hcju/isabelleps huggingface dataset:
inline_gpv_stop_obsf_converter: fixes gpv :: "('a, 'b, 'c) gpv"   and conv :: "('b, 'c, 'd, 'e) converter" shows "bind_gpv (inline run_converter (map_gpv' id id option_of_exception (gpv_stop gpv)) (obsf_converter conv)) (\<lambda>(x, conv'). case x of None \<Rightarrow> Fail | Some x' \<Rightarrow> Done (x, conv')) = bind_gpv (map_gpv' id id option_of_exception (gpv_stop (inline run_converter gpv conv))) (\<lambda>x. case x of None \<Rightarrow> Fail | Some (x', conv) \<Rightarrow> Done (Some x', obsf_converter conv))"
cproper_interval_finite_3_def: shows "cproper_interval = proper_interval"
Areflassump1: shows "Arefl RV \<inter> Arefl Rc \<subseteq> {[]}"
outstanding_non_volatile_Read\<^sub>s\<^sub>b_acquired_or_read_only_reads: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat" assumes "non_volatile_owned_or_read_only pending_write \<S> \<O> sb"   and "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb" shows "a \<in> acquired_reads True sb \<O> \<or> a \<in> read_only_reads \<O> sb"
ord_eq_0: fixes x :: "'a" assumes "x \<in> carrier G" shows "(ord x = 0) = (\<forall>n. n \<noteq> 0 \<longrightarrow> x [^] n \<noteq> \<one>)"
conf_def: fixes P :: "'b prog"   and h :: "'heap"   and v :: "'addr val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
1: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" assumes "b = send\<langle>t\<rangle>" shows "unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A) \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>) = receive\<langle>t \<cdot> \<theta>\<rangle> # unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
nat_power_less_imp_less: fixes i :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < i"   and "i ^ m < i ^ n" shows "m < n"
zero_order_2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and n :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
the_default_param: fixes R :: "('a \<times> 'b) set" shows "(the_default, the_default) \<in> R \<rightarrow> \<langle>R\<rangle>option_rel \<rightarrow> R"
self_finance_def: fixes Mkt :: "('c, 'b) discrete_market"   and v :: "real"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and asset :: "'b" shows "self_finance Mkt v pf asset = qty_sum pf (qty_single asset (remaining_qty Mkt v pf asset))"
F_le_f: fixes i :: "nat"   and x :: "'a" shows "enn2real (F i x) \<le> f x"
max2_greater: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(a, max2 a b) \<in> r \<and> (b, max2 a b) \<in> r"
prems: shows "Subobjs P C (Cs @ rev (D # Ds))"
add_le_cancel_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus c a) (plus c b) = less_eq a b"
bounded_insert: fixes x :: "'b"   and S :: "'b set" shows "bounded (insert x S) = bounded S"
comp_in_homE: fixes g :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>g \<cdot>\<^sub>B f : a \<rightarrow>\<^sub>B c\<guillemotright>" obtains b :: "'a" where "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow>\<^sub>B c\<guillemotright>"
order_unique_lemma: fixes a :: "'a"   and n :: "nat"   and p :: "'a poly" assumes "[:- a, 1::'a:] ^ n dvd p"   and "\<not> [:- a, 1::'a:] ^ Suc n dvd p" shows "order a p = n"
subst_poly_lec_sum_list: fixes \<sigma> :: "nat \<Rightarrow> linear_poly"   and ps :: "'a le_constraint list" shows "subst_poly_lec \<sigma> (sum_list ps) = sum_list (map (subst_poly_lec \<sigma>) ps)"
neg_add1: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D) + local.ddh.advantage n (local.ddh.DDH_\<A>' n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D)))"
less_le_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.preorder less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"
infer_def: fixes A :: "'a set"   and \<phi> :: "'a" shows "A \<turnstile> \<phi> \<equiv> \<phi> \<in> Cn A"
val_poss_mem_Vr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 < v x" shows "x \<in> carrier (Vr K v)"
