hcju/isabelleps huggingface dataset:
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c"   and list :: "'d list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
unfolding_complete: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_call \<Gamma> rs, s\<rangle> \<Rightarrow> t"
epred_iadd1: fixes a :: "enat"   and b :: "enat" assumes "a \<noteq> 0" shows "epred (a + b) = epred a + b"
characteristic_formula_is_characteristic: fixes Q :: "'state"   and P :: "'state" shows "Q \<Turnstile> characteristic_formula P = P \<sim>\<cdot> Q"
wf_finite_less: fixes C :: "'b set" assumes "finite C" shows "wf {(x, y). {x, y} \<subseteq> C \<and> x < y}"
assms: shows "\<forall>M\<in>Chains leq. finite M"
closedin_locally_finite_Union: fixes \<A> :: "'c set set"   and X :: "'c topology" assumes "\<And>S. S \<in> \<A> \<Longrightarrow> closedin X S"   and "locally_finite_in X \<A>" shows "closedin X (\<Union> \<A>)"
igSwapAllIPresIGWlsAllSTR_imp_igSwapAllIPresIGWlsAll: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igSwapAllIPresIGWlsAllSTR MOD" shows "igSwapAllIPresIGWlsAll MOD"
assms_3: fixes w :: "'a list" assumes "w \<in> G" shows "P w"
help'''': fixes ram :: "'c"   and xs :: "'c list"   and b :: "'c list"   and ys :: "'c list" shows "snd (splitAtRec ram xs b) = snd (splitAtRec ram ys b)"
one_is_same: fixes H :: "('c \<Rightarrow> 'a) set" assumes "subgroup H (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)" shows "\<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<lparr>carrier := H\<rparr>\<^esub> = \<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub>"
minus_mod_eq_mult_div: 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 a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (modulo a b) = times b (divide a b)"
sum_list_map_filter: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a" assumes "class.monoid_add plus zero"   and "\<And>x. \<lbrakk>x \<in> set xs; \<not> P x\<rbrakk> \<Longrightarrow> f x = zero" shows "monoid_add.sum_list plus zero (map f (filter P xs)) = monoid_add.sum_list plus zero (map f xs)"
Arg2pi_inverse: fixes z :: "complex" shows "Arg2pi (inverse z) = (if z \<in> \<real> then Arg2pi z else 2 * pi - Arg2pi z)"
char_std_normal_distribution: shows "char std_normal_distribution = (\<lambda>t. complex_of_real (exp (- t\<^sup>2 / 2)))"
scale_left_distrib: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and x :: "'c" assumes "module scale" shows "scale (a + b) x = scale a x + scale b x"
prems_1: shows "A \<subseteq> \<Union> (insert X \<A>)"
tdghm_dghm_comp_NTMap_app: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and x :: "V"   and \<NN> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "x \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>NTMap\<rparr>\<lparr>x\<rparr> = \<NN>\<lparr>NTMap\<rparr>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr>\<rparr>"
b_on_path_1: shows "(x, b) \<in> condensation_edges g P"
not_zero_Hermite_reduce_above: fixes a :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and i :: "'a"   and ass :: "'b \<Rightarrow> 'b"   and n :: "nat"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<not> is_zero_row a A"   and "\<not> is_zero_row i A"   and "echelon_form A"   and "ass_function ass"   and "n \<le> mod_type_class.to_nat i" shows "\<not> is_zero_row a (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)))) n i (LEAST n. A $ i $ n \<noteq> (0::'b)) res)"
f_Exec_Stream_Init_strictly_causal: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "xs \<down> n = ys \<down> n" shows "f_Exec_Comp_Stream_Init trans_fun xs c \<down> Suc n = f_Exec_Comp_Stream_Init trans_fun ys c \<down> Suc n"
