hcju/isabelleps huggingface dataset:
LI_infinity: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "(LI K v I = \<infinity>) = (I = {\<zero>\<^bsub>Vr K v\<^esub>})"
isFun_def: fixes f :: "ZF" shows "isFun f \<equiv> \<forall>x y1 y2. Opair x y1 |\<in>| f \<and> Opair x y2 |\<in>| f \<longrightarrow> y1 = y2"
\<tau>external'_def: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal" shows "\<tau>external' P h a M = (\<exists>hT. typeof_addr h a = \<lfloor>hT\<rfloor> \<and> \<tau>external P hT M)"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('a, 'b) atm" shows "F.satL \<xi> (Neg a) = (\<not> F.satA \<xi> a)"
kill_short_uwellformed: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "finite (uverts G)"   and "uwellformed G" shows "uwellformed (kill_short G k)"
ih1: fixes a :: "coname"   and M :: "trm"   and M' :: "trm" assumes "<a>:M \<in> \<parallel><A>\<parallel>"   and "M \<longrightarrow>\<^sub>a* M'" shows "<a>:M' \<in> \<parallel><A>\<parallel>"
cut_greater_subset_mono: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" assumes "A \<subseteq> B" shows "A \<down>> t \<subseteq> B \<down>> t"
add_right_mono_le: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and z :: "'a" assumes "a \<le> b"   and "b + c \<le> z" shows "a + c \<le> z"
SUCC_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm" assumes "H \<turnstile> t EQ t'" shows "H \<turnstile> SUCC t EQ SUCC t'"
a_rcos_module: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> carrier G"   and "x' \<in> carrier G" shows "(x' \<in> H +> x) = (x' \<oplus> \<ominus> x \<in> H)"
gval_In_cons: fixes v :: "'a"   and a :: "value"   and as :: "value list"   and s :: "'a \<Rightarrow> value option" shows "gval (In v (a # as)) s = gval (Eq (V v) (L a)) s \<or>? gval (In v as) s"
Act_eqvt: fixes p :: "perm"   and \<alpha> :: "'d"   and x :: "('b, 'c, 'd) formula" shows "p \<bullet> Act \<alpha> x = Act (p \<bullet> \<alpha>) (p \<bullet> x)"
assms_1: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
ivl_integral_has_vector_derivative: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and x :: "real" assumes "continuous_on (closed_segment a b) f"   and "x \<in> closed_segment a b" shows "((\<lambda>u. ivl_integral a u f) has_vector_derivative f x) (at x within closed_segment a b)"
distinct_path_intro: fixes x1 :: "'v"   and x2 :: "'w"   and x3 :: "'v"   and v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "valid_unMultigraph.is_trail (del_unEdge x1 x2 x3 G) v ps v'" shows "is_trail v ps v'"
fds_eq_fds_iff: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "(fds f = fds g) = (\<forall>n>0. f n = g n)"
mk_lossless_idem: fixes p :: "'h spmf" shows "mk_lossless (mk_lossless p) = mk_lossless p"
somegcd_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" shows "somegcd G a b = (SOME x. x \<in> carrier G \<and> x gcdof\<^bsub>G\<^esub> a b)"
run_put_writer: fixes put :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'d"   and m :: "('b, 'c, 'a) writerT" shows "run_writer (put_writer put s m) = put s (run_writer m)"
phis'_codem_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val"   and next' :: "'val"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" shows "phis'_codem g next next' nodes_of_phis = fold (\<lambda>n. Mapping.update n (map (id(snd next := next')) (the (Mapping.lookup (phis g) n)))) (sorted_list_of_set (case Mapping.lookup nodes_of_phis (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> Set.remove next x)) (Mapping.delete next (phis g))"
