hcju/isabelleps huggingface dataset:
eint_int_minus_distr: fixes n :: "int"   and c :: "eint"   and m :: "int" shows "eint n * c - eint m * c = eint (n - m) * c"
IH_2: shows "(\<exists>a\<in>set (dnf \<phi>2). case a of (al, fl) \<Rightarrow> (\<forall>a\<in>set al. aEval a xs) \<and> (\<forall>f\<in>set fl. eval f xs)) = eval \<phi>2 xs"
permute_zero: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.pt permute" shows "permute 0 x = x"
of_real_Re: fixes z :: "complex" assumes "z \<in> \<real>" shows "complex_of_real (Re z) = z"
that_2: fixes i :: "'b" assumes "i \<in> I" shows "closure (U i) \<subseteq> X"
infsubsetall: fixes x :: "'b llist"   and A :: "'b set" assumes "x \<in> A\<^sup>\<omega>" shows "x \<in> A\<^sup>\<infinity>"
agree: shows "Vagree \<nu> \<nu>' (FVDiff (Plus t1 t2))"
Union_rel_br: fixes l :: "'d \<Rightarrow> 'f set"   and lI :: "'d \<Rightarrow> bool"   and s :: "'f \<Rightarrow> 'e set"   and sI :: "'f \<Rightarrow> bool" shows "\<langle>br l lI, br s sI\<rangle>Union_rel = br (\<lambda>a. \<Union> (s ` l a)) (\<lambda>a. lI a \<and> (\<forall>c\<in>l a. sI c))"
equiv_utransI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> t P = u P" shows "equiv_utrans t u"
adjoint_minus: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat n m" shows "adjoint (A - B) = adjoint A - adjoint B"
delete_edges_simp2: fixes G :: "'c graph"   and E :: "('c \<times> 'c) set" shows "delete_edges G E = \<lparr>nodes = nodes G, edges = edges G - E\<rparr>"
funci_agree_def: fixes fs :: "'a list"   and n :: "nat"   and f :: "'a" shows "funci_agree fs n f = (n < length fs \<and> fs ! n = f)"
mset_x_eq_poly_roots_Q: fixes x :: "nat \<Rightarrow> complex"   and n :: "nat" shows "image_mset x (mset_set {..<n}) = poly_roots (power_sum_puzzle.Q x n)"
iMODb_cut_le: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and t :: "nat" shows "[ r, mod m, c ] \<down>\<le> t = (if t < r then {} else if t < r + m * c then [ r, mod m, (t - r) div m ] else [ r, mod m, c ])"
simps_6: fixes \<sigma> :: "nat \<Rightarrow> val"   and b :: "expr"   and e1 :: "expr"   and e2 :: "expr" shows "expr_sem \<sigma> (IF b THEN e1 ELSE e2) = expr_sem \<sigma> b \<bind> (\<lambda>b'. if b' = TRUE then expr_sem \<sigma> e1 else expr_sem \<sigma> e2)"
tconf_def: fixes P :: "'a prog"   and h :: "'heap"   and t :: "'thread_id" shows "P,h \<turnstile> t \<surd>t \<equiv> \<exists>C. typeof_addr h (thread_id2addr t) = \<lfloor>Class_type C\<rfloor> \<and> P \<turnstile> C \<preceq>\<^sup>* Thread"
sat_def: fixes S :: "'a formula set" shows "sat S \<equiv> \<exists>\<A>. \<forall>F\<in>S. \<A> \<Turnstile> F"
cmp_simps_5: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (\<Phi>\<^sub>H (f, g)) = H (f \<star>\<^sub>C g)"
DT01: shows "\<forall>x\<in>carrier. \<forall>y\<in>carrier. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds x. y \<notin> u) = (\<exists>v\<in>nhds y. x \<notin> v)"
exhaust: fixes y :: "'d strand_label" obtains x1 :: "'d" where "y = ln x1"   | "y = \<star>"
arrow_of_spans_data_ext_def: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c" shows "\<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons arrow_of_spans_data_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple Chn (Record.iso_tuple_cons Record.tuple_iso_tuple Dom Cod)) more"
ext_inject: fixes access :: "acc_modi"   and more :: "'b"   and access' :: "acc_modi"   and more' :: "'b" shows "(\<lparr>access = access, \<dots> = more\<rparr> = \<lparr>access = access', \<dots> = more'\<rparr>) = (access = access' \<and> more = more')"
