hcju/isabelleps huggingface dataset:
tendsto_arcosh: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "1 < a" shows "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F"
assms_3: shows "w\<^sub>r = p\<^sub>1 \<star> w"
int: fixes K :: "'a set" assumes "K \<in> \<D>" shows "interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
composition_closed: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<FF> z'"   and "b \<in> \<FF> z'" shows "+\<^bsub>z'\<^esub> a b \<in> \<FF> z'"
before_in_index: fixes l :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "x < y in l = (index l x = 0)"
valid_iff_valid_S_transform: fixes P :: "'state"   and x :: "('a, unit, ('act, 'pred) S_action) formula" shows "P \<Turnstile>\<^sub>S x = nominal_ts.valid (\<turnstile>\<^sub>S) S_transform.S_transition P (S_transform x)"
llist_of_inject: fixes xs :: "'b list"   and ys :: "'b list" shows "(llist_of xs = llist_of ys) = (xs = ys)"
is_nth_power_nth_power: fixes n :: "nat"   and x :: "'b" shows "is_nth_power n (x ^ n)"
octo_of_real_power: fixes x :: "real"   and y :: "nat" shows "octo_of_real (x ^ y) = octo_of_real x ^ y"
split_list_last: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
smult_zero_mat: fixes k :: "'b"   and nr :: "nat"   and nc :: "nat" shows "k \<cdot>\<^sub>m 0\<^sub>m nr nc = 0\<^sub>m nr nc"
select_convs_2: fixes precondition_of :: "'c list"   and add_effects_of :: "'c list"   and delete_effects_of :: "'c list"   and more :: "'d" shows "add_effects_of \<lparr>strips_operator.precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = add_effects_of"
in_hom_char: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" shows "C.VV.in_hom f a b = (C.VV.arr a \<and> C.VV.arr b \<and> C.VV.arr f \<and> C.VxV.in_hom f a b)"
square_free_factorization_last_coeff_nz: fixes f :: "'a poly"   and a :: "'a"   and fs :: "('a poly \<times> nat) list"   and fi :: "'a poly"   and i :: "nat" assumes "square_free_factorization f (a, fs)"   and "(fi, i) \<in> set fs"   and "coeff f 0 \<noteq> (0::'a)" shows "coeff fi 0 \<noteq> (0::'a)"
simps: fixes p :: "'a poly"   and k :: "nat" shows "decompose_prod_root_unity_main p k = (if k = 0 then ([], p) else let q = root_unity k in if q dvd p then if p = 0 then ([], 0) else map_prod ((#) k) id (decompose_prod_root_unity_main (p div q) k) else decompose_prod_root_unity_main p (k - 1))"
to_stream_stake: fixes k :: "nat"   and w :: "nat \<Rightarrow> 'b" shows "stake k (to_stream w) = prefix k w"
startsby_zero_power: fixes a :: "'a fps"   and n :: "nat" assumes "fps_nth a 0 = (0::'a)"   and "0 < n" shows "fps_nth (a ^ n) 0 = (0::'a)"
merge_conv_merge_it': fixes acc :: "'c list"   and f :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and ys :: "'c list" shows "rev acc @ Common.merge f xs ys = merge_it' f acc xs ys"
simps_2: fixes f :: "nat \<Rightarrow> nat"   and x :: "nat" shows "map_vars f (CVar x) = CVar (f x)"
gbijec_def: fixes F :: "('c, 'd) Group_scheme"   and G :: "('e, 'f) Group_scheme"   and f :: "'c \<Rightarrow> 'e" shows "gbij\<^bsub>F,G\<^esub> f = (gsurj\<^bsub>F,G\<^esub> f \<and> ginj\<^bsub>F,G\<^esub> f)"
no_cycles: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list" assumes "Tree G" shows "\<not> Graph.cycle G xs"
extended_Gromov_distance_triangle: fixes x :: "'a Gromov_completion"   and z :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_distance x z \<le> extended_Gromov_distance x y + extended_Gromov_distance y z"
UINF_alt_def: fixes A :: "'b \<Rightarrow> 'a upred"   and P :: "'b \<Rightarrow> 'a upred" shows "UINF A P = (\<Sqinter> i \<bullet> A i \<and> P i)"
