hcju/isabelleps huggingface dataset:
takeWhile_cong: fixes l :: "'a list"   and k :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> list.set l \<Longrightarrow> P x = Q x" shows "takeWhile P l = takeWhile Q k"
finite_Psegments_constI: fixes a :: "real"   and b :: "real"   and P :: "real \<Rightarrow> bool"   and c :: "bool" assumes "\<And>t. \<lbrakk>a < t; t < b\<rbrakk> \<Longrightarrow> P t = c" shows "finite_Psegments P a b"
bl_and_aux_bin: fixes n :: "nat"   and v :: "int"   and bs :: "bool list"   and w :: "int"   and cs :: "bool list" shows "map2 (\<and>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = bin_to_bl_aux n (v AND w) (map2 (\<and>) bs cs)"
Pi_I': fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'g"   and B :: "'f \<Rightarrow> 'g set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
triangle_right_implies_left: assumes "(S.UP g \<star>\<^sub>S S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g)) \<cdot>\<^sub>S (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f) \<star>\<^sub>S S.UP g) = S.UP g" shows "(S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) = S.UP f"
blockingpart: shows "(\<Sum>i<length qs. \<Sum>e\<in>set init. ALG e qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ())) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALGxy_det Strat qs init x y)"
\<omega>_limit_point_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b"   and p :: "'b" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.\<omega>_limit_point f X x p = ({0..} \<subseteq> auto_ll_on_open.existence_ivl0 f X x \<and> (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity> \<and> (auto_ll_on_open.flow0 f X x \<circ> s) \<longlonglongrightarrow> p))"
coeff_closed: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P" shows "coeff P p n \<in> carrier R"
Units_inv_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> Units (relative_homology_group (int n) (nsphere n) (equator n))" shows "inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
emeasure_subadditive: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
simps_5: fixes j :: "name"   and i :: "name"   and x :: "tm"   and A :: "fm" assumes "atom j \<sharp> (i, x)" shows "(SyntaxN.Ex j A)(i::=x) = SyntaxN.Ex j (A(i::=x))"
simps_2: fixes a :: "'b"   and b :: "'b" shows "bothNet (DenyAllFromTo a b) = {a, b}"
T_F_spec: fixes t :: "'b event list"   and P :: "'b process" shows "((t, {}) \<in> F P) = (t \<in> T P)"
WhileTrue: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com" assumes "s \<in> b" shows "\<Gamma>\<turnstile> (While b c, Normal s) \<rightarrow> (Seq c (While b c), Normal s)"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
equality: fixes r :: "('a, 'b, 'c) Propagate.configuration_scheme"   and r' :: "('a, 'b, 'c) Propagate.configuration_scheme" assumes "c_work r = c_work r'"   and "c_pts r = c_pts r'"   and "c_imp r = c_imp r'"   and "Propagate.configuration.more r = Propagate.configuration.more r'" shows "r = r'"
NotSwap: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> G" shows "\<^bold>\<not> G \<triangleright> \<Gamma> \<turnstile> F"
edge_antiparallel: fixes x :: "'v"   and y :: "'v" assumes "edge \<Gamma> x y" shows "\<not> edge \<Gamma> y x"
assms: shows "(\<sigma>, NodeS i \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"
simps_3: fixes xs :: "('a, 'b) Combinators list" shows "singleCombinators (DenyAll # xs) = singleCombinators xs"
