hcju/isabelleps huggingface dataset:
arcs_H: shows "arcs rev.H = arcs G - {a', rev.a'}"
awalk_to_path_no_neg_cyc_cost: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and f :: "'d \<Rightarrow> real" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "\<nexists>w c. pre_digraph.awalk G w c w \<and> w \<in> set (pre_digraph.awalk_verts G u p) \<and> wf_digraph.awalk_cost f c < 0" shows "wf_digraph.awalk_cost f (wf_digraph.awalk_to_apath G p) \<le> wf_digraph.awalk_cost f p"
history_consistent_appendI: fixes xs :: "'a memref list"   and \<theta> :: "nat \<Rightarrow> nat option"   and ys :: "'a memref list"   and p :: "'a" assumes "\<forall>sop\<in>write_sops xs. valid_sop sop"   and "history_consistent (\<theta> |` (dom \<theta> - read_tmps ys)) p xs"   and "history_consistent \<theta> (last_prog p xs) ys"   and "read_tmps ys \<inter> \<Union> (fst ` write_sops xs) = {}"   and "valid_prog p" shows "history_consistent \<theta> p (xs @ ys)"
trans: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and c :: "'b \<times> 'c" assumes "less_eq_prod b a"   and "less_eq_prod c b" shows "less_eq_prod c a"
simps_4: fixes F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "strip_guards F (Seq c\<^sub>1 c\<^sub>2) = Seq (strip_guards F c\<^sub>1) (strip_guards F c\<^sub>2)"
wp_eval_24: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "wlp (Bind g f) = (\<lambda>P s. wlp (f (g s)) P s)"
one_more_dom: fixes l :: "'a"   and f :: "'a \<Rightarrow> 'b option" assumes "l \<in> dom f" shows "\<exists>f'. f = f'(l \<mapsto> the (f l)) \<and> l \<notin> dom f'"
CompFun_ChiRel: fixes S1 :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and SA :: "('d, 'e, 'f) seqauto"   and S2 :: "'d" assumes "S1 \<in> HAStates A"   and "SA \<in> the (CompFun A S1)"   and "S2 \<in> States SA" shows "(S1, S2) \<in> ChiRel A"
pred_cong: fixes x :: "'b option"   and ya :: "'b option"   and P :: "'b \<Rightarrow> bool"   and Pa :: "'b \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set_option ya \<Longrightarrow> P z = Pa z" shows "pred_option P x = pred_option Pa ya"
cont_eff_ParT_UNlift01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and II :: "nat set"   and P :: "nat \<Rightarrow> nat set set"   and ii :: "nat"   and jj :: "nat"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and s :: "'state"   and t :: "'state" assumes "length cl = length dl"   and "properL cl"   and "properL dl"   and "SbisL cl dl"   and "II \<in> UNpart01 cl dl P - {BrnFT cl dl}"   and "ii \<in> II"   and "jj \<in> UNlift01 cl dl P F II"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n)"   and "\<And>n I i j. \<lbrakk>n < length cl; I \<in> P n; i \<in> I; j \<in> F n I\<rbrakk> \<Longrightarrow> eff (cl ! n) s i \<approx> eff (dl ! n) t j \<and> local.cont (cl ! n) s i \<approx>s local.cont (dl ! n) t j"   and "s \<approx> t" shows "eff (ParT cl) s ii \<approx> eff (ParT dl) t jj \<and> (local.cont (ParT cl) s ii, local.cont (ParT dl) t jj) \<in> thetaZOParT"
\<alpha>_def: shows "\<alpha> = radiant_of (A - center)"
diff_Union_pairwise_disjoint: fixes \<A> :: "'c set set"   and \<B> :: "'c set set" assumes "disjoint \<A>"   and "\<B> \<subseteq> \<A>" shows "\<Union> \<A> - \<Union> \<B> = \<Union> (\<A> - \<B>)"