col_addcol: fixes k :: "nat"   and A :: "'b mat"   and l :: "nat"   and a :: "'b" assumes "k < dim_col A"   and "l < dim_col A" shows "col (addcol a k l A) k = a \<cdot>\<^sub>v col A l + col A k"
bi_differentiable_Fourier_convergence_periodic: fixes f :: "real \<Rightarrow> real"   and t :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "f differentiable at_left t"   and "f differentiable at_right t"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> f t"
def_ms_l: shows "mset_lit L = {#t, s#} \<or> mset_lit L = {#t, t, s, s#}"
\<delta>_2: shows "set (bvars\<^sub>s\<^sub>t\<^sub>p x) \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
in_set_remove1: fixes a :: "'c"   and b :: "'c"   and xs :: "'c list" assumes "a \<noteq> b" shows "(a \<in> set (remove1 b xs)) = (a \<in> set xs)"
rank_succ: fixes x :: "V" shows "rank (ZFC_in_HOL.succ x) = ZFC_in_HOL.succ (rank x)"
assms_2: shows "v \<rightarrow>\<^sup>* w"
type_aexpr_Const: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and c :: "nat" shows "type_aexpr \<Gamma> (Const c) {}"
fresh_5: fixes a :: "atom"   and name :: "name"   and fm :: "fm" shows "a \<sharp> SyntaxN.Ex name fm = (a \<sharp> fm \<or> a \<in> set [atom name])"
abs_dif_fiX_bound_C: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and x :: "real"   and y :: "real"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "\<forall>l\<in>C. \<bar>f l - g l\<bar> \<le> x"   and "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> y"   and "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>g l - g m\<bar> \<le> y"   and "p \<in> C"   and "q \<in> C"   and "r \<in> C" shows "\<bar>fiX f p r - fiX g q r\<bar> \<le> x + (if y \<le> \<Delta> then 0 else y)"
exhaust: fixes y :: "label" obtains x1 :: "nat" where "y = Label x1"   | "y = Entry"   | "y = Exit"
step_is_path: fixes c :: "'c"   and c' :: "'c" assumes "(c, c') \<in> step\<^sup>*" shows "\<exists>p. lts.path c p c'"
ord_eq_le_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a = b"   and "prefix b c" shows "prefix a c"
SUP_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
psimps: fixes 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" assumes "shift1_dom (qs, B)" shows "shift1 (qs, B) = (if 1 < card B then let (h, U) = SOME b. b \<in> B \<and> card (snd b) = Suc m; x = SOME y. y \<in> U in shift1 (shift_list (h, U) x qs, B - {(h, U)}) else (qs, B))"
filt_equiv_borel_AE_eq: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> real"   and t :: "'c"   and g :: "'b \<Rightarrow> real" assumes "prob_space M"   and "filt_equiv F M N"   and "f \<in> borel_measurable (F t)"   and "g \<in> borel_measurable (F t)"   and "AE w in M. f w = g w" shows "AE w in N. f w = g w"
substitutes_iia: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "finite A" shows "substitutes_on A f = iia_on A f"
linear_homeomorphism_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f"   and "inj f" obtains g :: "'b \<Rightarrow> 'a" where "homeomorphism (f ` S) S g f"
residual_r_refl: fixes x :: "'a" shows "(1::'a) \<le> x \<rightarrow> x"
assms_1: shows "d = nat (fst (bezw e ((P - 1) * (Q - 1))) mod int ((P - 1) * (Q - 1)))"
transpose_mat1: fixes nr :: "nat"   and ze :: "'c"   and on :: "'c" shows "Matrix_Legacy.transpose nr (mat1I ze on nr) = mat1I ze on nr"
bind_rule_complete: fixes M :: "'f nres"   and f :: "'f \<Rightarrow> 'e nres"   and \<Phi> :: "'e \<Rightarrow> bool" shows "(M \<bind> f \<le> SPEC \<Phi>) = (M \<le> SPEC (\<lambda>x. f x \<le> SPEC \<Phi>))"
exponential_order_mono: fixes M :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a"   and b :: "real"   and N :: "real" assumes "exponential_order M a f"   and "a \<le> b"   and "M \<le> N" shows "exponential_order N b f"
singular_face_simplex_map: fixes p :: "nat"   and k :: "nat"   and f :: "'d \<Rightarrow> 'c"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'd" assumes "1 \<le> p"   and "k \<le> p" shows "singular_face p k (simplex_map p f c) = simplex_map (p - Suc 0) f (c \<circ> simplical_face k)"
