hcju/isabelleps huggingface dataset:
continuous_within_csqrt: fixes z :: "complex"   and s :: "complex set" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "continuous (at z within s) csqrt"
intros: fixes n :: "nat"   and g :: "nat" assumes "0 < n"   and "coprime n g"   and "ord n g = totient n" shows "residue_primroot n g"
nth_prime_def: fixes n :: "nat" shows "nth_prime n = (THE p. prime p \<and> card {q. prime q \<and> q < p} = n)"
assms_3: shows "y \<noteq> \<infinity>\<^sub>h"
one_FactGroup: fixes G :: "('b, 'c) monoid_scheme"   and N :: "'b set" shows "\<one>\<^bsub>G Mod N\<^esub> = N"
subgroup_subgroup_generated_iff: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set"   and B :: "'b set" assumes "Group.group G" shows "subgroup H (subgroup_generated G B) = (subgroup H G \<and> H \<subseteq> carrier (subgroup_generated G B))"
hf_of_unit_def: fixes u :: "unit" shows "hf_of u \<equiv> 0"
coprime_mult_self_left_iff: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "coprime (c * a) (c * b) = (is_unit c \<and> coprime a b)"
intruder_synth_induct: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term"   and P :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set \<Rightarrow> (('fun, 'atom, 'sets) prot_fun, 'a) Term.term \<Rightarrow> bool" assumes "M \<turnstile>\<^sub>c t"   and "\<And>t. t \<in> M \<Longrightarrow> P M t"   and "\<And>T f. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)" shows "P M t"
filter_eq_ConsD: fixes P :: "'g \<Rightarrow> bool"   and ys :: "'g list"   and x :: "'g"   and xs :: "'g list" assumes "filter P ys = x # xs" shows "\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
is_free: fixes D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T\<^sub>D :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha>\<^sub>D :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota>\<^sub>D :: "'b"   and V :: "'c \<Rightarrow> 'b"   and F :: "'c local.term set \<Rightarrow> 'b" assumes "strict_monoidal_extension_to_free_monoidal_category C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V F" shows "F = FreeMonoidalCategory.evaluation_functor.map C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"
returns_result_select_result: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog" assumes "h \<turnstile> ok f" shows "h \<turnstile> f \<rightarrow>\<^sub>r |h \<turnstile> f|\<^sub>r"
GreenThm_typeI_divisible_region_boundary_gen: fixes \<gamma> :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "valid_typeI_division s two_chain_typeI"   and "\<forall>twoC\<in>two_chain_typeI. analytically_valid (cubeImage twoC) (\<lambda>a. F a \<bullet> i) j"   and "only_vertical_division \<gamma> two_chain_typeI" shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
f3: fixes l :: "'a \<Longrightarrow> 'b"   and b :: "'b" shows "put\<^bsub>l\<^esub> b (get\<^bsub>l\<^esub> b) = b \<or> \<not> vwb_lens l"
P4_a: fixes x :: "'a" shows "x l\<rightarrow> (1::'a) = (1::'a)"
that: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets M"   and "\<Union> (range A) = space M"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "thesis"
radiant_of: fixes p :: "(real, 2) vec" shows "0 \<le> radiant_of p \<and> radiant_of p < 2 * pi \<and> of_radiant (radiant_of p) = normalize p"
standard_maintains_rules: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and L :: "'b Standard_Constant set" assumes "standard' C G" shows "maintainedA (standard_rules C L) G"
assms_2: shows "appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB F0 decisionVars \<or> appliedBackjump stateA stateB"