AA21: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "|~ \<diamond>\<langle>\<circle>F\<rangle>_v \<longrightarrow> \<circle>\<diamond>F"
vu_pos: fixes X :: "'M set" assumes "X \<in> \<D>" shows "0 < prj1 (vf X - uf X)"
zle_diff1_eq: fixes w :: "int"   and z :: "int" shows "(w \<le> z - 1) = (w < z)"
multiplicity_eq_imp_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero"   and "\<And>p. normalization_semidom.prime zero times one normalize p \<Longrightarrow> factorial_semiring.multiplicity times one p x = factorial_semiring.multiplicity times one p y" shows "normalize x = normalize y"
assms_2: assumes "rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2"   and "\<not> rt1 \<approx>\<^bsub>i\<^esub> rt2" shows "P rt1 rt2 i"
assms_2: shows "min_maxsimpchain (x # ys @ [y])"
allDefs_finite: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "finite (allDefs g n)"
simps_9: fixes x :: "nat" shows "extN.N x + extN.Bot = extN.N x"
push_bit_Suc: 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 n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit (Suc n) a = push_bit n (times a (numeral.numeral one plus (num.Bit0 num.One)))"
concurrent_set_ConsE: fixes a :: "'a"   and x :: "'a"   and xs :: "'a list" assumes "concurrent_set a (x # xs)" obtains "concurrent_set a xs" and "x \<parallel> a"
fps_coeff_residues_bigo: fixes A :: "complex set"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "nat \<Rightarrow> complex" assumes "open A"   and "connected A"   and "cball 0 r \<subseteq> A"   and "0 < r"   and "f holomorphic_on A - S"   and "S \<subseteq> ball 0 r"   and "finite S"   and "0 \<notin> S"   and "\<forall>\<^sub>F n in sequentially. g n = - sum (residue (\<lambda>z. f z / z ^ Suc n)) S" shows "(\<lambda>n. (deriv ^^ n) f 0 / fact n - g n) \<in> O(\<lambda>x. complex_of_real (1 / r ^ x))"
assms_2: shows "w \<le>p u \<cdot> [a]"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "(case allow x1 of allow x \<Rightarrow> f1 x | deny x \<Rightarrow> f2 x) = f1 x1"
Mp_ident_iff: fixes f :: "int poly" shows "(Mp f = f) = (\<forall>n. poly.coeff f n \<in> {0..<m})"
phi_top: shows "stone_phi top = Filters.filter.Abs_filter UNIV"
sorted_map_same: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b list \<Rightarrow> 'a"   and xs :: "'b list" shows "sorted (map f (filter (\<lambda>x. f x = g xs) xs))"
hom_compose: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and g :: "'d \<Rightarrow> 'g"   and I :: "('g, 'h) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H I" shows "g \<circ> f \<in> hom G I"
guardedStatEq_1: fixes P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b" assumes "guarded P"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>" shows "\<Psi>\<^sub>P \<simeq> \<one> \<and> supp \<Psi>\<^sub>P = {}"
Un_Int_eq_4: fixes T :: "'k set"   and S :: "'k set" shows "T \<inter> (S \<union> T) = T"
is_empty_empty: fixes t :: "('a, 'b) RBT.rbt" shows "RBT.is_empty t = (t = RBT.empty)"
accept_def: fixes w :: "nat \<Rightarrow> 'L" shows "igb.accept w \<equiv> \<exists>r. igb.is_acc_run r \<and> (\<forall>i. igb.L (r i) (w i))"
process_prop_5_def: fixes P :: "('b list \<times> 'b set) set \<times> 'b list set" shows "process_prop_5 P \<equiv> \<forall>xs x. xs \<in> snd P \<longrightarrow> xs @ [x] \<in> snd P"
