hcju/isabelleps huggingface dataset:
map_bset_abs_rep_Tree\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['h]" shows "map_bset abs_Tree\<^sub>\<alpha> (map_bset rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>) = tset\<^sub>\<alpha>"
Liminf_llist_subset_Sup_llist: fixes Xs :: "'a set llist" shows "Liminf_llist Xs \<subseteq> Sup_llist Xs"
dom_locals_avar_mono: fixes s :: "state"   and G :: "prog"   and i :: "val"   and a :: "val" shows "dom (locals (snd s)) \<subseteq> dom (locals (snd (snd (avar G i a s))))"
DERIV_deriv_iff_real_differentiable: fixes f :: "real \<Rightarrow> real"   and x :: "real" shows "(f has_real_derivative deriv f x) (at x) = (f differentiable at x)"
asInp_def: fixes qinp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm option" shows "asInp qinp \<equiv> lift asTerm qinp"
derived_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" shows "derived G H = generate G (derived_set G H)"
get_dom_component_separates_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'"   and "ptr' \<notin> set c" shows "set to \<inter> set c' = {}"
Int_def: fixes A :: "'f set"   and B :: "'f set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
comp_in_carr: fixes x :: "'e \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and i :: "'e" assumes "x \<in> carrier (DirProds G I)"   and "i \<in> I" shows "x i \<in> carrier (G i)"
has_vector_derivative_scaleR: fixes f :: "real \<Rightarrow> real"   and f' :: "real"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_real_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at x within s)" shows "((\<lambda>x. f x *\<^sub>R g x) has_vector_derivative f x *\<^sub>R g' + f' *\<^sub>R g x) (at x within s)"
strict_ext_def: fixes base_r :: "('b \<times> 'b) set"   and r :: "('b \<times> 'b) set" shows "strict_ext base_r r \<equiv> strict_partial_order r \<and> base_r \<subseteq> r"
thread_ok_init_fin_descend_thr: fixes ts :: "'c \<Rightarrow> (('h \<times> 'd) \<times> 'e \<Rightarrow>f nat) option"   and nta :: "('c, 'f, 'g) new_thread_action" shows "thread_ok (init_fin_descend_thr ts) nta = thread_ok ts nta"
factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "factorization_m f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> f =m Polynomial.smult c (\<Prod>\<^sub># fs) \<and> (\<forall>f\<in>#fs. irreducible\<^sub>d_m f \<and> monic (Mp f))"
inv_closed: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<ominus>\<^bsub>M\<^esub> x \<in> carrier M"
r_def: shows "r = prime_sum_upto (\<lambda>p. ln (real p) * frac (real n / real p)) (real n)"
enum_cycle_def: shows "enum_cycle = from_nat_into (cycles \<Delta>)"
A_to_prodag2_hom: fixes A :: "'a set"   and B :: "'a set"   and S :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set" assumes "ideal R A"   and "ideal R B"   and "S 0 = pj R A"   and "S (Suc 0) = pj R B" shows "A_to_prodag R {j. j \<le> Suc 0} S (prodB1 (R /\<^sub>r A) (R /\<^sub>r B)) \<in> rHom R (R /\<^sub>r A \<Oplus>\<^sub>r R /\<^sub>r B)"
RG_aezdelta0fun_closed: fixes r :: "'a" shows "r \<delta>\<delta> (0::'g) \<in> Supgroup.group_ring"
bv_to_int_msb1: fixes w1 :: "bit list" assumes "bv_to_int w1 < 0" shows "bv_msb w1 = \<one>"
weakPsiCongSym: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> Q \<doteq> P"
bsmap_rev_xlate: fixes q :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (rev q) = rev (map bsmapt q)"
convex_hull_next_dim: fixes X :: "'a vec set"   and y :: "'a vec" assumes "n + 1 = n + 1"   and "X \<subseteq> carrier_vec (n + 1)"   and "finite X"   and "\<forall>y\<in>X. y $ n = (1::'a)"   and "y \<in> carrier_vec (n + 1)"   and "y $ n = (1::'a)" shows "(vec_first y n \<in> m.convex_hull {vec_first y n |y. y \<in> X}) = (y \<in> next_dim.cone X)"
connected_contains_Icc: fixes A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "connected A"   and "a \<in> A"   and "b \<in> A" shows "{a..b} \<subseteq> A"
r_zero: fixes x :: "'c" assumes "x \<in> carrier M" shows "x \<oplus>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = x"
