hcju/isabelleps huggingface dataset:
mpoly_to_poly_add: fixes v :: "nat"   and p1 :: "'b mpoly"   and p2 :: "'b mpoly" shows "mpoly_to_poly v (p1 + p2) = mpoly_to_poly v p1 + mpoly_to_poly v p2"
index_add_mat_3: fixes A :: "'d mat"   and B :: "'d mat" shows "dim_col (A + B) = dim_col B"
B_def: shows "B \<equiv> {b \<in> A. bound X (\<lambda>x y. y \<sqsubseteq> x) b}"
normalize_power: 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 a :: "'a"   and n :: "nat" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize" shows "normalize (power.power one times a n) = power.power one times (normalize a) n"
indmhom_injec: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "injec\<^bsub>M /\<^sub>m (ker\<^bsub>M,N\<^esub> f),N\<^esub> f\<^sup>\<flat>\<^bsub>R M, N\<^esub>"
symmetric_reachable: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c"   and w :: "'c" assumes "symmetric G"   and "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w" shows "w \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
assms_3: assumes "\<not> stable_no_blocking_on ds X" shows "P"
sees_method_compP: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and f :: "'a \<Rightarrow> 'b" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "compP f P \<turnstile> C sees M: Ts\<rightarrow>T = f m in D"
zassenhaus_reconstruction_generic: fixes p :: "int"   and f :: "int poly"   and n :: "nat"   and hs :: "int poly list"   and sl_impl :: "(int poly, int \<times> int poly list, 'a) subseqs_foldr_impl"   and sli :: "int \<times> int poly list \<Rightarrow> int poly list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> bool"   and fs :: "int poly list" assumes "prime p"   and "comm_monoid_mult_class.coprime (lead_coeff f) p"   and "poly_mod.square_free_m p f"   and "0 < degree f"   and "berlekamp_hensel p n f = hs"   and "2 * \<bar>lead_coeff f\<bar> * factor_bound f (degree_bound hs) < p ^ n"   and "correct_subseqs_foldr_impl (\<lambda>v. map_prod (poly_mod.mul_const (p ^ n) v) ((#) v)) sl_impl sli"   and "zassenhaus_reconstruction_generic sl_impl hs p n f = fs" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
less_vec_sum_list_less: fixes v :: "nat list"   and w :: "nat list" assumes "v <\<^sub>v w" shows "sum_list v < sum_list w"
sym_symcl: fixes R :: "('b \<times> 'b) set" shows "sym (symcl R)"
D': shows "[D']\<^bsub>v,n\<^esub> = {u. u \<turnstile> inv_of A l'}"
singleton_well_ordered: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "less_eq a a" shows "well_ordered_set {a} less_eq"
vec_of_scal_dim_2: fixes x :: "'b" shows "vec_of_scal x \<in> carrier_vec 1"
bc_epsilon: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>"   and "\<epsilon> < 1" shows "\<forall>\<^sub>F n in sequentially. (real_of_int (c (n + 1)) - \<epsilon>) / real_of_int (c n) < real (nth_prime (n + 1)) / real (nth_prime n)"
gt_diff: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t >\<^sub>h\<^sub>d head s"   and "chkvar t s"   and "chkargs (>\<^sub>t) t s" shows "t >\<^sub>t s"
exec: shows "exec_meth ci P ins xt t h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"
map_poly_comm_ring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_comm_ring_hom hom \<equiv> comm_ring_hom hom"
frameImpEmpty: fixes \<Psi> :: "'b"   and \<phi> :: "'c" shows "\<langle>\<epsilon>, \<Psi>\<rangle> \<turnstile>\<^sub>F \<phi> = \<Psi> \<turnstile> \<phi>"
open_map_imp_subset_topspace: fixes X1 :: "'c topology"   and X2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "open_map X1 X2 f" shows "f ` topspace X1 \<subseteq> topspace X2"
nat_pred_predicate_change_exists2: fixes a :: "nat"   and b :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "\<not> P b"   and "P a" shows "\<exists>n>a. n \<le> b \<and> \<not> P n \<and> (\<forall>k\<ge>a. k < n \<longrightarrow> P k)"