left_right_inverse_eq: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd"   and h :: "'d \<Rightarrow> 'c" assumes "f \<circ> g = id"   and "g \<circ> h = id" shows "f = h"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> 'b"   and BL :: "nat \<Rightarrow> 'b option"   and BU :: "nat \<Rightarrow> 'b option"   and IL :: "nat \<Rightarrow> 'a"   and IU :: "nat \<Rightarrow> 'a" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>b\<^sub>e ((BL, BU), IL, IU) = ((\<forall>x c. BL x = Some c \<longrightarrow> IL x \<in> I \<longrightarrow> v x = c) \<and> (\<forall>x c. BU x = Some c \<longrightarrow> IU x \<in> I \<longrightarrow> v x = c))"
finite_OUT_g: fixes x :: "'v" assumes "x \<noteq> sink \<Delta>" shows "d_OUT g x \<noteq> \<top>"
leg0_commutes: shows "f.cod.leg0 \<cdot> f.chine = f.leg0"
tiny_digraph_def: fixes \<alpha> :: "V"   and \<CC> :: "V" shows "tiny_digraph \<alpha> \<CC> \<equiv> (\<Z> \<alpha> \<and> vfsequence \<CC>) \<and> vsv (\<CC>\<lparr>Dom\<rparr>) \<and> vsv (\<CC>\<lparr>Cod\<rparr>) \<and> tiny_digraph_axioms \<alpha> \<CC>"
simps: fixes a1 :: "'d list"   and a2 :: "'d list" shows "cancel1 a1 a2 = (\<exists>xs gs gs'. a1 = gs @ xs @ xs @ gs' \<and> a2 = gs @ xs @ gs' \<and> xs \<noteq> [])"
simps_1: fixes uu :: "'a" shows "fold_In uu [] = Bc False"
Perp_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B Perp C D \<equiv> \<exists>X. X PerpAt A B C D "
2: shows "\<guillemotleft>\<nu> : f \<star>\<^sub>B src\<^sub>B f \<rightarrow>\<^sub>B f' \<star>\<^sub>B src\<^sub>B f'\<guillemotright>"
monom_one: shows "MPoly_Type.monom 0 (1::'a) = 1"
U: shows "UNIV = (\<lambda>(((a, b), c), d). \<lparr>es = a, ps = b, pubActs = c, \<dots> = d\<rparr>) ` (((UNIV \<times> UNIV) \<times> UNIV) \<times> UNIV)"
invertible_Gauss_Jordan_up_to_k: fixes A :: "(('a, 'c) vec, 'b) vec"   and k :: "nat" shows "\<exists>P. invertible P \<and> Gauss_Jordan_upt_k A k = P ** A"
antisym_type: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<le> \<sigma>"   and "\<sigma> \<le> \<tau>" shows "\<tau> = \<sigma>"
widen1: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.WN le join Top widen narrow" shows "le x (widen x y)"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (vc C\<^sub>1 Q \<and> vc C\<^sub>2 Q)"
downset_prop: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "ord.downset less_eq x = {y. less_eq y x}"
accessible_imp_states: fixes q :: "hf" assumes "q \<in> accessible" shows "q \<in> states M"
set_ConsD: fixes y :: "'f"   and x :: "'f"   and xs :: "'f list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_mult_isom hom" shows "comm_monoid_mult_isom hom"
scalar_prod_left_unit: fixes v :: "'a Matrix.vec"   and n :: "nat"   and i :: "nat" assumes "v \<in> carrier_vec n"   and "i < n" shows "unit_vec n i \<bullet> v = v $v i"
rec_newleft1_def: shows "rec_newleft1 = recf.id 2 0"
lborel_distr_mult': fixes c :: "real" assumes "c \<noteq> 0" shows "lborel = density (distr lborel borel ((*) c)) (\<lambda>x. ennreal \<bar>c\<bar>)"
left_idem: fixes a :: "'c option"   and b :: "'c option" shows "least a (least a b) = least a b"
set_iterator_linord_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'c set" assumes "class.linorder less_eq less" shows "linorder.set_iterator_linord less_eq it S0 \<equiv> set_iterator_genord it S0 less_eq"
P''''Trans: shows "\<Psi> \<rhd> P'''' \<longmapsto> \<alpha> \<prec> P'''"
rebase_id: fixes x :: "'a mod_ring" assumes "rebase_ge TYPE('a) TYPE('b)" shows "@@x = @x"
WTrt_elim_cases_10: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and v :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> C\<bullet>\<^sub>sF{D} := v : T" obtains T\<^sub>2 :: "ty" and T :: "ty" where "Ta__ = Void" and "P,E,h,sh \<turnstile> v : T\<^sub>2" and "P \<turnstile> C has F,Static:T in D" and "P \<turnstile> T\<^sub>2 \<le> T"
disjnt_empty1: fixes A :: "'b set" shows "disjnt {} A"