max_Var_floatariths_mapI: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> floatarith"   and m :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> max_Var_floatarith (f x) \<le> m" shows "max_Var_floatariths (map f xs) \<le> m"
finite_pderivs_lang: fixes A :: "'a list set"   and r :: "'a rexp" shows "finite (pderivs_lang A r)"
monoid_homomorphism_axioms_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" shows "monoid_homomorphism_axioms \<eta> M composition unit composition' unit' \<equiv> (\<forall>x y. x \<in> M \<longrightarrow> y \<in> M \<longrightarrow> \<eta> (composition x y) = composition' (\<eta> x) (\<eta> y)) \<and> \<eta> unit = unit'"
assms_3: shows "\<chi>\<^sub>1 \<equiv> principal_dchar n"
select: fixes xs :: "'a list"   and s :: "natural \<times> natural" assumes "xs \<noteq> []" shows "fst (Random.select xs s) \<in> set xs"
exhaust: fixes y :: "'d DBMEntry" obtains x1 :: "'d" where "y = Le x1"   | x2 :: "'d" where "y = Lt x2"   | "y = \<infinity>"
mod_prop1: fixes a :: "nat"   and b :: "nat"   and c :: "nat" shows "a mod (b * c) mod c = a mod c"
unless_def: fixes P :: "bool"   and m :: "'c \<Rightarrow> (unit \<times> 'c) set \<times> bool" shows "unless P m \<equiv> when (\<not> P) m"
HC_intros_12: fixes F :: "'b formula" shows "AX10 \<turnstile>\<^sub>H \<^bold>\<not> (F::'b formula) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> \<bottom>"
simps_1: fixes x :: "quat"   and y :: "quat" shows "quat.Re (x + y) = quat.Re x + quat.Re y"
inf_sup_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inf_class.inf x (sup_class.sup y z) = sup_class.sup (inf_class.inf x y) (inf_class.inf x z)"
simps: fixes a1 :: "pi"   and a2 :: "pi" shows "bangPred a1 a2 = ((\<exists>P. a1 = P \<and> a2 = !P) \<or> (\<exists>P. a1 = P \<and> a2 = P \<parallel> !P))"
Un_subset_iff: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "(A \<union> B \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
elims_3: fixes d :: "nat"   and x :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "\<not> shift1_inv d x" obtains qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" where "x = (qs, B)" and "\<not> (B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)} \<and> shift2_inv d qs)"
Max_wrt_among_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "Max_wrt_among R A = {x \<in> A. R x x \<and> (\<forall>y\<in>A. R x y \<longrightarrow> R y x)}"
product_finite: fixes A :: "'b fset set"   and B :: "'b fset set" assumes "finite A"   and "finite B" shows "finite (A \<otimes> B)"
branchTerminalPropagates: fixes subs :: "'a \<Rightarrow> 'a set"   and Gamma :: "'a"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "branch subs Gamma f"   and "terminal subs (f m)" shows "terminal subs (f (m + n))"
restrict_indets_ideal: fixes F :: "(('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "restrict_indets ` ideal F = ideal (restrict_indets ` F)"
simps_1: fixes v :: "val" shows "fin\<^sub>1 (Val v) = Val v"
sqrt_int_floor_pos_def: fixes x :: "int" shows "sqrt_int_floor_pos x = root_int_floor_pos 2 x"
min_arg_not_ge_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c"   and n :: "'c" assumes "class.linorder less_eq less" shows "(\<not> less (ord.min less_eq m n) m) = (ord.min less_eq m n = m)"
insertCI: fixes a :: "'d"   and B :: "'d set"   and b :: "'d" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
sorted_mergesort_by_rel: fixes xs :: "'a list" shows "sorted (mergesort_by_rel (\<le>) xs)"
assms_4: shows "\<not> (\<exists>b\<in>pin (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>). length (bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>)) < length b)"
alloc_one: fixes h :: "loc \<Rightarrow> obj option"   and a :: "loc"   and n :: "nat"   and obj :: "obj" assumes "the (new_Addr h) = a"   and "atleast_free h (Suc n)" shows "new_Addr h = Some a \<and> atleast_free (h(a \<mapsto> obj)) n"
simps_1: fixes sel :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option"   and Rs :: "'f"   and init :: "nat \<times> ('d \<times> nat \<times> nat) set" shows "mk_chain sel Rs init 0 = init"