mask_twice: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" shows "(x AND mask n) AND mask m = x AND mask (min m n)"
nonneg_Reals_cases: fixes x :: "'a" assumes "x \<in> \<real>\<^sub>\<ge>\<^sub>0" obtains r :: "real" where "x = of_real r" and "0 \<le> r"
dfs_invariant': fixes succs :: "'a \<Rightarrow> 'a list"   and is_node :: "'a \<Rightarrow> bool"   and invariant :: "'b \<Rightarrow> bool"   and ins :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and memb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and empt :: "'b"   and S :: "'b"   and xs :: "'a list" assumes "DFS succs is_node invariant ins memb empt"   and "invariant S"   and "list_all is_node xs" shows "invariant (gen_dfs succs ins memb S xs)"
closed_eucl_atLeastAtMost: fixes a :: "'b"   and b :: "'b" shows "closed {a..b}"
simps_2: fixes x :: "'b"   and xs :: "['b]" shows "eq\<cdot>(x : xs)\<cdot>[] = FF"
shift_cons_preserves_T: fixes \<tau> :: "ty"   and \<rho> :: "val set list"   and b :: "val set" shows "T \<tau> \<rho> = T (shift (Suc 0) 0 \<tau>) (b # \<rho>)"
vsv_vrange_vsubset: fixes A :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_inverse \<Longrightarrow> x\<inverse>\<^sub>\<real> \<in>\<^sub>\<circ> A" shows "\<R>\<^sub>\<circ> vreal_inverse \<subseteq>\<^sub>\<circ> A"
ideal_G: shows "ideal (punit.reduced_GB (monomial (1::'a) ` S)) = ideal (monomial (1::'a) ` S)"
linordered_field_no_lb: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less y x"
iTILL_iT_Div_Int: fixes k :: "nat"   and A :: "nat set"   and n :: "nat" assumes "0 < k"   and "\<forall>x\<in>A. x div k \<le> n div k \<longrightarrow> x \<le> n" shows "A \<inter> [\<dots>n] \<oslash> k = (A \<oslash> k) \<inter> ([\<dots>n] \<oslash> k)"
poly_roots_finite: fixes p :: "'a list" shows "(poly p \<noteq> poly []) = finite {x. poly p x = (0::'a)}"
ordIso_iff_ordLeq: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set" shows "(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
unitary11_gen_div_sm: fixes k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "k \<noteq> 0"   and "unitary11_gen (k *\<^sub>s\<^sub>m M)" shows "unitary11_gen M"
arc_to_ends_eq: shows "arc_to_ends H = arc_to_ends G"
right_add_zero_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A + 0\<^sub>m nr nc = A"
weight_gpv_def: fixes \<I> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv" shows "weight_gpv \<I> gpv = enn2real (expectation_gpv 0 \<I> (\<lambda>_. 1) gpv)"
bin_rsplit_len_indep: fixes n :: "nat"   and nw :: "nat"   and v :: "int"   and w :: "int" assumes "n \<noteq> 0" shows "length (bin_rsplit n (nw, v)) = length (bin_rsplit n (nw, w))"
the1_list: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>!x. x \<in> set xs \<and> P x" shows "(THE x. x \<in> set xs \<and> P x) = hd (filter P xs)"
prime_product_factor_def: fixes n :: "nat" shows "prime_product_factor n = (case sqrt_nat n of [] \<Rightarrow> prime_product_factor_main 1 1 (nat (root_nat_floor 3 n)) n 2 | s # x \<Rightarrow> (s, 1))"
strict_domD: fixes i :: "nat"   and j :: "nat"   and a :: "nat list" assumes "strict_dom i j"   and "dom_kildall start ! j = a" shows "i \<in> set a"
FunctorMapsTo: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and f :: "'f" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## f maps\<^bsub>CatCod F\<^esub> F @@ dom\<^bsub>CatDom F\<^esub> f to F @@ cod\<^bsub>CatDom F\<^esub> f"
first_dilated_instant: fixes f :: "nat \<Rightarrow> nat" assumes "strict_mono f"   and "f 0 = 0" shows "Max {i. f i \<le> 0} = 0"
zone_delay_mono: fixes A :: "('d \<Rightarrow> 'e) set"   and B :: "('d \<Rightarrow> 'e) set" assumes "A \<subseteq> B" shows "A\<^sup>\<up> \<subseteq> B\<^sup>\<up>"
set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes uu :: "(_) character_data_ptr"   and uv :: "(_) node_ptr list" shows "set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r uu uv = error HierarchyRequestError"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.CC.arr f" shows "T\<^sub>DoFF.map f = D.null"
