hcju/isabelleps huggingface dataset:
left_right_impl_times: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "times (imp_l a b) a = times a (imp_r a b)"
horner_bounds_2: fixes x :: "float"   and f :: "nat \<Rightarrow> nat"   and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and F :: "nat \<Rightarrow> nat"   and s :: "nat"   and lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"   and prec :: "nat"   and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"   and j' :: "nat"   and n :: "nat" assumes "0 \<le> real_of_float x"   and "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"   and "\<And>i k x. lb 0 i k x = 0"   and "\<And>n i k x. lb (Suc n) i k x = float_plus_down prec (lapprox_rat prec 1 (int k)) (- float_round_up prec (x * ub n (F i) (G i k) x))"   and "\<And>i k x. ub 0 i k x = 0"   and "\<And>n i k x. ub (Suc n) i k x = float_plus_up prec (rapprox_rat prec 1 (int k)) (- float_round_down prec (x * lb n (F i) (G i k) x))" shows "(\<Sum>j = 0..<n. (- 1) ^ j * (1 / real (f (j' + j))) * real_of_float (x ^ j)) \<le> real_of_float (ub n ((F ^^ j') s) (f j') x)"
emeasure_space_1: fixes M :: "'a measure" assumes "prob_space M" shows "emeasure M (space M) = 1"
sentence_iff: fixes G :: "('b, 'a) labeled_graph"   and e\<^sub>1 :: "'b allegorical_term"   and e\<^sub>2 :: "'b allegorical_term" shows "G \<Turnstile> e\<^sub>1 \<sqsubseteq> e\<^sub>2 = (:G:\<lbrakk>e\<^sub>1\<rbrakk> \<subseteq> :G:\<lbrakk>e\<^sub>2\<rbrakk>)"
emeasure_ball: shows "emeasure lborel (ball c r) = ennreal (unit_ball_vol (real DIM('a)) * r ^ DIM('a))"
ext_cf_len: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and m :: "nat" assumes "pol_coeff S c" shows "fst (ext_cf S m c) = m + fst c"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom \<nu>)"
IH: fixes x :: "'a" assumes "x \<in> M" shows "x = a \<or> x \<in> iterates_above (f a)"
nat_mult_dvd_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "(k * m dvd k * n) = (m dvd n)"
assms_1: shows "agree (fmrestrict_set X f) (fmrestrict_set X g)"
filter_distinct_at3: fixes xs :: "'a list"   and as :: "'a list"   and u :: "'a"   and bs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and us :: "'a list"   and zs :: "'a list" assumes "distinct xs"   and "xs = as @ u # bs"   and "filter (\<lambda>v. v = u \<or> P v) xs = u # us"   and "\<forall>z\<in>set zs. z \<in> set as \<or> \<not> P z" shows "filter P (zs @ bs) = us"
degree_proots_count: fixes p :: "complex poly" shows "degree p = proots_count p UNIV"
RETURN_RES_refine: fixes x :: "'b"   and R :: "('b \<times> 'a) set"   and X :: "'a set" assumes "\<exists>x'. (x, x') \<in> R \<and> x' \<in> X" shows "RETURN x \<le> \<Down> R (RES X)"
exhaust_sel: fixes transaction :: "transaction" obtains "transaction = Transaction (un_Transaction1 transaction) (un_Transaction2 transaction) (views transaction)"
rep_eq: fixes x :: "nat \<Rightarrow> 'b"   and xa :: "'b mpoly" shows "insertion x xa = insertion_aux x (mapping_of xa)"
cases: fixes x :: "('b, 'c) Combinators" obtains xa :: "'b" and y :: "'b" where "x = DenyAllFromTo xa__ y__"   | xa :: "'b" and y :: "'b" and p :: "'c" where "x = AllowPortFromTo xa y p"   | "x = DenyAll"   | v :: "('b, 'c) Combinators" and va :: "('b, 'c) Combinators" where "x = v \<oplus> va"
exhaust: fixes \<Gamma> :: "('k, 'l) web_scheme"   and y :: "'m bi.vertex" assumes "countable_bipartite_web \<Gamma>" obtains "y = bi.SOURCE"   | "y = bi.SINK"   | x3 :: "'m" where "y = bi.Inner x3"