inv_right: fixes a :: "'a" shows "f (inv f a) = a"
map_family_triangle1: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'f"   and B :: "'f set"   and S :: "('g, 'h) aGroup_scheme"   and g :: "'c \<Rightarrow> 'g \<Rightarrow> 'd" assumes "\<forall>j\<in>I. aGroup (A j)"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to f (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "aGroup S"   and "\<forall>j\<in>I. g j \<in> aHom S (A j)" shows "\<exists>!h. h \<in> aHom S (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f) \<and> (\<forall>j\<in>I. compos S (ProjInd I A f j) h = g j)"
diff_le_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a - b \<le> c) = (a \<le> c + b)"
minus_SucSuc: fixes a :: "nat" shows "a - Suc (Suc 0) = a - Suc 0 - Suc 0"
fun_merge_def: fixes fs :: "('d \<Rightarrow> 'c) list"   and as :: "'d set list"   and a :: "'d" shows "fun_merge fs as a \<equiv> (fs ! (LEAST i. i < length as \<and> a \<in> as ! i)) a"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier A" shows "inv\<^bsub>A\<^esub> x \<in> carrier A"
sum_eq: fixes x :: "i"   and y :: "i" shows "x \<oplus> y = (THE z. \<forall>v. O v z = (O v x \<or> O v y))"
assms_3: shows "binop (\<SS>\<lparr>\<A>\<rparr>) (\<SS>\<lparr>mbinop\<rparr>)"
length_append: fixes xs :: "'a list"   and ys :: "'a list" shows "length (xs @ ys) = length xs + length ys"
1: shows "{s. timpls_transformable_to TI t s} \<subseteq> {s. timpls_transformable_to_pred (the_Abs ` {f \<in> funs_term t. is_Abs f} \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) t s}"
lprefix_lfiniteD: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lprefix xs ys"   and "lfinite ys" shows "lfinite xs"
finite_lessThan_enatI: fixes m :: "enat" assumes "m \<noteq> \<infinity>" shows "finite {..<m}"
lowner_is_lub_def: fixes f :: "nat \<Rightarrow> complex mat"   and M :: "complex mat" shows "lowner_is_lub f M = ((\<forall>n. f n \<le>\<^sub>L M) \<and> (\<forall>M'. (\<forall>n. f n \<le>\<^sub>L M') \<longrightarrow> M \<le>\<^sub>L M'))"
optimize_primitive_univ_correct_matchexpr: fixes \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and m :: "'c common_primitive match_expr" shows "matches (common_matcher, \<alpha>) m = matches (common_matcher, \<alpha>) (optimize_primitive_univ m)"
assms_4: shows "\<forall>n. 1 \<le> b n"
invalid_def: shows "invalid \<equiv> \<lambda>\<tau>. UML_Types.bot_class.bot"
coprime_iff_coprime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "comm_monoid_mult.coprime times one = algebraic_semidom.coprime times one"
not_LCons_le_LNil: fixes a :: "'b"   and l :: "'b llist" shows "\<not> a ## l \<le> LNil"
simps: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'c" shows "r\<inverse>\<inverse> a1 a2 = (\<exists>a b. a1 = b \<and> a2 = a \<and> r a b)"
even_xor_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (xor a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
A_connected: fixes u :: "'v"   and v :: "'v" assumes "(u, v) \<in> A" shows "(v, r) \<in> A\<^sup>*"
noninert_gauss_int_prime_axioms: shows "noninert_gauss_int_prime p"
simps_1: fixes f :: "'c \<Rightarrow> ('c, 'd) Nested_Datatype.term" shows "subst_term_list f [] = []"
fun_complete1: fixes a :: "'s"   and b :: "'s"   and F :: "'s \<Rightarrow> 's" assumes "ide a"   and "ide b"   and "F \<in> hom unity a \<rightarrow> hom unity b" shows "\<exists>f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> (\<forall>x. \<guillemotleft>x : unity \<rightarrow> local.dom f\<guillemotright> \<longrightarrow> f \<cdot> x = F x)"
I_lift_dnfeq_qe: fixes qe :: "atom list \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and xs :: "real list" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>R a \<Longrightarrow> qfree (qe as)"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>R a \<and> \<not> (case a of Less real list \<Rightarrow> False | Eq x [] \<Rightarrow> False | Eq x (c # xa) \<Rightarrow> c \<noteq> 0) \<Longrightarrow> R.is_dnf_qe qe as" shows "R.I (R\<^sub>e.lift_dnfeq_qe qe \<phi>) xs = R.I \<phi> xs"
