hcju/isabelleps huggingface dataset:
EM_remainder'_cong: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat"   and a' :: "real"   and b' :: "real" assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> f x = g x"   and "n = n'"   and "a = a'"   and "b = b'" shows "EM_remainder' n f a b = EM_remainder' n' g a' b'"
always_eventually: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d filter" assumes "\<forall>x. P x" shows "eventually P F"
Int_def: fixes A :: "'k set"   and B :: "'k set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
assms_1: shows "(f has_vector_derivative f') (at x within s)"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<tau> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q Q'"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (P \<parallel> Q')"
gd_term_axioms: shows "gd_term pair_of_term term_of_pair (\<preceq>) (\<prec>) (\<preceq>\<^sub>t) (\<prec>\<^sub>t)"
a_subset: fixes M :: "('c, 'd) message set"   and N :: "('c, 'd) message set"   and a :: "'c extended \<Rightarrow> 'c sort_key extended" assumes "M \<subseteq> N"   and "a_conditions (insert_messages N) a" shows "a_conditions (insert_messages M) a"
cast_ok_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val" shows "cast_ok P C h v \<equiv> v = Null \<or> P \<turnstile> cname_of h (the_Addr v) \<preceq>\<^sup>* C"
prems: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs__ @ ys__ = xs @ ys" shows "set xs \<inter> set ys = {}"
pred_prod_beta: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and xy :: "'a \<times> 'b" shows "pred_prod P Q xy = (P (fst xy) \<and> Q (snd xy))"
simps_2: fixes c :: "color"   and lt :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and rt :: "('b, 'c) RBT_Impl.rbt" shows "inv2 (Branch c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bheight lt = bheight rt)"
pr_user_authenticity: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and X :: "agent"   and K :: "rat"   and m :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "Says n run 5 X (Card n) (Crypt (sesK K) (Passwd m)) \<in> set evs" shows "Says n run 5 (User m) (Card n) (Crypt (sesK K) (Passwd m)) \<in> set evs"
prime_divisor_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant" shows "P\<^bsub> K v\<^esub> = {v'. valuation K v' \<and> v_equiv K v v'}"
ipresIGCons_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'f, 'opSym, 'var, 'd, 'e, 'g) model_scheme" shows "ipresIGCons h hA termMOD MOD = ipresCons h hA MOD"
fmdom'I: fixes m :: "('b, 'a) fmap"   and x :: "'b"   and y :: "'a" assumes "m $$ x = Some y" shows "x \<in> fmdom' m"
eq: fixes x :: "'a" assumes "x \<le> lfp f" shows "\<alpha> (f x) = g (\<alpha> x)"
simps: fixes dmus :: "'b iarray iarray"   and dmusi :: "'b iarray"   and dmusj :: "'b iarray"   and dll :: "'b"   and l :: "nat" shows "sigma_array_cost dmus dmusi dmusj dll l = (if l = 0 then (dmusi !! l * dmusj !! l, 1) else let l1 = l - 1; dll1 = dmus !! l1 !! l1; (sig, cost_rec) = sigma_array_cost dmus dmusi dmusj dll1 l1; res = (dll * sig + dmusi !! l * dmusj !! l) div dll1; local_cost = 4 in (res, local_cost + cost_rec))"
diff_eq2: fixes V :: "'b set"   and x :: "'b"   and y :: "'b" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "x + - y = x - y"
check_eqv_def: fixes n :: "nat"   and \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "slow.check_eqv n \<phi> \<psi> = (wf_formula n (FOr \<phi> \<psi>) \<and> slow.check_eqvRE enum_class.enum n (rexp_of'' n (norm \<phi>)) (rexp_of'' n (norm \<psi>)))"
updatable_imp_ex_se_suc: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label" assumes "updatable c" shows "\<exists>c'. se c l c'"
swapEnv_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var" shows "swapEnv xs x y \<equiv> swapEnvIm xs x y \<circ> swapEnvDom xs x y"
