hcju/isabelleps huggingface dataset:
elims_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and x :: "('a, 'c) efsym" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw"   and "ProblemIkTpart.TE_wtFsym wtFsym resOf x" obtains f :: "'a" where "x = Oldf f" and "wtFsym f"   | \<sigma> :: "'c" where "x = Tag \<sigma>__"   | \<sigma> :: "'c" where "x = Wit \<sigma>" and "\<not> ProblemIkTpart.isRes wtFsym resOf \<sigma>"
x_def: shows "x \<equiv> getFr [out, inp, zz] [] [\<sigma>, \<tau>]"
size_0: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "euclidean_size zero = 0"
equivD2: fixes x :: "'a"   and y :: "'a" assumes "x \<approx> y" shows "y \<le> x"
prems_1: shows "are_all_operators_non_interfering (op # ops)"
differing_vars_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.differing_vars mem\<^sub>1 mem\<^sub>2 = {x. mem\<^sub>1 x \<noteq> mem\<^sub>2 x}"
ltl_to_generalized_rabin_af_wellformed: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set" assumes "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>" shows "ltl_to_rabin_af \<Sigma> w"
map_upds_Nil1: fixes m :: "'a \<Rightarrow> 'b option"   and bs :: "'b list" shows "m([] [\<mapsto>] bs) = m"
assms_2: shows "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])"
user_reg_mod_def: fixes data_w32 :: "word32"   and win :: "'b word"   and ur :: "word5"   and state :: "'b sparc_state" shows "user_reg_mod data_w32 win ur state \<equiv> if ur = 0 then state else if 0 < ur \<and> ur < 8 then global_reg_mod data_w32 (nat NWINDOWS) ur state else if 7 < ur \<and> ur < 16 then out_reg_mod data_w32 win ur state else if 15 < ur \<and> ur < 24 then state\<lparr>user_reg := (user_reg state)(win := (user_reg state win)(ur := data_w32))\<rparr> else in_reg_mod data_w32 win ur state"
linear_zero: shows "linear (*a) (*b) (\<lambda>x. 0::'c)"
intro: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "\<And>x y z. \<lbrakk>less_eq x y; less_eq y z; x \<in> A; y \<in> A; z \<in> A\<rbrakk> \<Longrightarrow> less_eq x z" shows "transitive A less_eq"
follow_stl_smap_fst: fixes bs :: "bool stream"   and Ms :: "'marker stream"   and stepSs :: "(('sequent \<times> 'rule) \<times> 'a) stream" assumes "follow bs Ms (smap fst stepSs)" shows "follow (stl bs) (stl Ms) (smap fst (stl stepSs))"
Dom_comp: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'a \<times> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "partial_magma.arr (product_category.comp C1 C2) (product_category.comp C1 C2 g f)" shows "(if partial_magma.arr C1 (fst (product_category.comp C1 C2 g f)) \<and> partial_magma.arr C2 (snd (product_category.comp C1 C2 g f)) then (partial_magma.dom C1 (fst (product_category.comp C1 C2 g f)), partial_magma.dom C2 (snd (product_category.comp C1 C2 g f))) else (partial_magma.null C1, partial_magma.null C2)) = (if partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f) then (partial_magma.dom C1 (fst f), partial_magma.dom C2 (snd f)) else (partial_magma.null C1, partial_magma.null C2))"
get_root_node_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok Shadow_DOM.get_root_node ptr"
transE: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" obtains "(x, z) \<in> r"
lhd_def: fixes llist :: "'c llist" shows "lhd llist = (case llist of LCons x21 x22 \<Rightarrow> x21)"
X_3: assumes "list_all2 (\<lambda>x ys. x \<in> set ys) yss (map permutations xss)" shows "map set xss = map set yss"
dim_subset_UNIV: fixes S :: "'c set" shows "vs2.dim S \<le> vs2.dimension"
nadds_red: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>q. q \<in> B \<Longrightarrow> \<not> lt q adds\<^sub>t lt p"   and "red B p r" shows "r \<noteq> 0 \<and> lt r = lt p"
substInp_def2: fixes ys :: "'varSort"   and Y :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and y :: "'var" shows "substInp ys Y y = lift (subst ys Y y)"
