hcju/isabelleps huggingface dataset:
inj_xxts: fixes x :: "'a" assumes "is_inj x" shows "x ; x\<^sup>T\<^sup>\<star> \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
1: shows "u \<le>\<^sub>F\<^sub>I w"
l7_22: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A1 :: "'a"   and C :: "'a"   and A2 :: "'a"   and B1 :: "'a"   and B2 :: "'a"   and M1 :: "'a"   and M2 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A1 C A2"   and "Bet B1 C B2"   and "Cong C A1 C B1"   and "Cong C A2 C B2"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M1 A1 B1"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M2 A2 B2" shows "Bet M1 C M2"
pderivs_lang_Times_aux1: fixes s :: "'b list"   and r :: "'b rexp" assumes "s \<in> UNIV1" shows "pderivs_lang (PSuf s) r \<subseteq> pderivs_lang UNIV1 r"
Q: shows "Q = (\<lambda>(s, l, r). inc_inv n (as, lm) (s, l, r) ires)"
arev_eq: fixes a :: "'b" assumes "a \<notin> arcs G" shows "arev a = a"
part12_pred_def: fixes V :: "'b set"   and V1_V2 :: "'b set \<times> 'b set" shows "part12_pred V V1_V2 \<equiv> V = fst V1_V2 \<union> snd V1_V2 \<and> fst V1_V2 \<inter> snd V1_V2 = {} \<and> infinite (fst V1_V2) \<and> infinite (snd V1_V2)"
fbind_iff: fixes x :: "'c"   and S :: "'d fset"   and f :: "'d \<Rightarrow> 'c fset" shows "(x |\<in>| fbind S f) = (\<exists>s. x |\<in>| f s \<and> s |\<in>| S)"
ball_UN: fixes B :: "'e \<Rightarrow> 'd set"   and A :: "'e set"   and P :: "'d \<Rightarrow> bool" shows "(\<forall>z\<in>\<Union> (B ` A). P z) = (\<forall>x\<in>A. \<forall>z\<in>B x. P z)"
is_longest_prefix_routing_rule_exclusion: fixes r1 :: "('c, 'd) routing_rule_scheme"   and rn :: "('c, 'd) routing_rule_scheme"   and rss :: "('c, 'd) routing_rule_scheme list" assumes "is_longest_prefix_routing (r1 # rn # rss)" shows "is_longest_prefix_routing (r1 # rss)"
ys: shows "range ys \<subseteq> integral\<^sup>N M ` G \<and> Sup (integral\<^sup>N M ` G) = Sup (range ys)"
linorder_ow_axioms_def: fixes U :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "linorder_ow_axioms U le \<equiv> \<forall>x y. x \<in> U \<longrightarrow> y \<in> U \<longrightarrow> le x y \<or> le y x"
verticesFrom_split: fixes v :: "nat"   and f :: "face" shows "v # tl (verticesFrom f v) = verticesFrom f v"
simps_2: fixes variables_of :: "'c list"   and operators_of :: "'c strips_operator list"   and initial_of :: "'c \<Rightarrow> bool option"   and goal_of :: "'c \<Rightarrow> bool option"   and more :: "'d" shows "\<lparr>strips_problem.variables_of = variables_of, operators_of = operators_of, initial_of = initial_of, goal_of = goal_of, \<dots> = more\<rparr>\<^sub>\<O> = operators_of"
assms: fixes r :: "char list"   and s :: "char list" shows "a r @ s = a (r @ s)"
nodes_sim: fixes a :: "'a"   and b :: "'b" assumes "a \<in> s1.a.V"   and "(a, b) \<in> R1" shows "b \<in> s1.b.V"
foldli_snoc_id: fixes l :: "'a list"   and l0 :: "'a list" shows "foldli l (\<lambda>_. True) (\<lambda>x l. l @ [x]) l0 = l0 @ l"
in_homE: fixes f :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and a :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and b :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] b\<guillemotright>" obtains "J_C.arr f" and "J_C.dom f = a" and "J_C.cod f = b"
BoxE: fixes va :: "i"   and \<phi> :: "\<o>"   and v :: "i" assumes "[\<^bold>\<box>\<phi> in va]" shows "[\<phi> in v]"
