hcju/isabelleps huggingface dataset:
pseudonatural_transformation_def: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" shows "pseudonatural_transformation 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 \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (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>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 \<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 \<and> pseudonatural_transformation_axioms V\<^sub>C H\<^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 \<tau>\<^sub>0 \<tau>\<^sub>1"
simps_7: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game" shows "usubstappp \<sigma> U \<alpha>^d = (let V\<alpha> = usubstappp \<sigma> U \<alpha> in (fst V\<alpha>, Dualo (snd V\<alpha>)))"
invar_def: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.invar q \<equiv> BinomialHeapStruc.queue_invar q \<and> BinomialHeapStruc.rank_invar q"
intro: fixes \<chi> :: "'c \<Rightarrow> complex"   and G :: "('c, 'd) monoid_scheme" assumes "\<chi> \<one>\<^bsub>G\<^esub> \<noteq> 0"   and "\<And>a. a \<notin> carrier G \<Longrightarrow> \<chi> a = 0"   and "\<And>a b. \<lbrakk>a \<in> carrier G; b \<in> carrier G\<rbrakk> \<Longrightarrow> \<chi> (a \<otimes>\<^bsub>G\<^esub> b) = \<chi> a * \<chi> b" shows "character_axioms G \<chi>"
vsv_vlrestriction_vinsert: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r \<restriction>\<^sup>l\<^sub>\<circ> vinsert a A = vinsert \<langle>a, r\<lparr>a\<rparr>\<rangle> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
us: shows "us = getFrN (map snd txs) (n # map fst txs) [] (length txs)"
max_ideal_shapeI: fixes X :: "'c set"   and a :: "'c \<Rightarrow> 'd" shows "generates_max_ideal X ((\<lambda>x. monomial (1::'d) (monomial 1 x) - monomial (a x) 0) ` X)"
p: shows "\<one> = bv_msb (tl (norm_signed (\<one> # xs)))"
ipresVar_def: fixes h :: "('c, 'd, 'varSort, 'var, 'e) qTerm set \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'i, 'j, 'var, 'f, 'k, 'l) model_scheme" shows "ipresVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = igVar MOD xs x"
nn_integral_pmf: fixes A :: "'c set"   and p :: "'c pmf" shows "(\<Sum>\<^sup>+ x\<in>A. ennreal (pmf p x)) = emeasure (measure_pmf p) A"
rotater1_def: fixes ys :: "'b list" shows "rotater1 ys = (case ys of [] \<Rightarrow> [] | x # xs \<Rightarrow> last ys # butlast ys)"
Suc: shows "matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n \<in> carrier_mat d d"
assms: shows "\<Gamma> \<turnstile> Var x : \<tau>"
zero_diff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
relcomp_distrib: fixes R :: "('a \<times> 'c) set"   and S :: "('c \<times> 'b) set"   and T :: "('c \<times> 'b) set" shows "R O (S \<union> T) = R O S \<union> R O T"
Suc: fixes xss :: "'a llist llist" shows "lconcat (ltake (enat n) xss) = ltake (\<Sum>i<n. llength (lnth xss i)) (lconcat xss)"
assms: shows "B \<in> set (tr\<^sub>p\<^sub>c A D)"
frag_cmul_distrib2: fixes c :: "int"   and a :: "'b \<Rightarrow>\<^sub>0 int"   and b :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul c (a + b) = frag_cmul c a + frag_cmul c b"
f_inv_into_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and b :: "'b" assumes "bij_betw f A B"   and "b \<in> B" shows "f (inv_into A f b) = b"
hyps: fixes i :: "nat"   and j :: "nat"   and thesis :: "bool" assumes "\<And>i' j'. \<lbrakk>i' < j'; i' < length xs; j' < length xs; filter P xs ! i = xs ! i'; filter P xs ! j = xs ! j'\<rbrakk> \<Longrightarrow> thesis"   and "i < j"   and "i < length (filter P xs)"   and "j < length (filter P xs)" shows "thesis"
IH: assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   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>set cs. v c \<le> n"   and "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> M" shows "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d"
