hcju/isabelleps huggingface dataset:
infs_snth: fixes P :: "'c \<Rightarrow> bool"   and w :: "'c stream" shows "infs P w = (\<forall>n. \<exists>k\<ge>n. P (w !! k))"
len1_h: shows "length qs = 1 \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p qs (get_subsets (calculate_data p qs)) (get_signs (calculate_data p qs)) (get_matrix (calculate_data p qs))"
content_cbox': fixes a :: "'a"   and b :: "'a" assumes "cbox a b \<noteq> {}" shows "content (cbox a b) = (\<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i)"
simps_5: fixes S :: "(char list \<Rightarrow> int) set"   and Inv :: "(char list \<Rightarrow> int) set"   and b :: "bexp"   and c :: "(char list \<Rightarrow> int) set acom"   and P :: "(char list \<Rightarrow> int) set" shows "step S ({Inv} WHILE b DO c {P}) = {S \<union> post c} WHILE b DO step {s \<in> Inv. bval b s} c {{s \<in> Inv. \<not> bval b s}}"
tranclp_unfold: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "r\<^sup>+\<^sup>+ a b = ((a, b) \<in> {(x, y). r x y}\<^sup>+)"
set_of_vector_def: fixes A :: "('a, 'b) vec" shows "set_of_vector A = {A $ i |i. i \<in> UNIV}"
Collect_ball_eq: fixes A :: "'b set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
zero_uexpr_def: shows "0 = \<guillemotleft>0::'e\<guillemotright>"
invar_def: fixes t :: "'t" shows "invar t \<equiv> inv t \<and> sorted1 (inorder t)"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('w \<Rightarrow> 'w) \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme"   and v :: "'w" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
uniformly_continuous_on_uniformity: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "uniformly_continuous_on s f = (LIM (x, y) uniformity_on s. (f x, f y) :> uniformity)"
card_Suc_eq: fixes A :: "'e set"   and k :: "nat" shows "(card A = Suc k) = (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
trms_list\<^sub>s\<^sub>s\<^sub>t_is_trms\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('e, 'f) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t S = set (trms_list\<^sub>s\<^sub>s\<^sub>t S)"
alpha_def: shows "alpha = PZ.prob {x. a < \<bar>x\<bar>} / 4"
nb: shows "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi>"
irreducible_connect_field: fixes f :: "'c poly" shows "irreducible\<^sub>d f = irreducible f"
sup_mono: fixes c :: "'c"   and a :: "'c"   and d :: "'c"   and b :: "'c" assumes "c \<le> a"   and "d \<le> b" shows "c \<sqinter> d \<le> a \<sqinter> b"
sum_subtractf: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "(\<Sum>x\<in>A. f x - g x) = sum f A - sum g A"
interleavings_inject_1: fixes xs :: "'a list"   and xs' :: "'a list"   and ys :: "'a list"   and ys' :: "'a list"   and zs :: "'a list"   and zs' :: "'a list" assumes "(set xs \<union> set xs') \<inter> (set ys \<union> set ys') = {}"   and "interleavings xs ys zs"   and "interleavings xs' ys' zs'"   and "zs = zs'" shows "xs = xs'"
vlrestriction_VLambda: fixes A :: "V"   and f :: "V \<Rightarrow> V"   and B :: "V" shows "VLambda A f \<restriction>\<^sup>l\<^sub>\<circ> B = VLambda (A \<inter>\<^sub>\<circ> B) f"
zmset_of_add_mset: fixes x :: "'a"   and M :: "'a multiset" shows "zmset_of (add_mset x M) = add_zmset x (zmset_of M)"
completeness': fixes x0 :: "'g"   and x1 :: "'a"   and w :: "'h + 'b" assumes "((x0, x1), w) \<in> Rel_OR" shows "\<forall>e\<in>carrier L. spmf (\<Sigma>_OR.completeness_game (x0, x1) w e) True = 1"