asprod_mult: fixes m :: "int"   and n :: "int" shows "m *\<^sub>a ant n = ant (m * n)"
carrier_mat_triv: fixes m :: "'d mat" shows "m \<in> carrier_mat (dim_row m) (dim_col m)"
red2_in_Hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a monoidal_language.term"   and b :: "'a monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a"   and "monoidal_language.Diag C a"   and "monoidal_language.Ide C b"   and "monoidal_language.Diag C b" shows "monoidal_language.red2 C a b \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = monoidal_language.Tensor a b \<and> monoidal_language.Cod C t = monoidal_language.Diagonalize C (monoidal_language.Tensor a b)}"
openin_subtopology_self: fixes S :: "'a set" shows "openin (top_of_set S) S"
simps: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and x :: "'x"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "shift_list (h, U) x ps = (local.punit.monom_mult (1::'b) (monomial 1 x) h, U) # (h, U - {x}) # removeAll (h, U) ps"
full_component: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and i :: "nat"   and t :: "(nat, nat) pp" assumes "class.nat_term rep_nat_term splus"   and "snd (rep_nat_term x) = i" shows "\<exists>y. rep_nat_term y = (t, i)"
length_adjust: fixes p :: "(action \<times> nat) list"   and n :: "nat" shows "length (adjust p n) = length p"
vfsequence_vcard_vcons: fixes x :: "V" shows "vcard (xs' #\<^sub>\<circ> x) = ZFC_in_HOL.succ (vcard xs')"
proj_verts_H_in_G: fixes x :: "'a" assumes "x \<in> verts H" shows "proj_verts_H x \<in> verts G"
dgrad_set_le_trans: fixes d :: "'b \<Rightarrow> nat"   and S :: "'b set"   and T :: "'b set"   and U :: "'b set" assumes "dgrad_set_le d S T"   and "dgrad_set_le d T U" shows "dgrad_set_le d S U"
onllI: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'h, 'f, 'e) seqp"   and p :: "('g, 'h, 'f, 'e) seqp"   and p' :: "('g, 'h, 'f, 'e) seqp"   and P :: "('i \<times> 'e) \<times> 'j \<times> 'i \<times> 'e \<Rightarrow> bool"   and \<xi> :: "'i"   and a :: "'j"   and \<xi>' :: "'i" assumes "\<And>l l'. \<lbrakk>l \<in> labels \<Gamma> p; l' \<in> labels \<Gamma> p'\<rbrakk> \<Longrightarrow> P ((\<xi>, l), a, \<xi>', l')" shows "onll \<Gamma> P ((\<xi>, p), a, \<xi>', p')"
scc_of_empty_conv: fixes u :: "'a" shows "(S.scc_of u = {}) = (u \<notin> pverts (subdivide G e w))"
sgn1_eq_1_sub_sgn2: shows "sgn1 = (\<lambda>x. 1 - sgn2 x)"
induct: fixes P :: "'c list \<Rightarrow> bool"   and list :: "'c list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
three_SPRAY_def: fixes x :: "'a" shows "three_SPRAY x \<equiv> \<exists>S1\<in>\<P>. \<exists>S2\<in>\<P>. \<exists>S3\<in>\<P>. \<exists>S4\<in>\<P>. S1 \<noteq> S2 \<and> S1 \<noteq> S3 \<and> S1 \<noteq> S4 \<and> S2 \<noteq> S3 \<and> S2 \<noteq> S4 \<and> S3 \<noteq> S4 \<and> S1 \<in> SPRAY x \<and> S2 \<in> SPRAY x \<and> S3 \<in> SPRAY x \<and> S4 \<in> SPRAY x \<and> indep_set {S1, S2, S3, S4} \<and> (\<forall>S\<in>SPRAY x. dep_path S {S1, S2, S3, S4} x)"
preserves_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.cod (B.HoHV f) = B.HoHV (B.VVV.cod f)"
nb\<^sub>1: shows "{v \<in> set (\<Pi>\<^sub>\<V>). (\<Pi>\<^sub>I) v \<noteq> None} = set (\<Pi>\<^sub>\<V>)"
showsp_real_def: fixes p :: "nat"   and x :: "real"   and y :: "char list" shows "showsp_real p x y = show_real x @ y"
from_nat_mono': fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "i \<le> j"   and "j < CARD('a)" shows "less_eq (mod_type.from_nat Abs i) (mod_type.from_nat Abs j)"