sortingSameSet: fixes X :: "'c set" assumes "finite X" shows "set (sorted_list_of_set X) = X"
trans_r: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Semilat A r f"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
strict_order_transitive_acyclic: fixes x :: "'a" shows "(x \<le> - top \<and> x \<sqinter> x \<le> x) = (x \<sqinter> x \<le> x \<and> x \<sqinter> top \<le> - top)"
sum_subspace: fixes U :: "'c set"   and E :: "'c set"   and V :: "'c set" assumes "Subspace.subspace U E"   and "vectorspace E"   and "Subspace.subspace V E" shows "Subspace.subspace (U + V) E"
path_to_then_lookup_Some: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "nat \<Rightarrow> nat option"   and \<pi>s :: "char list list" assumes "\<exists>s'\<in>ast_problem.G problem. ast_problem.path_to problem s \<pi>s s'" shows "\<forall>\<pi>\<in>set \<pi>s. ast_problem.lookup_operator problem \<pi> \<noteq> None"
not_High_Low: fixes V :: "'var" assumes "V \<notin> H" shows "V \<in> L"
the_\<theta>_simps_5: shows "cod \<chi>.the_\<theta> = s\<^sub>0"
measure_linear_image: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set" assumes "linear f"   and "S \<in> lmeasurable" shows "Sigma_Algebra.measure lebesgue (f ` S) = \<bar>det (matrix f)\<bar> * Sigma_Algebra.measure lebesgue S"
dvd_gcd_0_iff_2: fixes x :: "'a"   and a :: "'a" shows "(x dvd some_gcd a (0::'a)) = (x dvd a)"
rel_gpv''_eq: shows "rel_gpv'' (=) (=) (=) = (=)"
B: shows "B \<equiv> snd (Gauss_Jordan_column_k (ia, A) k)"
partUnfold: fixes P :: "pi"   and Q :: "pi"   and s :: "(name \<times> name) list" assumes "P \<simeq>\<^sup>s Q" shows "P[
] \<simeq>\<^sup>s Q[
]"
Units_cong: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "Group.comm_monoid G"   and "a \<in> Units G"   and "a \<sim>\<^bsub>G\<^esub> b"   and "b \<in> carrier G" shows "b \<in> Units G"
assms_2: shows "F \<subseteq> dgrad_max_set d"
noguards_\<Gamma>: shows "\<forall>p\<in>dom \<Gamma>. noguards (the (\<Gamma> p))"
ccpo_Sup_singleton: fixes Sup :: "'f set \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f" assumes "class.ccpo Sup less_eq less" shows "Sup {x} = x"
that: assumes "nop_v11 A (2\<^sub>\<nat>) f" shows "thesis"
lift_preserves_beta: fixes r :: "dB"   and s :: "dB"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "lift r i \<rightarrow>\<^sub>\<beta> lift s i"
Fourier_Fejer_Cesaro_summable: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and t :: "real"   and r :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "(f \<longlongrightarrow> l) (at t within {..t})"   and "(f \<longlongrightarrow> r) (at t within {t..})" shows "(\<lambda>n. (\<Sum>m<n. \<Sum>k\<le>2 * m. Fourier_coefficient f k * trigonometric_set k t) / real n) \<longlonglongrightarrow> (l + r) / 2"
GSpan_single_nonzero: fixes m :: "'c" assumes "m \<in> V"   and "m \<noteq> (0::'c)" shows "GSpan [m] \<noteq> 0"
assms_1: shows "f ^ Suc m = g ^ Suc m"
app_not_fn: fixes A :: "'b trm"   and B :: "'b trm"   and y :: "'b"   and T :: "PreSimplyTyped.type"   and X :: "'b trm" shows "App A B \<noteq> Fn y T X"
sup_left_commute: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
DataOverride_def: fixes D1 :: "'b pdata"   and D2 :: "'b data" shows "(D1 [D+] D2) = (let (L1, DP1) = Rep_pdata D1; (L2, DP2) = Rep_data D2; L = map OptionOverride (zip L1 L2) in Abs_data (L, DP2))"
that: fixes G :: "V set" assumes "G \<subseteq> H"   and "ordertype G VWF = \<beta>" shows "thesis"
dvd_sum: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and A :: "'b set"   and d :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.comm_semiring_1 times one plus zero"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times d (f a)" shows "dvd.dvd times d (comm_monoid_add.sum plus zero f A)"
gt: shows "eval_poly \<alpha> p1 \<succ> eval_poly \<alpha> p2"