prems_2: shows "last_prog p ((r # sb\<^sub>1) @ sb') = p"
static_cenv_lookup: fixes i :: "char list"   and len :: "nat"   and b :: "tid_or_exn" assumes "cupcake_nsLookup static_cenv i = Some (len, b)" obtains name :: "char list" where "b = TypeId (Short name)"
assoc'_eq_inv_assoc: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = D.inv \<a>\<^sub>D[f, g, h]"
equiv_trans: fixes V :: "'b set"   and R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "equiv V R"   and "(x, y) \<in> R"   and "(y, z) \<in> R" shows "(x, z) \<in> R"
le_zero_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and n :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
splitAt_distinct_ram_fst: fixes vs :: "'c list"   and ram :: "'c" assumes "distinct vs" shows "ram \<notin> set (fst (splitAt ram vs))"
bin_last_numeral_simps_6: fixes w :: "num" shows "odd (numeral (num.Bit1 w))"
has_codomain_iff_arr: fixes f :: "'a" shows "(B.codomains f \<noteq> {}) = B.arr f"
concat_take_sing: fixes k :: "nat"   and l :: "nat"   and r :: "'b list" assumes "k \<le> l" shows "concat (take k ([r] \<^sup>@ l)) = r \<^sup>@ k"
lNact0: fixes c :: "'nid"   and t :: "nat \<Rightarrow> cnf" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>0\<^esub> = 0"
lnull_ltlI: fixes xs :: "'a llist" assumes "lnull xs" shows "lnull (ltl xs)"
append_exec_meth_xt: 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 ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   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 stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and n :: "nat"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and ins' :: "'addr instr list" assumes "exec_meth ci P ins xt t h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"   and "jump_ok ins 0 n"   and "pcs xt' \<subseteq> {0..<length ins'}" shows "exec_meth ci P (ins' @ ins) (xt' @ shift (length ins') xt) t h (stk, loc, length ins' + pc, xcp) ta h' (stk', loc', length ins' + pc', xcp')"
arr_char: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.arr f = (\<F>C.arr f \<and> (\<F>C.ide (\<F>C.dom f) \<and> Diag (\<F>C.DOM (\<F>C.dom f))) \<and> \<F>C.ide (\<F>C.cod f) \<and> Diag (\<F>C.DOM (\<F>C.cod f)))"
gcs_minus: fixes u :: "'a"   and s :: "'a"   and t :: "'a" assumes "u adds s"   and "u adds t" shows "gcs (s - u) (t - u) = gcs s t - u"
mat_assoc_test_13: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "A + B = B + A"
currentD_outside': fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "current \<Gamma> f"   and "e \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub>" shows "f e = 0"
openin_subtopology_empty: fixes U :: "'c topology"   and S :: "'c set" shows "openin (subtopology U {}) S = (S = {})"
trancl_unfold_left: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>+ = r O r\<^sup>*"
vars_power: fixes p :: "'a mpoly"   and n :: "nat" shows "vars (p ^ n) \<subseteq> vars p"
a: shows "\<forall>x\<in>X. \<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> [(a, b)] \<bullet> x \<in> X"
npeneg_def: fixes e :: "pexpr" shows "npeneg e = (case e of PExpr1 (PCnst c) \<Rightarrow> PExpr1 (PCnst (- c)) | PExpr1 _ \<Rightarrow> PExpr1 (PNeg e) | PExpr2 pexpr2 \<Rightarrow> PExpr1 (PNeg e))"
fls_times_fps_to_fls: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_to_fls (f * g) = fps_to_fls f * fps_to_fls g"
