hcju/isabelleps huggingface dataset:
scale_sum_right: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "a * sum f A = (\<Sum>x\<in>A. a * f x)"
intros_3: fixes n :: "nat"   and l :: "'b tree23"   and m :: "'b tree23"   and r :: "'b tree23"   and p :: "'b"   and q :: "'b" assumes "full n l"   and "full n m"   and "full n r" shows "full (Suc n) \<langle>l, p, m, q, r\<rangle>"
mult_le_cancel_left_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less zero c" shows "less_eq (times c a) (times c b) = less_eq a b"
inj_on_iff_surj: fixes A :: "'b set"   and A' :: "'c set" assumes "A \<noteq> {}" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> A') = (\<exists>g. g ` A' = A)"
natset: shows "{w \<in> space M. w !! 0} \<in> sets (nat_filtration (Suc 0))"
fun_type: fixes f :: "'fun"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "0 < arity f" shows "\<Gamma> (Fun f T) = Fun f (map \<Gamma> T)"
norm_beta_sound_multi: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N>* \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" shows "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l', [D']\<^bsub>v,n\<^esub>\<rangle> \<and> valid_dbm D'"
linear_compose: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and s3 :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "linear s1 s2 f"   and "linear s2 s3 g" shows "linear s1 s3 (g \<circ> f)"
Spy_see_private_Key: fixes evs :: "event list"   and b :: "bool"   and A :: "agent" assumes "evs \<in> set_pur" shows "(Key (invKey (publicKey b A)) \<in> parts (knows Spy evs)) = (A \<in> bad)"
proj2_Col_distinct_coeff_non_zero: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "proj2_Col p q r"   and "p \<noteq> q"   and "r \<noteq> p"   and "r \<noteq> q" shows "proj2_Col_coeff p q r \<noteq> 0"
Seq\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>0 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>1 :: "nat exp"   and e\<^sub>2 :: "nat exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>"
lc_subst': fixes s :: "(nat \<times> nat list \<times> form) list"   and f :: "form" shows "lc (subst' s f) \<subseteq> lc f \<union> lc_subst s"
sg_inc_set_mult: fixes L :: "'a set"   and H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> L "   and "H \<subseteq> L"   and "K \<subseteq> L" shows "H \<diamondop> K \<subseteq> L"
abc_list_crsp_step: fixes lma :: "nat list"   and lmb :: "nat list"   and aa :: "nat"   and i :: "abc_inst option"   and a :: "nat"   and lma' :: "nat list"   and a' :: "nat"   and lmb' :: "nat list" assumes "abc_list_crsp lma lmb"   and "abc_step_l (aa, lma) i = (a, lma')"   and "abc_step_l (aa, lmb) i = (a', lmb')" shows "a' = a \<and> abc_list_crsp lma' lmb'"
adv: assumes "\<And>\<eta>. exception_\<I> (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) \<turnstile>g absorb (\<A> \<eta>) (obsf_converter (1\<^sub>C |\<^sub>= conv \<eta>)) \<surd>" shows "Negligible.negligible (\<lambda>\<eta>. advantage (absorb (\<A> \<eta>) (obsf_converter (1\<^sub>C |\<^sub>= conv \<eta>))) (obsf_resource (sim \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal_resource \<eta>)) (obsf_resource (real_resource \<eta>)))"
SUP_insert: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and a :: "'c::type"   and A :: "'c::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Sup::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` insert (a::'c::type) (A::'c::type set)) = (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (f a) (Sup (f ` A))"
assms: shows "\<forall>r'\<ge>r. v_hist r' = Map.empty"
is_infinity_def: fixes a :: "('a, 'b) IEEE.float" shows "is_infinity a = (IEEE.exponent a = emax TYPE(('a, 'b) IEEE.float) \<and> fraction a = 0)"