add_mono_thms_linordered_field_5: fixes i :: "'g"   and j :: "'g"   and k :: "'g"   and l :: "'g" assumes "i < j \<and> k < l" shows "i + k < j + l"
isAbr_simps_3: fixes f :: "'f" shows "isAbr (Fault f) = False"
prio_match_matcher_alt: fixes flow_entries :: "('d, 'e) flow_entry_match list"   and \<gamma> :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and packet :: "'f" shows "{f \<in> set flow_entries. \<gamma> (ofe_fields f) packet \<and> (\<forall>fo\<in>set flow_entries. ofe_prio f < ofe_prio fo \<longrightarrow> \<not> \<gamma> (ofe_fields fo) packet)} = (let matching = {f \<in> set flow_entries. \<gamma> (ofe_fields f) packet} in {f \<in> matching. \<forall>fo\<in>matching. ofe_prio fo \<le> ofe_prio f})"
vcons_eqD_2: fixes xs\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "vfsequence_pair xs\<^sub>1 xs\<^sub>2"   and "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2" shows "x\<^sub>1 = x\<^sub>2"
simps_2: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)"   and s :: "'a \<times> 'b"   and r :: "'c"   and rs :: "'c list" shows "On_Off.config' step A s (r # rs) = On_Off.config' step A (On_Off.Step step A s r) rs"
intros_2: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> af A (\<lambda>y z. P y z \<or> P x y)" shows "af A P"
infs_UNIV: fixes w :: "'a stream" shows "infs UNIV w"
index_finsum_vec: fixes F :: "'b set"   and i :: "nat"   and n :: "nat"   and vs :: "'b \<Rightarrow> 'c Matrix.vec" assumes "finite F"   and "i < n"   and "vs \<in> F \<rightarrow> carrier_vec n" shows "finsum_vec TYPE('c) n vs F $v i = (\<Sum>f\<in>F. vs f $v i)"
restricted_agree_imp_agree: fixes s2 :: "('c, 'd) fmap"   and vs :: "'c set"   and s1 :: "('c, 'd) fmap" assumes "fmdom' s2 \<subseteq> vs"   and "agree (fmrestrict_set vs s1) s2" shows "agree s1 s2"
vconn_implies_hpar_4: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "S.in_hom \<mu> f g" shows "S.trg g = S.trg f"
sigmas: shows "{\<sigma>1, \<sigma>2, \<sigma>3', \<gamma>1'''} \<subseteq> seq ars"
stutter_reduced_suffix: fixes k :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "\<natural>suffix k (\<natural>\<sigma>) = suffix k (\<natural>\<sigma>)"
set_choose_thm: fixes s :: "'a set" assumes "s \<noteq> {}" shows "set_choose s \<in> s"
eqspanT: shows "affine hull (+) (- b) ` T = span ((+) (- b) ` T)"
filterlim_compose: fixes g :: "'c \<Rightarrow> 'd"   and F3 :: "'d filter"   and F2 :: "'c filter"   and f :: "'e \<Rightarrow> 'c"   and F1 :: "'e filter" assumes "filterlim g F3 F2"   and "filterlim f F2 F1" shows "LIM x F1. g (f x) :> F3"
ltrm_of_X: shows "monom P (X_poly R (deg R (X_poly R))) (deg R (X_poly R)) = X_poly R"
\<psi>_in_terms_of_\<Psi>': fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<psi> x g = (\<Phi>\<Psi>.\<psi>C (F y, x) \<circ> \<Phi>\<Psi>.\<Psi>.FUN (y, x) \<circ> inD) g"
strict_ex_prop_def: fixes X :: "'b program set" shows "strict_ex_prop X \<equiv> \<forall>F G. F ok G \<longrightarrow> (F \<in> X \<or> G \<in> X) = (F \<squnion> G \<in> X)"
hyp: fixes c' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> (c', s')" shows "\<Gamma>\<turnstile>c' \<down> s'"
IH: fixes y :: "nat" assumes "y < m" shows "sum \<chi> {..<y} = sum \<chi> {..<y mod n}"
sv01_converg_V01: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> V\<^sub>0\<^sub>1" shows "s\<^bsub>01\<^esub> (f \<triangleright> n) \<down>"
prenum_encode: fixes vs :: "nat list"   and x :: "nat" shows "prenum (Partial_Recursive.list_encode vs) x \<down>= (if x < length vs then vs ! x else 0)"
