hcju/isabelleps huggingface dataset:
converse_pseudonatural_equivalence_def: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" shows "converse_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G \<and> pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
eq0: shows "(x - a)\<^sup>2 + (y - b)\<^sup>2 = c"
consistent: fixes s :: "real" assumes "s \<in> {t0..t1}" shows "consistent x s t1 B p incr"
perp_foot_eq_implies_drop_perp_eq: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line"   and c :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident a l"   and "perp_foot b l = perp_foot c l" shows "drop_perp b l = drop_perp c l"
map_simp: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" shows "CopxC.map f = f"
lookup_hom_components: fixes q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "q \<in> hom_components p"   and "t \<in> keys q" shows "lookup q t = lookup p t"
better_CutL_intro: fixes M :: "trm"   and M' :: "trm"   and a :: "coname"   and x :: "name"   and N :: "trm" assumes "M \<longrightarrow>\<^sub>a M'" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>a Cut <a>.M' x.N"
p_n: shows "(\<lambda>x. real (nth_prime x) / (real x * ln (real x))) \<longlonglongrightarrow> 1"
r_right_minus_eq: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a" assumes "ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<ominus>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>) = (a = b)"
length_drop: fixes n :: "nat"   and xs :: "'d list" shows "length (drop n xs) = length xs - n"
is_link_diagram_def: fixes x :: "wall" shows "is_link_diagram x \<equiv> if is_tangle_diagram x then Preliminaries.abs (domain_wall x) + Preliminaries.abs (codomain_wall x) = 0 else False"
set_iterator_image_def: fixes g :: "'h \<Rightarrow> 'i" shows "set_iterator_image g \<equiv> set_iterator_image_filter (\<lambda>x. Some (g x))"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "P F"   and "P T"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Le v)"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (And v va)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
IN_k_i: fixes i :: "nat"   and x :: "'v" assumes "flow_network \<Delta> (f_i i)"   and "value_flow \<Delta> (f_i i) = value_flow \<Delta> (f_aux i)" shows "d_IN (k_i i) x \<le> value_flow (residual_network (f_i i)) (k_i i)"
list_seq_nth: fixes seq :: "nat \<Rightarrow> 'c list"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "\<And>i. \<exists>x. seq (Suc i) = x # seq i"   and "i < length (seq j)"   and "j \<le> k" shows "rev (seq k) ! i = rev (seq j) ! i"
tl_last: fixes xs :: "'a list" assumes "tl xs \<noteq> []" shows "last xs = last (tl xs)"
ih: fixes C :: "'a literal list" assumes "L \<in> set C'"   and "is_reducible_lit Ds (C @ remove1 L C') L" shows "reduce Ds C C' \<noteq> C'"
Jacobi_0_eq_0: fixes n :: "int" assumes "\<not> is_unit n" shows "Jacobi 0 n = 0"
assms_2: shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) (A_pos @ A_neg) L"
rank_singleton: fixes a :: "hf" shows "rank \<lbrace>a\<rbrace> = succ (rank a)"
ipresIGWlsAll_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('l, 'm, 'n, 'o, 'p, 'q, 'h, 'j, 'r) model_scheme"   and MOD' :: "('s, 't, 'n, 'o, 'u, 'v, 'i, 'k, 'w) model_scheme" shows "ipresIGWlsAll h hA MOD MOD' \<equiv> ipresIGWls h MOD MOD' \<and> ipresIGWlsAbs hA MOD MOD'"
cases: fixes x :: "typ" obtains uu_ :: "variable" and S :: "String.literal set" where "x = Tv uu_ S"   | uv_ :: "String.literal" and Ts :: "typ list" where "x = Ty uv_ Ts"
monadic_nfoldli_eq: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool nres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "monadic_nfoldli l c f s = (case l of [] \<Rightarrow> RETURN s | x # ls \<Rightarrow> c s \<bind> (\<lambda>b. if b then f x s \<bind> monadic_nfoldli ls c f else RETURN s))"
