hcju/isabelleps huggingface dataset:
intros: fixes F :: "'fenv_std"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op) Std.instr fundef"   and pc :: "nat"   and H :: "'henv"   and \<Sigma> :: "'dyn list" assumes "Fstd_get F f = Some fd"   and "pc = length (body fd)" shows "Sstd.final (State F H [Frame f pc \<Sigma>])"
LLL_factorization: fixes f :: "int poly"   and gs :: "int poly list" assumes "LLL_factorization f = gs"   and "square_free f"   and "degree f \<noteq> 0" shows "f = prod_list gs \<and> (\<forall>g\<in>set gs. irreducible\<^sub>d g)"
split_face_edges_f12_vs: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and f12 :: "face"   and f21 :: "face"   and vs1 :: "nat list" assumes "pre_split_face f ram1 ram2 []"   and "(f12, f21) = split_face f ram1 ram2 []"   and "vs1 = between (vertices f) ram1 ram2"   and "vs1 \<noteq> []" shows "\<E> f12 = {(ram2, ram1), (ram1, hd vs1), (last vs1, ram2)} \<union> Edges vs1"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"   and "surjective_map \<eta> R R'" shows "ring_epimorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"
gcd_proj1_if_dvd_int: fixes x :: "int"   and y :: "int" assumes "x dvd y" shows "gcd x y = \<bar>x\<bar>"
skip: fixes mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simplep Skip mem Stop mem"
VecEnd_GMap_is_FGModuleEnd: fixes T :: "'v \<Rightarrow> 'v" assumes "VectorSpaceEnd (\<sharp>\<cdot>) V T"   and "\<And>g v. \<lbrakk>g \<in> G; v \<in> V\<rbrakk> \<Longrightarrow> T (g *\<cdot> v) = g *\<cdot> T v" shows "GRepEnd T"
point_path_def: fixes \<gamma> :: "'b \<Rightarrow> 'c" shows "point_path \<gamma> \<equiv> \<exists>c. \<gamma> = (\<lambda>x. c)"
emeasure_eq_ennreal_measure: fixes M :: "'c measure"   and A :: "'c set" assumes "emeasure M A \<noteq> top" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
residual_countable_network: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "RES.wf_residual_network"   and "flow (residual_network f) f" shows "countable_network (RES.residual_network f)"
transfer_conj: fixes p :: "bool"   and P :: "nat \<Rightarrow> bool"   and q :: "bool"   and Q :: "nat \<Rightarrow> bool" assumes "p \<equiv> eventually P \<U>"   and "q \<equiv> eventually Q \<U>" shows "p \<and> q \<equiv> \<forall>\<^sub>F n in \<U>. P n \<and> Q n"
greatestP_def: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "greatestP P x = (P x \<and> Collect P *<= x)"
epath_to_depath: fixes p :: "'a uprod list"   and E :: "'a uprod set"   and u :: "'a"   and v :: "'a" assumes "set p \<subseteq> set p"   and "epath E u p v" shows "\<exists>dp. depath E u dp v \<and> set dp \<subseteq> set p"
unit_ball_vol_def: fixes n :: "real" shows "unit_ball_vol n = pi powr (n / 2) / Gamma (n / 2 + 1)"
vinsert_insert: fixes a :: "V"   and A :: "V" shows "vinsert a A = ZFC_in_HOL.set (insert a (elts A))"
ffb_def: shows "fb\<^sub>\<F> = \<partial>\<^sub>F \<circ> bd\<^sub>\<F> \<circ> op\<^sub>K"
assm_5: fixes i :: "'a set"   and x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set" assumes "i \<in> I"   and "(x1, K1) \<in> pfn i"   and "(x2, K2) \<in> pfn i"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
factors_wfactors: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and a :: "'c" assumes "Group.monoid G"   and "factors G as a"   and "set as \<subseteq> carrier G" shows "wfactors G as a"
real_arch_simple: fixes x :: "'b" shows "\<exists>n. x \<le> of_nat n"
close_opnet: fixes n :: "net_tree"   and U :: "(nat \<Rightarrow> state) \<Rightarrow> (nat \<Rightarrow> state) \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> state) \<Rightarrow> bool" assumes "wf_net_tree n"   and "oclosed (opnet (\<lambda>i. optoy i \<langle>\<langle>\<^bsub>i\<^esub> qmsg) n) \<Turnstile> (\<lambda>_ _ _. True, U \<rightarrow>) global P" shows "closed (pnet (\<lambda>i. ptoy i \<langle>\<langle> qmsg) n) \<TTurnstile> netglobal P"
list_of_oalist_ntm_cases: fixes y :: "('a \<times> 'b) list \<times> 'a nat_term_order" assumes "y \<in> Collect ko_ntm.oalist_inv" obtains x :: "('a, 'b) oalist_ntm" where "y = list_of_oalist_ntm x"