HAStates_CompFun_SAs_mem: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and T :: "('a, 'b, 'c) seqauto" assumes "S \<in> HAStates A"   and "T \<in> the (CompFun A S)" shows "T \<in> SAs A"
domain_eq: shows "Domainp (pcr_fmap (=) (=)) = (\<lambda>m. finite (dom m))"
simps_9: fixes x11 :: "'a"   and x12 :: "'b"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x11 \<leftarrow> x12 \<noteq> x31@[x32]"
multiplicity_same_power: fixes p :: "'a"   and n :: "nat" assumes "p \<noteq> (0::'a)"   and "\<not> is_unit p" shows "multiplicity p (p ^ n) = n"
T1_3: fixes w :: "i"   and F :: "\<Pi>\<^sub>3"   and x :: "\<kappa>"   and y :: "\<kappa>"   and z :: "\<kappa>" shows "[\<lparr>F,x,y,z\<rparr> in w] = (\<exists>r o\<^sub>1 o\<^sub>2 o\<^sub>3. Some r = d\<^sub>3 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> Some o\<^sub>3 = d\<^sub>\<kappa> z \<and> (o\<^sub>1, o\<^sub>2, o\<^sub>3) \<in> ex3 r w)"
mono_on_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "Fun.mono_on f A"   and "B \<subseteq> A" shows "Fun.mono_on f B"
aux_lemma_lose_3a: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat"   and ka :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"   and "ka \<le> Suc d" shows "lose (d + (t + (3 + k)) + ka) = [False]"
extends_subst_cong_term: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and t :: "('b, 'a) Term.term" assumes "extends_subst \<sigma> \<tau>"   and "vars_term t \<subseteq> dom \<sigma>" shows "subst_atm_abbrev t (subst_of_map Var \<sigma>) = subst_atm_abbrev t (subst_of_map Var \<tau>)"
measurable_distr_eq1: fixes Mf :: "'f measure"   and Nf :: "'d measure"   and f :: "'f \<Rightarrow> 'd"   and Mf' :: "'e measure" shows "distr Mf Nf f \<rightarrow>\<^sub>M Mf' = Nf \<rightarrow>\<^sub>M Mf'"
Dom_dom: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f" shows "J_C.Dom (J_C.dom f) = J_C.Dom f"
simps: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and defbot :: "'c"   and receiver_violation :: "bool"   and nP :: "'b \<Rightarrow> 'c" shows "new_configured_SecurityInvariant (sinvar, defbot, receiver_violation, nP) = (if SecurityInvariant sinvar defbot receiver_violation then Some \<lparr>c_sinvar = \<lambda>G. sinvar G nP, c_offending_flows = \<lambda>G. SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP, c_isIFS = receiver_violation\<rparr> else None)"
all_s: shows "\<forall>s. s \<le>s w \<and> Lyndon s \<longrightarrow> \<^bold>|s\<^bold>| \<le> \<^bold>|q\<^bold>|"
wf_no_direct_calls: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp" assumes "\<And>pn. \<forall>pn'. call(pn') \<notin> stermsl (\<Gamma> pn)" shows "wellformed \<Gamma>"
tdghm_NTDGCod: shows "\<NN>\<lparr>NTDGCod\<rparr> = \<CC>"
coeff_code: fixes lp :: "linear_poly" shows "coeff lp = get_var_coeff (linear_poly_map lp)"
IH: fixes gpv :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv' gpv \<le> max c fail"
assms_2: shows "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
for_rec3'_for_rec: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat" shows "for_rec3' f a n n n n = for_rec (\<lambda>a k. for_rec (\<lambda>a i. for_rec (\<lambda>a. f a k i) a n) a n) a n"
cont_def: shows "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * of_int (zorder f p) * h p) (circlepath p e)"
snd_comp_apfst: fixes f :: "'c \<Rightarrow> 'e" shows "snd \<circ> apfst f = snd"
tab_simps_2: shows "src \<sigma> = src s\<^sub>0"
T_BIT_ub: fixes n :: "nat" assumes "\<forall>i<n. qs ! i \<in> set init" shows "T_BIT n \<le> real (n * length init)"