continuous_closed_preimage: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "closed S"   and "closed T" shows "closed (S \<inter> f -` T)"
inner_mult_right: fixes a :: "'c"   and m :: "real"   and b :: "'c" shows "inner a (of_real m * b) = m * inner a b"
ts_lose_ack_st2send: fixes lose :: "'a \<Rightarrow> bool list"   and t :: "'a"   and ack :: "'a \<Rightarrow> 'b list"   and x :: "'b"   and i1 :: "'a \<Rightarrow> 'c list"   and vc :: "'a \<Rightarrow> 'd list"   and st_out :: "'a \<Rightarrow> 'b"   and y :: "'b"   and b :: "'a \<Rightarrow> 'c list" assumes "lose t = [True] \<or> lose t = [False]"   and "lose t = [True] \<longrightarrow> ack t = [x] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = x"   and "lose t = [False] \<longrightarrow> ack t = [y] \<and> i1 t = b t \<and> vc t = [] \<and> st_out t = y" shows "ack t = [st_out t]"
cs_when_recording: fixes i :: "nat"   and j :: "nat"   and t :: "('a, 'b, 'c) event list"   and p :: "'a"   and cid :: "nat"   and q :: "'a" assumes "i < j"   and "j \<le> length t"   and "trace init t final"   and "ps (S t i) p \<noteq> None"   and "snd (cs (S t i) cid) = Recording"   and "snd (cs (S t j) cid) = Done"   and "channel cid = Some (p, q)" shows "map Msg (fst (cs (S t j) cid)) = map Msg (fst (cs (S t i) cid)) @ takeWhile ((\<noteq>) Marker) (msgs (S t i) cid)"
simps_1: fixes a :: "'a"   and as :: "'a list"   and b :: "'a"   and bs :: "'a list"   and accs :: "'a list" shows "merge_tailrec (a # as) (b # bs) accs = (if a < b then merge_tailrec as (b # bs) (a # accs) else if a = b then merge_tailrec as bs (a # accs) else merge_tailrec (a # as) bs (b # accs))"
state_list_length_lemma: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and s :: "('a, 'b) fmap" shows "length as = length (state_list s as) - 1"
vwalk_arcs_butlast: fixes xs :: "'a list" shows "vwalk_arcs (butlast xs) = butlast (vwalk_arcs xs)"
combine: shows "subst_bv (Fv x T) (subst_bvs1' s 1 (map (\<lambda>(x, y). Fv x y) vs)) = subst_bvs (map (\<lambda>(x, y). Fv x y) ((x, T) # vs)) s"
ec': shows "spr_simAbs ec = spr_jkbpCSt t"
unit_div_mult_swap: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c one" shows "times a (divide b c) = divide (times a b) c"
accepts_atom: fixes a :: "'b"   and w :: "'b list" shows "accepts (atom a) w = (w = [a])"
rep_eq: fixes x :: "'b set"   and xa :: "('b \<times> 'b) set" shows "Rep_ugraph (graph_internal x xa) = (if finite x \<and> finite xa then (x \<union> fst ` xa \<union> snd ` xa, xa \<union> xa\<inverse> - Id) else ({}, {}))"
bin_to_bl_zero_aux: fixes n :: "nat"   and bl :: "bool list" shows "bin_to_bl_aux n 0 bl = replicate n False @ bl"
Skip: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "'b" shows "\<Gamma>\<turnstile>Skip \<down> Normal s"
dX_def: fixes X :: "'a set"   and d :: "'b" shows "Y.dX X d = {x \<in> X. Xd x = d}"
run_altc_nondet: fixes mergec :: "'f cset \<Rightarrow> ('f \<Rightarrow> 'd) \<Rightarrow> 'd"   and A :: "'f cset"   and f :: "'f \<Rightarrow> ('e, 'd) nondetT" shows "run_nondet (cset_nondetM_base.altc_nondet mergec A f) = mergec A (run_nondet \<circ> f)"
OF_match_fields_unsafe_def: fixes m :: "of_match_field set"   and p :: "(32, 'a) simple_packet_ext_scheme" shows "OF_match_fields_unsafe m p = (\<forall>f\<in>m. match_no_prereq f p)"
isoI: fixes f :: "'c \<times> 'c \<times> 'c" assumes "C.iso (fst f)"   and "C.VxV.iso (snd f)" shows "C.VxVxV.iso f"
eval_fds_higher_deriv: fixes f :: "'a fds"   and s :: "'a"   and n :: "nat" assumes "conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "eval_fds ((fds_deriv ^^ n) f) s = (deriv ^^ n) (eval_fds f) s"
ennreal_SUP_add_left: fixes I :: "'b set"   and f :: "'b \<Rightarrow> ennreal"   and c :: "ennreal" assumes "I \<noteq> {}" shows "(SUP i\<in>I. f i + c) = Sup (f ` I) + c"