rel_map_2: fixes Sa :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g list"   and g :: "'i \<Rightarrow> 'h"   and y :: "'i list" shows "list_all2 Sa x (map g y) = list_all2 (\<lambda>x y. Sa x (g y)) x y"
Spair_def: fixes s :: "'a permutation" shows "Spair s \<equiv> SOME fg. fg \<in> fundfoldpairs \<and> s = (case fg of (f, g) \<Rightarrow> Abs_induced_automorph f g)"
Mf: shows "f \<in> M \<rightarrow>\<^sub>M stock_measure BOOL"
sum_pCons_0_commute: fixes f :: "'b \<Rightarrow> 'a poly"   and S :: "'b set" shows "(\<Sum>i\<in>S. pCons (0::'a) (f i)) = pCons (0::'a) (sum f S)"
assms_2: assumes "invariant_1 rai"   and "1 < degree (poly_real_alg_1 rai)" shows "thesis"
s1BC: fixes l :: "hf" assumes "l \<^bold>\<in> succ k1" shows "Builds B C s1 l"
le_lt_convs_4: fixes x :: "'a"   and y :: "'a" shows "(comp x y = Lt) = lt x y"
z_neq_v1_solves: assumes "z \<noteq> v1" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
closed_in_fresh: fixes X :: "tyvrs"   and \<Gamma> :: "binding list"   and T :: "ty" assumes "X \<sharp> ty_dom \<Gamma>"   and "T closed_in \<Gamma>" shows "X \<sharp> T"
PRJ\<^sub>1_def: shows "PRJ\<^sub>1 \<equiv> \<lambda>K H. if cospan K H then \<lbrakk>\<lbrakk>B.prj\<^sub>1 (REP H) (REP K)\<rbrakk>\<rbrakk> else null"
completeness: fixes p :: "'a fm" assumes "\<forall>M s. M, s \<Turnstile> p" shows "\<turnstile> p"
nat_ex_greater_infinite_finite_Max_conv': fixes A :: "nat set"   and n :: "nat" shows "(\<exists>x\<in>A. n < x) = (finite A \<and> A \<noteq> {} \<and> n < Max A \<or> infinite A)"
inf_bot_left: fixes x :: "'a" shows "bot \<sqinter> x = bot"
split_head_repeat_1: fixes list1 :: "cell list"   and j :: "nat"   and list2 :: "cell list" shows "(Oc # list1 = Bk \<up> j @ list2) = (j = 0 \<and> Oc # list1 = list2)"
isomorphic_unit_left: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f" shows "Span.isomorphic (Span.hcomp (Span.trg f) f) f"
minus_diff_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "uminus (minus a b) = minus b a"
keys_extend_indets_subst_aux: fixes t :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (\<Sum>y\<in>keys t. monomial (lookup t y) (Some y)) = Some ` keys t"
real_chop: shows "\<exists>x1 x2. R_Chop(ext v,x1,x2)"
hull_unique: fixes s :: "'b set"   and t :: "'b set"   and S :: "'b set \<Rightarrow> bool" assumes "s \<subseteq> t"   and "S t"   and "\<And>t'. \<lbrakk>s \<subseteq> t'; S t'\<rbrakk> \<Longrightarrow> t \<subseteq> t'" shows "S hull s = t"
keys_ins: fixes k :: "'a"   and p :: "'b"   and t :: "('a \<times> 'b) tree" shows "keys (ins k p t) = insert k (keys t)"
rep_eq: fixes x :: "'c set" shows "rcset ` csets x = {B. B \<subseteq> x \<and> countable B}"
rep_eq: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xb :: "('b, 'd) commute"   and xc :: "('c, 'e) commute" shows "rel_commute x xa xb xc = rel_fun x (rel_fun x xa) (apply_commute xb) (apply_commute xc)"
ordered_resolvent_upon_is_resolvent: fixes P1 :: "'at Literal set"   and A :: "'at"   and P2 :: "'at Literal set" assumes "strictly_maximal_literal P1 (Pos A)"   and "strictly_maximal_literal P2 (Neg A)" shows "ordered_resolvent P1 P2 (resolvent_upon P1 P2 A)"
bit_iff_odd_drop_bit: 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 push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit a n = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (drop_bit n a))"