distinct_5: fixes x21 :: "nat"   and x22 :: "pol"   and x31 :: "pol"   and x32 :: "nat"   and x33 :: "pol" shows "Pinj x21 x22 \<noteq> PX x31 x32 x33"
triangle_right: shows "(g \<star> \<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g)) \<cdot> (\<eta> \<star> g) = g"
d_def: shows "d = Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
approx_with_finite_emeasure: fixes W :: "'a set"   and C :: "ennreal" assumes "W \<in> sets M"   and "C < emeasure M W" obtains Z :: "'a set" where "Z \<in> sets M" and "Z \<subseteq> W" and "emeasure M Z < \<infinity>" and "C < emeasure M Z"
assms_1: shows "\<forall>m. is_zero_row_upt_k m k A"
srtc_symmetric: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "relation.srtc r a b" shows "relation.srtc r b a"
simps: fixes a :: "'d"   and v :: "'c"   and uu :: "('d, 'c) bintree list" shows "alist (Node a v uu) = (v, a)"
\<S>: fixes S' :: "('fun, 'var) strand_step list" assumes "S' \<in> update\<^sub>s\<^sub>t \<S> (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)" shows "S' = S \<or> S' \<in> \<S>"
mstep_output_iff: fixes \<phi> :: "'a MFOTL.formula"   and \<pi> :: "(char list \<times> 'a list) prefix"   and R :: "'a list set"   and st :: "'a mstate"   and tdb :: "(char list \<times> 'a list) set \<times> nat"   and \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a option list"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "last_ts \<pi> \<le> snd tdb"   and "prefix_of (psnoc \<pi> tdb) \<sigma>"   and "mem_restr R v" shows "((i, v) \<in> fst (mstep tdb st)) = (Monitor.progress \<sigma> \<phi> (plen \<pi>) \<le> i \<and> i < Monitor.progress \<sigma> \<phi> (Suc (plen \<pi>)) \<and> wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v \<and> MFOTL.sat \<sigma> (map the v) i \<phi>)"
order_refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "order r A"   and "x \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x"
induct: fixes P :: "'b t_type \<Rightarrow> bool"   and a0 :: "'b t_type" assumes "\<And>x y yl yr ts. \<lbrakk>x \<le> y \<Longrightarrow> P \<lparr>t_type.folding = False, item = x, subtrees = yl # Branch y yl yr # ts\<rparr>; \<not> x \<le> y \<Longrightarrow> P \<lparr>t_type.folding = False, item = x, subtrees = yr # Branch y yl yr # ts\<rparr>\<rbrakk> \<Longrightarrow> P \<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr>"   and "\<And>x ts. P \<lparr>t_type.folding = True, item = x, subtrees = Branch x Leaf Leaf # ts\<rparr> \<Longrightarrow> P \<lparr>t_type.folding = False, item = x, subtrees = Leaf # ts\<rparr>"   and "\<And>x xt y yl yr ts. \<lbrakk>x \<le> y \<Longrightarrow> P \<lparr>t_type.folding = True, item = x, subtrees = Branch y xt yr # ts\<rparr>; \<not> x \<le> y \<Longrightarrow> P \<lparr>t_type.folding = True, item = x, subtrees = Branch y yl xt # ts\<rparr>\<rbrakk> \<Longrightarrow> P \<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr>"   and "\<And>va vc. P \<lparr>t_type.folding = True, item = va, subtrees = [], \<dots> = vc\<rparr>"   and "\<And>va v vc. P \<lparr>t_type.folding = True, item = va, subtrees = [v], \<dots> = vc\<rparr>"   and "\<And>va v vf vc. P \<lparr>t_type.folding = True, item = va, subtrees = v # Leaf # vf, \<dots> = vc\<rparr>"   and "\<And>v va vc. P \<lparr>t_type.folding = v, item = va, subtrees = [], \<dots> = vc\<rparr>"   and "\<And>va vc. P \<lparr>t_type.folding = True, item = va, subtrees = [Leaf], \<dots> = vc\<rparr>"   and "\<And>va vb vc. P \<lparr>t_type.folding = True, item = va, subtrees = Leaf # Leaf # vb, \<dots> = vc\<rparr>" shows "P a0"
CompDiag_Ide_Diag: fixes t :: "'a local.term"   and a :: "'a local.term" assumes "Diag t"   and "Ide a"   and "Dom a = Cod t" shows "a \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
tensor_preserves_iso: fixes f :: "'a"   and g :: "'a" assumes "local.iso f"   and "local.iso g" shows "local.iso (f \<otimes> g)"