IntD1: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
cos_of_real_pi_half: shows "cos (of_real pi / (2::'a)) = (0::'a)"
leg1_simps_3: shows "src g\<^sub>0h\<^sub>1.p\<^sub>1 = src g\<^sub>0h\<^sub>1.p\<^sub>0"
pseudominimal_in_pow: fixes A :: "'b set"   and x :: "'b set" assumes "order.pseudominimal_in (Pow A) x" shows "\<exists>a\<in>A. x = {a}"
lnth_ltl: fixes xs :: "'c llist"   and n :: "nat" assumes "\<not> lnull xs" shows "lnth (ltl xs) n = lnth xs (Suc n)"
starS: fixes x :: "'a" assumes "x \<in> S" shows "open_segment a x \<subseteq> rel_interior S"
sum_list_distinct_conv_sum_set: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and xs :: "'e list"   and f :: "'e \<Rightarrow> 'f" assumes "class.monoid_add plus zero"   and "distinct xs" shows "sum_list (map f xs) = sum f (set xs)"
assms_2: shows "\<not> list_all2 intT (parOf p) al"
assms_4: fixes x :: "real" assumes "x \<in> {a..b}" shows "(F has_real_derivative f x) (at x within {a..b})"
asymp_equivD_strong: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g"   and "\<forall>\<^sub>F x in F. f x \<noteq> (0::'c) \<or> g x \<noteq> (0::'c)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> (1::'c)) F"
m_c2: fixes C1 :: "(char list \<Rightarrow> 'av) option acom"   and C2 :: "(char list \<Rightarrow> 'av) option acom" assumes "top_on\<^sub>c C1 (- vars C1)"   and "top_on\<^sub>c C2 (- vars C2)"   and "C1 < C2" shows "m\<^sub>c C2 < m\<^sub>c C1"
ne: shows "(\<lambda>i. if i = n then 1 else 0) \<in> topspace (subtopology (nsphere (Suc n)) {x. 0 \<le> x (Suc n)}) \<inter> {x. x (Suc n) = 0}"
member_mset_neg_iff_zcount: fixes x :: "'a"   and M :: "'a zmultiset" shows "(x \<in># mset_neg M) = (zcount M x < 0)"
infinite_Ioo: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.dense_linorder less_eq less"   and "less a b" shows "infinite (ord.greaterThanLessThan less a b)"
no_initial_channel_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration" assumes "computation channel trans send recv init final" shows "\<forall>i. cs init i = ([], NotStarted)"
assms_3: shows "x (Suc n) \<noteq> \<zero>\<^bsub>residue_ring (p ^ Suc n)\<^esub>"
eq_pred_parent1: fixes i :: "nat"   and k :: "nat" assumes "eq_pred i (s1 k)"   and "i \<noteq> s1 k" shows "eq_pred i k"
sumCases: fixes P :: "ccs"   and Q :: "ccs"   and \<alpha> :: "act"   and R :: "ccs"   and Prop :: "ccs \<Rightarrow> bool" assumes "P \<oplus> Q \<longmapsto>\<alpha> \<prec> R"   and "\<And>P'. P \<longmapsto>\<alpha> \<prec> P' \<Longrightarrow> Prop P'"   and "\<And>Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<Longrightarrow> Prop Q'" shows "Prop R"
d_sub_c: shows "strictly_subsumes (fst Dl) (fst Cl) \<or> subsumes (fst Dl) (fst Cl) \<and> snd Dl \<sqsubset>l snd Cl"
image_map_indets_radical: fixes f :: "'c \<Rightarrow> 'd"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e) set" assumes "inj f" shows "map_indets f ` \<surd>F = \<surd>(map_indets f ` F) \<inter> P[range f]"
parent_child_rel_impl: shows "parent_child_rel = local.a_parent_child_rel"
stationary_distribution_exists: fixes A :: "'a st_mat" shows "\<exists>v. A *st v = v"
set_pmf_bind_spmf: fixes M :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "set_pmf (M \<bind> f) = set_pmf M \<bind> map_option_set (set_pmf \<circ> f)"
integral_by_parts': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and F :: "real \<Rightarrow> real"   and G :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> isCont f x"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> isCont g x"   and "\<And>x. (F has_real_derivative f x) (at x)"   and "\<And>x. (G has_real_derivative g x) (at x)" shows "LBINT x. indicat_real {a..b} x *\<^sub>R (F x * g x) = F b * G b - F a * G a - (LBINT x. indicat_real {a..b} x *\<^sub>R (f x * G x))"