wtC_def: shows "GE.wtC \<equiv> list_all GE.wtL"
id_dom_cod_1: fixes CC :: "('b, 'c, 'd) category_scheme"   and A :: "'b" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Dom\<^bsub>CC\<^esub> (Id\<^bsub>CC\<^esub> A) = A"
c5: shows "1\<^sub>\<sigma> \<parallel> 1\<^sub>\<sigma> = 1\<^sub>\<sigma>"
aaa: shows "count_list (rev (take i cs) @ [x] @ rev bs @ [x]) z \<le> 1"
af_ltl_continuation: fixes w :: "'a set list"   and w' :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl" shows "(w \<frown> w') \<Turnstile> \<phi> = w' \<Turnstile> af \<phi> w"
simps_1: fixes b :: "'a fps"   and a :: "'a fps" shows "gcompinv b a 0 = fps_nth b 0"
part_equiv_refl_2: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "part_equiv R"   and "(x, y) \<in> R" shows "(y, y) \<in> R"
add_point_def: fixes p :: "'a" shows "del_block_sys.add_point p \<equiv> insert p \<V>"
width_ge: shows "\<forall>ts v. (\<exists>va u. v=va--u \<and> |lan va| = 0 + 1 \<and> True) \<longrightarrow> (\<exists>x. |lan v| = x \<and> 0 < x)"
i_append_Nil: fixes f :: "nat \<Rightarrow> 'b" shows "[] \<frown> f = f"
divides_ff_mult_right: fixes a :: "'b fract"   and x :: "'b fract"   and y :: "'b fract" assumes "a \<noteq> 0"   and "divides_ff (x * inverse a) y" shows "divides_ff x (a * y)"
nsmulextp_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and m :: "'b multiset"   and n :: "'b multiset" shows "nsmulextp f m n = ((m, n) \<in> ns_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)})"
vmc_path_conforms: fixes n :: "nat" assumes "enat (Suc n) < llength P"   and "P $ n \<in> VV p" shows "override_on \<sigma>' \<sigma> (A - W) (P $ n) = P $ Suc n"
new_subc: fixes d :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form"   and c :: "'a" assumes "new_term d s"   and "new d p" shows "new d (subc c s p)"
subfield_iff_2: fixes K :: "'c set" assumes "subfield K S" shows "field (S\<lparr>carrier := K\<rparr>)"
rsp: shows "rel_sum (=) (eq_onp (\<lambda>x. x)) (Inr True) (Inr True)"
vconn_implies_hpar_3: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "src\<^sub>D g = src\<^sub>D f"
trms\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('d, 'e) stateful_strand_step list"   and N :: "('d, 'e) stateful_strand_step list" assumes "set M \<subseteq> set N" shows "trms\<^sub>s\<^sub>s\<^sub>t M \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t N"
xivp: shows "X \<in> Sols (\<lambda>t. f) U S 0 s"
two: shows "x ; 1 \<cdot> (- x \<cdot> 1') \<le> - x"
counterM_def: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "counterM f = range obj"
list_of_dlist_insert: fixes dxs :: "'c set_dlist"   and x :: "'c" assumes "\<not> member dxs x" shows "list_of_dlist (DList_Set.insert x dxs) = x # list_of_dlist dxs"
filter_pdevs_list_def: fixes I :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and X :: "'b pdevs list" shows "filter_pdevs_list I X = map (filter_pdevs (\<lambda>i _. I i (pdevs_applys X i))) X"
strict_boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "ord.max (\<sqsubseteq>) b c \<sqsubset> a" obtains "b \<sqsubset> a" and "c \<sqsubset> a"
is_extensional: fixes f :: "'j" assumes "\<not> \<chi>.J.arr f" shows "\<chi>.A.map f = A.null"