remdups_adj_append: fixes xs\<^sub>1 :: "'b list"   and x :: "'b"   and xs\<^sub>2 :: "'b list" shows "remdups_adj (xs\<^sub>1 @ x # xs\<^sub>2) = remdups_adj (xs\<^sub>1 @ [x]) @ tl (remdups_adj (x # xs\<^sub>2))"
finite_hom_components: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "finite (hom_components p)"
ordIso_symmetric: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set" assumes "r =o r'" shows "r' =o r"
denyAllDom: fixes x :: "'a" shows "x \<in> dom D\<^sub>U"
swap_id_eq: fixes a :: "'d"   and b :: "'d"   and x :: "'d" shows "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
Ok_in_err: fixes x :: "'a"   and A :: "'a set" shows "(OK x \<in> err A) = (x \<in> A)"
adjugate_def: fixes A :: "('c, 'd) sq_matrix" shows "adjugate A = Square_Matrix.transpose (cofactor A)"
index_uminus_vec_1: fixes i :: "nat"   and v :: "'b vec" assumes "i < dim_vec v" shows "(- v) $ i = - v $ i"
ex_replicate_mset_if_all_elems_eq: fixes M :: "'a multiset"   and y :: "'a" assumes "\<forall>x\<in>#M. x = y" shows "\<exists>n. M = replicate_mset n y"
finite_emeasure_space: shows "emeasure M (space M) \<noteq> \<top>"
real_abs_dist: fixes x :: "'b"   and y :: "'b" shows "\<bar>dist x y\<bar> = dist x y"
intros: fixes L1 :: "'n \<Rightarrow> 't \<Rightarrow> bool"   and Co1 :: "'o \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'v \<Rightarrow> bool"   and Contra1 :: "'q \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2 :: "'r \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "('n, 'o, 'p, 'q, 'r, 's) U"   and y :: "('t, 'u, 'v, 'w, 'x, 's) U" assumes "rel_G (rel_U L1 Co1 Co2 Contra1 Contra2) L1 Co1 Co2 Contra1 Contra2 (D_U x) (D_U y)" shows "rel_U L1 Co1 Co2 Contra1 Contra2 x y"
reachable_in_verts_2: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "v \<in> verts G"
rSym: fixes P :: "ccs"   and Q :: "ccs" assumes "(P, Q) \<in> X" shows "(Q, P) \<in> X"
new_trace: shows "trace init (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) final"
S_3: fixes i :: "nat" assumes "i < length T" shows "timpls_transformable_to_pred A (T ! i) (S ! i)"
simulation_silent1: fixes s1 :: "'s1"   and s2 :: "'s2"   and s1' :: "'s1" assumes "s1 \<approx> s2"   and "s1 -\<tau>1\<rightarrow> s1'" shows "\<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> s1' \<approx> s2'"
not_eq_order_implies_strict: fixes a :: "'c list"   and b :: "'c list" assumes "a \<noteq> b"   and "suffix a b" shows "strict_suffix a b"
plus_pratt1_def: fixes x :: "pratt1"   and y :: "pratt1" shows "x + y \<equiv> pratt1_max x y"
map_minus_def: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "f -- g = (\<lambda>x. if f x = g x then None else f x)"
less_N: fixes n :: "nat" assumes "n < N" shows "fds_nth h n = (0::'a)"
ex1E: fixes P :: "'e \<Rightarrow> bool" assumes "\<exists>!x. P x" obtains x :: "'e" where "P x" and "\<forall>y. P y \<longrightarrow> y = x"
check_mult_l_mult_err_def: fixes p :: "(char list list \<times> int) list"   and q :: "(char list list \<times> int) list"   and pq :: "(char list list \<times> int) list"   and r :: "(char list list \<times> int) list" shows "check_mult_l_mult_err p q pq r = SPEC (\<lambda>_. True)"
abs_validSet: fixes S :: "msg set" shows "local.abs (S \<inter> valid) = local.abs S"
intros_2: fixes F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula" shows "(F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H \<in> AX0"
assms_2: shows "f \<in>\<^sub>\<circ> A"
proper_interval_list_aux_correct: fixes xs :: "'b list"   and ys :: "'b list" shows "List_Proper_Interval.proper_interval_list_aux xs ys = (\<exists>zs>xs. zs < ys)"
