hcju/isabelleps huggingface dataset:
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog" assumes "known_ptr = DocumentClass.known_ptr"   and "create_element = l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_element get_disconnected_nodes set_disconnected_nodes set_tag_name" shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_tag_name create_element known_ptr"
cyclic_dnf_def: fixes ass :: "atom list list" shows "cyclic_dnf ass = (\<exists>iss. list_all2 cycle ass iss)"
simps: fixes p :: "char list"   and ts :: "fterm list" shows "hterm_of_fterm (Fun p ts) = HFun p (map hterm_of_fterm ts)"
dependent_explicit: fixes s :: "'d set" shows "cdependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *\<^sub>C v) = (0::'d) \<and> (\<exists>v\<in>t. u v \<noteq> 0))"
exhaust: fixes y :: "('b, 'c) concrete_category.arr" obtains "y = Null"   | x21 :: "'b" and x22 :: "'b" and x23 :: "'c" where "y = MkArr x21 x22 x23"
zdvd_mono: fixes k :: "int"   and m :: "int"   and t :: "int" assumes "k \<noteq> 0" shows "m dvd t \<equiv> k * m dvd k * t"
cmult: fixes c :: "'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "L F (\<lambda>x. c * f x) = L F f"
exhaust: fixes y :: "point" obtains x1 :: "real" and x2 :: "real" where "y = Point x1 x2"
disjuptoR0: shows "disj_dlHPP_Bisimulation_Up_To_R' d PP (R1 \<union> R2) R0"
lappend_LNil2: fixes xs :: "'a llist" shows "lappend xs LNil = xs"
Max_insert: fixes A :: "'t set"   and x :: "'t" assumes "finite A"   and "A \<noteq> {}" shows "ord_term_lin.Max (insert x A) = ord_term_lin.max x (ord_term_lin.Max A)"
simps_3: fixes u :: "nat \<Rightarrow> 'a Bonk_Schramm_extension_unfolded" shows "map_aux (would_be_Cauchy u) = suc wo_aux (range (map_aux \<circ> u))"
eff\<^sub>i_mono: fixes p :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and \<tau> :: "ty list \<times> ty err list"   and \<tau>' :: "ty list \<times> ty err list"   and i :: "instr"   and m :: "nat"   and rT :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'"   and "app i P m rT pc mpc xt \<lfloor>\<tau>'\<rfloor>"   and "succs i \<tau> pc \<noteq> []"   and "succs i \<tau>' pc \<noteq> []" shows "P \<turnstile> eff\<^sub>i (i, P, \<tau>) \<le>\<^sub>i eff\<^sub>i (i, P, \<tau>')"
card_mono: fixes B :: "'c set"   and A :: "'c set" assumes "finite B"   and "A \<subseteq> B" shows "card A \<le> card B"
erH_rHom_cf: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and s :: "'a" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B"   and "s \<in> carrier S" shows "erH R S X A B Y h s = h s"
hsplit_def: fixes c :: "hf \<Rightarrow> hf \<Rightarrow> 'b" shows "hsplit c \<equiv> \<lambda>p. c (hfst p) (hsnd p)"
set2_FGl_def: fixes x :: "(('b, 'a, 'c, 'd, 'h, 'i, 'j) G, 'b, 'k, 'c, 'l, 'm, 'h, 'n, 'o, 'p) F" shows "set2_FGl x = \<Union> (set2_G ` set1_F x)"
expansion_of_sum1: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "(1\<^sub>r \<plusminus> x)^\<^bsup>R n\<^esup> = \<Sigma>\<^sub>e R (\<lambda>i.  \<^bsub>n\<^esub>C\<^bsub>i\<^esub> \<times>\<^bsub>R\<^esub> x^\<^bsup>R i\<^esup>) n"
lax_aux2: fixes X :: "'b set"   and x :: "'b"   and y :: "'b" assumes "Sup_closed_set X"   and "comp_closed_set X" shows "\<Squnion> {z \<in> X. z \<le> x} \<cdot> \<Squnion> {z \<in> X. z \<le> y} \<le> \<Squnion> {z \<in> X. z \<le> x \<cdot> y}"
Cond_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (if (e) e1 else e2, xs) (if (e') e1 else e2, xs')"
finite_monomials: fixes m :: "'a mpoly" shows "finite (monomials m)"
step_3: shows "(\<Sum>a\<in>keys p. monomial (lookup p a) a) = p"
qGoodBinp_def: fixes qbinp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs option" shows "qGoodBinp qbinp \<equiv> liftAll qGoodAbs qbinp \<and> |{i. qbinp i \<noteq> None}| <o |UNIV|"
