hcju/isabelleps huggingface dataset:
eventually_compose_filterlim: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter"   and f :: "'d \<Rightarrow> 'c"   and G :: "'d filter" assumes "eventually P F"   and "filterlim f F G" shows "\<forall>\<^sub>F x in G. P (f x)"
0: shows "tensors_from_net (insert_weights shared_weights (deep_model_l rs) wd) = tensors_from_net (insert_weights shared_weights (shallow_model (rs ! 0) Z (last rs) (2 * N_half - 1)) ws)"
freshAbs_substAbs_ident: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodAbs A"   and "freshAbs xs x A" shows "A $[X / x]_xs = A"
Sup_upper: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "x \<le> Sup A"
ideD_3: fixes a :: "('o, 'a) arr" assumes "ide a" shows "cod a = a"
simps_1: fixes l2 :: "'b list" shows "diff_sorted [] l2 = []"
Rep_quot_inject: fixes x :: "'b quot"   and y :: "'b quot" shows "(Rep_quot x = Rep_quot y) = (x = y)"
monad_altc_stateT: assumes "monad_altc return bind altc" shows "monad_altc local.return_state local.bind_state local.altc_state"
deg_of_monoI: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "deg_of x \<le> deg_of y"
that: fixes T :: "'n set" assumes "\<not> affine_dependent T"   and "S \<subseteq> T"   and "T \<subseteq> V"   and "affine hull T = affine hull V" shows "thesis"
list2FWpolicyconc: fixes a :: "('b, 'c) Combinators list"   and xa :: "('b, 'c) Combinators" assumes "a \<noteq> []" shows "list2FWpolicy (xa # a) = xa \<oplus> list2FWpolicy a"
PPsD: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set" assumes "t \<in> .[X]" shows "keys t \<subseteq> X"
add_diff_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.group_add minus plus zero uminus" shows "plus a (minus b c) = minus (plus a b) c"
presheaf_of_rings_axioms: shows "presheaf_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str'"
inext_nth_mono2: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" assumes "\<exists>x\<in>I. I \<rightarrow> a < x" shows "(I \<rightarrow> a < I \<rightarrow> b) = (a < b)"
noLowDPs_NEW: fixes s :: "Var \<Rightarrow> Val"   and h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var"   and l :: "Location"   and C :: "Class" assumes "noLowDPs (s, h)" shows "noLowDPs (update s x (RVal (Loc l)), (l, C, []) # h)"
assms: shows "N \<subseteq> timpl_closure_set M TI"
flrestriction_vsingleton_nin: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "set {[a, b]\<^sub>\<circ>} \<restriction>\<^sup>l\<^sub>\<bullet> A = []\<^sub>\<circ>"
analz_idem: fixes H :: "msg set" shows "analz (analz H) = analz H"
sccs_verts_def: shows "G.sccs_verts = {S. S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u)}"
ordinal_number_mem_carrier_ODnods: fixes D :: "'c Order" assumes "Worder D" shows "ordinal_number D \<in> carrier ODnods"
comm_semiring_0_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "class.comm_semiring_0 plus zero times \<equiv> class.comm_monoid_add plus zero \<and> class.comm_semiring plus times \<and> class.mult_zero times zero"
space_PiM_empty_iff: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" shows "(space (Pi\<^sub>M I M) = {}) = (\<exists>i\<in>I. space (M i) = {})"
countable_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "countable B" shows "countable A"
mono_body_fixp: fixes F :: "('c \<Rightarrow> 'd \<Rightarrow> 'e llist) \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e llist" assumes "\<And>x. mono_llist_lift (\<lambda>f. F f x)" shows "llist_lift.fixp_fun F = F (llist_lift.fixp_fun F)"
bounded_linear_inner_right: fixes a :: "'b" shows "bounded_linear ((\<bullet>) a)"