CH_irc_idem: fixes A :: "'a set" shows "Pdds'.CH (Pdds'.CH A) = Pdds'.CH A"
induct: fixes P :: "opinl \<Rightarrow> Unboxed.type option list \<Rightarrow> bool"   and a0 :: "opinl"   and a1 :: "Unboxed.type option list" assumes "P OpAddNum [Some Ubx1, Some Ubx1]"   and "\<And>uv_. P OpMulNum uv_"   and "\<And>uu_. P uu_ []"   and "\<And>uu_ va. P uu_ (None # va)"   and "\<And>uu_ va. P uu_ (Some Ubx2 # va)"   and "\<And>uu_ v. P uu_ [v]"   and "\<And>uu_ v vc. P uu_ (v # None # vc)"   and "\<And>uu_ v vc. P uu_ (v # Some Ubx2 # vc)"   and "\<And>uu_ v vb vd ve. P uu_ (v # vb # vd # ve)" shows "P a0 a1"
monad_left_unit: fixes x :: "'g"   and f :: "'g \<rightarrow> 'e\<cdot>'f" shows "return\<cdot>x \<bind> f = f\<cdot>x"
nonzero_coeffs_nonzero_poly: fixes as :: "'a list" assumes "as \<noteq> []"   and "set as \<noteq> 0" shows "Poly as \<noteq> 0"
pderiv_monom': fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "pderiv (a \<odot>\<^bsub>UP Zp\<^esub> X_poly Zp [^]\<^bsub>UP Zp\<^esub> n) = [n] \<cdot> a \<odot>\<^bsub>UP Zp\<^esub> X_poly Zp [^]\<^bsub>UP Zp\<^esub> (n - 1)"
split_relation_alt: fixes as :: "'b list"   and ls :: "'b list"   and rs :: "'b list"   and i :: "nat" shows "split_relation as (ls, rs) i = (as = ls @ rs \<and> i = length ls)"
simps_4: fixes x :: "'b" shows "norm_instr (Inca.instr.IStore x) = Std.instr.IStore x"
cases: fixes x :: "'a list" obtains xa :: "'a" and xs :: "'a list" where "x = xa # xs"   | "x = []"
poincare_mapsto_def: fixes P :: "'a set"   and X0 :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and S :: "'a set"   and CX :: "'a set"   and Y :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set" shows "Y.poincare_mapsto P X0 S CX Y = (\<forall>(x, d)\<in>X0. Y.returns_to P x \<and> fst ` X0 \<subseteq> S \<and> Y.return_time P differentiable at x within S \<and> (\<exists>D. (Y.poincare_map P has_derivative blinfun_apply D) (at x within S) \<and> (Y.poincare_map P x, D o\<^sub>L d) \<in> Y) \<and> (\<forall>t\<in>{0<..<Y.return_time P x}. Y.flow0 x t \<in> CX))"
harm_remainder_bigo: fixes N :: "nat" assumes "0 < N" shows "harm_remainder N \<in> O(\<lambda>n. 1 / real n ^ (2 * N + 1))"
cycle_def: fixes fs :: "atom list"   and "is" :: "nat list" shows "cycle fs is = ((\<forall>i\<in>set is. i < length fs) \<and> (case map ((!) fs) is of [] \<Rightarrow> False | Less i j # fs' \<Rightarrow> cyclerec i j fs' | Eq nat1 nat2 # fs' \<Rightarrow> False))"
If: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and d :: "com" assumes "C \<turnstile> {\<lambda>z s. P z s \<and> b s} c {Q}"   and "C \<turnstile> {\<lambda>z s. P z s \<and> \<not> b s} d {Q}" shows "C \<turnstile> {P} IF b THEN c ELSE d {Q}"
preserves_cod: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.cod (V.map f) = V.map (V.cod f)"
summable_Suc_iff: fixes f :: "nat \<Rightarrow> 'b" shows "summable (\<lambda>n. f (Suc n)) = summable f"
mat_extension_def: fixes m :: "'b mat" shows "ps_P.mat_extension m = ps_P.tensor_mat m (1\<^sub>m ps_P.d2)"
ok2_nul_nullable: fixes r :: "'b mrexp2" assumes "ok2 r" shows "nul r = nullable (strip2 r)"
hd_filter_in_list: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" assumes "filter P xs \<noteq> []" shows "hd (filter P xs) \<in> set xs"
closure_constraint_mono': fixes g :: "('c \<Rightarrow> real) set"   and Z :: "('c \<Rightarrow> real) set" assumes "Closure\<^sub>\<alpha> g = g \<inter> V" shows "g \<inter> Closure\<^sub>\<alpha> Z \<subseteq> Closure\<^sub>\<alpha> (g \<inter> Z)"
erf_real_less_cancel: fixes a :: "real"   and b :: "real" shows "(erf a < erf b) = (a < b)"