AudioPlayer_CTRL_Delta_def: shows "AudioPlayer_CTRL_Delta = {(''Off'', (En ''O'', defaultguard, defaultaction), ''On''), (''On'', (En ''O'', defaultguard, defaultaction), ''Off'')}"
iadd_le_eint_iff: fixes x :: "eint"   and y :: "eint"   and n :: "int" shows "(x + y \<le> eint n) = (\<exists>y' x'. x = eint x' \<and> y = eint y' \<and> x' + y' \<le> n)"
strict_mono_monoD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a" assumes "OrdinalInduct.strict_mono f"   and "A \<le> B" shows "f A \<le> f B"
subr: shows "{flow0 x t1<--<flow0 x t2} \<subseteq> {flow0 x t1<--<b}"
assms_1: fixes P :: "'state"   and Q :: "'state" assumes "P \<approx>\<cdot> Q" shows "P \<Turnstile> x = Q \<Turnstile> x"
cnt_def: fixes S :: "'b set set"   and T :: "'c set set" shows "cnt S T = {f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>. \<forall>m. m open\<^bsub>T\<^esub> \<longrightarrow> carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>}"
max_absorb1: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "order_ow U le ls"   and "y \<in> U"   and "x \<in> U"   and "le y x" shows "(with le : \<guillemotleft>max\<guillemotright> x y) = x"
total_def: fixes valuation :: "'a"   and variables :: "nat set" shows "CNF.total valuation variables \<equiv> variables \<subseteq> vars valuation"
subst_floatarith_eq_self: fixes f :: "floatarith"   and s :: "nat \<Rightarrow> floatarith" assumes "max_Var_floatarith f = 0" shows "subst_floatarith s f = f"
ttree_restr_ccTTree: fixes S' :: "Vars.var set"   and S :: "Vars.var set"   and G :: "CoCalls" shows "ttree_restr S' (ccTTree S G) = ccTTree (S \<inter> S') G"
lt_p: shows "lt (rep_list a \<odot> b) = punit.lt (rep_list a) \<oplus> lt b"
cond_action_oks_conv_set: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and cts :: "'f conditional_action list" shows "cond_action_oks s t cts = (\<forall>ct\<in>set cts. cond_action_ok s t ct)"
Pi_split_insert_domain: fixes x :: "'a \<Rightarrow> 'b"   and i :: "'a"   and I :: "'a set"   and X :: "'a \<Rightarrow> 'b set" shows "(x \<in> Pi (insert i I) X) = (x \<in> Pi I X \<and> x i \<in> X i)"
simps_2: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and accu :: "('b \<times> 'b) list"   and e :: "'b \<times> 'b"   and Es :: "('b \<times> 'b) list" shows "filter_compliant_stateful_ACS_accu G M accu (e # Es) = (if e \<notin> backflows (edges G) \<and> (\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (e # accu)\<rparr>). F \<subseteq> backflows (set (e # accu))) then filter_compliant_stateful_ACS_accu G M (e # accu) Es else filter_compliant_stateful_ACS_accu G M accu Es)"
fls_base_factor_zero_iff: fixes f :: "'a fls" shows "(fls_base_factor f = 0) = (f = 0)"
axioms_2: fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "LSCategory_axioms C"
hom: fixes g :: "'g"   and g' :: "'g" shows "T (g + g') = T g + T g'"
thm_noncont_e_e_1: fixes v :: "i" shows "[Necessary L in v]"
hyps: fixes tp' :: "cell list \<times> cell list" assumes "steps0 (1, tp) A n = (0, tp')" shows "\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"
directedE: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "directed A less_eq"   and "x \<in> A"   and "y \<in> A" obtains z :: "'c" where "z \<in> A" and "less_eq x z" and "less_eq y z"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.in_hom f a b" obtains "S.arr f" and "S.dom f = a" and "S.cod f = b"
cases: fixes a :: "'d tree"   and S :: "'d set" assumes "a \<in> trees S" obtains "a = \<langle>\<rangle>"   | l :: "'d tree" and r :: "'d tree" and v :: "'d" where "a = \<langle>l, v, r\<rangle>" and "l \<in> trees S" and "r \<in> trees S" and "v \<in> S"