unit_SE_def: fixes e :: "'b" shows "unit\<^sub>S\<^sub>E e = (\<lambda>\<sigma>. Some (e, \<sigma>))"
isometry_on_diff: fixes A :: "'b set"   and t :: "'b" shows "isometry_on A ((-) t)"
perm_map: fixes a :: "'c list"   and b :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "a <~~> b" shows "map f a <~~> map f b"
is_cadjoint_def: fixes F :: "'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'c" shows "is_cadjoint F G = (\<forall>x y. \<langle>F x, y\<rangle> = \<langle>x, G y\<rangle>)"
mult_le_cancel_right1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq c (times b c) = ((less zero c \<longrightarrow> less_eq one b) \<and> (less c zero \<longrightarrow> less_eq b one))"
simps_5: fixes f1 :: "'b"   and f2 :: "'b" shows "rec_sort_dir f1 f2 sort_dir.Left = f1"
is_nth_power_nth_power: fixes n :: "nat"   and x :: "'a" shows "is_nth_power n (x ^ n)"
eval\<^sub>l_ground\<^sub>t\<^sub>s: fixes ts :: "fterm list"   and E :: "char list \<Rightarrow> hterm"   and G :: "char list \<Rightarrow> hterm list \<Rightarrow> bool"   and P :: "char list" assumes "ground\<^sub>t\<^sub>s ts" shows "eval\<^sub>l E HFun G (Pos P ts) = G P (hterms_of_fterms ts)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "category C"   and "replete_set_category S"   and "set_valued_functor C S F"   and "set_valued_functor C S G"   and "natural_transformation C S F G \<tau>" shows "set_valued_transformation C S F G \<tau>"
simps_2: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "nat"   and v :: "nat"   and w :: "nat"   and ws :: "nat list" shows "linordered_monoid.len m u v (w # ws) = dbm_add (m u w) (linordered_monoid.len m w v ws)"
is_BaseDim_def: fixes x :: "(int, 'a) dimvec" shows "is_BaseDim x \<equiv> \<exists>i. x = dim_lambda ((\<lambda>x. 0)(i := 1))"
eq_dff_cancel_right: fixes x :: "'b fract"   and y :: "'b fract"   and z :: "'b fract" shows "(x * y =dff x * z) = (x = 0 \<or> y =dff z)"
prob1_def: fixes n :: "nat"   and v :: "complex Matrix.mat"   and i :: "nat" shows "prob1 n v i \<equiv> \<Sum>j\<in>{k |k. select_index n i k}. (cmod (v $$ (j, 0)))\<^sup>2"
inv_inv: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> (inv\<^bsub>G\<^esub> x) = x"
InitObjectReds: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and sh' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C = Object"   and "sh' = sh(C \<mapsto> (sfs, Processing))"   and "P \<turnstile> \<langle>INIT C' (C # Cs,True) \<leftarrow> e,(h, l, sh'),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
bdt_child: fixes llt :: "dag"   and l :: "ref"   and rlt :: "dag"   and p :: "ref"   and lrt :: "dag"   and r :: "ref"   and rrt :: "dag"   and var :: "ref \<Rightarrow> nat"   and bdt1 :: "bdt" assumes "bdt (Node (Node llt l rlt) p (Node lrt r rrt)) var = Some bdt1" shows "\<exists>lbdt rbdt. bdt (Node llt l rlt) var = Some lbdt \<and> bdt (Node lrt r rrt) var = Some rbdt"
continuous_on_s: fixes T :: "real set" shows "continuous_on T s"
chop: shows "v=v1--v' \<and> v'=vn--v2 \<and> (0 < \<parallel>ext v1\<parallel> \<and> len v1 ts c = ext v1 \<and> restrict v1 (res ts) c = lan v1 \<and> |lan v1| = 1) \<and> (\<exists>v u. vn=v--u \<and> |lan v| = 0 + 1 \<and> True) \<and> 0 < \<parallel>ext v2\<parallel> \<and> len v2 ts c = ext v2 \<and> restrict v2 (res ts) c = lan v2 \<and> |lan v2| = 1"