put_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) document_ptr"   and setter :: "('w \<Rightarrow> 'w) \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme"   and v :: "'w" assumes "h \<turnstile> ok put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v" shows "ptr |\<in>| document_ptr_kinds h"
on_tripleD_3: fixes l1 :: "'e"   and v1 :: "'f"   and v2 :: "'f"   and l2 :: "'e"   and v3 :: "'g"   and v4 :: "'g"   and R :: "('f \<times> 'g) set" assumes "((l1, v1, v2), l2, v3, v4) \<in> on_triple R" shows "(v2, v4) \<in> R"
regset_below: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "bounded d k"   and "i < k"   and "j < k" shows "regset d i j k = {xs. deltas d xs i = j}"
pseudominimal_inD2: fixes P :: "'b set"   and x :: "'b"   and z :: "'b" assumes "order.pseudominimal_in P x"   and "z \<in> P"   and "order.greater x z" shows "z = order.bottom P"
iIN_iprev_diff_const: fixes n :: "nat"   and t :: "nat"   and d :: "nat" assumes "n < t"   and "t \<le> n + d" shows "t - iprev t [n\<dots>,d] = Suc 0"
prefix_of_node_history_def: fixes xs :: "('msgid \<times> 'op) event list"   and i :: "nat" shows "xs prefix of i \<equiv> \<exists>ys. xs @ ys = history i"
widen_Class: fixes P :: "'b prog"   and T :: "ty"   and C :: "char list" shows "subtype P T (Class C) = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C))"
Gco: fixes X :: "'a set" assumes "X \<in> L" shows "G \<in> X - B co X"
ordinal_times_left_cancel: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "0 < w" shows "(w * x = w * y) = (x = y)"
BIT_pairwise': fixes qs :: "'b list"   and init :: "'b list"   and x :: "'b"   and y :: "'b" assumes "set qs \<subseteq> set init"   and "(x, y) \<in> {(x, y). x \<in> set init \<and> y \<in> set init \<and> x \<noteq> y}"   and "x \<noteq> y"   and "distinct init" shows "Pbefore_in x y BIT qs init = Pbefore_in x y BIT (Lxy qs {x, y}) (Lxy init {x, y})"
big_int_less: fixes a :: "int"   and N :: "int" shows "a - int (nat (\<bar>a\<bar> + \<bar>N\<bar> + 1)) < N"
gen_llength_cons'_def: shows "local.gen_llength_cons' = gen_llength_cons (lgenerator g)"
assms: shows "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"
suffix_conc_fst: fixes n :: "nat"   and u :: "'a list"   and v :: "nat \<Rightarrow> 'a" assumes "n \<le> length u" shows "suffix n (u \<frown> v) = drop n u \<frown> v"
inorder_insert: fixes m :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "HBT m"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (HBT.insert m x t) = ins_list x (Tree2.inorder t)"
homeomorphic_refl: fixes s :: "'c set" shows "s homeomorphic s"
scaleR_poly_mapping_def: fixes r :: "real"   and x :: "'c \<Rightarrow>\<^sub>0 'd" shows "r *\<^sub>R x \<equiv> Abs_poly_mapping (\<lambda>i. r *\<^sub>R poly_mapping.lookup x i)"
ENR_sphere: fixes a :: "'a"   and r :: "real" shows "ENR (sphere a r)"
weight_alt: fixes w :: "'b set \<Rightarrow> nat"   and g :: "'b ugraph" shows "weight w g = sum w (uedge ` edges g)"
Q_closed_2: shows "supp Q\<^sub>2 = {}"
fmsubset_drop_set: fixes S :: "'a set"   and m :: "('a, 'b) fmap" shows "fmdrop_set S m \<subseteq>\<^sub>f m"
borel_measurable_real_of_ereal: fixes f :: "'c \<Rightarrow> ereal"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
partial_object_ext_def: fixes carrier :: "'a set"   and more :: "'b" shows "\<lparr>carrier = carrier, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons partial_object_ext_Tuple_Iso carrier more"
simps_1: fixes v :: "val" shows "expr_rf_to_cexpr (Val v) = CVal v"
total_sur: fixes x :: "'a" shows "is_total x = is_sur (x\<^sup>\<smile>)"