tendstoD: fixes dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and l :: "'e"   and F :: "'f filter"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "filterlim f (topological_space.nhds open l) F"   and "0 < e" shows "\<forall>\<^sub>F x in F. dist (f x) l < e"
p: shows "p = c - (3::'a) / (8::'a) * b\<^sup>2"
consec_un_equality: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "consec i j \<and> k \<noteq> bot \<longrightarrow> minimum (i \<squnion> j) = minimum k \<and> maximum (i \<squnion> j) = maximum k \<longrightarrow> i \<squnion> j = k"
simps_1: shows "time SKIP = (\<lambda>s. Suc 0)"
divide_self: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "divide a a = one"
power_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "num"   and l :: "num" assumes "class.semiring_numeral one times plus" shows "power.power one times (numeral.numeral one plus k) (numeral l) = numeral.numeral one plus (Num.pow k l)"
assms_3: shows "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"
assms_1: shows "x = y *\<^sub>v\<^sub>m A"
reward_until_def: fixes X :: "'s set" shows "reward_until X = lfp (\<lambda>F s \<omega>. if s \<in> X then 0 else \<rho> s + \<iota> s (shd \<omega>) + F (shd \<omega>) (stl \<omega>))"
midpoint_collinear: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<noteq> c" shows "(b = midpoint a c) = (collinear {a, b, c} \<and> dist a b = dist b c)"
isint_CF: fixes s :: "MIR.num"   and bs :: "real list"   and c :: "int"   and t :: "MIR.num" assumes "isint s bs" shows "isint (CF c t s) bs"
times_sq_mtx_def: shows "(*) \<equiv> map_fun to_vec (map_fun to_vec to_mtx) (**)"
bac_def: shows "bac = b2a\<^sup>2 - c / a"
rowvector_def: fixes v :: "('a, 'b) vec" shows "rowvector v = (\<chi>i j. v $ j)"
simps_3: fixes x1 :: "'b btree"   and x21 :: "'b btree"   and x22 :: "'b"   and x23 :: "'b btree" shows "T\<^sub>i x1 \<noteq> Up\<^sub>i x21 x22 x23"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<sqinter> d y) \<cdot> z = d x \<cdot> z \<sqinter> d y \<cdot> z"
trace_prefix: shows "trace init (take (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) (S t (j - 1))"
affine_hull_finite_intersection_hyperplanes: fixes S :: "'b set" obtains \<F> :: "'b set set" where "finite \<F>" and "int (card \<F>) + aff_dim S = int DIM('b)" and "affine hull S = \<Inter> \<F>" and "\<And>h. h \<in> \<F> \<Longrightarrow> \<exists>a b. a \<noteq> (0::'b) \<and> h = {x. a \<bullet> x = b}"
from_nat_to_nat_id: fixes a :: "'a" shows "Bij_Nat.from_nat (Bij_Nat.to_nat a) = a"
divides_prod_l: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G"   and "a divides b" shows "a divides c \<otimes> b"
assms_1: shows "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"
simps: fixes a :: "'c list"   and A :: "'c set" shows "(a \<in> lists A) = (a = [] \<or> (\<exists>aa l. a = aa # l \<and> aa \<in> A \<and> l \<in> lists A))"
almost_full_on_Sigma: fixes P1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A1 :: "'b set"   and P2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A2 :: "'c set" assumes "almost_full_on P1 A1"   and "almost_full_on P2 A2" shows "almost_full_on (prod_le P1 P2) (A1 \<times> A2)"
orderE: fixes b :: "'a"   and a :: "'a" assumes "b \<preceq> a" obtains "a = ordered_powerprod_lin.max a b"