mini_mini: fixes F :: "'a formula" shows "to_mini_formula (to_mini_formula F) = to_mini_formula F"
select_convs_3: fixes ok :: "bool"   and wait :: "bool"   and tr :: "'c list"   and ref :: "'c set"   and more :: "'d" shows "ref \<lparr>ok = ok, wait = wait, tr = tr, ref = ref, \<dots> = more\<rparr> = ref"
atU_fmt: fixes U :: "'f set"   and c :: "'f list multiset" assumes "atU U c" obtains ui :: "'f" and r :: "'f list" where "ui # r \<in># c" and "ui \<in> U"
5: shows "[l] \<langle>star \<circ> f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms"
tllist_all2_conv_all_tnth: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" shows "tllist_all2 P Q xs ys = (tlength xs = tlength ys \<and> (\<forall>n. enat n < tlength xs \<longrightarrow> P (tnth xs n) (tnth ys n)) \<and> (tfinite xs \<longrightarrow> Q (terminal xs) (terminal ys)))"
complete_iff_size: fixes kdt :: "'b kdt" shows "KD_Tree.complete kdt = (KD_Tree.size_kdt kdt = 2 ^ height kdt)"
simps: fixes a1 :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a2 :: "Vars.var \<Rightarrow> Value"   and a3 :: "Vars.var \<Rightarrow> Value" shows "eq\<rho> a1 a2 a3 = (\<exists>ae \<rho>1 \<rho>2. a1 = ae \<and> a2 = \<rho>1 \<and> a3 = \<rho>2 \<and> (\<forall>x xa. ae x = up\<cdot>xa \<longrightarrow> eq xa (\<rho>1 x) (\<rho>2 x)))"
blocks_image_def: fixes B :: "'d set multiset"   and f :: "'d \<Rightarrow> 'c" shows "blocks_image B f \<equiv> image_mset ((`) f) B"
set_bind_option: fixes x :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "set_option (x \<bind> f) = \<Union> ((set_option \<circ> f) ` set_option x)"
quasi_measure_preserving_AE: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "f \<in> quasi_measure_preserving M N"   and "almost_everywhere N P" shows "AE x in M. P (f x)"
one_imp_carrier: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "\<one>\<^bsub>R\<^esub> \<in> I" shows "I = carrier R"
gt_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<succ> y"   and "y \<succ> z" shows "x \<succ> z"
measure_le_1: fixes s :: "'a cfg"   and X :: "'a cfg stream set" shows "emeasure (MC.T s) X \<le> 1"
simps_5: fixes p :: "'a pfm"   and q :: "'a pfm" shows "static (p \<^bold>\<longrightarrow>\<^sub>! q) = (static p \<and> static q)"
b_e_check_single_type_type: fixes \<C> :: "t_context"   and e :: "b_e"   and xs :: "checker_type"   and tm :: "t list" assumes "check_single \<C> e xs = Type tm" shows "\<exists>tn. xs = Type tn"
boundedI: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b"   and "a \<subseteq># c" shows "a \<subseteq># b \<inter># c"
Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and A :: "'e set"   and Q :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>A Throw Q,A"
stutter_freeD: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat" assumes "stutter_free \<sigma>"   and "\<sigma> (Suc k) = \<sigma> k"   and "k < n" shows "\<sigma> n = \<sigma> k"
project_coord_aform_lv_def: fixes xs :: "(real \<times> real pdevs) list"   and b :: "nat"   and n :: "real" shows "project_coord_aform_lv xs b n = xs[b := (n, zero_pdevs)]"
rel_coinduct: fixes P :: "'b stream \<Rightarrow> 'c stream \<Rightarrow> bool"   and x :: "'b stream"   and y :: "'c stream"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "P x y"   and "\<And>stream stream'. P stream stream' \<Longrightarrow> R (shd stream) (shd stream') \<and> P (stl stream) (stl stream')" shows "stream_all2 R x y"
op_dghm_components_4: fixes \<FF> :: "V" shows "op_dghm \<FF>\<lparr>HomCod\<rparr> = op_dg (\<FF>\<lparr>HomCod\<rparr>)"