def_scheme: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1" shows "f self a1 \<equiv> \<lambda>\<tau>. SOME res. let res = \<lambda>_. res in if \<tau> \<Turnstile> \<delta> self \<and> \<tau> \<Turnstile> \<upsilon> a1 then \<tau> \<Turnstile> PRE self a1 \<and> \<tau> \<Turnstile> POST self a1 res else \<tau> \<Turnstile> res \<triangleq> invalid"
subring_Ring: fixes S :: "('a, 'e) Ring_scheme" assumes "Subring R S" shows "Ring S"
focus_zero: fixes X :: "'c set" shows "focus X 0 = 0"
ldistinct_conv_lnth: fixes xs :: "'c llist" shows "ldistinct xs = (\<forall>i j. enat i < llength xs \<longrightarrow> enat j < llength xs \<longrightarrow> i \<noteq> j \<longrightarrow> lnth xs i \<noteq> lnth xs j)"
foldr_prs_aux: fixes R1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs1 :: "'c \<Rightarrow> 'd"   and rep1 :: "'d \<Rightarrow> 'c"   and R2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs2 :: "'e \<Rightarrow> 'f"   and rep2 :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and l :: "'d list"   and e :: "'f" assumes "Quotient3 R1 abs1 rep1"   and "Quotient3 R2 abs2 rep2" shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"
free_lift: fixes t :: "dB"   and k :: "nat"   and i :: "nat" shows "free (lift t k) i = (i < k \<and> free t i \<or> k < i \<and> free t (i - 1))"
eval_wlp_PC: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" shows "wlp (a \<^bsub>Q\<^esub>\<oplus> b) P = (\<lambda>s. Q s * wlp a P s + (1 - Q s) * wlp b P s)"
ideal_cont_Rxa: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> I" shows "R \<diamondsuit>\<^sub>p a \<subseteq> I"
divisors_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_no_zero_divisors plus zero times"   and "times a b = zero" shows "a = zero \<or> b = zero"
true_not_false: fixes d :: "'dyn" assumes "is_true d" shows "\<not> is_false d"
sorted_wrt_linorder_of_list: fixes xs :: "'b list" assumes "distinct xs" shows "Linorder_Relations.sorted_wrt (linorder_of_list xs) xs"
ordinal_exp_distrib: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" shows "x ** (y + z) = x ** y * x ** z"
has_bochner_integral_reflect_real_lemma: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and i :: "'a" assumes "has_bochner_integral (lebesgue_on {a..b}) f i" shows "has_bochner_integral (lebesgue_on {- b..- a}) (\<lambda>x. f (- x)) i"
HBL1_axioms: shows "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv enc P"
inv_inv: fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.inv (VoV.inv f) = f"
m_eq: shows "(map_of (kv # l) = m) = (map_of l = m(k := None) \<and> m k = Some v)"
param_fun_comp: fixes Ra :: "('a \<times> 'd) set"   and Rb :: "('b \<times> 'e) set"   and Rc :: "('c \<times> 'f) set" shows "((\<circ>), (\<circ>)) \<in> (Ra \<rightarrow> Rb) \<rightarrow> (Rc \<rightarrow> Ra) \<rightarrow> Rc \<rightarrow> Rb"
emeasure_SUP: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and N :: "'c measure"   and X :: "'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets N"   and "X \<in> sets N"   and "I \<noteq> {}" shows "emeasure (Sup (M ` I)) X = (SUP J\<in>{J. J \<noteq> {} \<and> finite J \<and> J \<subseteq> I}. emeasure (Sup (M ` J)) X)"
SubstFormP_vquot_dbfm: fixes w :: "name"   and V :: "name set"   and V' :: "name set"   and dbfm :: "dbfm" assumes "w \<in> Vs - V"   and "V \<subseteq> Vs"   and "V' = p \<bullet> V"   and "supp dbfm \<subseteq> atom ` Vs" shows "insert (ConstP (F w)) {ConstP (F i) |i. i \<in> V} \<turnstile> SubstFormP \<guillemotleft>Var w\<guillemotright> (F w) (ssubst (vquot_dbfm V dbfm) V F) (subst w (F w) (ssubst (vquot_dbfm (insert w V) dbfm) V F))"