partial_equiv_subst: fixes \<tau> :: "char list \<Rightarrow> fterm"   and C :: "fterm literal set"   and G :: "bool list" assumes "falsifies\<^sub>c G (C \<cdot>\<^sub>l\<^sub>s \<tau>)" shows "falsifies\<^sub>c G C"
assms: shows "\<exists>k. f k < g k \<and> (\<forall>k'<k. f k' = g k')"
measure_eq_compl: fixes s :: "'a set"   and t :: "'a set" assumes "s \<in> sets M"   and "t \<in> sets M"   and "Sigma_Algebra.measure M (space M - s) = Sigma_Algebra.measure M (space M - t)" shows "Sigma_Algebra.measure M s = Sigma_Algebra.measure M t"
sel_16: fixes x21 :: "nat"   and x22 :: "'b"   and x23 :: "'b"   and x24 :: "'c"   and x25 :: "'c"   and x26 :: "'a" shows "getMsg (Send x21 x22 x23 x24 x25 x26) = x26"
env_no_step_s: fixes s :: "'a list"   and c :: "'a list multiset"   and e :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> ntrs fg" obtains csp :: "'a list multiset" where "c' = csp + c"
conjugate_dist_add: fixes conjugate :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (plus a b) = plus (conjugate a) (conjugate b)"
add_increasing2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "(0::'a) \<le> c"   and "b \<le> a" shows "b \<le> a + c"
ih: fixes D :: "'a literal list" assumes "reduce [C] D D' \<noteq> D'" shows "wstate_of_dstate (N, P, Q @ (D @ D', j) # Q', n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, (D @ reduce [C] D D', j) # P, Q @ Q', n)"
mrbst_union_correct: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "random_bst A \<bind> (\<lambda>t1. random_bst B \<bind> mrbst_union t1) = random_bst (A \<union> B)"
fps_gcd_def: shows "gcd = normalization_euclidean_semiring_class.gcd"
assms: shows "\<bar>x - y\<bar> \<le> e"
igAbsIPresIGWls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gAbsPresGWls MOD" shows "igAbsIPresIGWls (fromMOD MOD)"
space_def: fixes ts :: "traffic"   and v :: "view"   and c :: "cars" shows "space ts v c \<equiv> Abs_real_int (pos ts c, pos ts c + regular (own v) ts c)"
sorted_by_cons_imp: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "sorted_by cmp (x # xs)" shows "sorted_by cmp xs"
fresh_atom: fixes a :: "atom"   and b :: "atom" shows "a \<sharp> b = (a \<noteq> b)"
composite_simps_1: shows "src ((\<mu> \<star> \<nu>) \<star> \<pi>) = src \<pi>"
union: fixes M :: "'c multiset"   and N :: "'c multiset" shows "\<Sum>\<^sub># (M + N) = \<Sum>\<^sub># M + \<Sum>\<^sub># N"
component_cycle_free: fixes g :: "'b ugraph"   and r :: "'b" assumes "cycle_free g" shows "cycle_free (component_of g r)"
Plus: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and M :: "'a multiset"   and N :: "'b multiset" assumes "R a b"   and "rel_mset' R M N" shows "rel_mset' R (add_mset a M) (add_mset b N)"
midpoint_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B" shows "Tarski_neutral_dimensionless.Col Bet M A B"
coeff_notin_vars: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'a mpoly" assumes "\<not> keys m \<subseteq> vars p" shows "MPoly_Type.coeff p m = (0::'a)"
SIDE_GEN_ALGO_D: fixes P :: "bool" assumes "SIDE_GEN_ALGO P" shows "P"
assoc'_simps\<^sub>A\<^sub>W\<^sub>C_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "associative_weak_composition V H \<a>"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "H f g \<noteq> partial_magma.null V"   and "H g h \<noteq> partial_magma.null V" shows "partial_magma.cod V (category.inv V (\<a> f g h)) = H (H f g) h"
d_ex: fixes y :: "'a"   and t :: "real" assumes "dist y x < d"   and "0 \<le> t"   and "t \<le> d" shows "t \<in> existence_ivl0 y"
h8a: assumes "0 < Av"   and "Bv \<le> 0"   and "Av \<le> - Bv * sqrt Cv" shows "Av\<^sup>2 \<le> Bv\<^sup>2 * Cv"
trans_less_equiv: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "compare cmp a b = Less"   and "compare cmp b c = Equiv" shows "compare cmp a c = Less"