sum_case_non: fixes a :: "hf"   and f :: "hf \<Rightarrow> 'b"   and g :: "hf \<Rightarrow> 'b" assumes "\<not> is_hsum a" shows "sum_case f g a = undefined"
v': shows "v = lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j)) - fs ! j \<odot> monomial (1::'b) (term_of_pair (0::'a, i)))"
1: shows "bprv (NN \<langle>\<phi>R\<rangle> \<langle>neg \<phi>R\<rangle>)"
sec_pubk_less_rel: fixes m :: "nat"   and m' :: "nat"   and n :: "nat" assumes "{m, m'} \<subseteq> bad_sec_prik"   and "(n, m) \<notin> bad_id"   and "(n, m') \<in> bad_id" shows "s\<^sub>0 \<Turnstile> sec_pubk_less n"
InvariantConsistentAfterApplyDecide: fixes Vbl :: "nat set"   and state :: "State" assumes "\<not> Vbl \<subseteq> vars (elements (getM state))"   and "InvariantConsistent (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)" shows "let state' = applyDecide state Vbl in InvariantConsistent (getM state')"
index_Basis_list_nth: fixes i :: "nat" assumes "i < DIM('a)" shows "index Basis_list (Basis_list ! i) = i"
swapidseq_even_even: fixes m :: "nat"   and p :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "swapidseq m p"   and "swapidseq n p" shows "even m = even n"
left_associated_sum: fixes w :: "i"   and x :: "i"   and y :: "i"   and z :: "i" shows "O w ((x \<oplus> y) \<oplus> z) = ((O w x \<or> O w y) \<or> O w z)"
const_subterm_subst: fixes c :: "'a"   and t :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "Fun c [] \<sqsubseteq> t" shows "Fun c [] \<sqsubseteq> t \<cdot> \<sigma>"
assms: shows "parallel i j x \<down>= prod_encode (1, v)"
inject_2: fixes x31 :: "cmd"   and x32 :: "cmd"   and y31 :: "cmd"   and y32 :: "cmd" shows "(x31;; x32 = y31;; y32) = (x31 = y31 \<and> x32 = y32)"
observations_def: shows "observations = (\<Union>f\<in>observe ` keys. f ` messages)"
sup_complement_intro: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y = - x \<squnion> - - x \<sqinter> - y"
Quotient: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "Quotient (list_all2 R) (map Abs) (map Rep) (list_all2 T)"
fmdomE: fixes x :: "'d"   and m :: "('d, 'e) fmap" assumes "x |\<in>| fmdom m" obtains y :: "'e" where "fmlookup m x = Some y"
alpha_preserves_qGood2: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "X #= X'"   and "FixVars.qGood X'" shows "FixVars.qGood X"
vlrestriction_vdomain: shows "f \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> f = f"
add_case: fixes a :: "'b"   and b :: "'b"   and p :: "'b point"   and q :: "'b point"   and P :: "'b point \<Rightarrow> 'b point \<Rightarrow> 'b point \<Rightarrow> bool" assumes "on_curve a b p"   and "on_curve a b q"   and "\<And>p. P Infinity p p"   and "\<And>p. P p Infinity p"   and "\<And>p. on_curve a b p \<Longrightarrow> P p (opp p) Infinity"   and "\<And>p\<^sub>1 x\<^sub>1 y\<^sub>1 p\<^sub>2 x\<^sub>2 y\<^sub>2 l. \<lbrakk>p\<^sub>1 = Point x\<^sub>1 y\<^sub>1; p\<^sub>2 = Point x\<^sub>2 y\<^sub>2; p\<^sub>2 = ell_field_class.add a p\<^sub>1 p\<^sub>1; y\<^sub>1 \<noteq> (0::'b); l = ((3::'b) * x\<^sub>1\<^sup>2 + a) / ((2::'b) * y\<^sub>1); x\<^sub>2 = l\<^sup>2 - (2::'b) * x\<^sub>1; y\<^sub>2 = - y\<^sub>1 - l * (x\<^sub>2 - x\<^sub>1)\<rbrakk> \<Longrightarrow> P p\<^sub>1 p\<^sub>1 p\<^sub>2"   and "\<And>p\<^sub>1 x\<^sub>1 y\<^sub>1 p\<^sub>2 x\<^sub>2 y\<^sub>2 p\<^sub>3 x\<^sub>3 y\<^sub>3 l. \<lbrakk>p\<^sub>1 = Point x\<^sub>1 y\<^sub>1; p\<^sub>2 = Point x\<^sub>2 y\<^sub>2; p\<^sub>3 = Point x\<^sub>3 y\<^sub>3; p\<^sub>3 = ell_field_class.add a p\<^sub>1 p\<^sub>2; x\<^sub>1 \<noteq> x\<^sub>2; l = (y\<^sub>2 - y\<^sub>1) / (x\<^sub>2 - x\<^sub>1); x\<^sub>3 = l\<^sup>2 - x\<^sub>1 - x\<^sub>2; y\<^sub>3 = - y\<^sub>1 - l * (x\<^sub>3 - x\<^sub>1)\<rbrakk> \<Longrightarrow> P p\<^sub>1 p\<^sub>2 p\<^sub>3" shows "P p q (ell_field_class.add a p q)"