product_sigma_finite_def: fixes M :: "'b \<Rightarrow> 'c measure" shows "product_sigma_finite M \<equiv> \<forall>i. sigma_finite_measure (M i)"
le_sup_iff: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "(x \<union># y \<subseteq># z) = (x \<subseteq># z \<and> y \<subseteq># z)"
lmerge_def: fixes return :: "'f list \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('f list \<Rightarrow> 'e) \<Rightarrow> 'e"   and A :: "'f list"   and f :: "'f \<Rightarrow> 'e" shows "lmerge return bind A f = foldr (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A @ B)))) (map f A) (return [])"
HFinite_diff_Infinitesimal_approx: fixes x :: "'a star"   and y :: "'a star" assumes "x \<approx> y"   and "y \<in> HFinite - Infinitesimal" shows "x \<in> HFinite - Infinitesimal"
cr_transaction\<^sub>h_def: shows "cr_transaction\<^sub>h \<equiv> \<lambda>x y. x = the_Transaction\<^sub>h y"
simps_2: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b"   and ms :: "'b list" assumes "R_scalar_mult R" shows "R_scalar_mult.RSpan R smult (m # ms) = {smult r m |r. r \<in> R} + R_scalar_mult.RSpan R smult ms"
squarefree_mono: fixes a :: "'a"   and b :: "'a" assumes "a dvd b"   and "squarefree b" shows "squarefree a"
check_eqv_soundness: fixes \<AA> :: "interp"   and idx :: "idx"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and \<psi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" assumes "#\<^sub>V \<AA> = idx"   and "WS1S.check_eqv idx \<phi> \<psi>" shows "WS1S.sat \<AA> \<phi> = WS1S.sat \<AA> \<psi>"
mk_vec_def: fixes f :: "'c \<Rightarrow> 'b" shows "local.mk_vec f = (\<chi>i. f i)"
seq_sim_sym: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and A :: "('b \<Rightarrow> bool) set"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "\<sigma> \<simeq>A\<simeq> \<tau>" shows "\<tau> \<simeq>A\<simeq> \<sigma>"
InDomConc: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "p \<noteq> []"   and "x \<in> dom (Cp (list2FWpolicy p))" shows "x \<in> dom (Cp (list2FWpolicy (a # p)))"
aha: fixes xa :: "'a"   and ya :: "'a" assumes "{xa, ya} \<subseteq> set xs"   and "xa \<noteq> ya"   and "xa < ya in xs" shows "xa < ya in Lxy xs {xa, ya}"
mann7: fixes x :: "'a" shows "- 3 \<times> x = - (- (- 3 \<times> x \<squnion> x) \<squnion> 5 \<times> x)"
left_neutral: fixes a :: "'a" shows "(1::'a) \<cdot> a = a"
allDefs_def: fixes g :: "'a"   and n :: "'node" shows "i.allDefs g n \<equiv> defs2 g n \<union> i.phiDefs g n"
arr_iff_in_hom: fixes f :: "'a \<times> 'a" shows "VV.arr f = VV.in_hom f (VV.dom f) (VV.cod f)"
fps_integral0_fps_X_power: fixes n :: "nat" shows "fps_integral0 (fps_X ^ n) = fps_const (inverse (of_nat (Suc n))) * fps_X ^ Suc n"
uncurry_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
complex_rotations_bij: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> complex_rotations" shows "bij r"
antisymD: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "antisym r"   and "(a, b) \<in> r"   and "(b, a) \<in> r" shows "a = b"
cnf_of_encode_interfering_operator_pair_exclusion_is_i: fixes \<Pi> :: "'b strips_problem"   and k :: "nat"   and op\<^sub>1 :: "'b strips_operator"   and op\<^sub>2 :: "'b strips_operator" shows "cnf (encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2) = {{(Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>1))\<inverse>, (Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>2))\<inverse>}}"
prems: shows "wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t (x # S) \<subseteq> V"
assms: shows "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"
toFunctionWithFallbackAlg_def: fixes R :: "('c \<times> 'd) set"   and fallback :: "'d" shows "R Elsee fallback \<equiv> \<lambda>x. if x \<in> Domain R then R ,, x else fallback"
viewR_correct_1: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t"   and "t = Empty" shows "FingerTreeStruc.viewR t = None"