simps: fixes a :: "('g \<times> 'g) set" shows "asym a = (\<exists>R. a = R \<and> (\<forall>x xa. (x, xa) \<in> R \<longrightarrow> (xa, x) \<notin> R))"
list_of_dlist_map: fixes f :: "'c \<Rightarrow> 'b"   and dxs :: "'c dlist" shows "list_of_dlist (DList.map f dxs) = remdups (list.map f (list_of_dlist dxs))"
inR3': fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F, G, H, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> H, F, G, \<Delta> \<down> n"
le20: shows "tensor_mat A m2 \<le>\<^sub>L tensor_mat A B"
ffilter_def: shows "ffilter \<equiv> map_fun id (map_fun fset Abs_fset) Set.filter"
trans_prec_F: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Q :: "'c set"   and entails_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G_q :: "'c \<Rightarrow> 'b inference set"   and Red_I_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b inference set"   and Red_F_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set"   and \<G>_F_q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and \<G>_I_q :: "'c \<Rightarrow> 'a inference \<Rightarrow> 'b inference set option"   and Inf_FL :: "('a \<times> 'd) inference set"   and Equiv_F :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Prec_F :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Prec_L :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and active :: "'d"   and C1 :: "'a"   and C2 :: "'a"   and C3 :: "'a" assumes "given_clause_basis Bot_F Inf_F Bot_G Q entails_q Inf_G_q Red_I_q Red_F_q \<G>_F_q \<G>_I_q Inf_FL Equiv_F Prec_F Prec_L active"   and "Prec_F C1 C2"   and "Prec_F C2 C3" shows "Prec_F C1 C3"
prod_list_minus_1_exp: fixes f :: "'c \<Rightarrow> nat"   and xs :: "'c list" shows "(\<Prod>i\<leftarrow>xs. (- (1::'b)) ^ f i) = (- (1::'b)) ^ sum_list (map f xs)"
that: fixes p :: "'a list"   and y :: "'a" assumes "path \<Gamma> x p y"   and "y \<in> B"   and "distinct (x # p)"   and "\<And>z. z \<in> set p \<Longrightarrow> z \<notin> roofed_gen \<Gamma> B S" shows "thesis"
predCC_eq: fixes S :: "Vars.var set"   and f :: "Arity \<rightarrow> CoCalls"   and a :: "Arity" shows "predCC S f\<cdot>a = (if a \<noteq> 0 then f\<cdot>(pred\<cdot>a) G|` S else S\<^sup>2)"
OclAllInstances_generic_init_empty: fixes pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" assumes "\<And>x. pre_post (x, x) = x" shows "\<tau>\<^sub>0 \<Turnstile> OclAllInstances_generic pre_post H \<triangleq> Set{}"
pos_discrim: shows "0 < b\<^sup>2 - 4 * a * c"
ord_resolve_empty_main_prem: fixes Cs :: "'a literal multiset list"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" shows "\<not> ord_resolve Cs {#} AAs As \<sigma> E"
differentiable_on_compose: fixes g :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and f :: "'b \<Rightarrow> 'c" assumes "g differentiable_on S"   and "f differentiable_on g ` S" shows "(\<lambda>x. f (g x)) differentiable_on S"
differentiable_on_def: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" shows "(f differentiable_on s) = (\<forall>x\<in>s. f differentiable at x within s)"
map_ide_simp: fixes a :: "'a \<times> 'a \<times> 'a" assumes "VVV.ide a" shows "\<alpha>' a = local.inv \<a>[fst a, fst (snd a), snd (snd a)]"
N_zero_iff: fixes q :: "real"   and n :: "int" assumes "fbnorm q" shows "(fbnorm.N q n = 0) = (n = 0)"
assms: fixes f :: "complex \<Rightarrow> 'a"   and g :: "complex \<Rightarrow> 'a" assumes "continuous_on (sphere 0 1) f"   and "f ` sphere 0 1 \<subseteq> S"   and "continuous_on (sphere 0 1) g"   and "g ` sphere 0 1 \<subseteq> S" shows "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g"
FCD2: shows "FCD \<Gamma>2 \<V>2 Tr\<^bsub>ES2\<^esub>"