add_nonneg_nonneg: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "{#} \<subseteq># a"   and "{#} \<subseteq># b" shows "{#} \<subseteq># a + b"
coeff: fixes n :: "nat" shows "coeff p n = nth_default (0::'a) xs n"
assms_4: shows "\<E> (TER g) \<subseteq> TER f"
sublist_appendI: fixes xs :: "'b list"   and ps :: "'b list"   and ss :: "'b list" shows "xs \<le>f ps \<cdot> xs \<cdot> ss"
Liminf_bounded: fixes C :: "'b"   and X :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F n in F. C \<le> X n" shows "C \<le> Liminf F X"
normalI: fixes H :: "'g set" assumes "Group H"   and "\<forall>g\<in>G. \<forall>h\<in>H. \<exists>h'\<in>H. g + h = h' + g"   and "\<forall>g\<in>G. \<forall>h\<in>H. \<exists>h'\<in>H. h + g = g + h'" shows "normal H"
fmadd_empty_1: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmempty ++\<^sub>f m = m"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "comm_monoid_add_hom hom"   and "monoid_mult_hom hom" shows "semiring_hom hom"
pw_ref_I: fixes S' :: "'b nres"   and S :: "'c nres"   and R :: "('c \<times> 'b) set" assumes "nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> (\<exists>s'. (x, s') \<in> R \<and> inres S' s'))" shows "S \<le> \<Down> R S'"
APP_def: fixes f :: "'f \<Rightarrow> 'e"   and a :: "'f" shows "f $ a \<equiv> f a"
ccpo_Sup_least: fixes A :: "'a set"   and z :: "'a" assumes "Complete_Partial_Order.chain (\<le>) A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "\<Squnion> A \<le> z"
wf_ws_prog: fixes G :: "prog" assumes "wf_prog G" shows "ws_prog G"
comp_fun_idem_remove: shows "comp_fun_idem Set.remove"
relax_outgoing''_def: fixes l :: "('c \<times> 'b) list"   and du :: "'c"   and V :: "'b \<Rightarrow> 'd option"   and Q :: "'b \<Rightarrow> 'c option" shows "relax_outgoing'' l du V Q = fold (\<lambda>(d, v) Q. case Q v of None \<Rightarrow> if v \<in> dom V then Q else Q(v \<mapsto> du + d) | Some d' \<Rightarrow> Q(v \<mapsto> min (du + d) d')) l Q"
bv_to_nat_nat_to_bv_length: fixes x :: "nat"   and y :: "nat" assumes "nat_to_bv_length x y \<noteq> []" shows "bv_to_nat (nat_to_bv_length x y) = x"
assms_3: fixes x :: "\<nu>"   and y :: "\<nu>" shows "[\<phi> (x\<^sup>P) (y\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) in v]"
wellformed_\<R>: fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "ltl_FG_to_rabin \<Sigma> \<G> w" shows "finite (reach \<Sigma> (semi_mojmir_def.step \<Sigma> \<up>af\<^sub>G (Abs \<phi>)) (semi_mojmir_def.initial (Abs \<phi>)))"
lnth_lappend: fixes xs :: "'a llist"   and ys :: "'a llist"   and n :: "nat" shows "lappend xs ys $ n = (if enat n < llength xs then xs $ n else ys $ n - the_enat (llength xs))"
powr_def: fixes x :: "'b"   and a :: "'b" shows "x powr a \<equiv> if x = (0::'b) then 0::'b else exp (a * ln x)"
isCont_s: fixes x :: "real" shows "isCont other.s x"
assms_2: shows "BuildSeq B C s2 k2 y2"
dagger_denest2: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> (y \<cdot> x\<^sup>\<star>)\<^sup>\<star>"
max_set_def: fixes X :: "'a set" shows "max_set X = {x \<in> X. \<forall>y\<in>X. x \<le> y \<longrightarrow> x = y}"
dghm_ArrMap_vsv: shows "vsv (\<GG>\<lparr>ArrMap\<rparr>)"
i_complex_of_real: fixes r :: "real" shows "\<i> * complex_of_real r = Complex 0 r"
finite_chain_with2_def: fixes x :: "'a"   and z :: "'a"   and X :: "'a set" shows "[[x .. z]X] \<equiv> \<exists>y\<in>X. [[x .. y .. z]X]"