tuple_def: fixes h :: "'a"   and f :: "'a"   and g :: "'a"   and k :: "'a" shows "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> \<equiv> if commutative_square f g h k then THE l. \<p>\<^sub>0[f, g] \<cdot> l = k \<and> \<p>\<^sub>1[f, g] \<cdot> l = h else null"
select_convs_4: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_dst \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = p_dst"
vdisjnt_sym: fixes A :: "V"   and B :: "V" assumes "vdisjnt A B" shows "vdisjnt B A"
cr_list_def: shows "cr_list \<equiv> \<lambda>x y. x = list_to_list_R1 y"
\<alpha>_def: fixes \<mu>\<nu>\<tau> :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.\<alpha> \<mu>\<nu>\<tau> \<equiv> Span.assoc (fst \<mu>\<nu>\<tau>) (fst (snd \<mu>\<nu>\<tau>)) (snd (snd \<mu>\<nu>\<tau>))"
e2ennreal_mono: fixes x :: "ereal"   and y :: "ereal" assumes "x \<le> y" shows "e2ennreal x \<le> e2ennreal y"
parts_valid_LtKeys_disjoint: fixes I :: "msg set" assumes "I \<subseteq> valid" shows "parts I \<inter> range LtK = {}"
eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(x = y) = (x \<le>s y \<and> y \<le>s x)"
N2_def: shows "N2 = {u. (y, u) \<in> E'\<^sup>*}"
swap_add_edge_in_path: fixes G :: "('c, 'd) graph"   and a :: "'c"   and w :: "'d"   and b :: "'c"   and v :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and v' :: "'c"   and w' :: "'d"   and a' :: "'c"   and w'' :: "'d" assumes "valid_graph G"   and "is_path_undir (add_edge a w b G) v p v'"   and "(a, w', a') \<in> edges G \<or> (a', w', a) \<in> edges G" shows "nodes_connected (add_edge a' w'' b G) v v'"
sup_ge1: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<subseteq># x \<union># y"
\<T>e_ide_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b"   and b :: "'a" assumes "yoneda_lemma_fixed_e C S \<phi> F a e"   and "partial_magma.ide (dual_category.comp C) b" shows "partial_magma.arr S (transformation_by_components.map (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) (yoneda_lemma.\<T>o C S \<phi> F a e) b)"
mult_less_iff1: fixes z :: "'b"   and x :: "'b"   and y :: "'b" assumes "(0::'b) < z" shows "(x * z < y * z) = (x < y)"
mset_nondetMs_2: fixes return :: "'a multiset \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a multiset \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind" shows "monad_fail (nondetM_base.return_nondet return (\<lambda>x. {#x#})) (nondetM_base.bind_nondet bind (mmerge return bind)) (nondetM_base.fail_nondet return {#})"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and w :: "'c"   and \<theta> :: "'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and u :: "'c" assumes "partial_magma.ide V w \<and> partial_magma.in_hom V \<theta> (H f w) u" shows "uw\<theta>_axioms V H f u w \<theta>"
from_list_def: shows "Subx.Fenv.from_list \<equiv> Subx.Fenv.add_list Fubx_empty"
reachable_Mn: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list"   and z :: "nat" assumes "valid ((Mn n f, xs, []) # rest)"   and "\<And>xs rest. valid ((f, xs, []) # rest) \<Longrightarrow> correct ((f, xs, []) # rest, None)"   and "\<forall>y<z. eval f (y # xs) \<notin> {None, Some 0}" shows "reachable ((Mn n f, xs, []) # rest, None) ((f, z # xs, []) # (Mn n f, xs, [z]) # rest, None)"
invimage_vlrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A) -`\<^sub>\<circ> B = A \<inter>\<^sub>\<circ> r -`\<^sub>\<circ> B"
B_def: shows "B = (\<Sum>n. norm (fds_nth f (n + 2)) / real (n + 2) powr c) * 2 powr c"
assms: fixes i :: "'b" shows "((\<lambda>x. blinfun_apply (f x) i) has_derivative (\<lambda>x. f' y x i)) (at y)"
Keys_mono: fixes A :: "('d \<Rightarrow>\<^sub>0 'e) set"   and B :: "('d \<Rightarrow>\<^sub>0 'e) set" assumes "A \<subseteq> B" shows "Keys A \<subseteq> Keys B"