valid_stmt_NormalI: fixes A :: "'b triple set"   and G :: "prog"   and c :: "stmt"   and P :: "'b assn"   and Q :: "'b assn" assumes "\<And>n s0 L accC C s1 Y Z. \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G, L); normal s0; \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>; \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>In1r c\<guillemotright> C; G\<turnstile>s0 \<midarrow>c\<midarrow>n\<rightarrow> s1; Normal P Y s0 Z\<rbrakk> \<Longrightarrow> Q \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G, L)" shows "G,A|\<Turnstile>\<Colon>{{Normal P} .c. {Q}}"
compact_eq_closed: fixes S :: "'b set" shows "compact S = closed S"
Z_good_fun_def: shows "Z_good_fun \<equiv> {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n} \<and> f ` {0..<n} = g ` {0..<n}}"
assms: shows "image_mset \<eta> C' = mset lC"
reduction_pres_fv: fixes e :: "exp"   and e' :: "exp" assumes "e \<longrightarrow>* e'"   and "FV e = {}" shows "FV e' = {}"
le_imp_less_or_eq: fixes D :: "('e, 'f) Order_scheme"   and a :: "'e"   and b :: "'e" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> b = (a \<prec>\<^bsub>D\<^esub> b \<or> a = b)"
gR_downset_def: fixes x :: "'c \<times> 'd" shows "ps_prod.gR_downset x \<equiv> {y. ps_prod.gR_rel y x}"
homotopic_loops_imp_path_component_value: fixes S :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b"   and t :: "real" assumes "homotopic_loops S p q"   and "0 \<le> t"   and "t \<le> 1" shows "path_component S (p t) (q t)"
exact_decompD_3: fixes m :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and h' :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U' :: "'x set" assumes "exact_decomp m ps"   and "(h, U) \<in> set ps"   and "(h', U') \<in> set ps"   and "poly_deg h = poly_deg h'"   and "m < card U"   and "m < card U'" shows "(h, U) = (h', U')"
hom_decomp_map_times: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "hom_decomp ps"   and "homogeneous f" shows "hom_decomp (map (apfst ((*) f)) ps)"
funion_upper1: fixes A :: "'a fset"   and B :: "'a fset" shows "A |\<subseteq>| A |\<union>| B"
Cons_1: fixes s :: "('a, 'b) fmap"   and vs :: "'a set"   and s' :: "('a, 'b) fmap" assumes "sat_precond_as s as"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "sat_precond_as s' (as_proj as vs)"
arg_cong2: fixes a :: "'h"   and b :: "'h"   and c :: "'i"   and d :: "'i"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'j" assumes "a = b"   and "c = d" shows "f a c = f b d"
boundedI: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> x \<le>s a" shows "x \<le>s signed.Min A"
Polys_closed_monom_mult: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and c :: "'b" assumes "t \<in> .[X]"   and "p \<in> P[X]" shows "MPoly_Type_Class.punit.monom_mult c t p \<in> P[X]"
\<alpha>nl_def': fixes fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme" shows "\<alpha>nl fg \<equiv> \<alpha>n fg \<circ> le_rem_s"
intro: fixes q :: "real" assumes "1 < q" shows "fbnorm q"
welldefined2: shows "list_constr (snd I) (length qs)"
induces_elementary_monoidal_category: shows "elementary_monoidal_category (\<cdot>) (\<otimes>) \<I> lunit runit assoc"
isinceI: fixes Q :: "nat \<Rightarrow> bool"   and t :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "Q t"   and "\<And>t'. t' \<in> I \<down>> t \<Longrightarrow> P t'"   and "t \<in> I" shows "P t'. t' \<S> t I. Q t"
dep3_event_permute_3: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and R :: "'b set"   and S :: "'b set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.dep3_event \<P> Q R S x" shows "MinkowskiPrimitive.dep3_event \<P> R S Q x"
assms_2: shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
intro: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e" assumes "bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G" shows "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G"
zero_lens_indep: fixes X :: "'d \<Longrightarrow> 'c" shows "0\<^sub>L \<bowtie> X"
N_def: shows "N = {n \<in> M. rejects \<F> (insert n S) M} \<inter> {Sup S<..}"
2192: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> (- y \<sqinter> x) = - y \<sqinter> x"