p_zero: fixes a :: "real"   and v :: "real"   and s0 :: "real" shows "movement.p a v s0 0 = s0"
snd_comp_apfst: fixes f :: "'a \<Rightarrow> 'c" shows "(snd \<circ>\<circ> apfst) f = snd"
is_unit_iff_degree: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "is_unit p = (Polynomial.degree p = 0)"
prod_decode_eq: fixes x :: "nat"   and y :: "nat" shows "(prod_decode x = prod_decode y) = (x = y)"
lexordp_conv_lexord: fixes xs :: "'d word list"   and ys :: "'d word list" shows "signed.lexordp xs ys = ((xs, ys) \<in> lexord {(x, y). x <s y})"
discI_2: fixes result :: "('k, 'l) result"   and x2 :: "'l" assumes "result = Ok x2" shows "is_ok result"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom (cod \<iota> \<otimes> f) = cod \<iota> \<otimes> local.dom f"
submodule_is_module: fixes N :: "'c set" assumes "LinearCombinations.submodule R N M" shows "module R (md N)"
sees_wf_native: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and D :: "String.literal" assumes "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = Native in D" shows "D\<bullet>M(Ts) :: T"
Let_def: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "??.HOL.Let s f \<equiv> f s"
Gamma_series_Weierstrass_nonpos_Ints: fixes n :: "nat" shows "\<forall>\<^sub>F k in sequentially. Gamma_series_Weierstrass (- of_nat n) k = (0::'a)"
simps_5: fixes h :: "'b \<Rightarrow> 'a fm"   and \<phi>\<^sub>1 :: "'b fm"   and \<phi>\<^sub>2 :: "'b fm" shows "amap\<^sub>f\<^sub>m h (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = or (amap\<^sub>f\<^sub>m h \<phi>\<^sub>1) (amap\<^sub>f\<^sub>m h \<phi>\<^sub>2)"
subset_helper: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list" shows "\<forall>x\<in>set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2)). \<exists>x1\<in>set (characterize_consistent_signs_at_roots_copr p qs1). \<exists>x2\<in>set (characterize_consistent_signs_at_roots_copr p qs2). x = x1 @ x2"
assms: assumes "I" shows "hn_refine \<Gamma> c \<Gamma>' R m"
h_eq: fixes t :: "complex" assumes "\<bar>Re t\<bar> < 1" shows "h t = h2 t"
antipar_1: shows "trg (g \<star> k) = src (h \<star> f)"
divide_le_0_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a b) zero = (less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b)"
A_def: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "GAUSS.A p = {0<..(int p - 1) div 2}"
box_oc: fixes a :: "'b"   and b :: "'b" shows "{x. a <e x \<and> x \<le> b} = {x. a <e x} \<inter> {..b}"
that: fixes s'' :: "char list \<Rightarrow> val option" assumes "\<langle>c\<^sub>1,s\<rangle> \<rightarrow>* \<langle>Skip,s''\<rangle>"   and "\<langle>c\<^sub>2,s''\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>" shows "thesis"
head_union: fixes G :: "('d, 'c) pre_digraph"   and H :: "('d, 'c) pre_digraph" shows "head (Digraph_Component.union G H) = head G"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "sim a b"   and "PA a"   and "PB b" shows "FA a = FB b"
pred0E: fixes n :: "nat" assumes "pred n = 0" obtains "n = 0"   | "n = Suc 0"
distinct_disc_2: fixes event :: "('a, 'b, 'c) event" assumes "isSend event" shows "\<not> isTrans event"
tanh_real_le_iff: fixes x :: "real"   and y :: "real" shows "(tanh x \<le> tanh y) = (x \<le> y)"
not_eq_extremum: fixes a :: "'a word" shows "(a \<noteq> 0) = (0 < a)"
converse_Id_on: fixes A :: "'a set" shows "(Id_on A)\<inverse> = Id_on A"
ring_axioms: shows "ring (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)"
stateful_matcher_eq: fixes \<sigma>' :: "'b" shows "(\<lambda>a b. stateful_matcher_tagged' a (packet_tagger' \<sigma>' b)) = stateful_matcher' \<sigma>'"
distinct_map_fst_filter: fixes xs :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool" assumes "distinct (map fst xs)" shows "distinct (map fst (filter P xs))"
iprev_neq_imp_iMin: fixes n :: "nat"   and I :: "nat set" assumes "iprev n I \<noteq> n" shows "iMin I < n"
weakBisimWeakCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<X> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<approx> Q"
rcosets_part_G: fixes H :: "nat set" assumes "subgroup H G" shows "\<Union> (rcosets\<^bsub>G\<^esub> H) = carrier G"
upper_rank_of_def: fixes carrier' :: "'a set" shows "\<E>.upper_rank_of carrier' \<equiv> Max {card B |B. \<E>.basis_in carrier' B}"
unit_2: fixes b :: "'b" shows "lift (prod_list ys) dvd b"
bounded_linear_vec: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<And>i. bounded_linear (f i)" shows "bounded_linear (\<lambda>x. \<chi>i. f i x)"
subcringI': fixes H :: "'a set" assumes "subring H R" shows "subcring H R"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_backto_standard_pos_O ires rs (l, r) = (\<exists>ml mr ln rn. l = Oc \<up> ml @ Bk # Bk # Bk \<up> ln @ Oc # ires \<and> r = Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc (Suc rs) \<and> 0 < mr)"
ipresSwap_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'b, 'g, 'h) model_scheme" shows "ipresSwap h MOD \<equiv> \<forall>zs z1 z2 s X. wls s X \<longrightarrow> h (X #[z1 \<and> z2]_zs) = igSwap MOD zs z1 z2 (h X)"
hg: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "g x \<in> T \<and> h (g x) = x"
minus_eq: fixes x :: "int"   and y :: "int" shows "x - y = x + - y"
card_2_iff': fixes S :: "'b set" shows "(card S = 2) = (\<exists>x\<in>S. \<exists>y\<in>S. x \<noteq> y \<and> (\<forall>z\<in>S. z = x \<or> z = y))"
minus_unique: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus"   and "plus a b = zero" shows "uminus a = b"
project_set_reachable_extend_eq: fixes F :: "'a program"   and G :: "'c program" shows "project_set h (reachable (extend h F \<squnion> G)) = reachable (F \<squnion> project h (reachable (extend h F \<squnion> G)) G)"
measure_point_measure: fixes \<Omega> :: "'a set"   and A :: "'a set"   and p :: "'a \<Rightarrow> real" assumes "finite \<Omega>"   and "A \<subseteq> \<Omega>"   and "\<And>x. x \<in> \<Omega> \<Longrightarrow> 0 \<le> p x" shows "Sigma_Algebra.measure (point_measure \<Omega> (\<lambda>x. ennreal (p x))) A = sum p A"
comparator_compare: fixes compare :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.compare compare" shows "comparator compare"
subst_term_preserves_typ_of: fixes x :: "variable"   and \<tau> :: "typ"   and y :: "variable"   and t :: "Core.term" shows "typ_of (subst_term [((x, \<tau>), Fv y \<tau>)] t) = typ_of t"
wf_abs_eff: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and pre :: "(nat \<times> nat) list"   and eff :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "abs_ast_prob problem"   and "ast_problem.wf_operator problem (name, pre, eff, cost)" shows "ast_problem.wf_partial_state problem (map (\<lambda>(uu_, v, uu_, a). (v, a)) eff)"
dgrad_p_set_le_spoly: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d" shows "dgrad_p_set_le d {spoly p q} {p, q}"
T'_mom_select_mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "T'_mom_select m \<le> T'_mom_select n"
infsetsum_def: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "infsetsum f A = integral\<^sup>L (count_space A) f"
proots_count_def: fixes p :: "'b poly"   and s :: "'b set" shows "proots_count p s = (\<Sum>r\<in>proots_within p s. order r p)"
ln_upper_11_pos: fixes x :: "real" assumes "1 \<le> x" shows "ln x \<le> ln_upper_11 x"
quot_dbfm_inject_lemma: fixes e :: "(name, hf) finfun"   and A :: "dbfm"   and B :: "dbfm" shows "(\<lbrakk>quot_dbfm A\<rbrakk>e = \<lbrakk>quot_dbfm B\<rbrakk>e) = (A = B)"
member_empty: shows "RBT_Set2.member RBT_Set2.empty = (\<lambda>_. False)"