prems_4: shows "\<forall>s l1 l2. f (f s l1) l2 = f s (l1 @ l2)"
le_add_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_cancel_comm_monoid_diff plus zero less_eq less minus"   and "less_eq a b" shows "less_eq c (minus (plus b c) a)"
hom_deltas: fixes w :: "'b list"   and br :: "bool \<times> (bool \<times> 'b) rexp" shows "hom_ab (fold delta_a w br) = fold delta_b w (hom_ab br)"
pause_state_def: fixes out :: "'o"   and c :: "'i \<Rightarrow> ('s, 'm) stateT" shows "local.pause_state out c = StateT (\<lambda>s. pause out (\<lambda>i. run_state (c i) s))"
simps_3: fixes bs :: "real list"   and a :: "MIR.num" shows "Ifm bs (Lt a) = (Inum bs a < 0)"
nhds_def: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.topological_space open" shows "topological_space.nhds open a = Inf (principal ` {S. open S \<and> a \<in> S})"
induct: fixes P1 :: "('c, 'd) Nested_Datatype.term \<Rightarrow> bool"   and P2 :: "('c, 'd) Nested_Datatype.term list \<Rightarrow> bool"   and term_list :: "('c, 'd) Nested_Datatype.term list" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (App x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 term_list"
spec: fixes default_node_properties :: "'c"   and sinvar_spec :: "'d graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> bool"   and sinvar_impl :: "'d list_graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> bool"   and receiver_violation :: "bool"   and offending_flows_impl :: "'d list_graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<times> 'd) list list"   and node_props_impl :: "('d, 'c) TopoS_Params \<Rightarrow> 'd \<Rightarrow> 'c"   and eval_impl :: "'d list_graph \<Rightarrow> ('d, 'c) TopoS_Params \<Rightarrow> bool" assumes "TopoS_List_Impl default_node_properties sinvar_spec sinvar_impl receiver_violation offending_flows_impl node_props_impl eval_impl" shows "SecurityInvariant sinvar_spec default_node_properties receiver_violation"
is_natural_1: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "T.ToCT f \<cdot> \<alpha> (CCC.dom f) = \<alpha> f"
in_incs: fixes ys :: "nat list"   and t :: "nat"   and a :: "nat"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" assumes "(ys, t) \<in> set (incs a x (xs, s))" shows "length ys = length xs + 1 \<and> t = s + hd ys * a \<and> tl ys = xs \<and> C ys t"
last_index_less_size_conv: fixes xs :: "'c list"   and x :: "'c" shows "(last_index xs x < length xs) = (x \<in> set xs)"
case_cong: fixes sum :: "'d + 'e"   and sum' :: "'d + 'e"   and f1 :: "'d \<Rightarrow> 'f"   and g1 :: "'d \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'f"   and g2 :: "'e \<Rightarrow> 'f" assumes "sum = sum'"   and "\<And>x1. sum' = Inl x1 \<Longrightarrow> f1 x1 = g1 x1"   and "\<And>x2. sum' = Inr x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case sum of Inl x1 \<Rightarrow> f1 x1 | Inr x2 \<Rightarrow> f2 x2) = (case sum' of Inl x1 \<Rightarrow> g1 x1 | Inr x2 \<Rightarrow> g2 x2)"
abs_conv_abscissa_MInf_iff: fixes f :: "'a fds" shows "(abs_conv_abscissa f = - \<infinity>) = (\<forall>s. fds_abs_converges f s)"
plus_vec_def: shows "(+) \<equiv> \<lambda>x y. \<chi>i. x $h i + y $h i"
div_base_i: fixes n :: "nat"   and d :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "\<And>i. i < n \<Longrightarrow> d i < base"   and "0 < n"   and "i < n" shows "from_digits n d div base ^ i = (\<Sum>j = i..<n. d j * base ^ (j - i))"
one_step_def: fixes ode_ops :: "ode_ops"   and X0 :: "'d set"   and h :: "real"   and m :: "'d set \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'd set \<Rightarrow> ('d set \<times> 'c) option nres" shows "one_step ode_ops X0 h m = CHECK (\<lambda>_. print_msg ''one_step nonneg'') (0 < h) \<bind> (\<lambda>_. iterations_spec \<bind> (\<lambda>its. halve_stepsizes_spec \<bind> (\<lambda>halvs. cert_stepsize ode_ops m X0 h its halvs \<bind> (\<lambda>(h, res, res_ivl, err). ASSERT (0 < h) \<bind> (\<lambda>_. RETURN (h, err, res_ivl, res))))))"
a_in_as: fixes i :: "nat" assumes "i < k" shows "as ! i \<in> set as"