equals_cod_if_value_is_ide: assumes "C.ide a'" shows "A'.map = A'.G.map"
b: shows "X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>"
af_prefix_true: fixes i :: "nat"   and j :: "nat"   and \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "i \<le> j"   and "af \<phi> (prefix i w) \<sim>\<^sub>C true\<^sub>n" shows "af \<phi> (prefix j w) \<sim>\<^sub>C true\<^sub>n"
map_ide_simp: fixes a :: "'b \<times> 'b \<times> 'b" assumes "D.VVV.ide a" shows "D.\<alpha>' a = D.inv \<a>\<^sub>D[fst a, fst (snd a), snd (snd a)]"
get_attribute_is_l_get_attribute: shows "l_get_attribute DocumentClass.type_wf get_attribute get_attribute_locs"
legal_passign_def: fixes A :: "'v pvar \<Rightarrow> hmultiset" shows "legal_passign A = (\<forall>x. min_passign x \<le> A x)"
unat_sub_if_size: shows "unat (2 ^ size x - x) = (if unat x \<le> unat (2 ^ size x) then unat (2 ^ size x) - unat x else unat (2 ^ size x) + 2 ^ size (2 ^ size x) - unat x)"
nmaps_false: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and x :: "name"   and c :: "coname"   and P :: "trm" assumes "\<theta>_n nmaps x to Some (c, P)"   and "x \<sharp> \<theta>_n" shows "False"
\<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic: fixes K :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K"   and "y \<in> \<omega>_limit_set x"   and "f y \<noteq> (0::'a)"   and "z \<in> \<omega>_limit_set y \<union> \<alpha>_limit_set y"   and "f z \<noteq> (0::'a)" shows "periodic_orbit y \<and> range (flow0 y) = \<omega>_limit_set x"
63947: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> y) \<sqinter> - (- y \<squnion> x) = bot"
simps_21: fixes undef' :: "bool \<Rightarrow> bool"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "undef_update undef' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef' undef, \<dots> = more\<rparr>"
eps_fv_cong: fixes r :: "Formula.formula Regex.regex"   and v :: "event_data list"   and v' :: "event_data list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat" assumes "\<forall>x\<in>fv_regex r. v ! x = v' ! x" shows "Regex.eps (Formula.sat \<sigma> V v) i r = Regex.eps (Formula.sat \<sigma> V v') i r"
GB_imp_reducibility: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_Groebner_basis G"   and "f \<noteq> 0"   and "f \<in> pmdl G" shows "is_red G f"
path_cons_simp': fixes v :: "'v"   and v' :: "'v"   and e :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and pth :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" shows "path v v' (e # pth) = (\<exists>b a ba. e = ((v, b), a, ba) \<and> ((v, b), a, ba) \<in> edges \<and> path a v' pth)"
dletrec2: fixes funs :: "(char list \<times> char list \<times> exp0) list"   and ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and locs :: "locn \<times> locn" assumes "\<not> allDistinct (map (\<lambda>x. case x of (x, y, z) \<Rightarrow> x) funs)" shows "evaluate_dec ck mn env s (Dletrec locs funs) (s, Rerr (Rabort Rtype_error))"
insert_compr: fixes a :: "'b"   and B :: "'b set" shows "(a::'b) \<triangleright> (B::'b set) = {x::'b. x = a \<or> x \<in> B}"
abs_dlist_def: shows "abs_dlist = Abs_dlist \<circ> remdups"
fa: fixes e :: "real" assumes "0 < e" shows "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> norm (g x - f a) < e"
pred_set: fixes P :: "'a \<Rightarrow> bool" shows "pred_stream P = (\<lambda>x. Ball (sset x) P)"
ImplLeadsto_gen: fixes F :: "'a \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" shows "\<turnstile> \<box>(Init F \<longrightarrow> Init G) \<longrightarrow> (F \<leadsto> G)"
Inf_UNIV: shows "\<Sqinter> UNIV = \<bottom>"