iam_of_list_def: fixes l :: "'a option list"   and i :: "nat" shows "iam_of_list l i \<equiv> if i < length l then l ! i else None"
DirProd_group: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" assumes "Group.group G"   and "Group.group H" shows "Group.group (G \<times>\<times> H)"
les_qe_forward: fixes les :: "(real \<times> real \<times> real) list" assumes "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> ((\<forall>(d, e, f)\<in>set les. \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a'). \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)))" shows "\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
compose_leftassociativity: fixes x :: "wall"   and y :: "wall"   and z :: "wall" shows "(x \<circ> y) \<circ> z = x \<circ> y \<circ> z"
CJNB_def: fixes f :: "fm \<Rightarrow> fm"   and p :: "fm" shows "CJNB f p \<equiv> let cjs = conjuncts p; (yes, no) = partition bound0 cjs in MIR.conj (decr (list_conj yes)) (f (list_conj no))"
tensor_preserves_iso: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and f :: "'d"   and g :: "'d" assumes "monoidal_category C T \<alpha> \<iota>"   and "category.iso C f"   and "category.iso C g" shows "category.iso C (T (f, g))"
mset_le_not_sym: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "M < N" shows "\<not> N < M"
lm020: fixes X :: "'a set"   and mx :: "'a"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "finite X"   and "mx \<in> X"   and "f x < f mx" shows "x \<notin> argmax f X"
preallocated_new: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and x :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set" assumes "preallocated h"   and "h a = None" shows "preallocated (h(a \<mapsto> x))"
induct: fixes P :: "key \<Rightarrow> bool"   and key :: "key" assumes "P SigK"   and "P VerK"   and "\<And>x. P (PriK x)"   and "\<And>x. P (PubK x)"   and "\<And>x. P (ShaK x)"   and "\<And>x. P (SesK x)" shows "P key"
right_total_rel_resumption: fixes R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "right_total R1"   and "right_total R2" shows "right_total (Resumption.resumption.rel_resumption R1 R2)"
topspace_subtopology: fixes U :: "'d topology"   and V :: "'d set" shows "topspace (subtopology U V) = topspace U \<inter> V"
fps_of_poly_def: fixes p :: "'b poly" shows "fps_of_poly p = Abs_fps (coeff p)"
Sum2_2: fixes Q :: "pi"   and u :: "name"   and Q'' :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>lu in Q''\<rightarrow>a<x> \<prec> Q'" shows "P \<oplus> Q \<Longrightarrow>\<^sub>lu in Q''\<rightarrow>a<x> \<prec> Q'"
BuildSeq_imp_LstSeq: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf" assumes "BuildSeq B C s k y" shows "LstSeq s k y"
inv_into_image_cancel: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and S :: "'c set" assumes "inj_on f A"   and "S \<subseteq> A" shows "inv_into A f ` f ` S = S"
distinct_nthsI: fixes xs :: "'b list"   and I :: "nat set" assumes "distinct xs" shows "distinct (nths xs I)"
objE: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" obtains "Span.ide a" and "Span.src a = a" and "Span.trg a = a"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'b"   and bool :: "bool" shows "(case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = (if bool = True then f1 else f2)"
whileLoop_cond_fail: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and s :: "'d"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool" assumes "\<not> C x s" shows "whileLoop C   B x s = return x s"