Skip: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option" shows "E,dt\<Turnstile>Skip\<Colon>\<surd>"
abs_ge_self: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq a (abs a)"
length_swap: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" shows "length (swap xs i j) = length xs"
ACI_norm_Plus: fixes r :: "'c rexp"   and s :: "'c rexp"   and t :: "'c rexp" assumes "\<guillemotleft>r\<guillemotright> = Plus s t" shows "\<exists>s t. r = Plus s t"
lessThan_subset_iff: fixes x :: "'c"   and y :: "'c" shows "({..<x} \<subseteq> {..<y}) = (x \<le> y)"
pairwise_neutrality: fixes a :: "alt"   and b :: "alt"   and a' :: "alt"   and b' :: "alt"   and P :: "indi \<Rightarrow> (alt \<times> alt) set"   and P' :: "indi \<Rightarrow> (alt \<times> alt) set" assumes "a \<noteq> b"   and "a' \<noteq> b'"   and "P \<in> Prof"   and "P' \<in> Prof"   and "\<forall>i. (a <\<^bsub>P i\<^esub> b) = (a' <\<^bsub>P' i\<^esub> b')" shows "(a <\<^bsub>F P\<^esub> b) = (a' <\<^bsub>F P'\<^esub> b')"
\<rho>_ide_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "(if partial_magma.arr C a then C a (monoidal_category.runit C T \<alpha> \<iota> (partial_magma.dom C a)) else partial_magma.null C) = monoidal_category.runit C T \<alpha> \<iota> a"
P2_cong: fixes tms :: "trm list"   and P :: "trm \<Rightarrow> trm \<Rightarrow> fmla"   and H :: "fmla set"   and x :: "trm"   and x' :: "trm"   and y :: "trm"   and y' :: "trm" assumes "\<And>i t u x. atom i \<sharp> tms \<Longrightarrow> (P t u)(i::=x) = P (subst i x t) (subst i x u)"   and "H \<turnstile> x EQ x'"   and "H \<turnstile> y EQ y'" shows "H \<turnstile> P x y IFF P x' y'"
simps: fixes F :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F = \<Union> (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r ` set F)"
admissible_conj: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "ccpo.admissible lub ord P"   and "ccpo.admissible lub ord Q" shows "ccpo.admissible lub ord (\<lambda>x. P x \<and> Q x)"
monofun_sprod: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and h :: "'a"   and x :: "'a"   and a :: "'b" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "h \<in> H"   and "x \<in> fgs R M {h}"   and "a \<in> carrier R" shows "monofun R M N f H h (a \<cdot>\<^sub>s x) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> monofun R M N f H h x"
f: shows "forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr>"
set_mset_def: fixes M :: "'a multiset" shows "set_mset M = {x. 0 < count M x}"
card2_eE: fixes S :: "'b set" assumes "finite S"   and "\<exists>x y. x \<noteq> y \<and> x \<in> S \<and> y \<in> S" shows "2 \<le> card S"
l2_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l2_step2 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> guessed_frame Rb xgnx = Some gnx \<and> guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny))) \<and> Insec A B gnx \<in> chan s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgny, xgnx, xsk}), chan := {Auth B A \<langle>Number 0, Exp Gen (NonceF (Rb $ ny)), gnx\<rangle>} \<union> chan s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Running A B (Exp gnx (NonceF (Rb $ ny)))) else signalsInit s\<rparr>}"
weak_arrow_of_homs_axioms: shows "weak_arrow_of_homs (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) src\<^sub>D trg\<^sub>D F"
bin_log_esimps_2: fixes w :: "int" shows "w AND - 1 = w"
h1: shows "all_list_constr [[], [0]] (length qs)"
mrbst_join_reduce: fixes t1 :: "'a tree"   and t2 :: "'a tree" assumes "t1 \<noteq> \<langle>\<rangle>"   and "t2 \<noteq> \<langle>\<rangle>" shows "mrbst_join t1 t2 = bernoulli_pmf (real (size t1) / real (size t1 + size t2)) \<bind> (\<lambda>b. if b then case t1 of \<langle>l, x, r\<rangle> \<Rightarrow> map_pmf (Node l x) (mrbst_join r t2) else case t2 of \<langle>l, x, r\<rangle> \<Rightarrow> map_pmf (\<lambda>l'. \<langle>l', x, r\<rangle>) (mrbst_join t1 l))"