refl: fixes x :: "'a" assumes "x \<in> carrier L" shows "x .= x"
individually_rational_on_def: fixes ds :: "'d set"   and X :: "'x set" shows "individually_rational_on ds X = (CD_on ds X = X \<and> CH X = X)"
Ri_indep: fixes N' :: "'b literal multiset set"   and N :: "'b literal multiset set"   and \<Gamma> :: "'b inference set" assumes "N' \<subseteq> standard_redundancy_criterion.Rf N" shows "standard_redundancy_criterion.Ri \<Gamma> N \<subseteq> standard_redundancy_criterion.Ri \<Gamma> (N - N')"
walk_2: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "Graph.walk G [v, w]"
prems_3: shows "a \<notin> \<O> \<union> all_acquired (x # sb)"
set_minus_finite: fixes Y :: "'a set"   and X :: "'a set" assumes "finite Y" shows "set_minus X Y = Finite_Set.fold (\<lambda>a X. X - {a}) X Y"
simps: fixes a1 :: "'a"   and a2 :: "'b"   and r :: "('b \<times> 'a) set" shows "((a1, a2) \<in> r\<inverse>) = (\<exists>a b. a1 = b \<and> a2 = a \<and> (a, b) \<in> r)"
foldr_assoc: fixes b :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'c list"   and y :: "'c"   and z :: "'c" assumes "\<And>f g h. b (b f g) h = b f (b g h)" shows "foldr b xs (b y z) = b (foldr b xs y) z"
5: shows "row_length (B1 \<circ> B2) = row_length B1"
Rep_JF1_inject: fixes x :: "'d JF1"   and y :: "'d JF1" shows "(Rep_JF1 x = Rep_JF1 y) = (x = y)"
converse_def: shows "converse \<equiv> map_fun set_of_rel rel_of_set Relation.converse"
prod_ord_pairI: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "\<And>k. k \<in> fst ` set xs \<union> fst ` set ys \<Longrightarrow> P k (lookup_pair_tc xs k) (lookup_pair_tc ys k)" shows "prod_ord_pair_tc P xs ys"
cnorm_eq: fixes x :: "'b"   and y :: "'c" shows "(norm x = norm y) = (\<langle>x, x\<rangle> = \<langle>y, y\<rangle>)"
genby_0_closed: fixes S :: "'b set" shows "(0::'b) \<in> \<langle>S\<rangle>"
prod_assn_ctxt: fixes A1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and A2 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and x :: "'a \<times> 'c"   and y :: "'b \<times> 'd"   and z :: "assn" assumes "(A1 \<times>\<^sub>a A2) x y = z" shows "hn_ctxt (A1 \<times>\<^sub>a A2) x y = z"
Lp_Lq_duality': fixes p :: "ennreal"   and q :: "ennreal"   and M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "1 / p + 1 / q = 1"   and "sigma_finite_measure M"   and "f \<in> borel_measurable M" shows "eNorm (\<LL> p M) f = (\<Squnion>g\<in>{g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1}. ennreal (LINT x|M. f x * g x))"
awalk_appendI: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and q :: "'d list"   and w :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "pre_digraph.awalk G v q w" shows "pre_digraph.awalk G u (p @ q) w"
share_augment_release_commute: fixes R :: "nat set"   and L :: "nat set"   and A :: "nat set"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option"   and W :: "nat set" assumes "(R \<union> L \<union> A) \<inter> (all_shared sb \<union> all_unshared sb \<union> all_acquired sb) = {}" shows "share sb \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L = share sb (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
hyps_2: fixes B :: "'a set" assumes "B \<subset> S" shows "P B"
linear_split_alt: fixes xs :: "('b \<times> 'c) list"   and x :: "'c" shows "linear_split xs x = (takeWhile (\<lambda>(uu_, s). s < x) xs, dropWhile (\<lambda>(uu_, s). s < x) xs)"
norm_signed11: fixes xs :: "bit list" shows "norm_signed (\<one> # \<one> # xs) = norm_signed (\<one> # xs)"
simps_1: fixes uu :: "'d"   and u :: "'c \<Rightarrow> 'd" shows "[[]\<rightarrow>uu]u = u"
