hcju/isabelleps huggingface dataset:
mult_isol: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and z :: "('a \<times> 'a) set" assumes "x \<subseteq> y" shows "x ; z \<subseteq> y ; z"
T_down_closed: fixes v :: "val"   and \<tau> :: "ty"   and \<eta> :: "val set list"   and v' :: "val" assumes "v \<in> T \<tau> \<eta>"   and "v' \<sqsubseteq> v" shows "v' \<in> T \<tau> \<eta>"
equalityE: fixes A :: "'e set"   and B :: "'e set" assumes "A = B" obtains "A \<subseteq> B" and "B \<subseteq> A"
unit_factor_idem: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "unit_factor (unit_factor a) = unit_factor a"
bounded_clinear_const_scaleR: fixes f :: "'a \<Rightarrow> 'b"   and c :: "real" assumes "bounded_clinear f" shows "bounded_clinear (\<lambda>x. c *\<^sub>R f x)"
SUP_subset_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a \<^bold>\<sqinter> a \<^bold>\<sqinter> b = a \<^bold>\<sqinter> b"
the_\<nu>_props_1: shows "\<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>"
le1: shows "le_vec (rowvector w *v (A *v y)) (mu *s (rowvector w *v y))"
has_locks_Suc_lock_lock_has_locks_Suc_Suc: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and n :: "nat" assumes "has_locks l t = Suc n" shows "has_locks (lock_lock l t) t = Suc (Suc n)"
in_arcsD1: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs (list_digraph G_list)" shows "u \<in> pverts (list_digraph G_list)"
opeT: shows "openin (top_of_set T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
left_unique_iff: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "left_unique R = (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1 x. R x z)"
eval_pred_of_set: fixes A :: "'a set"   and x :: "'a" shows "pred.eval (pred_of_set A) x = (x \<in> A)"
h_start_def: shows "h_start = start + int (l * sum steps {r2..<k})"
resolvent_upon_is_resolvent: fixes A :: "'at"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set" assumes "Pos A \<in> P1"   and "Neg A \<in> P2" shows "resolvent P1 P2 (resolvent_upon P1 P2 A)"
rel_simps_70: shows "\<not> (0::'d) < (0::'d)"
simps_1: fixes c :: "int" shows "fnorm (FCnst c) = (PExpr1 (PCnst c), PExpr1 (PCnst 1), [])"
2: shows "\<exists>x\<in>{\<parallel>diag_mat f *v x\<parallel> |x. \<parallel>x\<parallel> = 1}. Max {\<bar>f i\<bar> |i. i \<in> UNIV} \<le> x"
of_weak_ranking_map: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'c set list" shows "map_relation f (of_weak_ranking xs) = of_weak_ranking (map ((-`) f) xs)"
higher_deriv_diff: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and z :: "complex"   and n :: "nat" assumes "f holomorphic_on S"   and "g holomorphic_on S"   and "open S"   and "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
midpoint_plus_self: fixes a :: "'b"   and b :: "'b" shows "midpoint a b + midpoint a b = a + b"
simps_13: fixes p :: "fm"   and q :: "fm" shows "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
br'_inner_invar_initial: fixes q :: "'d"   and W :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" assumes "q \<in> W" shows "br'_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) {r \<in> \<delta>. q \<in> set (rhsq r)} (Q, W - {q}, rcm)"
self_finance_portfolio: fixes pf :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and v :: "real"   and asset :: "'c" assumes "portfolio pf" shows "portfolio (self_finance Mkt v pf asset)"
pred_it_correct: fixes g :: "'G"   and v :: "'V" assumes "invar g" shows "set_iterator (pred_it g v) (pred (\<alpha> g) v)"
matrix_matrix_mul_diag_matr: fixes A :: "(('a, 'b) vec, 'c) vec"   and f :: "'b \<Rightarrow> 'a" shows "A ** diag_mat f = (\<chi>i j. A $ i $ j * f j)"
coreflexive_comp_top_coreflexive_complement: fixes x :: "'a" assumes "coreflexive x" shows "(x * top) ' = x '"