monomial_power: fixes t :: "'b"   and c :: "'c"   and n :: "nat" shows "monomial c t ^ n = monomial (c ^ n) (\<Sum>i = 0..<n. t)"
greatest_fixpoint_char: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_greatest_fixpoint f x = (has_greatest_fixpoint f \<and> x = \<nu> f)"
Mf: shows "f \<in> count_space UNIV \<rightarrow>\<^sub>M count_space UNIV"
sym: fixes x :: "i"   and y :: "i" shows "x r y \<longrightarrow> y r x"
1: shows "R f = f \<star>\<^sub>B src\<^sub>B f \<and> R f' = f' \<star>\<^sub>B src\<^sub>B f'"
ext_hd_or_tl_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_hd_or_tl_axioms ext \<equiv> \<forall>gt ys xs y x. (\<forall>z y x. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x) \<longrightarrow> (\<forall>y x. gt y x \<or> gt x y \<or> y = x) \<longrightarrow> length ys = length xs \<longrightarrow> ext gt (y # ys) (x # xs) \<longrightarrow> gt y x \<or> ext gt ys xs"
prems_3: shows "\<forall>xs ys. a # as = xs @ ys \<and> ys \<noteq> [] \<longrightarrow> upd_cs cs xs \<noteq> []"
Wbis_trace: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and t :: "'a"   and tr :: "(('b, 'c) com \<times> 'a) list \<times> 'a" assumes "PL_Indis indis"   and "\<And>s. PL.mustT tval aval c s"   and "(c, c) \<in> PL_Indis.Wbis tval aval indis"   and "(s, t) \<in> indis"   and "PL_Indis.finTrace tval aval tr"   and "iconfig tr = (c, s)" shows "\<exists>tr'. PL_Indis.finTrace tval aval tr' \<and> iconfig tr' = (c, t) \<and> (fstate tr, fstate tr') \<in> indis"
kmin_prop: fixes f :: "nat \<Rightarrow> real" shows "kmin f PR \<subseteq> PR \<and> card (kmin f PR) = khl \<and> (\<forall>i\<in>kmin f PR. \<forall>j\<in>PR - kmin f PR. f i \<le> f j)"
assms_2: shows "ipresIGWlsAll h' hA' MOD' MOD''"
five_segment_axiom: fixes x :: "complex_homo"   and y :: "complex_homo"   and z :: "complex_homo"   and u :: "complex_homo"   and x' :: "complex_homo"   and y' :: "complex_homo"   and z' :: "complex_homo"   and u' :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "z \<in> unit_disc"   and "u \<in> unit_disc"   and "x' \<in> unit_disc"   and "y' \<in> unit_disc"   and "z' \<in> unit_disc"   and "u' \<in> unit_disc"   and "x \<noteq> y"   and "poincare_between x y z"   and "poincare_between x' y' z'"   and "poincare_distance x y = poincare_distance x' y'"   and "poincare_distance x u = poincare_distance x' u'"   and "poincare_distance y u = poincare_distance y' u'"   and "poincare_distance y z = poincare_distance y' z'" shows "poincare_distance z u = poincare_distance z' u'"
Un: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<union> b \<in> sets M"
\<phi>G_def: shows "\<phi>G \<equiv> diag (neg P)"
comp_assoc: fixes h :: "'b"   and g :: "'b"   and f :: "'b" shows "J.comp (J.comp h g) f = J.comp h (J.comp g f)"
omega2_vsubset_Limit: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<omega> \<in>\<^sub>\<circ> \<alpha>" shows "\<omega> + \<omega> \<subseteq>\<^sub>\<circ> \<alpha>"
grab_def: fixes N :: "nat set"   and n :: "nat" shows "grab N n \<equiv> (N \<inter> enum N ` {..<n}, N \<inter> {enum N n..})"
le_cases: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" obtains "x \<preceq> y"   | "y \<preceq> x"