c: fixes s :: "complex" assumes "0 < Re s" shows "cmod (stirling_integral (Suc n) s) \<le> c / Re s ^ n"
SN_rel_ext_def: fixes P :: "('b \<times> 'b) set"   and Pw :: "('b \<times> 'b) set"   and R :: "('b \<times> 'b) set"   and Rw :: "('b \<times> 'b) set"   and M :: "'b \<Rightarrow> bool" shows "SN_rel_ext P Pw R Rw M \<equiv> \<nexists>f t. (\<forall>i. (f i, f (Suc i)) \<in> SN_rel_ext_step P Pw R Rw (t i)) \<and> (\<forall>i. M (f i)) \<and> (\<exists>\<^sub>\<infinity>i. t i \<in> {top_s, top_ns}) \<and> (\<exists>\<^sub>\<infinity>i. t i \<in> {top_s, normal_s})"
b_2: shows "arith_prog_fb (int_of_fbint x) b \<subseteq> U"
range_permutation_def: fixes A :: "'a set"   and B :: "'b set" shows "range_permutation A B = {(f, f'). (f, f') \<in> (A \<rightarrow>\<^sub>E B) \<times> (A \<rightarrow>\<^sub>E B) \<and> (\<exists>p. p permutes B \<and> (\<forall>x\<in>A. f x = p (f' x)))}"
Compl_step4: fixes I :: "'a set"   and s :: "'a set" assumes "I \<noteq> {}"   and "finite I"   and "\<not> I \<subseteq> s"   and "\<exists>lI. set lI = {x \<in> I. x \<notin> s} \<and> (\<exists>Sj. length Sj = length lI \<and> nodup_all lI \<and> (\<forall>j<length Sj. Sj ! j \<noteq> [] \<and> tl (Sj ! j) \<noteq> [] \<and> (Sj ! j ! 0, Sj ! j ! (length (Sj ! j) - 1)) = ({lI ! j}, s) \<and> (\<forall>i<length (Sj ! j) - 1. \<turnstile>\<N>\<^bsub>(Sj ! j ! i, Sj ! j ! (i + 1))\<^esub>)))" shows "\<exists>A. \<turnstile>A \<and> attack A = (I, s)"
simps_16: fixes b :: "nat"   and I :: "\<I>"   and r :: "formula Regex.regex" shows "fvi b (MatchF I r) = Regex.fv_regex (fvi b) r"
in_progress_med_progress: fixes x :: "Runs.var"   and r :: "fid_t \<Rightarrow> Runs.var set option"   and R :: "fid_t" assumes "x \<in> {xnx, xny, xgnx, xgny, xsk, xEnd}" shows "in_progress (med_progress r R) x = in_progress (r R) x"
is_projection_on_in_image: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and h :: "'b" assumes "is_projection_on \<pi> M" shows "\<pi> h \<in> M"
function_zero_closed: shows "\<zero>\<^bsub>F\<^esub> \<in> carrier F"
diff_diff_add_hmset: fixes a :: "hmultiset"   and b :: "hmultiset"   and c :: "hmultiset" shows "a - b - c = a - (b + c)"
mod_mod_cancel: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c dvd b" shows "a mod b mod c = a mod c"
fiter_induct: fixes x :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" assumes "(x::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice) \<sqinter> (c::'a::refinement_lattice) ; x" shows "(x::'a::refinement_lattice) \<sqsubseteq> (c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice)"
Minimal_Solutions_length: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Minimal_Solutions a b" shows "length x = length a \<and> length y = length b"
assms: fixes x :: "'a" assumes "x \<in> A" shows "(0::'b) \<le> f x"
take_bit_Suc_from_most: fixes n :: "nat"   and k :: "int" shows "take_bit (Suc n) k = 2 ^ n * of_bool (bit k n) + take_bit n k"
path_connectedin_continuous_map_image: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "continuous_map X Y f"   and "path_connectedin X S" shows "path_connectedin Y (f ` S)"
not_reduced_word_for: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and bs :: "'b list"   and A :: "'b set"   and a :: "'b"   and as :: "'b list" assumes "class.monoid_add plus zero"   and "bs \<in> lists A"   and "monoid_add.sum_list plus zero bs = a"   and "order.greater (length as) (length bs)" shows "\<not> monoid_add.reduced_word_for plus zero A a as"
"2_1": shows "v \<le>\<^sub>F\<^sub>I w\<^sub>1"
cringI: fixes R :: "('a, 'b) ring_scheme" assumes "abelian_group R"   and "Group.comm_monoid R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z" shows "cring R"