rep_eq: fixes x :: "('d, 'e) mapping_rbt" shows "init x = rbt_init (impl_of x)"
simps_6: fixes pi :: "(name \<times> name) list"   and x :: "'d" shows "pi \<bullet> rev pi \<bullet> x = x"
weakly_bisimilar_symp: shows "symp (\<approx>\<cdot>\<^sub>S)"
untilI: fixes Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat"   and P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "Q (\<sigma> |\<^sub>s i)"   and "\<forall>k<i. P (\<sigma> |\<^sub>s k)" shows "(P \<U> Q) \<sigma>"
max_0_1_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq zero one = one"
Abs_eq_res_set: fixes bs :: "atom set"   and x :: "'e"   and cs :: "atom set"   and y :: "'e" shows "([bs]res. x = [cs]res. y) = ([bs \<inter> supp x]set. x = [cs \<inter> supp y]set. y)"
max_step1_axioms: shows "max_step1 t (t0 + e2') B' (euler_C TYPE('a)) ((r - b) / 2)"
rep_eq: fixes x :: "'b mod_ring"   and xa :: "'b mod_ring" shows "Rep_mod_ring (x * xa) = Rep_mod_ring x * Rep_mod_ring xa mod int CARD('b)"
partn_lst_VWF_\<omega>_2: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "partn_lst_VWF (\<omega> \<up> (1 + \<alpha>)) [ord_of_nat 2, \<omega> \<up> (1 + \<alpha>)] 2"
isPath_rtc: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "isPath u p v" shows "(u, v) \<in> E\<^sup>*"
r_unit_sg: fixes H :: "'a set"   and h :: "'a" assumes "G \<guillemotright> H "   and "h \<in> H" shows "h \<cdot> \<one> = h"
extension_valid_maximal_conforming: fixes v' :: "'a" assumes "v' \<rightarrow> v0"   and "v' \<in> VV p \<Longrightarrow> \<sigma> v' = v0" shows "vmc_path G (LCons v' P') v' p \<sigma>"
Rep_frame_inject: fixes x :: "'d frame"   and y :: "'d frame" shows "(Rep_frame x = Rep_frame y) = (x = y)"
linear_span_subModule: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and A :: "'f set"   and H :: "'e set" assumes "R module M"   and "ideal R A"   and "H \<subseteq> carrier M" shows "submodule R M (linear_span R M A H)"
rep_eq: fixes x :: "'a freeword"   and xa :: "'a freeword" shows "freeword (x - xa) = prappend_signed_list (freeword x) (rev (map flip_signed (freeword xa)))"
P_ltl_step_0: fixes P :: "nat option \<times> value option list \<times> nat \<Rightarrow>f value option \<Rightarrow> bool"   and l :: "String.literal"   and i :: "value list" assumes "P (None, [], <>)"   and "l = STR ''select'' \<longrightarrow> P (Some 1, [], <1 $:= Some (hd i), 2 $:= Some (Num 0)>)" shows "P (ltl_step drinks (Some 0) <> (l, i))"
cINF_superset_mono: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c multiset"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_below (g ` B)"   and "A \<subseteq> B"   and "\<And>x. x \<in> B \<Longrightarrow> g x \<subseteq># f x" shows "Inf (g ` B) \<subseteq># Inf (f ` A)"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "b \<le>f a"   and "c \<le>f b" shows "c \<le>f a"
comp_char: fixes g :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.comp g f = (if VV.arr f \<and> VV.arr g \<and> VxV.seq g f then VxV.comp g f else VxV.null)"
elims: fixes x :: "nat set"   and xa :: "(nat set \<times> 'a option list set) set"   and y :: "(nat set \<times> 'a option list set) set" assumes "projectQuery x xa = y" obtains V :: "nat set" and s :: "(nat set \<times> 'a option list set) set" where "x = V" and "xa = s" and "y = projectTable V ` s"