empty_correct_1: shows "lm.\<alpha> (lm.empty ()) = Map.empty"
dom_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.seq g f" shows "B.VV.dom (B.VV.comp g f) = B.VV.dom f"
g_inj_on: shows "inj_on g (map_to_set m \<inter> dom g)"
dia\<T>: fixes K :: "'a set" assumes "K \<in> \<T>" shows "diameter K < e"
ltln_StrongRelease_to_Until: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi> U\<^sub>n (\<phi> and\<^sub>n \<psi>)"
finite_graph: fixes es :: "nat set set" shows "finite (uverts (ES.edge_ugraph es))"
reciprocal_fcompose: fixes P :: "'b poly"   and p :: "nat" assumes "degree P \<le> p" shows "reciprocal_poly p P = monom (1::'b) (p - degree P) * fcompose P 1 [:0::'b, 1::'b:]"
rq_ssuf: fixes u :: "'b list"   and v :: "'b list" assumes "u <s v" shows "v\<^sup><\<inverse>u  \<noteq> \<epsilon>"
abc_list_crsp_lm_s: fixes lma :: "nat list"   and lmb :: "nat list"   and m :: "nat"   and n :: "nat" assumes "abc_list_crsp lma lmb" shows "abc_list_crsp (abc_lm_s lma m n) (abc_lm_s lmb m n)"
min_worst_chain_0: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "wf {(x, y). gt y x}"   and "bad p x" shows "\<not> gt (worst_chain p gt 0) x"
conc_assoc: fixes A :: "'a list set"   and B :: "'a list set"   and C :: "'a list set" shows "(A @@ B) @@ C = A @@ B @@ C"
divide_eq_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = one) = (b \<noteq> zero \<and> a = b)"
ln_le_cancel_iff: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "(ln x \<le> ln y) = (x \<le> y)"
rep_eq: fixes x :: "nat_int"   and xa :: "nat_int" shows "Rep_nat_int (x \<sqinter> xa) = Rep_nat_int x \<inter> Rep_nat_int xa"
is_aligned_neg_mask_weaken: fixes p :: "'b word"   and n :: "nat"   and m :: "nat" assumes "is_aligned p n"   and "m \<le> n" shows "p AND NOT (mask m) = p"
subset: shows "{a. \<exists>b. g a b \<noteq> \<^bold>1} \<times> {b. \<exists>a. g a b \<noteq> \<^bold>1} \<subseteq> C"
int_nat_eq: fixes a :: "nat"   and b :: "nat" assumes "int a = int b" shows "a = b"
Quotient3_dlist: shows "Quotient3 dlist_eq dlist rep_dlist"
insertAt_removeAt: fixes i :: "nat"   and ls :: "'a list" assumes "i < length ls" shows "insertAt i (ls ! i) (removeAt i ls) = ls"
noWhile_discr_WbisT: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "noWhile c1"   and "noWhile c2"   and "discr c1"   and "discr c2" shows "c1 \<approx>wT c2"
log_twice: fixes n :: "nat" assumes "n \<noteq> 0" shows "local.log (2 * n) = Suc (local.log n)"
fair_def: fixes rules :: "'a stream"   and rs :: "'a stream" shows "i.fair rules rs \<equiv> i.R rs \<subseteq> i.R rules \<and> (\<forall>r\<in>i.R rules. alw (ev (holds ((=) r))) rs)"
iT_mult_3: fixes n :: "nat"   and d :: "nat"   and k :: "nat" shows "[n\<dots>,d] \<otimes> k = [ n * k, mod k, d ]"
one_normh: fixes n :: "nat" shows "isnpolyh (1)\<^sub>p n"
eta_star_preserves_term_ok: fixes r :: "Core.term"   and s :: "Core.term"   and thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "r \<rightarrow>\<^sub>\<eta>\<^sup>* s"   and "term_ok thy r" shows "term_ok thy s"
sff_2: fixes fi :: "int poly"   and i :: "nat" assumes "(fi, i) \<in> set hs" shows "content fi = 1 \<and> 0 < lead_coeff fi"
binop: shows "binop \<rat>\<^sub>\<circ> vrat_mult"
ex_finite_subset_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'b set \<Rightarrow> bool" shows "(\<exists>B. finite B \<and> B \<subseteq> f ` A \<and> P B) = (\<exists>B. finite B \<and> B \<subseteq> A \<and> P (f ` B))"
diff_add_assoc: fixes b :: "nat"   and c :: "nat" assumes "b \<le> c" shows "c - (c - b) = c - c + b"
degree_aform_of_ivl: fixes a :: "'a"   and b :: "'a" shows "degree_aform (aform_of_ivl a b) \<le> length Basis_list"
at_within_Icc_at_left: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order_topology less_eq less open"   and "less a b" shows "topological_space.at_within open b (ord.atLeastAtMost less_eq a b) = topological_space.at_within open b (ord.lessThan less b)"
transp_trans: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "transp r = trans {(x, y). r x y}"