permutation_insert_bij_betw: fixes q :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "q permutes {0..<n}"   and "i < Suc n"   and "j < Suc n" shows "bij_betw (permutation_insert i j q) {0..<Suc n} {0..<Suc n}"
strand_subst_fv_bounded_if_img_bounded: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" assumes "range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S" shows "fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> fv\<^sub>s\<^sub>t S"
inj_on_conv_Ex1: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "inj_on f A = (\<forall>y\<in>f ` A. \<exists>!x. x \<in> A \<and> y = f x)"
swap_restrict: fixes A :: "'a set"   and B :: "'b set"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A"   and "finite B" shows "(\<Sum>x\<in>A. sum (g x) {y \<in> B. R x y}) = (\<Sum>y\<in>B. \<Sum>x | x \<in> A \<and> R x y. g x y)"
is_measure_trivial: fixes f :: "'g \<Rightarrow> nat" shows "is_measure f"
isLub_def: fixes S :: "'b set"   and po :: "'b potype" shows "isLub S po \<equiv> \<lambda>L. L \<in> pset po \<and> (\<forall>y\<in>S. (y, L) \<in> order po) \<and> (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y, z) \<in> order po) \<longrightarrow> (L, z) \<in> order po)"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "b \<le>p a"   and "c \<le>p b" shows "c \<le>p a"
Arg2pi_exp: fixes z :: "complex" assumes "0 \<le> Im z"   and "Im z < 2 * pi" shows "Arg2pi (exp z) = Im z"
axioms_3: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "symmetry_functor A1 A2" shows "product_category A1 A2"
exteq_infinite: fixes f :: "recf"   and a :: "nat" assumes "recfn a f" shows "infinite {g. recfn a g \<and> g \<simeq> f}"
fin_digraph_del_arc: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" assumes "fin_digraph G" shows "fin_digraph (pre_digraph.del_arc G a)"
scene_override_id: fixes S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a" shows "S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on \<top>\<^sub>S = S\<^sub>2"
fds_abs_converges_weak: fixes s :: "complex" assumes "1 < Re s" shows "fds_abs_converges (fds \<chi>) s"
xyz: shows "[[Q\<^sub>x Q\<^sub>y Q\<^sub>z]]"
hyps_4: fixes e'' :: "'a expr" assumes "is_lub,P,E \<turnstile> e2 \<leadsto> e''" shows "e2' = e''"
iFROM_cut_greater: fixes n :: "nat"   and t :: "nat" shows "[n\<dots>] \<down>> t = (if n \<le> t then [Suc t\<dots>] else [n\<dots>])"
inverse_fds_nonunit: fixes f :: "'b fds" assumes "fds_nth f 1 = (0::'b)" shows "inverse f = 0"
carrier_direct_limE: fixes X :: "('a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)) set" assumes "X \<in> dl.carrier_direct_lim" obtains U :: "'a set set" and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" where "U \<in> top.neighborhoods \<pp>" and "s \<in> \<O> U" and "X = dl.class_of U s"
snth_0: fixes a :: "'b"   and w :: "'b stream" shows "(a ## w) !! 0 = a"
assms_4: shows "\<exists>n w. a n w \<noteq> 0"
simps_14: fixes e :: "char list exp" shows "fv (throw e) = fv e"
distinct_39: fixes x51 :: "('c, 'd) Term.term"   and x52 :: "('c, 'd) Term.term"   and x71 :: "'d list"   and x72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "delete\<langle>x51,x52\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
countable_support: shows "countable {x. M2.prob {x} \<noteq> 0}"
sigma_sigma_algebra: fixes A :: "'b set set" assumes "sigma_algebra A" shows "sigma A = A"