assms_6: shows "rk {A, B, C, A', B', C'} \<le> 3"
induct: fixes P :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list \<Rightarrow> ('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list \<Rightarrow> bool"   and a0 :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and a1 :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "\<And>D. P [] D"   and "\<And>i t A D. P A D \<Longrightarrow> P ((i, send\<langle>t\<rangle>) # A) D"   and "\<And>i t A D. P A D \<Longrightarrow> P ((i, receive\<langle>t\<rangle>) # A) D"   and "\<And>i ac t t' A D. P A D \<Longrightarrow> P ((i, \<langle>ac: t \<doteq> t'\<rangle>) # A) D"   and "\<And>i t s A D. P A (List.insert (i, t, s) D) \<Longrightarrow> P ((i, insert\<langle>t,s\<rangle>) # A) D"   and "\<And>i t s A D. (\<And>x. x \<in> set (subseqs (dbproj i D)) \<Longrightarrow> P A (filter (\<lambda>d. d \<notin> set x) D)) \<Longrightarrow> P ((i, delete\<langle>t,s\<rangle>) # A) D"   and "\<And>i ac t s A D. P A D \<Longrightarrow> P ((i, \<langle>ac: t \<in> s\<rangle>) # A) D"   and "\<And>i X F F' A D. P A D \<Longrightarrow> P ((i, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle>) # A) D" shows "P a0 a1"
iteratei_rule_P: assumes "set_iterator_genord it S0 (\<lambda>_ _. True)"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {x}. True; \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
group_l_invI: assumes "\<And>x. x \<in> UNIV \<Longrightarrow> \<exists>y\<in>UNIV. y * x = 1" shows "Group.group \<Z>"
not_empty_irreducible: fixes C :: "'s set" assumes "C \<in> UNIV // communicating" shows "C \<noteq> {}"
ac_Zp_in_Zp: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "ac_Zp x \<in> carrier Zp"
isom_gch_units_transpTr5_1: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and h :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "Gchain n h"   and "i \<le> n"   and "j \<le> n"   and "i < j"   and "isom_Gchains n (transpos i j) g h" shows "g i \<cong> h j"
Interval'_eq_Some: fixes a :: "'a"   and b :: "'a"   and i :: "'a interval" assumes "Interval' a b = Some i" shows "lower i = a \<and> upper i = b"
rbt_entries_generator_def: shows "rbt_entries_generator \<equiv> Generator (rbt_has_next, rbt_entries_next)"
seqI: fixes f :: "'a1"   and g :: "'a1" assumes "C1.arr f"   and "C1.arr g"   and "C1.dom g = C1.cod f" shows "C1.seq g f"
Un_insert_left: fixes a :: "'h"   and B :: "'h set"   and C :: "'h set" shows "insert a B \<union> C = insert a (B \<union> C)"
index_size_conv: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "length xs = n" shows "(index xs x = n) = (x \<notin> set xs)"
mayDiverge_Seq_L: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.mayDiverge tval aval c1 s" shows "PL.mayDiverge tval aval (c1 ;; c2) s"
unbounded_components_complement_absolute_retract: fixes C :: "'a set"   and S :: "'a set" assumes "C \<in> components (- S)"   and "compact S"   and "AR S" shows "\<not> bounded C"
filter_leI: fixes F' :: "'a filter"   and F :: "'a filter" assumes "\<And>P. eventually P F' \<Longrightarrow> eventually P F" shows "F \<le> F'"
e: shows "fmul (0::'b) e = (0::'c)"
tagged_fun_relD_lhs: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'g \<Rightarrow> 'h"   and A :: "('e \<times> 'g) set"   and B :: "('f \<times> 'h) set"   and x :: "'e"   and x' :: "'g" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, f' x') \<in> B"