rderivs_replicate_mult: fixes i :: "nat"   and as :: "'a"   and r :: "'b rexp"   and m :: "nat" assumes "\<guillemotleft>rderivs (replicate i as) r\<guillemotright> = \<guillemotleft>r\<guillemotright>"   and "0 < i" shows "\<guillemotleft>rderivs (replicate (m * i) as) r\<guillemotright> = \<guillemotleft>r\<guillemotright>"
outputFresh_2: fixes xvec :: "name list"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* N \<prec>' P = (xvec \<sharp>* N \<and> xvec \<sharp>* P)"
subring_props_1: fixes K :: "'a list set" assumes "subfield K (K [X])" shows "K \<subseteq> carrier (K [X])"
step_3_correct: fixes p :: "prog" assumes "spec\<^sub>3 p" shows "spec\<^sub>2 p"
star_unfold_eq: fixes x :: "'a" shows "1\<^sub>\<sigma> + x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
simps: fixes a :: "loc" shows "the_Addr (Addr a) = a"
weak_ranking_unique: fixes As :: "'c set list" assumes "is_weak_ranking As"   and "of_weak_ranking As = (\<lambda>_ _. False)" shows "As = weak_ranking (\<lambda>_ _. False)"
Exit_edge_Entry_or_Low: fixes a :: "'edge" assumes "valid_edge a"   and "targetnode a = (_Exit_)" shows "sourcenode a = (_Entry_) \<or> sourcenode a = (_Low_)"
a: shows "M[a\<turnstile>c>b] \<longrightarrow>\<^sub>l M'"
eval_nembed_true: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c" assumes "P s" shows "\<guillemotleft> \<N> P \<guillemotright> s = 0"
rep_eq: fixes x :: "mini_alg" shows "ma_rat x = fst (Rep_mini_alg x)"
arctan_upper_12_def: shows "arctan_upper_12 \<equiv> \<lambda>x. 3 * x / (x\<^sup>2 + 3)"
lm083: fixes X :: "'a set"   and Q :: "('a \<times> 'b) set"   and P :: "('a \<times> 'b) set" assumes "X \<inter> Domain Q = {}" shows "(P +< Q) `` X = P `` X"
dom_in_domains: fixes f :: "'c" assumes "C.domains f \<noteq> {}" shows "C.dom f \<in> C.domains f"
pr_weak_refl: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and fn :: "'a \<times> nat" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "pr_weak fn fn"
list_relE3: fixes x :: "'b"   and xs :: "'b list"   and l' :: "'c list"   and R :: "('b \<times> 'c) set" assumes "(x # xs, l') \<in> \<langle>R\<rangle>list_rel" obtains x' :: "'c" and xs' :: "'c list" where "l' = x' # xs'" and "(x, x') \<in> R" and "(xs, xs') \<in> \<langle>R\<rangle>list_rel"
pointfree_idE: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f \<circ> g = id" shows "f (g x) = x"
path_cons_elim: fixes a :: "'action"   and r :: "'action list"   and p :: "'state" assumes "R.path (a # r) p" obtains "ren a p" and "R.path r (ex a p)"
continuous_on: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "bounded_linear f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f (g x))"
i_f_join_empty: fixes f :: "nat \<Rightarrow> 'b" shows "f \<Join>\<^sub> {} = []"
mult_minus_distrib_mat_vec: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'c vec"   and w :: "'c vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc"   and "w \<in> carrier_vec nc" shows "A *\<^sub>v (v - w) = A *\<^sub>v v - A *\<^sub>v w"
nonneg_sr_1_largest_jb: fixes A :: "real mat"   and n_as :: "(nat \<times> complex) list"   and m :: "nat"   and lam :: "complex" assumes "nonneg_mat A"   and "jordan_nf (of_real_hom.mat_hom A) n_as"   and "(m, lam) \<in> set n_as"   and "cmod lam = 1"   and "\<And>x. poly (char_poly A) x = 0 \<Longrightarrow> x \<le> 1" shows "\<exists>M\<ge>m. (M, 1) \<in> set n_as"
l_set_child_nodes_get_disconnected_nodes_axioms_def: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_disconnected_nodes_axioms set_child_nodes_locs get_disconnected_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
type_wf_defs: fixes h :: "(_) heap" shows "NodeClass.type_wf h = (ObjectClass.type_wf h \<and> (\<forall>node_ptr\<in>fset (node_ptr_kinds h). get node_ptr h \<noteq> None))"
finite_minimalE: fixes A :: "'b set"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A"   and "A \<noteq> {}"   and "irreflp rel"   and "transp rel" obtains a :: "'b" where "a \<in> A" and "\<And>b. rel b a \<Longrightarrow> b \<notin> A"