iIN_cut_less: fixes n :: "nat"   and d :: "nat"   and t :: "nat" shows "[n\<dots>,d] \<down>< t = (if t \<le> n then {} else if t \<le> n + d then [n\<dots>,t - Suc n] else [n\<dots>,d])"
cring_fieldI: assumes "Units R = carrier R - {\<zero>}" shows "field R"
power_int_def: fixes a :: "int"   and b :: "int" shows "Optimize_Integer.power_int a b = a ^ nat b"
projecting_Constrains: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "projecting (\<lambda>G. reachable (extend h F \<squnion> G)) h F (extend_set h A Co extend_set h B) (A Co B)"
exec_append: fixes \<delta> :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and s :: "'a"   and rs :: "('b \<times> 'c) list"   and rs' :: "('b \<times> 'c) list" shows "pre_RDR.exec \<delta> (pre_RDR.exec \<delta> s rs) rs' = pre_RDR.exec \<delta> s (append rs' rs)"
simps_2: fixes x :: "'a"   and e :: "'c"   and h :: "('a \<times> 'c) list"   and eval :: "'c \<Rightarrow> 'b" shows "evalHeap ((x, e) # h) eval = (evalHeap h eval)(x := eval e)"
G'_gb_fields: shows "gbg_F G' = {f ` A |A. A \<in> F}"
cong_mat: fixes nr :: "nat"   and nr' :: "nat"   and nc :: "nat"   and nc' :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and f' :: "nat \<times> nat \<Rightarrow> 'a" assumes "nr = nr'"   and "nc = nc'"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> f (i, j) = f' (i, j)" shows "mat nr nc f = mat nr' nc' f'"
some_eq_imp: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "Eps P = a"   and "P b" shows "P a"
mag_inverse: fixes x :: "('a, 'b, 'c) Quantity_scheme" shows "mag (inverse x) = inverse (mag x)"
real_eq_imaginary_iff: fixes y :: "complex"   and x :: "complex" assumes "y \<in> \<real>"   and "x \<in> \<real>" shows "(x = \<i> * y) = (x = 0 \<and> y = 0)"
asm_2: shows "\<gamma> \<in> elts (\<omega> \<up> \<delta>)"
rev_split_conv: fixes l :: "'b list" assumes "l \<noteq> []" shows "rev (tl l) @ [hd l] = rev l"
1: shows "prv (imp (eql t1 (Var y2)) (imp (eql t1 t3) (eql (Var y2) t3)))"
strip_comb_induct: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "\<And>x y. P x \<Longrightarrow> P (app x y)"   and "\<And>t. unapp t = None \<Longrightarrow> P t" shows "P t"
fds_const_of_nat: fixes c :: "nat" shows "fds_const (of_nat c) = of_nat c"
preserves_cod: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "cod (trg f) = trg (cod f)"
induced_automorphism_halfmorphism_fopp_to_fimage: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexMorphism (ChamberComplexFolding.opp_half_apartment X f) (f \<turnstile> X) (OpposedThinChamberComplexFoldings.induced_automorphism X f g)"
quot_bounds: fixes x :: "'b"   and x' :: "'b"   and y' :: "'b"   and y :: "'b" assumes "x \<le> x'"   and "y' \<le> y"   and "(0::'b) < y"   and "(0::'b) \<le> x"   and "(0::'b) < y'" shows "x / y \<le> x' / y'"
add: fixes b1 :: "complex"   and b2 :: "complex" shows "(D \<circ> cnj) (b1 + b2) = (D \<circ> cnj) b1 + (D \<circ> cnj) b2"
preserved_code: fixes f :: "('ap, 'aq, 'ar) prog"   and h :: "'ap"   and h' :: "'ap" shows "preserved f h h' = (h \<turnstile> ok f \<and> h' \<turnstile> ok f \<and> |h \<turnstile> f|\<^sub>r = |h' \<turnstile> f|\<^sub>r \<or> \<not> h \<turnstile> ok f \<and> \<not> h' \<turnstile> ok f)"
rtrancl_edges_iff_path: fixes u :: "'a"   and v :: "'a"   and g :: "'a ugraph" shows "((u, v) \<in> (edges g)\<^sup>*) = (\<exists>p. path g u p v)"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(to_fract x = to_fract y) = (x = y)"
set_subst_ident: fixes M :: "('d, 'c) Term.term set"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "fv\<^sub>s\<^sub>e\<^sub>t M \<inter> subst_domain \<theta> = {}" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> = M"
invc_upd_2: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c"   and y :: "'d" assumes "invc t" shows "invc2 (upd x y t)"