rbl_add_carry_Cons: fixes car :: "bool"   and x :: "bool"   and xs :: "bool list"   and y :: "bool"   and ys :: "bool list" shows "(if car then rbl_succ else id) (rbl_add (x # xs) (y # ys)) = xor3 x y car # (if carry x y car then rbl_succ else id) (rbl_add xs ys)"
card_of_Plus_ordLess_infinite: fixes C :: "'c set"   and A :: "'d set"   and B :: "'e set" assumes "infinite C"   and "|A| <o |C|"   and "|B| <o |C|" shows "|A <+> B| <o |C|"
set_Cons: fixes e :: "'a"   and a :: "'a"   and as :: "'a list" shows "(e \<in> set (a # as)) = (e = a \<or> e \<in> set as)"
Bernstein_Poly_01_span: fixes p :: "nat" shows "poly_vs.span {Bernstein_Poly_01 x p |x. x \<le> p} = {x. degree x \<le> p}"
assms: shows "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) f (\<lambda>x. a)"
ccHeap_simp2: fixes x :: "Vars.var"   and e :: "exp"   and exp :: "exp"   and n :: "Arity" assumes "x \<notin> fv e" shows "ccHeap [(x, e)] exp\<cdot>n = CCfix_nonrec x e\<cdot>(Aexp_syn' n exp, ccExp_syn n exp)"
sturm_firsttwo_signs_aux: fixes p :: "real poly"   and q :: "real poly"   and x\<^sub>0 :: "real" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly q x) = sgn (poly (pderiv p) x)"   and "poly p x\<^sub>0 = 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly (p * q) x) = (if x\<^sub>0 < x then 1 else - 1)"
Field_r_def: shows "Field_r = {\<P>. (\<forall>C\<in>\<P>. - C \<notin> \<P>) \<and> (\<forall>A. finite A \<longrightarrow> A \<in> \<P>)}"
elims_3: fixes x :: "'c arctic_delta"   and xa :: "'c arctic_delta" assumes "\<not> x \<le> xa" obtains uu_ :: "'c" where "x = Num_arc_delta uu_" and "xa = MinInfty_delta"   | y :: "'c" and xb :: "'c" where "x = Num_arc_delta y" and "xa = Num_arc_delta xb" and "\<not> y \<le> xb"
filterlim_at_infinity_imp_norm_at_top: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "filterlim f at_infinity F" shows "LIM x F. norm (f x) :> at_top"
maddux_21: fixes x :: "'a" shows "x \<le> 1 ; x"
Kbase_hom: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "\<forall>j\<le>n. (Kb\<^bsub>K n P\<^esub>) j \<in> carrier K"
can_step_def: fixes l :: "nat"   and s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" shows "A.can_step l s \<equiv> \<exists>s'. alstep l s s'"
iso_assoc': fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.iso \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h]"
finite_accessible_tableaus: fixes s :: "('i, 'a) state" shows "finite (\<T> ` {s'. (s, s') \<in> {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}\<^sup>*})"
InvDomCod_2: fixes f :: "'b" assumes "ciso f" shows "cod Cinv f = dom f"
OV_inv1I: fixes x :: "'a obsv_state_scheme" assumes "\<And>r Q v Q'. \<lbrakk>quorum_for Q v (votes x r); Q' \<in> Quorum\<rbrakk> \<Longrightarrow> quorum_for Q' v (obsv_state.obs x r)" shows "x \<in> OV_inv1"
Collect_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "rel_ext P \<subseteq> rel_ext Q"
U_equiv: fixes v :: "'a" assumes "v \<in> V" shows "(v \<in> U) = (v \<notin> winning_region p**)"
are_rules_3: fixes i :: "'b" shows "graph_rule (reflexivity_rule i)"
interval_integrable_continuous_on: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "continuous_on {a..b} f" shows "interval_lebesgue_integrable lborel (ereal a) (ereal b) f"