table_left': shows "table (args_n args) (args_L args) X"
norm_eq_sqrt_cinner: fixes x :: "'b" shows "norm x = sqrt (cmod \<langle>x, x\<rangle>)"
card_SigmaI: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A"   and "\<forall>a\<in>A. finite (B a)" shows "card (Sigma A B) = (\<Sum>a\<in>A. card (B a))"
cong: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "abel_semigroup_set.F f z g A = abel_semigroup_set.F f z h B"
arrI: fixes f :: "'a \<times> 'a" assumes "S.arr (fst f) \<and> S.arr (snd f) \<and> S.src (fst f) = S.trg (snd f)" shows "S.VV.arr f"
Entry_target: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and a :: "'a" assumes "CFG sourcenode targetnode valid_edge Entry"   and "valid_edge a"   and "targetnode a = Entry" shows "False"
power_commutes: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and n :: "nat" assumes "class.monoid_mult one times" shows "times (power.power one times a n) a = times a (power.power one times a n)"
gHomcomp: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and H :: "('i, 'j) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and g :: "'g \<Rightarrow> 'i" assumes "Group F"   and "Group G"   and "Group H"   and "f \<in> gHom F G"   and "g \<in> gHom G H" shows "g \<circ>\<^bsub>F\<^esub> f \<in> gHom F H"
lins2D_1: shows "distinct (Inr e # \<pi>)"
ms_closure_add: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and ys :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "ms_closure P xs f basis"   and "ms_closure P ys g basis" shows "ms_closure P (plus_ms_aux xs ys) (\<lambda>x. f x + g x) basis"
map_graph_homo: fixes f :: "('e \<times> 'f) set"   and G :: "('g, 'e) labeled_graph" assumes "univalent f"   and "vertices G = Domain f"   and "graph G" shows "graph_homomorphism G (map_graph f G) f"
shift_bounds_nat_ivl: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "sum g {m + k..<n + k} = (\<Sum>i = m..<n. g (i + k))"
a_in_aug_pm_set: fixes a :: "'e"   and A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e" assumes "a \<in> A" shows "a \<in> aug_pm_set z i A"
distinct_7: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm" shows "PUnit \<noteq> PPair x51 x52"
naturality: fixes f :: "'b" assumes "arr f" shows "\<ll> (cod f) \<cdot> L f = local.map f \<cdot> \<ll> (local.dom f)"
Var: fixes e :: "nat \<Rightarrow> LambdaType.type"   and x :: "nat"   and T :: "LambdaType.type" assumes "e x = T" shows "e \<turnstile>\<^sub>R dB.Var x : T"
pred_is_dep: fixes m :: "('c, 'd) insert_message"   and i :: "'c" shows "P m = \<lbrakk>i\<rbrakk> \<longrightarrow> i \<in> deps (Insert m)"
fgmodule_def: fixes R :: "('f, 'g) Ring_scheme"   and A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" shows "fgmodule R A z i f s = \<lparr>carrier = addition_set f (aug_pm_set z i (s_set R s A)), pop = \<lambda>x\<in>addition_set f (aug_pm_set z i (s_set R s A)). restrict (BOP x f) (addition_set f (aug_pm_set z i (s_set R s A))), mop = restrict (ipp i) (addition_set f (aug_pm_set z i (s_set R s A))), zero = z, sprod = \<lambda>r\<in>carrier R. restrict (SOP r s) (addition_set f (aug_pm_set z i (s_set R s A)))\<rparr>"
new_obj_def: fixes a :: "loc"   and C :: "cname" shows "new_obj a C \<equiv> hupd(a\<mapsto>(C, init_vars (field C)))"
left_totalE: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "left_total R" obtains "\<And>x. \<exists>y. R x y"
