hcju/isabelleps huggingface dataset:
adm_below: fixes u :: "'a \<Rightarrow> 'b"   and v :: "'a \<Rightarrow> 'b" assumes "cont u"   and "cont v" shows "adm (\<lambda>x. u x \<sqsubseteq> v x)"
Var_def: fixes xs :: "'varSort"   and x :: "'var" shows "Var xs x \<equiv> asTerm (qVar xs x)"
eval_fds_exp: fixes f :: "'a fds"   and s :: "'a" assumes "fds_abs_converges f s" shows "eval_fds (fds_exp f) s = exp (eval_fds f s)"
right_invertible_transpose: fixes A :: "(('a, 'b) vec, 'c) vec" shows "(\<exists>B. Finite_Cartesian_Product.transpose A ** B = mat (1::'a)) = (\<exists>B. B ** A = mat (1::'a))"
simps_26: fixes vb :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "(SKIP {vb} \<sqsubseteq> v;; va) = False"
h_props_2: shows "\<gamma> = \<H> \<union> two_chain_vertical_boundary two_chain"
set2_G_map: fixes l1 :: "'p \<Rightarrow> 'x"   and l2 :: "'q \<Rightarrow> 'w" shows "set2_G \<circ> mapl_G l1 l2 = (`) l2 \<circ> set2_G"
mult_sm_ex_id2: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "\<exists>k. k \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m M"
irreflp_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "irreflp R = (\<forall>a. \<not> R a a)"
discI: fixes tree :: "'b tree"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" assumes "tree = Node x21 x22 x23 x24 x25" shows "tree \<noteq> Leaf"
strict_trans1: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a \<le>s b"   and "b <s c" shows "a <s c"
prime_imp_not_one: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "ASSUMPTION (normalization_semidom.prime zero times one normalize x)" shows "x \<noteq> one"
llexord_LCons_LCons: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "llexord r (LCons x xs) (LCons y ys) = (x = y \<and> llexord r xs ys \<or> r x y)"
Class_commutes_with_inverse: fixes a :: "'a" assumes "a \<in> G" shows "quotient.inverse (Class a) = Class (inverse a)"
ps'_def: shows "ps' = perp_up (Rep_hyp2 c') l'"
lm073: fixes x :: "'b set"   and X :: "'b set set" assumes "x \<in> X" shows "finestpart x \<subseteq> finestpart (\<Union> X)"
Domainp_applyI: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'c"   and y :: "'d" assumes "rel_fun A B f g"   and "A x y" shows "Domainp B (f x)"
scaleC_nonneg_nonneg: fixes a :: "complex"   and x :: "'b" assumes "0 \<le> a"   and "(0::'b) \<le> x" shows "(0::'b) \<le> a *\<^sub>C x"
tdghm_NTCod: shows "\<NN>\<lparr>NTCod\<rparr> = \<GG>"
Extend2: shows "\<exists>l23. l3 = l2 @ l23 \<and> l23 \<noteq> []"
e: shows "e = (c - b\<^sup>2 / ((3::'a) * a)) / a"
fps_prod_nth: fixes a :: "nat \<Rightarrow> 'a fps"   and m :: "nat"   and n :: "nat" shows "fps_nth (prod a {0..m}) n = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j = 0..m. fps_nth (a j) (v ! j))"
sel: fixes x :: "('d \<Rightarrow> 'e) \<Rightarrow> 'e" shows "run_cont (ContT x) = x"
Cons_6: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M"
rdvd_minus: fixes d :: "int"   and t :: "real" shows "(real_of_int d rdvd t) = (real_of_int d rdvd - t)"
get_shadow_root_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "h \<turnstile> ok get_shadow_root element_ptr" shows "element_ptr |\<in>| element_ptr_kinds h"
sup_commute: fixes x :: "'b"   and y :: "'b" shows "x + y = y + x"