not_le: fixes x :: "'a"   and y :: "'a" shows "(\<not> le x y) = lt y x"
continuous_on_sequentiallyI: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>u a. \<lbrakk>\<forall>n. u n \<in> s; a \<in> s; u \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a" shows "continuous_on s f"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) C.map C.map C.map"
pmf_of_list_wf_def: fixes xs :: "('c \<times> 'd) list" shows "pmf_of_list_wf xs = ((\<forall>x\<in>set (map snd xs). (0::'d) \<le> x) \<and> sum_list (map snd xs) = (1::'d))"
meta_allE: fixes P :: "'a \<Rightarrow> prop"   and x :: "'a" assumes "\<And>x. PROP P x" obtains "PROP P x"
monic_degree_0: fixes p :: "'d poly" assumes "monic p" shows "(degree p = 0) = (p = 1)"
min_product_comm: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<otimes>\<^sub>m B = B \<otimes>\<^sub>m A"
subst_empty: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "subst t fmempty = t"
is_projection_on_idem: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and x :: "'b" assumes "is_projection_on \<pi> M" shows "\<pi> (\<pi> x) = \<pi> x"
absch: shows "\<forall>x\<in>set_pmf (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) (take n qs)). (case x of (s, is) \<Rightarrow> real (Suc (index s (qs ! n)))) \<le> real (case x of (s, is) \<Rightarrow> length init)"
simps_1: fixes uu :: "'d"   and uv :: "'d"   and F :: "'c" shows "A_B.Map (A_B.MkArr uu uv F) = F"
xfer_holds: fixes s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and cs :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)"   and t :: "nat" assumes "sim_rel s cs" shows "is_HOLD (tts cs t) = is_HOLD (tts s t)"
sin_ms_aux'_def: fixes xs :: "('a \<times> real) msllist" shows "sin_ms_aux' xs = times_ms_aux xs (powser_ms_aux (msllist_of_msstream sin_series_stream) (times_ms_aux xs xs))"
parId: fixes P :: "ccs" shows "P \<parallel> \<zero> \<sim> P"
classes_eq: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "A {.=} closure_of A"
4: shows "top * e \<le> top * e * (w\<^sup>T \<sqinter> - v\<^sup>T)\<^sup>\<star> \<squnion> - - v\<^sup>T"
continuous_on: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
mset_set_set_mset: fixes M :: "'a multiset"   and A :: "'a set" assumes "M \<subseteq># mset_set A" shows "mset_set (set_mset M) = M"
domStart: fixes t :: "'a"   and p1 :: "'a \<Rightarrow> 'b option"   and p2 :: "'a \<Rightarrow> 'b option" assumes "t \<in> dom p1" shows "(p1 \<Oplus> p2) t = p1 t"
t_def: shows "t = (\<lambda>y. (e / r) *\<^sub>R (y + - apply_chart c x) + x0)"
l2_init_def: shows "l2_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, chan = {}, bad = bad_init\<rparr>}"
path_entry_is_trancl: fixes E :: "(nat \<times> nat) set"   and l :: "nat list"   and v :: "nat" assumes "path_entry E l v"   and "l \<noteq> []" shows "(0, v) \<in> E\<^sup>+"
check_def: fixes h :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and e :: "nat"   and z :: "nat" shows "check h a e z = (fst a \<otimes> fst h [^] e = \<^bold>g [^] z \<and> snd a \<otimes> snd h [^] e = g' [^] z \<and> fst a \<in> carrier \<G> \<and> snd a \<in> carrier \<G>)"
THE_default_def: fixes d :: "'a"   and P :: "'a \<Rightarrow> bool" shows "THE_default d P = (if \<exists>!x. P x then THE x. P x else d)"
h3: shows "\<forall>i<deg + 1. MPoly_Type.degree (f i + g i) var \<le> 1"
picard_lindeloef_affine: fixes T :: "real set"   and A :: "real \<Rightarrow> (('b, 'c) vec, 'c) vec"   and B :: "real \<Rightarrow> ('b, 'c) vec"   and S :: "('b, 'c) vec set"   and t\<^sub>0 :: "real" assumes "matrix_continuous_on T A"   and "\<And>\<tau> \<epsilon>. \<lbrakk>\<tau> \<in> T; 0 < \<epsilon>\<rbrakk> \<Longrightarrow> bdd_above {\<parallel>A t\<parallel>\<^sub>o\<^sub>p |t. dist \<tau> t \<le> \<epsilon>}"   and "continuous_on T B"   and "open S"   and "t\<^sub>0 \<in> T"   and "open T"   and "is_interval T" shows "picard_lindeloef (\<lambda>t s. A t *v s + B t) T S t\<^sub>0"