DERIV_continuous: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c"   and s :: "'c set" assumes "(f has_field_derivative D) (at x within s)" shows "continuous (at x within s) f"
l2_inv8I: fixes x :: "l2_state" assumes "secret x \<subseteq> {Exp (Exp Gen (NonceF (R $ N))) (NonceF (R' $ N')) |N N' R R'. R = test \<and> R' \<in> partners}" shows "x \<in> l2_inv8"
incl_Rel_vfsequence: fixes A :: "V"   and B :: "V" shows "vfsequence (incl_Rel A B)"
g: shows "g = (+) (- a) \<circ> ((+) a \<circ> g)"
integrable_product_swap: fixes f :: "'b \<times> 'a \<Rightarrow> 'c" assumes "integrable (M2 \<Otimes>\<^sub>M M1) f" shows "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f (y, x))"
monom_1_dvd_iff': fixes n :: "nat"   and p :: "'a poly" shows "(monom (1::'a) n dvd p) = (\<forall>k<n. coeff p k = (0::'a))"
corthogonal_def: fixes vs :: "'b vec list" shows "corthogonal vs \<equiv> \<forall>i<length vs. \<forall>j<length vs. (vs ! i \<bullet>c vs ! j = (0::'b)) = (i \<noteq> j)"
assms: shows "\<forall>\<^sub>F x in sequentially. norm (f x) = norm (g x)"
deg_f2: fixes \<sigma> :: "nat \<Rightarrow> nat" assumes "\<sigma> permutes {0..<m + n}"   and "\<sigma> \<noteq> \<pi>" shows "degree (f \<sigma>) < m * n"
h_def: fixes a :: "nat"   and b :: "nat"   and c :: "nat" shows "h a b c = comp_by_index index_of_id b"
tendsto_lowerbound: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b"   and F :: "'c filter"   and a :: "'b" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq a (f i)"   and "F \<noteq> bot" shows "less_eq a x"
intros_2: fixes KS :: "key list"   and X :: "msg"   and H :: "msg set" assumes "foldr Crypt KS (Hash X) \<in> crypts H" shows "foldr Crypt KS X \<in> crypts H"
f: shows "distributed M lborel (\<lambda>x. (X x - t) / c) (\<lambda>x. ennreal \<bar>c\<bar> * f (x * c + t))"
Cauchy_theorem_homotopic: fixes atends :: "bool"   and S :: "complex set"   and g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "if atends then homotopic_paths S g h else homotopic_loops S g h"   and "open S"   and "f holomorphic_on S"   and "valid_path g"   and "valid_path h" shows "contour_integral g f = contour_integral h f"
hta_remap_def: fixes f :: "'c \<Rightarrow> 'a"   and H :: "('c, 'b) hashedTa" shows "hta_remap f H \<equiv> init_hta (hh_set_xy.g_image f (hta_Qi H)) (ll_set_xy.g_image (remap_rule f) (hta_\<delta> H))"
\<beta>: shows "\<beta> < \<omega> \<up> \<alpha>2"
AxiomC: fixes t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term set" assumes "t \<in> M" shows "M \<turnstile>\<^sub>c t"
eq_\<I>_comp_cong: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>2 :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and conv1' :: "('i, 'j, 'k, 'l) converter"   and \<I>3 :: "('m, 'n) \<I>"   and conv2 :: "('k, 'l, 'm, 'n) converter"   and conv2' :: "('k, 'l, 'm, 'n) converter" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>3 \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1,\<I>3 \<turnstile>\<^sub>C conv1 \<odot> conv2 \<sim> conv1' \<odot> conv2'"
isSimplePath_singelton: fixes u :: "nat"   and e :: "nat \<times> nat"   and v :: "nat" shows "isSimplePath u [e] v = (e = (u, v) \<and> u \<noteq> v \<and> (u, v) \<in> E)"
bounded_iff: fixes a :: "'g"   and b :: "'g"   and c :: "'g" shows "(a \<le> inf b c) = (a \<le> b \<and> a \<le> c)"