l_unit: fixes G :: "('c, 'd) Group_scheme"   and a :: "'c" assumes "Group G"   and "a \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<cdot>\<^bsub>G\<^esub> a = a"
elims: fixes x :: "'a"   and xa :: "'a list \<times> 'a list"   and y :: "'a list \<times> 'a list" assumes "replace_hd_t x xa = y" obtains a :: "'a" where "x = a__" and "xa = ([], [])" and "y = ([], [])"   | a :: "'a" and l :: "'a" where "x = aa__" and "xa = ([], [l__])" and "y = ([], [aa__])"   | a :: "'a" and l :: "'a" and v :: "'a" and va :: "'a list" where "x = ab__" and "xa = ([], la__ # v # va)" and "y = (let fs = rev (v # va) in (ab__ # tl fs, [la__]))"   | a :: "'a" and f :: "'a" and fs :: "'a list" and l :: "'a" and ls :: "'a list" where "x = ac__" and "xa = (f__ # fs__, l # ls)" and "y = (ac__ # fs__, l # ls)"   | a :: "'a" and f :: "'a" and fs :: "'a list" where "x = a" and "xa = (f # fs, [])" and "y = undefined"
semilattice_axioms: shows "semilattice (\<^bold>\<sqinter>)"
wlp_ite_weights: fixes l1 :: "'d list"   and P :: "'d \<Rightarrow> bool"   and x :: "'d"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and w1 :: "'d \<Rightarrow> 'e"   and w2 :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'f" assumes "\<forall>y. ListMem y l1 \<longrightarrow> P y"   and "P x" shows "wlp R (\<lambda>y. if P y then w1 y else w2 y) g f x l1 = wlp R w1 g f x l1"
is_word_terminals: fixes p :: "(symbol \<times> character list) list" assumes "wellformed_tokens p" shows "is_word (terminals p)"
thetaParTT_RetrT: shows "thetaParTT \<subseteq> RetrT (thetaParTT \<union> BisT)"
subst_cong: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<sigma>' :: "('b \<times> 'b trm) list"   and \<theta> :: "('b \<times> 'b trm) list"   and \<theta>' :: "('b \<times> 'b trm) list" assumes "\<sigma> \<doteq> \<sigma>'"   and "\<theta> \<doteq> \<theta>'" shows "\<sigma> \<lozenge> \<theta> \<doteq> \<sigma>' \<lozenge> \<theta>'"
HBL3: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (imp (PP \<langle>\<phi>\<rangle>) (PP \<langle>PP \<langle>\<phi>\<rangle>\<rangle>))"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat" shows "ET \<noteq> MKT x21 x22 x23 x24"
vrange_vconst_on_ne: fixes A :: "V"   and c :: "V" assumes "A \<noteq> []\<^sub>\<circ>" shows "\<R>\<^sub>\<circ> (vconst_on A c) = ZFC_in_HOL.set {c}"
density_RN_deriv: fixes M :: "'b measure"   and N :: "'b measure" assumes "sigma_finite_measure M"   and "absolutely_continuous M N"   and "sets N = sets M" shows "density M (RN_deriv M N) = N"
map_fst_mk_snd: fixes k :: "'b"   and l :: "'a list" shows "map fst (map (\<lambda>x. (x, k)) l) = l"
Bex_cong: fixes B :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>b. b \<in> B \<Longrightarrow> P b = Q b" shows "Bex B P = Bex B Q"
tauTransitionChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P'"
hyps: assumes "(V, asx, asx) \<in> dependent_live_vars n'"   and "asx \<noteq> [] \<longrightarrow> n' = last (targetnodes asx)" shows "V \<in> Use n' \<and> (\<forall>as' a as''. asx = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n')"
clT: shows "closedin (top_of_set (S \<union> T)) T"
to_show: assumes "R" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
sub2: fixes B :: "'c"   and k :: "'c"   and b :: "'c"   and a :: "'c" assumes "B \<equiv> k + b" shows "a - B \<equiv> - k + (a - b)"
rhs: shows "map_spmf (\<lambda>b. b) coin_spmf = spmf_of_set {True, False}"
vrestr_Union: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set set" shows "f --` \<Union> A = \<Union> ((--`) f ` A)"