comp_fun_commute_axioms: shows "comp_fun_commute (\<otimes>)"
le_iff_add: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq a b = (\<exists>c. b = plus a c)"
rep_eq: fixes x :: "'b set"   and xa :: "'b ttree" shows "Rep_ttree (ttree_restr x xa) = filter (\<lambda>x'. x' \<in> x) ` Rep_ttree xa"
disconnected_nodes_eq: fixes document_ptr :: "(_) document_ptr"   and disconnected_nodes :: "(_) node_ptr list" assumes "document_ptr \<noteq> owner_document" shows "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disconnected_nodes = h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disconnected_nodes"
sign_nz: fixes p :: "'b \<Rightarrow> 'b" shows "sign p \<noteq> 0"
Par_transT_invert: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and phi :: "bool" assumes "(c1 | c2, s) \<rightarrow>t s'" shows "phi"
ring_simprules_6: fixes R :: "('e, 'f) ring_scheme" assumes "ring R" shows "\<one>\<^bsub>R\<^esub> \<in> carrier R"
maxdeg_homogenize: fixes x :: "'c"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "x \<notin> \<Union> (indets ` F)" shows "maxdeg (homogenize x ` F) = maxdeg F"
compat_snoc: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list"   and x :: "'a" assumes "ext gt ys xs" shows "ext gt (ys @ [x]) (xs @ [x])"
incomplete_designI: assumes "\<k> < int (card \<V>\<^sup>+)" shows "incomplete_design \<V>\<^sup>+ \<B>\<^sup>+ \<k>"
less_imp_neq: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "x \<noteq> y"
eq_inf_concat_end: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and w :: "'action stream" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 @- w =\<^sub>I w\<^sub>2 @- w"
first_def: fixes s :: "nat \<Rightarrow> 'a" shows "first s \<equiv> s 0"
is_natural_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'c"   and G :: "'a \<Rightarrow> 'c"   and \<tau> :: "'a \<Rightarrow> 'c"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = \<tau> f"
not_in_member: fixes a :: "((int \<times> int) set set, int) Combinators"   and b :: "((int \<times> int) set set, int) Combinators"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "member a b"   and "x \<notin> dom (C b)" shows "x \<notin> dom (C a)"
unit_factor_eq_0_iff: fixes a :: "'a" shows "(unit_factor a = (0::'a)) = (a = (0::'a))"
inf_eq_top_iff: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "(inf x y = top) = (x = top \<and> y = top)"
tllist_all2_eq: shows "tllist_all2 (=) (=) = (=)"
bs\<^sub>0_def: shows "bs\<^sub>0 \<equiv> (0::'a, 0::'b, \<lambda>_. INIT)"
category_with_finite_products_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "category_with_finite_products_axioms C \<equiv> \<forall>I. finite I \<longrightarrow> category.has_products C I"
Suc: fixes l :: "bit list" shows "length (selecthelp l x) = x + 1"
emeasure_UN_countable: fixes I :: "'a set"   and X :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M"   and "countable I"   and "disjoint_family_on X I" shows "emeasure M (\<Union> (X ` I)) = \<integral>\<^sup>+ i. emeasure M (X i) \<partial>count_space I"
Gr_univalent: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "univalent (BNF_Def.Gr A f)"
oInv_equality': fixes a :: "ordinal"   and x :: "ordinal" assumes "a = F x" shows "oInv F a = x"
cInf_superset_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_below less_eq B"   and "A \<subseteq> B" shows "less_eq (Inf B) (Inf A)"