transp_on_def: fixes U :: "'c set" shows "transp_on U = (\<lambda>r. \<forall>x\<in>U. \<forall>y\<in>U. \<forall>z\<in>U. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
apex_def: shows "H\<mu>H\<nu>\<pi>.apex \<equiv> C.dom H\<mu>H\<nu>\<pi>.leg0"
da0: shows "fps_deriv a $ 0 \<noteq> (0::'a)"
assms: shows "\<forall>x. x |\<in>| A \<longrightarrow> f x |\<in>| B"
empty_in_Fpow: fixes A :: "'h set" shows "{} \<in> Fpow A"
sat_inter_names_cong: fixes \<phi> :: "'c MFOTL.formula"   and E :: "(char list \<times> 'c list) set"   and F :: "(char list \<times> 'c list) set"   and \<sigma> :: "(char list \<times> 'c list) trace"   and v :: "'c list"   and i :: "nat" assumes "\<And>e. e \<in> names \<phi> \<Longrightarrow> {xs. (e, xs) \<in> E} = {xs. (e, xs) \<in> F}" shows "MFOTL.sat (map_\<Gamma> (\<lambda>D. D \<inter> E) \<sigma>) v i \<phi> = MFOTL.sat (map_\<Gamma> (\<lambda>D. D \<inter> F) \<sigma>) v i \<phi>"
i: shows "is_zero_row i (echelon_form_of A bezout)"
SFAccInitThrow: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and h :: "heap"   and l :: "sfields"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h, l, sh)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
list_remdups_def: fixes equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "equal_base.list_remdups equal \<equiv> rec_list [] (\<lambda>x xs xsa. if equal_base.list_member equal xs x then xsa else x # xsa)"
rel_gpv_lossless_gpvD1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'j) gpv"   and \<I> :: "('h, 'j) \<I>"   and \<I>' :: "('i, 'j) \<I>" assumes "rel_gpv A C gpv gpv'"   and "lossless_gpv \<I> gpv"   and "rel_\<I> C (=) \<I> \<I>'" shows "lossless_gpv \<I>' gpv'"
dim_col_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "dim_col (mat nr nc f) = nc"
exp_conj_mono_left: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real"   and R :: "'c \<Rightarrow> real" assumes "P \<tturnstile> Q" shows "P && R \<tturnstile> Q && R"
intro: fixes remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" assumes "remove_child = l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child get_child_nodes set_child_nodes get_owner_document get_disconnected_nodes set_disconnected_nodes"   and "remove_child_locs = l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child_locs set_child_nodes_locs set_disconnected_nodes_locs"   and "remove = l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove get_child_nodes set_child_nodes get_parent get_owner_document get_disconnected_nodes set_disconnected_nodes" shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes set_child_nodes set_child_nodes_locs get_parent get_owner_document get_disconnected_nodes set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove"
invar_l_le_k: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt" assumes "invar (Node k v l r)" shows "\<forall>p\<in>set_kdt l. p $ k \<le> v"
sum_ideals: fixes R :: "('c, 'd) Ring_scheme"   and I1 :: "'c set"   and I2 :: "'c set" assumes "Ring R"   and "ideal R I1"   and "ideal R I2" shows "ideal R (I1 \<minusplus>\<^bsub>R\<^esub> I2)"
t_wise_balance_axioms: shows "t_wise_balance \<V> \<B> 2 1"
sigma_sets_Int: fixes A :: "'c set"   and sp :: "'c set"   and st :: "'c set set" assumes "A \<in> sigma_sets sp st"   and "A \<subseteq> sp" shows "(\<inter>) A ` sigma_sets sp st = sigma_sets A ((\<inter>) A ` st)"
fromQ_inject: fixes x :: "'a['c, 'b]"   and y :: "'a['c, 'b]" shows "(fromQ x = fromQ y) = (x = y)"
map_minsky_comp: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and f' :: "'e \<Rightarrow> 'c"   and g' :: "'f \<Rightarrow> 'd"   and M :: "('e \<times> ('e, 'f) Op) set" shows "map_minsky f g (map_minsky f' g' M) = map_minsky (f \<circ> f') (g \<circ> g') M"
le_of_key_order_def: shows "le_of_key_order \<equiv> map_fun key_compare id le_of_comp"