P1_assert_correct1: fixes \<alpha>0 :: "nat"   and r :: "nat"   and \<alpha>1 :: "nat" shows "(\<^bold>g [^] \<alpha>0) [^] r \<otimes> \<^bold>g \<otimes> inv ((\<^bold>g [^] \<alpha>1) [^] r \<otimes> \<^bold>g) = (\<^bold>g [^] \<alpha>0 \<otimes> inv (\<^bold>g [^] \<alpha>1)) [^] r"
that: fixes g :: "'a \<Rightarrow> real" assumes "real_polynomial_function g"   and "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x - g x\<bar> < e" shows "thesis"
Units_one_closed: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> \<in> Units G"
Int_emptyI: fixes A :: "'g set"   and B :: "'g set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
measurable_V: fixes as :: "'c list"   and os :: "'d list" shows "Measurable.pred (stream_space (count_space UNIV)) (\<lambda>\<omega>. \<exists>\<omega>'. \<omega> = zip as os @- \<omega>')"
assms: shows "x \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A"
setwiseI: fixes X :: "'b set"   and Y :: "'c set"   and r :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x xa. \<lbrakk>x \<in> X; xa \<in> Y\<rbrakk> \<Longrightarrow> r x xa" shows "setwise r X Y"
Type_System_axioms: shows "Type_System E BMap DA AssignSideCondition WhileSideCondition IfSideCondition"
vsv_vrestriction: fixes r :: "V"   and A :: "V" assumes "vsv r" shows "vsv (r \<restriction>\<^sub>\<circ> A)"
usi_2: shows "us ! i \<notin> set us - {us ! i}"
subst_syntactic_and_def: fixes P :: "'b \<Rightarrow> bool" shows "subst_syntactic_and P \<equiv> simple_syntactic_and P \<and> subst_syntactic_and_axioms P"
success_def: fixes f :: "'c Heap"   and h :: "heap" shows "success f h = (execute f h \<noteq> None)"
start_pres_conj: fixes x :: "'a" assumes "x \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)" shows "x ^ * \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)"
Mu_Sim: fixes S :: "'e \<Rightarrow> 'd"   and P :: "('c, 'd) action \<Rightarrow> ('c, 'd) action"   and Q :: "('c, 'e) action \<Rightarrow> ('c, 'e) action" assumes "\<And>X Y. X \<preceq>S Y \<Longrightarrow> P X \<preceq>S Q Y"   and "mono P"   and "mono Q" shows "lfp P \<preceq>S lfp Q"
mono_on_ctble_discont: fixes f :: "real \<Rightarrow> real"   and A :: "real set" assumes "mono_on f A" shows "countable {a \<in> A. \<not> continuous (at a within A) f}"
ys_def: shows "ys = coeffs g @ [0::'a]"
isc: shows "iscale ` {- 1..1} \<subseteq> {0..1}"
SubstForm_quot_unique: fixes i :: "name"   and e :: "(name, hf) finfun"   and t :: "tm"   and A :: "fm"   and x' :: "hf" shows "SubstForm (q_Var i) \<lbrakk>\<guillemotleft>t\<guillemotright>\<rbrakk>e \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e x' = (x' = \<lbrakk>\<guillemotleft>A(i::=t)\<guillemotright>\<rbrakk>e)"
rel_intros: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'b"   and y1 :: "'c"   and x2 :: "'b stream"   and y2 :: "'c stream" assumes "R x1 y1"   and "stream_all2 R x2 y2" shows "stream_all2 R (x1 ## x2) (y1 ## y2)"
lossless_R1_12: fixes m :: "bool \<times> bool"   and c :: "bool" shows "lossless_spmf (R1_OT12 m c)"
emb_hsize: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s" shows "hsize s \<le> hsize t"
HomFtorContraMor_2: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "dom\<^bsub>SET\<^esub> (HomC\<^bsub>C\<^esub>[f,X]) = Hom\<^bsub>C\<^esub> cod\<^bsub>C\<^esub> f X"