integrable_mult_left_iff: fixes M :: "'a measure"   and c :: "real"   and f :: "'a \<Rightarrow> real" shows "integrable M (\<lambda>x. c * f x) = (c = 0 \<or> integrable M f)"
coeff_poly_x_mult_y: fixes p :: "'a poly"   and i :: "nat" shows "coeff (poly_x_mult_y p) i = monom (coeff p i) i"
span_subspace: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B"   and "B \<subseteq> vs2.span A"   and "vs2.subspace B" shows "vs2.span A = B"
path_entry_prepend: fixes u :: "nat"   and v :: "nat"   and E :: "(nat \<times> nat) set"   and l :: "nat list" assumes "(u, v) \<in> E"   and "path_entry E l u" shows "path_entry E (u # l) v"
finite_set_singleton_contra: fixes S :: "'a set" assumes "finite S"   and "S \<noteq> {}"   and "1 < card S \<Longrightarrow> False" shows "\<exists>j. S = {j}"
arrow_from_functor_def: fixes D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c"   and e :: "'c" shows "arrow_from_functor D C F x y e \<equiv> (category C \<and> category D) \<and> functor D C F \<and> arrow_from_functor_axioms D C F x y e"
not_eq_order_implies_strict: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<noteq> b"   and "a \<subseteq># b" shows "a \<subset># b"
vfsequence_axioms: shows "vfsequence \<SS>\<^sub>\<int>"
interp_ins_distinct: fixes op_list :: "('c \<times> 'c option) list" assumes "insert_ops op_list" shows "distinct (interp_ins op_list)"
Act_univ: shows "UNIV = {SayIKnow, SayNothing}"
sylvester_sum_mat_upper: fixes i :: "nat"   and q :: "'b poly"   and p :: "'b poly" assumes "i < degree q" shows "(\<Sum>j<degree p + degree q. monom (sylvester_mat p q $$ (i, j)) (degree p + degree q - Suc j)) = monom (1::'b) (degree q - Suc i) * p"
2: shows "sorted_list_of_set (insert a I) ! i = insort a (sorted_list_of_set I) ! i"
sel: fixes x :: "(((common_metadata, common_metadata) blindable\<^sub>m \<times> (participant_metadata, participant_metadata) blindable\<^sub>m) \<times> view\<^sub>m list, (common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>m" shows "the_Transaction\<^sub>m (Transaction\<^sub>m x) = x"
cinv: shows "opnet np p \<Turnstile> (otherwith (=) (net_tree_ips p) inoclosed, U \<rightarrow>) P"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "S.par f f'"   and "L f = L f'" shows "f = f'"
simps_5: fixes w :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "w \<Turnstile>\<^sub>p \<phi> U\<^sub>p \<psi> = (\<exists>i. w [i..] \<Turnstile>\<^sub>p \<psi> \<and> (\<forall>j<i. w [j..] \<Turnstile>\<^sub>p \<phi>))"
dual_inverse: fixes y :: "'a" assumes "y \<in> UNIV" shows "undual (dual y) = y"
prod_3: fixes b :: "'a N"   and a :: "'a N" shows "b l\<rightarrow> b * b \<le> a \<sqinter> (a l\<rightarrow> b) l\<rightarrow> b"
f_Exec_Stream_Acc_Output_Cons: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'f message_af"   and xs :: "'f message_af list"   and c :: "'e" assumes "0 < k" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun (x # xs) c = last_message (map output_fun (f_Exec_Comp_Stream trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c)) # f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs (f_Exec_Comp trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c)"
functor_axioms: shows "functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
orientable_1_2: fixes x :: "'a" shows "orientable_1 x = orientable_2 x"
assms_2: shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
abs_fun_pi_ineq: fixes pi :: "('b \<times> 'b) list"   and a :: "'c"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "cp TYPE('a) TYPE('b) TYPE('c)" shows "pi \<bullet> [a].x = [(pi \<bullet> a)].(pi \<bullet> x)"
left_neutral: fixes a :: "'a" shows "top \<sqinter> a = a"