lookup_lift_poly_fun_2_homogenous: fixes f :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "f 0 0 = 0" shows "lookup (lift_poly_fun_2 f p q) v = lookup (f (proj_poly (component_of_term v) p) (proj_poly (component_of_term v) q)) (pp_of_term v)"
chain_Field: fixes M :: "'a set" assumes "M \<in> Chains (restrict_rel A leq)"   and "M \<noteq> {}" shows "lub M \<in> A"
OUTfromChCorrect_data13: shows "OUTfromChCorrect data13"
ccSup_least: 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 bot :: "'b"   and top :: "'b"   and A :: "'b set"   and z :: "'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
valid_result_msaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "'msaux"   and auxlist :: "(nat \<times> event_data option list set) list" assumes "valid_msaux args cur aux auxlist" shows "result_msaux args aux = foldr (\<union>) (concat (map (\<lambda>(t, rel). if left (args_ivl args) \<le> cur - t then [rel] else []) auxlist)) {}"
vsv_vimage_iff: fixes b :: "V"   and A :: "V" shows "(b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A) = (\<exists>a. a \<in>\<^sub>\<circ> A \<and> a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<and> r\<lparr>a\<rparr> = b)"
flat_ordI: fixes x :: "'a"   and a :: "'a"   and y :: "'a" assumes "x \<noteq> a \<Longrightarrow> x = y" shows "flat_ord a x y"
snd_set_path_verties: fixes v :: "'vertex"   and v' :: "'vertex"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "path v v' pth" shows "fst ` snd ` set pth \<subseteq> fset vertices"
card_rcosets_equal: fixes R :: "nat set"   and H :: "nat set" assumes "R \<in> rcosets\<^bsub>G\<^esub> H"   and "H \<subseteq> carrier G" shows "card H = card R"
simps_2: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and x :: "name"   and \<Psi>' :: "'d"   and F :: "'d frame" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "x \<sharp> \<Psi>'" shows "assertionAux.insertAssertion SCompose (\<lparr>\<nu>x\<rparr>F) \<Psi>' = \<lparr>\<nu>x\<rparr>assertionAux.insertAssertion SCompose F \<Psi>'"
s2: shows "s2 = (x * y + q - s1) mod q"
to_Gromov_completion_inj: shows "inj to_Gromov_completion"
elims_3: fixes x :: "'d"   and xa :: "'e"   and xb :: "('f, 'g, 'd, 'e, 'h) qAbs" assumes "\<not> qFreshAbs x xa xb" obtains xs :: "'d" and x :: "'e" and ys :: "'d" and y :: "'e" and X :: "('f, 'g, 'd, 'e, 'h) qTerm" where "xa__ = xs" and "xa = x" and "xb = qAbs ys y X" and "\<not> (xs = ys \<and> x = y \<or> qFresh xs x X)"
iIN_0: fixes n :: "nat" shows "[n\<dots>,0] = {n}"
path_component_of_equiv: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "path_component_of X x y = (x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y)"
less_eq_def: fixes \<delta> :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and s :: "'a"   and s' :: "'a" shows "pre_RDR.less_eq \<delta> s s' \<equiv> \<exists>rs. s' = pre_RDR.exec \<delta> s rs"
uw\<theta>: shows "ide \<omega>.chine \<and> \<guillemotleft>\<omega>.the_\<theta> : t\<^sub>0 \<star> \<omega>.chine \<Rightarrow> r\<^sub>0\<guillemotright>"
solve_loop_unfold: fixes state :: "State"   and Vbl :: "nat set" shows "solve_loop state Vbl = (if getSATFlag state \<noteq> UNDEF then state else let state' = solve_loop_body state Vbl in solve_loop state' Vbl)"
check_tableau_valuated: fixes s :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<nabla> (check s)"
sol_dim_in_sat_dual: fixes x :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" assumes "x \<in> sat_dual A c" shows "dim_vec x = dim_row A"