supt_Fun: fixes s :: "('d, 'e) Term.term"   and f :: "'d"   and ss :: "('d, 'e) Term.term list" assumes "s \<rhd> Fun f ss"   and "s \<in> set ss" shows "False"
get_all_matching_src_ips_def: fixes iface :: "iface"   and m :: "'b common_primitive match_expr" shows "get_all_matching_src_ips iface m \<equiv> let (i_matches, rest1) = primitive_extractor (is_Iiface, iiface_sel) m in if \<forall>is\<in>set i_matches. case is of Pos i \<Rightarrow> match_iface i (iface_sel iface) | Neg i \<Rightarrow> \<not> match_iface i (iface_sel iface) then let (ip_matches, rest2) = primitive_extractor (is_Src, src_sel) rest1 in if \<not> has_primitive rest2 \<and> matcheq_matchAny rest2 then if ip_matches = [] then UNIV else \<Inter>ips\<in>set ip_matches. case ips of Pos ip \<Rightarrow> ipt_iprange_to_set ip | Neg ip \<Rightarrow> - ipt_iprange_to_set ip else {} else {}"
sep_disj_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj" shows "sep_disj x zero"
assms_9: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
nList_def: fixes A :: "'a set"   and n :: "nat" shows "nList A n \<equiv> {l. set l \<subseteq> A \<and> length l = n}"
1: shows "x (t + h) - x t - h *\<^sub>R f (t, x t) - (h\<^sup>2 / 2) *\<^sub>R f' (t, x t) $ (1, f (t, x t)) \<in> (*\<^sub>R) (1 / 3) ` closure (convex hull (\<lambda>xa. (h ^ 3 / 2) *\<^sub>R (f'' (h * xa + t, x (h * xa + t)) $ (1, f (h * xa + t, x (h * xa + t))) $ (1, f (h * xa + t, x (h * xa + t))) + f' (h * xa + t, x (h * xa + t)) $ (0, f' (h * xa + t, x (h * xa + t)) $ (1, f (h * xa + t, x (h * xa + t)))))) ` cbox 0 1)"
lookup_eq_and_update_lemma: fixes m :: "'b \<Rightarrow>\<^sub>0 'a"   and var :: "'b"   and deg :: "'a"   and x :: "'b \<Rightarrow>\<^sub>0 'a" shows "(lookup m var = deg \<and> Poly_Mapping.update var (0::'a) m = x) = (m = Poly_Mapping.update var deg x \<and> lookup x var = (0::'a))"
strict_order_iff: fixes b :: "'a"   and a :: "'a" shows "(b < a) = (a = max a b \<and> a \<noteq> b)"
lens_indep_left_comp: fixes z :: "'d \<Longrightarrow> 'e"   and x :: "'f \<Longrightarrow> 'd"   and y :: "'g \<Longrightarrow> 'd" assumes "mwb_lens z"   and "x \<bowtie> y" shows "x ;\<^sub>L z \<bowtie> y ;\<^sub>L z"
not_iWeakUntil: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<not> (P t1. t1 \<W> t2 I. Q t2)) = ((\<box> t I. Q t \<longrightarrow> (\<diamond> t' I \<down>< t. \<not> P t')) \<and> (\<diamond> t I. \<not> P t))"
M_def: shows "M = (\<lambda>i j. if j < dg then coeff_int f (int (degree f) - int i + int j) else coeff_int g (int (degree g) - int i + int (j - dg)))"
simps_23: shows "OUTfromCh data23 = {data21}"
locally_essentially_surjective: fixes a :: "'a"   and b :: "'a"   and g :: "'b" assumes "C.obj a"   and "C.obj b"   and "\<guillemotleft>g : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 b\<guillemotright>"   and "D.ide g" shows "\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) g"
Iff_cong: fixes H :: "fmla set"   and A :: "fmla"   and A' :: "fmla"   and B :: "fmla"   and B' :: "fmla" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> (A IFF B) IFF A' IFF B'"
natural_transformation_axioms: shows "natural_transformation \<F>\<^sub>SC.comp \<F>\<^sub>SC.comp \<F>\<^sub>SC.map \<F>\<^sub>SC.map \<F>\<^sub>SC.map"
zero_less_Limit: fixes \<beta> :: "V" assumes "Limit \<beta>" shows "0 < \<beta>"
maintain_invar2_ctd_2: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option"   and u :: "'v"   and d :: "nat" assumes "prim_invar2_ctd Q \<pi>"   and "Q u = enat d"   and "\<forall>v. enat d \<le> Q v" shows "T_measure2 (Q' Q \<pi> u) (\<pi>' Q \<pi> u) < T_measure2 Q \<pi>"