intros_3: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<longmapsto> a[b] \<prec> P"
os_it_init_def: fixes l :: "'b node ref option" shows "os_it_init l = return l"
step_def: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" shows "step e s r l i = (case random_member (possible_steps e s r l i) of None \<Rightarrow> None | Some (s', t) \<Rightarrow> Some (t, s', evaluate_outputs t i r, evaluate_updates t i r))"
inst_ok_code1: fixes \<Theta> :: "exetheory"   and insts :: "('a \<times> typ) list" assumes "exe_wf_theory' \<Theta>" shows "list_all (exetyp_ok \<Theta>) (map snd insts) = list_all (typ_ok (translate_theory \<Theta>)) (map snd insts)"
agent_abs_rel_inv: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a"   and x :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and y :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t"   and "x \<in> agent_abs a t"   and "(x, y) \<in> relations (spr_repMC (toSet (fst ec))) a" shows "y \<in> agent_abs a t"
generate_valid_topology_some_edges: fixes M :: "'a SecurityInvariant_configured list"   and V :: "'a set"   and E :: "('a \<times> 'a) set" shows "edges (generate_valid_topology_some M Es \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> E"
le_ereal_le: fixes a :: "ereal"   and x :: "real"   and y :: "real" assumes "a \<le> ereal x"   and "x \<le> y" shows "a \<le> ereal y"
right_hom_with_unit_axioms: shows "right_hom_with_unit (\<cdot>) (\<star>) \<a> (some_unit a) a"
cyclic_on_orbit': fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "permutation f" shows "cyclic_on f (orbit f x)"
swap_apply_2: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "Fun.swap a b f b = f a"
Hausdorff_space_subtopology: fixes X :: "'a topology"   and S :: "'a set" assumes "Hausdorff_space X" shows "Hausdorff_space (subtopology X S)"
simps_1: fixes uu :: "unit"   and x :: "unit + ('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" shows "BsSkewBinomialHeapStruc.bs_meld (Inl uu) x = x"
points_index_count_bl: fixes n :: "nat"   and bl :: "'a set"   and ps :: "'a set" assumes "n \<le> multiplicity bl"   and "ps \<subseteq> bl" shows "n \<le> count (filter_mset ((\<subseteq>) ps) \<B>) bl"
eq_divide_eq_1: fixes b :: "'c"   and a :: "'c" shows "((1::'c) = b / a) = (a \<noteq> (0::'c) \<and> a = b)"
negchecks_model_subst: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a list"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and D :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and F' :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> set X = {}" shows "negchecks_model (\<delta> \<circ>\<^sub>s \<theta>) D X F F' = negchecks_model \<theta> D X (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) (F' \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)"
pair_star: fixes x :: "('b, 'c) boffa_mon" assumes "conway_assm2 x" shows "x\<^bsub>1::'b\<^esub>\<^sup>\<star> = x\<^bsub>1::'b\<^esub>"
sub_K_sender: fixes A' :: "agent"   and n :: "msg"   and b :: "msg"   and l :: "msg"   and k :: "msg"   and A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A' TTP \<lbrace>n, b, l, k, Crypt (priEK A) X\<rbrace> \<in> set evs"   and "evs \<in> zg" shows "A' \<in> {A, Spy}"
mult_div_mono_left: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> c"   and "c \<noteq> 0"   and "a \<le> inverse c * b" shows "c * a \<le> b"
prem: shows "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (mk_eq' \<tau>' (f $ Bv 0) (g $ Bv 0))"
snd_edge_at: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and "is" :: "nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "snd (Solved_Task.edge_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is) = Solved_Task.edge_to freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is"
