sub>C g"
simps_2: fixes n :: "'a" shows "tape_of_nat_list [n] = <n>"
map_upds_index: fixes xs :: "'a list"   and As :: "'b list"   and x :: "'a"   and Ai :: "'b" assumes "length xs = length As"   and "[xs [\<mapsto>] As] x = Some Ai" shows "\<exists>i. As ! i = Ai \<and> i < length As \<and> (\<forall>Bs. length Bs = length As \<longrightarrow> [xs [\<mapsto>] Bs] x = Some (Bs ! i))"
paths_with_new_path: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and xs :: "'c list" assumes "DisjointPathsPlusOne G v0 v1 paths P_new"   and "xs \<in> DisjointPathsPlusOne.paths_with_new paths P_new" shows "Digraph.walk G xs \<and> distinct xs"
5: shows "x \<in> D (v (\<Squnion>i. Y i))"
concat_join: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> []"   and "ys \<noteq> []"   and "last xs = hd ys" shows "butlast xs @ ys = xs @ tl ys"
eqvtI_1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and p :: "(name \<times> name) list" assumes "P \<Longrightarrow>a<\<nu>x> \<prec> P'" shows "p \<bullet> P \<Longrightarrow>p \<bullet> a<\<nu>p \<bullet> x> \<prec> p \<bullet> P'"
trancl_list_impl_def: shows "trancl_list_impl = trancl_impl (\<lambda>r as. remdups (map snd (filter (\<lambda>(a, b). a \<in> set as) r))) (\<lambda>xs ys. filter (\<lambda>x. x \<notin> set ys) xs @ ys) (\<lambda>x xs. x \<in> set xs) []"
ceiling_unique: fixes z :: "int"   and x :: "'d" assumes "of_int z - (1::'d) < x"   and "x \<le> of_int z" shows "\<lceil>x\<rceil> = z"
integrable_scaleR_right: fixes c :: "real"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "c \<noteq> 0 \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. c *\<^sub>R f x)"
trivial_group: shows "trivial_group (relative_homology_group p (discrete_topology {()}) {()}) = (\<exists>a. carrier (relative_homology_group p (discrete_topology {()}) {()}) = {a})"
lnth_inf_llist': fixes f :: "nat \<Rightarrow> 'a" shows "(?!) (inf_llist f) = f"
l_comb_jointfun_jf: fixes H :: "'a set"   and A :: "'b set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and t :: "nat \<Rightarrow> 'b" assumes "H \<subseteq> carrier M"   and "ideal R A"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> Suc (n + m)} \<rightarrow> H"   and "t \<in> {j. j \<le> m} \<rightarrow> A" shows "\<Sigma>\<^sub>e M (\<lambda>j. jointfun n s m t j \<cdot>\<^sub>s f j) n = \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n"
cmp_simps'_1: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (\<Phi>\<^sub>F (\<mu>, \<nu>))"
interval_lebesgue_integral_mult_right: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and c :: "'b"   and f :: "real \<Rightarrow> 'b" shows "LINT x=a..b|M. (c * f x) = c * interval_lebesgue_integral M a b f"
id: shows "floor_1 x = (if \<lfloor>r'\<rfloor> = \<lfloor>l'\<rfloor> then \<lfloor>r'\<rfloor> else if rat_of_int \<lfloor>r'\<rfloor> < l'' then \<lfloor>r'\<rfloor> else \<lfloor>l'\<rfloor>)"
plus_1: fixes x :: "'b" shows "cos (x + (2::'b) * of_real pi) = cos x"
vimage_mono: fixes r' :: "V"   and r :: "V"   and A' :: "V"   and A :: "V" assumes "r' \<subseteq>\<^sub>\<circ> r"   and "A' \<subseteq>\<^sub>\<circ> A" shows "r' `\<^sub>\<circ> A' \<subseteq>\<^sub>\<circ> r `\<^sub>\<circ> A"
irreducibleD': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.irreducible times one zero a"   and "dvd.dvd times b a" shows "dvd.dvd times a b \<or> dvd.dvd times b one"