thm_2_3_6: fixes X :: "'a option set"   and fs :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) list"   and b :: "nat" assumes "finite X"   and "set fs \<subseteq> P[X]"   and "extended_ord.is_GB_cofactor_bound (set fs) b" shows "extended_ord.punit.is_Groebner_basis (set (extended_ord.punit.Macaulay_list (extended_ord.deg_shifts X b fs)))"
free_strict_monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "free_strict_monoidal_category C \<equiv> monoidal_language C \<and> free_monoidal_category C \<and> full_subcategory (free_monoidal_category.comp C) (\<lambda>f. partial_magma.ide (free_monoidal_category.comp C) f \<and> monoidal_language.Diag C (monoidal_language.Dom C (free_monoidal_category.rep C f)))"
congruentI: fixes r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z" shows "f respects r"
sublist_primrec: shows "sublist = (\<lambda>xs ys. rec_list (\<lambda>xs. xs = []) (\<lambda>y ys ysa xs. prefix xs (y # ys) \<or> ysa xs) ys xs)"
assms: shows "inj_on f {0..<m} \<and> f ` {0..<m} \<subseteq> {0..<n}"
is_cd: shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> k'"
Basis: fixes s :: "'c set"   and S :: "'c set set" assumes "s \<in> S" shows "generate_topology_on S s"
X: shows "X = u\<^sub>X *\<^sub>R S + (1 - u\<^sub>X) *\<^sub>R T"
nvPB_def: shows "nvPB \<equiv> \<Union> (nvC ` \<Phi>)"
simps: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and a1 :: "'b list"   and a2 :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.eligible S a1 a2 = (\<exists>DA As. a1 = As \<and> a2 = DA \<and> (S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> ground_resolution_with_selection.maximal_wrt (As ! 0) DA))"
Cod_Src: fixes t :: "'a local.term" assumes "Arr t" shows "Cod (Src t) = Src t"
numbound0_gen: fixes t :: "MIR.num"   and x :: "real"   and bs :: "real list" assumes "numbound0 t"   and "isint t (x # bs)" shows "\<forall>y. isint t (y # bs)"
P_accD_1: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b"   and K1 :: "'a \<Rightarrow> 'a pmf"   and K2 :: "'b \<Rightarrow> 'b pmf" assumes "((x1, x2), y1, y2) \<in> (SIGMA x:UNIV. set_pmf (MC_pair.Kp K1 K2 x))\<^sup>*" shows "(x1, y1) \<in> (SIGMA x:UNIV. set_pmf (K1 x))\<^sup>*"
L_3: shows "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
dg_prod_3_ArrI: fixes h :: "V"   and g :: "V"   and f :: "V" assumes "h \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "g \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"   and "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "[h, g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Arr\<rparr>"
power_set_Suc: fixes A :: "'a set"   and n :: "nat" shows "A *^ Suc n = A ** A *^ n"
inv_ordermap_VWF_mono_le: fixes M :: "V set"   and \<alpha> :: "V"   and \<beta> :: "V" assumes "inv_into M (ordermap M VWF) \<alpha> \<le> inv_into M (ordermap M VWF) \<beta>"   and "M \<subseteq> ON"   and "small M"   and "\<alpha> \<in> elts (tp M)"   and "\<beta> \<in> elts (tp M)" shows "\<alpha> \<le> \<beta>"
def_body': fixes x :: "'a"   and y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" assumes "x \<noteq> \<bottom>"   and "x \<noteq> null"   and "d\<^sub>y y \<tau> = true \<tau>" shows "g x (y \<tau>) \<noteq> \<bottom> \<and> g x (y \<tau>) \<noteq> null"
m_closed: fixes x :: "'b"   and y :: "'b" assumes "x \<in> carrier (G i)"   and "y \<in> carrier (G i)" shows "x \<otimes>\<^bsub>G i\<^esub> y \<in> carrier (G i)"
simps: fixes gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" shows "local.inline1 gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). local.inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))"
b'_def: shows "b' = Cn 3 b [recf.Id 3 2, recf.Id 3 0, recf.Id 3 1]"
assms_1: shows "x \<in>\<^sub>\<circ> A \<union>\<^sub>\<circ> B"