chamber_in_other_half_fg: fixes C :: "'a set" assumes "folding_g.chamber C"   and "C \<notin> f \<turnstile> folding_g.\<C>" shows "C \<in> g \<turnstile> folding_g.\<C>"
quot_domain_imp_primeideal: fixes R :: "('e, 'f) ring_scheme"   and P :: "'e set" assumes "cring R"   and "ideal P R"   and "domain (R Quot P)" shows "primeideal P R"
vwelldefined_alt_def_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "vwelldefined (Vrecabs css name \<Gamma>) = (fmpred (\<lambda>_. vwelldefined) \<Gamma> \<and> pred_fmap (list_all (\<lambda>(uu_, t). consts t |\<subseteq>| all_consts)) css \<and> name |\<in>| fmdom css \<and> fmdom css |\<subseteq>| fst |`| fset_of_list rs)"
rSim: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto><(bisim O X O bisim)> Q"
adjoint_of_jozsa_transform: shows "U\<^sub>f\<^sup>\<dagger> = U\<^sub>f"
open_iff: fixes t :: "nat set" shows "(t open) = (t = {} \<or> t = carrier)"
input_sizes_deep_model: shows "input_sizes (deep_model_l rs) = replicate (2 * N_half) (last rs)"
tree_emb_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "tree_emb A P = kruskal_tree_tree.emb A (prod_le P (\<lambda>_ _. True))"
dependent_zero: fixes A :: "'b set" assumes "(0::'b) \<in> A" shows "local.dependent A"
interpretLow2: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and e :: "expr"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option" assumes "\<Gamma> \<turnstile> e : Low"   and "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2" shows "\<lbrakk>e\<rbrakk>s1 = \<lbrakk>e\<rbrakk>s2"
assms_1: shows "\<forall>vs\<in>A. distinct vs"
strict_order_iff: fixes b :: "'c"   and a :: "'c" shows "(b < a) = (a = max a b \<and> a \<noteq> b)"
strictly_preserves_\<alpha>_ide: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.ide b"   and "\<F>C.ide c" shows "F (\<F>C.assoc a b c) = \<a>\<^sub>D[F a, F b, F c]"
m_lcomm: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z) = y \<otimes>\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> z)"
uint8_sdiv_def: fixes x :: "uint8"   and y :: "uint8" shows "uint8_sdiv x y = (if y = 0 then undefined (div) x 0 else Abs_uint8 (Rep_uint8 x sdiv Rep_uint8 y))"
edges_are_in_V_2: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^bsub>subgame V'\<^esub> w" shows "w \<in> V\<^bsub>subgame V'\<^esub>"
inf2: fixes B :: "'a"   and k :: "'a"   and b :: "'a"   and a :: "'a" assumes "B \<equiv> inf k b" shows "inf a B \<equiv> inf k (inf a b)"
sgn_finite2: fixes a :: "ereal"   and b :: "ereal" shows "\<bar>sgn a * sgn b\<bar> \<noteq> \<infinity>"
func_space_is_module: fixes S :: "'e set" shows "module R (func_space S)"
in_keys_iff: fixes x :: "'c"   and s :: "'c \<Rightarrow>\<^sub>0 'd" shows "(x \<in> keys s) = (lookup s x \<noteq> (0::'d))"
cod_def: fixes f :: "'a" shows "V.cod f = (if V.codomains f \<noteq> {} then SOME b. b \<in> V.codomains f else V.null)"
mm_equiv_step: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms\<^sub>1 :: "'c \<times> (Mode \<Rightarrow> 'a set)"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and cms\<^sub>2 :: "'c \<times> (Mode \<Rightarrow> 'a set)"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b"   and cms\<^sub>1' :: "'c \<times> (Mode \<Rightarrow> 'a set)"   and mem\<^sub>1' :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "((cms\<^sub>1, mem\<^sub>1), cms\<^sub>2, mem\<^sub>2) \<in> sifum_security_init.mm_equiv dma \<C>_vars eval"   and "snd cms\<^sub>1 = snd cms\<^sub>2"   and "((cms\<^sub>1, mem\<^sub>1), cms\<^sub>1', mem\<^sub>1') \<in> eval" shows "\<exists>c\<^sub>2' mem\<^sub>2'. ((cms\<^sub>2, mem\<^sub>2), (c\<^sub>2', snd cms\<^sub>1'), mem\<^sub>2') \<in> eval \<and> ((cms\<^sub>1', mem\<^sub>1'), (c\<^sub>2', snd cms\<^sub>1'), mem\<^sub>2') \<in> sifum_security_init.mm_equiv dma \<C>_vars eval"