Rep_quasinorm: fixes x :: "'c quasinorm" shows "Rep_quasinorm x \<in> {(x, N). 1 \<le> ennreal x \<and> (\<forall>x c. N (c *\<^sub>R x) = ennreal \<bar>c\<bar> * N x) \<and> (\<forall>xa y. N (xa + y) \<le> ennreal x * N xa + ennreal x * N y)}"
transfer_disconnect: fixes s :: "'s net_state"   and i :: "nat"   and i' :: "nat"   and s' :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state" assumes "(s, disconnect(i, i'), s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n" obtains \<sigma>'' :: "nat \<Rightarrow> 'g" and \<zeta>'' :: "'l net_state" where "((\<sigma>, \<zeta>), disconnect(i, i'), \<sigma>'', \<zeta>'') \<in> automaton.trans (opnet onp n)" and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>'' j = \<sigma> j" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>'', \<zeta>'')"
enum_idx_correct: fixes i :: "'c" shows "enum_class.enum ! enum_idx i = i"
wfP_iff_no_inf_chain: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wfP (\<lambda>x y. p y x) = (\<nexists>f. inf_chain p f)"
increasing_conv_subseq_fun_0: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "\<exists>s'. s' = convergent_subseq s"   and "a = acc_point s" shows "convergent_subseq_fun s a n < convergent_subseq_fun s a (Suc n)"
mult_1s_ring_1_2: fixes b :: "'b" shows "b * - Numeral1 = - b"
pat_freshs: fixes p :: "pat"   and T :: "ty"   and \<Delta> :: "(name \<times> ty) list"   and c :: "'a" assumes "\<turnstile> p : T \<Rightarrow> \<Delta>" shows "supp p \<sharp>* c = supp \<Delta> \<sharp>* c"
assms: fixes x :: "'a"   and y :: "'b" shows "\<langle>F x, y\<rangle> = \<langle>x, G y\<rangle>"
R35_1: shows "pmf (sds R35) a = pmf (sds R21) a"
full_componentE: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and i :: "nat"   and t :: "(nat, nat) pp" assumes "class.nat_term rep_nat_term splus"   and "snd (rep_nat_term x) = i" obtains y :: "'a" where "rep_nat_term y = (t, i)"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "directional_derivative k a (b1 + b2) = directional_derivative k a b1 + directional_derivative k a b2"
set_map_pmf: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d pmf" shows "set_pmf (map_pmf f M) = f ` set_pmf M"
birkhoff_sum_def: fixes f :: "'a \<Rightarrow> 'c"   and n :: "nat"   and x :: "'a" shows "birkhoff_sum f n x = (\<Sum>i<n. f ((T ^^ i) x))"
isomorphic_unit_left: fixes f :: "'a" assumes "ide f" shows "trg f \<star> f \<cong> f"
cindex_linear_comp: fixes c :: "real"   and lb :: "real"   and ub :: "real"   and f :: "real \<Rightarrow> real"   and b :: "real" assumes "c \<noteq> 0" shows "cindex lb ub (f \<circ> (\<lambda>x. c * x + b)) = (if 0 < c then cindex (c * lb + b) (c * ub + b) f else - cindex (c * ub + b) (c * lb + b) f)"
P_k_2: shows "hd (tl P_k) \<notin> Q'.second_vertices"
DERIV_mult: fixes f :: "'a \<Rightarrow> 'a"   and Da :: "'a"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and Db :: "'a" assumes "(f has_field_derivative Da) (at x within s)"   and "(g has_field_derivative Db) (at x within s)" shows "((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
simple_match: fixes \<mu> :: "('m \<times> 'w) set" assumes "simple \<mu>" shows "match \<mu>"
assms_2: shows "x ; 1 \<cdot> y\<^sup>T ; 1 = 0"
shift_bounds_cl_Suc_ivl: fixes g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..Suc n} = (\<Sum>i = m..n. g (Suc i))"
am6_lem: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set" shows "(rel_ad (x O y) O x) O rel_ad y = rel_ad (x O y) O x"