that: fixes as :: "'edge list" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S"   and "n' \<in> S" shows "thesis"
poly_prime_factorization_exists: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> normalize (\<Prod>\<^sub># A) = normalize p"
fun_complete: fixes a :: "'s"   and b :: "'s"   and F :: "'s \<Rightarrow> 's" assumes "ide a"   and "ide b"   and "F \<in> local.set a \<rightarrow> local.set b" shows "\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> Fun f = restrict F (local.set a)"
part_lift: fixes I0 :: "'a set"   and P :: "'a set set"   and Q :: "'a set set"   and F :: "'a set \<Rightarrow> 'b set"   and J0 :: "'b set" assumes "part I0 P"   and "part I0 Q"   and "finer P Q"   and "inj_on F P"   and "part J0 (F ` P)"   and "{} \<notin> P"   and "{} \<notin> F ` P" shows "part J0 (lift P F ` Q)"
cnj_numeral: fixes w :: "num" shows "Quaternions.cnj (numeral w) = numeral w"
arr_def: fixes f :: "'a1" shows "C1.arr f \<equiv> C1.domains f \<noteq> {} \<or> C1.codomains f \<noteq> {}"
P1: fixes a :: "'a common_primitive match_expr" assumes "normalized_nnf_match a"   and "a \<in> set (normalize_match (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one m))" shows "P (disc2, sel2) a"
assms_4: shows "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>"
int_not_BIT: fixes w :: "int"   and b :: "bool" shows "NOT (w BIT b) = NOT w BIT (\<not> b)"
i_take_0: fixes f :: "nat \<Rightarrow> 'b" shows "f \<Down> 0 = []"
z2: shows "case z2 of (x, y) \<Rightarrow> x \<le> y \<and> x\<^sup>2 + y\<^sup>2 = p"
comp_red_basis_is_red: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and f :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red (set (comp_red_basis xs)) f = is_red (set xs) f"
piecewise_C1_differentiable_on_imp_continuous_on: fixes f :: "real \<Rightarrow> 'a"   and s :: "real set" assumes "f piecewise_C1_differentiable_on s" shows "continuous_on s f"
ZN_add: shows "(ZN ===> ZN ===> ZN) (+) (+)"
projecting_def: fixes C :: "'d program \<Rightarrow> 'd set"   and h :: "'e \<times> 'f \<Rightarrow> 'd"   and F :: "'e program"   and X' :: "'d program set"   and X :: "'e program set" shows "projecting C h F X' X \<equiv> \<forall>G. extend h F \<squnion> G \<in> X' \<longrightarrow> F \<squnion> project h (C G) G \<in> X"
vec_lambda_cases: fixes x :: "('c, 'b) vec" obtains y :: "'b \<Rightarrow> 'c" where "x = vec_lambda y" and "y \<in> UNIV"
invc2I: fixes t :: "('c \<times> color) tree" assumes "invc t" shows "invc2 t"
k_1: shows "\<forall>b\<ge>k. norm (I b - J) < e"
eq': shows "(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])"
sorted_list_of_set_def: shows "compare_linorder.sorted_list_of_set = folding.F compare_linorder.insort []"
Cons_2: shows "list_all (\<lambda>A. A \<noteq> {}) (A # XS)"
f1: shows "\<forall>j<length l. length (l ! j) = mult.row_length l"
i_Exec_Stream_Init_nth_Suc_calc: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d"   and n :: "nat" shows "i_Exec_Comp_Stream_Init trans_fun input c (Suc n) = trans_fun (input n) (i_Exec_Comp_Stream_Init trans_fun input c n)"
Impl: fixes A :: "triple set"   and P :: "'a \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Q :: "'a \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Ms :: "(cname \<times> mname) set"   and Z :: "'a" assumes "\<forall>Z. A \<union> (\<Union>Z. (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm)) ` Ms) |\<turnstile> (\<lambda>Cm. (P Z Cm, body Cm, Q Z Cm)) ` Ms" shows "A |\<turnstile> (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm)) ` Ms"