rotate_conv_mod: fixes n :: "nat"   and xs :: "'b list" shows "rotate n xs = rotate (n mod \<^bold>|xs\<^bold>|) xs"
wsumT: shows "(\<Sum>t\<in>T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
rep_eq: fixes x :: "'b fpxs" shows "fls_nth (fls_of_fpxs x) = (\<lambda>n. fpxs_nth x (rat_of_int n / rat_of_int (LCM r\<in>Formal_Puiseux_Series.supp (fpxs_nth x). snd (quotient_of r))))"
inf1D2: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "inf A B x" shows "B x"
sup_commute: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
graph_eq: fixes g :: "'a ugraph" shows "graph (nodes g) (edges g) = g"
resl_subdist_var: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x \<leftarrow> z \<le> x \<squnion> y \<leftarrow> z"
trans: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "blinding_of_on A h bo"   and "bo x y"   and "bo y z"   and "x \<in> A" shows "bo x z"
while_def: fixes guard :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf" shows "loop_spmf.while guard body \<equiv> spmf.fixp_fun (\<lambda>while s. if guard s then body s \<bind> while else return_spmf s)"
w1: assumes "h - k \<in> orthogonal_complement M \<and> k \<in> M" shows "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) k"
lt_of_comp_def: fixes acomp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" shows "lt_of_comp acomp x y = (case acomp x y of Lt \<Rightarrow> True | _ \<Rightarrow> False)"
is_zariski_open_Int: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set"   and V :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U"   and "comm_ring.is_zariski_open R addition multiplication zero unit V" shows "comm_ring.is_zariski_open R addition multiplication zero unit (U \<inter> V)"
congruent2I: fixes A1 :: "'c set"   and r1 :: "('c \<times> 'c) set"   and A2 :: "'d set"   and r2 :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "equiv A1 r1"   and "equiv A2 r2"   and "\<And>y z w. \<lbrakk>w \<in> A2; (y, z) \<in> r1\<rbrakk> \<Longrightarrow> f y w = f z w"   and "\<And>y z w. \<lbrakk>w \<in> A1; (y, z) \<in> r2\<rbrakk> \<Longrightarrow> f w y = f w z" shows "congruent2 r1 r2 f"
suc_inField: fixes B :: "(nat + 'a Bonk_Schramm_extension_unfolded set) set" assumes "B \<subseteq> Field wo_aux"   and "AboveS wo_aux B \<noteq> {}" shows "suc wo_aux B \<in> Field wo_aux"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a" shows "x # xs \<Squnion>\<^bsub>f\<^esub> y = xs \<Squnion>\<^bsub>f\<^esub> (x \<squnion>\<^bsub>f\<^esub> y)"
inner_commute: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'d"   and y :: "'d" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x y = inner y x"
length_remdups_card_conv: fixes xs :: "'c list" shows "List.length (remdups xs) = card (set xs)"
r_neg1: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier M"   and "y \<in> carrier M" shows "\<ominus>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = y"
decr_grading_p_zero: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat" shows "decr_grading_p d n 0 = 0"
num_loops_def: shows "num_loops = m + 2 * m * m * nat \<lceil>log base (real N)\<rceil>"
assms_1: shows "v \<in> fmdom' (action_needed_asses a (f ++ g))"
sum_nonneg_eq_0_iff: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> (0::'c) \<le> f x" shows "(sum f A = (0::'c)) = (\<forall>x\<in>A. f x = (0::'c))"
inorder_join: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree" shows "Tree2.inorder (join l r) = Tree2.inorder l @ Tree2.inorder r"
that: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y"   and "A = {x, y}" shows "thesis"
project_sub: fixes lmbd :: "char list \<Rightarrow> fterm"   and C :: "fterm literal set"   and C' :: "fterm literal set"   and L' :: "fterm literal set" assumes "C \<cdot>\<^sub>l\<^sub>s lmbd = C'"   and "L' \<subseteq> C'" shows "\<exists>L\<subseteq>C. L \<cdot>\<^sub>l\<^sub>s lmbd = L' \<and> C - L \<cdot>\<^sub>l\<^sub>s lmbd = C' - L'"
finsum_Suc2: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> {..Suc n} \<rightarrow> carrier R" shows "finsum R f {..Suc n} = (\<Oplus>i\<in>{..n}. f (Suc i)) \<oplus> f 0"