cases: fixes a1 :: "state"   and a2 :: "com"   and a3 :: "state" assumes "a1 -a2\<rightarrow> a3" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = sa__" and "a2 = Do f__" and "a3 = ta__" and "ta__ \<in> f__ sa__"   | s0 :: "state" and c1 :: "com" and s1 :: "state" and c2 :: "com" and s2 :: "state" where "a1 = s0" and "a2 = (c1__; c2__)" and "a3 = s2" and "s0 -c1__\<rightarrow> s1" and "s1 -c2__\<rightarrow> s2"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and t :: "state" and c2 :: "com" where "a1 = sb__" and "a2 = IF ba__ THEN c1a__ ELSE c2a__" and "a3 = tb__" and "ba__ sb__" and "sb__ -c1a__\<rightarrow> tb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and t :: "state" and c1 :: "com" where "a1 = sc__" and "a2 = IF bb__ THEN c1 ELSE c2" and "a3 = tc__" and "\<not> bb__ sc__" and "sc__ -c2\<rightarrow> tc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = sd__" and "a2 = WHILE bc__ DO ca__" and "a3 = sd__" and "\<not> bc__ sd__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and t :: "state" and u :: "state" where "a1 = se__" and "a2 = WHILE b DO cb__" and "a3 = u" and "b se__" and "se__ -cb__\<rightarrow> td__" and "td__ -WHILE b DO cb__\<rightarrow> u"   | s :: "state" and p :: "pname" and t :: "state" where "a1 = sf__" and "a2 = CALL p" and "a3 = te__" and "sf__ -body p\<rightarrow> te__"   | f :: "state \<Rightarrow> state" and s :: "state" and c :: "com" and t :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = s" and "a2 = LOCAL f; c; g" and "a3 = g s t" and "f s -c\<rightarrow> t"
pref_prod_cancel': fixes u :: "'b list"   and p :: "'b list"   and w :: "'b list"   and q :: "'b list" assumes "u \<le>p p \<cdot> w \<cdot> q"   and "\<^bold>|p\<^bold>| < \<^bold>|u\<^bold>|"   and "\<^bold>|u\<^bold>| \<le> \<^bold>|p \<cdot> w\<^bold>|" obtains r :: "'b list" where "u = p \<cdot> r" and "r \<le>p w" and "r \<noteq> \<epsilon>"
comp_weak_lens: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'b \<Longrightarrow> 'c" assumes "weak_lens x"   and "weak_lens y" shows "weak_lens (x ;\<^sub>L y)"
green_typeII_cube_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real" shows "green_typeII_cube i j twoC F \<equiv> R2 i j \<and> green_typeII_cube_axioms i j twoC F"
Throw: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and tn :: "qtname" assumes "E,dt\<Turnstile>e\<Colon>-Class tn"   and "prg E\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable" shows "E,dt\<Turnstile>Throw e\<Colon>\<surd>"
nn_integral_monotone_convergence_SUP: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "incseq f"   and "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Squnion>i. f i x) \<partial>M = (\<Squnion>i. integral\<^sup>N M (f i))"
simps: fixes a1 :: "'a OCL_Types.type"   and a2 :: "'a OCL_Types.type" shows "element_type a1 a2 = ((\<exists>\<tau>. a1 = Collection \<tau> \<and> a2 = \<tau>) \<or> (\<exists>\<tau>. a1 = Set \<tau> \<and> a2 = \<tau>) \<or> (\<exists>\<tau>. a1 = OrderedSet \<tau> \<and> a2 = \<tau>) \<or> (\<exists>\<tau>. a1 = Bag \<tau> \<and> a2 = \<tau>) \<or> (\<exists>\<tau>. a1 = Sequence \<tau> \<and> a2 = \<tau>))"
inPorts_fset_of: 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 n :: "('b, 'd) graph_node" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "Abstract_Task.inPorts anyP antecedent n = fset_from_list (Abstract_Task.inPorts' anyP antecedent n)"
prod_star_closure: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<le> z\<^sup>\<star>"   and "y \<le> z\<^sup>\<star>" shows "x \<cdot> y \<le> z\<^sup>\<star>"