float_le_topfloat: fixes a :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "1 < LENGTH('a)" shows "a \<le> topfloat"
inter: fixes S :: "'a set set"   and T :: "'a set set" assumes "S \<in> fba"   and "T \<in> fba" shows "S \<inter> T \<in> fba"
gt_zero: fixes x :: "'a" assumes "x \<in> V"   and "x \<noteq> (0::'a)" shows "0 < \<parallel>x\<parallel>"
take_eq_take_append: fixes i :: "nat"   and j :: "nat"   and a :: "'a list" assumes "i \<le> j"   and "j \<le> length a" shows "\<exists>u. take j a = take i a @ u"
col_smult: fixes i :: "nat"   and A :: "'c mat"   and k :: "'c" assumes "i < dim_col A" shows "col (k \<cdot>\<^sub>m A) i = k \<cdot>\<^sub>v col A i"
Network_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat" shows "Network c s t \<equiv> ((s \<in> Graph.V c \<and> t \<in> Graph.V c) \<and> s \<noteq> t \<and> (\<forall>u v. (0::'a) \<le> c (u, v))) \<and> ((\<forall>u. (u, s) \<notin> Graph.E c) \<and> (\<forall>u. (t, u) \<notin> Graph.E c)) \<and> (\<forall>u v. (u, v) \<in> Graph.E c \<longrightarrow> (v, u) \<notin> Graph.E c) \<and> (\<forall>v\<in>Graph.V c. Graph.connected c s v \<and> Graph.connected c v t) \<and> finite (Graph.reachableNodes c s)"
FAll_1: fixes i :: "nat"   and \<AA> :: "'i"   and P :: "'v" assumes "\<not> local.find k i \<phi>"   and "LEQ k i (#\<^sub>V \<AA>)"   and "len P \<le> Length \<AA>" shows "(Extend k i \<AA> P \<Turnstile>\<^sub>b \<phi>) = (\<AA> \<Turnstile>\<^sub>b decr k i \<phi>)"
oapp: shows "order a (pderiv p) = order a e + order a d"
mfinalI: fixes s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('x1 \<times> 'b \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'd wait_set_status option) \<times> 'a set" assumes "\<And>t x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> final1 x \<and> ln = no_wait_locks \<and> wset s t = None" shows "r1.mfinal s"
L_subset_vars: fixes c :: "com"   and X :: "char list set" shows "L c X \<subseteq> rvars c \<union> X"
unfold_thesis: shows "vec_space.rank (length signs) (mat_of_cols (dim_row (M_mat_R signs subsets)) (take_indices (cols (M_mat_R signs subsets)) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))))) = length (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
simps_1: fixes I :: "\<I>"   and ts :: "nat list" shows "mprev_next I [] ts = ([], [], ts)"
Pi_mem: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and x :: "'d" assumes "f \<in> Pi A B"   and "x \<in> A" shows "f x \<in> B x"
scc_of_empty_conv: fixes u :: "'a" shows "(H.scc_of u = {}) = (u \<notin> verts H)"
converse_rtrancl_into_rtrancl: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>*"
replace_basis_completely: fixes vs :: "'b list"   and us :: "'b list" assumes "set vs \<subseteq> V"   and "R_lin_independent vs"   and "length us = length vs"   and "set us \<subseteq> RSpan vs"   and "R_lin_independent us" shows "set us \<subseteq> RSpan vs \<and> RSpan vs = RSpan us \<and> R_lin_independent us"
distrib_lincomb: fixes rs :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M" shows "T (rs \<bullet>\<cdot> ms) = rs \<bullet>\<star> map T ms"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Inca.instr.IPush x1 = Inca.instr.IPush y1) = (x1 = y1)"
u_in_nton: shows "\<forall>u\<in>set Seq. length u = length ys"
exec_meth_length_compE2D: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and e :: "'addr expr1"   and d :: "nat"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P (compE2 e) (compxE2 e 0 d) t h (stk, loc, pc, xcp) ta h' s'" shows "pc < length (compE2 e)"
conj_simps_6: fixes P :: "fm"   and Q :: "fm" assumes "P \<noteq> T"   and "P \<noteq> F"   and "Q \<noteq> T"   and "Q \<noteq> F"   and "P \<noteq> Q" shows "Ferrack.conj P Q = And P Q"
RG_mult_closed: fixes x :: "('a, 'g) aezfun"   and y :: "('a, 'g) aezfun" assumes "x \<in> group_ring"   and "y \<in> group_ring" shows "x * y \<in> group_ring"