assoc'_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and h :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.ide (span_vertical_category.vcomp C) g"   and "partial_magma.ide (span_vertical_category.vcomp C) h"   and "span_vertical_category.src C f = span_vertical_category.trg C g"   and "span_vertical_category.src C g = span_vertical_category.trg C h" shows "partial_magma.in_hom (span_vertical_category.vcomp C) \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' C prj0 prj1 f g h, Dom = Cod (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h)), Cod = Dom (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h)\<rparr> (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h)) (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h)"
Some_eq_map_of_iff: fixes xys :: "('c \<times> 'd) list"   and y :: "'d"   and x :: "'c" assumes "distinct (map fst xys)" shows "(Some y = map_of xys x) = ((x, y) \<in> set xys)"
WT_gpv_ContD: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv"   and input :: "'e" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out" shows "\<Gamma> \<turnstile>g c input \<surd>"
normal_inv_iff: fixes N :: "'a set" shows "N \<lhd> G\<lparr>carrier := H\<rparr> = (subgroup N (G\<lparr>carrier := H\<rparr>) \<and> (\<forall>x\<in>carrier (G\<lparr>carrier := H\<rparr>). \<forall>h\<in>N. x \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> h \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> x \<in> N))"
simps_2: fixes x :: "'h" shows "option_of_exception (OK x) = Some x"
map_of_eq: shows "map_of (\<Delta> @ \<Gamma>) = map_of (\<Gamma> @ \<Delta>)"
qtable_empty: fixes n :: "nat"   and A :: "nat set"   and P :: "'b option list \<Rightarrow> bool"   and Q :: "'b option list \<Rightarrow> bool" assumes "\<And>x. \<lbrakk>wf_tuple n A x; P x; Q x\<rbrakk> \<Longrightarrow> False" shows "qtable n A P Q empty_table"
distinct_9: fixes x1 :: "Vars.var"   and x61 :: "exp"   and x62 :: "exp"   and x63 :: "exp" shows "Var x1 \<noteq> (x61 ? x62 : x63)"
abs_if_raw: shows "abs \<equiv> \<lambda>a. if a < (0::'a) then - a else a"
ack_less_ack_Suc1: fixes i :: "nat"   and j :: "nat" shows "ack i j < ack (Suc i) j"
CDMode_CTRL_Init_def: shows "CDMode_CTRL_Init = ''Playing''"
subset_helper_R: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list" shows "\<forall>x\<in>set (characterize_consistent_signs_at_roots p (qs1 @ qs2)). \<exists>x1\<in>set (characterize_consistent_signs_at_roots p qs1). \<exists>x2\<in>set (characterize_consistent_signs_at_roots p qs2). x = x1 @ x2"
ord_eq_le_trans: fixes a :: "'e"   and b :: "'e"   and c :: "'e" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
assms_2: fixes \<sigma> :: "'b \<Rightarrow> 'a"   and a :: "'c" assumes "PO \<sigma> a" shows "QO \<sigma> a"
ennreal_numeral_less_top: fixes i :: "num" shows "numeral i < \<top>"
simps: fixes i :: "int" shows "the_Intg (Intg i) = i"
Lin1: assumes "Lin s s' \<and> t = t'"   and "a = Linearize 0" shows "thesis"
left_total_rel_converter: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "left_unique A"   and "right_total A"   and "left_total B"   and "left_total C"   and "left_unique R"   and "right_total R" shows "left_total (rel_converter A B C R)"
finite_Lim_measure_decseq: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> events"   and "decseq A" shows "(\<lambda>n. prob (A n)) \<longlonglongrightarrow> prob (\<Inter> (range A))"
FAssThrow1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
b_e_type_value2: fixes \<C> :: "t_context"   and v1 :: "v"   and v2 :: "v"   and t1s :: "t list"   and t2s :: "t list" assumes "\<C> \<turnstile> [C v1, C v2] : t1s _> t2s" shows "t2s = t1s @ [typeof v1, typeof v2]"
value_of_zero: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v \<zero> = \<infinity>"
to_hma\<^sub>v_def: fixes v :: "'b Matrix.vec" shows "to_hma\<^sub>v v = (\<chi>i. v $v Bij_Nat.to_nat i)"