subcocycle_AE: fixes u :: "nat \<Rightarrow> (int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> real" assumes "\<And>m n. AE x in MI. u (n + m) x \<le> u n x + u m ((TI ^^ n) x)"   and "\<And>n. integrable MI (u n)" shows "\<exists>w. I.subcocycle w \<and> (AE x in MI. \<forall>n. w n x = u n x)"
SeqWRP_SUCC_E: fixes y' :: "name"   and s :: "tm"   and k :: "tm"   and y :: "tm"   and H :: "fm set"   and k' :: "tm" assumes "atom y' \<sharp> (s, k, y)"   and "H \<turnstile> k' EQ SUCC k" shows "insert (SeqWRP s k' y) H \<turnstile> SyntaxN.Ex y' (SeqWRP s k (Var y') AND y EQ Q_Succ (Var y'))"
l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"
right_impl_one: fixes a :: "'a" shows "a l\<rightarrow> a = (1::'a)"
semiconnex_axioms: shows "semiconnex A (\<sqsubset>)"
less_dres_def: fixes a :: "'a dres"   and b :: "'a dres" shows "(a < b) = (a \<le> b \<and> \<not> b \<le> a)"
pre_zeta_aux_conv_zeta: fixes s :: "complex"   and a :: "real"   and N :: "nat" assumes "1 < Re s"   and "0 < a" shows "pre_zeta_aux N a s = (\<Sum>n. (of_nat n + complex_of_real a) powr - s) + complex_of_real a powr (1 - s) / (1 - s)"
proves_eq_abstract_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and g :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> g"   and "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (mk_eq' \<tau>' (f $ Bv 0) (g $ Bv 0)) \<longmapsto> mk_eq (Abs \<tau> (f $ Bv 0)) (Abs \<tau> (g $ Bv 0))"
finsum_closed: fixes f :: "'f \<Rightarrow> 'a"   and A :: "'f set" assumes "f \<in> A \<rightarrow> carrier R" shows "finsum R f A \<in> carrier R"
invariant_1D_7: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" shows "primitive (poly_real_alg_1 plr)"
lDown_tDouble: shows "invar l \<and> invar r \<and> lv = lvl r \<and> lv = Suc (Suc (lvl l)) \<and> sngl r"
order_iff: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "(a \<preceq> b) = (a = linordered_monoid.min a b)"
split_def: shows "case_prod = (\<lambda>c p. c (fst p) (final_b p))"
that: fixes W :: "'i fm set" assumes "V \<subseteq> W"   and "W \<in> C"   and "maximal W C" shows "thesis"
finite_UN: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" shows "\<Union> (A ` I) \<in> M"
mult_n_nrows: fixes fmul :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and fadd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and A :: "'d matrix"   and B :: "'b matrix" assumes "\<forall>a. fmul (0::'d) a = (0::'c)"   and "\<forall>a. fmul a (0::'b) = (0::'c)"   and "fadd (0::'c) (0::'c) = (0::'c)" shows "nrows (mult_matrix_n n fmul fadd A B) \<le> nrows A"
called_by_chain_converse: fixes \<Gamma> :: "char list \<Rightarrow> 'b rule list option" shows "calls_chain \<Gamma> = (called_by_chain \<Gamma>)\<inverse>"
fds_exp_ln_strong: fixes f :: "'a fds"   and l :: "'a" assumes "fds_nth f (Suc 0) \<noteq> (0::'a)" shows "fds_exp (fds_ln l f) = fds_const (exp l / fds_nth f (Suc 0)) * f"
goedel_first_classic_strong: assumes "consistent" shows "\<not> prv \<phi>G \<and> \<not> prv (neg \<phi>G) \<and> isTrue \<phi>G"
p2: shows "directly_subsumes e1 e2 s1 s2 t2 t3"
Hv_def: fixes v :: "(real, 3) vec" shows "Hv v \<equiv> Quat 0 (v $ 1) (v $ 2) (v $ 3)"
less_le_not_le: fixes x :: "char list"   and y :: "char list" shows "ord.lexordp (\<lambda>c d. of_char c < of_char d) x y = (ord.lexordp_eq (\<lambda>c d. of_char c < of_char d) x y \<and> \<not> ord.lexordp_eq (\<lambda>c d. of_char c < of_char d) y x)"
sorted_filter: fixes f :: "'b \<Rightarrow> 'a word"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "signed.sorted (map f xs)" shows "signed.sorted (map f (filter P xs))"