swap_at_base_simps_3: fixes x :: "'a"   and a :: "atom"   and b :: "atom" assumes "atom x \<noteq> a"   and "atom x \<noteq> b" shows "(a \<rightleftharpoons> b) \<bullet> x = x"
respects_imp_unique_image_rel: fixes r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and a :: "'b" assumes "f respects r"   and "y \<in> f ` r `` {a}" shows "y = f a"
CMUD_def: shows "CMUD \<equiv> CMU \<and> CMD \<and> CMUD_axioms"
INF1_D: fixes B :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set"   and b :: "'a"   and a :: "'b" assumes "(\<Sqinter> (B ` A)) b"   and "a \<in> A" shows "B a b"
sim': shows "ts' \<sim>\<^sub>h ts\<^sub>h' "
map_upd_upds_conv_if: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" shows "f(x \<mapsto> y, xs [\<mapsto>] ys) = (if x \<in> set (take (length ys) xs) then f(xs [\<mapsto>] ys) else f(xs [\<mapsto>] ys, x \<mapsto> y))"
indis_sym: fixes indis :: "('a \<times> 'a) set"   and s :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "(s, s') \<in> indis" shows "(s', s) \<in> indis"
is_success_alt_def: fixes a :: "status" shows "is_success a = (a = SUCCESS)"
C: shows "mat n n (\<lambda>(i, j). if i = 0 \<and> j = 0 then e else (0::'a)) \<in> carrier_mat n n"
filterlim_at_top_linear_iff: fixes c :: "'a"   and f :: "'a \<Rightarrow> 'b"   and b :: "'a"   and F2 :: "'b filter" assumes "c \<noteq> (0::'a)" shows "(LIM x at_top. f (x * c + b) :> F2) = (if (0::'a) < c then filterlim f F2 at_top else filterlim f F2 at_bot)"
nondegenerate: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "distinct3 p q r"   and "in3 S p q r" shows "ccw p q r \<or> ccw p r q"
tiTable_i1_4b: fixes req :: "nat \<Rightarrow> reqType list"   and x :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat"   and t2 :: "nat" assumes "tiTable_SampleT req x stop lose (fin_inf_append [init_state] st) b ack i1 vc st"   and "ts lose"   and "msg (Suc 0) x"   and "msg (Suc 0) stop"   and "\<forall>t1\<le>t. req t1 = []"   and "req (Suc t) = [init]"   and "\<forall>m<k + 3. req (t + m) \<noteq> [send]"   and "\<forall>m\<le>k. ack (Suc (Suc (t + m))) = [connection_ok]"   and "\<forall>j\<le>k + 3. lose (t + j) = [False]"   and "t2 < t + 3 + k" shows "i1 t2 = []"
mod_type_axioms: shows "mod_type (int CARD('a bit0)) Rep_bit0 Abs_bit0"
assms_5: shows "\<forall>t1\<le>t. req t1 = []"
atS_mono: fixes p :: "'e"   and ls :: "'f set"   and s :: "('g, 'f, 'e, 'h, 'i) system_state"   and ls' :: "'f set" assumes "atS p ls s"   and "ls \<subseteq> ls'" shows "atS p ls' s"
bit_not_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (not a) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> \<not> bit a n)"
a_heap_is_wellformed_def: fixes h :: "(_) heap" shows "local.CD.a_heap_is_wellformed h = (local.CD.a_acyclic_heap h \<and> local.CD.a_all_ptrs_in_heap h \<and> local.CD.a_distinct_lists h \<and> local.CD.a_owner_document_valid h)"
measurable_id: fixes M :: "'d measure" shows "(\<lambda>x. x) \<in> M \<rightarrow>\<^sub>M M"
hyp1_unfolded: shows "matrix_to_iarray (fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])) = fst (foldl echelon_form_of_column_k_iarrays (matrix_to_iarray A, 0, bezout) [0..<Suc k])"
id: shows "step_3_c x l k ((i_begin, i_end) # blocks) A = step_3_c x l k blocks B"