push_forward_def: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real" shows "d13.push_forward X = restrict0 d23.dest.diff_fun_space (\<lambda>ga. X (restrict0 d12.src.carrier (ga \<circ> (g \<circ> f))))"
Ord_trans: fixes i :: "hf"   and j :: "hf"   and k :: "hf" assumes "i \<^bold>\<in> j"   and "j \<^bold>\<in> k"   and "Ord k" shows "i \<^bold>\<in> k"
flow_of_c: shows "flow_of_cf c = (\<lambda>_. 0::'capacity)"
field_def: fixes R :: "('g, 'h) ring_scheme" shows "field R \<equiv> domain R \<and> field_axioms R"
Ord_Inter: fixes A :: "hf" assumes "\<And>i. i \<^bold>\<in> A \<Longrightarrow> Ord i" shows "Ord (\<Sqinter>A)"
minus_multiset_def: shows "(-) \<equiv> map_fun count (map_fun count Abs_multiset) (\<lambda>M N a. M a - N a)"
gb_rename_precond_axioms: shows "gb_rename_precond G f"
RSpan_single_nonzero: fixes m :: "'c" assumes "m \<in> V"   and "m \<noteq> (0::'c)" shows "GSpan [m] \<noteq> 0"
PiE: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and x :: "'c" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
ge: shows "\<Lambda> \<le> 2 * (\<Lambda> * (\<v> - 1) div (\<k> - 1))"
vec_carrier: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "vec n f \<in> carrier_vec n"
q: fixes z :: "'a" shows "pmf q z = (if z = y then 0 else pmf p z / a)"
TSL: shows "SIDE_GEN_ALGO (is_map_to_list Rk Rv Rs tsl)"
param_zip: fixes Ra :: "('a \<times> 'c) set"   and Rb :: "('b \<times> 'd) set" shows "(zip, zip) \<in> \<langle>Ra\<rangle>list_rel \<rightarrow> \<langle>Rb\<rangle>list_rel \<rightarrow> \<langle>Ra \<times>\<^sub>r Rb\<rangle>list_rel"
invariant_1_2_realI: fixes plr :: "int poly \<times> rat \<times> rat"   and x :: "real" assumes "root_cond plr x"   and "sgn (rai_lb plr) = sgn (rai_ub plr)"   and "\<exists>!x. root_cond plr x"   and "poly_cond2 (poly_real_alg_1 plr)" shows "invariant_1_2 plr \<and> real_of_1 plr = x"
simps_2: fixes y :: "'d" shows "flat_1 (allow (deny y)) = allow y"
cadjoint_univ_prop: fixes G :: "'a \<Rightarrow> 'b" assumes "bounded_clinear G" shows "\<forall>x y. (G\<^sup>\<dagger>) x \<bullet>\<^sub>C y = x \<bullet>\<^sub>C G y"
R'_def: shows "R' \<equiv> region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'}"
subset_Collect_conv: fixes S :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "((S::'a set) \<subseteq> Collect (P::'a \<Rightarrow> bool)) = (\<forall>x::'a\<in>S. P x)"
le_less_trans: fixes x :: "'c word"   and y :: "'c word"   and z :: "'c word" assumes "x \<le>s y"   and "y <s z" shows "x <s z"
pderiv_eq_0_iff: fixes p :: "'c poly" shows "(pderiv p = 0) = (degree p = 0)"
solve_iarrays_def: fixes A :: "'d iarray iarray"   and b :: "'d iarray" shows "solve_iarrays A b = (let GJ_P = Gauss_Jordan_iarrays_PA A; P_mult_b = fst GJ_P *iv b; rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (snd GJ_P))); consistent_Ab = (if \<not> is_zero_iarray P_mult_b then greatest_not_zero P_mult_b + 1 else 0) \<le> rank_A; GJ_transpose = Gauss_Jordan_iarrays_PA (transpose_iarray A); basis = set (map (\<lambda>i. row_iarray i (fst GJ_transpose)) [rank_A..<ncols_iarray A]) in if consistent_Ab then Some (solve_consistent_rref_iarrays (snd GJ_P) P_mult_b, basis) else None)"
case_two_not_learn_V: fixes j :: "nat" assumes "\<exists>t. local.prefixes t j \<up>"   and "2 \<le> j"   and "\<phi> j = \<psi> j" shows "\<not> learn_total \<phi> V_constotal s"
fold_fun_left_comm: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "comp_fun_commute f"   and "finite A" shows "f x (Finite_Set.fold f z A) = Finite_Set.fold f (f x z) A"