eq: shows "complex_of_real (sqrt (real n)) * sum \<chi> {1..x} = \<tau> * (\<Sum>k = 1..<n. cnj (\<chi> k) * (\<Sum>m = 1..x. unity_root n (- int m * int k)))"
qSwap_sym: fixes zs :: "'f"   and x :: "'g"   and y :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "X #[[x \<and> y]]_zs = X #[[y \<and> x]]_zs"
uminus_poly_def: fixes p' :: "(char list list \<times> int) list" shows "uminus_poly p' = map (\<lambda>(a, b). (a, - b)) p'"
R\<^sub>e_def: fixes r :: "rat" shows "R\<^sub>e r = (case quotient_of r of (n, d) \<Rightarrow> Num (of_int n) / Num (of_int d))"
one_is_polynomial: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "domain R"   and "subring K R" shows "polynomial\<^bsub>R\<^esub> K [\<one>\<^bsub>R\<^esub>]"
assms_3: shows "a' \<equiv> fst (normalize_quot (a, b))"
ring_hom_uminus: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and f :: "'e \<Rightarrow> 'c"   and a :: "'e" assumes "UP_cring R"   and "ring S"   and "f \<in> ring_hom S R"   and "a \<in> carrier S" shows "f (\<ominus>\<^bsub>S\<^esub> a) = \<ominus>\<^bsub>R\<^esub> f a"
prefix_nth: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "k < Suc n" shows "prefix f n ! k = the (f k)"
lim_h_plus_le_g: fixes e :: "'v \<times> 'v" shows "(\<Squnion>n. h_plus n e) \<le> g e"
get_root_node_locs_def: shows "Shadow_DOM.get_root_node_locs \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_locs Shadow_DOM.get_parent_locs"
fix_eq: shows "P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>f i,Up\<^sub>f i\<rbrakk> = (\<Lambda> X. (\<lambda>s. Mprefix (Tr\<^sub>f i s) (\<lambda>e. X (Up\<^sub>f i s e))))\<cdot>(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>f i,Up\<^sub>f i\<rbrakk>)"
compact: fixes R :: "('a \<Rightarrow> real) set"   and S :: "'a set" assumes "function_ring_on R S" shows "compact S"
wait_def: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" shows "wait \<phi> xs \<equiv> LEAST n. (nxt ^^ n) \<phi> xs"
game2c_2d_bad: shows "map_spmf snd game2_c = map_spmf snd game2_d"
F2: shows "\<forall>x\<^sub>2 x\<^sub>3. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>"
L_arr: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and D :: "'a \<times> 'b \<Rightarrow> 'd"   and l :: "'b \<Rightarrow> 'd"   and \<chi> :: "'b \<Rightarrow> 'a \<Rightarrow> 'd"   and a :: "'b" assumes "parametrized_diagram J A B D"   and "\<forall>a. partial_magma.ide A a \<longrightarrow> limit_cone J B (\<lambda>j. D (j, a)) (l a) (\<chi> a)"   and "partial_magma.arr A a" shows "(\<exists>!f. partial_magma.in_hom B f (l (partial_magma.dom A a)) (l (partial_magma.cod A a)) \<and> (\<lambda>\<chi>\<in>{\<chi>. Limit.cone J B (\<lambda>j. D (j, partial_magma.cod A a)) (partial_magma.cod B f) \<chi>}. \<lambda>j. if partial_magma.arr J j then B (\<chi> j) f else partial_magma.null B) (\<chi> (partial_magma.cod A a)) = vertical_composite.map J B (\<chi> (partial_magma.dom A a)) (\<lambda>j. D (j, a))) \<and> partial_magma.in_hom B (if partial_magma.arr A a then limit_cone.induced_arrow J B (\<lambda>j. D (j, partial_magma.cod A a)) (l (partial_magma.cod A a)) (\<chi> (partial_magma.cod A a)) (l (partial_magma.dom A a)) (vertical_composite.map J B (\<chi> (partial_magma.dom A a)) (\<lambda>j. D (j, a))) else partial_magma.null B) (l (partial_magma.dom A a)) (l (partial_magma.cod A a)) \<and> (\<lambda>\<chi>\<in>{\<chi>. Limit.cone J B (\<lambda>j. D (j, partial_magma.cod A a)) (partial_magma.cod B (if partial_magma.arr A a then limit_cone.induced_arrow J B (\<lambda>j. D (j, partial_magma.cod A a)) (l (partial_magma.cod A a)) (\<chi> (partial_magma.cod A a)) (l (partial_magma.dom A a)) (vertical_composite.map J B (\<chi> (partial_magma.dom A a)) (\<lambda>j. D (j, a))) else partial_magma.null B)) \<chi>}. \<lambda>j. if partial_magma.arr J j then B (\<chi> j) (if partial_magma.arr A a then limit_cone.induced_arrow J B (\<lambda>j. D (j, partial_magma.cod A a)) (l (partial_magma.cod A a)) (\<chi> (partial_magma.cod A a)) (l (partial_magma.dom A a)) (vertical_composite.map J B (\<chi> (partial_magma.dom A a)) (\<lambda>j. D (j, a))) else partial_magma.null B) else partial_magma.null B) (\<chi> (partial_magma.cod A a)) = vertical_composite.map J B (\<chi> (partial_magma.dom A a)) (\<lambda>j. D (j, a))"
of_nat_hmset: fixes n :: "nat" shows "of_nat n = HMSet (replicate_mset n 0)"
weak_SN_both_mono_ordered_semiring_1_axioms: shows "weak_SN_both_mono_ordered_semiring_1 weak_gt_arctic_delta 1 pos_arctic_delta"
one_imp_carrier: assumes "\<one> \<in> I" shows "I = carrier R"
int_Idl_subset_ideal: fixes k :: "int"   and l :: "int" shows "(Idl\<^bsub>\<Z>\<^esub> {k} \<subseteq> Idl\<^bsub>\<Z>\<^esub> {l}) = (k \<in> Idl\<^bsub>\<Z>\<^esub> {l})"
obtain_rep: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "good X"   and "fresh xs x' X" shows "\<exists>X'. (X, X') \<in> swapped \<and> good X' \<and> Abs xs x X = Abs xs x' X'"
set_plus_mono2: fixes C :: "'c set"   and D :: "'c set"   and E :: "'c set"   and F :: "'c set" assumes "C \<subseteq> D"   and "E \<subseteq> F" shows "C + E \<subseteq> D + F"
assms_2: assumes "arr f"   and "local.dom f = a"   and "cod f = b" shows "T"
check_eqv_def: fixes idx :: "nat"   and r :: "(presb, unit) aformula"   and s :: "(presb, unit) aformula" shows "Presb.check_eqv idx r s = ((presb_wf idx r \<and> presb_lformula r) \<and> (presb_wf idx s \<and> presb_lformula s) \<and> (case Presb.closure idx (norm (RESTRICT r), norm (RESTRICT s)) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False))"
UN_ball_bex_simps_2: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>\<Union> (B ` A). P x) = (\<forall>a\<in>A. \<forall>x\<in>B a. P x)"
\<phi>: shows "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright> \<and> local.iso \<phi>"
lt_plus_distinct_eq_max: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ordered_term.lt term_of_pair ord_term p \<noteq> ordered_term.lt term_of_pair ord_term q" shows "ordered_term.lt term_of_pair ord_term (p + q) = ord.max ord_term (ordered_term.lt term_of_pair ord_term p) (ordered_term.lt term_of_pair ord_term q)"
I_intP_def: fixes p :: "'psym"   and al :: "univ list" shows "I_intP p al \<equiv> Ik_intP p (map2 proj (parOf p) al)"
H\<^sub>N_def: fixes n :: "nat" shows "H\<^sub>N n = H {..<n}"
i_p: shows "ground\<^sub>l l \<and> nat_of_fatom (get_atom l) < length (ds @ d) \<and> (ds @ d) ! nat_of_fatom (get_atom l) = (\<not> sign l)"
hseq_leg\<^sub>0: shows "hseq u\<^sub>0 \<chi>.chine"
key_sets_crypts_fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "key_sets \<lbrace>X, Y\<rbrace> (crypts H) \<subseteq> key_sets X (crypts H)"
lF_char_1: fixes f :: "'a" assumes "C.ide f" shows "\<guillemotleft>lF f : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright>"
Borsuk_separation_theorem_gen: fixes S :: "'a set" assumes "compact S" shows "(\<forall>c\<in>components (- S). \<not> bounded c) = (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1 \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere (0::'a) 1) f (\<lambda>x. c)))"