ntou_uton: fixes x :: "U" assumes "x \<in> range ntou" shows "ntou (uton x) = x"
P1: shows "Bet P A Q \<and> Cong A Q A P"
zeta_real': fixes z :: "complex" assumes "z \<in> \<real>" shows "zeta z \<in> \<real>"
tj_stack_pop_set_leof_rule: fixes u :: "'v"   and tjs :: "'v list"   and \<Psi> :: "'v list \<times> 'v set \<Rightarrow> bool" assumes "\<And>x. case x of (tjs', scc) \<Rightarrow> u \<in> set tjs \<longrightarrow> set tjs = set tjs' \<union> scc \<and> u \<in> scc \<Longrightarrow> \<Psi> x" shows "tj_stack_pop tjs u \<le>\<^sub>n SPEC \<Psi>"
intro: fixes T :: "'b set set" assumes "\<forall>A. \<forall>x\<in>carrier\<^bsub>T\<^esub> - A. A \<subseteq> carrier\<^bsub>T\<^esub> \<and> A closed\<^bsub>T\<^esub> \<longrightarrow> (\<exists>B. \<exists>U\<in>nhds\<^bsub>T\<^esub> x. B open\<^bsub>T\<^esub> \<and> A \<subseteq> B \<and> B \<inter> U = {})" shows "T3_axioms T"
l_remove_child_wf2_axioms: shows "l_remove_child_wf2 type_wf known_ptr known_ptrs remove_child heap_is_wellformed get_child_nodes remove"
vars_monom_single: fixes v :: "nat"   and k :: "nat"   and a :: "'b" shows "vars (monom (Poly_Mapping.single v k) a) \<subseteq> {v}"
prj'_joint_monic: fixes h :: "'a"   and a :: "'a"   and h' :: "'a" assumes "\<guillemotleft>h : a \<rightarrow>\<^sub>C Chn (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>)\<guillemotright>"   and "\<guillemotleft>h' : a \<rightarrow>\<^sub>C Chn (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>)\<guillemotright>"   and "cods.Prj\<^sub>1 \<cdot> h = cods.Prj\<^sub>1 \<cdot> h'"   and "cods.Prj\<^sub>1\<^sub>0 \<cdot> h = cods.Prj\<^sub>1\<^sub>0 \<cdot> h'"   and "cods.Prj\<^sub>0\<^sub>0 \<cdot> h = cods.Prj\<^sub>0\<^sub>0 \<cdot> h'" shows "h = h'"
leB: fixes x :: "'a" assumes "x \<in> cbox a b - S" shows "norm (f x) \<le> B"
silent_move_iff: shows "trsys1.silent_move = (\<lambda>s s'. \<exists>tl. s -1-tl\<rightarrow> s' \<and> \<tau>move1 s tl s')"
h1: shows "minimal A (\<lambda>S. S \<subseteq> carrier V \<and> gen_set S)"
integral_mult_right_zero: fixes M :: "'e measure"   and c :: "'d"   and f :: "'e \<Rightarrow> 'd" shows "LINT x|M. c * f x = c * integral\<^sup>L M f"
Diag_TensorE_5: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "u \<noteq> \<^bold>\<I>"
P'S'Q': shows "(P', S', Q') = Smith_mxn (Mod_Type_Connect.from_hma\<^sub>m A)"
bilinear_continuous_on_compose: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "continuous_on S f"   and "continuous_on S g"   and "bilinear h" shows "continuous_on S (\<lambda>x. h (f x) (g x))"
lowEquivTransitive: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option"   and s3 :: "char list \<Rightarrow> val option" assumes "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"   and "\<Gamma> \<turnstile> s2 \<approx>\<^sub>L s3" shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s3"
is_aligned_iff_dvd_int: fixes ptr :: "'b word"   and n :: "nat" shows "is_aligned ptr n = (2 ^ n dvd uint ptr)"
assms_4: shows "g -` {g w} \<in> sets (F n)"
fx: shows "((\<lambda>x. f x y) has_derivative fx) (at x within X)"
indicator_leI: fixes x :: "'c"   and A :: "'c set"   and y :: "'d"   and B :: "'d set" assumes "x \<in> A \<Longrightarrow> y \<in> B" shows "indicator A x \<le> indicator B y"
SETCompAt: fixes f :: "ZF"   and g :: "ZF"   and x :: "ZF" assumes "f \<approx>>\<^bsub>SET\<^esub> g"   and "x |\<in>| |dom|f" shows "(f ;;\<^bsub>SET\<^esub> g) |@| x = g |@| (f |@| x)"
p: shows "\<forall>p\<in>num. prv (subst \<phi> p zz)"
Diff_wens_constrains: fixes F :: "'b program"   and act :: "('b \<times> 'b) set"   and A :: "'b set" shows "F \<in> wens F act A - A co wens F act A"