coeff_not_in_monomials: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and m :: "'a mpoly" assumes "x \<notin> monomials m" shows "coeff m x = (0::'a)"
WT_resource_of_oracle: fixes s :: "'s_core \<times> 'e" assumes "pred_prod I_core I_rest s" shows "(\<I>_adv_core \<oplus>\<^sub>\<I> \<I>_adv_rest) \<oplus>\<^sub>\<I> (\<I>_usr_core \<oplus>\<^sub>\<I> \<I>_usr_rest) \<turnstile>res RES (fuse rest) s \<surd>"
inject_1: fixes x3 :: "name"   and x1 :: "name"   and x2 :: "pi"   and y3 :: "name"   and y1 :: "name"   and y2 :: "pi" shows "(x3<\<nu>x1> \<prec> x2 = y3<\<nu>y1> \<prec> y2) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
lub_bin: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "lub {x, y} = y"
group_l_invI: assumes "\<And>x. x \<in> carrier S \<Longrightarrow> \<exists>y\<in>carrier S. y \<oplus>\<^bsub>S\<^esub> x = \<zero>\<^bsub>S\<^esub>" shows "Group.group (add_monoid S)"
Tgs_Issues_A: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV" shows "Tgs Issues A with Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> on evs"
Q_def: fixes i :: "V" shows "Q i = (if i = []\<^sub>\<circ> \<Rightarrow> VPow (\<Union>\<^sub>\<circ> X \<times>\<^sub>\<circ> \<Union>\<^sub>\<circ> Y) | i = 1\<^sub>\<nat> \<Rightarrow> X | otherwise \<Rightarrow> Y)"
convex_epigraph: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real" shows "convex (epigraph S f) = (convex_on S f \<and> convex S)"
inject: fixes x3 :: "'c"   and x2 :: "('a, 'b, 'c) psi"   and x1 :: "('a, 'b, 'c) psiCase"   and y3 :: "'c"   and y2 :: "('a, 'b, 'c) psi"   and y1 :: "('a, 'b, 'c) psiCase" shows "(\<box> x3 \<Rightarrow> x2 x1  = \<box> y3 \<Rightarrow> y2 y1 ) = (x3 = y3 \<and> x2 = y2 \<and> x1 = y1)"
cases: fixes x :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" obtains vars :: "(char list \<times> nat option \<times> char list list) list" and init :: "nat list" and goal :: "(nat \<times> nat) list" and ops :: "(char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" where "x = (vars, init, goal, ops)"
at_least_at_most_member: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> carrier L"   and "a \<sqsubseteq>\<^bsub>L\<^esub> x"   and "x \<sqsubseteq>\<^bsub>L\<^esub> b" shows "x \<in> \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>"
simps_1: fixes uu :: "unit" shows "BsSkewBinomialHeapStruc.bs_to_mset (Inl uu) = {#}"
Id_in_AllowedActs: fixes F :: "'b program" shows "Id \<in> AllowedActs F"
differing_vars_elim: fixes x :: "'Var"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat" assumes "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "fst (mems ! i) x \<noteq> mem\<^sub>1 x \<or> snd (mems ! i) x \<noteq> mem\<^sub>2 x"
is_node_ptr_kind_def: fixes ptr :: "(_) object_ptr" shows "is_node_ptr_kind ptr = (cast ptr \<noteq> None)"
Gram_Schmidt_matrix_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "Gram_Schmidt_matrix A = Gram_Schmidt_upt_k A (ncols A - 1)"
gso_norm_beta: fixes j :: "nat" assumes "j < length fs" shows "\<beta> fs j = \<parallel>gso j\<parallel>\<^sup>2"
cases: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f" assumes "free_thread_id ts t" obtains "ts t = None"
wf_listn_termination_rel: shows "wf ({(y, x). x [\<sqsubset>\<^bsub>r\<^esub>] y} <*lex*> (\<lambda>(x, y). (sorted_list_of_set x, sorted_list_of_set y)) ` finite_psubset)"