election: fixes agents :: "'a set"   and alts :: "'b set" assumes "election agents alts" shows "election agents alts"
lexordp_eq_trans: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "lexordp_eq xs ys"   and "lexordp_eq ys zs" shows "lexordp_eq xs zs"
comb: fixes rs :: "(name \<times> sterm) list"   and \<Gamma> :: "(name, sterm) fmap"   and t :: "sterm"   and cs :: "(Term_Class.term \<times> sterm) list"   and u :: "sterm"   and u' :: "sterm"   and env :: "(name, sterm) fmap"   and uu :: "Term_Class.term"   and rhs :: "sterm"   and val :: "sterm" assumes "rs, \<Gamma> \<turnstile>\<^sub>s t \<down> Sabs cs"   and "rs, \<Gamma> \<turnstile>\<^sub>s u \<down> u'"   and "find_match cs u' = Some (env, uu, rhs)"   and "rs, \<Gamma> ++\<^sub>f env \<turnstile>\<^sub>s rhs \<down> val" shows "rs, \<Gamma> \<turnstile>\<^sub>s t $\<^sub>s u \<down> val"
principal_char_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a" shows "principal_char G a = (if a \<in> carrier G then 1 else 0)"
permute_plus: fixes p :: "'a perm"   and q :: "'a perm"   and x :: "'b" shows "perm1 (p + q) x = perm1 p (perm1 q x)"
is_true_def: fixes d :: "dynamic" shows "is_true d = (d = DBool True)"
abs_eq: fixes xb :: "nat"   and xa :: "int"   and x :: "int" shows "lapprox_rat xb xa x = float_of (truncate_down xb (real_of_int xa / real_of_int x))"
simps_2: fixes x31 :: "('a, 'b) MWLfCom"   and x32 :: "('a, 'b) MWLfCom"   and y31 :: "('a, 'b) MWLfCom"   and y32 :: "('a, 'b) MWLfCom" shows "(x31;x32 = y31;y32) = (x31 = y31 \<and> x32 = y32)"
index_f_def: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.index_f M \<equiv> SOME h. bij_betw h (UNIV // eq_app_right (dfa.language M)) (hfset (ord_of (dfa.min_states M)))"
z: shows "z \<equiv> getFr [x] [t1, t2] [\<phi>]"
pos_mod_bound: fixes b :: "'c"   and a :: "'c" assumes "(0::'c) < b" shows "a mod b < b"
is_measure_trivial: fixes f :: "'b \<Rightarrow> nat" shows "is_measure f"
\<iota>_in_hom: shows "\<guillemotleft>\<iota> : MC.tensor MC.unity MC.unity \<rightarrow> MC.unity\<guillemotright>"
induct: fixes x1 :: "expr list"   and x2 :: "val list"   and x3 :: "expr list"   and P :: "expr list \<Rightarrow> val list \<Rightarrow> expr list \<Rightarrow> bool" assumes "map_val2 x1 x2 x3"   and "P [] [] []"   and "\<And>xs ys zs y. \<lbrakk>map_val2 xs ys zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (Val y # xs) (y # ys) zs"   and "\<And>e xs. P (throw e # xs) [] (throw e # xs)" shows "P x1 x2 x3"
AGM_Contraction_axioms: shows "AGM_Contraction Cn (\<lambda>A. meet_contraction A \<gamma>\<^sub>T\<^sub>R)"
facetrelI: fixes v :: "'b"   and y :: "'b set"   and x :: "'b set" assumes "v \<notin> y"   and "x = insert v y" shows "y \<lhd> x"
star_power_def: shows "(^) \<equiv> \<lambda>x n. (*f* (\<lambda>x. x ^ n)) x"
1: shows "t = \<^bold>\<langle>un_Prim t\<^bold>\<rangle> \<and> arr (un_Prim t) \<and> Nml t \<and> Nml u \<and> \<not> is_Prim\<^sub>0 u \<and> \<^bold>\<langle>src (un_Prim t)\<^bold>\<rangle>\<^sub>0 = Trg u"
process_rule_5_general: fixes xs :: "'e list"   and P :: "'e process"   and ys :: "'e list" assumes "xs \<in> divergences P" shows "xs @ ys \<in> divergences P"
l_get_owner_document_wf_get_root_node_wf_axioms: shows "l_get_owner_document_wf_get_root_node_wf heap_is_wellformed type_wf known_ptr known_ptrs get_root_node get_owner_document"
simps_2: fixes uv :: "'a" shows "(extB.Val uv \<le> extB.Bot) = False"