Exit_not_dyn_standard_control_dependent: fixes n :: "'b"   and as :: "'a list" assumes "n controls\<^sub>s Exit via as" shows "False"
path_continuous_image: fixes g :: "real \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "path g"   and "continuous_on (path_image g) f" shows "path (f \<circ> g)"
iso_cancel_left: fixes f :: "'s"   and g :: "'s"   and g' :: "'s" assumes "local.iso f"   and "f \<cdot> g = f \<cdot> g'"   and "seq f g" shows "g = g'"
upt_append_sing: fixes xs :: "nat list"   and x :: "nat"   and a :: "nat"   and n_vars :: "nat" assumes "xs @ [x] = [a..<n_vars]"   and "a < n_vars" shows "xs = [a..<n_vars - 1] \<and> x = n_vars - 1 \<and> 0 < n_vars"
pred_rel: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c pmf" shows "pred_pmf P x = rel_pmf (eq_onp P) x x"
Inf_pres_map_dual: shows "Inf_pres = Sup_pres \<circ> \<partial>\<^sub>F"
weak_preservation_of_barbs_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_preserves_barbs Rel CWB" shows "rel_weakly_preserves_barbs (Rel\<^sup>=) CWB"
jnf_vector_def: fixes A :: "'b mat" shows "jnf_vector A = ??.Jordan_Normal_Form_Existence.jnf_vector_main (dim_row A) A"
False: shows "vs @ xs \<notin> RM M2 M1 \<Omega> V m i"
eq_fin_range: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "set w\<^sub>1 = set w\<^sub>2"
lexordp_simps_2: fixes xs :: "'a list" shows "lexordp xs [] = False"
Dp_Mp_eq: fixes f :: "int poly" shows "f = q.Mp f + Polynomial.smult (p ^ n) (q.Dp f)"
uminus_add_minus_mat: fixes l :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and r :: "'b mat" assumes "l \<in> carrier_mat nr nc"   and "r \<in> carrier_mat nr nc" shows "- (l + r) = - l - r"
base: assumes "u = v"   and "u \<in> verts G" shows "thesis"
wf_mult: fixes r :: "('b \<times> 'b) set" assumes "wf r" shows "wf (mult r)"
\<phi>: shows "local.iso \<phi> \<and> \<guillemotleft>\<phi> : a \<star> f \<Rightarrow> a \<star> g\<guillemotright>"
nonempty_imp_ex_least: fixes a :: "hf" assumes "a \<noteq> 0" shows "\<exists>x. x \<^bold>\<in> a \<and> (\<forall>y. y \<^bold>\<in> a \<longrightarrow> Rep_hf x \<le> Rep_hf y)"
iso_iff_section_and_epi: fixes f :: "'a \<times> 'a" shows "CC.iso f = (CC.section f \<and> CC.epi f)"
SEQ_iff: fixes f :: "nat \<Rightarrow> 'c"   and A :: "'c set" shows "(f \<in> SEQ A) = (\<forall>i. f i \<in> A)"
zcount_image_zmset_zero: fixes M :: "'a zmultiset"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "\<And>m. m \<in>#\<^sub>z M \<Longrightarrow> f m \<noteq> x" shows "x \<notin>#\<^sub>z Auxiliary.image_zmset f M"
init: fixes \<xi> :: "'b"   and p :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d" assumes "(\<xi>, p) \<in> init A"   and "l \<in> labels \<Gamma> p" shows "P (\<xi>, l)"
hyps_1: fixes G :: "('v, 'w) graph" assumes "n = degree v G"   and "finite (edges G)"   and "valid_unSimpGraph G" shows "degree v G = card (Collect (valid_unMultigraph.adjacent G v))"
map_def: fixes f :: "'a" shows "Cop.map f = (if Cop.arr f then f else Cop.null)"
IH: fixes g :: "complex \<Rightarrow> complex" assumes "i \<le> n"   and "g holomorphic_on t" shows "(deriv ^^ i) (g \<circ> f) z = (deriv ^^ i) g (f z)"
quadratic_hensel_main_def: fixes j :: "nat" shows "local.quadratic_hensel_main j = (case local.quadratic_hensel_loop j of (qq, S, T, D, H) \<Rightarrow> (D, H))"
simps_2: fixes m :: "nat"   and n :: "nat" shows "Suc m + n = Suc (m + n)"