transitive_closure_of_TRel_to_indRelRTPO: fixes TP :: "'procT"   and TQ :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(TP, TQ) \<in> TRel\<^sup>+" shows "TargetTerm TP \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm TQ"
prime_elem_normalize_iff: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "comm_semiring_1.prime_elem times one zero (normalize x) = comm_semiring_1.prime_elem times one zero x"
f_def: fixes x :: "nat" shows "f x = (c_fst x, c_snd x)"
classes_eq: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set" assumes "equivalence S"   and "A \<subseteq> carrier S" shows "A {.=}\<^bsub>S\<^esub> closure_of\<^bsub>S\<^esub> A"
initState_Spy: shows "initState Spy = Key ` shrK ` bad"
ereal_top: fixes x :: "ereal" assumes "\<And>B. ereal B \<le> x" shows "x = \<infinity>"
g_def: shows "g \<equiv> \<lambda>x. if x \<in> box c d then f x else (0::'b)"
arith_prog_def: fixes start :: "int"   and step :: "nat"   and i :: "nat" shows "arith_prog start step i = start + int (i * step)"
step: fixes \<xi> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and a :: "'f"   and \<xi>' :: "'a"   and p' :: "('b, 'c, 'd, 'e) seqp"   and l' :: "'e" assumes "(\<xi>, p) \<in> reachable A I"   and "\<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)"   and "((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> p'"   and "I a" shows "P (\<xi>', l')"
phis'_all_def: fixes g :: "'g" shows "phis'_all g \<equiv> snd (substAll g)"
dot_square_norm: fixes x :: "'a" shows "inner x x = (norm x)\<^sup>2"
None_com: fixes x :: "'c option" shows "(None = x) = (x = None)"
dbl_inc_simps_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit1 k)"
one_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "one_hom hom \<equiv> hom (1::'a) = (1::'b)"
scene_compat_sym: fixes X :: "'a scene"   and Y :: "'a scene" assumes "X ##\<^sub>S Y" shows "Y ##\<^sub>S X"
term_ok_eta_norm: fixes thy :: "((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 t :: "Core.term" assumes "term_ok thy t" shows "term_ok thy (eta_norm t)"
simps_55: fixes x41 :: "stmt"   and x42 :: "stmt"   and x31 :: "jump"   and x32 :: "stmt" shows "x41;; x42 \<noteq> x31\<bullet> x32"
simps_2: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and x :: "'b memref"   and xs :: "'b memref list" shows "non_volatile_owned_or_read_only pending_write \<S> \<O> (x # xs) = (case x of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> if volatile then non_volatile_owned_or_read_only True (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) (\<O> \<union> A - R) xs else a \<in> \<O> \<and> non_volatile_owned_or_read_only pending_write \<S> \<O> xs | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> (\<not> volatile \<longrightarrow> pending_write \<longrightarrow> a \<in> \<O> \<or> a \<in> read_only \<S>) \<and> non_volatile_owned_or_read_only pending_write \<S> \<O> xs | Prog\<^sub>s\<^sub>b p1 p2 list \<Rightarrow> non_volatile_owned_or_read_only pending_write \<S> \<O> xs | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> non_volatile_owned_or_read_only pending_write (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) (\<O> \<union> A - R) xs)"
threshold_suffix: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and \<G> :: "'b ltl set"   and k :: "nat"   and i :: "nat"   and k' :: "nat" assumes "threshold \<phi> w \<G> = Some k"   and "threshold \<phi> (suffix i w) \<G> = Some k'" shows "k \<le> k' + i"
min_enat_simps_3: fixes q :: "enat" shows "min 0 q = 0"
vwb_lens_iff_mwb_UNIV_src: fixes X :: "'d \<Longrightarrow> 'e" shows "vwb_lens X = (mwb_lens X \<and> \<S>\<^bsub>X\<^esub> = UNIV)"
strip_while_0_nnil: fixes as :: "'a list" assumes "as \<noteq> []"   and "set as \<noteq> 0" shows "strip_while ((=) (0::'a)) as \<noteq> []"