lower_on_lit_eq: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and L :: "'a literal" assumes "lower_on \<sigma> \<eta> (vars_of_lit L)" shows "subst_lit L \<sigma> = subst_lit L \<eta> \<or> (subst_lit L \<sigma>, subst_lit L \<eta>) \<in> lit_ord"
dagger_plus_one: fixes x :: "('a \<times> 'a) set" shows "Id \<union> x\<^sup>* = x\<^sup>*"
non_empty_cycle_root_msc: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "many_strongly_connected x"
associative_rat_poly_mat: fixes m1 :: "int poly fract list list"   and m2 :: "int poly fract list list"   and m3 :: "int poly fract list list"   and nc :: "nat" assumes "mat (rat_poly.row_length m1) (rat_poly.row_length m2) m1"   and "mat (rat_poly.row_length m2) (rat_poly.row_length m3) m2"   and "mat (rat_poly.row_length m3) nc m3" shows "rat_poly.matrix_mult m1 (rat_poly.matrix_mult m2 m3) = rat_poly.matrix_mult (rat_poly.matrix_mult m1 m2) m3"
funion_fminus_cancel: fixes A :: "'a fset"   and B :: "'a fset" shows "A |\<union>| (B |-| A) = A |\<union>| B"
rep_eq: fixes x :: "'a reg_lan"   and xa :: "'a reg_lan" shows "Rep_reg_lan (x + xa) = Rep_reg_lan x + Rep_reg_lan xa"
leI: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "\<not> lex_pm_strict x y" shows "lex_pm y x"
list_induct3: fixes xs :: "'d list"   and ys :: "'e list"   and zs :: "'f list"   and P :: "'d list \<Rightarrow> 'e list \<Rightarrow> 'f list \<Rightarrow> bool" assumes "length xs = length ys"   and "length ys = length zs"   and "P [] [] []"   and "\<And>x xs y ys z zs. \<lbrakk>length xs = length ys; length ys = length zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys) (z # zs)" shows "P xs ys zs"
compl_pmdl_def: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h" shows "compl_pmdl compl = (\<forall>gs bs ps sps data. is_Groebner_basis (fst ` set gs) \<longrightarrow> sps \<noteq> [] \<longrightarrow> set sps \<subseteq> set ps \<longrightarrow> unique_idx (gs @ bs) data \<longrightarrow> fst ` set (fst (compl gs bs (ps -- sps) sps data)) \<subseteq> pmdl (args_to_set (gs, bs, ps)))"
R_REF: fixes x :: "'b"   and x' :: "'a" assumes "(x, x') \<in> R'"   and "\<not> b x"   and "\<not> b' x'"   and "I x"   and "I' x'" shows "(x, x') \<in> R"
\<R>_divergent_divergent_aux: fixes xs :: "('c \<Rightarrow> real) stream"   and i :: "nat"   and c1 :: "'c"   and j :: "nat"   and c2 :: "'c" assumes "stream_trans xs"   and "stream_all (\<lambda>u. u \<in> V) xs"   and "(xs !! i) c1 = 0"   and "\<exists>k>i. k \<le> j \<and> (xs !! k) c2 = 0"   and "\<forall>k>i. k \<le> j \<longrightarrow> (xs !! k) c1 \<noteq> 0"   and "c1 \<in> \<X>"   and "c2 \<in> \<X>" shows "(xs !! j) c2 \<le> (xs !! j) c1"
set_empty: fixes xs :: "'e list" shows "(set xs = {}) = (xs = [])"
lossless: shows "gen_lossless_gpv b \<I> gpv"
simps_5: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and C :: "char list" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Class C) = (P,E,h \<turnstile> e : Class C \<or> P,E,h \<turnstile> e : NT)"
nprv_impE0: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla"   and F :: "'fmla set"   and \<chi> :: "'fmla" assumes "imp \<phi> \<psi> \<in> F"   and "nprv F \<phi>"   and "nprv (insert \<psi> F) \<chi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "\<chi> \<in> fmla" shows "nprv F \<chi>"
limit_g: shows "(g \<longlongrightarrow> G) at_top"