et_op: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and oper :: "pdf_operator"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile> e : t"   and "op_type oper t = Some t'" shows "\<Gamma> \<turnstile> oper $$ e : t'"
is_sty_many_def: fixes P :: "cld list"   and tys :: "ty list"   and tys' :: "ty list" shows "is_sty_many P tys tys' = (if length tys \<noteq> length tys' then None else case lift_opts (map2 (is_sty_one P) tys tys') of None \<Rightarrow> None | Some bools \<Rightarrow> Some (list_all id bools))"
proj_def: fixes \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "proj \<phi> x \<equiv> {y. \<phi> x y}"
RMWReadOnly: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O>s :: "nat set list"   and i :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool" assumes "\<not> cond (\<theta>(t \<mapsto> m a))"   and "a \<in> \<O> \<or> a \<in> dom \<S>" shows "\<O>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
prefix_fininf_eq_inf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "traces.eq_inf ind w\<^sub>2 w\<^sub>3" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>3"
fminus_partition: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subseteq>| B" shows "A |\<union>| (B |-| A) = B"
inv_translationD_3: fixes r :: "nat set" assumes "inv_translation r" shows "0 \<in> r"
Abs_Node_inj: fixes x :: "(nat \<Rightarrow> 'b + nat) \<times> ('a + nat)"   and y :: "(nat \<Rightarrow> 'b + nat) \<times> ('a + nat)" assumes "Abs_Node x = Abs_Node y"   and "x \<in> Node"   and "y \<in> Node" shows "x = y"
cltn2_transpose_def: fixes A :: "cltn2" shows "cltn2_transpose A \<equiv> cltn2_abs (Finite_Cartesian_Product.transpose (cltn2_rep A))"
simps_2: fixes v :: "nat" shows "index (Suc v) = (if odd (Suc v) then 0 else Suc (index (Suc v div 2)))"
alpha_abs_set_abs_res: fixes as :: "atom set"   and x :: "'b"   and p :: "perm"   and as' :: "atom set"   and x' :: "'b" assumes "(as \<inter> supp x, x) \<approx>set (=) supp p (as' \<inter> supp x', x')" shows "(as, x) \<approx>res (=) supp p (as', x')"
option_assn_alt_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'a option"   and b :: "'b option" shows "option_assn R a b = (case (a, b) of (None, None) \<Rightarrow> emp | (None, Some a) \<Rightarrow> false | (Some x, None) \<Rightarrow> false | (Some x, Some y) \<Rightarrow> R x y)"
simps_3: fixes v :: "'c" shows "cast_Ubx1 (OpUbx2 v) = None"
winning_strategyI: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "\<And>P. vmc_path G P v0 p \<sigma> \<Longrightarrow> winning_path p P" shows "winning_strategy p \<sigma> v0"
span_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale" shows "module.span scale S = module.subspace scale hull S"
C: fixes y :: "'a" assumes "y < xa" shows "hoare (p y) x q"
simps_5: fixes var :: "'c"   and p :: "('a, 'b, 'c) formula" shows "SIGF (Exists var p) = SIGF p"
winding_number_reversepath: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "winding_number (reversepath \<gamma>) z = - winding_number \<gamma> z"
subset_li_is_li: fixes A :: "'a vec set"   and B :: "'a vec set" assumes "lin_indpt A"   and "B \<subseteq> A" shows "lin_indpt B"
sup2I2: fixes B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "B x y" shows "sup A B x y"
Inf_in: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "Inf A \<in> A"
PAC_checker_l'_def: fixes p :: "(char list list \<times> int) list"   and \<V> :: "char list set"   and A :: "(nat, (char list list \<times> int) list) fmap"   and status :: "char list code_status"   and steps :: "((char list list \<times> int) list, char list, nat) pac_step list" shows "PAC_checker_l' p \<V> A status steps = PAC_checker_l p (\<V>, A) status steps"
imp_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.imp \<phi> \<psi> \<equiv> Shallow.dis (Shallow.neg \<phi>) \<psi>"