int_cases3: fixes k :: "int" obtains "k = 0"   | n :: "nat" where "k = int n__" and "0 < n__"   | n :: "nat" where "k = - int n" and "0 < n"
abs_ok_1: shows "term_ok \<Theta> (Abs \<tau>' s)"
nn_integral_scale_measure: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and r :: "ennreal" assumes "f \<in> borel_measurable M" shows "integral\<^sup>N (scale_measure r M) f = r * integral\<^sup>N M f"
disj_def: fixes p :: "fm"   and q :: "fm" shows "Cooper.disj p q = (if p = T \<or> q = T then T else if p = F then q else if q = F then p else Or p q)"
assms_2: shows "dgrad_set_le d (pp_of_term ` Keys F) (Keys (set fs))"
iLastStrong_False: fixes t0 :: "nat"   and I :: "nat set" shows "\<not> (\<ominus>\<^sub>S t t0 I. False)"
Y_def: fixes n :: "nat" shows "Y n = \<Union> (Y2 n ` Pow (A_n n))"
prime_ge_2_int: fixes p :: "int" assumes "prime p" shows "2 \<le> p"
ok_0_atms: fixes mr :: "mregex" assumes "ok 0 mr" shows "regex.atms (from_mregex mr []) = {}"
Bang: fixes P :: "pi"   and Rs :: "residual" assumes "P \<parallel> !P \<Longrightarrow>\<^sub>l\<^sup>^Rs"   and "Rs \<noteq> \<tau> \<prec> P \<parallel> !P" shows "!P \<Longrightarrow>\<^sub>l\<^sup>^Rs"
sets_Collect_countable_Ex: fixes M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. {x \<in> space M. P i x} \<in> sets M" shows "{x \<in> space M. \<exists>i. P i x} \<in> sets M"
deriv_anno: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P anno Q,A"
assms_2: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "2 \<le> card bl"
conf_Null: fixes P :: "'b prog"   and h :: "heap"   and T :: "ty" shows "(P,h \<turnstile> Null :\<le> T) = P \<turnstile> NT \<le> T"
additiveD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "Transformers.additive t"   and "sound P"   and "sound Q" shows "t (\<lambda>s. P s + Q s) = (\<lambda>s. t P s + t Q s)"
M: shows "M = {#m \<in># M. m = n#} + {#m \<in># M. m \<noteq> n#}"
take_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" shows "take n (take m xs) = take (min n m) xs"
assms_1: shows "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"
ennreal_mult: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "0 \<le> b" shows "ennreal (a * b) = ennreal a * ennreal b"
healthyI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "feasible t"   and "mono_trans t"   and "scaling t" shows "healthy t"
fbrelation_vsingleton: fixes a :: "V"   and b :: "V" shows "fbrelation (set {[a, b]\<^sub>\<circ>})"
choose_3: fixes S :: "'a set" assumes "card S = 3" shows "\<exists>x y z. S = {x, y, z}"
reflp_list_all2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "reflp R" shows "reflp (list_all2 R)"
iT_Plus_neg_inext2: fixes n :: "nat"   and I :: "nat set"   and k :: "nat" shows "inext n (I \<oplus>- k) = inext (n + k) I - k"
state_update_vs_allInstances_generic_ntc: fixes pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and mk :: "'d state \<Rightarrow> 'd state \<times> 'd state"   and oid :: "nat"   and \<sigma>' :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and Object :: "'d"   and P :: "('d state \<times> 'd state \<Rightarrow> Set(\<langle>\<langle>'e\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)) \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>" assumes "\<And>a. pre_post (mk a) = a"   and "oid \<notin> dom \<sigma>'"   and "Type Object = None"   and "cp P"   and "\<And>X. const X \<Longrightarrow> const (P X)" shows "(mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr> \<Turnstile> P (OclAllInstances_generic pre_post Type)) = (mk \<lparr>heap = \<sigma>', assocs = A\<rparr> \<Turnstile> P (OclAllInstances_generic pre_post Type))"
split_cong: fixes q :: "'d \<times> 'e"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and p :: "'d \<times> 'e" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
IH1: shows "continuous_on UNIV (blin_frechet (good_interp I) (simple_term \<theta>\<^sub>1))"
Powp_def: fixes A :: "'a \<Rightarrow> bool" shows "Powp A = (\<lambda>B. \<forall>x\<in>B. A x)"