apply_ncrit_append: fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and xs :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ys :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "apply_ncrit crit data gs bs hs (xs @ ys) = rev (fold (\<lambda>(ic, q_in_bs, p, q) ps'. if \<not> ic \<and> crit data gs bs hs q_in_bs ps' p q then ps' else (ic, p, q) # ps') ys (rev (apply_ncrit crit data gs bs hs xs)))"
Norm_cmult: fixes N :: "'b quasinorm"   and c :: "real"   and x :: "'b" shows "Norm N (c *\<^sub>R x) = \<bar>c\<bar> * Norm N x"
Prog\<^sub>s\<^sub>b: fixes m :: "nat \<Rightarrow> nat"   and p :: "'b"   and p' :: "'b"   and "is" :: "instr list"   and rs :: "'b memref list"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(m, Prog\<^sub>s\<^sub>b p p' is # rs, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m, rs, \<O>, \<R>, \<S>)"
id1: fixes i :: "nat" shows "m + i - (n + m - Suc j) = i + Suc j - n"
card_order_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
map_of_eqI: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" assumes "set (map fst xs) = set (map fst ys)"   and "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k" shows "map_of xs = map_of ys"
emp_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and dl :: "('a, 'b, 'c) cmd list" assumes "\<And>n. n < length cl \<Longrightarrow> {} \<notin> P n" shows "{} \<notin> UNpart1 cl dl P"
lexordp_conv_lexord: fixes xs :: "'a word list"   and ys :: "'a word list" shows "signed.lexordp xs ys = ((xs, ys) \<in> lexord {(x, y). x <s y})"
less_restrictive_def: fixes atom_ordering :: "('a \<times> 'a) set"   and R1 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool"   and R2 :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.less_restrictive R1 R2 = (\<forall>P1 P2 C. R2 P1 P2 C \<longrightarrow> R1 P1 P2 C \<or> R1 P2 P1 C)"
mirror_D: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list" assumes "trans r"   and "irrefl r"   and "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "Decreasing_Diagrams.D r \<sigma> \<tau> \<tau>' \<sigma>'"
decisionD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and X :: "final_decision" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "s = Decision X" shows "t = Decision X"
hom: shows "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) f g"
ty: shows "\<Gamma> \<turnstile> Lam [x].t : T"
check_disjoint_def: fixes xs :: "'b list"   and ys :: "'b list" shows "check_disjoint xs ys = (try forallM (\<lambda>x. if x \<notin> set ys then Inr () else Inl x) xs catch (\<lambda>x. Inl (snd x)))"
F_def: shows "F = A\<^sup>* `` {(N, M)}"
lookup_HSem_heap: fixes y :: "'var"   and h :: "('var \<times> 'exp) list"   and \<rho> :: "'var \<Rightarrow> 'value" assumes "y \<in> domA h" shows "(\<lbrace> h \<rbrace>\<rho>) y = \<lbrakk> the (map_of h y) \<rbrakk>\<^bsub>\<lbrace> h \<rbrace>\<rho>\<^esub>"
mult_aux: fixes a :: "nat"   and b :: "nat" assumes "1 < a"   and "1 < b" shows "f (a * b) = f a * f b"
cmod_complex_polar: fixes r :: "real"   and a :: "real" shows "cmod (complex_of_real r * (complex_of_real (cos a) + \<i> * complex_of_real (sin a))) = \<bar>r\<bar>"
satisfies_bounded_NFOR: fixes \<Phi> :: "('a, 'k) aformula set"   and \<AA> :: "'i" assumes "finite \<Phi>" shows "(\<AA> \<Turnstile>\<^sub>b NFOR \<Phi>) = (\<exists>\<phi>\<in>\<Phi>. \<AA> \<Turnstile>\<^sub>b \<phi>)"
malicious_sec_P2: fixes x1 :: "'grp"   and n :: "nat"   and x0 :: "'grp"   and \<sigma> :: "bool"   and z :: "'d"   and \<A> :: "(bool \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'e) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> 'f spmf)" assumes "x1 \<in> carrier (\<G> n)"   and "x0 \<in> carrier (\<G> n)" shows "mal_def.perfect_sec_P2 n (x0, x1) \<sigma> z (P2_S1 n, P2_S2 n) \<A>"
\<Delta>_naturality: shows "(\<omega> \<star> r\<^sub>0) \<cdot> \<rho> = (t \<star> \<omega>.the_\<theta>) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine] \<cdot> (\<tau> \<star> \<omega>.chine) \<cdot> \<omega>.the_\<nu>"