word_neq_0_conv: fixes w :: "'a word" shows "(w \<noteq> 0) = (0 < w)"
Inf_nat_def1: fixes K :: "nat set" assumes "K \<noteq> {}" shows "\<Sqinter> K \<in> K"
i_m_giving_this_a_funny_name_so_i_can_thank_my_future_me_when_sledgehammer_will_find_this_one_day: fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b match_expr" assumes "\<not> has_disc_negated disc neg m" shows "\<forall>m_DNF\<in>set (normalize_match m). \<not> has_disc_negated disc neg m_DNF"
hfset_hdiff: fixes x :: "hf"   and y :: "hf" shows "hfset (x - y) = hfset x - hfset y"
nprv_cnjH: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "nprv (insert \<phi>1 (insert \<phi>2 F)) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv (insert (cnj \<phi>1 \<phi>2) F) \<psi>"
extremum_uniqueI: fixes a :: "'k" assumes "\<top> \<le> a" shows "a = \<top>"
configuration_rev_step': fixes x :: "nat"   and n :: "nat"   and q :: "'b" assumes "x \<noteq> Suc n"   and "x \<in> configuration q (Suc n)" obtains q' :: "'b" where "q = \<delta> q' (w n)" and "x \<in> configuration q' n"
cases_scheme: fixes r :: "'a methodDef_scheme" obtains mReturn :: "nat" and mName :: "nat" and mParams :: "varDef list" and mBody :: "exp" and more :: "'a" where "r = \<lparr>mReturn = mReturn, mName = mName, mParams = mParams, mBody = mBody, \<dots> = more\<rparr>"
d1_3: fixes K :: "'a set" assumes "K \<in> p1" shows "K \<noteq> {}"
synth_simps_1: fixes n :: "nat"   and H :: "msg set" shows "(Nonce n \<in> synth H) = (Nonce n \<in> H)"
eventually_nonzero_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "(\<forall>\<^sub>F x in F. f x \<noteq> (0::'c)) = (\<forall>\<^sub>F x in F. g x \<noteq> (0::'c))"
step_valid: fixes stack :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "valid stack" shows "valid (fst (step (stack, rv)))"
Key_image_eq: fixes x :: "nat"   and A :: "nat set" shows "(Key x \<in> Key ` A) = (x \<in> A)"
preserves_arr: fixes f :: "'d \<times> 'c" assumes "DopxC.arr f" shows "S.arr (Hom_FopxC.map f)"
atom_ordering_trans: shows "\<forall>x y z. (x, y) \<in> atom_ordering \<longrightarrow> (y, z) \<in> atom_ordering \<longrightarrow> (x, z) \<in> atom_ordering"
decl: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mu :: "'Var ModeUpd"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and E :: "('Var, 'AExp, 'BExp) Stmt list" assumes "\<langle>c, update_modes mu mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "\<langle>cxt_to_stmt E (c@[mu]), mds, mem\<rangle> \<leadsto> \<langle>cxt_to_stmt E c', mds', mem'\<rangle>"
factgroup_subgroup_union_normal: fixes A :: "'a set set" assumes "A \<lhd> G Mod H" shows "\<Union> A \<lhd> G"
assms_1: shows "II1 \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}"
bind_le_nofailI: fixes m :: "'b nres"   and f :: "'b \<Rightarrow> 'c nres"   and m' :: "'c nres" assumes "nofail m"   and "\<And>x. RETURN x \<le> m \<Longrightarrow> f x \<le> m'" shows "m \<bind> f \<le> m'"
hyps: assumes "a \<in> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs" shows "\<exists>sop v ys zs A L R W. xs = ys @ Write\<^sub>s\<^sub>b True a sop v A L R W # zs"
eq: shows "0 = Sigma_Algebra.measure M (\<Union>n. fst (A n)) - Sigma_Algebra.measure M (\<Inter>n. snd (A n))"