field_Units: shows "Units R = carrier R - {\<zero>\<^bsub>R\<^esub>}"
poly_mod_2_def: fixes m :: "int" shows "poly_mod_2 m \<equiv> 1 < m"
sconst_alt: fixes s :: "'d stream"   and x :: "'d" shows "(s = sconst x) = (sset s = {x})"
finite_imageI: fixes F :: "'g set"   and h :: "'g \<Rightarrow> 'h" assumes "finite F" shows "finite (h ` F)"
spies_Gets_rev: fixes evs :: "event list"   and A :: "agent"   and X :: "msg" shows "knows Spy (evs @ [Gets A X]) = knows Spy evs"
onA: shows "on_triple (f_a \<union> f_b) `` edges A = on_triple (Id_on (vertices A) O (f_a \<union> f_b)) `` edges A"
a1: shows "snd (execute_instr_sub1 i s) = False \<and> s' = snd (fst (execute_instr_sub1 i s)) \<and> get_S (cpu_reg_val PSR s) = 0"
mult_1_left: fixes a :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "Var \<circ>\<^sub>s a = a"
expind_def: shows "expind = (\<lambda>x. f x * indicat_real (g -` {g z}) x)"
simply_connected_homotopic_loops: fixes S :: "'a set" shows "simply_connected S = (\<forall>p q. homotopic_loops S p p \<and> homotopic_loops S q q \<longrightarrow> homotopic_loops S p q)"
reach_reach\<^sub>t_fst: fixes \<Sigma> :: "'d set"   and \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c" shows "reach \<Sigma> \<delta> q\<^sub>0 = fst ` reach\<^sub>t \<Sigma> \<delta> q\<^sub>0"
L: shows "(\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - (1::'a))) \<longlonglongrightarrow> L"
A\<^sub>A_of_def: fixes A :: "'Var\<^sub>C \<Rightarrow> ('Val \<times> 'Val) option" shows "A\<^sub>A_of A \<equiv> \<lambda>x\<^sub>A. case A (var\<^sub>C_of x\<^sub>A) of None \<Rightarrow> None | Some (v, v') \<Rightarrow> Some (v, v')"
1: fixes m :: "int" shows "poly_mod_type TYPE('a) m = (m = int CARD('a))"
Copied_4: shows "descendants k i ((ps' @ ps, a) # branch) xs"
cases: fixes x :: "'b list" obtains "xa__ = []"   | x :: "'b" and xs :: "'b list" where "xa__ = x # xs"
unitl_ex: fixes x :: "'c" shows "\<exists>e\<in>E. D e x \<and> e \<oplus> x = x"
simple_extension_mem: fixes K :: "'a set"   and x :: "'a" assumes "subring K R"   and "x \<in> carrier R" shows "x \<in> simple_extension K x"
auth_guard_step4: fixes s :: "l2_state"   and A :: "agent"   and B :: "agent"   and Rb :: "fid_t"   and gnx :: "msg"   and Ni :: "msg" assumes "s \<in> l2_inv5"   and "s \<in> l2_inv1"   and "Insec 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> chan s"   and "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "can_signal s A B" shows "\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress s Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Ra xgnx = Some gnx \<and> sklvl1.guessed_frame Ra xni = Some Ni \<and> sklvl1.guessed_frame Ra xnr = Some (NonceF (Rb $ nr)) \<and> sklvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny)))"
commuting_imp_normal1: assumes "carrier G \<subseteq> A <#> B"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" shows "A \<lhd> G"
wf_lderiv: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and x :: "'x" assumes "local.wf idx \<phi>"   and "lformula \<phi>" shows "local.wf idx (lderiv x \<phi>)"
B_def: shows "B = step_3_c_inner_loop (A $$ (i_end, k) / x) l i_end (Suc i_end - i_begin) A"
continuous_on_final_topologyI2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Y :: "'c \<Rightarrow> 'd topology"   and X :: "'e set"   and i :: "'c" assumes "\<And>i. f i \<in> topspace (Y i) \<rightarrow> X" shows "continuous_map (Y i) (final_topology X Y f) (f i)"
LNil_def: shows "LNil \<equiv> llist_abs\<cdot>(sinl\<cdot>ONE)"