subset: shows "additive.Class ` R \<subseteq> additive.Partition"
conf_PrimT: fixes v :: "val"   and t :: "prim_ty"   and G :: "prog"   and s :: "st" assumes "\<forall>dt. typeof dt v = Some (PrimT t)" shows "G,s\<turnstile>v\<Colon>\<preceq>PrimT t"
vec_is_zero_def: fixes n :: "nat"   and v :: "'b vec" shows "vec_is_zero n v = (\<forall>i<n. v $ i = (0::'b))"
inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "A \<subseteq> carrier L"   and "z \<in> carrier L"   and "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq>\<^bsub>L\<^esub> x" shows "z \<sqsubseteq>\<^bsub>L\<^esub> \<Sqinter>\<^bsub>L\<^esub>A"
prognosis_Var2: fixes e :: "exp"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and S :: "stack_elem list" assumes "isVal e"   and "x \<notin> domA \<Gamma>" shows "prognosis ae as 0 ((x, e) # \<Gamma>, e, S) \<sqsubseteq> prognosis ae as 0 (\<Gamma>, e, Upd x # S)"
inf_exists: fixes A :: "'a hrel set" assumes "A \<subseteq> carrier (fpl \<P> \<H>)" shows "\<exists>i. is_glb (fpl \<P> \<H>) i A"
greaterThanLessThan_empty_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b = {}) = less_eq b a"
is_SDG_path_matched: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'" obtains ns' :: "'node SDG_node list" where "matched n ns' n'" and "set ns \<subseteq> set ns'"
order_less_asym: fixes x :: "'a"   and y :: "'a"   and P :: "bool" assumes "x < y"   and "\<not> P \<Longrightarrow> y < x" shows "P"
differing_finite: 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 mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval" shows "finite (sifum_security_init.differing_vars mem\<^sub>1 mem\<^sub>2)"
h: shows "\<guillemotleft>h : y \<rightarrow>\<^sub>D G x\<guillemotright>"
eq: shows "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}"
ta_lang_def: fixes TA :: "('b, 'a, 'c) tree_automaton_rec_scheme" shows "ta_lang TA \<equiv> {t. \<exists>q\<in>ta_initial TA. accs (ta_rules TA) t q}"
simps_2: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_to_set (RangeUnion r1 r2) = wordinterval_to_set r1 \<union> wordinterval_to_set r2"
intr: shows "(intr, op_inter_ivl) \<in> \<langle>A\<rangle>ivl_rel \<rightarrow> \<langle>A\<rangle>ivl_rel \<rightarrow> \<langle>A\<rangle>ivl_rel"
strong_induct: fixes P :: "'a \<Rightarrow> trmI \<Rightarrow> bool"   and z :: "'a"   and trmI :: "trmI" assumes "\<And>name z. P z (IVar name)"   and "\<And>name trmI z. \<lbrakk>name \<sharp> z; \<And>z. P z trmI\<rbrakk> \<Longrightarrow> P z (ILam [name].trmI)"   and "\<And>trmI1 trmI2 z. \<lbrakk>\<And>z. P z trmI1; \<And>z. P z trmI2\<rbrakk> \<Longrightarrow> P z (IApp trmI1 trmI2)"   and "\<And>z. P z IUnit"   and "\<And>nat z. P z (INat nat)"   and "\<And>trmI z. (\<And>z. P z trmI) \<Longrightarrow> P z (ISucc trmI)"   and "\<And>trmI1 trmI2 z. \<lbrakk>\<And>z. P z trmI1; \<And>z. P z trmI2\<rbrakk> \<Longrightarrow> P z (trmI1\<mapsto>trmI2)"   and "\<And>trmI z. (\<And>z. P z trmI) \<Longrightarrow> P z (IRef trmI)"   and "\<And>trmI1 trmI2 z. \<lbrakk>\<And>z. P z trmI1; \<And>z. P z trmI2\<rbrakk> \<Longrightarrow> P z (trmI1;;trmI2)"   and "\<And>trmI1 trmI2 trmI3 z. \<lbrakk>\<And>z. P z trmI1; \<And>z. P z trmI2; \<And>z. P z trmI3\<rbrakk> \<Longrightarrow> P z (Iif trmI1 trmI2 trmI3)" shows "P z trmI"
to_int_mod_ring_of_int_M: fixes x :: "int" shows "to_int_mod_ring (of_int x) = M x"
SUC_SUC: fixes k :: "'k"   and k' :: "'k"   and idx :: "'n" shows "SUC k (SUC k' idx) = SUC k' (SUC k idx)"
simps: fixes u :: "'a Point" shows "onAxisT u = (xval u = (0::'a) \<and> yval u = (0::'a) \<and> zval u = (0::'a))"
path_conforms_LCons_LNil: fixes p :: "Player"   and v :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "path_conforms_with_strategy p (LCons v LNil) \<sigma>"