filtermap_filtermap: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and F :: "'d filter" shows "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
simps_2: fixes v :: "Vars.var"   and e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and z :: "Vars.var" shows "((v, e) # \<Gamma>)[y::h=z] = (v, e[y::=z]) # \<Gamma>[y::h=z]"
that: fixes y :: "('idx, unit, ('act, 'pred) S_action) formula" assumes "S_transform.weak_formula y"   and "\<forall>P. P \<Turnstile> x = P \<Turnstile>\<^sub>S y" shows "thesis"
bangExt: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "guarded P" shows "\<Psi> \<rhd> !P \<sim> P \<parallel> !P"
eval_fds_norm_nonneg: fixes f :: "'a fds"   and s :: "real" assumes "fds_abs_converges (fds_norm f) s" shows "0 \<le> eval_fds (fds_norm f) s"
ladj_radj_dual: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'b" shows "ladj f x = \<partial> (radj (\<partial>\<^sub>F f) (\<partial> x))"
nonneg_Reals_Real: fixes x :: "'a" assumes "x \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "x \<in> \<real>"
assms: shows "C \<in> Chains (relation_of (\<lesssim>) \<S>)"
removeLastMarkedFromMarkedElementsToLastMarkedAreAllMarkedElementsInPrefixLastMarked: fixes M :: "('b \<times> bool) list" assumes "markedElements M \<noteq> []" shows "removeAll (lastMarked M) (markedElementsTo (lastMarked M) M) = markedElements (prefixBeforeElement (lastMarked M) M)"
hd_replicate: fixes n :: "nat"   and x :: "'a" assumes "n \<noteq> 0" shows "hd (replicate n x) = x"
frechet_derivative_const: fixes c :: "'d"   and a :: "'c" shows "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0::'d)"
row_space_subset_phull: fixes ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and ts :: "'c list" assumes "Keys (set ps) \<subseteq> set ts" shows "row_to_poly ts ` row_space (polys_to_mat ts ps) \<subseteq> phull (set ps)"
cs_done_implies_has_snapshotted: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "snd (cs (S t i) cid) = Done"   and "channel cid = Some (p, q)" shows "ps (S t i) q \<noteq> None"
distinct_1: fixes x2 :: "final_decision" shows "Undecided \<noteq> Decision x2"
compile2: shows "rec_ci (gs ! i) = (gap, gar, gft) \<and> gar = length args"
cross_edges_target_finished: fixes u :: "'v"   and v :: "'v" assumes "(u, v) \<in> cross_edges s" shows "v \<in> dom (finished s)"
assms: shows "H = {x \<in> A. (x, f x) \<in> r}"
Q_fin: fixes i :: "nat" shows "\<integral>\<^sup>+ x. f x * indicator (Q i) x \<partial>M \<noteq> \<infinity>"
iT_Minus_cut_less2: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" shows "(k \<ominus> I) \<down>< (k - t) = k \<ominus> I \<down>> t"
map_kE_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and Tl :: "'fsym trm list" assumes "I.wtE (kE \<xi>)"   and "list_all wt Tl" shows "map (I.int (kE \<xi>)) Tl = map2 fromFull (map tpOf Tl) (map (F.int \<xi>) Tl)"
arsinh_real_has_field_derivative: fixes x :: "real"   and A :: "real set" shows "(arsinh has_real_derivative 1 / sqrt (x\<^sup>2 + 1)) (at x within A)"
S_sound_precise: fixes v :: "'n"   and M :: "'m set"   and P :: "'p multiset"   and k :: "nat" shows "((v, M, P) \<in> S_cs fg k) = (\<exists>p c' w. (([entry fg p], {#}), w, [v], c') \<in> trcl (trss fg) \<and> size P \<le> k \<and> {#[entry fg p]. p \<in># P#} \<subseteq># c' \<and> M = mon_w fg w)"