vdiff_def: fixes w :: "nat list"   and v :: "nat list" shows "w -\<^sub>v v = map (\<lambda>i. w ! i - v ! i) [0..<length w]"
bij_betw_inv_into_left: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and a :: "'a" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A f (f a) = a"
elims: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list"   and y :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "BsSkewBinomialHeapStruc.uniqify x = y" obtains "x = []" and "y = []"   | t :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree" and bq :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" where "x = t # bq" and "y = BsSkewBinomialHeapStruc.ins t bq"
empty_all_roots: fixes r :: "'b list" shows "\<epsilon> \<in> r*"
pdec2'_def: fixes e :: "nat" shows "pdec2' e \<equiv> (GREATEST y. triangle y \<le> e) - pdec1' e"
simps_3: fixes Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "transfer (Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs) (cf # cfs) = (let ins = THE ins. \<exists>outs. (p, ins, outs) \<in> set procs in (Map.empty(ins [:=] params fs (fst cf)), r) # cf # cfs)"
divide_less_cancel: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a c) (divide b c) = ((less zero c \<longrightarrow> less a b) \<and> (less c zero \<longrightarrow> less b a) \<and> c \<noteq> zero)"
is_div_op_def: fixes div_op :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "is_div_op div_op = (\<forall>a b. b dvd a \<longrightarrow> div_op a b * b = a)"
language_eq_right_lang: shows "ST.language = ST.right_lang (dfa.init \<lparr>dfa.states = dfa.states MS, init = dfa.init MS, final = dfa.states MS - dfa.final MS, nxt = dfa.nxt MS\<rparr>)"
induct: fixes P :: "'a set \<Rightarrow> ('a, 'b) fm \<Rightarrow> bool"   and a0 :: "'a set"   and a1 :: "('a, 'b) fm" assumes "\<And>used p. P used (\<^bold>\<not> \<^bold>\<forall> p)"   and "\<And>uu_. P uu_ \<^bold>\<bottom>"   and "\<And>uu_ v va. P uu_ (\<^bold>\<ddagger> v va)"   and "\<And>uu_ va. P uu_ (\<^bold>\<bottom> \<^bold>\<longrightarrow> va)"   and "\<And>uu_ vb vc va. P uu_ (\<^bold>\<ddagger> vb vc \<^bold>\<longrightarrow> va)"   and "\<And>uu_ vb vc va. P uu_ ((vb \<^bold>\<longrightarrow> vc) \<^bold>\<longrightarrow> va)"   and "\<And>uu_ v vb vc. P uu_ (v \<^bold>\<longrightarrow> \<^bold>\<ddagger> vb vc)"   and "\<And>uu_ v vb vc. P uu_ (v \<^bold>\<longrightarrow> vb \<^bold>\<longrightarrow> vc)"   and "\<And>uu_ v vb. P uu_ (v \<^bold>\<longrightarrow> \<^bold>\<forall> vb)"   and "\<And>uu_ v. P uu_ (\<^bold>\<forall> v)" shows "P a0 a1"
less_bot_def: fixes x :: "'b bot"   and y :: "'b bot" shows "(x < y) = (case y of bot.Value y \<Rightarrow> case x of bot.Value x \<Rightarrow> x < y | bot.Bot \<Rightarrow> True | bot.Bot \<Rightarrow> False)"
pmf_eqI: fixes M :: "'e pmf"   and N :: "'e pmf" assumes "\<And>i. pmf M i = pmf N i" shows "M = N"
apf_def: fixes x :: "'f \<Rightarrow> 'h \<Rightarrow> 'g"   and y :: "'f \<Rightarrow> 'h" shows "Applicative.ap x y = (\<lambda>z. x z (y z))"
execn_blockFault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and n :: "nat"   and f :: "'f"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> =n\<Rightarrow> Fault f"
simps_1: fixes p :: "'b list \<Rightarrow> bool" shows "\<box> p [] = True"
not_conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C CongA A' B' C'"   and "\<not> A B C CongA D E F" shows "\<not> A' B' C' CongA D E F"
permutes_def: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set" shows "(p permutes S) = ((\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y))"
optimize_matches_option_append: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr option"   and rs1 :: "'b rule list"   and rs2 :: "'b rule list" shows "optimize_matches_option f (rs1 @ rs2) = optimize_matches_option f rs1 @ optimize_matches_option f rs2"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F (\<Phi>'.Y a') \<Psi>'.map"
next_prime_bound: fixes n :: "nat" shows "\<exists>p. prime p \<and> n < p \<and> p \<le> fact n + 1"