a1_def: shows "a1 = [:- a, 1::'a:]"
cindex_pathE_shiftpath: fixes g :: "real \<Rightarrow> complex"   and z :: "complex"   and s :: "real" assumes "finite_ReZ_segments g z"   and "s \<in> {0..1}"   and "path g"   and "pathfinish g = pathstart g" shows "cindex_pathE (shiftpath s g) z = cindex_pathE g z"
ext_is_act: fixes a :: "'e"   and S :: "'e set \<times> 'e set \<times> 'e set" assumes "a \<in> externals S" shows "a \<in> actions S"
fps_compose_linear: fixes f :: "'a fps"   and c :: "'a" shows "f oo fps_const c * fps_X = Abs_fps (\<lambda>n. c ^ n * fps_nth f n)"
order_tendstoD_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and y :: "'b"   and F :: "'c filter"   and a :: "'b" assumes "class.order_topology less_eq less open"   and "filterlim f (topological_space.nhds open y) F"   and "less y a" shows "\<forall>\<^sub>F x in F. less (f x) a"
views: shows "v=vl\<parallel>v1 \<and> v1=v2\<parallel>vr \<and> v2=vd--v3 \<and> v3=v'--vu"
lcs_comm: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lcs :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "class.lcs_powerprod plus minus zero lcs" shows "lcs s t = lcs t s"
is_extensional: fixes f :: "('a, 'a) CC.arr" assumes "\<not> CC.arr f" shows "CC.map f = CC.null"
reflection_and_closures_2: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> bool" assumes "rel_reflects_pred Rel Pred" shows "rel_reflects_pred (Rel\<^sup>+) Pred"
eq_Integer_bottom_iff: fixes x :: "Integer"   and y :: "Integer" shows "(eq\<cdot>x\<cdot>y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
trm_def: fixes f :: "'a" shows "\<t>[f] \<equiv> if arr f then THE t. \<guillemotleft>t : local.dom f \<rightarrow> \<one>\<guillemotright> else null"
distinct_insort_insert: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 't"   and x :: "'c" assumes "distinct xs" shows "distinct (ord_term_lin.insort_insert_key f x xs)"
hyps_2: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList (applyUnitPropagate state')) (getF (applyUnitPropagate state'))"   and "InvariantWatchListsUniq (getWatchList (applyUnitPropagate state'))"   and "InvariantWatchListsCharacterization (getWatchList (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesEl (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesDiffer (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))" shows "let state'a = exhaustiveUnitPropagate (applyUnitPropagate state') in currentLevel (getM state'a) = currentLevel (getM (applyUnitPropagate state'))"
Y: shows "Y = acc_err p (aform_to_aform_err X1 (degree_aform_err X)) e1"
det_non_zero_imp_unit: fixes A :: "'a mat"   and n :: "nat"   and b :: "'b" assumes "A \<in> carrier_mat n n"   and "det A \<noteq> (0::'a)" shows "A \<in> Units (ring_mat TYPE('a) n b)"
transitive_coinduct_weak: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "eqvt X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<(bisim O X O bisim)> Q \<and> (Q, P) \<in> X" shows "P \<approx> Q"
of_int_code_1: fixes k :: "num" shows "of_int (Int.Neg k) = - numeral k"
finite_subcls1: fixes P :: "'b prog" shows "finite (subcls1 P)"
select_convs_1: fixes sprFst :: "('b, 'c, 'd) BEState"   and sprLst :: "('b, 'c, 'd) BEState"   and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set"   and more :: "'e" shows "sprFst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprFst"