word_log2_unfold: fixes w :: "'b word" shows "word_log2 w = (if w = 0 then 0 else Max {n. bit w n})"
add_diff_add: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" shows "a + c - (b + d) = a - b + (c - d)"
mk_eqcl'_nth: fixes xs :: "nat option list"   and T :: "bool list list"   and jj :: "nat"   and l :: "nat"   and ii :: "nat"   and i :: "nat"   and m :: "nat"   and j :: "nat" assumes "\<And>i' j' k. \<lbrakk>i' < length xs; j' < length xs; xs ! i' = Some k\<rbrakk> \<Longrightarrow> (xs ! j' = Some k) = (\<not> tr_lookup T (i' + jj) (j' + jj))"   and "\<And>a b c. \<lbrakk>a \<le> length T; b \<le> length T; c \<le> length T; \<not> tr_lookup T a b; \<not> tr_lookup T b c\<rbrakk> \<Longrightarrow> \<not> tr_lookup T a c"   and "length xs + jj = length T + 1"   and "\<And>x k. \<lbrakk>x \<in> set xs; x = Some k\<rbrakk> \<Longrightarrow> k < l"   and "\<And>i'. \<lbrakk>i' < length xs; \<not> tr_lookup T (i' + jj) ii\<rbrakk> \<Longrightarrow> xs ! i' = None"   and "ii < jj"   and "i < length xs"   and "mk_eqcl' xs ii jj l T ! i = Some m"   and "j < length xs" shows "(mk_eqcl' xs ii jj l T ! j = Some m) = (\<not> tr_lookup T (i + jj) (j + jj))"
singleton: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "semilattice_set f" shows "semilattice_set.F f {x} = x"
simps_1: fixes v :: "'c" shows "in_tree v E = False"
lossless_gpv_lossless_spmfD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "lossless_gpv \<I> gpv" shows "lossless_spmf (the_gpv gpv)"
pair_graphI_graph: fixes G :: "'b pair_pre_digraph" assumes "graph (with_proj G)" shows "pair_graph G"
M_car: shows "(mat_of_rows n G)\<^sup>T \<in> carrier_mat n n"
no_comm_bin_code: fixes u\<^sub>0 :: "'b list"   and u\<^sub>1 :: "'b list"   and us :: "'b list list"   and vs :: "'b list list" assumes "u\<^sub>0 \<cdot> u\<^sub>1 \<noteq> u\<^sub>1 \<cdot> u\<^sub>0"   and "us \<in> lists {u\<^sub>0, u\<^sub>1}"   and "vs \<in> lists {u\<^sub>0, u\<^sub>1}"   and "concat us = concat vs" shows "us = vs"
f5: shows "ucast (get_CWP (cpu_reg_val PSR s2)) = fst (fst (get_curr_win () s2)) \<and> s2 = snd (fst (get_curr_win () s2))"
transform2_rev: fixes P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<forall>\<tau>. \<tau> \<Turnstile> \<delta> P \<and> \<tau> \<Turnstile> \<delta> Q \<and> (\<tau> \<Turnstile> P) = (\<tau> \<Turnstile> Q)" shows "P = Q"
Polygamma_converges': fixes z :: "'a"   and n :: "nat" assumes "z \<noteq> (0::'a)"   and "2 \<le> n" shows "summable (\<lambda>k. inverse ((z + of_nat k) ^ n))"
INT_decseq_offset: fixes F :: "nat \<Rightarrow> 'b set"   and n :: "nat" assumes "decseq F" shows "\<Inter> (range F) = \<Inter> (F ` {n..})"
lunit_in_hom_2: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>B.lunit f : trg\<^sub>B f \<star>\<^sub>B f \<rightarrow>\<^sub>B f\<guillemotright>"
simps_1: fixes x :: "arctic" shows "(MinInfty \<le> x) = True"
sub_def: fixes as :: "'d iarray"   and n :: "nat" shows "as !! n = IArray.list_of as ! n"
Mp_product_modulus: fixes m :: "int"   and m' :: "int"   and k :: "int"   and f :: "int poly" assumes "poly_mod_2 m"   and "m' = m * k"   and "0 < k" shows "poly_mod.Mp m (poly_mod.Mp m' f) = poly_mod.Mp m f"
ik_rcv_map': fixes t :: "('a, 'b) Term.term"   and L :: "('a, 'b) Term.term list" assumes "t \<in> ik\<^sub>s\<^sub>t (map Receive L)" shows "t \<in> set L"
vars_mult: fixes p :: "'b mpoly"   and q :: "'b mpoly" shows "vars (p * q) \<subseteq> vars p \<union> vars q"
left_unique_alt_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "left_unique R = (R OO R\<inverse>\<inverse> \<le> (=))"