intro: fixes A1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and A2 :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and B2 :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and F1 :: "'d \<Rightarrow> 'f"   and F2 :: "'e \<Rightarrow> 'g" assumes "category A1"   and "category A2"   and "category B1"   and "category B2"   and "functor A1 B1 F1"   and "functor A2 B2 F2"   and "product_category A1 A2"   and "product_category B1 B2" shows "product_functor A1 A2 B1 B2 F1 F2"
subterm_of_iff_subtermeq: fixes t :: "('a, 'b) Term.term"   and t' :: "('a, 'b) Term.term" shows "(t \<sqsubseteq> t') = subterm_of t t'"
assms_2: shows "m2 \<in> smaps_over A B"
DDH_advantage_def: fixes \<G> :: "'d cyclic_group"   and \<A> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool spmf" shows "DH_Ext.ddh.DDH_advantage \<G> \<A> = \<bar>spmf (DH_Ext.ddh.DDH0 \<G> \<A>) True - spmf (DH_Ext.ddh.DDH1 \<G> \<A>) True\<bar>"
W_cmpser_is_w_cmpser: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "W_cmpser G n f" shows "w_cmpser G n f"
simps_2: fixes uu :: "'b"   and a :: "('a, 'b) Node"   and b :: "('a, 'b) Node" shows "FingerTreeStruc.nodeToList (Node2 uu a b) = FingerTreeStruc.nodeToList a @ FingerTreeStruc.nodeToList b"
assms_3: shows "k \<in> dom (map_of (l ! bounded_hashcode_nat (length l) k))"
dghm_ArrMap_vsv: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv (\<FF>\<lparr>ArrMap\<rparr>)"
times_inf: fixes x :: "'a"   and y :: "'a" shows "x * y = x \<sqinter> y"
ccompatible2: fixes c :: "real"   and k :: "'d \<Rightarrow> nat"   and x :: "'d"   and X :: "'d set" assumes "c \<le> real (k x)"   and "c \<in> \<nat>"   and "x \<in> X" shows "ccompatible {region X I r |I r. valid_region X k I r} (LT x c)"
rep_eq: fixes x :: "real"   and xa :: "'b nonzero" shows "Rep_nonzero (x *\<^sub>R xa) = (if x = 0 then One else x *\<^sub>R Rep_nonzero xa)"
critical_set_oLimit: fixes A :: "ordinal set"   and f :: "nat \<Rightarrow> ordinal" shows "critical_set A (oLimit f) = (\<Inter>n. critical_set A (f n))"
induct: fixes x1 :: "'b"   and x2 :: "'b list"   and init :: "'b set"   and acts :: "('b \<times> 'b) set set"   and P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" assumes "(x1, x2) \<in> traces init acts"   and "\<And>s. s \<in> init \<Longrightarrow> P s []"   and "\<And>act s evs s'. \<lbrakk>act \<in> acts; (s, evs) \<in> traces init acts; P s evs; (s, s') \<in> act\<rbrakk> \<Longrightarrow> P s' (s # evs)" shows "P x1 x2"
ParamUses_call_source_length: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and ins :: "'c list"   and outs :: "'c list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "length (ParamUses (sourcenode a)) = length ins"
inj_on_QuadRes: fixes p :: "int" assumes "prime p" shows "inj_on (\<lambda>x. x\<^sup>2 mod p) {0..(p - 1) div 2}"
dual_Meet: fixes A :: "'b set" shows "dual (\<Sqinter>A) = \<Squnion>dual ` A"
left_unique_rel_resumption: fixes R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "left_unique R1"   and "left_unique R2" shows "left_unique (Resumption.resumption.rel_resumption R1 R2)"
interval_lebesgue_integral_gt_eq': fixes b :: "ereal"   and a :: "ereal"   and M :: "real measure"   and f :: "real \<Rightarrow> 'a" assumes "b < a" shows "interval_lebesgue_integral M a b f = - interval_lebesgue_integral M b a f"
assms_2: shows "sorted_snd (p\<^sub>0 # ps)"
simps_3: fixes var :: "'c"   and t :: "('a, 'c) trm" shows "SIGP (DiffAssign var t) = {Inl x |x. x \<in> SIGT t}"
ffold_fun_left_comm: fixes x :: "'a"   and z :: "interface"   and A :: "'a fset" shows "((\<otimes>) \<circ> p) x (ffold ((\<otimes>) \<circ> p) z A) = ffold ((\<otimes>) \<circ> p) (((\<otimes>) \<circ> p) x z) A"
subset_wens: fixes A :: "'b set"   and B :: "'b set"   and act :: "('b \<times> 'b) set"   and F :: "'b program" assumes "A - B \<subseteq> wp act B \<inter> awp F (B \<union> A)" shows "A \<subseteq> wens F act B"
ofilter_def: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "wo_rel r" shows "ofilter r A \<equiv> A \<subseteq> Field r \<and> (\<forall>a\<in>A. under r a \<subseteq> A)"