sub_sharp_homTr: fixes N :: "('i, 'b, 'j) Module_scheme"   and L :: "('k, 'b, 'l) Module_scheme"   and u :: "'a \<Rightarrow> 'i"   and f :: "'k \<Rightarrow> 'a" assumes "R module N"   and "R module L"   and "u \<in> mHom R M N"   and "f \<in> mHom R L M" shows "sub_sharp R L M N u f \<in> mHom R L N"
simps_3: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "fwi m n k i (Suc j) = fw_upd (fwi m n k i j) k i (Suc j)"
ideal_Int_Polys_eq_cone: shows "ideal {f} \<inter> P[X] = cone (f, X)"
new_negative_cycle_aux': fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and xs :: "nat list"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and m :: "nat"   and d :: "'c" assumes "i \<le> n"   and "j \<le> n"   and "set xs \<subseteq> {0..n}"   and "cycle_free M n"   and "length xs = m"   and "len (\<lambda>i' j'. if i' = i \<and> j' = j then Le d else if i' = j \<and> j' = i then Le (- d) else M i' j') i i (j # xs) < \<one> \<or> len (\<lambda>i' j'. if i' = i \<and> j' = j then Le d else if i' = j \<and> j' = i then Le (- d) else M i' j') j j (i # xs) < \<one>"   and "i \<noteq> j" shows "\<exists>xs. set xs \<subseteq> {0..n} \<and> j \<notin> set xs \<and> i \<notin> set xs \<and> (len (\<lambda>i' j'. if i' = i \<and> j' = j then Le d else if i' = j \<and> j' = i then Le (- d) else M i' j') i i (j # xs) < \<one> \<or> len (\<lambda>i' j'. if i' = i \<and> j' = j then Le d else if i' = j \<and> j' = i then Le (- d) else M i' j') j j (i # xs) < \<one>)"
wp_inv_stdD: fixes G :: "'a \<Rightarrow> bool"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and I :: "'a \<Rightarrow> bool" assumes "wp_inv G body \<guillemotleft> I \<guillemotright>"   and "healthy (wp body)" shows "\<guillemotleft> G \<guillemotright> && \<guillemotleft> I \<guillemotright> \<tturnstile> wp body \<guillemotleft> I \<guillemotright>"
dist_commute: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.metric_space dist uniformity open" shows "dist x y = dist y x"
collapse_2: fixes llist :: "'b llist" assumes "\<not> lnull llist" shows "LCons (lhd llist) (ltl llist) = llist"
subgroup_action: fixes H :: "'a set" assumes "subgroup H G" shows "group_action (G\<lparr>carrier := H\<rparr>) (conjugation_action p) (subgroups_of_size p)"
map\<^sub>0_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (G.map\<^sub>0 a) = G.map\<^sub>0 a"
counit_simps_5: shows "trg \<tau> = trg h"
all_coplanar: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" shows "Coplanar A B C D"
half_st_def: shows "half_st = Abs_freelist (pair_relator_halflist s t)"
rcosets_carrier: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and X :: "'c set" assumes "subgroup H G"   and "Group.group G"   and "X \<in> rcosets\<^bsub>G\<^esub> H" shows "X \<subseteq> carrier G"
lappend_LNil2: fixes xs :: "'b llist" shows "xs $ <> = xs"
dirichlet_prod_fds_nth_fds_right: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "dirichlet_prod f (fds_nth (fds g)) = dirichlet_prod f g"
arr_def: fixes f :: "'a \<times> 'a" shows "CC.arr f \<equiv> CC.domains f \<noteq> {} \<or> CC.codomains f \<noteq> {}"
mantissa_noteq_0: fixes f :: "float" assumes "f \<noteq> 0" shows "mantissa f \<noteq> 0"
\<Z>_srng_def: fixes \<alpha> :: "V"   and \<SS> :: "V" shows "\<Z>_srng \<alpha> \<SS> \<equiv> \<Z>_vfsequence \<alpha> \<SS> \<and> \<Z>_srng_axioms \<alpha> \<SS>"
exp_conj_rzero: fixes P :: "'a \<Rightarrow> real" assumes "bounded_by 1 P" shows "P && (\<lambda>s. 0) = (\<lambda>s. 0)"