blinfun_ext: fixes x :: "'b \<Rightarrow>\<^sub>L 'c"   and y :: "'b \<Rightarrow>\<^sub>L 'c" shows "(x = y) = (\<forall>i. blinfun_apply x i = blinfun_apply y i)"
Diff_idemp: fixes A :: "'d set"   and B :: "'d set" shows "A - B - B = A - B"
inputPres: fixes P :: "pi"   and x :: "name"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name" assumes "\<forall>y. (P[x::=y], Q[x::=y]) \<in> Rel"   and "eqvt Rel" shows "a<x>.P \<leadsto>[Rel] a<x>.Q"
count_list_gr_1_E: fixes xs :: "'b list"   and x :: "'b" assumes "1 < count_list xs x" obtains i :: "nat" and j :: "nat" where "i < j" and "j < length xs" and "xs ! i = x" and "xs ! j = x"
corestr_def: fixes f :: "'s" shows "corestr f = mkArr (Dom f) (Img f) (Fun f)"
lm11: fixes x :: "'a"   and P :: "('a \<times> 'b) set" shows "{x} \<times> UNIV \<inter> P = {x} \<times> P `` {x}"
simply_connected_eq_contractible_loop_all: fixes S :: "'b set" shows "simply_connected S = (S = {} \<or> (\<exists>a\<in>S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> homotopic_loops S p (linepath a a)))"
WT_gpv_full: fixes gpv :: "('d, 'b, 'c) gpv" shows "\<I>_full \<turnstile>g gpv \<surd>"
list_rel_split_left_iff: fixes l :: "'d list"   and y :: "'e"   and ys :: "'e list"   and R :: "('d \<times> 'e) set" shows "((l, y # ys) \<in> \<langle>R\<rangle>list_rel) = (\<exists>x xs. l = x # xs \<and> (x, y) \<in> R \<and> (xs, ys) \<in> \<langle>R\<rangle>list_rel)"
ident_has_integral: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "((\<lambda>x. x) has_integral (b\<^sup>2 - a\<^sup>2) / 2) {a..b}"
comm_neutral: fixes a :: "'b" shows "a * (1::'b) = a"
HFinite_inverse2: fixes x :: "'a star" assumes "x \<in> HFinite - Infinitesimal" shows "inverse x \<in> HFinite"
of_nat_ceiling: fixes r :: "'c" shows "r \<le> of_nat (nat \<lceil>r\<rceil>)"
fbd_ffb: fixes f :: "'d \<Rightarrow> 'c set"   and X :: "'d set" shows "bd\<^sub>\<F> f X = \<Inter> {Y. X \<subseteq> fb\<^sub>\<F> f Y}"
toplevel_summands_ACI_norm: fixes r :: "'c rexp" shows "toplevel_summands \<guillemotleft>r\<guillemotright> = ACI_norm ` toplevel_summands r"
v1: shows "V1 = \<lparr>ext = x1, lan = lan v, own = own v\<rparr>"
root3_def: shows "root3 \<equiv> prod.fst (fix_syn (\<lambda>(root3, op3, rep3, grep3). (\<Lambda> pat. rep3\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root3\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op3\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. grep3\<cdot>pat\<cdot>(case cfst\<cdot>usvs of [::] \<Rightarrow> Null | u :# us \<Rightarrow> sfoldl\<cdot>(op3\<cdot>pat)\<cdot>(root3\<cdot>pat)\<cdot>us)\<cdot>usvs, \<Lambda> pat l usvs. Node\<cdot>usvs\<cdot>(next\<cdot>(csnd\<cdot>usvs)\<cdot>l)\<cdot>(case csnd\<cdot>usvs of [::] \<Rightarrow> Null | v :# vs \<Rightarrow> rep3\<cdot>pat\<cdot>(cfst\<cdot>usvs :@ [:v:], vs)))))"
eNorm_Norm: fixes x :: "'c"   and N :: "'c quasinorm" assumes "x \<in> space\<^sub>N N" shows "eNorm N x = ennreal (Norm N x)"
assms_4: fixes \<sigma> :: "'\<sigma>" assumes "I (dom (\<alpha> m)) \<sigma>" shows "P \<sigma>"
assms_2: shows "(g \<longlongrightarrow> 0) F"
subcs_psubst: fixes z :: "('a, 'b) form list"   and c :: "'a"   and f :: "'a \<Rightarrow> 'c"   and s :: "'a FOL_Fitting.term" assumes "\<forall>x\<in>\<Union> (params ` set z). x \<noteq> c \<longrightarrow> f x \<noteq> f c" shows "map (psubst f) (subcs c s z) = subcs (f c) (psubstt f s) (map (psubst f) z)"
mat_one_def: fixes i :: "'b"   and j :: "'b" shows "\<epsilon> i j \<equiv> if i = j then 1::'a else (0::'a)"
less_SucE: fixes m :: "nat"   and n :: "nat" assumes "m < Suc n" obtains "m < n"   | "m = n"
vproduct_vdoubletonD_3: fixes f :: "V"   and a :: "V"   and b :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct (set {a, b}) A" shows "f\<lparr>b\<rparr> \<in>\<^sub>\<circ> A b"