vcons_vsingleton: fixes a :: "V" shows "[a]\<^sub>\<circ> = ZFC_in_HOL.set {\<langle>0\<^sub>\<nat>, a\<rangle>}"
deriv_c1: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 I,A"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
l_p: shows "D \<in> P_of_state (lnth Sts l) \<and> D \<notin> P_of_state (lnth Sts (Suc l)) \<and> i \<le> l \<and> enat (Suc l) < llength Sts"
regex_atms_size: fixes x :: "'a"   and r :: "'a Regex.regex" assumes "x \<in> regex.atms r" shows "size x < regex.size_regex size r"
onl_invariant_sterms_weaken: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "('i \<times> ('f, 'g, 'e, 'h) seqp, 'j) automaton"   and I :: "'j \<Rightarrow> bool"   and P :: "'i \<times> 'h \<Rightarrow> bool"   and \<xi> :: "'i"   and p :: "('f, 'g, 'e, 'h) seqp"   and I' :: "'j \<Rightarrow> bool"   and p' :: "('f, 'g, 'e, 'h) seqp"   and l :: "'h" assumes "wellformed \<Gamma>"   and "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I'"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'"   and "\<And>a. I' a \<Longrightarrow> I a" shows "P (\<xi>, l)"
wls_psubstEnv_substEnv_compose_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" assumes "wlsEnv rho"   and "wlsEnv rho'"   and "wls (asSort ys) Y"   and "freshEnv ys y rho'" shows "rho &[Y / y]_ys &[rho'] = rho &[rho'] &[Y #[rho'] / y]_ys"
pt_fresh_fresh: fixes a :: "'b"   and x :: "'a"   and b :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
left_null_space_eq_null_space_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "left_null_space A = null_space (Finite_Cartesian_Product.transpose A)"
map_disc_iff: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b msllist" shows "(mslmap f a = MSLNil) = (a = MSLNil)"
N: shows "norm x / (1 - norm x) < real_of_int N"
sign_minus_float: fixes f :: "('a, 'b) IEEE.float" shows "sign (- f) = 1 - sign f"
k: shows "k \<in> {..<nat \<lfloor>primes_pi x\<rfloor>}"
top_on_inv_bval': fixes S :: "'av st option"   and X :: "char list set"   and b :: "bexp"   and r :: "bool" assumes "top_on\<^sub>o S X"   and "vars b \<subseteq> - X" shows "top_on\<^sub>o (inv_bval' b r S) X"
simps_3: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat" shows "MKT x21 x22 x23 x24 \<noteq> ET"
in_set_inner_verts_appendI_r: fixes u :: "'a"   and q :: "('a \<times> 'a) list"   and p :: "('a \<times> 'a) list" assumes "u \<in> set (inner_verts q)" shows "u \<in> set (inner_verts (p @ q))"
dual_equivalence: shows "internal_equivalence d\<^sub>0 e\<^sub>0 (local.inv \<epsilon>\<^sub>0) (local.inv \<eta>\<^sub>0)"
cone_mkCone: fixes p0 :: "'c"   and p1 :: "'c" assumes "is_rendered_commutative_by p0 p1" shows "local.cone (C.dom p0) (mkCone p0 p1)"
numeral_Bit0: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
fac_len: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>f v" shows "\<^bold>|u\<^bold>| \<le> \<^bold>|v\<^bold>|"
f2: shows "traps (snd (fst (get_curr_win () s1))) = traps (snd (fst (get_curr_win () s2)))"
borel_Ici: shows "borel = sigma UNIV (range atLeast)"
length_initialization_list_le_length_start_addrs: shows "length start_addrs \<le> length initialization_list"
cSup_singleton: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Sup {x} = x"
bsoks_simps_2: fixes e :: "'a expr1"   and es :: "'a expr1 list"   and n :: "nat" shows "bsoks (e # es) n = (bsok e n \<and> bsoks es n)"
in_measure: fixes x :: "'d"   and y :: "'d"   and f :: "'d \<Rightarrow> nat" shows "((x, y) \<in> measure f) = (f x < f y)"
list_two_induct: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x1 x2 xs. \<lbrakk>P xs; P (x2 # xs)\<rbrakk> \<Longrightarrow> P (x1 # x2 # xs)" shows "P xs"
hom_mult: fixes x :: "'a poly"   and y :: "'a poly" shows "coeff (x * y) 0 = coeff x 0 * coeff y 0"
bourbaki_witt_fixpoint_axioms: shows "bourbaki_witt_fixpoint lub leq f"