abs_conv_abscissa_zeta: shows "abs_conv_abscissa fds_zeta = 1"
mono: fixes x :: "enat \<times> 'c llist" shows "mono_llist (\<lambda>ldrop. case x of (0, xs) \<Rightarrow> xs | (eSuc n', LNil) \<Rightarrow> LNil | (eSuc n', LCons x xs') \<Rightarrow> curry ldrop n' xs')"
simps_2: fixes F :: "'f set"   and f :: "'d \<Rightarrow> 'd" shows "strip_guards F (Basic f) = Basic f"
reorder_prop: fixes fs :: "('a \<times> 'b) list"   and P :: "'b \<Rightarrow> bool"   and fTs :: "('a \<times> 'c) list" assumes "\<forall>(l, t)\<in>set fs. P t"   and "\<forall>(l, U)\<in>set fTs. \<exists>u. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>u\<rfloor>" shows "\<forall>(l, t)\<in>set (map (\<lambda>(l, T). (l, the (fs\<langle>l\<rangle>\<^sub>?))) fTs). P t"
red_setE2: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red B p q" obtains b :: "'t \<Rightarrow>\<^sub>0 'c" where "b \<in> B" and "b \<noteq> 0" and "red {b} p q"
to_bl_2p: fixes n :: "nat" assumes "n < LENGTH('b)" shows "to_bl (2 ^ n) = replicate (LENGTH('b) - Suc n) False @ True # replicate n False"
that: fixes l :: "'x list" assumes "it = foldli l"   and "it' = foldli l" shows "thesis"
rHom_0_0: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f \<zero>\<^bsub>A\<^esub> = \<zero>\<^bsub>R\<^esub>"
induct: fixes x1 :: "msg"   and x2 :: "msg"   and x3 :: "nat"   and evs :: "event list"   and P :: "msg \<Rightarrow> msg \<Rightarrow> nat \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> respond evs"   and "\<And>KAB A B NA. Key KAB \<notin> used evs \<Longrightarrow> P Hash[Key (shrK A)] \<lbrace>Agent A, Agent B, Nonce NA, END\<rbrace> \<lbrace>Crypt (shrK A) \<lbrace>Key KAB, Agent B, Nonce NA\<rbrace>, END\<rbrace> KAB"   and "\<And>PA RA KAB KBC A B NA Pa C NB. \<lbrakk>(PA, RA, KAB) \<in> respond evs; P PA RA KAB; Key KBC \<notin> used evs; Key KBC \<notin> parts {RA}; PA = Hash[Key (shrK A)] \<lbrace>Agent A, Agent B, Nonce NA, Pa\<rbrace>\<rbrakk> \<Longrightarrow> P Hash[Key (shrK B)] \<lbrace>Agent B, Agent C, Nonce NB, PA\<rbrace> \<lbrace>Crypt (shrK B) \<lbrace>Key KBC, Agent C, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Key KAB, Agent A, Nonce NB\<rbrace>, RA\<rbrace> KBC" shows "P x1 x2 x3"
uminus_sum_list: fixes as :: "'a list" shows "- sum_list as = sum_list (map uminus (rev as))"
relhoare_abstract_Un_left: fixes pre :: "('a \<times> 'b) set"   and Ra1 :: "('a \<times> 'a) set"   and Rc :: "('b \<times> 'b) set"   and post :: "('a \<times> 'b) set"   and Ra2 :: "('a \<times> 'a) set" assumes "{pre} Ra1, Rc {> post}" shows "{pre} Ra1 \<union> Ra2, Rc {> post}"
set_iterator_map_linord_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "('c \<times> 'e) set" assumes "class.linorder less_eq less" shows "linorder.set_iterator_map_linord less_eq it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). less_eq k k')"
is_monom_monom_coeff_not_zero: fixes l :: "linear_poly" assumes "is_monom l" shows "monom_coeff l \<noteq> 0"
sync_gen_model_tLength: fixes T :: "'s Trace set"   and n :: "nat"   and T' :: "'s Trace set"   and t :: "'s Trace" assumes "{t \<in> T. tLength t = n} = {t \<in> T'. tLength t = n}"   and "t \<in> {t \<in> T. tLength t = n}" shows "gen_model (mkM T) t = gen_model (mkM T') t"
wit_G_set1: fixes x :: "'n"   and l2 :: "'o" assumes "x \<in> set1_G (wit_G l2)" shows "False"
spmf_of_set_singleton: fixes x :: "'a" shows "spmf_of_set {x} = return_spmf x"
del_point_def: fixes p :: "'a" shows "del_point p \<equiv> \<V> - {p}"
rreq_rrep_sn_def: fixes m :: "msg" shows "rreq_rrep_sn m \<equiv> case m of Rreq x xa xb xc xd xe osnc xf \<Rightarrow> 1 \<le> osnc | Rrep x xa dsnc xb xc \<Rightarrow> 1 \<le> dsnc | _ \<Rightarrow> True"
is_volatile_Write\<^sub>s\<^sub>b_address_of: fixes x :: "'b memref" assumes "is_volatile_Write\<^sub>s\<^sub>b x" shows "address_of x \<noteq> {}"