is_on_stack_impl_def: fixes SBIP :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list"   and v :: "'b" shows "GS.is_on_stack_impl SBIP v \<equiv> case GS.I SBIP v of None \<Rightarrow> False | Some (STACK x) \<Rightarrow> True | Some DONE \<Rightarrow> False"
InfCard_cdouble_eq: fixes \<kappa> :: "V" assumes "InfCard \<kappa>" shows "\<kappa> \<oplus> \<kappa> = \<kappa>"
split_minD: fixes t :: "'b tree23"   and x :: "'b"   and t' :: "'b upD" assumes "split_min t = (x, t')"   and "complete t"   and "0 < height t" shows "x # inorder (treeD t') = inorder t"
units_of_units: fixes G :: "('c, 'd) monoid_scheme" shows "Units (units_of G) = Units G"
reflpI: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
M_def: fixes k :: "nat" shows "gs1.M k = mat k k (\<lambda>(x, y). gs1.\<mu> x y)"
lists_accI: fixes xs :: "'a list"   and r :: "('a \<times> 'a) set" assumes "xs \<in> Wellfounded.acc (listrel1 r)" shows "xs \<in> lists (Wellfounded.acc r)"
remdups_eq_nil_right_iff: fixes x :: "'e list" shows "([] = remdups x) = (x = [])"
sup_top_left: fixes x :: "'a" shows "top \<squnion> x = top"
less_eq: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (\<exists>c. a = c * b)"
frac_less: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and w :: "'d"   and z :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less x y"   and "less zero w"   and "less_eq w z" shows "less (divide x z) (divide y w)"
sc_assoc: fixes a :: "'b"   and b :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "m \<in> carrier M" shows "(a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) \<cdot>\<^sub>s m = a \<cdot>\<^sub>s (b \<cdot>\<^sub>s m)"
norm_lt: fixes x :: "'a"   and y :: "'b" shows "(norm x < norm y) = (x \<bullet> x < y \<bullet> y)"
f_def: shows "f = restrict1 (canonical_retraction A D \<circ> canonical_retraction B C) (\<Union> A)"
coeff_Var: fixes p :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (Var p) m = ((1::'b) when m = monomial 1 p)"
mm_equiv_intro: 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 dma :: "'b \<Rightarrow> Sec"   and \<R> :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and lc\<^sub>1 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)"   and lc\<^sub>2 :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.strong_low_bisim_mm dma eval \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "(lc\<^sub>1, lc\<^sub>2) \<in> sifum_security.mm_equiv dma eval"
tObsC_tLength: fixes t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace" assumes "tObsC t = tObsC t'" shows "tLength t = tLength t'"
true_cls_union: fixes I :: "'b set"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "(I \<Turnstile> C + D) = (I \<Turnstile> C \<or> I \<Turnstile> D)"
lemma_4_5_vi: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion>2 b r\<rightarrow> c = (a r\<rightarrow> c) \<sqinter> (b r\<rightarrow> c)"
le_neq_trans: fixes a :: "'c list"   and b :: "'c list" assumes "suffix a b"   and "a \<noteq> b" shows "strict_suffix a b"
sum_inverse_squares_tail_bound_2: fixes d :: "real" assumes "0 < d" shows "(\<Sum>n. 1 / (real (Suc n) + d)\<^sup>2) \<le> 1 / d"
convex_polychain_ConsD: fixes x :: "(real \<times> real) \<times> real \<times> real"   and xs :: "((real \<times> real) \<times> real \<times> real) list" assumes "convex_polychain (x # xs)" shows "convex_polychain xs"
fun_eq_iff: fixes f :: "'k \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l" shows "(f = g) = (\<forall>x. f x = g x)"
diff_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs -- ys = fold removeAll ys xs"
le_TimesR: fixes r :: "'a language"   and s :: "'a language" assumes "\<oo> r" shows "s \<le> Times r s"
funpow_simps_right_1: fixes f :: "'c \<Rightarrow> 'c" shows "f ^^ 0 = id"
preservation_and_closures_1: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> bool" assumes "rel_preserves_pred Rel Pred" shows "rel_preserves_pred (Rel\<^sup>=) Pred"