powr_negD: fixes a :: "real"   and b :: "real" assumes "a powr b \<le> 0" shows "a = 0"
max_less_iff_conj: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "(signed.max x y <s z) = (x <s z \<and> y <s z)"
MapsToMorDomCod_1: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "f ;;\<^bsub>C\<^esub> g \<in> mor\<^bsub>C\<^esub>"
assms: shows "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<bullet> A"
Int_UNIV_left: fixes B :: "'a set" shows "UNIV \<inter> B = B"
interchange_same_rows: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and a :: "'f" shows "interchange_rows A a a = A"
assms_3: shows "ts t = \<lfloor>(x, ln)\<rfloor>"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(V x2 = V y2) = (x2 = y2)"
\<tau>move2Sync8: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and o' :: "'addr expr1"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (sync\<^bsub>V\<^esub> (o') e) (8 + length (compE2 o') + length (compE2 e)) None"
simps_5: fixes uu :: "('a, 'rule) graph_node"   and v :: "'form" shows "hyps uu (Reg v) = None"
product_run_finite_snd: fixes \<delta>\<^sub>1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'b"   and \<delta>\<^sub>2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>1 :: "'b"   and q\<^sub>2 :: "'c"   and w :: "nat \<Rightarrow> 'd" assumes "finite (range (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w))" shows "finite (range (run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w))"
locally_compact_Int: fixes S :: "'c set"   and t :: "'c set" assumes "locally compact S"   and "locally compact t" shows "locally compact (S \<inter> t)"
eventually_within_interior: fixes x :: "'b"   and S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> interior S" shows "eventually P (at x within S) = eventually P (at x)"
Ajb: shows "A $ j $ b \<noteq> (0::'a)"
spanning_subset_independent: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set"   and A :: "'b set" assumes "module scale"   and "B \<subseteq> A"   and "\<not> module.dependent scale A"   and "A \<subseteq> module.span scale B" shows "A = B"
TS_prefix_containment: fixes vs :: "'d list"   and xs :: "'d list"   and M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and i :: "nat"   and xs' :: "'d list" assumes "vs @ xs \<in> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "prefix xs' xs" shows "vs @ xs' \<in> TS M2 M1 \<Omega> V m i"
empty_Collect_eq: fixes P :: "'k \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
lm128: fixes l :: "'a list"   and x :: "'a"   and n :: "nat" assumes "\<forall>n. set (perm2 l n) = set l" shows "set (perm2 (x # l) n) = {x} \<union> set l"
mod_diff_self2: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "(n - m) mod m = n mod m"
pdouble_correct: fixes a :: "'b"   and p :: "'b \<times> 'b \<times> 'b" shows "make_affine (pdouble a p) = ell_field_class.add a (make_affine p) (make_affine p)"
assms_5: shows "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root_ptr"
part_fn_Suc_imp_part_fn: fixes Y :: "'b set"   and r :: "nat"   and s :: "nat"   and f :: "'b set \<Rightarrow> nat"   and y :: "'b" assumes "infinite Y"   and "part_fn (Suc r) s Y f"   and "y \<in> Y" shows "part_fn r s (Y - {y}) (\<lambda>u. f (insert y u))"
syzygy_crit: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "is_sig_GB_upt d G u"   and "is_syz_sig d u"   and "p \<in> dgrad_sig_set' (length fs) d"   and "lt p = u" shows "sig_red_zero (\<prec>\<^sub>t) G p"
gen_lossless_gpv_True: fixes co :: "bool"   and \<I> :: "('e, 'f) \<I>" assumes "co = True" shows "gen_lossless_gpv co \<I> \<equiv> gfp (\<lambda>gen_lossless_gpv gpv. \<exists>pa. gpv = GPV pa \<and> lossless_spmf pa \<and> (\<forall>out c input. IO out c \<in> set_spmf pa \<longrightarrow> input \<in> responses_\<I> \<I> out \<longrightarrow> gen_lossless_gpv (c input)))"