sorted_list_of_multiset_mset: fixes xs :: "'b list" shows "sorted_list_of_multiset (mset xs) = sort xs"
and_mask_dvd: fixes n :: "nat"   and w :: "'b word" shows "(2 ^ n dvd uint w) = (w AND mask n = 0)"
lits_of_cnf_finite: fixes cnf :: "lit uprod set" shows "finite (lits_of_cnf cnf) = finite (vars_of_cnf cnf)"
map_preserves_cod: fixes f :: "'c" assumes "Cop.arr f" shows "local.map (Cop.cod f) = S.cod (local.map f)"
lfp_trans_def: fixes T :: "(('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "lfp_trans T = Inf_trans {t. (\<forall>P. sound P \<longrightarrow> sound (t P)) \<and> le_trans (T t) t}"
restrict_chart_in_atlas: fixes c :: "('a, 'b) chart"   and S :: "'a set" assumes "c \<in> atlas" shows "restrict_chart S c \<in> atlas"
intvs_incdec_3: fixes l :: "int"   and h :: "int" shows "{l..<h - 1} = {l..<h} - {h - 1}"
assms_2: shows "U \<in> carrier_mat (dim_col A) (dim_col A)"
diff: fixes x :: "complex"   and y :: "complex"   and xa :: "'a" shows "(x - y) *\<^sub>C xa = x *\<^sub>C xa - y *\<^sub>C xa"
cut_le_Int_right: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>\<le> t = A \<inter> B \<down>\<le> t"
card_image: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "card (f ` A) = card A"
sub1_preserves_dimensions_1: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "sub1 mf q k l (r, A) = (r', A')" shows "dim_row A' = dim_row A"
af_G_letter_abs_opt_split: fixes \<Phi> :: "'b ltl_prop_equiv_quotient"   and \<nu> :: "'b set" shows "\<up>Unf\<^sub>G (\<up>step \<Phi> \<nu>) = \<up>af\<^sub>G\<^sub>\<UU> \<Phi> \<nu>"
case_prod_conv: fixes f :: "'n \<Rightarrow> 'o \<Rightarrow> 'm"   and a :: "'n"   and b :: "'o" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
ent_disjI1_direct: fixes A :: "assn"   and B :: "assn" shows "A \<Longrightarrow>\<^sub>A A \<or>\<^sub>A B"
mrMTNone: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" shows "applied_rule_rev Cp x [] = \<bottom>"
simps_3: fixes r :: "'form" shows "inPorts (Conclusion r) = {|plain_ant r|}"
\<alpha>edges_subset: fixes g :: "'m" assumes "invar g" shows "\<alpha>edges_aux g \<subseteq> \<alpha>nodes_aux g \<times> \<alpha>nodes_aux g"
HSem_def: fixes ESem :: "'f \<Rightarrow> ('d \<Rightarrow> 'e) \<rightarrow> 'e"   and \<Gamma> :: "('d \<times> 'f) list" shows "has_ESem.HSem ESem \<Gamma> = (\<Lambda> \<rho>. \<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> evalHeap \<Gamma> (\<lambda>e. ESem e\<cdot>x))"
sqrt_Max_power2_eq_max_abs: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "finite A"   and "A \<noteq> {}" shows "sqrt (Max {(f i)\<^sup>2 |i. i \<in> A}) = Max {\<bar>f i\<bar> |i. i \<in> A}"
maximal_ltl: fixes G :: "('c, 'd) Graph_scheme"   and P :: "'c llist" assumes "Digraph G"   and "Digraph.maximal_path G P" shows "Digraph.maximal_path G (ltl P)"
IH: assumes "range_vars \<delta> \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> \<circ>\<^sub>s \<theta> = (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>"
scale_right_imp_eq: fixes x :: "'c"   and a :: "'c"   and b :: "'c" assumes "x \<noteq> (0::'c)"   and "a * x = b * x" shows "a = b"
huets_us_simps_1: fixes y :: "nat list" shows "\<^bold>u y 0 = inc y 0 (zeroes (length y))"
groupF_tuple: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "groupF f xs = map (map fst) (groupF snd (map (\<lambda>x. (x, f x)) xs))"