subst_elimI': fixes v :: "'b"   and \<theta> :: "'a \<Rightarrow> ('c, 'b) Term.term" assumes "\<And>w. v \<notin> fv (Var w \<cdot> \<theta>)" shows "subst_elim \<theta> v"
ode_d_Suc_0_eq_ode_d2: fixes x :: "'c" assumes "x \<in> Csafe" shows "ode_d (Suc 0) x = blinfun_apply (ode_d2 x)"
linear_def: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" shows "Vector_Spaces.linear s1 s2 f \<equiv> vector_space s1 \<and> vector_space s2 \<and> module_hom s1 s2 f"
abs_fresh_6: fixes b :: "vrs"   and a :: "tyvrs"   and x :: "'a" shows "b \<sharp> [a].x = b \<sharp> x"
l3_step4_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "l3_step4 Rb A B Ni gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni \<and> implInsec A B (hmac \<langle>Number 1, Exp Gen (NonceF (Rb $ ny)), gnx, Messages.Agent A, Messages.Agent B\<rangle> (hmac Ni (NonceF (Rb $ nr)))) \<in> ik s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xni, xnr, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gnx (NonceF (Rb $ ny)) \<and> Rb = test} \<union> secret s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) else signalsResp s, signalsResp2 := if can_signal s A B then addSignal (signalsResp2 s) (Commit A B (Exp gnx (NonceF (Rb $ ny)))) else signalsResp2 s\<rparr>}"
ref_pop_hd: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<noteq> \<epsilon>"   and "us \<in> lists \<langle>G\<rangle>" shows "Ref G us = (Dec G hd us) \<cdot> Ref G tl us"
taylor_expansion_ring_hom: fixes c :: "'a" assumes "c \<in> carrier R" shows "taylor_expansion R c \<in> ring_hom P P"
simps_2: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat"   and ys :: "nat list"   and s :: "nat" shows "cond2 a b (y # ys) s = (y \<le> Max (set a) \<and> s \<le> a \<bullet> map (max_x_impl' a b (y # ys)) [0..<length a])"
drop_all: fixes xs :: "'d list"   and n :: "nat" assumes "length xs \<le> n" shows "drop n xs = []"
gs_hs: shows "image_mset pl.Mp (mset gs) = hs1 + hs2"
Rel_def: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "Transfer.Rel r \<equiv> r"
has_dom_perm_swap: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set" shows "has_dom (perm_swap a b f) S = has_dom f ((a \<rightleftharpoons>\<^sub>F b) ` S)"
pf_short_count_def: fixes n :: "nat" shows "pf_short_count n = probGn p n (\<lambda>es. real n / 2 \<le> real (short_count (edge_space.edge_ugraph n es)))"
3: shows "App (chop (apps (Hd (head t)) (butlast (args t)))) (last (args t)) = chop t"
round_up_correct: fixes e :: "int"   and f :: "real" shows "round_up e f - f \<in> {0..2 powr - real_of_int e}"
vsv_Limit_vrange_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "vfinite (\<D>\<^sub>\<circ> r)" shows "\<R>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"
HVZK_unfold2_alt: assumes "\<Sigma>0.\<Sigma>_protocol" shows "\<forall>h a e z. e \<in> carrier L \<longrightarrow> h \<in> valid_pub0 \<longrightarrow> (a, z) \<in> set_spmf (S0_raw h e) \<longrightarrow> check0 h a e z"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs, Undecided\<rangle> \<Rightarrow> t"
cprank_max1_order0: fixes B :: "'b tensor"   and A :: "'b tensor" assumes "cprank_max1 B"   and "order A = 0" shows "cprank_max1 (A \<otimes> B)"
round_up_mono: fixes x :: "real"   and y :: "real"   and p :: "int" assumes "x \<le> y" shows "round_up p x \<le> round_up p y"
closure_mono: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "closure S \<subseteq> closure T"
deriv_preserves: fixes r :: "'b rexp"   and s :: "'b rexp"   and as :: "'b" assumes "\<guillemotleft>r\<guillemotright> = \<guillemotleft>s\<guillemotright>" shows "\<guillemotleft>deriv as r\<guillemotright> = \<guillemotleft>deriv as s\<guillemotright>"
feasible_bounded_by: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and b :: "real" assumes "feasible t"   and "sound P"   and "bounded_by b P" shows "bounded_by b (t P)"