right_neutral: fixes a :: "'e" shows "a \<cdot> (1::'e) = a"
base_length: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set" assumes "p \<in> sparsegrid' dm" shows "length (base ds p) = dm"
aval_updated: fixes r :: "nat"   and u :: "vname aexp"   and U :: "(nat \<times> vname aexp) list"   and s :: "vname \<Rightarrow> value option"   and c :: "nat \<Rightarrow>f value option" assumes "(r, u) \<in> set U"   and "r \<notin> set (map fst (removeAll (r, u) U))" shows "apply_updates U s c $ r = aval u s"
code: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "iterates f x = LCons x (iterates f (f x))"
dvd_power: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat"   and x :: "'c" assumes "class.comm_semiring_1 times one plus zero"   and "0 < n \<or> x = one" shows "dvd.dvd times x (power.power one times x n)"
pac_step_rel_raw_def: fixes K :: "('c \<times> 'f) set"   and V :: "('a \<times> 'd) set"   and R :: "('b \<times> 'e) set" shows "\<langle>K, V, R\<rangle>pac_step_rel_raw = pac_step_rel_raw K V R"
mset_take_subseteq: fixes n :: "nat"   and xs :: "'b list" shows "mset (take n xs) \<subseteq># mset xs"
assms_6: shows "(_Entry_) -as'\<rightarrow>* (_Low_)"
simps_1: fixes x :: "'a" shows "x \<up> 0 = []"
join_return_pmf: fixes M :: "'c pmf" shows "join_pmf (return_pmf M) = M"
member_remove: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" shows "(x \<in> Set.remove y A) = (x \<in> A \<and> x \<noteq> y)"
phl_n_cond: fixes v :: "'a"   and w :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "\<lbrace>!v \<cdot> !w\<rbrace>x\<lbrace>z\<rbrace>"   and "\<lbrace>!v \<cdot> t w\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>!v\<rbrace>!w \<cdot> x + t w \<cdot> y\<lbrace>z\<rbrace>"
tllist_ord_TCons_TCons: fixes b :: "'f"   and x :: "'g"   and xs :: "('g, 'f) tllist"   and y :: "'g"   and ys :: "('g, 'f) tllist" shows "tllist_ord b (TCons x xs) (TCons y ys) = (x = y \<and> tllist_ord b xs ys)"
PhiWhile_def: fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" shows "PhiWhile b \<Phi> = FIX (PhiWhileOp b \<Phi>)"
dube_eq_2: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and j :: "nat" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F"   and "j < card X" shows "(\<lambda>z. z + int (card X) - int j - 1 gchoose (card X - j - 1)) = (\<lambda>z. z - int (poly_deg f) + int (card X) - int j - 1 gchoose (card X - j - 1) + (z - int (poly_deg f) + int (card X) - int j gchoose (card X - j)) + (z + int (card X) - int j gchoose (card X - j)) - 2 - (\<Sum>i = Suc j..card X. z - int (pm_powerprod.aa X f F i) + int i - int j - 1 gchoose (i - j) + (z - int (pm_powerprod.bb ord X F i) + int i - int j - 1 gchoose (i - j))))"
extends_to_prebicategory_with_homs: shows "prebicategory_with_homs (\<cdot>) (\<star>) \<a> some_src some_trg"
filternew_subseteq_flows_state: fixes \<T> :: "'b stateful_policy" shows "filternew_flows_state \<T> \<subseteq> flows_state \<T>"
pr_simps_1: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "arr (pr0 a b)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext_irrefl_before_trans ext"   and "ext_hd_or_tl ext" shows "ext_wf_bounded ext"
distinct_1: fixes x1 :: "(char list, val, char list \<times> label, char list) edge_kind"   and x2 :: "char list \<times> expr list \<times> char list list" shows "IEdge x1 \<noteq> CEdge x2"
a: shows "\<bar>a\<bar> * \<bar>b\<bar> = pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