image_poly_iso_incl: fixes L :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and i :: "'c"   and q :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = M.indexed_const \<zero>\<^bsub>M\<^esub>"   and "q \<in> carrier (poly_ring L)"   and "0 < degree q" shows "id \<in> ring_hom L (image_ring (M.eval_pmod L i q) (poly_ring L))"
exhaust: fixes y :: "'m option" obtains "y = None"   | x2 :: "'m" where "y = Some x2"
left_commute: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "b * (a * c) = a * (b * c)"
induct: fixes P :: "primfun \<times> 'a \<Rightarrow> bool"   and a0 :: "primfun \<times> 'a" assumes "\<And>x a. P (x, a)" shows "P a0"
cos_one_2pi: fixes x :: "real" shows "(cos x = 1) = ((\<exists>x. x = real x * 2 * pi) \<or> (\<exists>x. x = - (real x * 2 * pi)))"
composable: shows "src (cod \<mu>) = trg (cod \<nu>)"
conservative: fixes A :: "'a set" assumes "A \<in> sets M"   and "0 < emeasure M A" shows "\<exists>n>0. 0 < emeasure M ((T ^^ n) -` A \<inter> A)"
gcdI: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c dvd a"   and "c dvd b"   and "\<And>d. \<lbrakk>d dvd a; d dvd b\<rbrakk> \<Longrightarrow> d dvd c"   and "normalize c = c" shows "c = gcd a b"
SUP_combine: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>a b c d. \<lbrakk>a \<le> b; c \<le> d\<rbrakk> \<Longrightarrow> f a c \<le> f b d" shows "(SUP i. Sup (range (f i))) = (SUP i. f i i)"
below_trans: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.po below"   and "below x y"   and "below y z" shows "below x z"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>D)"
permutes_compose: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and q :: "'a \<Rightarrow> 'a" assumes "p permutes S"   and "q permutes S" shows "q \<circ> p permutes S"
89: fixes x :: "'a"   and y :: "'a" shows "top \<squnion> (x \<squnion> - y) = x \<squnion> top"
types_preserved_b_e1: fixes es :: "e list"   and es' :: "e list"   and s :: "s"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>es\<rparr> \<leadsto> \<lparr>es'\<rparr>"   and "store_typing s \<S>"   and "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> es' : ts _> ts'"
no_vars_in_ty: fixes x :: "Syntax.var"   and \<tau> :: "ty" shows "atom x \<sharp> \<tau>"
Hx: shows "x \<in> synth (analz (insert K (ik s)))"
eij_def: fixes i :: "nat"   and j :: "nat" shows "local.eij i j = lcm (a ! i) (b ! j) div b ! j"
SET_CR2: fixes evs2 :: "event list"   and i :: "nat"   and C :: "agent"   and NC1 :: "nat" assumes "evs2 \<in> set_cr"   and "Gets (CA i) \<lbrace>Agent C, Nonce NC1\<rbrace> \<in> set evs2" shows "Says (CA i) C \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce NC1\<rbrace>, cert (CA i) (pubEK (CA i)) onlyEnc (priSK RCA), cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace> # evs2 \<in> set_cr"
UniqueSucStates_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "UniqueSucStates (SAs (HA ST)) (CompFun (HA ST)) (StepConf (HA ST) (Conf ST) TS)"
delete_node_def: fixes v :: "'b"   and G :: "'b list_graph" shows "FiniteListGraph.delete_node v G = \<lparr>nodesL = remove1 v (nodesL G), edgesL = filter (\<lambda>(e1, e2). e1 \<noteq> v \<and> e2 \<noteq> v) (edgesL G)\<rparr>"
measurable_stopping_time_stream: fixes M :: "'b measure"   and T :: "'b stream \<Rightarrow> enat" assumes "stopping_time (stream_filtration M) T" shows "T \<in> stream_space M \<rightarrow>\<^sub>M count_space UNIV"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "category A"   and "category B"   and "functor_axioms A B F" shows "functor A B F"
simps_2: fixes v :: "vname aexp"   and va :: "vname aexp" shows "enumerate_gexp_inputs (Eq v va) = enumerate_aexp_inputs v \<union> enumerate_aexp_inputs va"