converse_rtranclE2: fixes xa :: "'a"   and xb :: "'b"   and za :: "'a"   and zb :: "'b"   and r :: "(('a \<times> 'b) \<times> 'a \<times> 'b) set" assumes "((xa, xb), za, zb) \<in> r\<^sup>*" obtains "(xa, xb) = (za, zb)"   | a :: "'a" and b :: "'b" where "((xa, xb), a, b) \<in> r" and "((a, b), za, zb) \<in> r\<^sup>*"
vec_lambda_inverse: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> UNIV" shows "($h) (vec_lambda y) = y"
exhaust: fixes y :: "('f, 'g, 'd, 'e, 'h) qTerm" obtains x11 :: "'d" and x12 :: "'e" where "y = qVar x11 x12"   | x21 :: "'h" and x22 :: "'f \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm option" and x23 :: "'g \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qAbs option" where "y = qOp x21 x22 x23"
split_face_empty_ram1_ram2_in_f21': fixes oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat" assumes "pre_split_face oldF ram1 ram2 []" shows "(ram1, ram2) \<in> \<E> (snd (split_face oldF ram1 ram2 []))"
east_west_on_equator_1: shows "proj2_incident east equator"
"4_1": shows "v = v\<^sub>1 @ [a] @ v\<^sub>2"
q_decomp_1: shows "q = q1 @ (u, w) # (w, v) # q2"
transTerminationLessState1: fixes Vbl :: "nat set" shows "trans (terminationLessState1 Vbl)"
prod_lub_def: fixes Sup_a :: "'a set \<Rightarrow> 'a"   and Sup_b :: "'b set \<Rightarrow> 'b"   and Y :: "('a \<times> 'b) set" shows "prod_lub Sup_a Sup_b Y = (Sup_a (fst ` Y), Sup_b (snd ` Y))"
simps_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and zs :: "'a list" shows "linord_of_listP x y (z # zs) = (z = y \<and> x \<in> set (z # zs) \<or> linord_of_listP x y zs)"
lm06: fixes f :: "('c \<times> 'd) set" assumes "trivial f" shows "runiq f"
has_derivative_within: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> ((\<lambda>y. (1 / norm (y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> (0::'c)) (at x within s))"
mk_deriv_intro_n: fixes i :: "nat"   and n :: "nat"   and j :: "nat" shows "deriv_n (mk_deriv_intro i n j) = n"
enum_char_unfold: shows "enum_class.enum = map char_of [0..<256]"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "rev (x # xs) = rev xs @ [x]"
split_list_first_prop: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>y\<in>set ys. \<not> P y)"
WeakF_def: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "WF(F)_v \<equiv> \<diamond>\<box>Enabled \<langle>F\<rangle>_v \<longrightarrow> \<box>\<diamond>\<langle>F\<rangle>_v"
9: shows "Ind (set [a]) (set v\<^sub>1)"
cComm2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "distinct xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'"
calc1: shows "CLBINT s=0..ereal x. f s n * (iexp s - 1) = (CLBINT s=0..ereal x. f s n * iexp s) - (CLBINT s=0..ereal x. f s n)"
inner_prod_distrib_left: fixes u :: "'c vec"   and n :: "nat"   and v :: "'c vec"   and w :: "'c vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "inner_prod (v + w) u = inner_prod v u + inner_prod w u"
per_eq: fixes x :: "'b list"   and r :: "'b list" shows "x \<le>p r\<^sup>\<omega> = ((\<exists>k z. r \<^sup>@ k \<cdot> z = x \<and> z \<le>p r) \<and> r \<noteq> \<epsilon>)"
dgrad_p_set_le_subset: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat" assumes "F \<subseteq> G" shows "punit.dgrad_p_set_le d F G"
divisor_sigma_0_left: fixes n :: "nat" shows "divisor_sigma (0::'b) n = of_nat (divisor_count n)"
foldl_def: fixes f :: "'d \<Rightarrow> 'e \<times> 'f \<Rightarrow> 'd"   and \<sigma> :: "'d"   and t :: "('e, 'f) FingerTree" shows "local.foldl f \<sigma> t \<equiv> FingerTreeStruc.foldl f \<sigma> (Rep_FingerTree t)"
finite_ordLess_infinite2: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "infinite B" shows "|A| <o |B|"
oalist_inv_raw_map_val_pair: fixes xs :: "('a \<times> 'd) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e" assumes "oalist_inv_raw xs" shows "oalist_inv_raw (map_val_pair f xs)"
pwD1: fixes S :: "'a nres"   and S' :: "'a nres" assumes "S \<le> S'"   and "nofail S'" shows "nofail S"
assms_3: shows "b \<in> hyp2 \<union> S"