factors_closed: fixes fs :: "'a list"   and a :: "'a" assumes "factors G fs a"   and "set fs \<subseteq> carrier G" shows "a \<in> carrier G"
assms_2: shows "b \<in>\<^sub>\<circ> B"
add_left_idem: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "x \<union> (x \<union> y) = x \<union> y"
expand: fixes stream :: "'a stream"   and stream' :: "'a stream" assumes "shd stream = shd stream' \<and> stl stream = stl stream'" shows "stream = stream'"
select_convs_2: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c" shows "dfa.init \<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = init"
set_disconnected_nodes_get_host_is_l_set_disconnected_nodes_get_host: shows "l_set_disconnected_nodes_get_host set_disconnected_nodes_locs get_host_locs"
merge_fun_def: fixes m :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "merge_fun m f g = (if \<forall>x. m (f x) (g x) \<noteq> None then Some (\<lambda>x. the (m (f x) (g x))) else None)"
deg_smult: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
cis_conv_exp: fixes b :: "real" shows "cis b = iexp b"
rel_fun_eq_onp_rel: fixes R :: "'d \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "eq_onp R ===> S = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
elims_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a list"   and xa :: "'a list" assumes "proper_intrvl.proper_interval_set_aux less proper_interval x xa" obtains y :: "'a" and ys :: "'a list" where "x = []" and "xa = y__ # ys__" and "ys__ \<noteq> [] \<or> proper_interval (Some y__) None"   | xb :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = xb # xs" and "xa = y # ys" and "if less xb y then False else if less y xb then proper_interval (Some y) (Some xb) \<or> ys \<noteq> [] \<or> \<not> proper_intrvl.exhaustive_above proper_interval xb xs else proper_intrvl.proper_interval_set_aux less proper_interval xs ys"
r_no_call: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and c :: "char list"   and rs :: "'c rule list"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> Rule MatchAny (Call c) # rs \<Rightarrow>\<^sub>r t"   and "\<Gamma> c = None" shows "False"
lm13: fixes N :: "'c set"   and \<Omega> :: "'d set"   and a :: "('c \<times> 'd set) set" shows "summedBidVectorRel (pseudoAllocation a <| (N \<times> finestpart \<Omega>)) N \<Omega> = summedBidVectorRel (bidMaximizedBy a N \<Omega> Elsee 0) N \<Omega>"
eq_iff: fixes x :: "'c multiset"   and y :: "'c multiset" shows "(x = y) = (x \<subseteq># y \<and> y \<subseteq># x)"
emeasure_eq_measure: fixes A :: "'a stream set" shows "emeasure (stream_space M) A = ennreal (S.prob A)"
nonempty: fixes x :: "'h"   and A :: "'h set" assumes "x \<in> A" shows "A \<noteq> {}"
totatives_subset: fixes n :: "nat" shows "totatives n \<subseteq> {0<..n}"
assms: shows "\<lless>\<pi>, e\<ggreater> I\<rightarrow> \<lless>\<pi>', e'\<ggreater>"
dim_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "dim_vec (vec n f) = n"
adj_sub_geq1_fact: fixes i :: "'d"   and x1 :: "('d, 'e) trm"   and j :: "'f + 'f"   and \<sigma> :: "('d, 'g, 'f, 'h) subst_scheme"   and x2 :: "('d, 'i) trm" assumes "i \<in> SIGT x1"   and "j \<in> (case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Inl i \<in> ({Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}) \<inter> {Inl x |x. x \<in> SIGT x1 \<or> x \<in> SIGT x2}"
\<theta>_1: shows "subterm_inj_on \<theta> (subst_domain \<theta>)"