span_alt: fixes B :: "'b set" assumes "B \<subseteq> U\<^sub>V\<^sub>S" shows "local.span B = {x \<in> U\<^sub>V\<^sub>S. \<exists>f. x = (\<Sum>x | x \<in> U\<^sub>V\<^sub>S \<and> f x \<noteq> (0::'a). f x *s x) \<and> finite {x \<in> U\<^sub>V\<^sub>S. f x \<noteq> (0::'a)} \<and> {x \<in> U\<^sub>V\<^sub>S. f x \<noteq> (0::'a)} \<subseteq> B}"
eta_esque: shows "(\<lambda>t'. \<chi>i. t' * (if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0)) = (\<lambda>t'. t' *\<^sub>R (\<chi>i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0))"
e0_def: shows "e0 = exp (- epsilonG TYPE('a) * C0)"
dual_iff: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "(dual_prod x = y) = (x = dual_prod y)"
le_cases: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
monad_state_contT: fixes return' :: "'a \<Rightarrow> 'b"   and bind' :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state return' bind' get put" shows "monad_state return_cont bind_cont (get_cont get) (put_cont put)"
Perp_in_cases: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D  \<or> X PerpAt B A C D  \<or> X PerpAt A B D C  \<or> X PerpAt B A D C  \<or> X PerpAt C D A B  \<or> X PerpAt C D B A  \<or> X PerpAt D C A B  \<or> X PerpAt D C B A " shows "X PerpAt A B C D "
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "pure (f x) h"
assms_1: shows "simplex (mat_leqb_eqc A b c) = Inr x"
inj: shows "inj_on (get_map S) (set (sorted_list_of_set (fst ` S)))"
sheaf_spec_def: fixes U :: "'a set set" shows "\<O> U \<equiv> {s \<in> \<Pi>\<^sub>E \<pp>\<in>U. R \<^bsub>\<pp> (+) (\<cdot>) \<zero>\<^esub>. is_regular s U}"
two_set: fixes ls :: "'b list"   and a :: "'b"   and b :: "'b" assumes "distinct ls"   and "set ls = set [a, b]"   and "a \<noteq> b" shows "ls = [a, b] \<or> ls = [b, a]"
intro: fixes B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a" assumes "partial_magma.ide B b" shows "constant_functor_axioms B b"
r2: shows "k-smooth_on (apply_chart c2 ` (domain d2 \<inter> domain c2)) (apply_chart d2 \<circ> inv_chart c2)"
point_rep_number_alt_def: fixes x :: "'a" shows "arcs_blocks rep x = int (size (filter_mset ((\<in>) x) arcs_blocks))"
object_ptr_kinds_preserved: fixes SW :: "((_) heap, 'c, 'd) prog set"   and setter :: "((_) heap, 'c, 'e) prog"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w object_ptr. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing) h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
max_fun_diff_in: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> FINFUNC"   and "g \<in> FINFUNC" shows "s.max_fun_diff f g \<in> {a \<in> Field s. f a \<noteq> g a}"
times_monomial_left: fixes t :: "'a"   and c :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "monomial c t * p = MPoly_Type_Class.punit.monom_mult c t p"
orbits: shows "pmf (sds R26) c = pmf (sds R26) b"
cod_cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.cod C (partial_magma.cod C f) = partial_magma.cod C f"
abs_eq: fixes x :: "('c, 'd) phi \<Rightarrow> 'c" assumes "eq_onp (\<lambda>xf. True) x x" shows "- Abs_lifted_boolean_algebra x = Abs_lifted_boolean_algebra (\<lambda>f. - x f)"
borel_measurable_nn_integral: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> ennreal"   and N :: "'b measure"   and i :: "nat" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M (case i of 0 \<Rightarrow> count_space {} | Suc x \<Rightarrow> N x))" shows "(\<lambda>x. integral\<^sup>N (case i of 0 \<Rightarrow> count_space {} | Suc x \<Rightarrow> N x) (f x)) \<in> borel_measurable N"