Tw_eq_ord_isom: fixes E :: "('e, 'f) Order_scheme"   and a :: "'a"   and f :: "'a \<Rightarrow> 'e"   and x :: "'a" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))"   and "a \<in> carrier D"   and "ord_isom (Iod D (segment D a)) (Iod E (segment E ((Tw\<^bsub>D,E\<^esub>) a))) f"   and "x \<in> segment D a" shows "f x = (Tw\<^bsub>D,E\<^esub>) x"
square_integrable_def: fixes f :: "real \<Rightarrow> real"   and S :: "real set" shows "f square_integrable S \<equiv> S \<in> sets lebesgue \<and> f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (\<lambda>x. (f x)\<^sup>2)"
measure_preserving_comp: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'c \<Rightarrow> 'd"   and P :: "'d measure" assumes "f \<in> measure_preserving M N"   and "g \<in> measure_preserving N P" shows "g \<circ> f \<in> measure_preserving M P"
simps: fixes a1 :: "'b"   and a2 :: "'b"   and Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" shows "((a1, a2) \<in> cSim_cs Rel Cal) = ((\<exists>Q Q' P'. a1 = P' \<and> a2 = Q \<and> Q \<longmapsto>Cal* Q' \<and> (P', Q') \<in> Rel) \<or> (\<exists>P P' Q. a1 = P' \<and> a2 = Q \<and> P \<longmapsto>Cal* P' \<and> (Q, P) \<in> Rel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> (P, Q) \<in> cSim_cs Rel Cal \<and> (Q, R) \<in> cSim_cs Rel Cal))"
IH_1: assumes "linears (xs @ ys)" shows "linears xs"
shorter_falsifies\<^sub>l: fixes ds :: "bool list"   and d :: "bool list"   and l :: "fterm literal" assumes "falsifies\<^sub>l (ds @ d) l"   and "nat_of_fatom (get_atom l) < length ds" shows "falsifies\<^sub>l ds l"
upd_left_comm: fixes f :: "'a \<Rightarrow> 'b" shows "comp_fun_commute (\<lambda>a. upd a (f a))"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "GE.wtE \<xi>" shows "Ik.satPB \<xi> \<Phi>"
finter_iff: fixes c :: "'d"   and A :: "'d fset"   and B :: "'d fset" shows "(c |\<in>| A |\<inter>| B) = (c |\<in>| A \<and> c |\<in>| B)"
dense_le: fixes y :: "'a"   and z :: "'a" assumes "\<And>x. x < y \<Longrightarrow> x \<le> z" shows "y \<le> z"
l2_inv6D: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and x :: "l2_state"   and gny :: "msg" assumes "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "in_progress (progress x Ra) xsk"   and "sklvl1.guessed_frame Ra xgny = Some gny"   and "x \<in> l2_inv6" shows "sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))"
eq': shows "{(y, f). y \<in> S \<and> y \<noteq> x \<and> f \<in> derangements (S - {x, y})} = (SIGMA y:S - {x}. derangements (S - {x, y}))"
Gramian_determinant_0: fixes n :: "nat"   and G :: "'a vec list" shows "gram_schmidt.Gramian_determinant n G 0 = (1::'a)"
allow_pfun_def: fixes pf :: "'b \<Rightarrow> 'c option" shows "A\<^sub>p pf \<equiv> \<lambda>x. case pf x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>allow y\<rfloor>"
tanh_real_lt_1: fixes x :: "real" shows "tanh x < 1"
IH: fixes t :: "('fun, 'var) Term.term" assumes "t \<in> ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"   and "t \<notin> ik\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>" shows "intruder_deduct (ik\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>) t"
relations_1: fixes \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and v :: "i" assumes "IsProperInX \<phi>" shows "[\<^bold>\<exists>F. \<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P)) in v]"
single_non_qAFreshAbs_ordLess_var: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and xs :: "'varSort" assumes "qGoodAbs A" shows "|{x. \<not> qAFreshAbs xs x A}| <o |UNIV|"
fls_subdegree_of_nat: fixes n :: "nat" shows "fls_subdegree (of_nat n) = 0"
assms_2: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto>\<^sup>^<(X \<union> weakBisim)> Q"