even_of_nat_iff: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (semiring_1.of_nat one plus zero n) = even n"
cases: fixes x :: "'b sort_key extended \<times> 'b sort_key extended" obtains l :: "'b sort_key extended" and u :: "'b sort_key extended" where "x = (l, u)"
is_intersection_def: fixes M :: "point"   and A :: "point"   and B :: "point"   and C :: "point"   and D :: "point" shows "is_intersection M A B C D = (collinear A M B \<and> collinear C M D)"
assign_doesnt_modify: fixes x :: "'Var"   and y :: "'Var"   and e :: "'AExp"   and annos :: "'Var ModeUpd list" assumes "x \<noteq> y"   and "x \<notin> \<C>_vars y" shows "doesnt_modify ((x \<leftarrow> e) \<otimes> annos) y"
reducible: shows "\<exists>Us. (\<forall>U\<in>set Us. IrrFinGroupRepresentation G (\<cdot>) U) \<and> 0 \<notin> set Us \<and> V = (\<Oplus>U\<leftarrow>Us. U)"
l1_inv6_derived: shows "l1_inv4 \<inter> l1_inv5 \<subseteq> l1_inv6"
a0i_agreementI: fixes x :: "a0n_state" assumes "\<And>A B M. signals x (Commit A B M) \<le> signals x (Running A B M)" shows "x \<in> a0i_agreement"
cr_id_prob_Grp: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "cr_id_prob (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_id x \<subseteq> A} (return_pmf \<circ> f \<circ> extract)"
sub_negative: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (neg_numeral.sub minus plus one n m) zero = (n < m)"
entails_concl_compact: fixes EE :: "'f set"   and CC :: "'f set" assumes "finite EE"   and "CC \<Turnstile> EE" shows "\<exists>CC'\<subseteq>CC. finite CC' \<and> CC' \<Turnstile> EE"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> kerbIV_gets" shows "\<exists>A. Says A B X \<in> set evs"
self_codomain_iff_ide: fixes a :: "'a2" shows "(a \<in> C2.codomains a) = C2.ide a"
\<tau>move2_Seq2D: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "\<tau>move2 P h stk (e1;; e2) (Suc (length (compE2 e1) + pc')) xcp'" shows "\<tau>move2 P h stk e2 pc' xcp'"
parts: shows "\<forall>b\<in>B. card {x \<in> A. f x = b} = 1 \<or> {x \<in> A. f x = b} = {}"
simps: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" shows "ord_resolve S a1 a2 a3 a4 a5 a6 = (\<exists>CAs n Cs AAs As \<sigma> D. a1 = CAs \<and> a2 = D + negs (mset As) \<and> a3 = AAs \<and> a4 = As \<and> a5 = \<sigma> \<and> a6 = (\<Sum>\<^sub># (mset Cs) + D) \<cdot> \<sigma> \<and> length CAs = n \<and> length Cs = n \<and> length AAs = n \<and> length As = n \<and> n \<noteq> 0 \<and> (\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)) \<and> (\<forall>i<n. AAs ! i \<noteq> {#}) \<and> Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs)) \<and> eligible S \<sigma> As (D + negs (mset As)) \<and> (\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)) \<and> (\<forall>i<n. S (CAs ! i) = {#}))"
smult_zero: fixes r :: "('f, 'g) aezfun" assumes "r \<in> FG" shows "r \<cdot> (0::'v) = (0::'v)"
converse_mono: fixes r :: "('d \<times> 'c) set"   and s :: "('d \<times> 'c) set" shows "(r\<inverse> \<subseteq> s\<inverse>) = (r \<subseteq> s)"
id: shows "(\<Prod>(a, i)\<in>set bs. a ^ Suc i) = q ^ Suc i * (b ^ Suc i * (\<Prod>(a, i)\<in>set bs - {(a, i)}. a ^ Suc i))"
open_Compl: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open"   and "topological_space.closed open S" shows "open (- S)"