iteration_denest: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<infinity> = x\<^sup>\<infinity> \<cdot> (y \<cdot> x\<^sup>\<infinity>)\<^sup>\<infinity>"
bounded_linear_const_scaleR: fixes g :: "'d \<Rightarrow> 'e"   and r :: "real" assumes "bounded_linear g" shows "bounded_linear (\<lambda>x. r *\<^sub>R g x)"
pure_cong: fixes x :: "dB"   and y :: "dB"   and R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool" assumes "x \<leftrightarrow> y" shows "itrm_cong R (Pure x) (Pure y)"
completeness_game_def: fixes h :: "'a"   and w :: "'b"   and e :: "'e" shows "\<Sigma>1.completeness_game h w e = init1 h w \<bind> (\<lambda>(r, a). response1 r w e \<bind> (\<lambda>z. return_spmf (check1 h a e z)))"
Func_upd: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and x :: "'d"   and y :: "'e" assumes "f \<in> Func A B"   and "x \<in> A"   and "y \<in> B" shows "f(x := y) \<in> Func A B"
trivial_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and v :: "'val" shows "uninst_code.ssa.trivial u p g v \<equiv> \<exists>v'\<in>uninst_code.ssa.allVars u p g. uninst_code.ssa.isTrivialPhi p g v v'"
summable_sums: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "f sums suminf f"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "abel_semigroup f"   and "comm_monoid_axioms f z" shows "comm_monoid f z"
cr_dual_ord_def: shows "cr_dual_ord \<equiv> \<lambda>x y. x = of_dual_ord y"
domintros_4: fixes m :: "nat" shows "ackloop_dom [m]"
sgn_mult_self_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "times (sgn a) (sgn a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
2: shows "wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (\<A> @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)]) = wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t \<A> \<union> fv t"
empty: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and p :: "'d" assumes "Matching_Ternary.matches \<gamma> m Empty p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m Empty], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided"
orthogonal_Gram_Schmidt_matrix: fixes A :: "(('a, 'c) vec, 'b) vec" shows "pairwise orthogonal {column i (Gram_Schmidt_matrix A) |i. i \<in> UNIV}"
waveD_separating: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "wave \<Gamma> f" shows "separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f)"
update_convs_1: fixes carrier' :: "'c set \<Rightarrow> 'c set"   and carrier :: "'c set"   and more :: "'d" shows "carrier_update carrier' \<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier' carrier, \<dots> = more\<rparr>"
assms_5: shows "state_rank (token_run x n) n = Some j"
omit_branch_mem_diff_opening: fixes i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and ps :: "('c, 'b) fm list"   and a :: "'b" assumes "only_touches i branch xs"   and "(ps, a) \<in>. branch"   and "i \<noteq> a" shows "(ps, a) \<in>. omit_branch xs branch"
fps_mult_fps_X_commute: fixes a :: "'a fps" shows "fps_X * a = a * fps_X"
prefix_fin_append: fixes u :: "'a list"   and v :: "'a list" shows "u \<le> u @ v"
S': shows "S' = poly_mod.Mp r (S - Polynomial.smult p A')"
Complete_set_residues_def: fixes g :: "'b \<Rightarrow> 'b set" shows "Complete_set_residues g = (\<exists>f. res_function f \<and> (\<forall>c. pairwise (\<lambda>a b. \<not> cong a b c) (range (f c)) \<and> g c = range (f c)))"
singleI_unit: fixes x :: "unit" shows "pred.eval (Predicate.single ()) x"