simps_17: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and l :: "floatarith"   and r :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Powr l r) env = None"
induced_vector_apply2: fixes v :: "'v"   and x :: "('f, 'g) aezfun" assumes "v \<in> V"   and "x \<notin> FH" shows "induced_vector v x = (0::'v)"
g_has_sgnx2: assumes "g tt = 0"   and "0 < sin tt \<or> cos tt = - 1" shows "(g has_sgnx 1) (at_left tt)"
comp_ide_self: fixes a :: "'s" assumes "ide a" shows "a \<cdot> a = a"
iso: shows "reduced_homology_group p (nsphere (Suc n)) \<cong> reduced_homology_group (p - 1) (nsphere n)"
test_bit_eq_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.test_bit bit = bit"
a: shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. aEvalUni (EqUni (a, b, c)) x \<or> aEvalUni (LessUni (a, b, c)) x"
sign_nz: fixes p :: "'a \<Rightarrow> 'a" shows "sign p \<noteq> 0"
mult_stalk_def: shows "stfx.mult_stalk \<equiv> stfx.mult_rel"
d_below: fixes x :: "'a"   and s :: "real" assumes "x \<in> {a--b}"   and "- d \<le> s"   and "s < 0" shows "(flow0 x s - x) \<bullet> rot n < 0"
unique: fixes active :: "'b \<Rightarrow> cnf \<Rightarrow> bool" assumes "singleton active" shows "\<exists>id. \<forall>k id'. active id' k \<longrightarrow> id = id'"
enat_ile: fixes n :: "enat"   and m :: "nat" assumes "n \<le> enat m" shows "\<exists>k. n = enat k"
apex_def: shows "\<nu>'.cod.apex \<equiv> C.dom \<nu>'.cod.leg0"
Q: shows "Q = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (n - 1)) (0\<^sub>m (n - 1) 1) Q' * W'"
gp: fixes a :: "'a fps" assumes "fps_nth a 0 = (0::'a)" shows "Abs_fps (\<lambda>n. 1::'a) oo a = 1 / (1 - a)"
stable_no_blocking_onI: fixes X :: "X4 set"   and ds :: "D2 set" assumes "\<And>h X''. \<lbrakk>X'' = CX4h h (X \<union> X''); X'' \<noteq> CX4h h X; X'' \<subseteq> StableNoDecomp.CD_on ds (X \<union> X'')\<rbrakk> \<Longrightarrow> False" shows "StableNoDecomp.stable_no_blocking_on ds X"
valid_appD_1: fixes \<Gamma> :: "(name \<times> ty) list"   and \<Delta> :: "(name \<times> ty) list" assumes "valid (\<Gamma> @ \<Delta>)" shows "valid \<Gamma>"
intros_1: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set" assumes "(a, b) \<in> r" shows "(a, b) \<in> r\<^sup>+"
gauss_int_code_post_5: shows "Gauss_Int 1 1 = 1 + \<i>\<^sub>\<int>"
K_3: shows "Y \<in> sets (Pi\<^sub>M K M)"
f_nonneg: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a"   and n :: "nat" shows "0 \<le> MC_syntax.f K x y n"
Rep_real_int_cases: fixes y :: "real \<times> real" assumes "y \<in> {r. fst r \<le> snd r}" obtains x :: "real_int" where "y = Rep_real_int x"
simps_21: fixes \<C> :: "t_context"   and ts :: "checker_type" shows "check_single \<C> Return ts = (case return \<C> of None \<Rightarrow> Bot | Some tls \<Rightarrow> type_update ts (to_ct_list tls) (TopType []))"
ex_mset_zip_right: fixes xs :: "'d list"   and ys :: "'e list"   and ys' :: "'e list" assumes "length xs = length ys"   and "mset ys' = mset ys" shows "\<exists>xs'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
assms: shows "\<langle>a, b\<rangle> \<^bold>\<in> HSigma A B"
tr_Delete_constr_iff_aux1: fixes Di :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and M :: "('fun, 'var) Term.term set" assumes "\<forall>d\<in>set Di. (t, s) \<cdot>\<^sub>p \<I> = d \<cdot>\<^sub>p \<I>"   and "\<forall>d\<in>set D - set Di. (t, s) \<cdot>\<^sub>p \<I> \<noteq> d \<cdot>\<^sub>p \<I>" shows "\<lbrakk>M; map (\<lambda>d. \<langle>check: pair (t, s) \<doteq> pair d\<rangle>\<^sub>s\<^sub>t) Di @ map (\<lambda>d. \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>"
functor_axioms: shows "functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R"