Mul_Redirect_Edge_def: fixes j :: "nat"   and n :: "nat" shows "Mul_Redirect_Edge j n \<equiv> \<lbrace>\<acute>Mul_mut_init n \<and> Z (\<acute>Muts ! j)\<rbrace> \<langle>IF T (\<acute>Muts ! j) \<in> Reach \<acute>E THEN \<acute>E := \<acute>E[R (\<acute>Muts ! j) := (fst (\<acute>E ! R (\<acute>Muts ! j)), T (\<acute>Muts ! j))] FI,, \<acute>Muts := \<acute>Muts[j := (\<acute>Muts ! j)\<lparr>Z := False\<rparr>]\<rangle>"
Ex_I: fixes H :: "fm set"   and A :: "fm"   and i :: "name"   and x :: "tm" assumes "H \<turnstile> A(i::=x)" shows "H \<turnstile> SyntaxN.Ex i A"
chain_imp_trancl: fixes S :: "nat \<Rightarrow> 'b"   and r :: "('b \<times> 'b) set"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. (S i, S (Suc i)) \<in> r"   and "i < j" shows "(S i, S j) \<in> r\<^sup>+"
cong_2: fixes x :: "'a" assumes "x \<in> Y" shows "f x = g x"
simps_2: fixes r :: "'form" shows "inPorts (Assumption r) = {||}"
ide_def: fixes a :: "'a" shows "Left_a.ide a \<equiv> Left_a.comp a a \<noteq> Left_a.null \<and> (\<forall>f. (Left_a.comp f a \<noteq> Left_a.null \<longrightarrow> Left_a.comp f a = f) \<and> (Left_a.comp a f \<noteq> Left_a.null \<longrightarrow> Left_a.comp a f = f))"
wf_bs_insensitive': fixes x :: "'b"   and bs :: "'b list"   and p :: "'b poly"   and y :: "'b" shows "wf_bs (x # bs) p = wf_bs (y # bs) p"
Y_seq'_AE: fixes n :: "nat" shows "AE \<omega> in restrict_space lborel {0<..<1}. Y_seq' n \<omega> = \<mu>.I n \<omega>"
negligible_mono: fixes g :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "Negligible.negligible g"   and "f \<in> O(g)" shows "Negligible.negligible f"
case_1: fixes f1 :: "'b"   and f2 :: "'c tree \<Rightarrow> 'b"   and f3 :: "'c tree \<Rightarrow> 'b" shows "(case Same of Same \<Rightarrow> f1 | Bal x \<Rightarrow> f2 x | Unbal x \<Rightarrow> f3 x) = f1"
length_list_of_conv_the_enat: fixes xs :: "'c llist" assumes "lfinite xs" shows "length (list_of xs) = the_enat (llength xs)"
left_right_simps_1: fixes l :: "'a"   and r :: "'a" shows "left (app l r) = l"
inverse_unique: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.inverse_arrows f g" shows "Maps.inv f = g"
of_real_poly_map_poly: fixes p :: "real poly"   and x :: "real" shows "of_real (poly p x) = poly (map_poly of_real p) (of_real x)"
mod_nat_int_pow_eq: fixes a :: "int"   and p :: "int"   and n :: "nat" assumes "0 \<le> a"   and "0 \<le> p" shows "nat a ^ n mod nat p = nat (a ^ n mod p)"
natural_isomorphism_axioms: shows "natural_isomorphism VVV.comp (\<bullet>) HoHV HoVH (\<lambda>\<mu>\<nu>\<pi>. \<a>[fst \<mu>\<nu>\<pi>, fst (snd \<mu>\<nu>\<pi>), snd (snd \<mu>\<nu>\<pi>)])"
no_multi: fixes v :: "'v"   and w :: "'w"   and u :: "'v"   and w' :: "'w" assumes "(v, w, u) \<in> E"   and "(v, w', u) \<in> E" shows "w = w'"
abs_eq_iff: fixes x :: "real_alg_3"   and y :: "real_alg_3" shows "(Real_Alg_Quotient x = Real_Alg_Quotient y) = (real_of_3 x = real_of_3 y)"
drop_ext: fixes k :: "nat"   and xs :: "'c list"   and xs' :: "'c list" assumes "\<forall>i. k \<le> i \<and> i < length xs \<longrightarrow> xs' ! i = xs ! i"   and "length xs' = length xs" shows "drop k xs' = drop k xs"
test_s_prod_is_meet: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" assumes "R \<subseteq> 1\<^sub>\<sigma>"   and "S \<subseteq> 1\<^sub>\<sigma>" shows "R \<cdot> S = R \<inter> S"
last_map: fixes xs :: "'f list"   and f :: "'f \<Rightarrow> 'g" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
fminusD2: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset"   and P :: "bool" assumes "c |\<in>| A |-| B"   and "c |\<in>| B" shows "P"