s_ge: fixes i :: "nat" shows "real i \<le> s i"
induct: fixes P :: "nat list \<Rightarrow> ('var \<Rightarrow> 'var) \<Rightarrow> ('form, 'rule, 'subst, 'var) itnode rose_tree \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "'var \<Rightarrow> 'var"   and a2 :: "('form, 'rule, 'subst, 'var) itnode rose_tree" assumes "\<And>is f r ants. (\<And>x i. x \<in> set ants \<Longrightarrow> P (is @ [i]) (rerename (a_fresh (inPorts' (iNodeOf (RNode r ants)) ! i)) (iAnnot (RNode r ants)) (isidx is) f) x) \<Longrightarrow> P is f (RNode r ants)" shows "P a0 a1 a2"
coregular_tcsigs_imp_exe_coregular_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss"   and "coregular_tcsigs (set cs) (translate_ars arss)" shows "exe_coregular_tcsigs cs arss"
sorted2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and zs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # y # zs) = (less_eq x y \<and> linorder.sorted less_eq (y # zs))"
M_def: shows "M \<equiv> \<lambda>k. let (dk, ak, bk, M) = DF k in M"
length_subseqs: fixes xs :: "'a list" shows "length (subseqs xs) = 2 ^ length xs"
length_subset_card: fixes X :: "'c set"   and x :: "'c"   and xs :: "'c list" assumes "finite X"   and "distinct (x # xs)"   and "set (x # xs) \<subseteq> X" shows "length xs < card X"
us: shows "us = [] @ berlekamp_factorization_main d [f] (berlekamp_basis f) (length (berlekamp_basis f))"
butlast_append_Cons: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "butlast (xs @ y # ys) = xs @ butlast (y # ys)"
VLambda_mono: fixes A :: "V"   and B :: "V"   and f :: "V \<Rightarrow> V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "VLambda A f \<subseteq>\<^sub>\<circ> VLambda B f"
from_to_Gromov_completion: fixes x :: "'c" shows "from_Gromov_completion (to_Gromov_completion x) = x"
ind: shows "add_independentS [G, \<Oplus>H\<leftarrow>Gs. H]"
simulation_reaches: fixes a :: "'b"   and a' :: "'b"   and b :: "'a" assumes "B.reaches a a'"   and "b \<sim> a" shows "\<exists>b'. A.reaches b b' \<and> b' \<sim> a'"
vsv_axioms: shows "vsv (F\<lparr>ArrVal\<rparr>)"
aux5: fixes a :: "'a"   and b :: "'a"   and p :: "('a, 'b) Combinators list" assumes "DenyAllFromTo a b \<in> set p" shows "a \<in> set (net_list p)"
is_lub_below_iff: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and u :: "'a" assumes "class.po below"   and "po.is_lub below S x" shows "below x u = po.is_ub below S u"
Retr_incl_1: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "Sretr theta \<subseteq> ZOretrT theta"
simps_8: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "fol fm" shows "semantics e f g (fm.Not p) = (\<not> semantics e f g p)"
Total_subset_Id: fixes r :: "('d \<times> 'd) set" assumes "Total r"   and "r \<subseteq> Id" shows "r = {} \<or> (\<exists>a. r = {(a, a)})"
\<tau>Red': shows "s' -\<triangleright>ttas'\<rightarrow>* s''"
msetext_huet_singleton: fixes y :: "'b"   and x :: "'b"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "y \<noteq> x" shows "msetext_huet gt [y] [x] = gt y x"
variants_def: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "variants C D = (generalizes C D \<and> generalizes D C)"
carrier_vs_is_self: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and N :: "'g set" assumes "module R M" shows "carrier (M\<lparr>carrier := N\<rparr>) = N"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "Y.Cd d X = greatest (Pd_singletons_for_ds Y ds d) (X \<inter> Field (Pd_singletons_for_ds Y ds d))"
assms: fixes z :: "'a" assumes "z \<in> set xs" shows "f z = f x"